A Taxonomy of Networks arXiv:1006.5731v1 - Semantic Scholar

arXiv:1006.5731v1 [physics.data-an] 29 Jun 2010

A Taxonomy of Networks Jukka-Pekka Onnela1,2,3,4†,∗ , Daniel J. Fenn5,4,† , Stephen Reid3 Mason A. Porter6,4 , Peter J. Mucha7 , Mark D. Fricker8,4 , Nick S. Jones3,9,4 1 Harvard

Medical School, Harvard University, Boston MA 02115, USA Kennedy School, Harvard University, Cambridge, MA 02138, USA 3 Department of Physics, University of Oxford, Oxford OX1 3PU, U.K. 4 CABDyN Complexity Centre, University of Oxford, Oxford OX1 1HP, U.K. 5 Mathematical and Computational Finance Group, University of Oxford, Oxford OX1 3LB, U.K. 6 Oxford Centre for Industrial and Applied Mathematics, Mathematical Institute, University of Oxford, OX1 3LB, U.K. 7 Carolina Center for Interdisciplinary Applied Mathematics, Department of Mathematics and Institute for Advanced Materials, Nanoscience & Technology, University of North Carolina, Chapel Hill, NC 27599, USA 8 Department of Plant Sciences, University of Oxford, South Parks Road, Oxford, OX1 3RB, U.K. 9 Oxford Centre for Integrative Systems Biology, Department of Biochemistry, University of Oxford, Oxford, OX1 3QU, U.K. 2 Harvard

† ∗

These authors contributed equally. To whom correspondence should be addressed: [email protected] The study of networks has grown into a substantial interdisciplinary endeavor across the natural, social, and information sciences. Yet there have been very few attempts to investigate the interrelatedness of the different classes of networks studied by different disciplines. Here, we introduced a framework to establish a taxonomy of networks from various origins. The provision of this family tree not only helps understand the kinship of networks, but also facilitates the transfer of empirical analysis, theoretical modeling, and conceptual developments across disciplinary boundaries. The framework is based on probing the mesoscopic properties of networks, an important source of heterogeneity for their structure and function. Using our method, we computed 1

a taxonomy for 752 individual networks and a separate taxonomy for 12 network classes. We also computed three within-class taxonomies for political, fungal, and financial networks, and found them to be insightful in each case.

Introduction Although there is a long tradition of scholarship on networks, the last fifteen years have witnessed advances in network science arising from developments in physics, mathematics, computer science, sociology, and numerous other disciplines (1, 2, 3). Many of the questions asked by researchers in different fields are surprisingly similar, yet investigative methods have often had great difficulty penetrating disciplinary boundaries. This suggests that it would be beneficial to have a taxonomy of networks to enable the identification of problems from different disciplines that might be approached similarly in terms of empirical analyses and theoretical modeling. Apart from praxis, however, there is another historically important reason for having a taxonomy. Classification of objects of study has often been central to the progress of science, two supreme examples of which are the periodic table of elements in chemistry and phylogenetic trees of organisms in biology. Having such a classification might shed light on mechanisms for generating networks, reveal how an unknown network should be treated once its position in the hierarchy is known, or help identify an anomalous member of a family of networks expected to be similar by its origin. Further applications include unsupervised study of multiple realizations of a given process, model or empirical, as well as detecting atypical changes in time ordered series of networks. In this paper, we developed such a categorization framework, with accompanying diagnostic tools and examples of its benefits for understanding relationships within and between classes. In aiming to taxonomize networks, one has to consider the scales at which one wants to investigate their structures. Much of past research on networks has focused on the extremes, 2

either on microscopic (e.g. degree) or macroscopic (e.g. diameter) properties, typically finding that most empirically observed networks have heavy-tailed degree distributions and possess the small-world property (4, 2). Given the ubiquity of these findings across various real-world networks, it is clear that more nuanced approaches are needed to differentiate effectively between networks. Indeed, interpretations of both approaches often implicitly assume that networks are homogeneous and have no “mesoscopic” structure. To overcome some of these limitations, prior work has focused on the statistics of a priori specified modules (“motifs”) (61, 57), roleto-role connectivity profiles of nodes (7), isolating statistically significant structures (“backbones”) (8), and interrelations of modules (9). Our approach is based on the community structure of networks (7, 8), a principal feature of a network’s mesoscopic scale, where a community is a set of nodes with more links between them than expected by chance. Community detection is an active area of network science, in part because communities are thought to be related to functional units in many networks and because they have been shown to have strong effects on dynamical processes that operate on networks (7, 8). Here we exploit modularity maximization, possibly the most popular approach to detecting communities (4, 34, 1, 2, 7, 8). We start by superimposing on the network an infinite range q-state Potts model by assigning a spin state σi to each node i (1), where a spin can have one of q states (16). The Hamiltonian (energy function) for this system is given by the sum over all pairwise interactions between spins in the same state, as H(λ) = −

X

Jij δ(σi , σj ).

(1)

i6=j

Minimization of H favors aligning spins (placing them in the same state) that interact ferromagnetically (Jij > 0) and anti-aligning spins (placing them in different states) that interact antiferromagnetically (Jij < 0). At the minimum energy, the communities correspond to domains of neighboring nodes with identical states. We let Jij = Aij − λPij , where A is the 3

(weighted) adjacency matrix and Pij gives the expected weight of the edge between i and j under a specified null model. To avoid specifying the structures or sizes of communities a priori, the resolution parameter λ allows one to probe structures at multiple scales, avoiding the “resolution limit” problem (17). We chose the standard null model Pij = ki kj /(2W ), where P ki = j Aij is the strength of node i, and W is the total link weight in a weighted network (in an unweighted network W = L, the number of links) (2). This null model preserves the expected undirected strength distribution.

Mesoscopic response functions We generalize the Hamiltonian framework by introducing an unsupervised method for constructing a mesoscopic taxonomy of networks. The first step is to determine for each network the limiting values of λ, denoted by Λmin and Λmax , and then sweep λ from Λmin to Λmax so that the number of communities η increases from 1 to N , the number of nodes. For each pair of nodes, λ = Λij = Aij /Pij is the resolution above which the interaction Jij becomes antiferromagnetic (Jij < 0). As a larger fraction of interactions become antiferromagnetic, the single large community disintegrates into smaller communities, which then continue breaking up into smaller pieces as the value of λ is increased. The way this happens provides a wealth of information about the mesoscopic structure of the network (see Fig. 1A). For example, depending on the network, a community may persist over a range of λ-values, or it may break up after only a small increase in λ; when it does break, it may break into two or more pieces, and the sizes of these pieces may be evenly or unevenly distributed. To capture the nature of the disintegration process, we monitor three summary statistics throughout such sweeps, the first of which is the effective energy Heff =

H(λ) − Hmin = 1 − H(λ)/Hmin , Hmax − Hmin 4

(2)

where Hmin = H(Λmin ) and Hmax = H(Λmax ) = 0 (18). We let nk denote the size (number of nodes) of community k and define pk = nk /N as the probability to choose uniformly at random a member of cluster k. We then define the effective partition entropy Seff = [S(λ) − Pη(λ) S(Λmin )]/[S(Λmax )−S(Λmin )] = S(λ)/ log N , where S(λ) = − k=1 pk log pk is the Shannon entropy and η(λ) is the number of communities at resolution λ and S(Λmin ) = 0. Finally, we define the effective number of communities ηeff = [η(λ) − η(Λmin )]/[η(Λmax ) − η(Λmin )] = [η − 1]/[N − 1]. For a given network, Heff (λ), Seff (λ), and ηeff (λ) quantify the energy of the communities (a measure of how frustrated the spin system is), the disorder in the associated community size distribution, and the number of communities (also a measure of fragmentation). Although minimizing Eq. (1) is an NP-hard problem (19) and H can have a complicated landscape of local optima (33), an increasing number of computational heuristics make this approach computationally tractable (7, 8). We chose to use a greedy algorithm (21) which, due to its speed, enabled us to deal with large networks. Comparing outcomes with greedy, spectral and simulated annealing algorithms, we showed that the methods yield essentially identical summary statistics (see below) and taxonomies (see SOM Sec. 5). In general, Λmin and Λmax vary across networks and, more importantly, are strongly affected by the existence of links with very large Λij values. To overcome these problems, one needs a control parameter other than λ, and we used the fraction of effective antiferromagnetic interactions denoted by ξ (22). In sweeping ξ from 0 to 1 for each network, the measures Heff , Seff , and ηeff increase from 0 to 1 as the number of communities increases from 1 to N , leaving behind a signature that we call the mesoscopic response function (MRF) of that network (see SOM Sec. 2). Importantly, because the three measures and the control parameter ξ all lie in the unit interval, MRFs can be directly compared across networks (see Fig. 1B). The shapes of the MRFs are the non-trivial result of many factors, including the fraction of possible edges in the network; the relative weights of inter- versus intra-community edges; the 5

edge weights compared with the expected edge weights in the random null model; the number of edges that need to become antiferromagnetic for a community to fragment; and the way in which the communities fragment (e.g., whether a single node leaves a community if an edge becomes antiferromagntetic or a community splits in half). The effects of some of these factors on the shapes of the MRFs can be better understood by considering some examples (see SOM Fig. 1). Within our framework, comparing any two networks at the mesoscopic level amounts to characterizing the difference in the behavior of the corresponding MRFs. We quantified the distance between any two MRFs α and β by the area between the curves (see Fig. 1B) of the S c three distinct summary statistics, giving three distance measures dH αβ , dαβ , and dαβ for energy,

entropy and number, respectively. While one can define other plausible metrics, this compares MRFs across all scales (for all values of ξ), lies in the unit interval by construction, and a posteriori is effective in grouping networks. In some cases, however, it is practical to introduce a parsimonious measure, and we used principal component analysis (PCA) (30) to define a distance matrix with elements S c dαβ = wH dH αβ + wS dαβ + wc dαβ , after which we normalized the dαβ values to the unit interval.

The PCA distance dαβ was the most successful of our metrics at classifying networks. We then applied the average linkage clustering algorithm to the PCA distance matrix to construct our taxonomies (see SOM Sec. 3.2).

Taxonomy of networks and network classes To apply our method, we obtained data for 752 networks (listed in SOM Table 2). These included social (e.g. collaboration), biological (e.g. metabolic), political (e.g. roll-call voting), technological (e.g. internet), financial (NYSE) networks, and more. In addition to covering several fields of study, the networks encompassed temporal snapshots of time-dependent systems, different realizations of certain types of networks, and various observations of the same 6

system using different observational techniques (e.g. for protein interactions). While the set of 752 networks also contains synthetic and benchmark networks, we focus here on the empirical networks, and show the taxonomy for a representative subset of 192 networks in Fig. 2A. We confined ourselves to this subset to have a more balanced set of networks across different classes, to exclude classes with a very small number of members (fewer than five), and to keep the dendrogram more readable (see SOM Sec. 4 for the full dendrogram of 752 networks). We colored the leaves of the dendrogram according to their type (see SOM Table 1). If there were no correlation between network types and their mesoscopic classification, the color distribution of leaves would look randomly shuffled. Instead, we observed contiguous blocks of the same color, an indication of systematic structure in the dendrogram. To test the robustness of our taxonomy, we carried out a sensitivity analysis by perturbing the structure of the underlying networks, finding that the framework performed well for small and intermediate amounts of noise (see SOM Sec. 4). Our method distinguished political cosponsorship networks (yellow) from political voting networks (blue), and successfully separated cosponsorship networks from different two-year Congresses in the US House of Representatives (13 leftmost yellow leaves) from those in the Senate (13 rightmost yellow leaves). The apparently poorly grouped networks are also of interest. The left-most, outlying, social network is for Marvel comic book characters (arguably an atypical social network), and the scattered nature of the networks labelled as ‘language’ suggests that this label puts a weak constraint on the networks in the class (scrutiny of this class shows that it is very heterogenous in its sources). Both the collaboration and Protein Interaction Networks (PINs) are spread through the taxonomy and this is consistent with the diverse origins of the networks; indeed, the best way of characterizing PINs is hotly debated (25). Contextualized by substantial data uncertainty, we note that it is interesting to see human collaboration networks showing MRFs similar to some PIN data, given the collaborative nature of proteins in protein complexes. 7

We note that we tested other measures (cumulative degree distributions, weightedness, size, link density clustering coefficient and assortativity) but found that the separation into classes provided by these measures was less effective and also less robust (see SOM Sec. 7). After applying the framework to individual networks, we turned to establishing a taxonomy for network classes. We computed average within-class MRFs by taking the average of Heff (ξ), Seff (ξ), and ηeff (ξ) over networks within each of the given 12 classes. These MRFs are shown in Fig. 2B, where we also show our taxonomy of network classes, generated the same way as the one for individual networks in Fig. 2A. We found that the uppermost three classes comprise what might be called similarity networks, which were constructed from a measure characterizing the similarity of nodes in the network. As this measure could be computed between any two nodes, these networks were typically weighted and fully connected. The remaining classes consist of interaction networks, which are based on the existence of an interaction between any two nodes rather than a characterization of the similarity of the properties of the respective nodes. These networks were mostly sparse and typically unweighted. For an extended discussion of the different origins of the observed MRFs, see SOM Sec. 2 and Ref. (28).

Taxonomies within network classes We started the application of our method by comparing networks irrespective of their classes, then moved on to comparing networks across classes, and will now turn to comparing networks within classes. We chose political, fungal, and financial networks to exemplify the potential of the method to generate meaningful within-class taxonomies. Networks in each of these classes could, of course, be studied using field specific methods, but this approach would not make it possible to relate one class to another. The taxonomies we produce identify meaningful subclasses, help detect outliers, and reveal changes in time ordered sequences of networks. Using auxiliary information, we were able to assess the quality of our clusterings. 8

As the first example, we considered roll-call voting in the US Congress, which is the legislative branch of the federal government and consists of two chambers: the House of Representatives and the Senate. We analyzed roll-call voting for the 1st –110th Congresses, covering the period 1789–2008. We were interested in characterizing voting similarities between each pair of legislators, yielding a weighted adjacency matrix A with elements Aij ∈ [0, 1] determined by the similarity of their roll-call voting over a single two-year Congress (12). Since each such network gives a snapshot taken over a two-year period, we used the method to study the evolution of voting blocs in time. Fig. 3A shows the dendrogram obtained. Much research on the US Congress has been devoted to the extent of partisan polarization, the influence of party on rollcall voting, and the variations of both over time (12, 13, 31). One of the most popular measures of polarization is the difference between mean party “DW-Nominate scores” (14), which is simplest to calculate for a competitive two-party system and thus readily available from the 46th Congress onward. As Fig. 3A and B demonstrate, Congresses with similar levels of polarization usually appear in the same clusters and often sequentially. This suggests that our method was successful at grouping Congresses based on the polarization of roll-call votes. We suggestively colored the five uppermost branches of the dendrogram based on the level of polarization, from brown (highly polarized) to blue (lower polarization). It is also insightful to study the variation in the polarization as a function of time. For example, the House was highly polarized for the 5th –7th Congresses, as party politics emerged and became more important after the presidency passed from George Washington. The same cluster also includes the 38th Congress, which occurred during the Civil War, and the 56th –58th Congresses, when the Populist party was strong. Many other similar connections to US history can be made (see SOM Sec. 8.1). As our next example, we examined fungal mycelial networks extracted from time series of digitized images of colony growth. These are undirected, weighted networks with nodes representing hyphal tips, branch points or anastomoses (hyphal fusions), and links representing the 9

interconnecting hyphal cords weighted by their conductivity (34, 36, 35). Weighted networks of the acellular slime mold, Physarum polycephalum (37) were also digitized for comparison. We found that, in general, replicate networks from different species at comparable time points were grouped together (Fig. 3C). Furthermore, the overall clustering pattern reflected increasing levels of cross-linking, ranging from the relatively sparse networks formed by Resinicium bicolor, through the intermediate levels of interconnection in Phanerochaete velutina, to the dense networks formed by Phallus impudicus (Fig. 3E). By constructing a dendrogram for only one species, but including data from repeated experiments and over time, we see a progression from simple branching trees at early developmental times to an increasingly cross-linked network later in mycelium growth (Fig. 3D) (34, 33). In early growth, the developmental stage appeared to dominate the clustering pattern, with networks from different replicates but similar age grouping together. At later times, networks showed a high overall level of similarity, with the fine grained clustering predominantly reflecting the subtle changes in structure evolving within each replicate. As our final example, we considered a set of stock-return correlation networks for the NYSE, which is the largest stock exchange in the world as measured by the aggregate US dollar value of the securities listed on it. Each node represents a stock, and the strength of the link connecting stocks i and j is proportional to the correlation between the daily logarithmic returns of the stocks (5). We considered N = 100 stocks covering the time period 1985–2008, where we constructed a network for each 6 months of data. This resulted in a sequence of fullyconnected, weighted adjacency matrices whose elements quantify the similarity of two stocks, normalized in the unit interval for each time window (see SOM Sec. 8.2). We show the dendrogram for the NYSE networks in Fig. 3F. The first division of these networks classified them into two clusters, colored in blue and red. The red cluster on the right in the figure appears to correspond to periods of market turmoil, including the networks for the second half of 1987 (in10

cluding the Black Monday crash of October 1987), all of 2000–2002 (including and following the bursting of the dot-com bubble), and the second half of 2007 and all of 2008 (including the recent credit and liquidity crisis). Intuitively, this may be explained by the fact that during business as usual, different market sectors are subject to their own specific business environments, and mostly maintain their characteristic constellations. In times of crises, however, these constellations tend to break down, making the economy more strongly connected as a whole (5). We provide further support for the hypothesis that the red cluster is associated with periods of market turmoil by considering the NYSE composite index, which measures the aggregate performance of all common stocks listed on the NYSE (39), adjusted to eliminate the effects of capitalization changes, new listings, and delistings. In Fig. 3F, we indicate the volatility of the composite index over each 6-month period, demonstrating that the networks assigned to the red cluster correspond (with one or two exceptions) to the periods of highest volatility (40).

Conclusions We developed a network clustering framework based on the introduced mesoscopic response functions. Since prior work on network communities has concentrated almost exclusively on detecting communities and has not sought to harness their structure subsequently, our framework represents a novel methodological application of community detection. In fact, the development of community detection methods and their application to data are frequently motivated by the idea that the community structure of a network has a bearing on its function. While this has mostly been a presumption in the existing literature, it actually presents an empirically testable hypothesis. If different networks perform different functions, as can be fairly safely assumed, and if they do this at least to some degree by encompassing different structural building blocks, one should be able to start from the structural units and arrive at a functional classification. As shown here, by systematically utilizing information about the mesoscopic structure, 11

we arrived at a remarkably clear categorization of networks. In addition, we also derived a taxonomy for network classes which, as a first order approximation, may reflect their functional similarities. We found that the networks which were not grouped with members of their class were indeed unusual and found that we could detect historically noted financial and political changes from time-ordered sequences of networks. Overall, we believe that our framework has significant potential for exploring and exploiting similarities in network structures across classes and disciplinary boundaries.

References and Notes 1. Science 357, 405 (2009). 2. M. E. J. Newman, SIAM Rev. 45, 167 (2003). 3. M. E. J. Newman, Phys. Today 61 (2008). 4. R. Albert, A.-L. Barab´asi, Rev. Mod. Phys. 74, 47 (2002). 5. R. Milo, et al., Science 298, 824 (2002). 6. R. Milo, et al., Science 303, 1538 (2004). 7. R. Guimer`a, M. Sales-Pardo, L. A. N. Amaral, Nat. Phys. 3, 63 (2007). 8. M. A. Serrano, M. Bogun´a, A. Vespignani, Proc. Natl. Acad. Sci. U.S.A. 106, E67 (2009). 9. A. Arenas, J. Borge-Holthoefer, A. Gomez, G. Zamora, arXiv:0911.2651 (2009). 10. M. A. Porter, J.-P. Onnela, P. J. Mucha, Not. Am. Math.Soc. 56, 1082 (2009). 11. S. Fortunato, Phys. Rep. 486, 75-174 (2010).

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12. M. E. J. Newman, M. Girvan, Phys. Rev. E 69, 026113 (2004). 13. M. E. J. Newman, Proc. Natl. Acad. Sci. U.S.A. 103, 8577 (2006). 14. J. Reichardt, S. Bornholdt, Phys. Rev. Lett. 93, 218701 (2004). 15. J. Reichardt, S. Bornholdt, Phys. Rev. E 74, 016110 (2006). 16. F. Y. Wu, Rev. Mod. Phys. 54, 235 (1982). 17. S. Fortunato, M. Barth´elemy, Proc. Natl. Acad. Sci. U.S.A. 104, 36 (2007). 18. Although the energy H(λ) can be reduced by having λ < Λmin , there are no further changes in the partition and, hence, only the regime λ ∈ [Λmin , Λmax ] is of interest. 19. U. Brandes, et al., IEEE T. Knowl. Data En. 20, 172 (2008). 20. B. H. Good, Y. A. de Montjoye, A. Clauset, Phys. Rev. E 81, 046106 (2010) 21. V. D. Blondel, J. Guillaume, R. Lambiotte, E. Lefebvre, J. Stat. Mech. Theory E. P10008, 1742 (2008). 22. Note that ξ is a monotonically increasing function of λ. 23. I. T. Jolliffe, Principal Component Analysis (Springer-Verlag, New York, NY, USA, 1986). 24. D. J. Felleman, D. C. Van Essen, Cereb. Cortex 1, 1 (1991). 25. L. Hakes, J. W. Pinney, D. L. Robertson, S. C. Lovell, Nat. Biotechnol. 26(1), 69 (2008). 26. We removed self-edges when present and symmetrized all directed networks. 27. S. N. Dorogovtsev, A. V. Goltsev, J. F. F. Mendes, Rev. Mod. Phys. 80, 1275 (2008). 13

28. D. J. Fenn, thesis, University of Oxford (2010). 29. A. S. Waugh et al., arXiv:0907.3509 (2009). 30. K. T. Poole, H. Rosenthal, Congress: A Political-Economic Histor of Roll Call Voting (Oxford University Press, Oxford, UK, 1997). 31. K. T. Poole, Voteview (2009). http://voteview.com. 32. N. M. McCarty, K. T. Poole, H. Rosenthal, Polarized America: The Dance of Ideology and Unequal Riches (MIT Press, Cambridge, MA, USA, 2007). 33. L. L. M. Heaton, E. L´opez, P. K. Maini, M. D. Fricker, N. S. Jones, Proc. Roy. Soc. B, in press (2010). 34. D. Bebber, J. Hynes, P. R. Darrah, L. Boddy, M.D. Fricker, Proc. Roy. Soc. B 274, 2307 (2007). 35. M. D. Fricker, L. Boddy, T. Nakagaki, D. P. Bebber, in Adaptive Networks: Theory, Models and Applications. Eds T. Gross and H. Sayama, 51 (2009). 36. M. D. Fricker, L. Boddy, D. P. Bebber, in The Mycota. Vol VIII, Biology of the Fungal Cell (2nd Ed.), Eds. R. J. Howard and N. A. R. Gow. (Springer, Berlin) p. 307 (2007). 37. A. Tero et al., Science 327, 439 (2010). 38. J.-P. Onnela, A. C. K., Kaski, J. Kert´esz, A. Kanto, Phys. Rev. E 68, 056110 (2003). 39. See http://www.nyse.com/. 40. Note that a time window of half of a year might include shorter periods of high and low volatility, so some such exceptions are to be anticipated. 14

41. Acknowledgements: The source of each data set is indicated by corresponding citations. We particularly acknowledge A. D’Angelo, Facebook, A. Merdzanovic, M. E. J. Newman, E. Voeten, and Voteview (maintained by K. Poole and H. Rosenthal). We thank A. Lewis, G. Villar, S. Agarwal, and D. Smith for useful discussions, and S. Arbesman, A. Arenas, N. Christakis, J. Fowler and B. Fulcher for their comments on the manuscript. We adapted some of the code, originally written by R. Guimera and R. Lambiotte, available at http://www.lambiotte.be/codes.html. This research was partly supported by the Fulbright Program (JPO) and the NIH grant P01 AG-031093 (JPO), the James S. McDonnell Foundation (MAP: #220020177), and the NSF (PJM: DMS-0645369). NSJ acknowledges support from the EPSRC and BBSRC.

15

A

B

≥0

ξ

0 is small number, and consider a resolution Λmin such that Λmin = maxij {λ|η(λ = Λij ) = 1}, where η(λ) is the number of communities at resolution λ. In other words, Λmin is the largest value of the resolution parameter λ for which the network still forms a single community. Finally, we define a matrix Λ with entries Λij and examine the distribution P (Λ) of its elements, as well as the corresponding cumulative distribution F (x) = P (Λ ≤ x). The values of Λmin and Λmax are different across networks and, for the standard null model Pij = ki kj /(2W ), are determined by the node strengths. For example, consider a binary network in which nodes i and j are linked. This interaction will become antiferromagnetic when λ > Λij = Aij /Pij = 2LAij /ki kj . If there are a large number of links in the network, but both i and j have very low degrees, then λ will need to be large to make the interaction an21

tiferromagnetic. Therefore, for large networks, very large values of Λmax arise when two low strength nodes are connected. To avoid the issues arising from a few interactions requiring large resolutions to become antiferromagnetic, we work in terms of the effective number of antiferromagnetic interactions ξ defined as `A (λ) − `A (Λmin ) ξ = ξ(λ) = A , ` (Λmax ) − `A (Λmin )

(2)

where `A (λ) is the total number of antiferromagnetic interactions in the system for the given value of λ and `A (Λmin ) is the largest number of antiferromagnetic interactions for which the network still forms a single community. The effective number of antiferromagnetic interactions ξ(λ) is therefore the number of antiferromagnetic interactions in excess of `A (Λmin ), normalized to the unit interval. Note that ξ is a monotonically increasing function of λ. It is also useful to divide the elements of the adjacency matrix A into links (Aij > 0) and non-links (Aij = 0). Based on the values Λij , we further distinguish between two types of links: links with 0 < Λij ≤ Λmin are called Λ− -links, and links with Λij > Λmin are called Λ+ -links. The sum of the number of Λ− -links and Λ+ -links is then equal to L, the number of links in the network. When λ = Λmin all of the Λ− -links are antiferromagnetic, but the network nevertheless consists of a single community. Any increase of λ beyond this point will cause some of the Λ+ -links to become antiferromagnetic, resulting in the network disintegrating into communities. In the definition of ξ that we select, we sweep over values λ ∈ [Λmin , Λmax ], so that the number of communities varies between 1 and N . Although the regime λ < Λmin affects the energy H(λ) (see Eq. 1), there are no further changes in the partition into communities and, consequently, only the region Λmin ≤ λ ≤ Λmax is interesting. The normalization in our definition of ξ accounts for the existence of antiferromagnetic Λ− -links, which are insufficient by themselves to cause the network to break up into communities, and ensures that ξ ∈ [0, 1].

22

2

The Mesoscopic Response Functions

2.1

Definition

We now examine network summary statistics as a function of the effective number of antiferromagnetic links ξ. In particular, we choose to investigate the number of communities η, the energy H (given by Eq. 1) and the partition entropy

S=−

η(λ) X

pk log pk ,

(3)

k=1

where nk is the number of nodes in cluster k and pk = nk /N is the probability to choose cluster k uniformly at random. The values of these quantities vary across networks. For example, as we have already seen, the energy H is determined by the resolution λ, and Λmax depends strongly on the link structure of the network. We wish to use the profiles of the summary statistics versus ξ to compare networks; for this to be possible, we need to normalize H, S, and η. We define an effective energy for Λmin < λ < Λmax as

Heff (λ) =

H(λ) − Hmin H(λ) =1− , Hmax − Hmin Hmin

(4)

where Hmin = H(Λmin ) and Hmax = H(Λmax ). Similarly, we define an effective partition entropy

Seff (λ) =

S(λ) − Smin S(λ) = , S(Λmax ) − Smin log N

(5)

where Smin = S(Λmin ) and S(Λmax ) = S(Λmax ), and an effective number of communities

ηeff (λ) =

η(λ) − η(Λmin ) η(λ) − 1 = , η(Λmax ) − η(Λmin ) N −1

(6)

where η(Λmin ) = η(Λmin ) and η(Λmax ) = η(Λmax ). We can now consider the behavior Heff , Seff , and ηeff as a function of ξ. In sweeping ξ from 0 to 1, the number of communities 23

increases from η(ξ = 0) = 1 to η(ξ = 1) = N , with corresponding changes in energy and entropy, producing a signature that we call the mesoscopic response function (MRF). Because Heff ∈ [0, 1], Seff ∈ [0, 1], and ηeff ∈ [0, 1], while ξ ∈ [0, 1] for any network, we can compare the response functions across networks.

2.2

Examples of MRFs

In Fig. 1, we show example MRFs for a number of networks. Fig. 1 demonstrates that, although there are large variations in the shapes of the response functions, there are also common features. Of particular interest are plateaus in the ηeff and Seff curves, accompanied by large increases in Heff . The NYSE (1980–1999) network (5) provides an example of this behavior [see Fig. 1(b)]. These plateaus imply that as the resolution λ is increased (leading to an increase in Heff ), the number of antiferromagnetic interactions also increases even though the number of communities remains constant. As λ is increased, and more interactions become antiferromagnetic, there is an increased energy incentive for communities to break up. The plateaus demonstrate that, despite this incentive, the communities remain intact. Community partitions corresponding to these plateaus are therefore very robust and potentially represent interesting structures (6,2,7,8). The large increase in Heff shows that such partitions are robust over a large range of resolutions. The Fractal MRFs [Fig. 1(k)] also demonstrate that there can be plateaus in ηeff and Seff that are not accompanied by significant changes in Heff . Such plateaus can be explained by considering the distribution of Λij . If several interactions have identical Λij , then they will all become antiferromagnetic at exactly the same resolution, which leads to a significant increase in ξ, but only a small change in Heff . If these interactions do not lead to additional communities, then there are plateaus in the ηeff and Seff curves. Another common feature is a sharp increase in the Heff and Seff curves at ξ = 0. We define Λ∗ = minij (Λij |Λij > 0), i.e., the smallest non-zero element of Λ. Some networks initially

24

break into two communities at a resolution Λmin < Λ∗ . As λ is increased, the communities then continue to split before Λmin is reached, at which point another interaction becomes antiferromagnetic. In these networks, the number of communities increases to η ≥ 2 at ξ = 0. This usually occurs in sparse networks in which the non-links play a significant role in determining the community structure. The Biogrid D. melanogaster (9) and the Garfield Scientometrics citation (10) MRFs demonstrate this effect [see Figs. 1(e) and (m)]. The MRFs for the voting network of the U.K. House of Commons over the period 2001– 2005 (11) [see Fig. 1(g)] and the roll-call voting network for the 108th US House of Representatives (2003–2004) (12, 13, 14) [Fig. 1(q)] also reveal that sharp increases in Heff can be accompanied by small changes in ηeff and Seff . This observation can also be explained by considering the distribution of Λij . If the Λij distribution is multi-modal, then there can be a large separation between consecutive Λij values. A large increase in λ is then needed to increase ξ, which leads to a large change in Heff . However, because this only results in a single additional antiferromagnetic interaction, the changes in ηeff and Seff are small.

2.3

Simulated MRFs

The shapes of the MRFs reflect the manner in which the network splits into communities as the resolution is increased. To provide further insights into the mesoscopic heterogeneities leading to different MRFs it is therefore instructive to consider the effect of different community fragmentation mechanisms on the MRFs. In this section, we do not assume any network structure or detect communities; instead, we create synthetic ηeff and Seff MRFs by considering different rates of community fragmentation as a function of ξ, and different community size distributions. We begin by assuming a fixed shape for the ηeff response function. Figure 1 suggests that most of the ηeff curves are either approximately linear or some convex function so, as a first

25

(a) DIP: C. elegans

(b) NYSE: 1980−1999

(c) STRING: C. elegans

(d) BA: (500,1)

(e) Biogrid: D. melanogaster

(f) Human brain cortex: participant C

(g) U.K. House of Commons voting: 2001−2005

(h) Dolphins

(i) ER: (500,75)

(j) LF benchmark: (1000,15,50,05,01,2,2,2)

(k) Fractal: (10,2,8)

(l) Fungal: (17,8)

1 0.5 0

1 Heff Seff

0.5

ηeff 0 (m) Garfield: Scientometrics citations

(n) Zachary karate club

(o) Metabolic: DR

(q) Roll call: U.S. House 108

(p) U.S. airlines

(r) WS: (100,4,10)

1 0.5 0 0

0.5

1

0

0.5

1

0

0.5

1

0

0.5

1

0

0.5

1

0

0.5

1

ξ

Figure 1: Example mesoscopic response functions (MRFs). The curves show Heff (red), Seff (blue), and ηeff (green) as a function of the fraction of effective antiferromagnetic links ξ for the following networks: (a) DIP: C. elegans (15, 16), (b) NYSE: 1980–1999 (5), (c) STRING: C. elegans (17), (d), BA: (500,1) (18), (e) Biogrid: D. melanogaster (9), (f) Human brain cortex: participant C (19), (g) U.K. House of Commons Voting: 2001–2005 (11), (h) Dolphins (20), (i) ER: (500,75) (21), (j) LF benchmark: (1000,15,50,05,01,2,2,2) (22), (k) Fractal: (10,2,8) (23), (l) Fungal: (17,8) (24), (m) Garfield: Scientometrics citations (10), (n) Zachary karate club (25), (o) Metabolic: DR (26), (p) US airlines (27, 28), (q) Roll call: US House 108 (12, 13, 14), (r) WS: (100,4,10) (29).

26

approximation, we assume that the ηeff MRFs are either linear, quadratic, or cubic.1 We create synthetic ηeff MRFs for each of these cases as follows. We assume that as the resolution λ is increased, the number of communities increases as  ik η(i) = , N k−1 

(7)

where d· · ·e is the ceiling function, i ∈ {1, · · · , N }, where N is the number of nodes in the network and we investigate k ∈ {1, 2, 3}. The normalization N k−1 ensures that the number of communities does not exceed N and the ceiling function ensures that we have only integer numbers of communities. We normalize these values to effective numbers of communities ηeff lying in the unit interval through the transformation ηeff (i) =

η(i) − η(Λmin ) η(i) − 1 = . η(Λmax ) − η(Λmin ) N −1

(8)

We then construct synthetic Seff MRFs based on the number of communities η at each value of i in the ηeff MRFs. We investigate two extreme cases for the community fragmentation process: 1. We consider the case in which each increase in the number of communities η results from a single node leaving the largest community. For example, at η = 2 we assume that one community contains a single node and the other community contains N − 1 nodes; at η = 3, we assume that there are two communities containing single nodes and a third community containing N − 2 nodes; and so on. 2. We examine the case in which each increase in η results from the largest community splitting in half. For example, at η = 2 we assume that each community contains N/2 nodes; at η = 3, we assume that there are two communities containing N/4 nodes and a third community containing N/2 nodes; at η = 4, we assume that each community 1

Although this assumption is not strictly true for most networks, it is nevertheless a reasonable starting point.

27

contains N/4 nodes; and so on2 . To plot the MRFs we assume that the ξ are uniformly distributed over the interval [0, 1] such that the ith value is given by ξ(i) =

i−1 , N −1

(9)

where again i = 1, · · · , N . We show in Section 2.2 that the number of communities increases to η > 1 at ξ = 0 for many networks, so for each splitting regime we examine two behaviours for the MRFs at ξ = 0: 1. The number of communities does not exceed η = 1 at ξ = 0. 2. The number of communities initially increases without an increase in the effective fraction of antiferromagnetic interactions ξ i.e., η > 1 at ξ = 0. We create MRFs that represent the second type of behaviour by setting the first ι elements of the ξ vector to zero; increasing ι results in the MRFs reaching a higher values at ξ = 0. In Fig. 2, we show synthetic MRFs for networks with N = 500 nodes and ι = 20. For all of the curves where we assume that each increase in η results from a single node leaving the largest community, the Seff MRF closely tracks the ηeff MRF. For each example in which increases in η result from the largest community splitting in half, the entropy increases faster than in the equivalent MRF for single nodes splitting from the largest community. This is because in the former case there is greater uncertainty in the community membership of individual nodes. Figure 2 also demonstrates that, for the fragmentation mechanism in which communities split in half, the MRFs have very different shapes for the different assumptions. For example, there is a plateau in some, but not all, of the Seff MRFs and there is a large variation in the amount by which the Seff MRFs increase at ξ = 0. 2

When splitting a community into two, if N/2 is not an integer, we assume that one of the communities contains bN/2c nodes and that the other community contains dN/2e nodes. If two communities contain the same number of nodes, we choose one at random to split – this choice has no effect on the resulting MRF.

28

A

B

C

D

linear

1 0.5 0

quadratic

1 η

eff

0.5

Seff

0

cubic

1 0.5 0 0

0.5

1

0

0.5

1

ξ

0

0.5

1

0

0.5

1

Figure 2: Simulated MRFs for ηeff and Seff . We assume that the ηeff response functions are either linear, quadratic, or cubic. We also make additional assumptions to produce the observed curves: (A) Each increase in the number of communities η results from a single node leaving the largest community. (B) Again, each increase in η results from a single node leaving the largest community, but we make the additional assumption that, as the resolution is increased, η initially increases without there being an increase in the effective number of antiferromagnetic interactions ξ. (C) Each increase in the number of communities η results from the largest community splitting in half. (D) Each increase in the number of communities η results from the largest community splitting in half and there is an initial increase in η without an increase in ξ.

This is just a simple demonstration of how different fragmentation processes lead to different shaped MRFs. For real-world networks, the community splitting mechanism is likely to be somewhere between these two extreme cases, with single nodes leaving communities for some changes in ξ and communities splitting more equally at other values.

29

3

Details of Distance Measure

3.1

Definition

We use the differences in the behaviors of the MRFs of two networks to define a distance between the networks. There are several plausible choices of distance measure, but we add the constraint that the measure should compare the MRFs across all network scales, i.e., for all values of ξ. With this in mind, we define the pairwise distance between networks with respect to one of the investigated properties as the area between the corresponding MRFs. For example, consider the effective energy Heff MRF for two networks i and j. The distance between these networks with respect to Heff is given by

dH ij

1

Z

j i |Heff − Heff |dξ.

= 0

(10)

Similarly, for the effective entropy and effective number of communities, we obtain

dSij

Z

1

= 0

j i |Seff − Seff |dξ

(11)

j i |ηeff − ηeff |dξ.

(12)

and

dηij

Z = 0

1

We then represent the three metrics in matrix form as DH , DS , and Dη .

3.2

PCA Distance

We analyze MRFs for the energy H, entropy S, and number of communities η, but the techniques that we present work similarly for other summary statistics. We check whether the summary statistics that we investigate are sufficiently different for it to be worthwhile to include all of them in our analysis by calculating the correlation between their distance measures. 30

0.8

0.6

0.6

0.6

0.4 0.2

dη

0.8

dη

dS

0.8

0.4 0.2

0

0.2

0 0

0.2

0.4

dH

0.6

0.8

0.4

0 0

0.2

0.4

0.6

0.8

dS

0

0.2

0.4

0.6

0.8

dH

η S Figure 3: Scatter plots showing the correlation between the distances measures dH ij , dij , and dij . . . η S S The linear correlations ρ between the distances are: ρ(dH ij , dij ) = 0.36, ρ(dij , dij ) = 0.58, and . η ρ(dH ij , dij ) = 0.24.

η S In Fig. 3, we show scatter plots for each pairwise combination of the distances dH ij , dij , and dij .

None of these correlations are sufficiently high to justify excluding any, so we use all three. In the interests of parsimony, we reduce the number of distance measures using principal component analysis (PCA) (30). PCA is a standard dimensionality-reduction technique that transforms a number of correlated variables into uncorrelated variables in which the first component accounts for as much of the variance in the original data as possible. Subsequent components then account for as much of the remaining variance as possible. We perform PCA on the N (N − 1)/2 × 3 matrix in which each column corresponds to the vector representation of the upper triangle of one of the distance matrices DH , DS , Dη . We then define a distance η S matrix D from elements d˜ij = wH dH ij + wS dij + wc dij , where the weights are the principal-

component coefficients for the first component, normalizing the d˜ij to the unit interval to obtain . . . dij . The principal component coefficients are wH = 0.23, wS = 0.79, and wc = 0.56. With the first component accounting for 69% of the total variance, the distances D provide a reasonable single-variable representation of the distances DH , DS , and Dη .

31

4

Dendrogram and Network Classes

To apply our method, we obtained data for 752 networks (Table 2 below). However, in the main text, due to space constraints, we used a representative subset of 192 networks to generate the dendrogram. We show the full dendrogram encompassing all 752 networks below in Fig. 4. In contrast to the dendrogram in the main text in Fig. 2A, the dendrogram below also contains some synthetic (model) networks. Here, too, we observe contiguous blocks of the same color. To help assess the quality of the clusters that we identify, we assign each of the networks to a family group that describes the type of network (see Table 1). We also use this categorization to create a taxonomy of network classes, which we describe in the main text. The assignment of the networks to one of these groups is subjective because several of the networks could belong to more than one category. For example, one could classify the network of hyperlinks between weblogs on US politics as a WWW network or a citation network, and one could classify the network of jazz musicians as a collaboration network or a social network. The initial selection of network categories is of course also subjective. One could argue that if one has a social network category, then it is not necessary to have a collaboration network category as well because collaboration networks are merely a subset of social networks. However, in choosing the network categories, we have attempted to maintain a balance between having enough categories to make it possible to understand the differences that lead to the clusters of networks without having so many categories that it is impossible to discern the essential differences.

32

33

!

!"#

!"$

!"%

!"&

Figure 4: Dendrogram for 752 networks obtained using the method of Mesoscopic Response Functions (MRFs) as explained in the main text and the SOM. Note that the colors used to encode network classes here are not the same as those used in the main text.

'()*+,*-. '/.-01 20.,3//4 5/1-*-.01678/*-)9 !"#$%$&'#()&"*+,"-+".+/$, 5/1-*-.0167./::-**,, !."%0$-)$-%0.'&%$";,*03/1-. 1.'$2 1. The results that we present are very similar for different random stock selections.

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9

Appendix: Network Details

Finally, we describe the networks that we included in our study. In Table 2, we list all of the networks and give the network category, whether it is weighted or unweighted, the number of nodes N , the number of edges L, the fraction of possible edges present fe = 2L/[N (N − 1)], and a reference providing details of the data source. We highlight in bold all of the networks included in the subset of 25 networks used in Section 5, we indicate with an asterisk (∗ ) all of the networks used in Section 6, and we color red all of the networks included in the subset of 270 networks used to produce the results described in several Sections. We include a number of network models and a number of benchmark networks that were introduced to test community detection algorithms. For many of the model and benchmark networks, we include multiple realizations of the network, but with different parameter values. Below, we briefly describe these models and explain the notation we use to label them in Table 2. Erd˝os-R´enyi (ER): In an ER network of N nodes, each pair of nodes is connected by an unweighted edge with probability p and not connected with probability 1 − p (21). The degree of each node is distributed according to a binomial distribution. We label the ER networks using the notation “ER: (N ,p)”. Watts-Strogatz (WS): We consider the small-world network of Watts and Strogatz (29) for a one-dimensional lattice of N nodes with periodic boundary conditions. The network consists of a ring in which each node is connected with an unweighted edge to all of its neighbours that are k or fewer lattice spacings away. Each edge is then visited in turn and one end is rewired with probability p to a different node selected uniformly at random, subject to the constraint that there can be no self-edges or double-edges. We label each Watts-Strogatz network as “WS: (N ,k,p)”. Barab´asi-Albert (BA): BA networks (18) are obtained using a network growth mechanism in 60

which nodes with degree m are added to the network and the other end of each edge attaches to another node with a probability proportional to the degree of that node. We label each BA network “BA: (N ,k)”. Fractal: We generate fractal networks using the method described in (23). We begin by generating an isolated group of 2m fully connected nodes, where m gives the size of the clusters. These groups correspond to the hierarchical level h = 0. We then create a second identical group and we link the two groups with a link density of E −h (h = 1), where the link density is the number of links out of all possible links between the groups and E gives the connection density fall-off per hierarchical level. We then duplicate this network and connect the two duplicates at the level h = 2 with a link density E −2 . We repeat this until we reach the desired network size N = 2n , where n is the number of hierarchical levels. At each step the connection density is decreased, resulting in progressively sparser interconnectivity at higher hierarchical levels. The resulting network exhibits self-similar properties. We label each network “Fractal: (n, m, E)”. Random fully-connected: We produce random, fully connected networks of N nodes by linking every node to every other node with an edge whose weight is chosen uniformly at random on the unit interval. The networks have N (N − 1)/2 edges. We label each network “Random fully-connected: (N )”. Kumpula-Onnela-Saram¨aki-Kaski-Kert´esz (KOSKK) model: This is a weighted-network model of social networks with emphasis on community formation. We generate weighted networks containing communities using the model described in (40). We create links through two mechanisms: First, at each time-step, each node i selects a neighbour j with probability wij /si , where wij is the weight of of the link connecting i and j and P si = j wij is the strength of i. If j has other neighbours in addition to i, then one of 61

them is selected with probability wjk /(sj − wij ). If i and k are not connected, then a new link of weight wik = w0 is created with probability pn . If the link already exists, its weight is increased by an amount δ. In both cases, wij and wjk are also increased by δ. This process is termed local attachment. Second, if a node has no links, with probability pr , it creates a link of weight w0 to a randomly selected node, which is termed global attachment. A node can be deleted with probability pd , in which case all of its links are also removed and it is replaced by a new node, so that the total number of nodes N remains constant. The mechanism begins with an empty network, and links are added by running the local and global attachment mechanisms in parallel. We label each network “Weighted: (N, w0 , δ, pn , pr , pd , t)”, where t is the total number of simulation time steps. Lancichinetti-Fortunato-Radicchi (LFR) benchmark: The LFR benchmark, introduced in (22), are unweighted networks with non-overlapping communities. The network is constructed by assigning each node a degree from a power law distribution with exponent γ, where the extremes of the distribution kmin and kmax are chosen so that the mean degree is hki, and the nodes connected using the configuration model (41). Each node shares a fraction µ of its links with nodes in other communities and 1 − µ with nodes in its own community. The community sizes are taken from a power law distribution with exponent β, subject to the constraint that the sum of all of the community sizes equals the number of nodes N in the network. The minimum and maximum community sizes (qmin and qmax ) are then chosen to satisfy the additional constraint that qmin > kmin and qmax > kmax , which ensures that each node is included in at least one community. We label each network “LFR: (N, hki, kmax , γ, β, µ, qmin , qmax )”. Lancichinetti-Fortunato (LF) benchmark: The LF benchmark introduced in (42) allows the networks to be weighted and the communities to overlap. We only consider weighted

62

networks with non-overlapping communities. The node degrees are again taken from a power law degree distribution, but this time we label the exponent τ1 , and the community sizes are taken from a power law degree distribution with exponent τ2 . The strength si of each node is chosen so that si = kiβ , where ki again gives the degree of node i. There are also two mixing parameters: a topological mixing parameter µt , which measures the proportion of links outside a node’s community, and a mixing parameter µw , which measures the weight of a node’s links outside its community. We label each network “LF: (N, hki, kmax , µt , µw , β, τ1 , τ2 )”. For all of the LF networks, we set N = 1000. One can alternatively set the minimum and maximum community sizes qmin and qmax . We always use qmin = 20 and qmax = 50, so we do not include these parameters when we label the networks. LF-Newman-Girvan benchmark: We also include a network with parameters N = 128, hki = 16, kmax = 16, µw = 0.1, qmin = 32, qmax = 32, and β = 1, which is similar to the NG benchmark (4, 42). Table 2: Network summary statistics. We symmetrize all networks, remove self-edges, and only consider the largest connected component. We give the network category, whether it is weighted or unweighted, the number of nodes N in the largest connected component, the number of edges L, the fraction of possible edges present fe = 2L/[N (N − 1)], and a reference providing details of the data source. We highlight the 25 networks used in the randomizations in Section 5 in bold and the 270 networks used in the aggregate taxonomy in red. We indicate with an asterisk (∗ ) all of the networks used in Section 6. ID

Name

Category

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Human brain cortex: participant A1 Human brain cortex: participant A2 Human brain cortex: participant B Human brain cortex: participant D Human brain cortex: participant E Human brain cortex: participant C Cat brain: cortical∗ Cat brain: cortical/thalmic∗ Macaque brain: cortical∗ Macaque brain: visual/sensory cortex∗ Macaque brain: visual cortex 1∗ Macaque brain: visual cortex 2∗ Co-authorship: astrophysics Co-authorship: comp. geometry Co-authorship: condensed matter Co-authorship: Erd˝os Co-authorship: high energy theory

Brain Brain Brain Brain Brain Brain Brain Brain Brain Brain Brain Brain Collaboration Collaboration Collaboration Collaboration Collaboration

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Weighted

N

L

fe

References

Y Y Y Y Y Y Y Y N N N N Y Y Y N Y

994 987 980 996 992 996 52 95 47 71 30 32 14,845 3,621 13,861 6,927 5,835

13,520 14,865 14,222 14,851 14,372 14,933 515 1,170 313 438 190 194 119,652 9,461 44,619 11,850 13,815

0.0274 0.0305 0.0296 0.0300 0.0292 0.0301 0.3884 0.2620 0.2895 0.1763 0.4368 0.3911 0.0011 0.0014 0.0005 0.0005 0.0008

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ID

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18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100

Co-authorship: network science Hollywood film music∗ Jazz collaboration Facebook: American Facebook: Amherst Facebook: Auburn Facebook: Baylor Facebook: BC Facebook: Berkeley Facebook: Bingham Facebook: Bowdoin Facebook: Brandeis Facebook: Brown Facebook: BU Facebook: Bucknell Facebook: Cal Facebook: Caltech Facebook: Carnegie Facebook: Colgate Facebook: Columbia Facebook: Cornell Facebook: Dartmouth Facebook: Duke Facebook: Emory Facebook: FSU Facebook: Georgetown Facebook: GWU Facebook: Hamilton Facebook: Harvard Facebook: Haverford Facebook: Howard Facebook: Indiana Facebook: JMU Facebook: Johns Hopkins Facebook: Lehigh Facebook: Maine Facebook: Maryland Facebook: Mich Facebook: Michigan Facebook: Middlebury Facebook: Mississippi Facebook: MIT Facebook: MSU Facebook: MU Facebook: Northeastern Facebook: Northwestern Facebook: Notre Dame Facebook: NYU Facebook: Oberlin Facebook: Oklahoma Facebook: Penn Facebook: Pepperdine Facebook: Princeton Facebook: Reed Facebook: Rice Facebook: Rochester Facebook: Rutgers Facebook: Santa Facebook: Simmons Facebook: Smith Facebook: Stanford Facebook: Swarthmore Facebook: Syracuse Facebook: Temple Facebook: Tennessee Facebook: Texas80 Facebook: Texas84 Facebook: Trinity Facebook: Tufts Facebook: Tulane Facebook: U. Chicago Facebook: U. Conn. Facebook: U. Illinois Facebook: U. Mass. Facebook: U. Penn. Facebook: UC33 Facebook: UC61 Facebook: UC64 Facebook: UCF Facebook: UCLA Facebook: UCSB Facebook: UCSC Facebook: UCSD

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fe

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Y Y N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N Y N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N

379 39 198 6,370 2,235 18,448 12,799 11,498 22,900 10,001 2,250 3,887 8,586 19,666 3,824 11,243 762 6,621 3,482 11,706 18,621 7,677 9,885 7,449 27,731 9,388 12,164 2,312 15,086 1,446 4,047 29,732 14,070 5,157 5,073 9,065 20,829 3,745 30,106 3,069 10,519 6,402 32,361 15,425 13,868 10,537 12,149 21,623 2,920 17,420 41,536 3,440 6,575 962 4,083 4,561 24,568 3,578 1,510 2,970 11,586 1,657 13,640 13,653 16,977 31,538 36,364 2,613 6,672 7,740 6,561 17,206 30,795 16,502 14,888 16,800 13,736 6,810 14,936 20,453 14,917 8,979 14,936

914 219 2,742 217,654 90,954 973,918 679,815 486,961 852,419 362,892 84,386 137,561 384,519 637,509 158,863 351,356 16,651 249,959 155,043 444,295 790,753 304,065 506,437 330,008 1,034,799 425,619 469,511 96,393 824,595 59,589 204,850 1,305,757 485,564 186,572 198,346 243,245 744,832 81,901 1,176,489 124,607 610,910 251,230 1,118,767 649,441 381,919 488,318 541,336 715,673 89,912 892,524 1,362,220 152,003 293,307 18,812 184,826 161,403 784,596 151,747 32,984 97,133 568,309 61,049 543,975 360,774 770,658 1,219,639 1,590,651 111,996 249,722 283,912 208,088 604,867 1,264,421 519,376 686,485 522,141 442,169 155,320 428,987 747,604 482,215 224,578 443,215

0.0128 0.2955 0.1406 0.0107 0.0364 0.0057 0.0083 0.0074 0.0033 0.0073 0.0334 0.0182 0.0104 0.0033 0.0217 0.0056 0.0574 0.0114 0.0256 0.0065 0.0046 0.0103 0.0104 0.0119 0.0027 0.0097 0.0063 0.0361 0.0072 0.0570 0.0250 0.0030 0.0049 0.0140 0.0154 0.0059 0.0034 0.0117 0.0026 0.0265 0.0110 0.0123 0.0021 0.0055 0.0040 0.0088 0.0073 0.0031 0.0211 0.0059 0.0016 0.0257 0.0136 0.0407 0.0222 0.0155 0.0026 0.0237 0.0290 0.0220 0.0085 0.0445 0.0058 0.0039 0.0053 0.0025 0.0024 0.0328 0.0112 0.0095 0.0097 0.0041 0.0027 0.0038 0.0062 0.0037 0.0047 0.0067 0.0038 0.0036 0.0043 0.0056 0.0040

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ID

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101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183

Facebook: UF Facebook: UGA Facebook: UNC Facebook: USC Facebook: USF Facebook: USFCA Facebook: UVA Facebook: Vanderbilt Facebook: Vassar Facebook: Vermont Facebook: Villanova Facebook: Virginia Facebook: Wake Facebook: Wash. U. Facebook: Wellesley Facebook: Wesleyan Facebook: William Facebook: Williams Facebook: Wisconsin Facebook: Yale NYSE: 1980-1999 NYSE: 1980-1983 NYSE: 1984-1987 NYSE: 1988-1991 NYSE: 1992-1995 NYSE: 1996-1999 NYSE: H1 1985 NYSE: H2 1985 NYSE: H1 1986 NYSE: H2 1986 NYSE: H1 1987 NYSE: H2 1987 NYSE: H1 1988 NYSE: H2 1988 NYSE: H1 1989 NYSE: H2 1989 NYSE: H1 1990 NYSE: H2 1990 NYSE: H1 1991 NYSE: H2 1991 NYSE: H1 1992 NYSE: H2 1992 NYSE: H1 1993 NYSE: H2 1993 NYSE: H1 1994 NYSE: H2 1994 NYSE: H1 1995 NYSE: H2 1995 NYSE: H1 1996 NYSE: H2 1996 NYSE: H1 1997 NYSE: H2 1997 NYSE: H1 1998 NYSE: H2 1998 NYSE: H1 1999 NYSE: H2 1999 NYSE: H1 2000 NYSE: H2 2000 NYSE: H1 2001 NYSE: H2 2001 NYSE: H1 2002 NYSE: H2 2002 NYSE: H1 2003 NYSE: H2 2003 NYSE: H1 2004 NYSE: H2 2004 NYSE: H1 2005 NYSE: H2 2005 NYSE: H1 2006 NYSE: H2 2006 NYSE: H1 2007 NYSE: H2 2008 NYSE: H1 2008 NYSE: H2 2000 0126-bm06-wt-k2-1 0149-bm05-wt-k2-1 0157-bm03-wt-k2-1 0166-bm03-wt-k2-1 0181-bm02-wt-k2-1 0185-bm02-wt-k2-1 AG-1 PI113-1 PI120-1

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N N N N N N N N N N N N N N N N N N N N Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y

35,111 24,380 18,158 17,440 13,367 2,672 17,178 8,063 3,068 7,322 7,755 21,319 5,366 7,730 2,970 3,591 6,472 2,788 23,831 8,561 477 477 477 477 477 477 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 411 345 251 492 192 199 2366 543 483

1,465,654 1,174,051 766,796 801,851 321,209 65,244 789,308 427,829 119,161 191,220 314,980 698,175 279,186 367,526 94,899 138,034 266,378 112,985 835,946 405,440 113,526 113,526 113,526 113,526 113,526 113,526 4,950 4,950 4,950 4,950 4,950 4,950 4,950 4,950 4,950 4,950 4,950 4,950 4,950 4,950 4,950 4,950 4,950 4,950 4,950 4,950 4,950 4,950 4,950 4,950 4,950 4,950 4,950 4,950 4,950 4,950 4,950 4,950 4,950 4,950 4,950 4,950 4,950 4,950 4,950 4,950 4,950 4,950 4,950 4,950 4,950 4,950 4,950 4,950 645 548 399 778 307 311 3665 0.0013 725 559

0.0024 0.0040 0.0047 0.0053 0.0036 0.0183 0.0054 0.0132 0.0253 0.0071 0.0105 0.0031 0.0194 0.0123 0.0215 0.0214 0.0127 0.0291 0.0029 0.0111 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.0077 0.0092 0.0127 0.0064 0.0167 0.0158 (24) 0.0049 0.0048

(51) (51) (51) (51) (51) (51) (51) (51) (51) (51) (51) (51) (51) (51) (51) (51) (51) (51) (51) (51) (5) (5) (5) (5) (5) (5) (52) (52) (52) (52) (52) (52) (52) (52) (52) (52) (52) (52) (52) (52) (52) (52) (52) (52) (52) (52) (52) (52) (52) (52) (52) (52) (52) (52) (52) (52) (52) (52) (52) (52) (52) (52) (52) (52) (52) (52) (52) (52) (52) (52) (52) (52) (52) (52) (24) (24) (24) (24) (24) (24)

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184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266

PI37-1 PI40-1 PI-1 RB3ctl-3 RB4ctl-3 RB7ctl-3 SC-1 control11-1 control11-2 control11-3 control11-4 control11-5 control11-6 control11-7 control11-8 control11-9 control11-10 control11-11∗ control17-1 control17-2 control17-3 control17-4 control17-5 control17-6 control17-7 control17-8 control17-9 control17-10 control17-11 control4-1 control4-2 control4-3 control4-4 control4-5 control4-6 control4-7 control4-8 control4-9 control4-10 control4-11 pi150ctl-1 pv81-1 pv81-2 pv81-3 pv81-4 pv81-5 pv82-1 pv82-2 pv82-3 pv82-4 pv82-5 pv83-1 pv83-2 pv83-3 pv83-4 pv83-5 Online Dictionary of Computing Online Dictionary Of Information Science Reuters 9/11 news Roget’s thesaurus Word adjacency: English Word adjacency: French Word adjacency: Japanese Word adjacency: Spanish Metabolic: AA Metabolic: AB Metabolic: AG Metabolic: AP Metabolic: AT Metabolic: BB Metabolic: BS Metabolic: CA Metabolic: CE Metabolic: CJ Metabolic: CL Metabolic: CQ Metabolic: CT Metabolic: CY Metabolic: DR Metabolic: EC Metabolic: EF Metabolic: EN Metabolic: HI

Fungal Fungal Fungal Fungal Fungal Fungal Fungal Fungal Fungal Fungal Fungal Fungal Fungal Fungal Fungal Fungal Fungal Fungal Fungal Fungal Fungal Fungal Fungal Fungal Fungal Fungal Fungal Fungal Fungal Fungal Fungal Fungal Fungal Fungal Fungal Fungal Fungal Fungal Fungal Fungal Fungal Fungal Fungal Fungal Fungal Fungal Fungal Fungal Fungal Fungal Fungal Fungal Fungal Fungal Fungal Fungal Language Language Language Language Language Language Language Language Metabolic Metabolic Metabolic Metabolic Metabolic Metabolic Metabolic Metabolic Metabolic Metabolic Metabolic Metabolic Metabolic Metabolic Metabolic Metabolic Metabolic Metabolic Metabolic

Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y N N N N N N N N N N N N N N N N N N N N N N N N

644 550 1,357 202 426 380 536 65 117 240 403 526 591 690 721 772 789 823 16 232 502 703 816 950 1,047 1,113 1,142 1,160 1,205 200 461 826 1,044 1,380 1,623 1,756 1,869 1,992 2,086 2,190 1,810 75 653 911 1,064 986 111 467 630 644 551 129 424 671 708 551 13,356 2,898 13,308 994 7,377 8,308 2,698 11,558 411 386 494 201 296 175 772 483 453 370 382 187 211 537 800 762 375 374 505

826 748 1,858 233 545 458 689 71 136 273 458 588 661 772 821 884 907 954 15 240 539 754 874 1,058 1,182 1,303 1,347 1,384 1,469 213 490 862 1,087 1,476 1,767 1,923 2,061 2,196 2,301 2,431 2,537 82 897 1,255 1,467 1,351 112 523 726 749 627 142 510 857 905 668 91,471 16,376 148,035 3,640 44,205 23,832 7,995 43,050 1,818 1,691 2,173 857 1,231 628 3,611 2,274 2,025 1,631 1,646 663 772 2,503 3,789 3,683 1,721 1,617 2,325

0.0040 0.0050 0.0020 0.0115 0.0060 0.0064 0.0048 0.0341 0.0200 0.0095 0.0057 0.0043 0.0038 0.0032 0.0032 0.0030 0.0029 0.0028 0.1250 0.0090 0.0043 0.0031 0.0026 0.0023 0.0022 0.0021 0.0021 0.0021 0.0020 0.0107 0.0046 0.0025 0.0020 0.0016 0.0013 0.0012 0.0012 0.0011 0.0011 0.0010 0.0015 0.0295 0.0042 0.0030 0.0026 0.0028 0.0183 0.0048 0.0037 0.0036 0.0041 0.0172 0.0057 0.0038 0.0036 0.0044 0.0010 0.0039 0.0017 0.0074 0.0016 0.0007 0.0022 0.0006 0.0216 0.0228 0.0178 0.0426 0.0282 0.0412 0.0121 0.0195 0.0198 0.0239 0.0226 0.0381 0.0348 0.0174 0.0119 0.0127 0.0245 0.0232 0.0183

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267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349

Metabolic: HP Metabolic: MB Metabolic: MG Metabolic: MJ Metabolic: ML Metabolic: MP Metabolic: MT Metabolic: NG Metabolic: NM Metabolic: OS Metabolic: PA Metabolic: PF Metabolic: PG Metabolic: PH Metabolic: PN Metabolic: RC Metabolic: RP Metabolic: SC Metabolic: ST Metabolic: TH Metabolic: TM Metabolic: TP Metabolic: TY Metabolic: YP U.S. political books co-purchase∗ Power grid Slovenian magazine co-purchase Transcription: E. coli Transcription: Yeast U.S. airlines 2008 NCAA football schedule∗ Internet: autonomous systems Garfield: scientometrics citations Garfield: Small and Griffith citations Garfield: small-world citations Electronic circuit (s208)∗ Electronic circuit (s420) Electronic circuit (s838) Protein: serine protease inhibitor (1EAW)∗ Protein: immunoglobulin (1A4J)∗ Protein: oxidoreductase (1AOR)∗ AIDS blogs∗ Political blogs WWW (Stanford) Trade product proximity World trade in metal (1994): Net World trade in metal (1994): Total Bill cosponsorship: U.S. House 96 Bill cosponsorship: U.S. House 97 Bill cosponsorship: U.S. House 98 Bill cosponsorship: U.S. House 99 Bill cosponsorship: U.S. House 100 Bill cosponsorship: U.S. House 101 Bill cosponsorship: U.S. House 102 Bill cosponsorship: U.S. House 103 Bill cosponsorship: U.S. House 104 Bill cosponsorship: U.S. House 105 Bill cosponsorship: U.S. House 106 Bill cosponsorship: U.S. House 107 Bill cosponsorship: U.S. House 108 Bill cosponsorship: U.S. Senate 96 Bill cosponsorship: U.S. Senate 97 Bill cosponsorship: U.S. Senate 98 Bill cosponsorship: U.S. Senate 99 Bill cosponsorship: U.S. Senate 100 Bill cosponsorship: U.S. Senate 101 Bill cosponsorship: U.S. Senate 102 Bill cosponsorship: U.S. Senate 103 Bill cosponsorship: U.S. Senate 104 Bill cosponsorship: U.S. Senate 105 Bill cosponsorship: U.S. Senate 106 Bill cosponsorship: U.S. Senate 107 Bill cosponsorship: U.S. Senate 108 Committees: U.S. House 101, comms. Committees: U.S. House 102, comms. Committees: U.S. House 103, comms. Committees: U.S. House 104, comms. Committees: U.S. House 105, comms. Committees: U.S. House 106, comms. Committees: U.S. House 107, comms. Committees: U.S. House 108, comms. Committees: U.S. House 101, Reps. Committees: U.S. House 102, Reps.

Metabolic Metabolic Metabolic Metabolic Metabolic Metabolic Metabolic Metabolic Metabolic Metabolic Metabolic Metabolic Metabolic Metabolic Metabolic Metabolic Metabolic Metabolic Metabolic Metabolic Metabolic Metabolic Metabolic Metabolic Other Other Other Other Other Other Other Other Other Other Other Other Other Other Other Other Other Other Other Other Other Other Other Political: cosponsorship Political: cosponsorship Political: cosponsorship Political: cosponsorship Political: cosponsorship Political: cosponsorship Political: cosponsorship Political: cosponsorship Political: cosponsorship Political: cosponsorship Political: cosponsorship Political: cosponsorship Political: cosponsorship Political: cosponsorship Political: cosponsorship Political: cosponsorship Political: cosponsorship Political: cosponsorship Political: cosponsorship Political: cosponsorship Political: cosponsorship Political: cosponsorship Political: cosponsorship Political: cosponsorship Political: cosponsorship Political: cosponsorship Political: committee Political: committee Political: committee Political: committee Political: committee Political: committee Political: committee Political: committee Political: committee Political: committee

67

Weighted

N

L

fe

References

N N N N N N N N N N N N N N N N N N N N N N N N N N Y N N Y Y N Y Y N N N N N N N N Y N Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y N N N N N N N N N N

365 418 199 422 414 171 577 394 369 285 720 310 412 318 405 663 203 552 391 427 328 194 803 552 105 4,941 124 328 662 324 121 22,963 2,678 1,024 233 122 252 512 53 95 97 146 1,222 8,929 775 80 80 438 435 437 437 439 437 437 437 439 442 436 442 439 101 101 101 101 101 100 102 101 102 100 102 101 100 159 163 141 106 108 107 113 118 434 436

1,703 1,850 783 1,874 1,862 685 2,653 1,824 1,708 1,168 3,429 1,379 1,772 1,394 1,829 3,111 775 2,595 1,756 1,955 1,452 788 3,863 2,471 441 6,594 5,972 456 1,062 2,081 764 48,436 10,368 4,916 994 189 399 819 123 213 212 180 16,714 26,320 283,094 875 875 95,529 94,374 95,256 94,999 96,125 95,263 95,051 95,028 95,925 97,373 94,820 97,233 96,104 5,050 5,050 5,050 5,049 5,050 4,950 5,142 5,050 5,151 4,950 5,151 5,049 4,950 3,610 4,093 2,983 1,839 1,997 2,031 2,429 2,905 18,714 20,134

0.0256 0.0212 0.0397 0.0211 0.0218 0.0471 0.0160 0.0236 0.0252 0.0289 0.0132 0.0288 0.0209 0.0277 0.0224 0.0142 0.0378 0.0171 0.0230 0.0215 0.0271 0.0421 0.0120 0.0162 0.0808 0.0005 0.7831 0.0085 0.0049 0.0398 0.1052 0.0002 0.0029 0.0094 0.0368 0.0256 0.0126 0.0063 0.0893 0.0477 0.0455 0.0170 0.0224 0.0007 0.9439 0.2769 0.2769 0.9982 0.9998 0.9999 0.9972 0.9998 1.0000 0.9977 0.9975 0.9978 0.9991 0.9999 0.9977 0.9996 1.0000 1.0000 1.0000 0.9998 1.0000 1.0000 0.9983 1.0000 1.0000 1.0000 1.0000 0.9998 1.0000 0.2874 0.3100 0.3022 0.3305 0.3456 0.3581 0.3838 0.4208 0.1992 0.2123

(26) (26) (26) (26) (26) (26) (26) (26) (26) (26) (26) (26) (26) (26) (26) (26) (26) (26) (26) (26) (26) (26) (26) (26) (58) (29) (59) (60) (61) (27, 28) (62) (63) (10) (10) (10) (57) (57) (57) (57) (57) (57) (64) (65) (66) (67) (68, 28) (68, 28) (69, 70) (69, 70) (69, 70) (69, 70) (69, 70) (69, 70) (69, 70) (69, 70) (69, 70) (69, 70) (69, 70) (69, 70) (69, 70) (69, 70) (69, 70) (69, 70) (69, 70) (69, 70) (69, 70) (69, 70) (69, 70) (69, 70) (69, 70) (69, 70) (69, 70) (69, 70) (71, 72) (71, 72) (71, 72) (71, 72) (71, 72) (71, 72) (71, 72) (71, 72) (71, 72) (71, 72)

ID

Name

Category

350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432

Committees: U.S. House 103, Reps. Committees: U.S. House 104, Reps. Committees: U.S. House 105, Reps. Committees: U.S. House 106, Reps. Committees: U.S. House 107, Reps. Committees: U.S. House 108, Reps. Roll call: U.S. House 1 Roll call: U.S. House 2 Roll call: U.S. House 3 Roll call: U.S. House 4 Roll call: U.S. House 5 Roll call: U.S. House 6 Roll call: U.S. House 7 Roll call: U.S. House 8 Roll call: U.S. House 9 Roll call: U.S. House 10 Roll call: U.S. House 11 Roll call: U.S. House 12 Roll call: U.S. House 13 Roll call: U.S. House 14 Roll call: U.S. House 15 Roll call: U.S. House 16 Roll call: U.S. House 17 Roll call: U.S. House 18 Roll call: U.S. House 19 Roll call: U.S. House 20 Roll call: U.S. House 21 Roll call: U.S. House 22 Roll call: U.S. House 23 Roll call: U.S. House 24 Roll call: U.S. House 25 Roll call: U.S. House 26 Roll call: U.S. House 27 Roll call: U.S. House 28 Roll call: U.S. House 29 Roll call: U.S. House 30 Roll call: U.S. House 31 Roll call: U.S. House 32 Roll call: U.S. House 33 Roll call: U.S. House 34 Roll call: U.S. House 35 Roll call: U.S. House 36 Roll call: U.S. House 37 Roll call: U.S. House 38 Roll call: U.S. House 39 Roll call: U.S. House 40 Roll call: U.S. House 41 Roll call: U.S. House 42 Roll call: U.S. House 43 Roll call: U.S. House 44 Roll call: U.S. House 45 Roll call: U.S. House 46 Roll call: U.S. House 47 Roll call: U.S. House 48 Roll call: U.S. House 49 Roll call: U.S. House 50 Roll call: U.S. House 51 Roll call: U.S. House 52 Roll call: U.S. House 53 Roll call: U.S. House 54 Roll call: U.S. House 55 Roll call: U.S. House 56 Roll call: U.S. House 57 Roll call: U.S. House 58 Roll call: U.S. House 59 Roll call: U.S. House 60 Roll call: U.S. House 61 Roll call: U.S. House 62 Roll call: U.S. House 63 Roll call: U.S. House 64 Roll call: U.S. House 65 Roll call: U.S. House 66 Roll call: U.S. House 67 Roll call: U.S. House 68 Roll call: U.S. House 69 Roll call: U.S. House 70 Roll call: U.S. House 71 Roll call: U.S. House 72 Roll call: U.S. House 73 Roll call: U.S. House 74 Roll call: U.S. House 75 Roll call: U.S. House 76 Roll call: U.S. House 77

Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political:

committee committee committee committee committee committee voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting

68

Weighted

N

L

fe

References

N N N N N N Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y

437 432 435 435 434 437 66 71 108 114 117 113 110 149 147 149 153 146 195 195 195 197 199 221 220 219 220 217 257 255 256 255 257 234 236 236 241 239 240 236 245 243 197 187 199 233 256 253 302 308 302 301 306 338 330 326 347 340 376 368 371 369 371 397 397 398 402 408 452 441 454 453 452 442 437 443 455 447 445 440 445 456 450

18,212 17,130 18,297 18,832 19,824 21,214 2,122 2,428 5,669 6,342 6,600 6,222 5,921 10,888 10,582 10,857 11,482 10,535 18,723 18,540 18,666 19,118 19,429 23,812 23,993 23,666 23,985 23,404 32,502 32,062 32,366 32,067 32,743 26,788 27,562 27,669 28,804 28,318 28,570 27,545 29,630 29,312 18,735 17,326 19,593 26,605 32,109 31,626 45,151 46,723 45,315 44,987 46,214 56,484 54,160 52,907 59,303 57,285 69,943 67,085 68,270 67,059 67,383 75,891 76,299 77,921 80,174 82,442 101,498 96,780 102,108 101,199 101,482 96,885 95,226 97,497 102,502 99,028 98,647 96,170 98,474 102,495 99,956

0.1912 0.1840 0.1938 0.1995 0.2110 0.2227 0.9893 0.9771 0.9811 0.9846 0.9726 0.9832 0.9877 0.9875 0.9861 0.9847 0.9874 0.9953 0.9898 0.9802 0.9868 0.9903 0.9862 0.9795 0.9960 0.9914 0.9956 0.9986 0.9880 0.9900 0.9916 0.9902 0.9953 0.9826 0.9939 0.9978 0.9960 0.9957 0.9962 0.9933 0.9913 0.9969 0.9704 0.9963 0.9945 0.9843 0.9837 0.9921 0.9934 0.9883 0.9970 0.9964 0.9903 0.9918 0.9977 0.9987 0.9879 0.9940 0.9921 0.9934 0.9947 0.9877 0.9818 0.9655 0.9707 0.9863 0.9947 0.9929 0.9958 0.9975 0.9930 0.9885 0.9956 0.9941 0.9996 0.9959 0.9924 0.9934 0.9986 0.9958 0.9968 0.9880 0.9894

(71, 72) (71, 72) (71, 72) (71, 72) (71, 72) (71, 72) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14)

ID

Name

433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515

Roll call: Roll call: Roll call: Roll call: Roll call: Roll call: Roll call: Roll call: Roll call: Roll call: Roll call: Roll call: Roll call: Roll call: Roll call: Roll call: Roll call: Roll call: Roll call: Roll call: Roll call: Roll call: Roll call: Roll call: Roll call: Roll call: Roll call: Roll call: Roll call: Roll call: Roll call: Roll call: Roll call: Roll call: Roll call: Roll call: Roll call: Roll call: Roll call: Roll call: Roll call: Roll call: Roll call: Roll call: Roll call: Roll call: Roll call: Roll call: Roll call: Roll call: Roll call: Roll call: Roll call: Roll call: Roll call: Roll call: Roll call: Roll call: Roll call: Roll call: Roll call: Roll call: Roll call: Roll call: Roll call: Roll call: Roll call: Roll call: Roll call: Roll call: Roll call: Roll call: Roll call: Roll call: Roll call: Roll call: Roll call: Roll call: Roll call: Roll call: Roll call: Roll call: Roll call:

Category U.S. House 78 U.S. House 79 U.S. House 80 U.S. House 81 U.S. House 82 U.S. House 83 U.S. House 84 U.S. House 85 U.S. House 86 U.S. House 87 U.S. House 88 U.S. House 89 U.S. House 90 U.S. House 91 U.S. House 92 U.S. House 93 U.S. House 94 U.S. House 95 U.S. House 96 U.S. House 97 U.S. House 98 U.S. House 99 U.S. House 100 U.S. House 101 U.S. House 102 U.S. House 103 U.S. House 104 U.S. House 105 U.S. House 106 U.S. House 107 U.S. House 108 U.S. House 109 U.S. House 110 U.S. Senate 1 U.S. Senate 2 U.S. Senate 3 U.S. Senate 4 U.S. Senate 5 U.S. Senate 6 U.S. Senate 7 U.S. Senate 8 U.S. Senate 9 U.S. Senate 10 U.S. Senate 11 U.S. Senate 12 U.S. Senate 13 U.S. Senate 14 U.S. Senate 15 U.S. Senate 16 U.S. Senate 17 U.S. Senate 18 U.S. Senate 19 U.S. Senate 20 U.S. Senate 21 U.S. Senate 22 U.S. Senate 23 U.S. Senate 24 U.S. Senate 25 U.S. Senate 26 U.S. Senate 27 U.S. Senate 28 U.S. Senate 29 U.S. Senate 30 U.S. Senate 31 U.S. Senate 32 U.S. Senate 33 U.S. Senate 34 U.S. Senate 35 U.S. Senate 36 U.S. Senate 37 U.S. Senate 38 U.S. Senate 39 U.S. Senate 40 U.S. Senate 41 U.S. Senate 42 U.S. Senate 43 U.S. Senate 44 U.S. Senate 45 U.S. Senate 46 U.S. Senate 47 U.S. Senate 48 U.S. Senate 49 U.S. Senate 50

Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political:

voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting

69

Weighted

N

L

fe

References

Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y

450 448 448 444 447 440 437 444 443 449 443 442 437 448 443 443 441 441 440 442 439 439 440 440 441 441 445 443 440 443 440 440 448 29 31 32 43 44 37 35 44 37 37 44 37 46 44 46 51 52 52 59 53 54 53 54 61 58 60 59 57 63 72 70 73 70 64 73 70 70 54 59 69 80 75 79 82 82 81 83 78 81 76

100,513 99,246 99,902 98,054 99,281 96,506 95,253 97,955 97,377 99,774 97,842 97,139 95,251 99,815 97,579 97,848 96,837 96,493 96,379 96,761 95,922 95,875 96,544 96,505 96,811 96,348 98,720 97,841 96,557 97,816 96,561 96,549 99,603 393 449 472 760 808 644 537 864 645 660 855 663 947 898 977 1,249 1,294 1,304 1,589 1,343 1,339 1,348 1,378 1,732 1,627 1,689 1,662 1,575 1,895 2,320 2,341 2,511 2,308 2,002 2,542 2,370 2,051 1,402 1,610 2,274 3,084 2,773 3,041 3,261 3,265 3,219 3,362 2,998 3,210 2,850

0.9949 0.9912 0.9977 0.9970 0.9960 0.9992 0.9999 0.9960 0.9946 0.9920 0.9994 0.9967 0.9998 0.9969 0.9967 0.9994 0.9981 0.9946 0.9979 0.9928 0.9977 0.9972 0.9996 0.9992 0.9978 0.9931 0.9993 0.9994 0.9998 0.9991 0.9998 0.9997 0.9948 0.9680 0.9656 0.9516 0.8416 0.8541 0.9670 0.9025 0.9133 0.9685 0.9910 0.9038 0.9955 0.9150 0.9493 0.9440 0.9796 0.9759 0.9834 0.9287 0.9746 0.9357 0.9782 0.9630 0.9464 0.9843 0.9542 0.9714 0.9868 0.9703 0.9077 0.9694 0.9555 0.9557 0.9931 0.9673 0.9814 0.8493 0.9797 0.9410 0.9693 0.9759 0.9993 0.9870 0.9819 0.9831 0.9935 0.9880 0.9983 0.9907 1.0000

(12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14)

ID

Name

Category

516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 571 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598

Roll call: U.S. Senate 51 Roll call: U.S. Senate 52 Roll call: U.S. Senate 53 Roll call: U.S. Senate 54 Roll call: U.S. Senate 55 Roll call: U.S. Senate 56 Roll call: U.S. Senate 57 Roll call: U.S. Senate 58 Roll call: U.S. Senate 59 Roll call: U.S. Senate 60 Roll call: U.S. Senate 61 Roll call: U.S. Senate 62 Roll call: U.S. Senate 63 Roll call: U.S. Senate 64 Roll call: U.S. Senate 65 Roll call: U.S. Senate 66 Roll call: U.S. Senate 67 Roll call: U.S. Senate 68 Roll call: U.S. Senate 69 Roll call: U.S. Senate 70 Roll call: U.S. Senate 71 Roll call: U.S. Senate 72 Roll call: U.S. Senate 73 Roll call: U.S. Senate 74 Roll call: U.S. Senate 75 Roll call: U.S. Senate 76 Roll call: U.S. Senate 77 Roll call: U.S. Senate 78 Roll call: U.S. Senate 79 Roll call: U.S. Senate 80 Roll call: U.S. Senate 81 Roll call: U.S. Senate 82 Roll call: U.S. Senate 83 Roll call: U.S. Senate 84 Roll call: U.S. Senate 85 Roll call: U.S. Senate 86 Roll call: U.S. Senate 87 Roll call: U.S. Senate 88 Roll call: U.S. Senate 89 Roll call: U.S. Senate 90 Roll call: U.S. Senate 91 Roll call: U.S. Senate 92 Roll call: U.S. Senate 93 Roll call: U.S. Senate 94 Roll call: U.S. Senate 95 Roll call: U.S. Senate 96 Roll call: U.S. Senate 97 Roll call: U.S. Senate 98 Roll call: U.S. Senate 99 Roll call: U.S. Senate 100 Roll call: U.S. Senate 101 Roll call: U.S. Senate 102 Roll call: U.S. Senate 103 Roll call: U.S. Senate 104 Roll call: U.S. Senate 105 Roll call: U.S. Senate 106 Roll call: U.S. Senate 107 Roll call: U.S. Senate 108 Roll call: U.S. Senate 109 Roll call: U.S. Senate 110 U.K. House of Commons voting: 1992-1997 U.K. House of Commons voting: 1997-2001 U.K. House of Commons voting: 2001-2005 U.N. resolutions 1 U.N. resolutions 2 U.N. resolutions 3 U.N. resolutions 4 U.N. resolutions 5 U.N. resolutions 6 U.N. resolutions 7 U.N. resolutions 8 U.N. resolutions 9 U.N. resolutions 10 U.N. resolutions 11 U.N. resolutions 12 U.N. resolutions 13 U.N. resolutions 14 U.N. resolutions 15 U.N. resolutions 16 U.N. resolutions 17 U.N. resolutions 18 U.N. resolutions 20 U.N. resolutions 21

Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political: Political:

voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting voting

70

Weighted

N

L

fe

References

Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y

91 93 95 90 96 93 90 93 93 95 102 109 101 100 111 101 105 102 105 102 109 103 100 100 102 104 108 104 107 97 108 98 110 99 101 103 105 103 103 101 102 102 103 101 104 101 101 101 101 101 100 102 102 103 100 102 102 100 101 102 668 671 657 54 57 59 59 60 60 60 60 60 65 81 82 82 82 99 104 110 113 117 122

3,998 4,249 4,413 4,000 4,445 4,201 3,939 4,174 4,251 4,382 5,033 5,719 5,029 4,931 5,899 5,005 5,413 5,081 5,353 5,082 5,779 5,220 4,879 4,933 5,126 5,106 5,575 5,304 5,466 4,655 5,646 4,748 5,724 4,845 5,014 5,246 5,444 5,249 5,247 5,048 5,148 5,147 5,246 5,049 5,345 5,049 5,049 5,049 5,049 5,049 4,950 5,148 5,080 5,247 4,950 5,148 5,148 4,950 5,049 5,147 220,761 223,092 215,246 1,431 1,594 1,711 1,711 1,770 1,768 1,770 1,770 1,770 2,037 3,239 3,317 3,294 3,321 4,851 5,356 5,995 6,246 6,672 7,333

0.9763 0.9932 0.9884 0.9988 0.9748 0.9820 0.9835 0.9757 0.9937 0.9814 0.9771 0.9716 0.9958 0.9962 0.9663 0.9911 0.9914 0.9864 0.9804 0.9866 0.9818 0.9937 0.9857 0.9966 0.9951 0.9533 0.9649 0.9903 0.9639 0.9998 0.9772 0.9989 0.9548 0.9988 0.9929 0.9987 0.9971 0.9992 0.9989 0.9996 0.9994 0.9992 0.9987 0.9998 0.9979 0.9998 0.9998 0.9998 0.9998 0.9998 1.0000 0.9994 0.9862 0.9989 1.0000 0.9994 0.9994 1.0000 0.9998 0.9992 0.9909 0.9925 0.9988 1.0000 0.9987 1.0000 1.0000 1.0000 0.9989 1.0000 1.0000 1.0000 0.9793 0.9997 0.9988 0.9919 1.0000 1.0000 1.0000 1.0000 0.9870 0.9832 0.9935

(12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (12, 13, 14) (11) (11) (11) (73) (73) (73) (73) (73) (73) (73) (73) (73) (73) (73) (73) (73) (73) (73) (73) (73) (73) (73) (73)

ID

Name

Category

599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 651 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681

U.N. resolutions 22 U.N. resolutions 23 U.N. resolutions 24 U.N. resolutions 25 U.N. resolutions 26 U.N. resolutions 27 U.N. resolutions 28 U.N. resolutions 29 U.N. resolutions 30 U.N. resolutions 31 U.N. resolutions 32 U.N. resolutions 33 U.N. resolutions 34 U.N. resolutions 35 U.N. resolutions 36 U.N. resolutions 37 U.N. resolutions 38 U.N. resolutions 39 U.N. resolutions 40 U.N. resolutions 41 U.N. resolutions 42 U.N. resolutions 43 U.N. resolutions 44 U.N. resolutions 45 U.N. resolutions 46 U.N. resolutions 47 U.N. resolutions 48 U.N. resolutions 49 U.N. resolutions 50 U.N. resolutions 51 U.N. resolutions 52 U.N. resolutions 53 U.N. resolutions 54 U.N. resolutions 55 U.N. resolutions 56 U.N. resolutions 57 U.N. resolutions 58 U.N. resolutions 59 U.N. resolutions 60 U.N. resolutions 61 U.N. resolutions 62 U.N. resolutions 63 Biogrid: A. thaliana Biogrid: C. elegans Biogrid: D. melanogaster Biogrid: H. sapien Biogrid: M. musculus Biogrid: R. norvegicus∗ Biogrid: S. cerevisiae Biogrid: S. pombe DIP: H. pylori DIP: H. sapien DIP: M. musculus DIP: C. elegans Human: Ccsb Human: Ophid STRING: C. elegans STRING: S. cerevisiae Yeast: Oxford Statistics Yeast: DIP Yeast: DIPC Yeast: FHC Yeast: FYI Yeast: PCA Corporate directors in Scotland (1904-1905)∗ Corporate ownership (EVA) Dolphins∗ Family planning in Korea Unionization in a hi-tech firm∗ Communication within a sawmill on strike∗ Leadership course Les Miserables∗ Marvel comics Mexican political elite Pretty-good-privacy algorithm users Prisoners Bernard and Killworth fraternity: observed Bernard and Killworth fraternity: recalled Bernard and Killworth HAM radio: observed Bernard and Killworth HAM radio: recalled Bernard and Killworth office: observed Bernard and Killworth office: recalled Bernard and Killworth technical: observed

Political: voting Political: voting Political: voting Political: voting Political: voting Political: voting Political: voting Political: voting Political: voting Political: voting Political: voting Political: voting Political: voting Political: voting Political: voting Political: voting Political: voting Political: voting Political: voting Political: voting Political: voting Political: voting Political: voting Political: voting Political: voting Political: voting Political: voting Political: voting Political: voting Political: voting Political: voting Political: voting Political: voting Political: voting Political: voting Political: voting Political: voting Political: voting Political: voting Political: voting Political: voting Political: voting Protein interaction Protein interaction Protein interaction Protein interaction Protein interaction Protein interaction Protein interaction Protein interaction Protein interaction Protein interaction Protein interaction Protein interaction Protein interaction Protein interaction Protein interaction Protein interaction Protein interaction Protein interaction Protein interaction Protein interaction Protein interaction Protein interaction Social Social Social Social Social Social Social Social Social Social Social Social Social Social Social Social Social Social Social

71

Weighted

N

L

fe

References

Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y N N N N N N N N N N N N N N N N N N N N N N Y N N N N N N Y Y N N N Y Y Y Y Y Y Y

124 126 126 126 132 132 134 137 143 144 146 148 150 151 155 156 157 158 158 158 158 158 158 154 168 174 178 174 179 180 176 177 174 182 179 187 189 191 191 192 192 192 406 3,353 7,174 8,205 710 121 1,753 1,477 686 639 50 2,386 1,307 5,464 1,762 534 2,224 4,906 2,587 2,233 778 889 131 4,475 62 33 33 36 32 77 6,449 35 10,680 67 58 58 41 44 40 40 34

7,616 7,855 7,851 7,868 8,641 8,646 8,905 9,202 10,117 10,291 10,585 10,878 11,173 11,287 11,935 12,090 12,243 12,403 12,403 12,403 12,402 12,403 12,403 11,781 13,872 14,944 15,606 14,913 15,826 16,096 15,349 15,500 14,970 16,333 15,812 17,373 17,735 18,140 18,110 18,331 18,331 18,328 625 6,449 24,897 25,699 1,003 135 4,811 11,404 1,351 982 55 3,825 2,483 23,238 95,227 57,672 6,609 17,218 6,094 5,750 1,798 2,407 676 4,652 159 68 91 62 80 254 168,211 117 24,316 142 967 1,653 153 442 238 779 175

0.9987 0.9975 0.9970 0.9991 0.9994 1.0000 0.9993 0.9878 0.9965 0.9995 1.0000 1.0000 0.9998 0.9966 1.0000 1.0000 0.9998 1.0000 1.0000 1.0000 0.9999 1.0000 1.0000 1.0000 0.9889 0.9929 0.9907 0.9908 0.9934 0.9991 0.9967 0.9951 0.9946 0.9916 0.9925 0.9990 0.9983 0.9997 0.9981 0.9997 0.9997 0.9996 0.0076 0.0011 0.0010 0.0008 0.0040 0.0186 0.0031 0.0105 0.0058 0.0048 0.0449 0.0013 0.0029 0.0016 0.0614 0.4053 0.0027 0.0014 0.0018 0.0023 0.0059 0.0061 0.0794 0.0005 0.0841 0.1288 0.1723 0.0984 0.1613 0.0868 0.0081 0.1966 0.0004 0.0642 0.5850 1.0000 0.1866 0.4672 0.3051 0.9987 0.3119

(73) (73) (73) (73) (73) (73) (73) (73) (73) (73) (73) (73) (73) (73) (73) (73) (73) (73) (73) (73) (73) (73) (73) (73) (73) (73) (73) (73) (73) (73) (73) (73) (73) (73) (73) (73) (73) (73) (73) (73) (73) (73) (9) (9) (9) (9) (9) (9) (9) (9) (15, 16) (15, 16) (15, 16) (15, 16) (74) (75, 76) (17) (17) (77) (15, 16, 77) (15, 16, 77) (78, 77) (79, 77) (80, 77) (81, 28) (82) (20) (83) (84) (85) (57) (56) (86) (87) (88) (57) (89, 90, 91) (89, 90, 91) (92, 93, 94) (92, 93, 94) (92, 93, 94) (92, 93, 94) (92, 93, 94)

†

ID

Name

Category

Weighted

N

L

fe

References

681 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752

Bernard and Killworth technical: recalled Kapferer tailor shop: instrumental (t1) Kapferer tailor shop: instrumental (t2) Kapferer tailor shop: associational (t1) Kapferer tailor shop: associational (t2) University Rovira i Virgili (Tarragona) e-mail Zachary karate club∗ BA: (100,1)∗ BA: (100,2)∗ BA: (1000,1) BA: (1000,2) BA: (500,1) BA: (500,2) ER: (100,25)∗ ER: (100,50) ER: (100,75) ER: (1000,25) ER: (1000,50) ER: (1000,75) ER: (50,25) ER: (50,50) ER: (50,75) ER: (500,25) ER: (500,50) ER: (500,75) Fractal: (10,2,1) Fractal: (10,2,2) Fractal: (10,2,3) Fractal: (10,2,4) Fractal: (10,2,5) Fractal: (10,2,6) Fractal: (10,2,7) Fractal: (10,2,8) H13-4 benchmark LF benchmark: (1000,15,50,0.1,2,2) LF benchmark: (1000,15,50,0.1,3,1) LFR benchmark: (1000,15,50,0.5,2,2) LFR benchmark: (1000,15,50,0.5,3,1) LFR benchmark: (1000,25,50,0.1,2,2) LFR benchmark: (1000,25,50,0.1,3,1) LFR benchmark: (1000,25,50,0.5,2,2) LFR benchmark: (1000,25,50,0.5,3,1) LF benchmark: (1000,15,50,0.1,0.1,1,2,1) LF benchmark: (1000,15,50,0.1,0.1,1,2,2) LF benchmark: (1000,15,50,0.5,0.1,1,2,1) LF benchmark: (1000,15,50,0.5,0.1,2,2,2) LF benchmark: (1000,15,50,0.5,0.5,1,2,1) LF benchmark: (1000,15,50,0.5,0.5,1,2,2) LF benchmark: (1000,25,50,0.1,0.1,1,2,1) LF benchmark: (1000,25,50,0.1,0.1,2,2,2) LF benchmark: (1000,25,50,0.5,0.1,1,2,1) LF benchmark: (1000,25,50,0.5,0.1,2,2,2) LF benchmark: (1000,25,50,0.5,0.5,1,2,1) LF benchmark: (1000,25,50,0.5,0.5,2,2,2) LF-NG benchmark Random fully-connected: (100) Random fully-connected: (500) WS: (100,1,0.1) WS: (100,1,0.5) WS: (100,4,0.1) WS: (100,4,0.5) WS: (1000,1,0.1) WS: (1000,1,0.5) WS: (1000,4,0.1) WS: (1000,4,0.5) KOSKK:(1000,1,10,10,5 × 10−5 ,1 × 10−3 ,100) KOSKK:(1000,1,10,10,5 × 10−5 ,1 × 10−3 ,1000) KOSKK:(1000,1,100,10,5 × 10−5 ,1 × 10−3 ,1000) KOSKK:(1000,1,100,105 × 10−5 ,1 × 10−3 ,100) KOSKK:(1000,1,50,10,5 × 10−5 ,1 × 10−3 ,100) KOSKK:(1000,1,50,10,5 × 10−5 ,1 × 10−3 ,1000)

Social Social Social Social Social Social Social Synthetic Synthetic Synthetic Synthetic Synthetic Synthetic Synthetic Synthetic Synthetic Synthetic Synthetic Synthetic Synthetic Synthetic Synthetic Synthetic Synthetic Synthetic Synthetic Synthetic Synthetic Synthetic Synthetic Synthetic Synthetic Synthetic Synthetic Synthetic Synthetic Synthetic Synthetic Synthetic Synthetic Synthetic Synthetic Synthetic Synthetic Synthetic Synthetic Synthetic Synthetic Synthetic Synthetic Synthetic Synthetic Synthetic Synthetic Synthetic Synthetic Synthetic Synthetic Synthetic Synthetic Synthetic Synthetic Synthetic Synthetic Synthetic Synthetic Synthetic Synthetic Synthetic Synthetic Synthetic

Y N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y N N N N N N N N Y Y Y Y Y Y

34 35 34 39 39 1,133 34 100 100 1,000 1,000 500 500 100 100 100 1,000 1,000 1,000 50 50 50 500 500 500 1,024 1,024 1,024 1,024 1,024 1,024 1,024 1,024 256 1,000 1,000 1,000 1,000 1,000 1,000 1,000 1,000 1,000 1,000 1,000 1,000 1,000 1,000 1,000 1,000 1,000 1,000 1,000 1,000 128 100 500 100 73 100 100 850 877 1,000 1,000 519 895 870 652 459 851

561 76 93 158 223 5,451 78 99 197 999 1,997 499 997 1,264 2,436 3,697 124,455 249,512 374,846 287 589 936 31,148 62,301 93,780 9,256 16,875 30,344 53,009 89,812 147,784 232,794 343,563 2,311 7,573 7,447 7,624 7,177 12,739 12,523 12,744 12,662 7,680 7,791 7,657 7,912 7,693 7,906 12,660 12,641 12,771 12,772 12,962 12,881 1,024 4,950 124,750 100 73 407 522 850 877 4,053 5,138 2,096 7,682 4,725 2,125 1,554 4,960

1.0000 0.1277 0.1658 0.2132 0.3009 0.0085 0.1390 0.0200 0.0398 0.0020 0.0040 0.0040 0.0080 0.2554 0.4921 0.7469 0.2492 0.4995 0.7504 0.2343 0.4808 0.7641 0.2497 0.4994 0.7517 0.0177 0.0322 0.0579 0.1012 0.1715 0.2822 0.4445 0.6559 0.0708 0.0152 0.0149 0.0153 0.0144 0.0255 0.0251 0.0255 0.0253 0.0154 0.0156 0.0153 0.0158 0.0154 0.0158 0.0253 0.0253 0.0256 0.0256 0.0259 0.0258 0.1260 1.0000 1.0000 0.0202 0.0278 0.0822 0.1055 0.0024 0.0023 0.0081 0.0103 0.0156 0.0192 0.0125 0.0100 0.0148 0.0137

(92, 93, 94) (95) (95) (95) (95) (96) (25) (18) (18) (18) (18) (18) (18) (21) (21) (21) (21) (21) (21) (21) (21) (21) (21) (21) (21) (23) (23) (23) (23) (23) (23) (23) (23) (97) (22) (22) (22) (22) (22) (22) (22) (22) (42) (42) (42) (42) (42) (42) (42) (42) (42) (42) (42) (42) (4, 42) [† ] [† ] (29) (29) (29) (29) (29) (29) (29) (29) (40) (40) (40) (40) (40) (40)

See the description at the beginning of this appendix for details of this network.

72