Network models that reflect multiplex dynamics - arXiv

arXiv:1507.00695v1 [cs.SI] 2 Jul 2015

Network models that reflect multiplex dynamics Daryl DeFord

Scott Pauls

Department of Mathematics Dartmouth College Hanover, NH 03755 Email: [email protected]

Department of Mathematics Dartmouth College Hanover, NH 03755 Email: [email protected]

only express a single type of relationship, representing a multiplex as a single network can lead to distortion of the very topological properties we are trying to understand. A meaningful global dynamic model should respect the individual topological behaviors of the layers without introducing confounding structural effects. In this paper, we revisit some recent multiplex structural representations, examining them through the lens of dynamical consistency to show that in many cases the structural definitions introduce factors that confound the analysis of the multiplex dynamics. To remedy this, we develop a new method for modeling multiplex systems which respects dynamics and derive spectral results similar to those known for monoplex networks.

Abstract—Many of the systems that are traditionally analyzed as complex networks have natural interpretations as multiplex structures. While these formulations retain more information than standard network models, there is not yet a fully developed theory for computing network metrics and statistics on these objects. As many of the structural representations associated to these models can distort the underlying characteristics of dynamical process, we introduce an algebraic method for modeling arbitrary dynamics on multiplex networks. Since several network metrics are based on generalized notions of dynamical transfer, we can use this framework to extend many of the standard network metrics to multiplex structures in a consistent fashion. Index Terms—Multiplex Networks, Network Dynamics

I. I NTRODUCTION Mathematical analyses of simple network models of complex systems provide a surprising amount of information determining important components of the system [5], finding hidden communities [27], demonstrating the robustness of the system to failures [2], [15], among many others. But the simplest network models, where nodes represent components of the system and edges represent interaction or association between two nodes, are limited as they conflate different types of interaction between nodes. A nascent theory of multiplex networks (see [23] for a thorough review of the current research) deals with this limitation by allowing for an all-encompassing structure with multiple layers, one for each type of interaction, where each layer is a network on the same set of nodes. These structures arise naturally in many settings: trade networks in economics [3], [4], social networks [22], [25], and transportation networks [14], [17], among others. While researchers have recently focused on describing structural representations of multiplex networks and analyzing their properties [9], [11], only a relatively small portion of the literature engages questions about dynamics on multiplex networks. As structural representations of simple networks that reflect dynamics are essential to some aspects of network analysis - including diffusion processes, random walk probabilities, clustering, and percolation - transporting these ideas to the multiplex setting requires structural representations consistent with multiplex dynamics. Studying multiplex structures using tools from complex networks exposes a tension between structural representations of the entire multiplex and the dynamical interpretations of the individual layers. Since the edges in traditional networks

Related Work Kivel¨a et al. [23] provide a descriptive overview of many of the methods and techniques that are being developed for studying multiplex structures. Currently, these methods favor tensorial representations [11], as they contain all available data, in contrast to earlier attempts which aggregated multiplex data into a single network [35]. Many standard monoplex metrics and statistics have been extended to multiplex structures. For example Cozzo et al. describe a generalized notion of clustering coefficients for multiplex structures in [9], while De Domenico et al. generalized a wider variety of single network techniques to the multilayer setting [11]. A study of the redundant information captured by these representations is presented in [10]. A first approach, [26], studied dynamics on disaggregated multiplex structures via percolation theory. Buldyrev et al. [7] considered cascading failures on related networks. Building on this work, Gao et al. characterized robustness of interconnected networks [18] and developed models for studying the connectivity of interconnected networks [19]. Recently, a k−core approach to these problems in the multiplex setting was discussed in [1]. Several studies have extended monoplex centralities, such as eigenvector centrality, Katz centrality, and various geodesic centralities, to multiplex networks [11]–[13]. Our dynamical method presented in this paper is a generalization of the Khatri-Rao product approach to centrality problems introduced by Sol et al. in [32]. 1

To study distributed consensus problems on multiplex structures Trpevski et al. [34] proposed a supratransition matrix as an analogue of the standard stochastic random walk matrix for networks. Their stochastic model relates to De Domenico et al.’s model of random walks on multiplex structures [14]. Both of these constructions are special cases of the method we develop, see Section IV. Additionally, our work provides a general framework for interpreting similar results on arbitrary multiplex structures not arising from these specific applications. In [21], the authors describe a process for modeling diffusion in multiplex networks by constructing a generalized Laplacian. They used pertubation theory to characterize the eigenvalues of the supra–Laplacian in order to discuss rates of diffusion across multiplex networks [33]. Further spectral considerations of multiplex structures were studied in [31]. Additionally, the structural behavior of multiplex networks has been connected to the eigenvalues of this generalized Laplacian [29], [30], although these results may be artifacts of the composition of the operator under consideration [20].

We analyze the global network statistics of the Krackhardt High–Tech Managers data [25] and a collection of random multiplex networks, comparing the values derived from the original layers with the values associated to the single network representations. For the random networks, we follow [34] and use a mixture of Erd¨os–Renyi, Barabasi–Albert, and Watts–Strogatz networks on 100 nodes per layer. The results, summarized in Table 1 and Figure 2 respectively, show that these multiplex models obscure the heterogeneity of the layer statistics. The Krackhardt data gives rise to a multiplex with directed layers, which poses a particular problem for the matched sum model as the structural conflation can be particularly misleading. Each of the three layers in the Krackhardt data represents a particular type of association: advice relations, friendship relations, and direct report relations in the company. However, the matched sum does not distinguish between these types of connections. Additionally, the edges between copies of the same individual do not have a natural interpretation. TABLE I C OMPARISON OF M ULTIPLEX G LOBAL S TATISTICS

II. T OPOLOGICAL C ONSEQUENCES OF S TRUCTURAL R EPRESENTATIONS

Statistic Density Transitivity Reciprocity Mean Degree

There are two main constructions or structural models in the literature for generating such a network, aggregation and matched summation. Aggregation combines all edges between pairs of nodes into a single edge, while matched summation (called diagonal, categorical multilayer networks in [23]) creates a single network from the layers by introducing new edges connecting the nodes to copies of themselves in other layers. Figure 1 shows aggregate and matched sum networks for an example two-layer multiplex. Both of these structural models distort network statistics relative to those of the original layers. The main drawback to aggregation is that it loses information by conflating different types of interactions. The World Trade Web network provides an important example where aggregation obscures much of the topological heterogeneity of the individual layers, particularly with respect to clustering and reciprocity [3]. Unlike aggregate networks, no information is lost when forming the matched sum network of a multiplex. Matched summation corresponds to unfolding/flattening the tensorial representation of a multiplex structure [11], which underlies the centrality measures presented in [12], as well as many of the network metrics discussed in [11], [23]. Although the tensorial construction preserves all of the structural information about the multiplex, unfolding distorts the original data. There are two main problems with matched summation. First, the added interlayer edges have a different role than the intralayer edges, but the two are conflated. Second, connecting all the copies of a given node to one another creates dense cliques throughout the matched sum. Many of the dynamics based on flattening/unfolding tensorial representations of multiplex structures implicitly incorporate these distortions in a fundamental way.

Advice 0.452 0.465 0.236 18.1

Friend 0.243 0.276 0.225 9.71

Report 0.0476 0.000 0.000 1.90

Aggregate 0.552 0.561 0.314 22.1

Matched Sum 0.112 0.311 0.299 13.9

The distortions we witness in these examples are not isolated: given an arbitrary multiplex with n nodes and k layers, we compute bounds on many of the global statistics of the aggregate and matched sum networks in terms of the layer statistics. Density: The density of the aggregate monoplex is greater than or equal to the density of the densest layer. Matched sum constructions tend to be less dense than the original layers since the only inter–layer connections occur between copies of the same node. Thus, the maximum number of neighbors of a node is (n − 1) + (k − 1) out of the nk − 1 total nodes in the matched sum. Transitivity: The transitivity of the aggregate monoplex is greater than or equal to the transitivity of the most transitive layer. Matched sum networks tend to be less transitive than the original layers since the only inter layer triangles occur between the copies of each node. None of the connected triples formed by one inter–layer edge and one intra-layer edge can be closed, diluting the proportion of transitive triples. Reciprocity: The reciprocity of the aggregate monoplex is greater than or equal to the reciprocity of the most reciprocal layer. The reciprocity of the matched sum is greater than or equal to the reciprocity of the least reciprocal layer since all of the connecting arcs are bidirectional. Mean Degree: The mean degree of the aggregate monoplex is greater than or equal to the maximum mean degree of the layers. The mean degree of the aggregate monoplex is greater than or equal to the minimum mean degree of the layers because each node receives k−1 new incident edges. Although 2

(a) Layer 1

(b) Layer 2

(c) Aggregate Network

(d) Matched Sum

Fig. 1. A toy multiplex example with two layers and ten nodes. Figures (a) and (b) show the individual layers, while figure (c) shows the associated aggregate monoplex and figure (d) shows the associated matched sum.

(a) Density

(b) Transitivity

(c) Mean Degree

(d) Path Length

Fig. 2. Global statistic comparison for single network representations. We constructed a nine layer random multiplex on 100 nodes and computed global network statistics: (a) density, (b) transitivity, (c) mean degree, and (d) path length, for the individual layers as well as for the corresponding aggregate and matched sum networks. The layer values are reported as the blue bars, while the red line represents the matched sum statistic and the green line represents the aggregate statistic. Note that in addition to over or underestimating the layer values, the metrics associated to the single network representations do not capture the heterogeneity of the layer statistics.

III. DYNAMICAL C ONSEQUENCES OF S TRUCTURAL R EPRESENTATIONS

the mean degree and density capture the same information about the original layers, the matched sum perturbs each statistic differently.

In addition to distorting the basic topological properties, forming aggregate or matched sum networks from multiplex data also distorts multiplex dynamics. We identify two issues – instances where interpretations of the dynamics are difficult, and where these models provide incorrect generalizations of standard dynamics. The structural distortions we observed in the previous section interfere with such optimizations. As the problems with aggregate models have been previously noted by Barigozzi et al. in the context of the World Trade Web network [3], [4], and more generally by Kivel¨a et al. in their survey [23], we focus here on the effects of matched summation models and, consequently, of tensorial representations. To illustrate the problem of interpretability of tensorial representations, we demonstrate the consequences of conflating edge types by considering the simplest associated dynamical model. Here, the tensor acts on supra–vector of quantities, representing flow along the “edges” of the structure. The resulting quantity at node j layer i is the sum of the neighbors of j along layer i combined with the sum of the values at β for all other layers. Although the intra–layer component corresponds directly to familiar monoplex dynamics, the inter– layer component reflects the effect of adding edges between the node copies in the matched sum. Additionally, all of the copies of node j represent the same object: it is unclear what meaning should be attached to the edges connecting the layers. Together these problems of interpretation make it difficult to describe meaningful objective functions by directly viewing

Average Path Length: The average path length of the aggregate monoplex is less than or equal to the shortest average path length of the layers because the shortest paths from each layer are still viable in the aggregate network. In matched sum networks, the average path length also tends to decrease since it requires at most two interlayer steps to make use of the shortest intra–layer path between two nodes. Clique Numbers: The clique numbers of the aggregate and matched sum networks are both be greater than or equal to the maximum clique number among the layers. Except in the case of degenerate examples, such as duplicate or empty layers, the bounds above become strict inequalities. We observe that increasing the number of layers magnifies the affects on these statistics. For example, as the number of layers increases with a fixed number of nodes and average density, the aggregate density increases while the matched sum density decreases. Similar results hold for the other statistics. Taken together, these results emphasize the toplogical damage inherent in using aggregate or matched sum representations of multiplex networks. Although for some applications, such as the transportation networks studied in [14], these structural additions accurately model the system, researchers should consider possible confounding effects before applying these techniques. 3

Notationally, v(i−1)n+j is the quantity at node j in layer i.  T To extract layer quantities, we write v = v 1 , v 2 , · · · v k where each v i is a 1 × n vector associated to layer i. That is, vji ≡ v i (j) = v(i−1)n+j is the amount at node j in layer i.

the tensorial representation as an operator. This problem also arises when considering heterogeneous multiplex data, for example when some of the layers of the multiplex are directed while others are undirected. While it is true that an undirected edge may be considered as a pair of opposing directed edges with no loss of structural information, the dynamical interpretation of such relations can be quite different and is entirely obscured by a na¨ıve model. Indeed, the more heterogeneous the layers, the more important it is not to add confounding structural factors that obscure the effects of the topology on dynamical models. Questions of interpretation are not the only issue inherent in these tensorial dynamics. The structural distortions described in the previous section have clear dynamical consequences. In a matched sum, each node belongs to a clique representing all its copies across the layers, vastly distoring local topologies in networks with sparse layers. This in turn distorts metrics and dynamics computed with walks and geodesics, since the number of possible paths between two nodes is greatly magnified. For example, measures like Katz centrality and Pagerank penalize longer paths by an exponential factor, so most of the centrality value accrues directly from the other copies of the same node, overwhelming the actual connections of interest. The combinatorial supra–Laplacian, as defined in [11] and [23] suffers from these deficiencies. Interpreted as a linear differential operator on the tensor space, it represents the Laplacian of the matched sum of the multiplex and hence incorporates these structural distortions at a fundamental level. Although possibly a reasonable model in some applications, it is not a clear dynamical generalization of the standard graph Laplacian. Consequently, we cannot derive the relations between the eigenvalues of this structural representation and the multiplex properties analogously to the monoplex derivation. Finally, as discussed above as the number of layers increases, the effects of these distortions increase. This explains the behaviors noted in several papers based on dynamical models associated to tensorial or supra– constructions that for large portions of the parameter space, the dynamical process is controlled by the layer mixing [12]–[14], [21], [29]–[31], [33], [34]. It is unclear at this point what information about multiplex models is actually related to the spectral structures of the associated tensor representations since the corresponding dynamic operators do not always have a natural interpretation.

Motivation We want to interleave the intra–layer dynamics with an operation that gathers the quantity stored at each copy of a given node and redistributes among the other copies of that node. We model this as an iterative two step process: we first apply D to v and second enforce the interlayer dynamics by setting (v 0 )ij to a proportion of a convex combination of the {(Dv)`j }k`=1 , (v 0 )ij = αji

k X

` ci,` j (Dv)j .

(1)

`=1

The constants ci,` j provide the rate at which quantities pass from layer ` to layer i through node j, while the αji allow node j in layer i to send or receive quantities external to the network. The restriction to a convex combination of the (Dv)`j Pk i,` enforces that ci,` j ≥ 0 and `=1 cj = 1 for all 1 ≤ j ≤ n and 1 ≤ i, ` ≤ k. Scaling by αji allows any linear combination of the (Dv)ij with non–negative coefficients. As we will see, we can perform this second operation algebraically using a collection of scaled orthogonal projection operators, one for each node in the multiplex. This dynamic process preserves the intra–layer dynamics, while also allowing effects to pass between layers through the copies of each node. Unless the intra–layer dynamics have a component that incorporates the value at the node, the copies of a node do not interact directly. Instead the dynamical effects acting on any particular copy are passed through to the other copies, with proportions dictated by the coupling constraints, the αji and ci,` j . Consequently, we avoid the pitfalls of aggregate and matched sum networks described above. Derivation of the Operator We accomplish step one of our method with the block diagonal matrix D, so it suffices to construct a matrix that redistributes the values between the copies of the node as described in equation (1). Let Pji = P(i−1)n+j be the projection onto the (i − 1)n + j coordinate, Pji is a 1 × nk vector with a one in the (i − 1)n + j entry and zeros elsewhere. Then, Pji Dv = (Dv)ij . Using the αji and ci,` j , we proportionally distribute this value among all copies of node j using the nk × 1 column vector

IV. M ULTIPLEX DYNAMICS We present a model for multiplex dynamics that avoids the structural conflation of single network models by allowing effects to pass between layers without artificially adding edges to the network. For a multiplex structure on n nodes and k layers with a matrix dynamical operator Di associated to each layer 1 ≤ i ≤ k we model the intra–layer dynamics by defining the block diagonal matrix D = diag(D1 , D2 , . . . , Dk ). The matrix D acts on an nk × 1 vector v representing the “quantities” contained at k copies of each if the n nodes.

h iT k,i 1,i k,i . αji c1,i j δ1 (j), . . . , cj δn (j), . . . , cj δ1 (j), . . . , cj δn (j) The projections are pairwise orthogonal, so we perform the projection and redistribution steps concurrently using the 4

matrix

defined by the network structure or application. This is equivi,m alent to the condition ci,` for all 1 ≤ i, `, m ≤ k and j = cj 1 ≤ j ≤ n. For our second simplification, the hierarchical layer model, we assume there is a hierarchy of layers where the effect of layer i on layer ` is fixed for all the nodes. In this case, each of the matrices C i,` is just a linear homothety or scalar multiple i,` ` ` of the identity, i.e. ci,` a = cb and αa = αb for all layers a and b. We see two natural approaches to defining these coefficients in the absence of an application specific determination. The first is to use the density of the layers as the C i,` . Second, we set C i,` to be the ratio of the number of edges in layer i to the number of edges in layer ` or more globally define C i,` to be equal to the proportion of all of the edges in the multiplex that occur in layer `. This model is the assymmetric influence matrix W introduced in [32] for measuring eigenvector centrality in multiplex networks (we recover the asymmetric version by assuming ci,` = ci,` = c`,i = c`,i for all 1 ≤ a, b ≤ n). When a a b b the Di are the adjacency matrices for the layers, we recover their computation of the global heterogeneous eigenvector centrality. Thus, our model is a direct generalization which allows for arbitrary dynamics and more complex layer ranking behavior individualized to each node. In the next section, we use a still simpler version of this i model where we further assume that ci,` j ≡ c for 1 ≤ ` ≤ k, 1 ≤ j ≤ n. Our third and simplest version, the equidistribution model, occurs when D is closed and the quantities at each copy of a node are distributed equally among other copies. We set 1 ci,` j = k for all 1 ≤ j ≤ n and 1 ≤ i, ` ≤ k and name the resulting operator as E. This formulation is a natural model for applications where the dynamic flow is equally likely to move between layers, or when the quantities at each node represent the outcomes of a binary process. In the next section we will show that E has a particularly simple spectral structure related to that of the aggregate network.

 c1,i j δ1 (j)   ..   .   c1,i δ (j) n   j k n XX   i .. i   Pj . M= αj  .    k,i i=1 j=1  cj δ1 (j)    ..     . k,i cj δn (j) 

We realize M as block matrix,  1,1 C C 1,2 2,1 C C 2,2  M = . ..  .. . C k,1

C k,2

··· ··· .. . ···

 C 1,k C 2,k   ..  , . 

C k,k

` i,` ` i,` where C i,` = diag(α1` ci,` 1 , α2 c2 , . . . , αn cn ). The final multiplex dynamic operator is a product of the layer dynamics matrix D and the redistribution matrix M  1,1  C D1 C 1,2 D2 · · · C 1,k Dk C 2,1 D1 C 2,2 D2 · · · C 2,k Dk    D = MD =  . . .. . . . . .  .  . . .

C k,1 D1

C k,2 D2

···

C k,k Dk

Before examining the spectral properties of D we discuss three special cases of D reflecting natural modeling constraints. We begin with a useful definition, describing a hypothesis that enforces a conservation law for the quantities under consideration. Definition 1. The operator D is called closed if αji = 1 for all 1 ≤ i ≤ k and 1 ≤ j ≤ n When D is closed, the matrix M is stochastic, which we will use in the next section when analyzing multiplex random walk dynamics. When D is closed and the Di are random walk matrices we recover the models used in [14] and [34]. We motivate our first simplification, which we call the unified node model, with an example using economic exchange networks. Consider trade relationships between countries, where bilateral trade occurs within commodity based layers, but parties inside the countries redistribute money internally redistributed before international trade begins anew. This allows countries to use surpluses in one area to fund deficits in another. Further, the α coefficients represent unmodeled sources or sinks of income external to the trade network. To adjust our model, we enforce a homogeneity condition at each node: instead of computing Pk M from nk projections we use the n projections Qj = `=1 αj` Pj` , j ∈ {1, . . . , n}. The value of the product Qj Dv is the weighted sum of the quantities at node j on each layer. When D is closed, Qj simply becomes the uniform projection onto the subspace spanned by all of the copies of node j. For each node j, we redistribute Qj Dv proportionally among its copies according to their relative importance as

V. S PECTRAL R ESULTS We present basic spectral results for our multiplex formulation and show that our operators arise as natural multiplex interpretations of standard network dynamic models, such as random walks and heat diffusion, derived from first principles. One particularly valuable aspect of D is that under reasonable assumptions, it preserves matrix properties of the original layer dynamics, such as primitivity, stochasticness, and postive semi–definiteness. These properties provide the link between the spectrum of the operator and the multiplex structure. We begin with a simple result, relating the eigenvalues of the equidistribution operator E to the spectrum of the aggregate network. Proposition 1. Except for P zero, the eigenvalues of E are k exactly the eigenvalues of k1 `=1 D` . Similar results exist for the other formulations, but they are more complex. In the simplest version of the hierarchical layer 5

i model, where ci,` j ≡ c , we have that the non–zero eigenvalues Pk i of D are exactly the eigenvalues of i=1 c Di . Although determining the spectral structure of sums of matrices in terms of the spectra of the summands is difficult, when the matrices are symmetric, we can use standard results due to Weyl described in [16], [24] to obtain upper and lower bounds on the individual eigenvalues of the derived operators (as in Proposition 4).

its neighbors: k X X dvji ` (vj` − vm =K ). ci,` j dt ` ` `=1

nj ∼nm

Here K is the diffusion constant and the ci,` j represent the proportion of the effect on layer ` that passes through to nij . Linear algebraically, this is: " k # X i,` dvji =K cj L` v ` (2) dt

Stochastic Dynamics

`=1

A left stochastic matrix is a non–negative matrix where the entries in each column sum to one. Such a matrix defines a Markov process on a network, with the entries Di,j corresponding to the probability of moving from from node j to node i at each time step. Stochastic processes on networks are commonly used for determining centrality and community detection. If a stochastic operator is primitive or irreducible, the Perron–Fr¨obenius Theorem guarantees that the largest eigenvalue of that matrix is 1 and that the entries in one of the corresponding eigenvectors are non–negative. This vector represents the limiting state of the dynamical system for arbitrary input.

j

which agrees with (1) when the layer dynamics are given by the respective layer Laplacians, Li . This model implicitly assumes that each copy of each node can be assigned a separate temperature. This is a reasonable assumption for applications such as some economic exchange networks or the transportation models considered in [14]. However, in examples such as our social multiplex where the node copies are only representing different interaction types associated to the same individual, we enforce the additional condition that vji = vj` for all 1 ≤ j ≤ n and 1 ≤ i, ` ≤ k. Intuitively, at each time step, each individual has some fixed amount of information regardless of what types of interactions they are performing. The equidistribution operator E, encodes this additional condition. As we’ve reduced to n independent variables, we restate Equation (2) in terms of a n × q vector hPw recording i dw k ` . the temperature at each node as dtj = K `=1 L w k

Proposition 2. If each Di is stochastic and D is closed then D is stochastic. Additionally, if each Di is irreducible then D is irreducible and if each Di is primitive then D is primitive. The first statement follows from the fact that the hypotheses guarantee that D is the product of two stochastic operators, and the second two conclusions can be easily shown from the graph theoretic interpretation of the conditions. We view a stochasitic D as describing a random walk on the multiplex with the probability of transitioning from the copy of node j on layer ` to a neighbor of node j on any arbitrary layer i 1 i given by ci,` j deg(nij ) , where nj is the copy of node i in layer j. This Markov formulation is equivalent to the problem considered in [34] for modeling distributed concensus which provides additional confirmation of our methods. Under similar hypotheses to those in [14], [34], our operator reduces to a version of their supratransition matrix.

j

The ability to models these types of dynamics distinguishes our analysis from previous studies of diffusion on multiplex networks [21]. Using tools from the theory of Hermitian matrices we can prove the following bounds for the operator E when the intra– layer dynamics are Laplacians. Proposition 3. If each Di is the graph Laplacian then E is positive semi–definite and the eigenvectors of E corresponding to distinct non–zero eigenvalues are orthogonal. Consequently, the solutions to the differential equation ddtϕˆ + Eϕˆ = 0 have the same algebraic and analytic structure as the standard Laplacian: the solution consists of constant elements and terms that decay exponentially. We now find eigenvalue bounds for E in this case. Our first result shows that diffusion across the multiplex network should happen at least as fast as diffusion across the best connected layer. We assume that the layer networks are connected - although similar bounds exist in the case where the layer networks are disconnected, the formulas become more complex. We also introduce some additional notation. Let λij be the j th eigenvalue of Di written in descending order with λif representing the Fiedler value corresponding to Di . Let the eigenvalues of E be λ1 ≥ λ2 · · · ≥ λkn . As rank(E) = n − 1, for ` > n − 1 we have λ` = 0. Finally, let m be the index i such that λm 1 = maxi (λ1 ).

Diffusion Dynamics We often model diffusion on networks as a discretization of the continuous heat flow problem, which yields the graph Laplacian, L = D − A, whose spectral structure has strong connections to important graph properties such as connectivity, communities, and random walks [6], [8], [28]. To extend this model to the multiplex setting, we consider an initial vector v representing the temperature at each node copy and define the change in the value of vji with respect to time to be proportional to the sum of the differences in temperature between each copy of node j (including nij ) and 6

Proposition 4. If each Di is the graph Laplacian then we have the following eigenvalue bounds for the operator E P j i m • Fiedler Value: maxi (λf ) ≤ kλf ≤ λf + j6=` λ1 , P i i • Leading Value: maxi (λ1 ) ≤ kλ1 ≤ i λ1 ,

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These bounds are special cases of the following more general but less computationally feasible bounds: !! k X σ(i) i maxi (λn−` ) ≤ kλn−` ≤ min min λ ji , J`n+k−(`+1)

σ∈Sn

i=1

Pk

where J = (j1 , j2 , . . . , jk ) such that i=1 ji = n + k − (` + 1). As remarked after Proposition 1, we can give a similar characterization of the eigenvalues of the simplest case of the hierarchical layer model, where we obtain results equivalent to Propositions 3 and 4, replacing λij with (ci λi )j (after possible reordering) in each occurrence. VI. D ISCUSSION AND C ONCLUSIONS The field of multiplex networks is rapidly becoming an important segment of the complex systems community. Although many natural structural representations for multiplex structures have been presented, there is still a great deal left to learn about multiplex structures and their associated dynamics. To faithfully represent dynamical processes on multiplex networks, we have shown it is sometimes necessary to avoid the confounding structural features introduced by tensorial and supra– constructions. To obtain meaningful results in the multiplex setting we must verify that the correct analogy with the monoplex network case holds. While there may be particular applications for which these structurally motivated dynamics are well suited, in general we recommend that researchers choose dynamical models that do not introduce structural features in order to more accurately capture the dynamics of interest. It is particularly important when relying on spectral computations and invariants that the associated operator have a natural interpretation that allows the appropriate Rayleigh quotient to be derived as a solution to the given objective function. Our model avoids the structural defects inherent in single network multiplex representations and allows the inherent layer dynamics to interact without explicitly confounding structural effects, generalizing several successful models to a wide class of network dynamics. R EFERENCES [1] N. Azimi-Tafreshi, J. Gomez-Gardenes, and S. N. Dorogovtsev. k-core percolation on multiplex networks. Physical Review E, 90(3):032816, September 2014. [2] Albert-Laszlo Barabasi and Eric Bonabeau. Scale-free networks. Scientific American, 288(5):60–69, May 2003. [3] Matteo Barigozzi, Giorgio Fagiolo, and Diego Garlaschelli. Multinetwork of international trade: A commodity-specific analysis. Physical Review E, 81(4):046104, April 2010. [4] Matteo Barigozzi, Giorgio Fagiolo, and Giuseppe Mangioni. Community Structure in the Multi-network of International Trade. In Luciano da F. Costa, Alexandre Evsukoff, Giuseppe Mangioni, and Ronaldo Menezes, editors, Complex Networks, number 116 in Communications in Computer and Information Science, pages 163–175. Springer Berlin Heidelberg, 2011.

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