Discovering Community Structure in Dynamic Social Networks using ...

2014 ASE BIGDATA/SOCIALCOM/CYBERSECURITY Conference, Stanford University, May 27-31, 2014

Discovering Community Structure in Dynamic Social Networks using the Correlation Density Rank Zeynab Bahrami Bidoni 1, Roy George2 Department of Computer and Information Systems Clark Atlanta University, Atlanta, GA

[email protected] 1; [email protected] 2

changing organizational structure, such as who is becoming more powerful or the shifting of the power structure.

Abstract Recent research has produced advances in the understanding of communities within a dynamic social network. A “community” in this context is defined as a subgraph with a higher internal density and a lower crossing density with respect to other subgraphs. In this paper, we describe a novel and efficient distance based ranking algorithm, called the “Correlation Density Rank” (CDR), which is utilized to derive the community tree from the social network and to develop a tree learning algorithm that is employed to construct an evolving community tree. Also, we present an evolution graph of the organizational structure, through which new insights into the dynamic network may be obtained. The experiments, conducted on a datasets, both synthetic and real, demonstrate the feasibility and applicability of the framework. Keywords: Dynamic social network; Organizational structure; Community discovery; Evolution analysis; Web ranking; Crawling; Correlation Density Rank.

1. Introduction Community detection is an important research issue in social network analysis (SNA), where the objective is to recognize related sets of members such that intra-community associations are denser than inter-communities associations [110]. Researchers have presented various methods to extract communities from a Social Network (SN) data. . In particular, discovering the organizational structure of communities in an SN has been identified as an interesting but challenging problem [11,12] Examples of important applications include characterizing potential candidates for viral marketing, finding members of criminal groups, discovering affinity groups, etc. [12] While there has been research on finding key members in an SN [11-15] the results have limited power to supply a complete view of the organizational structure. In the real world, social networks are constantly changing and evolving. New members may join the network, existing members may quit from the network, and associations among members constantly change over time. Therefore, the approach should be capable of supporting the exploration of organizational structure dynamically. Some earlier research has provided approaches to detecting communities from a dynamic social network [16-19] These approaches discover changes of communities in an SN, but do not answer questions related to

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In the workflow area, organization mining has also been the focus of past research [20-22]. The workflow area research has emphasized the exploration of organizational structure in the unit from event logs of information systems. Research efforts [23-25] have also addressed determining the hierarchy in an SN, where hierarchy has a similarity to an organizational structure. In this paper we use the notion of a community tree data structure to represent organizational structure and its evolution. This approach is similar to that presented by Qui and Lin [26] where the ranking of nodes is based on PageRank algorithm with iteration complexity of the order of logN [27]. However, the technique presented here has no iteration complexity and is not susceptible to network anomalies [28] such as spider-traps, dead-ends, etc. and the “rich-get-richer” problem [29]. Consequently, the approach in this paper is algorithmically efficient and produces communities and organizational structures with better accuracy. We propose a novel density measure, the Correlation Density Rank (CDR), as the basis of community and organizational structure detection. We apply a density based method to describe the relationship between nodes linked by edges. In comparison with other earlier Density based algorithms [30-32], this approach offers several advantages (1) the CDR is a generalized formulation, which permits the weighting of different correlation types, . (2) a more realistic solution where with important neighbors have more priority than lesser ones is employed, and (3) optimization of the CDR is easier, since the number of parameters needed for tuning is smaller. The contributions of this paper are as follows: (1) Developing an algorithm to derive the community tree from the SN. (2) Developing a tree learning algorithm to generate the evolving community trees. (3) Proposing an approach for representing the evolution of the organizational structure based on the evolving community trees. The rest of the paper is organized as follows. Section 2 introduces the concept of community tree and proposes an approach for deriving the community tree from a static social network. Section 3 presents the methods for analyzing the evolution of organizational structure in a dynamic SN. Section 4 provides the experimental results. Section 5 offers concluding remarks.

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2014 ASE BIGDATA/SOCIALCOM/CYBERSECURITY Conference, Stanford University, May 27-31, 2014

2. Discovering the organizational structure in a static social network 2.1 Organization structure of a network Song and Van [33] developed a definition of organizational model, where the organizational model consists of organizational units (e.g. functional units), roles (e.g. duty), originators, and relationships (e.g., hierarchy). In this paper, we assume that the organizational structure of an SN is a hierarchy that represents communities (or units) and subordinations of members in the SN. However, the derived hierarchical organizational structure of an SN is not necessarily reflected in a real-world organization [26]. For instance, there exists real organizations that are cannot be characterized as an academic network, nor in a blogosphere. In this case for a network formed by e-mail communications, we may not be able to exactly determine to what extent the relationships can be mapped to the real-world organizational structure. Therefore, in deriving the organizational relationships, subordination describes the relationship between two members where the leader is the most likely and more important destination of information flow starting from the subordinate. i.e., the subordinate has a closest interaction with the leader in comparison with others while the leader has a higher possibility of attracting interaction with the nodes across the entire network in comparison to the subordinate nodes. The importance of members in the SN, is depicted through a score, the m-Score, which is equivalent to CDR value of a member. The m-Score value of a member would be its certainty on attracting interaction with all nodes through whole network. The higher the score is, the more important the member is. We use a data structure, called the community tree, to represent the SN organizational structure. Definition 1. (Community Tree): Let N  n1,..., n k  be a collection of members in an SN, CT is a tree, and NULL is the root of the tree, and every member in N is referred to as a node in the tree. Each member ni in CT it has a unique parent node

n j where m-Score ( n j ) > m-Score ( ni ). If the parent node of

ni is the root node NULL, ni is called a core of the tree. A core and its descendants compose a community. To derive the organizational structure, we calculate the mScore for every member and then attempt to find the immediate leader of every member in a network. Further, we construct the community tree using m-Score and subordinations. After having constructed the SN community tree, we can discover communities and obtain the SN organizational structure. An example of a community tree is illustrated in Fig. 1 where node 1 and node 5 are cores of community 1 and community 2, respectively.

This research is funded in part by the Army Research Laboratory under Grant No: W911NF-12-2-0067 and Army Research Office under Grant Number W911NF-11-1-0168. Any opinions, findings, conclusions or recommendations expressed here are those of the author(s) and do not necessarily reflect the views of the sponsor.

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Figure 1. An example of a community tree [26].

SN may be regarded as a graph indicating information flows among members. The relationship of SN members can be obtained by analyzing information flows [34]. The process of determining SN information flow is similar to a random walk on a graph [35,36]. Given a graph and a starting point, the starting point's neighbor is selected at random, and the next start point is moved to this neighbor then a neighbor of this new start point is selected again at random and so on. The random sequence of points generated in this process is a random walk on the graph. The expected lengths of random walks on the graph, can be used to derive randomized shortest paths (RSP) dissimilarity [37,38]. The RSP dissimilarity, which has its foundation in statistical physics, may be used to compute the shortest path distance for all pairs of nodes of a graph in closed form. Recently [39] has generalized distance from the RSP framework based on the Helmholtz free energy between two states of a thermodynamic system with a distance measure, the free energy distance. We employ the RSP measurement method in [39] as the distance between nodes, but with one major difference: we consider customized initial cost for edges such that, along with finding shortest path between nodes. The random walker intelligently selects the most important neighbor resulting in lower cost and smaller distance. Combining RSP with m-Score of every node, we can find an immediate leader for every member in an SN (see section 2.3). The SN community tree is derived in this way. Our framework includes the following steps to derive the community tree: a) Employ the novel “Correlation Density Rank” method to ranking nodes as the m-Score for every node in an SN; and, b) Combine RSP with m-Score of every node to derive a community tree. 2.2 Calculating m-Score To calculate the m-Score for every node, we need to investigate each node's importance in a network. Those nodes that link many important nodes are also themselves important. Such a process is very similar to PageRank based algorithms [40]. PageRank is a link analysis algorithm that produces a global “importance” ranking for every web page by analyzing

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links among web pages. A fast and efficient page ranking mechanism for web crawling and retrieval remains a challenging issue. Recently, several link based ranking algorithms like PageRank, HITS, OPIC and etc. [27] have been proposed. We propose a novel method, Correlation Density Rank, based on finding more frequent and influential RSP. The CDR considers the entropy of distance between nodes as punishment and is used to compute ranks of nodes. Hence, there will be a larger traffic amongst shortest path of nodes, if the distance becomes smaller. As proved in [41], if the distance between i and j was less than the distance between i and k, then, i’s rank effect on j is more than on k, in other words, the probability that a random surfer reach j from i is more than the probability to reach k. Therefore, the objective is to minimize punishment so that a node with less distance entropy to have a higher rank. The Shannon entropy is a measurement of system uncertainty [42]. The larger the Shannon entropy is, the more uncertainty the system will be. If the CDR value of a node in complex network is the smallest, then the uncertainty of its distance distribution from other nodes is the greatest. On the contrary, while the CDR value of a node in complex network is very high, then the uncertainty of its distance distribution from other nodes is the smallest. Moreover, the more popular nodes are, the more linkages other nodes tend to have to them or are linked to by them. The proposed algorithm is analogous to the weighted PageRank algorithm [43], assigning larger rank values to more important (popular) nodes instead of dividing the rank value of a node evenly among its out-link nodes. We assign each out-link node a value proportional to its popularity (its number of in-links and out-links). The popularity from the number of in-links and out-links is recorded as W ijin and W ijout , respectively.

On j

W ijout 



Op

p R ( ni )

Where O n j and O p represent the number of out-links of node n j and node p, respectively. R( ni ) denotes the reference node list of node ni . These equations has two exceptions, first, if node n j is a dead-end (which may be easily determined from the frequency matrix), we let W ijout   that  is a very small number less than 1. Second, W ijout , W ijin  1 , that means R( ni )={ n j } we add 

to sum of the reference nodes’

frequency out/in-link. An algorithm for calculating the mScore of members in a social network is described as follows. Algorithm 1. Calculating m-Score for members: Correlation Density Rank (CDR) Input: social network G Out: vector of m-Score for all members R 1.

Initialize cost distance matrix C

C [ i , j ]  lo g

(1  e x p (   f ij ))

1  w

in out ij w ij



(The logarithm of (1  exp( f ij )) based on (1  w ijinw ijout ) ) 2.

W ijin is the weight of link between node ni and n j

Finding the matrix of RSP dissimilarities by employ the algorithm of [43]: { W ← P ref  exp(  C ) 1

calculated based on the number of in-links of node n j and the

Z ←  I W  (Note that  I W 1  I

number of in-links of all reference nodes of node ni .

S ← Z C W  Z  (Z   )



I nj

W ijin 



Where I n j and I p represent the number of frequency inlinks of node n j and node p, respectively. R( ni ) denotes the reference node list of node ni .

2

W

3

 ...)



C ← S  ed sT T  R S P ← C  (1   )C

Ip

p R ( ni )

0   1

}

3.

M ← Normalize matrix  R S P on rows

4.

For each node ni ( 1  i  k ) compute the entropy of related row from matrix M: 1 k Ei ←   M ij Ln (M ij ) Lnk j 1

W ijout is the weight of link between node ni and n j calculated based on the number of frequency out-links of node n j and the number of out-links of all reference nodes of node

di ← 1 Ei Ri ← d i

ni . 5.

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W  W



k i 1

di

Return R

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Where f ij is the number of frequency from node ni to node

n j . if f ij  0 , let C [i , j ]   or a very big number. P ref is the transition probability matrix that Pijref is equal to the rate of f ij divided by sum of frequency between node ni and all its references nodes. k is the number of members in social network (or nodes on G). The parameters  ,  and  are input values determined by user.  controls the effect of frequency on the cost function which restrict cost ratio with respect to our defined infinite constant. β is the influence of the cost on the walker’s selection of a path, and is equal to inverse of temperature at Helmholtz free energy in thermodynamical system [39].  is the weight factor by which the leadership status of members will be distinguished by the amount of contact density comes in or goes out. Also, in step 2, d s  diag (S ) is the vector of diagonal elements of S, and e is the identity matrix. Note that A ◦ B and A ÷ B are elementwise product and division, respectively. For calculating step 2, we use the easier way of computing the matrix Z [38]. The values of Ri ( 1  i  k ) indicate the final m-Scores of members in the social network. 2.3 Deriving the Community Tree The m-Score of every member is combined with the normalized RSP matrix (M) on the graph to derive the community tree from an SN. The RSP matrix helps us to find the most likely and closest interaction for each node, and mScores determine whether there is the leadership relation between two nodes with closest interaction. Algorithm 2. Deriving Community Tree: CT_Deriving Input: Social network G Output: Community Tree CT 1. CT← [null,…,null] 2. R ← Correlation Density Rank (G); 3. For each member ni {

node of node i. After all the operations end, we obtain a community tree CT represented in an array where CT[i] indicates the parent node of node i. A null value of CT[i] indicates that there is no the immediate leader of node i. Hence node i is the core of community. If a node does not have both immediate leader and belongingness, is called private node. Node i and its descendent compose a community. The mScore of each member shows the member's importance in the community. The parent of each node is its immediate leader. 3. Exploring evolution of the organizational structure in a dynamic social network In Section 2, we develop the algorithms to derive a static community tree from a static SN. However, the static community tree does not do a good job in presenting the evolution of the organizational structure in the dynamic SN, because it does not consider intra time-step evolutions. To present the evolution of organizational structure in a dynamic SN, we aggregate the change of an SN during different time periods, and then derive community trees. We further construct the evolving community tree from the two closest community trees. Hence, the evolving community tree can accurately present evolution of organizational structure over time. The Best-first search algorithm explores a graph by expanding the most promising node chosen according to a specified rule [44]. Motivated by the idea of the Best-first search, [26] uses tree edit distance [45] (defined as the least cost of edit operation to change a tree to another tree) as a measure of distance (similarity) between the two trees. We propose a new algorithm to derive the evolving community tree with more efficiency and less complexity than previous ways. Because constructing community tree in our idea is based on RSP, we only need to compute linear combination of RSP of two static community trees and use the new RSP to drive evolving CT using algorithm (2) without any iteration. However, [26] generated a collection of candidate evolving community trees and then chose the candidate having the minimum ES score (by means of scoring function, which measures distance errors among evolving CT, previous period CT and current period CT) as a solution of each iteration. 3.1 Tree learning algorithm We propose a tree learning algorithm to derive an evolving community tree from two static community trees. The constructing process is as follows:

k← arg j min (M ij )

(1) Obtain a collection of members in the evolving community tree N ce  N pre  N cs where N pre and N cs are

if R[k] > R[i] CT[i] ← k } 4. Return CT

collections of members in the previous time period static community tree and current time periods one respectively;

Initiating a random walks at node i, we can find ending node j with the most likely correlation density, from the normalized RSP matrix M, by starting. If the m-Score of node j is much greater than that of node i, we regard j as the parent

(2) Compute the RSP for all pair members in the evolving community tree, where α is a smoothing factor, RSP ceRSP  (1   )   RSP . For those members that pre    cs

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appear in the evolving community tree but not in the current community tree, if their m-scores are less than a threshold , we regard them as retired members and remove them from the evolving community tree; (3) According to the definition of algorithm 1, having  matrix, we can compute Rce , M ce matrices. Then, by applying them into Algorithm 2, we construct the evolving community tree. RSP ce

Algorithm 3. Learning evolving ECT_Learning RSP Input: RSP matrices  RpreSP , cs

community

tree:

c i 1 is the evolving entity of ci . We say c1 ,...,c n  is a lifeline of the community c1 . Community c k C is called the entity of the life-line. A life-line depicts an evolving process of one community in the dynamic SN. Definition

3.

(supporter):

Given

a

life-line

CL  c1,c 2 ,...,c n  , we call the members as supporters of CL if they appear in CL not less than δ times with   n . In Definition 3,  is a parameter set by the user. If there is a life-line CL  c1,c 2 ,c3 ,c 4  and   3 , that means only members appearing in the life-line not less than 3 times are supporters of CL . Exploring life-lines in the evolution graph and their supporters helps to better understand dynamic social networks. For example, we can discover the backbone of criminal group or detect loyal members in a forum over time.

Output: Evolving community tree CT e 1.

N ce  N pre  N cs

2.

RSP  ceRSP  (1   )   RSP pre     cs

3.

N ce  N ce  n i  N ce  N cs m  score (n i )   

4. 5. 6.

Do step 3,4 of algorithm 1 to find Rce , M ce Employ algorithm2 to find CT ce Return CT ce

3.2 Exploring dynamic social network After deriving a series of evolving community trees, we exploit these trees to discover the evolution path of the organizational structure and to study the properties of the dynamic SN. We can consider four types of relationships among communities to generate an evolution graph that represent evolution of the organizational structure. Fig. 2 provides examples to illustrate the relationships among communities [26].

Definition 4. (Activeness of community): Let node p be the core of community c , we use the m -Score of p to indicate the activeness of community c . Activeness of community is a metric for the extent to which associations occur among members of the community over one period of time. We can use Activeness of community to reveal hot communities, which may reflect on-going hot topics in the forum or new activities in a criminal group. 4. Experiments In order to implement our approach, first, we consider a small dynamic, synthetic network (Fig. 3) during five time intervals.

Figure 2. Relationships between communities.

Combing evolution graph and dynamic SN properties, we can obtain insights into the dynamic SN. Definitions 2–4 define key properties of the dynamic SN [26]. Definition 2. (life-line): Let C  c1 ,...,c n  be a collection of communities. For each community c i C with i  n ,

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Figure 3. A small dynamic network with frequency contacts between nodes

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After employing algorithms 1 and 2, we find static community trees for each periods of time separately that shown in Fig. 4 The arrows indicate leadership, and The parameters let   0.1,   0.001,   0.5,   0.01,   0.7 .

Now, in this stage, we have information about leadership during each periods of time separately that is not enough for the complete analysis. So, with the purpose of achieve a good understanding of the network’s changing trends during intra time-step evolutions, we employ algorithms 3 to drive the evolving community trees shown in Fig. 6.

Figure 6. The evolving community trees for the synthetic network

The analysis of outputs clarifies some events that were happened during time:

Figure 4. Static community trees for each periods of time separately.

Also, the trend of member’s activity in Fig. 5 May be seen.

0.35 0.3 Activity

0.25

A

0.2

B

0.15

C

0.1

D

0.05

E

0 0

2 4 periods of time

6 Figure 7. Evolution of communities for the synthetic network.

Figure 5. Evolving member’s activity for the synthetic network

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a) From T1 to T2, the leader of member E changes from C to B. b) From T2 to T3, leader of members A and E change from B to D. c) From T3 to T4, 1) splitting on B, C. 2) evolving between C, D. 3) emerging the core D related to members B, E. d) From T4 to T5, splitting on member E.

Fig. 7 Shows the evolution map of communities in which Community that its core is D has most stability (number of

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supporters divided by the size, named stability. Size is the average number of members in each community in life-lines.) among all communities and its supporters are members B and C. While the straightforward application of this method is in social networks, this technique is appropriate for all type of complex networks, and the type of network does not influence the results. We have used the real frequency data from a computer network of 288 nodes to evaluate this approach. Data for a period of 100 seconds, divided into 10 equal periods of 10 seconds each, is used to construct evolving community trees and draw evolution map of communities they are. After employing algorithms 1, 2 and 3, we have reached evolving community trees as shown in Appendix. The arrows indicate hierarchical leadership between nodes, and the parameters let   0.1,   0.001,   0.5,   0.01,   0.7 .

and a real world datasets, and produce good results. The experiments show that the framework can well present the organizational structure of a social network. We obtained experimentally the following insights: (1) those communities with long life-line and great stability likely correspond to a real organization; and (2) the cores in an organizational structure, in general, are either the leaders of the organization or the agents of these leaders. Although it is possible that the organizational structure discovered from a social network is not perfectly in line with the real world organization, the approach described here helps reach new understandings of the organization based on the power of attracting information flow and the interaction closeness.

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[4]

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Figure 8. Evolution of communities for the real data set.

Fig. 8 Shows the evolution map of communities, in which the community that its core is node 82 has 185 size, longer lifeline (30 second) and the largest stability with amount of 0.7 among all communities. As expected, the algorithm identifies routers and hubs as the cores of communities.

[8]

[9]

[10]

5. Conclusions Exploring organizational structure in a dynamic social network has a broad range of applications, such as monitoring gang activities, fraud detection, and improving performance of viral marketing. In this paper, we present our research effort in extracting organizational structure from such data to obtain a better understanding of the social network. We formalize a community tree data structure for the purpose of representing the social network organizational structure, and propose a framework to explore the dynamic behavior of the participants of the community. The framework is composed of three main parts: (1) defining the “Correlation Density Rank”, to rank the nodes to acquire a community tree from the static social network; and, (2) a tree learning algorithm, which employs the tree edit distance as a scoring function, to generate the evolving community tree; These algorithms were applied to a synthetic

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2014 ASE BIGDATA/SOCIALCOM/CYBERSECURITY Conference, Stanford University, May 27-31, 2014 [17] Tang, L., et al. Community evolution in dynamic multi-mode networks. in Proceedings of the 14th ACM SIGKDD international conference on Knowledge discovery and data mining. 2008. ACM. [18] Tantipathananandh, C., T. Berger-Wolf, and D. Kempe. A framework for community identification in dynamic social networks. in Proceedings of the 13th ACM SIGKDD international conference on Knowledge discovery and data mining. 2007. ACM. [19] [19]. Zhou, D., et al. Discovering temporal communities from social network documents. in Data Mining, 2007. ICDM 2007. Seventh IEEE International Conference on. 2007. IEEE. [20] Bertino, E., E. Ferrari, and V. Atluri, The specification and enforcement of authorization constraints in workflow management systems. ACM Transactions on Information and System Security (TISSEC), 1999. 2(1): p. 65-104. [21] Song, M. and W.M. van der Aalst, Towards comprehensive support for organizational mining. Decision Support Systems, 2008. 46(1): p. 300317. [22] Zur Muehlen, M., Organizational management in workflow applications–issues and perspectives. Information Technology and Management, 2004. 5(3-4): p. 271-291. [23] Clauset, A., C. Moore, and M.E. Newman, Hierarchical structure and the prediction of missing links in networks. Nature, 2008. 453(7191): p. 98-101. [24] Grobelnik, M., D. Mladenic, and B. Fortuna, Semantic technology for capturing communication inside an organization. Internet Computing, IEEE, 2009. 13(4): p. 59-67. [25] Li, H., et al. Scalable community discovery on textual data with relations. in Proceedings of the 17th ACM conference on Information and knowledge management. 2008. ACM. [26] Qiu, J. and Z. Lin, A framework for exploring organizational structure in dynamic social networks. Decision Support Systems, 2011. 51(4): p. 760-771. [27] Singh, A.K. and P. Ravi Kumar, A comparative study of page ranking algorithms for information retrieval. International journal of electrical and computer engineering, 2009. 4(7): p. 469-480. [28] Rajaraman, A. and J.D. Ullman, Mining of massive datasets. 2012: Cambridge University Press. [29] Cho, J., S. Roy, and R.E. Adams. Page quality: In search of an unbiased web ranking. in Proceedings of the 2005 ACM SIGMOD international conference on Management of data. 2005. ACM. [30] Jin, Hong, Shuliang Wang, and Chenyang Li. "Community detection in complex networks by density-based clustering." Physica A: Statistical Mechanics and its Applications 392.19 (2013): 4606-4618. [31] Qi, X., et al., Optimal local community detection in social networks based on density drop of subgraphs. Pattern Recognition Letters, 2014. 36: p. 46-53. [32] Gong, M., et al., Novel heuristic density-based method for community detection in networks. Physica A: Statistical Mechanics and its Applications, 2014. 403: p. 71-84. [33] Song, M. and W.M. van der Aalst, Towards comprehensive support for organizational mining. Decision Support Systems, 2008. 46(1): p. 300317. [34] Gruhl, D., et al. Information diffusion through blogspace. in Proceedings of the 13th international conference on World Wide Web. 2004. ACM. [35] Craswell, N. and M. Szummer. Random walks on the click graph. in Proceedings of the 30th annual international ACM SIGIR conference on Research and development in information retrieval. 2007. ACM. [36] Lovász, L., Random walks on graphs: A survey. Combinatorics, Paul erdos is eighty, 1993. 2(1): p. 1-46. [37] Yen, L., et al. A family of dissimilarity measures between nodes generalizing both the shortest-path and the commute-time distances. in Proceedings of the 14th ACM SIGKDD international conference on Knowledge discovery and data mining. 2008. ACM. [38] Saerens, M., et al., Randomized shortest-path problems: Two related models. Neural Computation, 2009. 21(8): p. 2363-2404. [39] Kivimäki, I., M. Shimbo, and M. Saerens, Developments in the theory of randomized shortest paths with a comparison of graph node distances.

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2014 ASE BIGDATA/SOCIALCOM/CYBERSECURITY Conference, Stanford University, May 27-31, 2014

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Evolving between Time Periods 6 and 7

Evolving between Time Periods 7 and 8 The evolving community trees

ISBN: 978-1-62561-000-3

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Evolving between Time Periods 8 and 9

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Evolving between Time Periods 9 and 10

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