physical review e102, 062307 (2020)

PHYSICAL REVIEW E 102, 062307 (2020)

Inference of edge correlations in multilayer networks

A. Roxana Pamfil and Sam D. HowisonMathematical Institute, University of Oxford, Oxford OX2 6GG, United Kingdom

Mason A. PorterDepartment of Mathematics, University of California, Los Angeles, Los Angeles, California 90095, USA

and Mathematical Institute, University of Oxford, Oxford OX2 6GG, United Kingdom

(Received 10 August 2019; revised 8 September 2020; accepted 14 September 2020; published 16 December 2020)

Many recent developments in network analysis have focused on multilayer networks, which one can use toencode time-dependent interactions, multiple types of interactions, and other complications that arise in complexsystems. Like their monolayer counterparts, multilayer networks in applications often have mesoscale features,such as community structure. A prominent approach for inferring such structures is the employment of multilayerstochastic block models (SBMs). A common (but potentially inadequate) assumption of these models is thesampling of edges in different layers independently, conditioned on the community labels of the nodes. In thispaper, we relax this assumption of independence by incorporating edge correlations into an SBM-like model. Wederive maximum-likelihood estimates of the key parameters of our model, and we propose a measure of layercorrelation that reflects the similarity between the connectivity patterns in different layers. Finally, we explainhow to use correlated models for edge “prediction” (i.e., inference) in multilayer networks. By incorporating edgecorrelations, we find that prediction accuracy improves both in synthetic networks and in a temporal network ofshoppers who are connected to previously purchased grocery products.

DOI: 10.1103/PhysRevE.102.062307

I. INTRODUCTION

A network is an abstract representation of a system inwhich entities called “nodes” interact with each other viaconnections called “edges” [1]. Most traditionally, in a type ofnetwork called a “graph”, each edge encodes an interaction ora tie between two nodes. Networks arise in many domains andare useful for numerous practical problems, such as detectingbot accounts on Twitter [2], finding vulnerabilities in electri-cal grids [3], and identifying potentially harmful interactionsbetween drugs [4]. A common feature of many networks ismesoscale (i.e., intermediate-scale) structures. Detecting suchstructures in a network amounts to producing a coarse-graineddescription of the network that is more compact than listing allof its nodes and edges. Types of mesoscale structures includecommunity structure [5], core–periphery structure [6,7], rolesimilarity [8], and others. An increasingly popular approachfor modeling and detecting such structures is by using stochas-tic block models (SBMs) [9], which are generative modelsthat can produce networks with community structure or othermesoscale structures.

For many applications of network analysis, it is importantto move beyond ordinary graphs (i.e., “monolayer networks”)to examine more complicated network structures, such ascollections of interrelated networks. One can study such struc-tures through the flexible lens of multilayer networks [10–13].A multilayer network consists of a collection of “state nodes”that are connected pairwise by edges. A state node is a mani-festation of a “physical node” (which we will also sometimes

call simply a “node”), which represents some entity, in a spe-cific layer. Different layers may correspond to interactions indifferent time periods (yielding a temporal network), differenttypes of relations (yielding a multiplex network), or other pos-sibilities. As in the setting of monolayer networks, modelingand inferring mesoscale structures in multilayer networks is aprominent research area.

A key assumption of almost all existing models of mul-tilayer networks with mesoscale structure is that edges aregenerated independently, conditioned on a multilayer par-tition [14–21]. This independence assumption applies bothwithin each layer (which is inconsistent with the fact thatreal networks often include 3-cliques and other small-scalestructures) and across layers (which is inconsistent with thefact that the same nodes are often adjacent to each other inmultiple layers). In this paper, we focus on relaxing the edge-independence assumption that applies to edges between thesame two physical nodes in different layers. We still considereach pair of nodes independently.

In Fig. 1, we show an example of a two-layer networkwith both strong positive and strong negative edge correla-tions. Incorporating such correlations into a network modelis beneficial for many applications. For example, a multiplexnetwork of air routes in which each layer corresponds to oneairline is likely to include some popular routes that appear inmultiple layers (and unpopular routes may appear only in onelayer) [22]. In a temporal social network, we expect peopleto have repeated interactions with certain other people [23].This is a stronger assertion than just stating that people tend to

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PAMFIL, HOWISON, AND PORTER PHYSICAL REVIEW E 102, 062307 (2020)

FIG. 1. Example of a correlated multilayer network with twolayers and with two blocks of nodes in each layer. Edges between twoblack nodes are positively correlated across layers, edges betweentwo white nodes are negatively correlated across layers, and edgesbetween a black node and a white node are uncorrelated acrosslayers.

interact more within the same community over time than withpeople in other communities. Such edge persistence is alsocommon in many bipartite user–item networks: Shoppers tendto buy the same grocery products over time [24], customers ofa music-streaming platform listen repeatedly to their favoritesongs [25], and Wikipedia users edit specific pages severaltimes [26].

Multilayer network models that incorporate edge correla-tions have many important applications. One is the inferencetask of edge prediction (which is also called “link prediction”),in which one seeks to assign probabilities of occurrence tounobserved edges. Stochastic block models have been usedoften for edge prediction in both monolayer networks [4,9,27]and multilayer networks [15,20,28]. The models that we pro-pose should yield better performance than past efforts, asthey take advantage of interlayer edge correlations in data.We discuss this in more detail in Sec. III. Another appli-cation is graph matching [29], in which one seeks to infera latent correspondence between nodes in two different net-works when one does not know the identities of the nodes.For example, one may wish to match common users be-tween anonymized Twitter and Facebook networks. A seriesof papers [30–32] established conditions under which graphmatching is successful. These papers considered correlatedErdos–Rényi (ER) networks and correlated SBMs, which wealso examine in this paper (albeit for a different purpose). Wetake these models further by incorporating degree correction[33], which generates networks with heterogeneous degreedistributions and is important for inference using SBMs. Thisextension may allow one to study the graph-matching prob-lem on more realistic network models. A third application isefficient computation of pairwise correlations between layersof a multilayer network. One can use such estimates of corre-lations to quantify the similarity between different layers andpotentially to compress multilayer network data by discarding(or merging) layers that are strongly positively correlated withan existing layer [34]. Previous papers have focused primarilyon node-centric notions of layer similarity [35–37], whereasour correlated models yield edge-centric measures of similar-

ity. Benefits of our approach over related studies [34,38,39]include the fact that correlation values cover an intuitive range(between −1 and 1) and that they work equally well forquantifying layer similarity and dissimilarity.

Our edge-correlated network models are also useful forcommunity detection. Given a multilayer network, one candesign an inference algorithm that determines both the pa-rameters that describe the edge probabilities (and correlations)and a multilayer community structure that underlies theseprobabilities. Solving this inference problem enables the de-tection of correlated communities. Because of the additionalcomplexity in the resulting model, this is bound to be moredifficult than standard multilayer community detection, so weleave this inference problem for future work. Instead, for therest of our paper, we assume that we know the block structureof a network. We infer the remaining parameters, includingthe correlations that are the core element of our model [40].We also assume that the block assignments g are the samefor all layers. Situations in which communities can vary arbi-trarily across layers are significantly more difficult [41], andwe leave consideration of them for future work. With theserestrictions, one can determine g using any method of choice.

Some existing models of multilayer networks incorporateinterlayer dependencies by prescribing joint degree distribu-tions [36,42,43], by incorporating edge overlaps [44,45], or bymodeling the appearance of new edges through preferential-attachment mechanisms [46]. The models that we describein this paper are similar to those that were introduced in[30–32] for graph-matching purposes. Another noteworthypaper is one by Barucca et al. [47] that described a generalizedversion of the temporal SBM of Ghasemian et al. [17]. Thisgeneralization includes an “edge-persistence” parameter ξ ,which gives the probability that an edge from one layer alsooccurs in the next temporal layer. For several reasons, we takea different approach. First, the model in [47] is specific totemporal multilayer networks, whereas we are also interestedin other types of multilayer networks. Second, their modeldoes not easily incorporate degree correction. Third, we wantto include correlations explicitly in the model, rather thanimplicitly using the edge-persistence parameter ξ .

Our paper proceeds as follows. In Sec. II, we describeour models of multilayer networks with edge correlations.We start with a simple example of correlated ER graphsin Sec. II A to make it easier to follow our expositionfor more complicated models. In Sec. II B, we integratemesoscale structures by incorporating correlations into anSBM-like model. We then incorporate degree correction inSec. II C. For all of these models, we derive maximum-likelihood (ML) estimates both of the marginal edge-existenceprobabilities in each layer and of the interlayer correla-tions. Maximum-likelihood estimation is common for SBMsand degree-corrected SBMs (DCSBMs) in both monolayernetworks [33] and multilayer networks [16]. Although ML es-timation is less powerful than performing Bayesian inference[9], the former is consistent for both SBMs and DCSBMs[48] and it recovers many common techniques for detectingmesoscale structures in networks [49]. In Sec. III, we de-scribe how to use our models for edge prediction and givesome results for synthetic networks. We then proceed withapplications in Sec. IV. In Sec. IV A, we use our models

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INFERENCE OF EDGE CORRELATIONS IN MULTILAYER … PHYSICAL REVIEW E 102, 062307 (2020)

to estimate pairwise layer correlations in several empiricalnetworks. In Sec. IV B, we use our correlated models foredge prediction in a temporal network of grocery purchases.We summarize our results in Sec. V and discuss a few ideasfor future work. We give additional details in a quartet ofappendices.

II. CORRELATED MODELS

In our derivations, we consider just two network layersat a time. Although this may seem limiting, we can applyour framework to generate correlated networks with morethan two layers in a sequential manner (see the discussion inSec. II A) and we can determine pairwise layer correlations fora network with arbitrarily many layers (see the applications inSec. IV). In Sec. V, we briefly discuss the challenges that arisewhen considering three or more layers simultaneously, ratherthan in a sequential pairwise fashion.

Consider a network with two layers and identical sets ofnodes in each layer; this is known as a node-aligned multilayernetwork [10]. Let A1 and A2 denote the adjacency matricesof our two network layers. As in many generative modelsof networks, we assume that edges in these two layers aregenerated by some random process, so the entries A1

i j andA2

i j are random variables. Imposing some statistical correla-tion between these two sets of random variables introducesinterlayer correlations in the resulting multilayer networkstructure.

Our goal is to propose a model of correlated networksin which, marginally, each layer is a DCSBM [33]. How-ever, it is instructive to first consider the simpler cases inwhich, marginally, each layer is an ER random graph (seeSec. II A) or each layer is an SBM without degree correc-tion (see Sec. II B). Correlated ER models and correlatedSBMs have been studied previously, most notably in [30–32]in the graph-matching problem. Such prior efforts employedthese models to generate synthetic networks. By contrast,we fit these models to empirical data and use them to es-timate layer correlations. We also extend these models to adegree-corrected setting with the correlated DCSBMs that wepropose in Sec. II C.

In monolayer SBMs, it is common to use either Bernoullior Poisson random variables to generate edges between nodes.The former choice is generally more accurate because it doesnot yield multiedges; however, the latter is more common,as it often simplifies calculations considerably [48,50]. Nev-ertheless, we have found that Bernoulli models are simplerwhen incorporating correlations [24]. They also have severalother advantages, including the fact that they work for bothsparse and dense networks and that they can handle the entirecorrelation range between −1 and 1. Therefore, we consideronly Bernoulli models in this paper.

A. Correlated Erdos–Rényi layers

1. Forward model

Assume that the intralayer networks that correspond to A1

and A2 are ER graphs from the G(n, p) ensemble [1] with edge

FIG. 2. Visualization of the feasible region (gray area) for theprobabilities p1 and p2, given a value of q. The boundaries of thisregion are defined by the inequalities q � p1 � 1, q � p2 � 1, andp1 + p2 � 1 + q. The hyperbola p1 p2 = q specifies the boundarybetween regimes with a positive layer correlation and regimes with anegative layer correlation.

probabilities p1 and p2. For each node pair (i, j), we have

P(A1

i j = 1) = p1 , (1)

P(A2

i j = 1) = p2 . (2)

To couple edges that connect the same pair of nodes in differ-ent layers, let

q := P(A1

i j = 1, A2i j = 1

)(3)

denote the joint probability for an edge to occur in both layers.Unless q = p1 p2, this construction implies that the randomvariables A1

i j and A2i j are not independent.

The parameters p1, p2, and q (which lie in the interval[0,1]) fully specify a forward model of networks with cor-related ER layers. To generate a network from this model,one considers each node pair (i, j) and, independently of allother node pairs, assigns values to A1

i j and A2i j according to the

following probabilities:

P(A1

i j = 1, A2i j = 1

) = q ,

P(A1

i j = 1, A2i j = 0

) = p1 − q ,

P(A1

i j = 0, A2i j = 1

) = p2 − q ,

P(A1

i j = 0, A2i j = 0

) = 1 − p1 − p2 + q . (4)

These expressions follow from the laws of probability andfrom the definitions of p1, p2, and q. For these probabilitiesto be well-defined, it is both necessary and sufficient that 0 �q � min{p1, p2} and p1 + p2 � 1 + q. In Fig. 2, we illustratethe feasible region for p1 and p2, given a value of q.

It is also possible to generate a correlated ER network in asequential manner. First, one generates the adjacency matrixA1 by placing edges with probability p1. One then determinesthe probabilities of edges in the second layer by conditioning

062307-3


on the first layer:

P(A2

i j = 1|A1i j = 1

) = P(A1

i j = 1, A2i j = 1

)P

(A1

i j = 1) = q

p1,

P(A2

i j = 1|A1i j = 0

) = P(A1

i j = 0, A2i j = 1

)P

(A1

i j = 0) = p2 − q

1 − p1,

P(A2

i j = 0|A1i j = 1

) = P(A1

i j = 1, A2i j = 0

)P

(A1

i j = 1) = p1 − q

p1,

P(A2

i j = 0|A1i j = 0

) = P(A1

i j = 0, A2i j = 0

)P

(A1

i j = 0)

= 1 − p1 − p2 + q

1 − p1. (5)

With this approach, it is possible to generate networks witharbitrarily many layers by first sampling edges in the firstlayer and then sampling edges in each subsequent layer byconditioning on the previous one. This kind of process isespecially well-suited to temporal networks, in which layershave a natural ordering. For multiplex networks, it is moreappropriate to extend Eq. (4) to handle more than two layers.

It is also possible to parametrize correlated ER graphs interms of the marginal Bernoulli probabilities, p1 and p2, andthe Pearson correlation

ρ = E[A1

i jA2i j

] − E[A1

i j

]E

[A2

i j

]σ[A1

i j

]σ[A2

i j

]= q − p1 p2√

p1(1 − p1)p2(1 − p2), (6)

where E[·] and σ [·] denote, respectively, the mean and stan-dard deviation of a random variable. One benefit of using ρ,rather than q, as the third model parameter is that its value iseasier to interpret. A value of ρ that is close to 0 indicatesa weak correlation between layers, whereas values that areclose to the extremes of +1 and −1 indicate a strong positivecorrelation and a strong negative correlation, respectively.

We can gain further intuition by considering the cases ρ =0, ρ = 1, and ρ = −1. First, ρ = 0 if and only if P (A1

i j =1, A2

i j = 1) = P (A1i j = 1)P (A2

i j = 1). That is, the correlationis 0 if and only if edges are generated independently in the twolayers with marginal probabilities of p1 and p2. For ρ = 1,one can show (see [24]) that p1 = p2 = q, which correspondsto the two layers having identical network structure. Finally,for ρ = −1, we have q = 0 and p1 = 1 − p2 (see [24]), andtwo nodes are adjacent in one layer if and only if they are notadjacent in the other layer.

2. Maximum-likelihood parameter estimates

We now derive ML estimates of the parameters p1, p2, andq. Let E denote the set of node pairs and hence the set ofpossible edges in a layer. For undirected networks withoutself-edges, there are |E | = N (N − 1)/2 node pairs to con-sider, where N is the number of physical nodes. By contrast,|E | = N (N − 1) when generating directed networks withoutself-edges. With this general notation, all of our derivations inSec. II are valid for both directed and undirected networks,with or without self-edges. (They are also valid for bipar-tite networks [24].) We consider each node pair (i, j) ∈ Eindependently when generating edges, so the likelihood ofobserving adjacency matrices A1 and A2 is

P (A1, A2|p1, p2, q) =∏

(i, j)∈EqA1

i j A2i j (p1 − q)A1

i j (1−A2i j )(p2 − q)(1−A1

i j )A2i j (1 − p1 − p2 + q)(1−A1

i j )(1−A2i j ) . (7)

It is helpful to introduce the following notation:

e11 := |{(i, j) ∈ E : A1i j = 1, A2

i j = 1}| ,e10 := |{(i, j) ∈ E : A1

i j = 1, A2i j = 0}| ,

e01 := |{(i, j) ∈ E : A1i j = 0, A2

i j = 1}| ,e00 := |{(i, j) ∈ E : A1

i j = 0, A2i j = 0}| .

In order, these quantities count the numbers of node pairs thatare adjacent in both layers (e11), adjacent in the first layer butnot in the second (e10), adjacent in the second layer but notin the first (e01), and not adjacent in either layer (e00). Usingthis notation and taking the logarithm of (7), we arrive at thefollowing expression for the log-likelihood:

L = e11 log q + e10 log(p1 − q) + e01 log(p2 − q)

+ e00 log(1 − p1 − p2 + q) . (8)

When fitting our model to network data, the quantities e11,e10, e01, and e00 are all known; we seek to determine the valuesof p1, p2, and q that are most consistent with the data. To doso, we maximize the log-likelihood (8) by setting its partial

derivatives to 0 [51]. We obtain

p1 = e11 + e10

e11 + e10 + e01 + e00, (9)

p2 = e11 + e01

e11 + e10 + e01 + e00, (10)

q = e11

e11 + e10 + e01 + e00. (11)

In each of these three expressions, the denominator is equalto the number of potential edges (i.e., the cardinality of E).Additionally, let m1 = e11 + e10 and m2 = e11 + e01 denotethe number of observed edges in the first and second layers,respectively. It follows that the ML estimate p1 is equal to thenumber of observed edges in layer 1 divided by the numberof potential edges, and an analogous relation holds for p2.The estimate q is equal to the number of node pairs that areadjacent in both layers divided by the total number of nodepairs. These results match our intuition.

We obtain an estimate of the Pearson correlation ρ betweenthe two layers by substituting the ML estimates p1, p2, and q

062307-4


into Eq. (6) to obtain

ρ = e00e11 − e10e01√(e11 + e10)(e11 + e01)(e10 + e00)(e01 + e00)

. (12)

One can show that maximizing the log-likelihood (8) withrespect to p1, p2, and ρ (rather than with respect to p1, p2,and q) gives the same expression for ρ, confirming that this isindeed an ML estimate of the correlation. Note that ρ is notdefined when either layer is an empty or a complete graph, asthe associated Bernoulli random variable for such a layer hasa standard deviation of 0.

In Appendix A, we calculate the variances that are asso-ciated with the ML estimates p1, p2, and q. We then showusing a synthetic example that these scale as 1/N2, where Nis the number of physical nodes. These results quantify theuncertainty around the ML estimates for correlated ER modelsas a function of network size.

B. Correlated SBMs

One of the ways in which real-world networks differ fromER random graphs is that the former have mesoscale struc-tures, such as communities [5]. We use SBMs to incorporatesuch structures into our correlated models.

Let g be a vector of block assignments, which we taketo be identical in our two network layers, and let K denote

the number of blocks. As we explained in Sec. I, we assumethroughout the present paper that we are given g and weaim to estimate the remaining model parameters. Followingterminology from [27], let B = {1, . . . , K} × {1, . . . , K} bethe set of “edge bundles” (r, s), each of which is described byits own set of parameters p1

rs, p2rs, and qrs. The K × K matrices

p1, p2, and q play an analogous role to p1, p2, and q in the ERlayers.

Let gi denote the block assignment of node i. A correlatedtwo-layer SBM is described by the following set of equalities:

P(A1

i j = 1) = p1

gig j,

P(A2

i j = 1) = p2

gig j,

P(A1

i j = 1, A2i j = 1

) = qgig j .

Lyzinski et al. proposed this forward model in [31] to studygraph matching. By contrast, we focus on the inverse prob-lem of estimating the parameters p1, p2, and q, given somenetwork data.


As in Sec. II A, suppose that we consider each node pair(i, j) independently. The likelihood of observing adjacencymatrices A1 and A2 is then

P (A1, A2|g, p1, p2, q) =∏

(i, j)∈E

[q

A1i j A

2i j

gig j

(p1

gig j− qgig j

)A1i j (1−A2

i j )(p2

gig j− qgig j

)(1−A1i j )A

2i j

× (1 − p1

gig j− p2

gig j+ qgig j

)(1−A2i j )(1−A2

i j )]

. (13)

In the product in Eq. (13), each factor depends on i and j onlyvia their block memberships gi and g j , so we can combineseveral terms. First, define

eabrs := ∣∣{(i, j) ∈ E : A1

i j = a, A2i j = b, gi = r, g j = s

}∣∣for (a, b) ∈ {(1, 1), (1, 0), (0, 1), (0, 0)}, in analogy with e11,e10, e01, and e00 from Sec. II A. We can then write the log-likelihood as

L =∑

(r,s)∈B

[e11

rs log qrs

+ e10rs log

(p1

rs − qrs) + e01

rs log(p2

rs − qrs)

+ e00rs log

(1 − p1

rs − p2rs + qrs

)]. (14)

The advantage of writing the log-likelihood as in (14) isthat it clearly separates the contribution from different edgebundles. Using the results for ER layers from Sec. II A, weimmediately obtain (without further calculations) the ML pa-rameter estimates

p1rs = e11

rs + e10rs

e11rs + e10

rs + e01rs + e00

rs

= m1rs

ers, (15)

p2rs = e11

rs + e01rs

e11rs + e10

rs + e01rs + e00

rs

= m2rs

ers, (16)

qrs = e11rs

e11rs + e10

rs + e01rs + e00

rs

= e11rs

ers, (17)

where m1rs and m2

rs denote the number of edges between blocksr and s in layers 1 and 2, respectively, and ers is the number ofpossible edges between nodes in block r and nodes in blocks. When there is a single edge bundle (i.e., when we do notassume any block structure in a network), the ML estimates(15)–(17) recover those that we obtained for correlated ERnetworks in Sec. II A. Each edge bundle also has a correspond-ing Pearson correlation. The ML estimate of this correlationis

ρrs = e00rs e11

rs − e10rs e01

rs√(e11

rs + e10rs

)(e11

rs + e01rs

)(e10

rs + e00rs

)(e01

rs + e00rs

) . (18)

In applications to temporal consumer–product networks, wefind that different edge bundles have vastly different correla-tion values [24]. We anticipate that other empirical multilayernetworks have similar properties.

2. Effective correlation

Although having different correlation values for differentedge bundles can be useful, it is also helpful to have a singlecorrelation measurement for a given multilayer network. For

062307-5


example, one may wish to use such a network diagnostic forone of the purposes that we outlined in Sec. I. One way todefine an effective correlation is to first sample two nodeindices, I and J , uniformly at random and then compute thePearson correlation of the random variables A1

IJ and A2IJ . One

thereby obtains

corr(A1IJ , A2

IJ ) = E[A1

IJA2IJ

] − E[A1

IJ

]E

[A2

IJ

]σ[A1

IJ

]σ[A2

IJ

] , (19)

where we use capital letters for the node indices I and J toemphasize that they are random variables.

We can calculate each term on the right-hand side of (19)by conditioning on the block assignments of I and J . First, forl ∈ {1, 2}, we have

E[AlIJ ] = P

(Al

IJ = 1)

=∑

(r,s)∈BP

(Al

IJ = 1|gI = r, gJ = s)P (gI = r, gJ = s)

=∑

(r,s)∈Bp1

rs

ers

|E | =∑

(r,s)∈B

mlrs

ers

ers

|E | = ml

|E | , (20)

where ml denotes the number of edges in layer l . The expres-sion (20) is the same as the probability pl of generating anedge in layer l for the ER case. Because Al

IJ is a Bernoullirandom variable (in other words, it can only take the values 1or 0), its standard deviation is

σ[Al

IJ

] =√

ml

|E |(

1 − ml

|E |)

.

Finally,

E[A1

IJA2IJ

] = P(A1

IJ = 1, A2IJ = 1

)=

∑(r,s)∈B

qrsers

|E | =∑

(r,s)∈B

e11rs

ers

ers

|E | = e11

|E | .

The estimated value of the effective correlation is thus

ρ = corr(A1IJ , A2

IJ )

= e00e11 − e10e01√(e11 + e10)(e11 + e01)(e10 + e00)(e01 + e00)

, (21)

which is the same as the value in (12) for ER layers (i.e., with-out any block structure in the model). We stress that there is noreason a priori to expect this outcome. In fact, the analogousresult does not hold for Poisson models [24]. In the presentcase, the fact that there is such a correspondence betweenmodels is convenient for practical reasons, as it implies thatone can perform the simpler calculations from Sec. II A toobtain correlation estimates between network layers, even fornetworks with nontrivial mesoscale structure.

C. Correlated degree-corrected SBMs

The models that we have discussed thus far generate net-works in which nodes in the same block have the same

expected degree. Stochastic block models that make this kindof assumption tend to perform poorly when they are used toinfer mesoscale structure in real networks [9], many of whichhave highly heterogeneous degree distributions. This observa-tion led to the development of degree-corrected SBMs [33].We expect that such adjustments can also be important whenmodeling edge correlations, so we now extend the model fromSec. II B to incorporate degree correction.

We continue to work with two-layer networks, which weagain specify in terms of two intralayer adjacency matrices,A1 and A2, with a common block structure that we specifywith a vector g. For each node pair (i, j) ∈ E , we place edgesin the two layers according to the probabilities

P(A1

i j = 1) = θ1

i θ1j p1

gig j, (22)

P(A2

i j = 1) = θ2

i θ2j p2

gig j, (23)

P(A1

i j = 1, A2i j = 1

) =√

θ1i θ1

j θ2i θ2

j qgig j . (24)

We will soon justify the expression in (24). The quantities θ li

and θ lj (with l ∈ {1, 2}) are the (intralayer) degrees of nodes

i and j, normalized by the mean degrees. We calculate thesequantities directly from the input degree sequences, so theyare not model parameters. For undirected and unipartite net-works, θ l

i = dli /〈dl〉, where i ∈ N and 〈dl〉 is the mean degree

in layer l . This normalization recovers the model in Sec. II Bwhen θ l

i = 1 (i.e., when all nodes have the same degree). Themodel parameters p1

rs, p2rs, and qrs are now edge “propensi-

ties” that, together with the degrees, control the probabilitiesof the edges in the layers.

The probabilities in Eqs. (22) and (23) ensure that,marginally, A1 and A2 are generated according to monolayerDCSBMs [33]. It is not obvious how to model the joint prob-ability P (A1

i j = 1, A2i j = 1). In particular, it is not clear how

it should depend on the observed degrees of nodes i and j inlayers 1 and 2 [52]. Part of the complication is that there arefour such quantities for each node pair (i, j). The choice from(24) works particularly well when ρ = 1 and the normalizeddegree sequences θ1 and θ2 are the same, as it reduces toa single DCSBM that generates two identical network lay-ers. Another sensible option is to set P (A1

i j = 1, A2i j = 1) =

θ1i θ1

j θ2i θ2

j qgig j . This choice has the nice property that edges ina particular edge bundle (r, s) ∈ B are independent if and onlyif qrs = p1

rs p2rs, which matches the independence condition

from Sec. II B for the setting without degree correction. How-ever, this second model underperforms the one from (22)–(24)for edge prediction (see Sec. III and [24]). Consequently, forthe rest of this paper, we use the model from Eqs. (22)–(24)as our correlated DCSBM.


When writing the log-likelihood for a correlated DCSBM,we can ignore any additive terms that only involve knownquantities, such as the normalized degrees θ l

i . We can thus

062307-6


write

L =∑

(r,s)∈B

∑(i, j)∈E

[A1

i jA2i j log qrs + A1

i j

(1 − A2

i j

)log

⎛⎝p1rs −

√√√√θ2i θ2

j

θ1i θ1

j

qrs

⎞⎠ + (1 − A1

i j

)A2

i j log

⎛⎝p2rs −

√√√√θ1i θ1

j

θ2i θ2

j

qrs

⎞⎠+ (

1 − A1i j

)(1 − A2

i j

)log

(1 − θ1

i θ1j p1

rs − θ2i θ2

j p2rs +

√θ1

i θ1j θ

2i θ2

j qrs)]

δ(gi, r)δ(g j, s) + const . (25)

As in Sec. II B, we seek to maximize L with respect to the parameters p1rs, p2

rs, and qrs by setting the corresponding derivativesto 0. However, degree-corrected models have the crucial complication that node pairs (i, j) in the same edge bundle (r, s) are nolonger stochastically equivalent (i.e., the corresponding entries of the adjacency matrix are no longer sampled from independentand identically distributed random variables), so their contributions to the log-likelihood are no longer the same in general.Consequently, the ML equations for correlated DCSBMs that we obtain from (25) involve O(N2/K2) terms (where N is thenumber of physical nodes and K is the number of blocks), making them difficult to solve efficiently.

In certain cases, we are able to make some approximations that make these ML equations easier to solve. Recall that θ li = 1

if the degree of node i is equal to the mean degree in layer l . For (i, j) ∈ E , we write

θ1i θ1

j = 1 + ε1i j ,

θ2i θ2

j = 1 + ε2i j .

If the degree distribution is narrow, such that all node degrees are close to the mean degree, then ε1i j and ε2

i j are small parameters(which can be either positive or negative). In this case, a first-order Taylor expansion yields√

θ1i θ1

j θ2i θ2

j =√

(1 + ε1i j )(1 + ε2

i j ) ≈ 1 + ε1i j + ε2

i j

2. (26)

We also calculate √√√√θ1i θ1

j

θ2i θ2

j

=√√√√1 + ε1

i j

1 + ε2i j

≈ 1 + ε1i j − ε2

i j

2(27)

and √√√√θ2i θ2

j

θ1i θ1

j

≈ 1 + ε2i j − ε1

i j

2. (28)

Using the approximations (26)–(28), we expand the first derivatives of L to first order in ε1i j and ε2

i j . (See [24] for details.)This calculation yields the following system of equations:

e10rs

p1rs − qrs

− e00rs

1 − p1rs − p2

rs + qrs+ g10

rs

2

qrs(p1

rs − qrs)2 − f 1

rs

1 − p2rs + qrs/2(

1 − p1rs − p2

rs + qrs)2 − f 2

rs

p2rs − qrs/2(

1 − p1rs − p2

rs + qrs)2 = 0 ,

e01rs

p2rs − qrs

− e00rs

1 − p1rs − p2

rs + qrs+ g01

rs

2

qrs(p2

rs − qrs)2 − f 1

rs

p1rs − qrs/2(

1 − p1rs − p2

rs + qrs)2 − f 2

rs

1 − p1rs + qrs/2(

1 − p1rs − p2

rs + qrs)2 = 0 ,

e11rs

qrs− e10

rs

p1rs − qrs

− e01rs

p2rs − qrs

+ e00rs

1 − p1rs − p2

rs + qrs− g10

rs

2

p1rs(

p1rs − qrs

)2 − g01rs

2

p2rs(

p2rs − qrs

)2

+1

2

f 1rs

(1 + p1

rs − p2rs

) + f 2rs

(1 − p1

rs + p2rs

)(1 − p1

rs − p2rs + qrs

)2 = 0 . (29)

In Eqs. (29), e11rs , e10

rs , e01rs , and e00

rs have the same definitions asin Sec. II B. Additionally, we set

g10rs =

∑(i, j)∈E

A1i j

(1 − A2

i j

)(ε2

i j − ε1i j

)δ(gi, r)δ(g j, s) ,

g01rs =

∑(i, j)∈E

(1 − A1

i j

)A2

i j

(ε1

i j − ε2i j

)δ(gi, r)δ(g j, s) ,

f 1rs =

∑(i, j)∈E

(1 − A1

i j

)(1 − A2

i j

)ε1

i jδ(gi, r)δ(g j, s) ,

f 2rs =

∑(i, j)∈E

(1 − A1

i j

)(1 − A2

i j

)ε2

i jδ(gi, r)δ(g j, s) .

We can efficiently calculate all of these quantities from thematrices A1 and A2.

062307-7


The system of equations (29) reduces to the analogoussystem of equations for correlated SBMs if we ignore allof the terms that depend on εl

i j (i.e., the terms that corre-spond to perturbations of the degrees from their mean values).The zeroth-order solution that we obtain from ignoring theseterms provides a good initialization of a numerical algorithmto solve (29) for the parameters p1

rs, p2rs, and qrs. In prac-

tice, when using correlated DCSBMs for edge prediction(see Sec. III), we find that using a first-order approximationto determine p1

rs, p2rs, and qrs gives results that are almost

identical to those from the zeroth-order approximation [seeEqs. (15)–(17)]. Consequently, we suggest using these zeroth-order approximations for edge-prediction applications, giventhat they are straightforward to calculate and have negligibleimpact on the quality of the results. We also obtain a no-ticeable improvement in calculation speed when using theseapproximations.

In Appendix B, we compare the parameters that we es-timate using the approximate system of equations (29) withthose from the log-likelihood (25). As expected, the quality ofthe approximation depends on the shape of the degree distri-bution, with larger discrepancies between the two approachesfor broader degree distributions.

2. Correlation values

For the SBM without degree correction from Sec. II B,node pairs (i, j) from a given edge bundle (r, s) have thesame Pearson correlation ρrs. This no longer holds for degree-corrected models. Instead, each node pair (i, j) has its owncorrelation value

�i j = E[A1

i jA2i j

] − E[A1

i j

]E

[A2

i j

]σ[A1

i j

]σ[A2

i j

]=

√θ1

i θ1j θ

2i θ2

j qrs − θ1i θ1

j θ2i θ2

j p1rs p2

rs√θ1

i θ1j p1

rs

(1 − θ1

i θ1j p1

rs

)θ2

i θ2j p2

rs

(1 − θ2

i θ2j p2

rs

) . (30)

As in our earlier expansions of the ML equations, weapproximate �i j to first order in ε1

i j and ε2i j . We obtain

�i j ≈ ρrs + ρrs

(ε1

i j

2

p1rs

1 − p1rs

+ ε2i j

2

p2rs

1 − p2rs

− ε1i j + ε2

i j

2

p1rs p2

rs

qrs − p1rs p2

rs

). (31)

Ignoring terms that depend on ε1i j and ε2

i j (i.e., terms thatcorrespond to perturbations of the degrees from their meanvalues), we obtain �i j ≈ ρrs. This approximation works espe-cially well when p1

rs and p2rs are also small, such that their

corresponding network layers are sparse.The case qrs = p1

rs p2rs requires separate consideration [53]

to avoid dividing by 0. First-order approximations in ε1i j and

ε2i j for this case give

�i j ≈ −ε1i j + ε2

i j

2

√p1

rs

1 − p1rs

√p2

rs

1 − p2rs

. (32)

In particular, the zeroth-order solution gives � ≈ 0, in agree-ment with the SBM without degree correction from Sec. II B.

III. EDGE PREDICTION

The aim of edge prediction (which is also called “link pre-diction”) in networks is to infer likely missing edges and/orspurious edges [54]. Edge prediction is useful for filling inincomplete data sets, such as protein-interaction networks (inwhich edges are often established as a result of costly ex-periments) [55] or terrorist-association networks (which aretypically constructed based on partial knowledge) [56]. In thecontext of bipartite user–item networks (in which users areadjacent to items), edge-prediction techniques provide candi-dates for personalized recommendations.

One can perform edge prediction in either a supervisedor an unsupervised fashion. We briefly discuss each of thesetypes of approaches.

Supervised methods rely on models that learn how a speci-fied set of features relates to the presence or absence of edges.Existing methods that take advantage of multilayer structurefor edge prediction typically do so through the specification ofmultilayer features. These include collections of monolayerfeatures [57], as well as path-based [58] and neighborhood-based [59,60] features that consider multiple layers. Althoughmany of these features depend indirectly on the similaritiesbetween different layers, none of these methods quantify thelevel of correlation or use it for edge prediction.

With unsupervised methods, one obtains a ranking of nodepairs such that edges are more likely to occur between nodesin higher-ranked pairs. Common approaches include ones thatare based on probabilistic models and ones that are based onsimilarity indices (like the Jaccard index or the Adamic–Adarindex) [54]. An example of the former for multilayer networksis a method that maps each network layer independently to ahyperbolic space and then uses the hyperbolic distance be-tween nodes in one layer to predict edges in another layer[61]. This work used node-centric notions of correlationand thereby complements the edge-centric perspective of ourwork. Methods that rely on similarity indices include thosethat first generate latent states (i.e., so-called “embeddingvectors”) for the nodes and then rank node pairs according tothe similarities of these latent states [62,63]. Tillman et al. [64]used layer-level correlations to combine monolayer similarityindices into a single score. There are also approaches thatimplicitly take advantage of similarities across layers, suchas by extracting common higher-order structures (specifically,subgraphs with three or more nodes) and looking for patternsthat differ by exactly one edge [65].

Variants of SBMs are popular choices for unsupervisededge-prediction methods [4,9,55,56], including in multilayersettings [15,20]. As an example, in a monolayer degree-corrected Bernoulli SBM, the probability that nodes i and jare adjacent according to the model is P (Ai j = 1) = θiθ j pgig j .Pairs (i, j) for which these probabilities are relatively large butwhich are not adjacent (i.e., Ai j = 0) in the actual network ofinterest may be associated with missing edges. Similarly, pairs(i, j) for which these probabilities are small but which are ad-jacent (i.e., Ai j = 1) in the actual network may be associatedwith spurious edges.

062307-8


TABLE I. Edge-prediction probabilities for various correlated multilayer and monolayer network models.

Model P (A2i j = 1|A1

i j = 1) P (A2i j = 1|A1

i j = 0)

Correlated ER q/p1 (p2 − q)/(1 − p1)

Correlated SBM qrs/p1rs (p2

rs − qrs )/(1 − p1rs )

Correlated CM√

θ1i θ 1

j θ2i θ 2

j q/

(θ 1i θ 1

j p1) (θ 2i θ 2

j p2 −√

θ1i θ 1

j θ2i θ 2

j q)/

(1 − θ1i θ 1

j p1)

Correlated DCSBM√

θ1i θ 1

j θ2i θ 2

j qrs

/(θ 1

i θ 1j p1

rs ) (θ2i θ 2

j p2rs −

√θ1

i θ 1j θ

2i θ 2

j qrs )/

(1 − θ 1i θ 1

j p1rs )

SBM p2rs

DCSBM θ2i θ 2

j p2rs

A. Edge prediction using correlated models

There have been several recent attempts to perform edgeprediction in multilayer networks [15,20,28]. All of thesemethods use multilayer information to infer mesoscale struc-tures in networks, but then they perform edge predictionindependently in each layer, conditioned on the inferredmesoscale structure and the other model parameters. In par-ticular, when using one of these approaches, observing thattwo nodes are adjacent in one layer has no bearing on theirprobability to be adjacent in another layer. We aim to use ourcorrelated models to overcome this limitation.

As in our prior discussions, consider a network with twolayers with intralayer adjacency matrices A1 and A2 and letg be the shared block structure of these layers. Our goal isto predict edges in the second layer, conditioned on the adja-cency structure of the first layer. For each node pair (i, j) ∈ E ,the key quantities to calculate are the probabilities P (A2

i j =1|A1

i j = 1) and P (A2i j = 1|A1

i j = 0) for nodes i and j to beadjacent in the second layer, conditioned on their adjacencyor nonadjacency in the first layer. For example, using thecorrelated Bernoulli SBM from Sec. II B (which has no degreecorrection), we have

P(A2

i j = 1|A1i j = 1

) = qgig j

p1gig j

, (33)

P(A2

i j = 1|A1i j = 0

) =p2

gig j− qgig j

1 − p1gig j

. (34)

This set of probabilities is the same across all node pairs(i, j) from the same edge bundle (r, s). Now suppose thatwe have a positive correlation in this edge bundle, so ρrs >

0. From the definition of the Pearson correlation, it followsthat qrs > p1

rs p2rs. We then find that P (A2

i j = 1|A1i j = 1) > p2

rs

and P (A2i j = 1|A1

i j = 0) < p2rs, whereas using a monolayer

SBM would entail that P (A2i j = 1) = p2

rs. Therefore, the ef-fect of incorporating correlations into our edge-predictionmodel when these correlations are positive is (1) to increasethe probability that nodes i and j are adjacent in the secondlayer when the corresponding edge also exists in the first layerand (2) to decrease this probability when the correspondingedge is absent from the first layer. The effect is reversed fornegative correlations, as the aforementioned probability nowdecreases in situation (1) and increases in situation (2).

In Table I, we summarize the two key probabilities (33) and(34) for four different correlated models, along with the prob-

abilities P (A2i j = 1) for monolayer SBMs and DCSBMs. We

include a correlated configuration model (CM) [66], whichis a special case of the degree-corrected SBM from Sec. II Cwhen there is only one block. Alternatively, one can think ofcorrelated CMs as extensions of correlated ER models thatincorporate degree correction. We use all of these models foredge prediction in synthetic networks in Sec. III B and inconsumer–product networks in Sec. IV B. The two monolayermodels are baselines that we hope to outperform using ourcorrelated models.

B. Tests on synthetic networks

We use K-fold [67] cross-validation to assess the edge-prediction performance of the models in Table I. In machinelearning, this approach is an effective way to measure predic-tive performance [54]. After partitioning a given data set intoK parts, one fits a model to K − 1 of these subsets and uses itto make predictions on the remaining (i.e., “holdout”) subset.One uses each subset once as a holdout, so one does thisprocess K times in total. For our problem, we perform 5-foldcross-validation (which is a standard choice in the machine-learning literature) by splitting the data in the second layer ofa given network into five subsets. Effectively, this consists ofhiding 20% of the entries of the adjacency matrix A2, such thatwe do not know whether they are edges or not. We then train amodel on 100% of the entries of A1 and 80% of the entries ofA2, and we use it to make predictions about the 20% holdoutdata from A2. We do this five times to cover each choice ofholdout data.

A common way to assess the performance of a binary clas-sification model (i.e., a model that assigns one of two possiblevalues to test data) is by using a receiver operating charac-teristic (ROC) curve. An ROC curve plots the true-positiverate (TPR) of a classifier versus its false-positive rate (FPR)for various choices of a threshold. Many models—includingthose that are used for edge prediction in networks—makeprobabilistic predictions, so specifying a threshold is neces-sary to convert these into binary predictions. Lowering thethreshold increases both the TPR and the FPR. A model haspredictive power if the former grows faster than the latter,such that the entire ROC curve lies above the diagonal lineTPR = FPR, which gives the performance of a random clas-sifier. As a single summary measure of a model’s predictiveperformance, it is common to report the area under an ROCcurve (AUC). Larger AUC values are better, with a value of

062307-9


AUC = 0.5 indicating the same level of success as randomguessing and AUC = 1 corresponding to perfect prediction.Even in the latter case, one still needs to determine a choice ofthreshold that completely separates true positives from falsepositives.

The AUC is not the only possible quantity to assess edge-prediction performance, although it is very common [15,68].In many networks, the number of edges is much smaller thanthe number of nonedges (i.e., pairs of nodes that are notadjacent). In situations with such an imbalance, the area underthe precision–recall (PR) curve is more sensitive than the AUCto variations in model performance. Nevertheless, we use theAUC because it has an intuitive interpretation (specifically, asthe probability that the underlying model ranks a true positiveabove a true negative) that allows us to prove the propositionin Appendix C. Additionally, the conclusions that we drawin Sec. IV B about the performance of different models donot change significantly if we use PR curves instead of ROCcurves.

We now describe how to generate synthetic network bench-marks that are suitable for testing the models in Table I.We construct these networks so that they have two tun-able parameters: the Pearson correlation ρ ∈ [−1, 1] anda community-mixing parameter μ ∈ [0, 1] that controls thestrength of the planted mesoscale structure. (See Ref. [14]for more details about the definition of μ.) One can alsoexplicitly control the degree distribution, such as by includinga parameter ηk for the slope of a truncated power law (e.g.,as used in [14] to sample a degree sequence in each layer).For the experiments in this section, we fix ηk = −2 and usea minimum degree of kmin = 10 and a maximum degree ofkmax = 50. It would be interesting to explore the performancegap between degree-corrected models and models withoutdegree correction as one varies ηk , kmin, and kmax, althoughwe do not do so in the present paper. Finally, for our numericalexperiments in this section, we use networks with N = 2000nodes in each layer and nc = 5 communities, with communitysizes sampled from a flat Dirichlet distribution (i.e., one withθ = 1 in the notation of [14]).

We examine two versions, which we call CORRSBM andCORRDCSBM, of a correlated benchmark that is parametrizedby the correlation ρ and the community-mixing parameter μ.For both versions of the benchmark, we generate the (undi-rected and unipartite) adjacency matrix A1 of the first layer inthe same way. Specifically, given μ and degree-distributionparameters ηk , kmin, and kmax, we use the code from [69]to generate A1 and its associated block structure g. We fit amonolayer model—either an SBM or a DCSBM, dependingon the selected version of the benchmark—to A1 to obtainthe marginal edge propensities p1 for the first layer. We thenchoose p2 in one of two ways. For ρ ∈ [0, 1], we set p2 = p1,which ensures that we can generate networks with correlationsthat cover the entire range from 0 to 1. For ρ ∈ [−1, 0], weset p2 = 1 − p1, where 1 is a matrix with all entries equalto 1; this ensures that we can generate networks with correla-tions that cover the entire range from −1 to 0. Given p1, p2,and ρ, we then determine q using either the correlated SBMof Sec. II B or the correlated DCSBM of Sec. II C. For theCORRDCSBM benchmark with ρ � 0, we set the normalizeddegrees θ2

i to be equal to the corresponding quantities θ1i from

the first layer. Again, this choice ensures that we can gener-ate networks all the way to ρ = 1. (We also implemented aversion of the CORRDCSBM benchmark that samples degreesindependently in the second layer, and we found qualitativelysimilar results.) The final step consists of generating A2 givenA1, the propensities p2 and q, and (for the CORRDCSBMbenchmark only) the normalized degree sequences (θ1 andθ2) of the two layers. To perform this step, we first computeedge probabilities using either of the correlated models fromSecs. II B and II C and we then generate edges independentlyaccording to these probabilities.

In Fig. 3, we show sample ROC curves for one network thatwe create using the CORRDCSBM benchmark with μ = 0.3and ρ = 0.5. We compare the performance of our correlatedmodels with a monolayer DCSBM baseline, which performsedge prediction using only information from the second net-work layer. Two of the correlated models outperform thisbaseline and the correlated CM performs comparably well(i.e., it has a similar AUC value).

In Fig. 4, we show results for the CORRSBM benchmarkfor two choices of the community-mixing parameter μ andseveral values (both positive and negative) of the Pearson cor-relation ρ. As expected, the AUC values for monolayer SBMsare independent of ρ, whereas the predictive performance ofcorrelated ER models and correlated SBMs improves as weincrease |ρ|. In particular, when |ρ| = 1, the two correlatedmodels make perfect predictions. When ρ = 0, the perfor-mance of the correlated ER model is almost indistinguishablefrom chance (because AUC ≈ 0.5) and the correlated SBMsperform identically to monolayer SBMs. The gap betweenthe two correlated models is smaller for μ = 0.8 than forμ = 0.3, because the underlying block structure is weaker inthe former case than in the latter. The AUC value that weobtain with the monolayer baseline is also smaller for μ = 0.8than for μ = 0.3.

One striking feature in Fig. 4 is that all curves are ap-proximately straight lines (to within sampling error). It makessense that the performance of monolayer SBMs does not varywith ρ, as these models do not use multilayer information,but the linear dependence on ρ of the other two curves isless intuitive. For the correlated ER model, we can establishrigorously (see Appendix C) that the AUC is approximatelyequal to (1 + |ρ|)/2 when p1 ≈ p2 or when p1 ≈ 1 − p2.Given that the correlated SBM curves from Fig. 4 also exhibita linear dependence on ρ, we believe that it is possible toestablish similar results for correlated models that incorporatemesoscale structure. These results have practical importance,as they allow one to quickly estimate the additional benefits ofusing correlated models instead of monolayer SBMs for edgeprediction.

In Fig. 5, we show results for the CORRDCSBM bench-mark for two choices of the community-mixing parameterμ and non-negative [70] values of the Pearson correlationρ. As expected, when ρ = 0, correlated DCSBMs performsimilarly to monolayer DCSBM. As in Fig. 4, the perfor-mance of the monolayer DCSBM is roughly independentof ρ, whereas the two correlated models perform betteras ρ increases. The gap between the two correlated mod-els narrows substantially as one increases μ from 0.3 to0.8.

062307-10


FIG. 3. ROC curves from a 5-fold cross-validation for a network that we sample from the CORRDCSBM benchmark with community-mixing parameter μ = 0.3 and correlation ρ = 0.5. (a) We compare correlated models that do not incorporate any mesoscale structure toa monolayer DCSBM baseline. The baseline gives AUC ≈ 0.83, whereas the AUC values for the two correlated models are approximately0.76 (correlated ER) and 0.83 (correlated CM). (b) We compare correlated models that incorporate mesoscale structure to the same monolayerDCSBM baseline. The baseline again gives AUC ≈ 0.83, and the AUC values for the two correlated models are approximately 0.89 (correlatedSBM) and 0.91 (correlated DCSBM).

FIG. 4. Edge-prediction results on synthetic networks from the CORRSBM benchmark with parameter values (a) μ = 0.3 and ρ � 0,(b) μ = 0.3 and ρ � 0, (c) μ = 0.8 and ρ � 0, and (d) μ = 0.8 and ρ � 0. In all plots, along the horizontal axis, we vary the correlation ρ

that we use to generate network instantiations. On the vertical axis, we indicate the AUC for 5-fold cross-validation using a monolayer SBM(dashed curves) and correlated SBM and ER models (solid curves). Each data point is a mean across ten trials, and the error bars correspondto one standard deviation from that mean. As expected, the AUC does not change with ρ for the monolayer model, but it increases with |ρ|for the two correlated models. As we increase μ, such that the sampled networks have progressively weaker mesoscale structure, we observea substantial narrowing of the performance gap between correlated ER models and correlated SBMs.

062307-11


FIG. 5. Edge-prediction results on synthetic networks from the CORRDCSBM benchmark with correlation ρ � 0 and community-mixingparameters of (a) μ = 0.3 and (b) μ = 0.8. In both panels, along the horizontal axis, we vary the correlation ρ that we use to generate networkinstantiations. On the vertical axis, we indicate the AUC for 5-fold cross-validation using a monolayer DCSBM (dashed curves) and thecorrelated DCSBM and CM (solid curves). Each data point is a mean across ten trials, and the error bars correspond to one standard deviationfrom that mean. As expected, the AUC is roughly independent of ρ for the monolayer model, but it increases with ρ for the two correlatedmodels. As we increase μ, such that the sampled networks have progressively weaker mesoscale structure, we observe a substantial narrowingof the performance gap between the correlated CMs and the correlated DCSBMs.

IV. APPLICATIONS

We discuss two applications of correlated multilayer-network models to the analysis of empirical networks. InSec. IV A, we report pairwise layer correlations for severalmultiplex networks. In Sec. IV B, we consider a temporaland bipartite network of customers and products. Using anapproach similar to that from Sec. III, we demonstrate thatcorrelated multilayer models have a better edge-predictionperformance than monolayer baselines.

A. Layer correlations in empirical networks

We now calculate pairwise layer correlations using formula(21). Recall that this expression gives the effective correlationbetween two layers, assuming that they have identical blockstructures (although their edge-propensity parameters can bedifferent). Crucially, this calculation does not require thatwe first determine the underlying block structure. In fact, aswe demonstrated in Sec. II B, the effective correlation of thecorrelated SBM is the same as the correlation of the correlatedER model, and the latter is straightforward to compute. Ac-counting for node degrees, as we did in Sec. II C for correlatedDCSBMs, significantly increases the complexity of such a cal-culation. Additionally, as we showed in Sec. II C, correlationsusing a degree-corrected model are rather similar to those thatone obtains without degree correction.

In Table II, we report the mean pairwise layer correlationsfor nine multiplex networks. (See Appendix D for descriptionsof the associated data sets.) To provide additional insight intothese networks, we also report the two layers with the largesteffective correlations.

We make a few observations about some of the resultsin Table II. In the C. elegans connectome, the layers thatcorrespond to two types of chemical synapses are highlycorrelated with each other; their correlations with the layerof electrical synapses is comparatively lower. In the Euro-

pean Union (EU) air transportation network, the two mostcorrelated layers are those that correspond to ScandinavianAirlines and Norwegian Air Shuttle flights; this is consistentwith previous findings [38] (which were based on a differentmethod for quantifying layer similarity). In the network ofarXiv collaborations between network scientists, the two mostsimilar categories are physics.data-an (which stands for “DataAnalysis, Statistics and Probability”) and cs.SI (which standsfor “Social and Information Networks”). One hypothesis isthat these two labels are often used together in papers. Suchcommon usage leads to a large edge overlap between thecorresponding layers and hence to a large correlation value.

To quantify edge correlations at a more granular level, onecan first infer block assignments g and then calculate corre-lations ρrs for all edge bundles (r, s). One possible findingfrom such a calculation may be that correlations betweenScandinavian Airlines and Norwegian Air Shuttle routes aresignificantly larger in certain geographical regions than inothers.

B. Edge prediction in shopping networks

The data-science company dunnhumby gave us accessto “pseudonymized” transaction data from stores of a ma-jor grocery retailer in the United Kingdom. The data werepseudonymized by replacing personally identifiable informa-tion with numerical IDs, rendering it impossible to identifyindividual shoppers. For our analysis, we aggregate transac-tions over fixed time windows to construct bipartite networksof customers and products. We refer to these structuresas “shopping networks”. Because some purchases occur inhigher volumes than others, it is useful to incorporate edgeweights. Given a customer i and a product j, the item-penetration weight is equal to the fraction of all of the itemspurchased by customer i that are product j. The basket-penetration weight is equal to the fraction of all baskets (i.e.,

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TABLE II. Pairwise layer correlations in several multiplex networks.

Number MeanDomain Network of layers correlation Largest correlation (associated layers)

Social CS Aarhus [71] 5 0.27 0.45 (“work” and “lunch” layers)

Lazega law firm [72] 3 0.39 0.48 (“advice” and “co-work” layers)

YouTube [73] 5 0.12 0.20 (“shared subscriptions” and “shared subscribers”)

Biological C. elegans connectome [74] 3 0.47 0.85 (“MonoSyn” and “PolySyn” layers)

P. falciparum genes [75] 9 0.08 0.25 (“HVR7” and “HVR9” layers)

Homo sapiens proteins [76] 7 0.04 0.29 (“direct interaction” and “physical association” layers)

Other FAO international trade [34] 364 0.13 0.74 (“pastry” and “sugar confectionery”)

EU air transportation [22] 37 0.03 0.39 (“Scandinavian Airlines” and “Norwegian Air Shuttle”)

arXiv collaborations [77] 13 0.07 0.73 (“physics.data-an” and “cs.SI”)

distinct shopping trips) of customer i that include product j.See Ref. [24] for more details about these weighting schemes.

We now apply the edge-prediction methodology fromSec. III to temporal shopping networks, in which edges andedge weights can change from changes in shopping behavior,with a fixed set of customers and a fixed set of products. Weconstruct networks with two layers, which cover the three-month time periods of March to May 2013 and June to August2013, respectively. Using the same underlying transactiondata, we construct two shopping networks by determiningthe vector g of block assignments (the same vector for bothlayers [78]) in two different ways. For the first network (whichwe call SHOPPINGMOD), we use basket-penetration weightsfor the edges and apply multilayer modularity maximization[79,80] to the weighted network to determine the commu-nity assignments g [81]. For the second network (which wecall SHOPPINGSBM), we initially calculate item-penetrationweights and we then apply a threshold to remove edges whoseweight is below the median weight (i.e., approximately 50%of the edges). We fit a degree-corrected SBM to the resultingunweighted network using the belief-propagation algorithmof [82]. We expect better edge-prediction performance for thesecond network, because we detect its block structure using anSBM (as opposed to using modularity maximization, which ismore restrictive).

As with our tests on synthetic networks in Sec. III B, weuse 5-fold cross-validation to assess edge-prediction perfor-mance. We summarize the AUC values that we obtain withour various correlated multilayer models and the monolayerbaselines in Table III, and we show sample ROC curves inFig. 6. We make a few observations about these results. First,our approximation AUC ≈ (1 + ρ)/2 for correlated ER mod-els is very accurate for our two shopping networks, whosecorrelations are ρ ≈ 0.44 and ρ ≈ 0.48, respectively. Second,our correlated multilayer models outperform the monolayerbaselines for both networks. In particular, the very simplecorrelated ER model—which assigns one of two probabilitiesto edges, as indicated in Table I—performs about as wellas the more sophisticated monolayer DCSBM for the SHOP-PINGMOD network. Third, as expected, the AUC values aresystematically larger for the SHOPPINGSBM network than forthe SHOPPINGMOD network. Finally, although incorporating

mesoscale structure leads to better performance when thereis no degree correction, this does not seem to be the casefor degree-corrected models, as correlated DCSBMs do notperform significantly better than correlated CMs. This is alsoapparent in Figs. 6(b) and 6(d), where we observe almostidentical ROC curves for the two models. This result suggeststhat, for some networks, taking into account layer correlationsand degree heterogeneity alleviates the need to also considermesoscale structure when performing edge prediction. Thisobservation has practical implications, as a correlated CM ismuch easier than a correlated DCSBM to fit to data and to usefor edge prediction. However, for recommendation systems,there are situations in which fitting a correlated DCSBM isbeneficial, even if its edge-prediction performance is similarto that of a correlated CM. For instance, one may wish toidentify relevant customers for a chosen product, irrespectiveof how much they buy (i.e., their degree). Stochastic blockmodels are able to distinguish between customers with equaldegrees and identify those with the greatest potential predis-position to buy a particular product, whereas CMs are not.

We also note a result from [83] that an ROC curve liescompletely above (i.e., “dominates”) another ROC curve ifand only if the same relationship holds for the associated PRcurves. This result implies for almost all of the curves in Fig. 6that the rankings of our models based on AUC values arealmost identical to those that we would obtain if we insteadbased them on the areas under PR curves. The only curveswhose ranking when we use PR curves is unclear from this

TABLE III. Predictive performance of different models on theshopping data set, as measured by AUC values.

AUC AUCModel (SHOPPINGMOD) (SHOPPINGSBM)

Monolayer SBM 0.549 0.633Correlated ER 0.724 0.743Correlated SBM 0.742 0.793

Monolayer DCSBM 0.725 0.797Correlated CM 0.817 0.870Correlated DCSBM 0.818 0.875

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FIG. 6. ROC curves for our edge-prediction task using 5-foldcross-validation on (a) the SHOPPINGMOD network without degreecorrection, (b) the SHOPPINGMOD network with degree correction,(c) the SHOPPINGSBM network without degree correction, and (d) theSHOPPINGSBM network with degree correction. The dotted diagonalline in each plot indicates the expected ROC curve for a randomclassifier. All other curves lie above this diagonal line, suggestingthat they have predictive power. In all cases, correlated multilayermodels outperform their monolayer counterparts. For both networks,correlated SBMs outperform correlated ER models, whereas corre-lated DCSBMs perform similarly to correlated CMs [as illustratedby the almost overlapping curves in (b) and (d)].

result are the correlated ER and monolayer SBM curves inFig. 6(c).

It is informative to consider the source of false positivesin Fig. 6. Because the correlation between layers is positive,we are likely to predict an edge where none exists when all ofthe following conditions hold: (1) Nodes i and j are adjacentin the first temporal layer but not in the other one, (2) thenode pair (i, j) belongs to an edge bundle (r, s) with a largelayer correlation ρrs, and (3) nodes i and j have large degrees.Note that condition (3) applies only to correlated models withdegree correction.

V. CONCLUSIONS AND DISCUSSION

We introduced models of multilayer networks in which theedges that connect the same nodes in different layers are notindependent of each other. In comparison to models withoutedge correlations, our models offer an improved representa-tion of many empirical networks, as interlayer correlations area common phenomenon: Flights between major airports areserviced by multiple airlines, individuals interact repeatedlywith the same people, consumers often buy the same productsover time, and so on. Among other potential applications, onecan use our models to improve edge prediction, to study the

graph-matching problem on benchmark networks with highlyheterogeneous degree distributions, and to calculate layer cor-relations as insightful summary statistics for networks.

To model layer correlations, we used bivariate Bernoullirandom variables to generate edges simultaneously in two net-work layers. (See [24] for derivations using Poisson randomvariables.) Correlated Bernoulli stochastic block models wereproposed previously [31], although only as forward modelsfor generating networks, rather than for performing inferencegiven empirical data. Another key contribution of our work isa degree-corrected variant of such a model. The maximum-likelihood equations are significantly more difficult to solvein this case than without degree correction, but we were ableto make useful simplifications with suitable approximations.Notably, these simplified equations closely approximate theequations for models without degree correction for networkswith almost homogeneous degree distributions.

The models in the present paper that incorporate somemesoscale structure g assume that such structure is given. Thissetup has the benefit that one can use any desired algorithm toproduce a network partition, including ones that operate onweighted or annotated networks or that use nonstandard nullmodels in a modularity objective function. This makes ourapproach for analyzing correlations suitable for a wide varietyof applications.

Fitting a correlated SBM to network data yields a corre-lation value ρrs for each edge bundle (r, s). We have definedan effective correlation ρ that combines all of these valuesinto a single measure of similarity between two layers of amultilayer network. Notably, the value of the effective correla-tion is independent of a network’s mesoscale structure, so it isextremely easy to compute [see Eq. (12)]. We illustrated thismethod of assessing layer similarity for multiplex networksfrom social, biological, and other domains.

Another application of our work is to edge predictionin multilayer networks. Our numerical experiments revealedthat simple correlated models (e.g., a correlated configurationmodel or a correlated SBM without degree correction) canoutperform monolayer DCSBMs (as measured by AUC val-ues) even for moderate correlation values. We also observedsuch improved performance for consumer–product networks,which have significant layer correlations (with ρ > 0.4). Weexpect that a correlated multilayer DCSBM will typically out-perform a monolayer DCSBM for most empirical networks,even for small levels of correlation.

There are many interesting ways to build further on ourwork. For example, it would be useful to be able to modelall layers simultaneously, rather than in a pairwise fashion,especially for multiplex networks (in which layers do nothave a natural ordering). One challenge is that a multivariateBernoulli distribution of dimension L has 2L − 1 parameters;this grows quickly with the number L of layers. For a temporalsetting, we have proposed generating correlated networks ina sequential way by conditioning each layer on the previousone. In some cases, it will be useful to relax this memorylessassumption and condition a layer on all previous layers, ratherthan only on the most recent layer. For example, the purchasesof shoppers in December in one year are strongly related notonly to their purchases in November, but also to what theybought in December during the previous year.

062307-14


In the present paper, we have not considered the case ofnonidentical mesoscale structures across layers. (See [24] fora possible approach.) Additionally, although one can accountfor edge weights when fitting a block structure g to usewith our models, the rest of our derivations apply only tounweighted networks. Modeling correlated weighted net-works entails prescribing both edge-existence and edge-weight correlations. Yet another idea is to derive correlatedmodels for networks with overlapping communities. Previousresearch [15] has found that incorporating community over-laps can substantially improve edge prediction. Moreover, forcorrelated DCSBMs, it would be useful to learn the depen-dence of the joint probability P (A1

i j = 1, A2i j = 1) on the node

degrees, instead of assuming the parametric form in Eq. (24).This is a challenging problem for which maximum-likelihoodestimation is unlikely to be a suitable tool, so it falls outsidethe scope of the present paper.

Our work is also a starting point for designing algorithmsto detect correlated communities in networks by inferring galongside other model parameters. A practical outcome ofsuch an algorithm would be a set of communities that persistacross layers if and only if the edges in those communitiesare sufficiently highly correlated with each other. Such anapproach would offer a new notion of what it means for acommunity to span multiple layers [41,84].

Finally, although we performed edge prediction in anunsupervised manner, it is also possible to use estimated cor-relations as features in supervised models (see, e.g., [57–60])to improve their performance.

Our work highlights the importance of relaxing edge-independence assumptions in statistical models of networkdata. Doing so provides richer insights into the structureof empirical networks, improves edge prediction, and yieldsmore realistic models on which to test algorithms for commu-nity detection, graph matching, and other tasks.

ACKNOWLEDGMENTS

A.R.P. was funded by the EPSRC Centre for DoctoralTraining in Industrially Focused Mathematical Modelling(through Grant No. EP/L015803/1) in partnership withdunnhumby. We are grateful to dunnhumby for providing ac-cess to grocery-shopping data and to Shreena Patel and RosiePrior for many helpful discussions. We also thank Junbiao Lufor helpful comments.

APPENDIX A: VARIANCE OF ML ESTIMATES

In Sec. II, we derived ML parameter estimates for varioustypes of correlated network models. In this appendix, weshow how to obtain the variance of these estimates and weillustrate how these variances scale with network size (i.e.,the number of nodes). We focus on correlated ER models andthe corresponding log-likelihood (8). The same approach alsoworks for the other types of models that we examined.

Let β = [p1 p2 q]� denote the vector of parameters for acorrelated ER model. Under mild conditions [85], the MLestimate β converges in distribution to a multivariate normaldistribution as the number N of nodes tends to infinity. Forlarge but finite N , the quantity β is distributed approximately

FIG. 7. Maximum-likelihood estimates with 95% confidence in-tervals that we obtain from the inverse of the Fisher informationmatrix for different values of the number N of nodes. The true param-eter values (dashed horizontal lines) are p1 = 0.1, p2 = 0.085, andq = 0.05. The widths of the confidence intervals scale approximatelyas 1/N .

according to N (β∗, I−1(β∗)), where β∗ is the true-parametervector and I−1(β∗) is the inverse of the Fisher informationmatrix evaluated at the parameter values in β∗. The diagonalentries of I−1(β∗) give estimates of the variances of p1, p2,and q.

To illustrate how these variance estimates scale with thenumber N of nodes, we simulate correlated ER networks withp1 = 0.1, p2 = 0.085, and q = 0.05 (which correspond to acorrelation of ρ = 0.5) for different network sizes. In Fig. 7,we plot the 95% confidence intervals around the ML estimatesof the three parameters in our model. We find empirically thatthe variances of the parameters scale approximately as 1/N2,so the standard deviations (and thus the widths of the 95%confidence intervals) scale approximately as 1/N .

APPENDIX B: MAXIMIZING THE FULLLOG-LIKELIHOOD VERSUS OUR APPROXIMATELOG-LIKELIHOOD FOR CORRELATED DCSBMs

In Sec. II C, we derived an approximation (29) of thelog-likelihood for correlated DCSBMs that enables a more ef-ficient estimation of the parameters than maximizing the exactlog-likelihood (25). We now use simulated data to comparethe results that we obtain using this approximation to thosefrom maximizing the original log-likelihood.

We simulate networks using the CORRDCSBM benchmark(see Sec. III B) with N = 1000 nodes, K = 5 communities, amixing parameter of μ = 0.3, and a correlation of ρ = 0.5.For N = 2000 nodes, which we used in other experiments inour paper, we find that maximizing the full log-likelihood isprohibitively slow. Even for N = 1000, obtaining estimatesusing the full log-likelihood takes about 30 min on a typicallaptop computer, whereas the calculation runs in about 5 s ona typical laptop computer when we use the approximate log-likelihood.

The quality of our approximation depends strongly on theshape of the degree distribution. Specifically, we observe that

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FIG. 8. Maximum-likelihood estimates from the approximatelog-likelihood (29) versus ML estimates from the full log-likelihood(25). (a) Example of a network with a relatively narrow degreedistribution (ηk = 0, kmin = 18, and kmax = 20). For this example,the approximation works well. (b) Example of a network with a rela-tively wide degree distribution (ηk = −2, kmin = 10, and kmax = 50).For this example, we observe some discrepancies between the twosets of estimates.

accuracy degrades as we consider distributions with progres-sively larger variances. We illustrate this behavior in Fig. 8,where we plot the full-likelihood [see Eq. (25)] estimatesversus the approximate-likelihood [see Eq. (29)] estimates fortwo networks. For both of these networks, we sample degreesfrom truncated power-law distributions using the code in [69].In Fig. 8(a), we choose a relatively narrow degree distribution(with ηk = 0, a minimum degree of kmin = 18, and a maxi-mum degree of kmax = 22). In this case, the parameter valuesthat we estimate using the approximate log-likelihood closelymatch those from the full log-likelihood. For Fig. 8(b), wechoose a relatively wide degree distribution (with ηk = −2,a minimum degree of kmin = 10, and a maximum degree ofkmax = 50). In this case, the approximate log-likelihood tendsto overestimate parameter values, especially for larger valuesof these parameters. See the top-right corner of Fig. 8(b).

As this example illustrates, there is a trade-off betweenaccuracy and speed. It may be possible to derive better ap-proximations. For example, one can consider a second-orderexpansion (instead of a first-order expansion) with respect tothe quantities ε1

i j and ε2i j . One can also add corrections for

large-degree nodes to the approximate log-likelihood.

APPENDIX C: EDGE-PREDICTION AUC AS A FUNCTIONOF THE PEARSON CORRELATION ρ

We establish the following result.

Proposition. The AUC for a correlated ER model is anaffine function of the Pearson correlation ρ. In particu-lar, when p1 ≈ p2 or p1 ≈ 1 − p2, we have AUCER ≈ (1 +|ρ|)/2.

Proof. Suppose that ρ > 0. (The case ρ � 0 is similar.)With a correlated ER model, all unobserved interactions (i, j)in the second layer have one of two probabilities: q/p1 if A1

i j =1 and (p2 − q)/(1 − p1) if A1

i j = 0. Because ρ > 0, we haveq > p1 p2, which implies that q/p1 > (p2 − q)/(1 − p1). Se-lecting a threshold between these two probabilities amountsto predicting that everything that is an edge in the first layer

FIG. 9. Diagram of the ROC curve for our correlated ER model,which assigns one of two possible edge probabilities to each pair ofnodes.

is also an edge in the second layer and that everything that isnot an edge in the first layer is also not an edge in the secondlayer. Let a and b denote the TPR and FPR, respectively, atsuch an intermediate threshold. In this case, a and b are thecoordinates of the point at which the slope of the ROC curvechanges. See the illustration in Fig. 9.

By straightforward geometry,

AUCER = 1 − ab

2− (1 − a)(1 − b)

2− (1 − a)b

= 1

2+ a − b

2

is the area under this ROC curve. The next step is to estimatea and b. With a correlated ER model, the number of truepositives is proportional [86] to e11; the model’s predictionis correct every time that an edge that is present in the firstlayer is also present in the second layer. To find the TPR, oneneeds to divide this quantity by the number of edges (there aree11 + e01 of them) in the second layer. We obtain

a ≈ e11

e11 + e01≈ q

p2= p1 + ρ

√1 − p2

p2p1(1 − p1) . (C1)

Similarly, each time an edge that is present in the first layer isnot present in the second layer counts as an incorrect predic-tion of the model. Therefore, the number of false positives isproportional to e10. Dividing this by the number of nonedgesin the second layer yields

b ≈ e10

e10 + e00≈ p1 − q

1 − p2= p1 − ρ

√p2

1 − p2p1(1 − p1) .

(C2)From (C1) and (C2), it follows that

AUCER ≈ 1

2+ ρ

2

√p1(1 − p1)

(√1 − p2

p2+

√p2

1 − p2

),

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which is an affine function of ρ. When p1 ≈ p2 or p1 ≈ 1 −p2, as is the case in Fig. 3, we obtain AUCER ≈ (1 + |ρ|)/2,as desired. Using a similar argument, one can show that thesame result holds (with the same assumptions on p1 and p2)when ρ � 0. �

Given that the correlated SBM curves from Fig. 4 alsoappear to depend linearly on ρ, we believe that it is possible toestablish similar results for correlated models that incorporatemesoscale structure.

APPENDIX D: DATA SETS

We provide brief descriptions of the multiplex networksthat we analyzed in Sec. IV A. For weighted networks, wedisregard edge weights when calculating layer correlations.We downloaded these networks, aside from the YouTube andP. falciparum data sets, from [87].

1. CS Aarhus

This is an undirected and unweighted social network ofoffline and online relationships between N = 61 members ofthe Department of Computer Science at Aarhus University[71]. There are T = 5 layers: regularly eating lunch together,friendships on Facebook, coauthorship, leisure activities, andworking together.

2. Lazega law firm

This directed and unweighted network encompasses inter-actions between N = 71 partners and associates who work atthe same law firm [72]. The network has T = 3 layers thatencode co-work, friendship, and advice relationships.

3. YouTube

This is an undirected and weighted network of interactionsbetween N = 15 088 YouTube users [73]. There are T = 5types of interactions: direct contacts (“friendships”), sharedcontacts, shared subscriptions, shared subscribers, and sharedfavorites.

4. C. elegans connectome

This is a directed and unweighted network of synap-tic connections between N = 279 neurons of the nematode

C. elegans [74]. There are T = 3 layers, which correspondto electric, chemical monadic (“MonoSyn”), and chemicalpolyadic (“PolySyn”) junctions.

5. P. falciparum genes

This is an undirected and unweighted network of N = 307recombinant genes from the parasite P. falciparum, whichcauses malaria [75]. There are T = 9 layers that correspondto distinct highly variable regions (HVRs), in which theserecombinations occur. Two genes are adjacent in a layer if theyshare a substring whose length is statistically significant.

6. Homo sapiens proteins

This is a directed and unweighted network of interactionsbetween N = 18 222 proteins in Homo sapiens [88]. Thereare T = 7 layers, which correspond to the following typesof interactions: direct interactions, physical associations,suppressive genetic interactions, association, colocalization,additive genetic interactions, and synthetic genetic interac-tions. The original data set is from BioGRID [76], a publicdatabase of protein interactions (for humans as well as otherorganisms) that is curated from different types of experiments.

7. Food and Agriculture Organization (FAO) trade

This is a directed and weighted network of food importsand exports during the year 2010 between N = 214 countries[34]. There are T = 364 layers, which correspond to differentfood products.

8. European Union air transportation

This is an undirected and unweighted network of flightsbetween N = 450 airports in Europe [22]. There are T = 37layers, each of which corresponds to a different airline.

9. arXiv collaborations

This is an undirected and weighted coauthorship net-work between N = 14 489 network scientists [77]. Thereare T = 13 layers, which correspond to different arXiv sub-ject areas: physics.soc-ph, physics.data-an, physics.bio-ph,math-ph, math.OC, cond-mat.dis-nn, cond-mat.stat-mech, q-bio.MN, q-bio, q-bio.BM, nlin.AO, cs.SI, and cs.CV.

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