Mesoscale Structures in Networks

Mesoscale Structures in Networks

Lake Como School on Complex Networks, 2016

Expander?

– I do not expect to have time to cover all of my slides!

– Introduction and Overview

– Community Structure

– Core–Periphery Structure

– Roles and Positions

– Summary and Conclusions

– Note: I’ll occasionally mention other ideas from the advertised blurb along the way.

Studying mesoscale network structures goes very far beyond studying only community structure!

– Microscale structures: information centered on nodes, edges, or other substructures– Examples: degree of node i, centrality (various types) of node i, centrality (various

types) of edge (i,j), clustering coefficient of node i, etc.– Macroscale structures: properties of distributions of microscale properties

across all nodes– Examples: Is the degree distribution a power law? What is the relationship

between degree and local clustering coefficient? – Mesoscale structures: middle-scale properties

– Examples: cohesive social groups, core versus peripheral banks, functional roles of nodes in a network, etc.

– Note: Useful to examine distributions of microscale quantities separately within mesoscale structures

Puck Rombach

CENSORED!

– The paradigm, on which many methods have been developed, is that one finds densely-connected sets of nodes (called “communities”) with sparse connections between them.

– Important note: Most of these methods will return a community structure whether or not it is present.– Exercise: Try methods on Erdös–Rényi random graphs, which have no

inherent structure, and see what results you get.

– My view: We make an assumption when doing this, so there is an “if” statement in these calculations: If we view a network in this way (or, for that matter, in another way), what do we see? What, if anything, do we learn in an application by doing this?

– “We must be cautious.” (Obi Wan Kenobi, Star Wars)

– Sometimes it can be, but that intuitive is extremely naïve, and good low-dimensional structures are often (typically?) too much to expect.

– “Community detection: You will never find a more wretched hive of scum and villainy. We must be cautious.”– Inspired by the full quote from Obi

Wan Kenobi

– Figure: Jeub et al., Phys. Rev. E, 2015

– Other structures besides assortative structures: different types of block sructures– Bipartite, core–periphery structure, etc.

– Block Models– Roles and Positions

– Nodes that are “similar” (or, more strongly, the same) in some way, but they don’t have to be part of the same densely-connected set

– Example: Given network structure only at a university, who is a professor, who is a postdoc, who is a grad student, who is an undergrad, and who is staff? Perhaps the network structure near a mathematics graduate student looks similar to that near a physics graduate student?

– A different type of block model

– Stochastic Block Models– Statistically principled approach

– See the presentation by Tiago Peixoto

This is the “traditional” (assortative) type of mesoscale structure to study in networks (in the network-science community). There is a very large body of work on it.

– Survey article: MAP, J.-P. Onnela & P. J. Mucha [2009], “Communities in Networks”, Notices of the American Mathematical Society 56:1082–1097, 1164–1166

– Review article: S. Fortunato [2010], “Community Detection in Graphs”, Physics Reports 486:75–174

– Important: These articles are out of date in several respects. There have been significant developments since they were published. We need new reviews.

– 1. Much more emphasis on statistical inference and statistically principled methods. Significant development of these methods.

– Tiago’s presentation

– 2. Development of some methods for generalized situations (e.g., spatial networks, temporal networks, multilayer networks)– Introduction to a few of the available ideas towards the end of my presentation

– 3. Validation of results (e.g., with ”ground truth”) of methods applied to empirical studies?– More than there used to be, but there is still much more to do here. It will happen.

– Note: not just development of benchmarks

– Use of results of clustering method to do something

– Still much less focus here than on methods to cluster data in the first place

Traud et al., SIAM Review, 2011

– “Hard/rigid” versus “soft/fuzzy/overlapping” clustering

– A community should describe a “cohesive group” of nodes– Tons of methods available

– Usual notion: more intra-community edges than one would expect at random– But what does “at random” mean?

– Has a “low-dimensional” assortative (block diagonal) structure that has unduly influenced our intuition of what we should see. (Real life is usually more complicated.)

– We’re making a big assumption.

– Assortative structure

Puck Rombach

CENSORED!

– Network Scientists with Karate Trophies: http://networkkarate.tumblr.com

• Popular approach: Use a “modularity” quality function

• GOAL: Assign nodes to communities to maximize Q. (Use some computational heuristic.)

• Cannot guarantee optimal quality without full enumeration of possible partitions– NP-hard problem– Many algorithms available (spectral, Louvain, etc.)– Need to pick null model appropriate to problem– Extreme near-degeneracies in “good” local optima of Q

• (B. H. Good, Y.-A. de Montjoye, & A. Clauset, PRE 81:046106, 2010)

• Erdös–Rényi (Bernoulli) • Newman–Girvan*

• Arenas et al.*, Leicht–Newman* (directed)

• Barber* (bipartite)

• With additional resolution parameter γ• To try to take

“resolution limit” into account, although there are still some issues

• Examine multiple resolutions of assortative structure

– Directly from consideration of assortative structure: counting edges within communities versus edges between them

– Potts Hamiltonian with a particular choice of interaction energy– From random walks (Laplacian dynamics) on networks

– For some null models– R. Lambiotte, J.-C. Delvenne, &. M Barahona, arXiv:0812.1770 (now published,

with updates, in TNSE, 2015)– I like this derivation, because it provides a direct connection between community

structure and dynamical systems on networks. It suggests that one can think about community structure based not only on network adjacencies per se but also based on dynamical process of interest, such that one seeks bottlenecks in network to such dynamics starting from initial (seed) set of nodes.– This idea provides way to get to local community structure and overlapping communites.

It also leads to direct connections with spectral and expander properties of graphs.

– Nodes = individuals– Edges = self-identified friendships (1 or 0)– The data (“Facebook100”)

– 100 different universities (full networks)

– Single-time snapshot: September 2005 (Facebook was university-only)

– Self-reported demographics: Gender, class year, high school, major, dormitory/”House”

– Provided by Adam D’Angelo and Facebook

– We consider 4 types of networks for each school.– Largest connected component (LCC); “Full”

– Student-only subset of LCC; “Student”

– Female-only subset of LCC; “Female”

– Male-only subset of LCC; “Male”

– Full networks (single university, largest connected component)

� Related to other set distances, but applied to node pairs� w11 = # node pairs put in the same group in 1st and also in the same

group in 2nd partition� w10 = # node pairs put in the same group in 1st partition but different

groups in 2nd partition� w01 and w00 defined analogously� M = total node pairs = Σijwij

1. Z-scores for Rand, Adjusted Rand, Fowlkes-Mallows, & gamma indices are provably identical

2. Analytical formulas exist for the above indices (need permutation tests for Jaccardand Minkowski)

Legends gives disk size as a function of maximum distance d between the 6 different partitions

Full

– We visualize social organization using barycentriccoordinates.

– Center around Year vertex because of importance of that category.

– Compute coordinates for each of 6 partition methods and for each institution plot a disk whose radius is proportional to maximum difference between the 6 coordinates.

– Dormitory residence dominates organization at Rice (31), Caltech (36), and UC Santa Cruz (68).

“Angel, it's not like this is the first time I've had sex under a mystical influence. I went to U.C. Santa Cruz.”

Full networks

Female Networks Male Networks

– Greater importance of High School vertex in many Female networks versus corresponding Full networks

– Residence vertex very important for Males at Michigan and Notre Dame, in contrast to Full, Student, and Female networks at those institutions.

– Male networks seem to have a larger variation among second-most important factor (after Year) than the Female networks.– Suggestive of possibly interesting differences in friendship patterns between

the two genders?

– Relative ordering of Major at a given institution is sometimes gender-dependent

– L. G. S. Jeub, P. Balachandran, MAP, P. J. Mucha, & M. W. Mahoney [2015], Phys. Rev. E 91(1):012821

– L. G. S. Jeub, MWM, PJM, & MAP [2015], arXiv:1510.05185

– Code available at http://github.com/LJeub/LocalCommunities

THINK LOCALLY, ACT LOCALLY: DETECTION OF . . . PHYSICAL REVIEW E 91, 012821 (2015)

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FIG. 6. (Color online) NCP plots [in panel (a)] and conductance ratio profile (CRP) plots [in panel (b)] for CA-GRQC, FB-JOHNS55, andUS-SENATE (i.e., the smaller network from each of the three pairs of networks from Table I) generated using the ACLCUT method. In panels(c)–(e), we show modified Kamada-Kawai [86] spring-embedding visualizations that emphasize community structure [87] of corresponding(color-coded) communities and their neighborhoods (a 2-neighborhood for CA-GRQC, a 1-neighborhood for FB-JOHNS55, and all Senatesthat have at least one senator in common with those in the communities for US-SENATE). We find good small communities but no good largecommunities in CA-GRQC, some weak large-scale structure in FB-JOHNS55 that does not create substantial bottlenecks for the random-walkdynamics, and signatures of low-dimensional structure (i.e., good large communities but no good small communities) for US-SENATE. Thelow-dimensional structure in US-SENATE results from the multilayer structure that encapsulates the network’s temporal properties. [The dashedline in panel (b) indicates a conductance ratio of 1.]

reason for the downward-sloping shape is that US-SENATE andUS-HOUSE each consist of a low-dimensional structure that isevolving along a one-dimensional scaffolding (i.e., time), uponwhich the detailed structure of individual Congresses (i.e., agood partition that is nearly along party lines) is superimposed.(One can examine such structures by using smaller values ofthe interlayer coupling parameter ω; see Ref. [74].) This isconsistent with previous results [90].

These results, which illustrate that community qualitychanges very differently with size in each of the three pairsof networks, also indicate that these three types of networkshave very different properties with respect to large-scale versussmall-scale community structure. Moreover, the qualitativesimilarity in behavior between the two networks in each pairsuggests that the coarse behavior of an NCP (downward-sloping, upward-sloping, or flat) is indicative of large classesof networks and not an artifact of our particular choice ofexample networks. One obtains similar insights about global

structure using the MOVCUT (see Appendix C) and EGONET(see Appendix D) methods, although they can exhibit ratherdifferent local behavior. We investigate these differences inlocal behavior in Sec. IV C.

C. Comparison of results from ACLCUT, MOVCUT,and EGONET

The NCPs generated using either ACLCUT or MOVCUT(see Appendix C), and to a somewhat lesser extent thosegenerated using EGONET (see Appendix D), have similarglobal features—i.e., they exhibit the same general trendsand have dips at small size scales that correspond to nearlyidentical communities—indicating that we obtain a broadlysimilar picture of the large-scale community structure by usingany of the methods. However, the detailed local behavior of thethree methods can differ considerably. Such behavior dependssensitively on the choice of seed node, the choice of the

012821-11

– Upper left plot of previous slide: highest-conductance community for each community size (isoperimetric structure)

– Smaller conductance è better communities (i.e., more ”community-like”)

JEUB, BALACHANDRAN, PORTER, MUCHA, AND MAHONEY PHYSICAL REVIEW E 91, 012821 (2015)

we then discuss our extensions of such ideas. For more detailson conductance and NCPs, see Refs. [25,37,67,68]. If G =(V,E,w) is a graph with weighted adjacency matrix A, then the“volume” between two sets S1 and S2 of nodes (i.e., Si ⊂ V )equals the total weight of edges with one end in S1 and oneend in S2. That is,

vol(S1,S2) =!

i∈S1

!

j∈S2

Aij . (1)

In this case, the “volume” of a set S ⊂ V of nodes is

vol(S) = vol(S,V ) =!

i∈S

!

j∈V

Aij . (2)

In other words, the set volume equals the total weight ofedges that are attached to nodes in the set. The volumevol(S,S) between a set S and its complement S has anatural interpretation as the “surface area” of the “boundary”between S and S. In this study, a set S is a hypothesizedcommunity. Informally, the conductance of a set S of nodes isthe surface area of that hypothesized community divided by“volume” (i.e., size) of that community. From this perspective,studying community structure amounts to an exploration of theisoperimetric structure of G.

Somewhat more formally, the conductance of a set of nodesS ⊂ V is

φ(S) = vol(S,S)

min (vol(S),vol(S)). (3)

Thus, smaller values of conductance correspond to bettercommunities. The conductance of a graph G is the minimumconductance of any subset of nodes:

φ(G) = minS⊂V

φ(S). (4)

Computing the conductance φ(G) of an arbitrary graph isan intractable problem (in the sense that the associateddecision problem is NP-hard [69]), but this quantity can beapproximated by the second-smallest eigenvalue λ2 of thenormalized Laplacian [67,68].

If the “surface area to volume” (i.e., isoperimetric) inter-pretation captures the notion of a good community as a set ofnodes that is connected more densely internally than with theremainder of a network, then computing the solution to Eq. (4)leads to the “best” (in this sense) community of any size in thenetwork.

Instead of defining a community quality score in termsof the best community of any size, it is useful to define acommunity quality score in terms of the best community of agiven size k as a function of the size k. To do this, Ref. [25]introduced the idea of a network community profile (NCP) asthe lower envelope of the conductance values of communitiesof a given size:

φk(G) = minS⊂V,|S|=k

φ(S). (5)

An NCP plots a community quality score (which, as inRef. [25], we take to be the set conductance of communities)of the best possible community of size k as a function of k.Clearly, it is also intractable to compute the quantity φk(G)in Eq. (5) exactly. Previous work has used spectral-based andflow-based algorithms to approximate it [24–26].

To gain insight into how to understand an NCP and what itreveals about network structure, consider Fig. 2. In Fig. 2(a),we illustrate three possible ways that an NCP can behave. Ineach case, we use conductance as a measure of communityquality. The three cases are the following ones.

(1) Upward-sloping NCP. In this case, small communitiesare “better” than large communities.

(2) Flat NCP. In this case, community quality is indepen-dent of size. (As illustrated in this figure, the quality tends tobe comparably poor for all sizes.)

(3) Downward-sloping NCP. In this case, large communi-ties are better than small communities.

For ease of visualization and computational considerations,we only show NCPs for communities up to half of the size ofa network. An NCP for very large communities, which we donot show in figures as a result of this choice, roughly mirrorsthat for small communities, as the complement of a good smallcommunity is a good large community because of the inherentsymmetry in conductance [see Eq. (3)].

In Fig. 2(b), we show an NCP of a LIVEJOURNAL networkfrom Ref. [25]. It demonstrates an empirical fact about alarge variety of large social and information networks: Thereexist good small conductance-based communities, but theredo not exist any good large conductance-based communitiesin many such networks. (See Refs. [24–26,37,67,68] for moreempirical evidence that large social and information networkstend not to have large communities with low conductances.)On the contrary, Fig. 2(c) illustrates a small toy network—aso-called “caveman network”—formed from several smallcliques connected by rewiring one edge from each clique tocreate a ring [70]. As illustrated by the downward-sloping NCPin Fig. 2(d), this network possesses good conductance-basedcommunities, and large communities are better than smallones. One obtains a similar downward-sloping NCP for theZachary Karate Club network [59] as well as for many othernetworks for which there exist meaningful visualizations [25].The wide use of networks that have interpretable visualizations(such as the Zachary Karate Club and planted-partitionmodels [71] with balanced communities) to help developand evaluate methods for community detection and otherprocedures can lead to a strong selection bias when evaluatingthe quality of those methods.

We now consider the relationship between the phenomenaillustrated in Fig. 2 and the idealized block models of Fig. 1.As a concrete example, Fig. 3 shows the NCPs for the examplenetworks in Fig. 1.

First, note that the best partitions consist roughly ofwell-balanced communities in the low-dimensional case ofFigs. 1(a) and 3(a), and the “lowest” point on an NCP tendsto be for large community sizes. Thus, an NCP tends to bedownward sloping for low-dimensional examples.

Networks with pronounced core-periphery structure—e.g.,networks that look like the example in Fig. 1(b)—tend tohave many good small communities but no comparably goodor better large communities. This situation arises in manylarge, extremely sparse networks [24–26]. The good smallcommunities in such networks are sets of connected nodes inthe extremely sparse periphery, and they do not combine to

012821-6



vol(S1,S2) =!

i∈S1

!

j∈S2

Aij . (1)



i∈S

!

j∈V

Aij . (2)



φ(S) = vol(S,S)



φ(G) = minS⊂V

φ(S). (4)





φ(S). (5)











012821-6

Network Community Profile (NCP)

– Upper right plot from two slides ago: ratio of conductance to internal conductance

– Smaller ratio è better communities


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FIG. 3. Network community profiles of the idealized examplenetworks from Fig. 1. (a) NCP for the Zachary Karate Club network.(b) NCP for an example network generated from a block model withcore-periphery structure. (c) NCP for an Erdos-Renyi graph. (d) NCPfor an example network generated from a bipartite block model.

form good, large communities, as they are only connected viaa set of core nodes with denser connections than the periphery.Thus, an NCP of a network with core-periphery structure tendsto be upward sloping, as illustrated in Figs. 1(b) and 3(b).However, this observation does not apply to all networkswith well-defined density-based core-periphery structure. Ifthe periphery is sufficiently well connected (though still muchsparser than the core), then one no longer observes good,small communities. Such networks act like expanders fromthe perspective of the behavior of random walkers, so theyhave a flat NCP. One can generate examples of such networksby modifying the parameters of the block model that we usedto generate the example network in Fig. 1(b) [61].

For a complete graph or a degree-homogeneous expander[see Figs. 1(c) and 3(c)], all communities tend to have poorquality, so an NCP is roughly flat. (See Appendix A for adiscussion of expander graphs.)

Finally, bipartite structure itself does not have any charac-teristic influence on an NCP. Instead, an NCP of a bipartitenetwork reveals other structure present in the network. Forthe example network in Fig. 1(d), the two types of nodes areconnected uniformly at random, so its NCP [see Fig. 3(d)] hasthe characteristic flat shape of an expander.

B. Robustness and information content of NCPs

It is important to discuss the robustness properties ofNCPs. Such properties are not obvious a priori, as an NCP isan extremal diagnostic. Importantly, however, the qualitativeproperty of being downward-sloping, upward-sloping, orroughly flat is very robust to the removal of nodes and edges,variations in data generation and preprocessing decisions,and similar sources of perturbation [24–26]. For example,upward-sloping NCPs typically have many small communities

of good quality, so losing some communities via noise orsome other perturbations has little effect on a realistic NCP.Naturally, whether a particular set of nodes achieves a localminimum is not robust to such modifications. In addition, onecan easily construct pathological networks whose NCPs arenot robust.

It is also important to consider the robustness of a network’sNCP with respect to the use of conductance versus othermeasures of community quality. (Recall that many othermeasures have been proposed to capture the criteria that agood community should be densely connected internally butsparsely connected to the rest of a network [5,25].) Indeed,it has been shown that measures that capture both criteriaof community quality (internal density and external sparsity)behave in a roughly similar manner to conductance-basedNCPs, whereas measures that capture only one of the twocriteria exhibit qualitatively different behavior (typically forrather trivial reasons) [26].

Although the basic NCP that we have been discussingyields numerous insights about both small-scale and large-scale network structure, it also has important limitations.For example, an NCP gives no information on the numberor density of communities with different community qualityscores. (This contributes to the robustness properties of NCPswith respect to perturbations of a network.) Accordingly,the communities that are revealed by an NCP need not berepresentative of the majority of communities in a network.However, the extremal features that are revealed by an NCPhave important system-level implications for the behavior ofdynamical processes on a network: They are responsible forthe most severe bottlenecks for associated dynamical processeson networks [72].

Another property that is not revealed by an NCP is theinternal structure of communities. Recall from Eq. (3) thatthe conductance of a community measures how well (relativeto its size) it is separated from the remainder of a network,but it does not consider the internal structure of a community(except for size and edge density). In an extreme case, a com-munity with good conductance might even consist of severaldisjoint pieces. Recent work has addressed how spectral-basedapproximations to optimizing conductance also approximatelyoptimize measures of internal connectivity [73].

We augment the information from basic NCPs withsome additional computations. To obtain an indication ofa community’s internal structure, we compute the internalconductance of the communities that form an NCP. Theinternal conductance φin(S) of a community S is

φin(S) = φ(G|S), (6)

where G|S is the subgraph of G induced by the nodes inthe community S. The internal conductance is equal to theconductance of the best partition into two communities of thenetwork G|S viewed as a graph in isolation. Because a goodcommunity should be well separated from the remainder ofa network and also relatively well connected internally, weexpect good communities to have low conductance but highinternal conductance. We thus compute the conductance ratio

"(S) = φ(S)φin(S)

(7)

012821-7



vol(S1,S2) =!

i∈S1

!

j∈S2

Aij . (1)



i∈S

!

j∈V

Aij . (2)



φ(S) = vol(S,S)



φ(G) = minS⊂V

φ(S). (4)





φ(S). (5)











012821-6


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FIG. 3. Network community profiles of the idealized examplenetworks from Fig. 1. (a) NCP for the Zachary Karate Club network.(b) NCP for an example network generated from a block model withcore-periphery structure. (c) NCP for an Erdos-Renyi graph. (d) NCPfor an example network generated from a bipartite block model.

form good, large communities, as they are only connected viaa set of core nodes with denser connections than the periphery.Thus, an NCP of a network with core-periphery structure tendsto be upward sloping, as illustrated in Figs. 1(b) and 3(b).However, this observation does not apply to all networkswith well-defined density-based core-periphery structure. Ifthe periphery is sufficiently well connected (though still muchsparser than the core), then one no longer observes good,small communities. Such networks act like expanders fromthe perspective of the behavior of random walkers, so theyhave a flat NCP. One can generate examples of such networksby modifying the parameters of the block model that we usedto generate the example network in Fig. 1(b) [61].

For a complete graph or a degree-homogeneous expander[see Figs. 1(c) and 3(c)], all communities tend to have poorquality, so an NCP is roughly flat. (See Appendix A for adiscussion of expander graphs.)

Finally, bipartite structure itself does not have any charac-teristic influence on an NCP. Instead, an NCP of a bipartitenetwork reveals other structure present in the network. Forthe example network in Fig. 1(d), the two types of nodes areconnected uniformly at random, so its NCP [see Fig. 3(d)] hasthe characteristic flat shape of an expander.

B. Robustness and information content of NCPs

It is important to discuss the robustness properties ofNCPs. Such properties are not obvious a priori, as an NCP isan extremal diagnostic. Importantly, however, the qualitativeproperty of being downward-sloping, upward-sloping, orroughly flat is very robust to the removal of nodes and edges,variations in data generation and preprocessing decisions,and similar sources of perturbation [24–26]. For example,upward-sloping NCPs typically have many small communities

of good quality, so losing some communities via noise orsome other perturbations has little effect on a realistic NCP.Naturally, whether a particular set of nodes achieves a localminimum is not robust to such modifications. In addition, onecan easily construct pathological networks whose NCPs arenot robust.

It is also important to consider the robustness of a network’sNCP with respect to the use of conductance versus othermeasures of community quality. (Recall that many othermeasures have been proposed to capture the criteria that agood community should be densely connected internally butsparsely connected to the rest of a network [5,25].) Indeed,it has been shown that measures that capture both criteriaof community quality (internal density and external sparsity)behave in a roughly similar manner to conductance-basedNCPs, whereas measures that capture only one of the twocriteria exhibit qualitatively different behavior (typically forrather trivial reasons) [26].

Although the basic NCP that we have been discussingyields numerous insights about both small-scale and large-scale network structure, it also has important limitations.For example, an NCP gives no information on the numberor density of communities with different community qualityscores. (This contributes to the robustness properties of NCPswith respect to perturbations of a network.) Accordingly,the communities that are revealed by an NCP need not berepresentative of the majority of communities in a network.However, the extremal features that are revealed by an NCPhave important system-level implications for the behavior ofdynamical processes on a network: They are responsible forthe most severe bottlenecks for associated dynamical processeson networks [72].

Another property that is not revealed by an NCP is theinternal structure of communities. Recall from Eq. (3) thatthe conductance of a community measures how well (relativeto its size) it is separated from the remainder of a network,but it does not consider the internal structure of a community(except for size and edge density). In an extreme case, a com-munity with good conductance might even consist of severaldisjoint pieces. Recent work has addressed how spectral-basedapproximations to optimizing conductance also approximatelyoptimize measures of internal connectivity [73].

We augment the information from basic NCPs withsome additional computations. To obtain an indication ofa community’s internal structure, we compute the internalconductance of the communities that form an NCP. Theinternal conductance φin(S) of a community S is

φin(S) = φ(G|S), (6)

where G|S is the subgraph of G induced by the nodes inthe community S. The internal conductance is equal to theconductance of the best partition into two communities of thenetwork G|S viewed as a graph in isolation. Because a goodcommunity should be well separated from the remainder ofa network and also relatively well connected internally, weexpect good communities to have low conductance but highinternal conductance. We thus compute the conductance ratio

"(S) = φ(S)φin(S)

(7)

012821-7

Conductance Ratio Profile (CRP)


other low-dimensional space. Spectral clustering or otherclustering methods often find meaningful communities in suchnetworks, and one can often readily construct meaningful andinterpretable visualizations of network structure.

(2) Core-periphery structure. In Fig. 1(b), we illustratethe case in which α11 ≫ α12 ≫ α22. This is an exampleof a network with a density-based “core-periphery” struc-ture [24,25,62–64]. There is a core set of nodes that arerelatively well connected both among themselves and to a setof peripheral nodes that interact very little among themselves.

(3) Expander or complete graph. In Fig. 1(c), we illustratethe case in which α11 ≈ α12 ≈ α22. This corresponds to anetwork with little or no discernible structure. For example,if α11 = α12 = α22 = 1, then the graph is a clique (i.e., thecomplete graph). Alternatively, if the graph is a constant-degree expander, then α11 ≈ α12 ≈ α22 ≪ 1. As discussedin Appendix A, constant-degree expanders yield the metricspaces that embed least well in low-dimensional Euclideanspaces. In terms of the idealized block model in Fig. 1, theylook like complete graphs, and partitioning them would notyield network structure that one should expect to construe asmeaningful. Informally, they are largely unstructured whenviewed at large size scales.

(4) Bipartite structure. In Fig. 1(d), we illustrate the casein which α12 ≫ α11 ≈ α22. This corresponds to a bipartite ornearly bipartite graph. Such networks arise, e.g., when thereare two different types of nodes, such that one type of nodeconnects only to (or predominantly to) nodes of the othertype [65].

Most methods for algorithmic detection of communitieshave been developed and validated using the intuition that net-works have some sort of low-dimensional structure [5,25,36].As an example, consider the infamous Zachary Karate Clubnetwork [59], which we show in Fig. 1(a). This well-knownbenchmark graph, which seems to be an almost obligatoryexample to include in papers that discuss community structure,clearly looks like it has a nice low-dimensional structure. Forexample, there is a clearly identifiable left half and right half,and two-dimensional visualizations of the network [such asthat in Fig. 1(a)] highlight that bipartition. Indeed, the ZacharyKarate Club network possesses well-balanced and (quotingSimon [66]) “nearly decomposable” communities; and thenodes in each community are more densely connected to nodesin the same community than they are to nodes in the othercommunity. Relatedly, appropriately reordering the nodes ofthe Zachary Karate Club network yields an adjacency-matrixrepresentation with an almost block-diagonal structure withtwo blocks [as typified by the cartoon in Fig. 1(a)]; and anyreasonable community-detection algorithm should be able tofind (exactly or approximately) the two communities.

Another well-known network that (slightly less obviously)looks like it has a low-dimensional structure is a so-calledcaveman network, which we illustrate later [in Fig. 2(c)].Arguably, a caveman network has many more communitiesthan the Zachary Karate Club, so details such as whetheran algorithm “should” split it into two or a somewhat largernumber of reasonably well-balanced communities might bedifferent than in the Zachary Karate Club network. However,a caveman network also has a natural well-balanced parti-tion that respects intuitive community structure. Reasonable

(a) Three possible NCPs (b) Realistic NCP from [25]

(c) A caveman network

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FIG. 2. (Color online) Illustration of network community pro-files (NCPs) of conductance versus community size. (a) Stylizedversions of possible shapes for an NCP: downward-sloping (black,solid curve), upward-sloping (red, dotted curve), and flat (blue,dashed curve). (b) NCP of a LIVEJOURNAL network that illustratesthe characteristic upward-sloping NCP that is typical for many largeempirical social and information networks [25]. (c) A toy “cavemannetwork” with 10 cliques of 10 nodes each, where one edge from eachclique has been rewired to create a ring [70]. (d) NCP for a similarcaveman network with 100 cliques of 10 nodes each [the NCP for thenetwork in panel (c) is identical for communities with fewer than 50nodes], illustrating the characteristic downward-sloping NCP that istypical of networks that are embedded in a low-dimensional space.

two-dimensional visualizations of this network [such as theone that we present in Fig. 2(c)] shed light on that structure;and any reasonable community-detection algorithm can beadjusted to find (exactly or approximately) the expectedcommunities. In this paper, we demonstrate that most realisticnetworks do not look like these small examples. Instead,realistic networks are often poorly approximated by low-dimensional structures (e.g., with a small number of relativelywell-balanced communities, each of which is more denselyconnected internally than it is with the rest of the network).Realistic networks often include substructures that moreclosely resemble core-periphery graphs or expander graphs[see Figs. 1(b) and 1(c)]; and networks that partition intonice, nearly decomposable communities tend to be exceptionalrather than typical [24,25,36].

III. NETWORK COMMUNITY PROFILESAND THEIR INTERPRETATION

Recall from Sec. I that an NCP measures the quality of thebest possible community of a given size as a function of thesize of the purported community [24–26]. In this section, weprovide a brief description of NCPs and how we use them.

A. The basic NCP: Measuring size-resolved community quality

We start with the definition of conductance and the originalconductance-based definition of an NCP from Ref. [25], and

012821-5

– We examine a few different processes (a community S reflects a roadblock to the dynamics of a given process).

– Example: Personalized PageRank

– The dynamics is a random walk with teleportation. Look at which nodes get visited as it unfolds. Sample over different seed nodes. Use approximate PPR vector in estimation of conductance.


We also define the edge expansion of a set of nodesS ⊂ V as

h(S) = |E(S,S)||S|

. (A2)

The edge expansion of a graph G is the minimum edgeexpansion of any subset (of size no greater than n/2) of nodes:

h(G) = minS⊂V :|S|! n

2

h(S). (A3)

A sequence of d-regular graphs {Gt }t∈N is a family ofexpander graphs if there exists an ϵ > 0 such that h(Gt ) " ϵfor all t ∈ N. Informally, a given graph G is an expander if itsedge expansion is large.

As reviewed in Ref. [114], one can view expanders fromseveral complementary viewpoints. From a combinatorialperspective, expanders are graphs that are highly connectedin the sense that one has to sever many edges to disconnect alarge part of an expander graph. From a geometric perspective,this disconnection difficulty implies that every set of nodes hasa very large boundary relative to its size. From a probabilisticperspective, expanders are graphs for which the naturalrandom-walk process converges to its limiting distribution asrapidly as possible. Finally, from an algebraic perspective,expanders are graphs in which the first nontrivial eigenvalueof the Laplacian operator is bounded away from 0. (Becausewe are discussing d-regular graphs, note that this statementholds for both the combinatorial Laplacian and the normalizedLaplacian.) In addition, constant-degree (i.e., d-regular, forsome fixed value of d) expanders are the metric spaces that(in a very precise and strong sense [114]) embed least well inlow-dimensional spaces (such as those discussed informally inSec. II B). All of these interpretations imply that smaller valuesof expansion correspond more closely to the intuitive notionof better communities (whereas larger values of expansioncorrespond, by definition, to better expanders).

Note the similarities between Eq. (A2) and Eq. (A3), whichdefine expansion, and Eq. (3) and Eq. (4), which defineconductance. These equations make it clear that the differencebetween expansion and conductance simply amounts to adifferent notion of the size (or volume) of sets of nodes and thesize of the boundary (or surface area) between a set of nodesand its complement. This difference is inconsequential ford-regular graphs. However, because of the deep connectionsbetween expansion and rapidly mixing random walks, the latternotion (i.e., conductance) is much more natural for graphs witha substantial degree of heterogeneity. The interpretation offailing to embed well in low-dimensional spaces (like lines orplanes) is not extremal in the case of conductance and degree-heterogeneous graphs (as it is in the case of expansion anddegree-homogeneous graphs). However, the interpretationsof many other properties, such as being well-connected andfailing to provide bottlenecks to random walks, also holdfor conductance and degree-heterogeneous graphs (such asthose that we consider in the main text of the present paper).Accordingly, it is insightful to interpret our empirical resultson small-scale versus large-scale structures in networks in thelight of known facts about expander (and expanderlike) graphs.

APPENDIX B: COMMUNITY QUALITY, DYNAMICSON GRAPHS, AND BOTTLENECKS TO DYNAMICS

In this section, we describe in more detail how we algo-rithmically identify possible communities in graphs. Becausewe are interested in local properties and how they relateto meso-scale and global properties, we take an operationalapproach and view communities as the output of variousdynamical processes (e.g., diffusions or geodesic hops), andwe discuss the relationship between the output of such pro-cedures and well-defined optimization problems. The idea ofusing dynamics on a network has been exploited successfullyby many methods for finding “traditional” communities (ofdensely connected nodes) [9,32,53,115–118] as well as forfinding sets of nodes that are related to each other in otherways [48,54,115,119,120].

In this paper, we build on the idea that random walks andrelated diffusion-based dynamics, as well as other types oflocal dynamics (e.g., ones, like geodesic hops, that depend onideas based on egocentric networks), should get “trapped” ingood communities. We examine three dynamical methods forcommunity identification.

1. Dynamics type 1: Local diffusions (the “ACLCUT” method)

In this procedure, we consider a random walk that starts at agiven seed node s and runs for some small number of steps. Wetake advantage of the idea that if a random walk starts inside agood community and takes only a small number of steps, thenit should become trapped inside that community. To do this,we use the locally biased PPR procedure of Refs. [121,122].Recall that a PPR vector is defined implicitly as the solutionpr(α,s) of the equation

pr(α,s) = αD−1A pr(α,s) + (1 − α)s, (B1)

where 1 − α is a “teleportation” probability and s is a seedvector. From the perspective of random walks, evolution occurseither by the walker moving to a neighbor of the current node orby the walker “teleporting” to a random node (e.g., determineduniformly at random as in the usual PageRank procedure, or toa random node that is biased towards s in the PPR procedure).The PPR vector pr(α,s) represents the stationary distributionof this random walk. In general, teleportation results in a biasto the random walk, and one usually tries to minimize such abias when detecting communities. (See Ref. [123] for cleverways to choose s with this goal in mind.)

The algorithm of Refs. [121,122] deliberately exploits thebias from teleportation to achieve localized results. It computesan approximation to the solution of Eq. (B1) (i.e., it computesan approximate PPR vector) by strategically “pushing” massbetween the iteratively updated approximate solution vectorand a residual vector in such a way that most of the nodesin the original network are not reached. Consequently, thisalgorithm is typically much faster for moderately large tovery large graphs than is the naive algorithm to compute asolution to Eq. (B1). The algorithm is parametrized in terms ofa “truncation” parameter ϵ, where larger values of ϵ correspondto more locally biased solutions. We refer to this procedure asthe ACLCUT method.

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– L. G. S. Jeub, P. Balachandran, MAP, P. J. Mucha, & M. W. Mahoney [2015], Phys. Rev. E 91(1):012821

– L. G. S. Jeub, MWM, PJM, & MAP [2015], arXiv:1510.05185

– Code available at http://github.com/LJeub/LocalCommunities


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CA-GrQc

FB-Johns55

US-Senate

(a) NCP

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(b) CRP

(c) CA-GrQc (d) FB-Johns55

00.51

(e) US-Senate

FIG. 6. (Color online) NCP plots [in panel (a)] and conductance ratio profile (CRP) plots [in panel (b)] for CA-GRQC, FB-JOHNS55, andUS-SENATE (i.e., the smaller network from each of the three pairs of networks from Table I) generated using the ACLCUT method. In panels(c)–(e), we show modified Kamada-Kawai [86] spring-embedding visualizations that emphasize community structure [87] of corresponding(color-coded) communities and their neighborhoods (a 2-neighborhood for CA-GRQC, a 1-neighborhood for FB-JOHNS55, and all Senatesthat have at least one senator in common with those in the communities for US-SENATE). We find good small communities but no good largecommunities in CA-GRQC, some weak large-scale structure in FB-JOHNS55 that does not create substantial bottlenecks for the random-walkdynamics, and signatures of low-dimensional structure (i.e., good large communities but no good small communities) for US-SENATE. Thelow-dimensional structure in US-SENATE results from the multilayer structure that encapsulates the network’s temporal properties. [The dashedline in panel (b) indicates a conductance ratio of 1.]

reason for the downward-sloping shape is that US-SENATE andUS-HOUSE each consist of a low-dimensional structure that isevolving along a one-dimensional scaffolding (i.e., time), uponwhich the detailed structure of individual Congresses (i.e., agood partition that is nearly along party lines) is superimposed.(One can examine such structures by using smaller values ofthe interlayer coupling parameter ω; see Ref. [74].) This isconsistent with previous results [90].

These results, which illustrate that community qualitychanges very differently with size in each of the three pairsof networks, also indicate that these three types of networkshave very different properties with respect to large-scale versussmall-scale community structure. Moreover, the qualitativesimilarity in behavior between the two networks in each pairsuggests that the coarse behavior of an NCP (downward-sloping, upward-sloping, or flat) is indicative of large classesof networks and not an artifact of our particular choice ofexample networks. One obtains similar insights about global

structure using the MOVCUT (see Appendix C) and EGONET(see Appendix D) methods, although they can exhibit ratherdifferent local behavior. We investigate these differences inlocal behavior in Sec. IV C.

C. Comparison of results from ACLCUT, MOVCUT,and EGONET

The NCPs generated using either ACLCUT or MOVCUT(see Appendix C), and to a somewhat lesser extent thosegenerated using EGONET (see Appendix D), have similarglobal features—i.e., they exhibit the same general trendsand have dips at small size scales that correspond to nearlyidentical communities—indicating that we obtain a broadlysimilar picture of the large-scale community structure by usingany of the methods. However, the detailed local behavior of thethree methods can differ considerably. Such behavior dependssensitively on the choice of seed node, the choice of the

012821-11

– You’re making an assumption by saying you are looking for assortative structures.

– Other structures may be more informative and/or more appropriate.

– Modularity maximization has well-studied issues. They include:– Numerous near-degeneracies in the optimization landscape (Good et al., 2010)

– Resolution limit (Fortunato & Barthelemy, 2007)

– Statistical inconsistency (Bickel & Chen, 2009)

– Always gives you an answer as output, but is it meaningful?

– Other methods have unknown issues. They haven’t been as well-studied, so their problems are less well appreciated. Don’t assume that they don’t have problems.

🤔

– Studying community structure can be very insightful—I spend time doing this, after all!—but one has to use such tools carefully.

😉

Another important type of mesoscale structure, which is becoming increasingly

popular to study.

– P. Csermely, A. London, L.-Y. Wu, & B. Uzzi [2013], “Structure and Dynamics of Core–Periphery Networks”, Journal of Complex Networks1:93–123.

– Note: We also included extensive discussion of the background to studying core–periphery structure in the following article:

– M. P. Rombach, MAP, J. H. Fowler, & P. J. Mucha [2014], “Core–Periphery Structure in Networks”, SIAM J. App. Math. 74(1):167–190.

Core–Periphery StructureCommunity Structure

ì Note: Intuitive that many networks have such structure, but how to examine it?

ì Core versus peripheral countries in international relations (seems to be origin of the notion), social networks, core versus peripheral banks, transportation networks, etc.

ì Borgatti–Everett (1999):ì Discrete notions: simpler one is to compare network to an ideal block model consisting of a fully

connected core and a periphery with no internal connections but fully connected to the coreì Continuous notion: start with above idea and determine a “core value” for each node

ì A subset of other notions of core-periphery structure ì Holme (2005): Defined a core-periphery coefficient in terms of the k-core of a graphì Da Silva et al (2008): Defined a core coefficient using closeness centrality and a measure of

shortest pathsì Leskovec et al. (2009): Onions and whiskersì Leskovec and collaborators (2013): Core regions from overlap of communities

– Origin in international relations (political, economical, etc.)– First-world countries = “core” countries

– Second-world countries = “semi-peripheral” countries

– Third-world countries = “peripheral” countries

– Discrete versus continuous core–periphery structure

– Debates and discussions date back to the early qualitative work several decades ago– Continuous method gives a centrality measure. One can then obtain a discrete

classification starting from a continuous spread of values.

– Intuition: Peeling an onion

– Remark: “nestedness” in ecology is a bipartite analog of core–periphery structure (see, e.g., discussion in S. H. Lee, PRE 93:022306, 2016)

New York & Erie Railroad, diagram from about 160 years ago

The London Underground (“The Tube”)

– Given k, remove all nodes of degree k-1 or less. After this, some nodes that previously had degree k now have degree k-1 or less, so remove those too. Iterate until all nodes have at least degree k. That is the k-core.

– Good points:– Very fast algorithm, captures intuition of onion peeling, mathematically tractable (e.g.,

analysis of k-core percolation), probably does a reasonable job of getting high-degree nodes in the core

– Bad points:– Low-degree nodes can be core nodes, so there are false negatives (in the most interesting

situation, so it’s not really solving the problem. It’s also “too coarse” in other respects.

– Example: Should all nodes in k-shell be in the same level of the core?

– Example: How deep is the core? The largest k for a given network may not be satisfactorily deep to study a problem in this way.

ì Core–periphery coefficient:

ì Average over all undirected, unweighted graphs with the same degree sequence (configuration model)

ì P(i,j) = number of edges in shortest path between i and j

ì A k-core is a maximal connected subgraph in which all nodes have degree at least k

ì Aij = element of weighted, undirected adjacency matrix

ì Seek a value of ρC that is large compared to expected value of ρC obtained if entries of vector C are shuffledì Output = core vector C giving core and periphery nodes

ì Continuous notion: node i is assigned a ‘coreness’ value and Cij = Ci x Cj = a

Maximize

where Cij= 1 if i or j is in the core and Cij= 0 otherwise.

Find the best fit to a core–periphery block model.

1 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 0 0 0 01 1 1 1 0 0 0 01 1 1 1 0 0 0 01 1 1 1 0 0 0 0

Find the best fit to a core–periphery block model.

1 1 1 1 a a a a1 1 1 1 a a a a1 1 1 1 a a a a1 1 1 1 a a a aa a a a 0 0 0 0a a a a 0 0 0 0a a a a 0 0 0 0a a a a 0 0 0 0

Maximize

where Cij= 1 if i and j are in the core, Cij= a if i xor j is in the core, and Cij= 0 otherwise.

Let C be some vector of values between 0 and 1, and maximize

This method does not assume any “shape” of the core–periphery structure beforehand.

Approach by Rombach et al. [2014] builds on this idea.

v Interpolates between continuous and discrete notions of core-periphery structurev We consider weighted, undirected networks

v Entries of core vector C can take non-negative values (e.g. Cij = Ci x Cj)

v Seek C that is normalized and is a shuffle of the vector C* whose components specify local core values [N = total number of nodes] via a transition function.

v Example transition function:

v C is chosen to maximize the core quality R:

v Parameters: α [where 0≤α≤1] sets sharpness of boundary between core and periphery; and β [where 0≤β≤1] sets the size of the core

v Another transition function:

ì We obtain a core score for each node by averaging results over different values of α and β:

ì Z = normalization constant; ensures that max[CS(i)] = 1

ì It would be interesting to develop more sophisticated procedures for sampling values of α and β.

α = 0.1

β = 0.5

α = 0.9

β = 0.5

α = 0.5

β = 0.9

α = 0.5

β = 0.1

Picture courtesy of Aaron Clauset

Tube data available from my website.

– The desire to be able to have both a continuous core-periphery spectrum and discrete core/periphery (or core/semi-periphery/periphery) partition was already recognized in old work in international relations and sociology.

2006 Network

Important note: The lists depend strongly on which papers are in the data sets, is based on coauthorship only, etc. (Therefore: Don’t take them too seriously!)

Richardson, Porter, Mucha (PRE, 2009)

– For applications such as transportation networks, perhaps we should look directly at path-based notions to determine core junctions (nodes) and core edges?

– For example, one can use modified notions of centrality measures like betweenness, and now one can easily define notions for directed networks (which is difficult for the density-based approaches discussed earlier).

– Theory: M. Cucuringu, P. Rombach, S. H. Lee, & MAP [2016], “Detection of Core–Periphery Structure in Networks Using Spectral Methods and Geodesic Paths”, Eur. J. App. Math., in press (arXiv:1410.6572)

– Applications: S. H. Lee, M. Cucuringu, & MAP [2014], “Density-Based and Transport-Based Core–Periphery Structures in Networks”, Phys. Rev. E89:032810

ì Rank nodes by a “participation score”, which is computed as follows: For each edge (i,j) in a graph G, compute the shortest path in G with that edge removed. All nodes participating in such a path have a +1 added to their participation score.

ì This method (“edge-removed betweenness centrality”) rewards nodes for being part of cycles.

ì Similar for alternative measures of short paths (need not consider only geodesic paths)

ì Similar definition for path-based core values for edges

Data available at https://sites.google.com/site/lshlj82/

S. H. Lee & P. Holme, Phys. Rev. Lett. 108, 128701 (2012)Data (100 cites) available at https://sites.google.com/site/lshlj82/

S. H. Lee, M. D. Fricker, and MAP [2016], J. Cplx. Networks, advance access [http://comnet.oxfordjournals.org/content/early/2016/04/29/comnet.cnv034.abstract]; includes release of large fungal data set

Uses variant of taxonomy method from Onnela et al., PRE, Vol. 86, 036104 (2012).(Mesoscopic response functions based on community structure at multiple scales.)

• Fungi = living networks

• Edges are fundamental (nodes are placeholders)

Given a network, can we assign “roles” (i.e., colors) to nodes to identify their type?(Not based on density of connections!)

– S. Wasserman & K. Faust [1994], Social Network Analysis: Methods and Applications, Cambridge University Press

– P. Doreian, V. Batagelj, and A. Ferligoj [2004], Generalized Blockmodeling, Cambridge University Press

– R. A. Rossi and N. K. Ahmed, “Role discovery in Networks” [2015], IEEE Transactions on Knowledge and Data Engineering 27(4):1112–1131

– M. G. Everett and S. B. Borgatti [1994], “Regular equivalence: General theory”, Journal of Mathematical Sociology 19(1):29–52

– One can examine roles in networks by looking at types of block structure that are based on things other than density

– Role equivalence/assignment/coloring

– Define an equivalence relation between nodes, such that two nodes are in the same equivalence class (i.e., colored in the same way) if they are the same in some respect.

– Loosely speaking, “role equivalence” is trying to find nodes that are playing similar roles (e.g., social roles, etc.) in a network. These nodes are supposed to have the same network environment (or, more generally, similar ones), such as a social environment, as measured in some way.

– Rearrange the nodes so that each color indicates a set of successive nodes. Then the adjacency matrix shows a block structure.

Some parts (and snapshots!) of my presentation on roles and positions are taken or adapted from slides by Tom Snijders (5/2/2012): http://www.stats.ox.ac.uk/~snijders/Equivalences.pdf

Each type of coloring is a member of the class specified above it. (Each type corresponds to a different way of what it means for a pair of nodes to be “equivalent”.)

– F. Lorrain and H. C. White [1971], “Structural equivalence of individuals in social networks” Journal of Mathematical Sociology 1:49–80

– Written in language of “category theory”

– Nodes i and j are structurally equivalent if they relate to other nodes in the same way.

– Consider the following example from Borgatti and Everett:

Tom Snijders

Tom Snijders

– L. D. Sailer [1978], “Structural equivalence: Meaning and Definition, Computation and Application”, Social Networks 1:73–90

– D. R. White & K. P. Reitz [1983], “Graph and Semigroup Homomorphismson Networks of Relations”, Social Networks 5:193–234

Acoloring isaregular equivalence iftwonodes of thesamecolor also haveneighbors of thesame color.

Tom Snijders

– For empirical data, asking for exact equivalence is too stringent a demand. It is necessary to relax this idea.

– One way to do this is to examine stochastic equivalence between nodes.

– For a probability distribution of edges in a graph, a coloring is a stochastic equivalence if nodes with the same color have the same probability distribution of edges with other nodes.

– That is, the probability distribution of the network has to remain the same when (stochastically-)equivalent nodes are exchanged. This probability distribution is a stochastic block model.

➞

– Another way to loosen notions of exact equivalence is to compute similarities between nodes that play similar roles in a network.

– One can then study community structure (i.e., assortative cohesive groups) of a network, and an associated adjacency matrix, that encodes these similarities.

– Example similarity from the following paper:

– E. A. Leicht, P. Holme, and M. E. J. Newman [2006], “Vertex Similarity in Networks” Physical Review E 73:026120

– α is a parameter

– λ1 is the largest eigenvalue of A

– Then you can detect communities (i.e., assortative structures) in the similarity matrix S

– M. Beguerisse-Díaz, G. Garduño-Hernández, B. Vangelov, S. N. Yaliraki, & M. Barahona[2014], “Interest Communities and Flow Roles in Directed Networks: The Twitter Network of the UK Riots”, Journal of the Royal Society Interface 11:20140940

Some illustrative examples and basic ideas for examining community structure in more

general types of networks.

– Multilayer Networks– M. Kivelä, A. Arenas, M. Barthelemy, J. P. Gleeson, Y. Moreno, & MAP [2014],

“Multilayer Networks”, Journal of Complex Networks, 2(3):203–271– S. Boccaletti et al. [2014], “Structure and Function of Multilayer Networks”,

Physics Reports, 544(1):1–122

– Temporal Networks– P. Holme & J. Saramäki [2012], “Temporal Networks”, Phys. Rep. 519:97–125– P. Holme [2015], “Modern Temporal Network Theory: A Colloquium”, Eur. Phys. J.

B 88(9):234– Spatial Networks

– M. Barthelemy [2011], “Spatial Networks”, Phys. Rep. 499:1–101

– We’ll discuss extending community structure to these situations, but of course one also wants to extend other ways of examining mesoscale structures in these networks.

– Example: Using stochastic block models (see Tiago’s presentation)

– I am only giving examples and will focus mostly on the context of modularity optimization (though I’ll also show an example with extending the Jeub et al. local approach). One can also generalize other approaches, and there is a lot more work to do.

– Many networks are either explicitly embedded in space (e.g., road networks, granular materials) or have structures that are affected by space (e.g., due to mobility).

– This has a large effect on network structure (e.g., see Marc Barthelemy’slecture and review article).

– Useful to develop and consider null models that incorporate spatial information.

– 2D, vertical, 1 layer aggregate of photoelastic disks

– Internal stress pattern in compressed packing manifests as network of force chains (panel B)

– Force network is a weighted graph in which an edge between 2 particles (nodes) exists if the two particles are in contact with each other; the forces give the weights

– D. S. Bassett, E. T. Owens, K. E. Daniels, and MAP [2012], Physical Review E 86:041306

– 2D granular medium of photoelastic disks

– Two networks

– Underlying topology (unweighted)

– Forces (weighted)

– Both types of networks are needed for characterizing sound propagation

Ø Use a null model that includes more information

Ø Fix topology (i.e., connectivity) but scramble geometry (i.e., edge weights)› Wij = weighted adjacency-matrix element =

force network› Aij = binary adjacency-matrix element =

contact networkØ Communities obtained from optimization of

modularity (with “physical null model”) match well with empirical granular force networks in both laboratory and computational experiments

thereby construct a force-weighted contact network W from alist of all inter-particle forces. If particle i and j are in contact,then Wij ¼ fij/mean (f), where fij is the normal force betweenthem. If two particles are not in contact, then Wij ¼ 0. In addi-tion, we let Wii ¼ 0. We also construct an unweighted (i.e.,binary) matrix B whose elements are

Bij ¼!1; Wijs0;0; Wij ¼ 0:

The matrix B is oen called an “adjacency matrix”,24 and thematrix W is oen called a “weight matrix.”

To obtain force chains fromW, we want to determine sets ofparticles for which strong inter-particle forces occur amidstdensely connected sets of particles. We can obtain a solution tothis problem via “community detection”,34,35,44 in which we seeksets of densely connected nodes called “modules” or “commu-nities.” A popular way to identify communities in a network isby maximizing a quality function known as modularity withrespect to the assignment of particles to sets called “commu-nities.” Modularity Q is dened as

Q ¼X

i;j

"Wij " gPij

#d$ci; cj

%; (1)

where node i is assigned to community ci, node j is assigned tocommunity cj, the Kronecker delta d(ci, cj) ¼ 1 if and only if ci ¼cj, the quantity g is a resolution parameter, and Pij is theexpected weight of the edge that connects node i and nodej under a specied null model.

One can use the maximum value of modularity to quantifythe quality of a partition of a force network into sets of particlesthat are more densely interconnected by strong forces thanexpected under a given null model. The resolution parameterg provides a means of probing the organization of inter-particleforces across a range of spatial resolutions. To provide someintuition, we note that a perfectly hexagonal packing with non-uniform forces should still possess a single community forsmall values of g and should consist of a collection of single-particle (i.e., singleton) communities for large values of g. Atintermediate values of g, we expect maximizing modularity toyield a roughly homogeneous assignment of particles intocommunities of some size (i.e., number of particles) between 1and the total number of particles. (The exact size depends onthe value of g.) The strongly inhomogeneous communityassignments that we observe in the laboratory and numericalpackings (see Section IV) are a direct consequence of thedisorder in the packings.

An important choice in maximizing modularity optimizationis the null model Pij.45,46 The most common null model formodularity optimization is the Newman–Girvan (NG) nullmodel34,35,47,48

PNGij ¼ kikj

2m; (2)

where ki ¼X

j

Wij is the strength (i.e., weighted degree) of node

i and m ¼ 12

X

ij

Wij. The NG null model is most appropriate for

networks in which a connection between any pair of nodes ispossible. Importantly, many networks include (explicit orimplicit) spatial constraints that exert a strong inuence onwhich edges are present.12 For particulate systems, numerousedges are simply physically impossible, so it is important toimprove upon the NG null model for such applications. We usethe term geographical constraints to describe the explicit spatialconstraints in such systems. These constraints exert a signi-cant effect on network structure, so it is important to take theminto account when choosing a null model. For granular mate-rials (and other particulate systems), each particle can only be incontact (i.e., Wij s 0) with its immediate neighbors. We there-fore use a null model, which we call the geographical null model,to account for this constraint. The geographical null model is

Pij ¼ rBij, (3)

where r is themean edge weight in a network and B is the binaryadjacency matrix of the network (such a null model was usedpreviously for applications in neuroscience46). Recall that theadjacency matrix encapsulates the presence or absence ofcontact between each pair of particles. For a granular material,r ¼ !f :¼ hfiji is the mean inter-particle force. Because we havenormalized the edge weights in the force network (Wij¼ fij/mean(f)), we note that in our case r ¼ 1.

Maximizing Q yields a so-called “hard partition” of anetwork into communities in which the total edge weightinside of modules is as large as possible relative to the chosennull model. A hard partition assigns each node to exactly onecommunity. (An alternative is a “so partition”,49 whichallows each node in a network to be associated with multiplecommunities.) For the geographical null model in (3), maxi-mizing Q assigns the particles into communities that haveinter-module particle forces that are larger than the meanforce. Such communities represent the force chains in agranular system.

Because maximizing Q is NP-hard,50 the success of themaximization is subject to the limitations of the employedcomputational methods. In the present paper, we use a Lou-vain-like locally greedy algorithm.51 Additionally, given thenumerous near-degeneracies in the modularity landscape thattends to inict networks that are constructed from empiricaldata52 (i.e., many different partitions oen yield comparablylarge values of Q), we report community-detection results thatare ensemble averages over 20 optimizations.

B. Properties of force chains

We characterize communities using several diagnostics: size,network force, and a gap factor (a novel notion that we introducein the present paper). The size sc of a community c is simply thenumber of particles in that community. The systemic size s isgiven by the mean of sc over all communities. The modularity Qis composed of sums of magnitudes of bond forces and there-fore also has units of force. Therefore, we use the term networkforce to indicate the contribution of a community c to modu-larity. The network force of a community is given by the formula

2734 | Soft Matter, 2015, 11, 2731–2744 This journal is © The Royal Society of Chemistry 2015

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we observe a maximum of guniform at g¼ 0.9 (for g ˛ {0.1, 0.3,.,2.1}) in high-pressure packings (5.9 " 10#3E) and at g ¼ 1.5 forlow-pressure packings (2.7" 10#4E). In the numerical packings,we observe a maximum of guniform at g¼ 1.1 for all pressures. Incomparison to our observations in the main text from employ-ing the size-weighted systemic gap factor g, we nd that theoptimal value of g is larger when we instead employ guniform(compare Fig. 10 to Fig. 5 and 7). We also observe that the curvesof the systemic gap factor versus resolution parameter exhibitlarger variation for the uniformly-weighted gap factor than forthe size-weighted gap factor.

Optimal value of the resolution parameter

The large variation in the maximum of guniform over packingsand pressures makes it difficult to choose an optimal resolu-tion-parameter value. We choose to take gopt¼ 1.1 because (1) itcorresponds to the maximum of guniform in the numericalpackings and (2) it corresponds to the mean of the maximum ofguniform in the laboratory packings. To facilitate the comparisonof optimal values of g from the two weighting schemes, wedenote gopt for g as g and we denote gopt for guniform as guniform.Note that guniform ¼ 1.1 differs from (and is larger than) g ¼ 0.9.

Force-chain structure at the optimal value of the resolutionparameter

The force chains that we identify for the optimal value for theuniformly-weighted gap factor (at guniform ¼ 1.1) differ fromthose that we identied in the main text for the optimal value ofthe size-weighted gap factor (at g ¼ 0.9). We show ourcomparison in Fig. 11. For both laboratory and numericalpackings, the force chains that we identify at g ¼ 0.9 are largerand more branched than the ones that we identify at g ¼ 1.1(which are smaller and more linear). Indeed, the communitiesthat we identify at g ¼ 1.1 have more singletons than thecommunities that we identify at g ¼ 0.9. These results followfrom the difference in the two weighting schemes for calcu-lating a systemic gap factor. The size-weighted systemic gapfactor g weights larger communities more heavily than smallerones, and the larger communities tend to be the more branchedcommunities that we identify at smaller values of the resolutionparameter (e.g., at g ¼ 0.9). In contrast, the uniformly-weightedsystemic gap factor guniform gives equal weight to small and largecommunities, and it therefore uncovers the linear communitiesthat are evident at larger values of the resolution parameter(e.g., at g ¼ 1.1). We can therefore use the size-weighted gapfactor to identify larger, more branched force chains and theuniformly-weighted gap factor to identify smaller, more linearforce chains.

Appendix 2: MethodologicalconsiderationsRobustness of community structure to errors in theestimation of contact forces

In our frictional laboratory experiments, we estimate that errorsin the force measurements could be as large as $30% of the

contact force fij; the high variability arises from the nonlineartting process. (Recall that we take particles to be in contact ifthe force between them is measurable by our photoelasticcalculations. We then determine the particle contact forces bysolving the inverse photoelastic problem using images takenwith polarizers39). Somewhat surprisingly, we nd that theerrors in the force estimates are independent of both the localforce and the global pressure. To ensure that our results arequalitatively robust to these variations, we construct 20 simu-lated force networks for each experimental network (21 pack-ings, seven pressures) by adding Gaussian noise with width fij/3to each contact. For each of these simulated networks, wereevaluate the estimated community structure (from which weinfer the force chains).

To determine whether the estimates are robust to thisamount of noise, we compare the community structure of theactual force networks with those of the simulated forcenetworks using the z-score of the Rand coefficient.62 Forcomparing two partitions a and b, we calculate the Rand z-scorein terms of the network's total number M of pairs of nodes, thenumberMa of pairs that are in the same community in partition

Fig. 11 In both (A) (frictional) laboratory and (B) (frictionless) numericalpackings, we identify larger and more branched force chains at theoptimal resolution determined by (left; g ¼ 0.9) the size-weighted gapfactor g, and we identify smaller and less branched force chains at theoptimal resolution determined by (right; g ¼ 1.1) the uniformly-weighted gap factor guniform. These observations are consistent acrossall pressure values, but they are especially evident at high pressures inthe laboratory packings and are least evident at low pressures in thelaboratory packings. In the numerical packings, we observe littlevariation for different values of pressure. In both panels, we highlightthe network force for the force chains that we identify in examplepackings.

This journal is © The Royal Society of Chemistry 2015 Soft Matter, 2015, 11, 2731–2744 | 2741

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– For larger pressures, we obtain larger and more branched force chains in both the (frictional) laboratory packings described earlier and in (frictionless) numerical packings

– One can also incorporate mobility models into the construction of null models Pij

– P. Expert, T. Evans, V. Blondel, R. Lambiotte [2011], “Uncovering Space-Independent Communities in Spatial Networks”, PNAS 108:7663–7668– Introduced null model based on gravity model

– Found French vs. Flemish communities in mobile phone network in Belgium

– M. Sarzynska, E. A. Leicht, G. Chowell, & MAP [2016], “Null Models for Community Detection in Spatially-Embedded, Temporal Networks”, J. Cplx. Networks, advance access (doi:10.1093/comnet/cnv027)– Comparison of results using Newman–Girvan, gravity, and (newly introduced in null-model

form) radiation null models

– The situation is much more complicated than in the example studied by Expert et al.

– Introduction of new generative benchmarks (e.g., one based on distance, one based on flux) and also empirical example from weekly cases of dengue fever in Peru over 15 years

– Example: importance based on node strength (weighted degree):

– You can construct a radiation null model in a similar way, and both the gravity and radiation null models can be generalized to temporal networks using a multilayer representation (see later).

! 38!

LN 2000 Spa 2000

Fig. 10. Circular plots of migration community structures in 2000. The size of the ribbons corresponds to the amount of migration stock that remains in a community or is directed to other communities. The color of the ribbons indicates the source communities. We create the plots using Circos Table Viewer (Krzywinski et al., 2009), which is available at http://mkweb.bcgsc.ca/tableviewer/visualize/.

7. Continuity and Change in Migration Communities Migration communities are involved in complex processes of emerging, splitting,

merging, and dissolving. In Fig. 11, we map continuity and change in migration

communities using alluvial diagrams (Rosvall and Bergstrom, 2010). Instead of

processes of integration, we observe a split in bridging communities since 1960,

with a noticeable effect in the last decade. However, there are also instances of

merging of newly industrialized areas, e.g., the CHN community in 1960 joins the

largest bridging community in 2000 (spatial null model). Such processes of

consolidation are likely to result from direct foreign investments and

manufacturing export, which induce relationships between distant regions (e.g.,

North America and South Asia) in a network of socio-economic relationships

(Sassen, 2007, Castells, 1996). Cave communities (e.g., RUS, CIV) are relatively

isolated from the temporal dynamics in the WMN.

– M. Kivelä et al., “Multilayer Networks”, JCN, 2014

– Use multilayer representations of temporal (e.g., with ordinary coupling) and multiplex networks (with categorical coupling).

• P. J. Mucha, T. Richardson, K. Macon, MAP, & J.-P. Onnela [2010], “Community Structure in Time-Dependent, Multiscale, and Multiplex Networks”, Science 328(5980):876–878 (2010)• Code available at

http://netwiki.amath.unc.edu/GenLouvain/GenLouvain

• Schematic from M. Bazzi, MAP, S. Williams, M. McDonald, D. J. Fenn, & S. D. Howison [2016] Multiscale Modeling and Simulation: A SIAM Interdisciplinary Journal, 14(1):1–41

13

Layer 1

11 21

31

Layer 2

12 22

32

Layer 3

13 23

33

!

2

6666666666664

0 1 1 ! 0 0 0 0 01 0 0 0 ! 0 0 0 01 0 0 0 0 ! 0 0 0! 0 0 0 1 1 ! 0 00 ! 0 1 0 1 0 ! 00 0 ! 1 1 0 0 0 !0 0 0 ! 0 0 0 1 00 0 0 0 ! 0 1 0 10 0 0 0 0 ! 0 1 0

3

7777777777775

Fig. 3.1. Example of (left) a multilayer network with unweighted intra-layer connections (solidlines) and uniformly weighted inter-layer connections (dashed curves) and (right) its correspondingadjacency matrix. (The adjacency matrix that corresponds to a multilayer network is sometimescalled a “supra-adjacency matrix” in the network-science literature [39].)

or an adjacency matrix to represent a multilayer network.) The generalization in [49]consists of applying the function in (2.16) to the N |T |-node multilayer network:

r(C, t) =

N |T |X

i,j=1

✓⇡i

⇥�ij + t⇤ii(Mij � �ij)

⇤� ⇡i⇢i|j

◆�(ci, cj) , (3.1)

where C is now a multilayer partition (i.e., a partition of an N |T |-node multilayernetwork), ⇤ is the N |T | ⇥N |T | diagonal matrix with the rates of the exponentiallydistributed waiting times at each node of each layer on its diagonal, M (with en-tries Mij := Aij/

Pj Aij) is the N |T | ⇥ N |T | transition matrix for the N |T |-node

multilayer network with adjacency matrix A, ⇡i is the corresponding stationary dis-tribution (with the strength of a node and the total edge weight now computed fromthe multilayer adjacency matrix A), and ⇢i|j is the probability of jumping from nodei to node j at stationarity in one step conditional on the structure of the networkwithin and between layers. The authors’ choice of ⇢i|j , which accounts for the “spar-sity pattern”10 of inter-layer edges in the multilayer adjacency matrix, motivates themultilayer modularity-maximization problem

maxC2C

N |T |X

i,j=1

Bij�(ci, cj) , (3.2)

which we can also write as maxC2C Q(C|B), where B is the multilayer modularitymatrix

B =

2

66666664

B1

!I 0 . . . 0

!I. . .

. . .. . .

...

0. . .

. . .. . . 0

.... . .

. . .. . . !I

0 . . . 0 !I B|T |

3

77777775

, (3.3)

10The sparsity pattern of a matrix X is a matrix Y with entries Yij = 1 when Xij 6= 0 andYij = 0 when Xij = 0.

• Find communities algorithmically by optimizing “multislicemodularity”– We derived this function in Mucha et al, 2010

• Laplacian dynamics: find communities based on how long random walkers are trapped there. Exponentiate and then linearize to derive modularity.

• Generalizes derivation of ordinary modularity from R. Lambiotte, J.-C. Delvenne, &. M Barahona, arXiv:0812.1770 (now published, with updates, in TNSE, 2015)• Different spreading weights on different types of edges

– Recall: Node x in layer r is a different node-layer from node x in layer s

Remark: One can generalize the null model to incorporate space (as discussed previously). See, e.g., M. Sarzynska et al. [2016].

• A. S. Waugh, L. Pei, J. H. Fowler, P. J. Mucha, & MAP, “Party Polarization in CongressL A Network Science Approach”, arXiv:0907.3509 (processed data available via figshare; original data from Voteview)

• One network layer for each two-year Congress• Intralayer edges given by number of bills in which two legislators voted the same way divided by the total

number of bills on which they both voted• Interlayer edges of weight ω = constant between a legislator and him/herself in consecutive Congresses if

a member for both (all other interlayer edges are 0)• Each node-layer (i,s) assigned to a community by maximizing multislice modularity

with a single parameter controlling the interslicecorrespondence of communities.

Important to our method is the equivalencebetween themodularity quality function (12) [witha resolution parameter (5)] and stability of com-munities under Laplacian dynamics (13), whichwe have generalized to recover the null models forbipartite, directed, and signed networks (14). First,we obtained the resolution-parameter generaliza-

tion of Barber’s null model for bipartite networks(15) by requiring the independent joint probabilitycontribution to stability in (13) to be conditionalon the type of connection necessary to stepbetween two nodes. Second, we recovered thestandard null model for directed networks (16, 17)(again with a resolution parameter) by generaliz-ing the Laplacian dynamics to include motionalong different kinds of connections—in this case,

both with and against the direction of a link. Bythis generalization, we similarly recovered a nullmodel for signed networks (18). Third, weinterpreted the stability under Laplacian dynamicsflexibly to permit different spreading weights onthe different types of links, giving multiple reso-lution parameters to recover a general null modelfor signed networks (19).

We applied these generalizations to derive nullmodels for multislice networks that extend theexisting quality-function methodology, includingan additional parameter w to control the couplingbetween slices. Representing each network slice sby adjacencies Aijs between nodes i and j, withinterslice couplingsCjrs that connect node j in slicer to itself in slice s (Fig. 1), we have restricted ourattention to unipartite, undirected network slices(Aijs = Ajis) and couplings (Cjrs = Cjsr), but we canincorporate additional structure in the slices andcouplings in the same manner as demonstrated forsingle-slice null models. Notating the strengths ofeach node individually in each slice by kjs =∑iAijsand across slices by cjs = ∑rCjsr, we define themultislice strength by kjs = kjs + cjs. The continuous-time Laplacian dynamics given by

pis ¼ ∑jr

ðAijsdsr þ dijCjsrÞpjrkjr

− pis ð1Þ

respects the intraslice nature of Aijs and theinterslice couplings of Cjsr. Using the steady-stateprobability distribution p∗jr ¼ kjr=2m, where 2m =∑ jrkjr, we obtained the multislice null model interms of the probability ris| jr of sampling node i inslice s conditional on whether the multislice struc-ture allowsone to step from ( j, r) to (i, s), accountingfor intra- and interslice steps separately as

risj jrp∗jr ¼

kis2ms

kjrkjr

dsr þCjsr

cjr

cjrkjr

dij

! "kjr2m

ð2Þ

where ms = ∑jkjs. The second term in parentheses,which describes the conditional probability ofmotion between two slices, leverages the definitionof the Cjsr coupling. That is, the conditionalprobability of stepping from ( j, r) to (i, s) alongan interslice coupling is nonzero if and only if i = j,and it is proportional to the probability Cjsr/kjr ofselecting the precise interslice link that connects toslice s. Subtracting this conditional joint probabilityfrom the linear (in time) approximation of theexponential describing the Laplacian dynamics,weobtained a multislice generalization of modularity(14):

Qmultislice ¼12m

∑ijsr

h#Aijs − gs

kiskjs2ms

dsr$þ

dijCjsr

idðgis,gjrÞ ð3Þ

where we have used reweighting of the conditionalprobabilities, which allows a different resolution gsin each slice. We have absorbed the resolution pa-rameter for the interslice couplings into the mag-nitude of the elements ofCjsr, which, for simplicity,we presume to take binary values {0,w} indicatingthe absence (0) or presence (w) of interslice links.

1800 1820 1840 1860 1880 1900 1920 1940 1960 1980 2000

40PA, 24F, 8AA

151DR, 30AA, 14PA, 5F141F, 43DR

44D, 2R

1784R, 276D, 149DR, 162J, 53W, 84other

176W, 97AJ, 61DR, 49A,24D, 19F, 13J, 37other

3168D, 252R, 73other

222D, 6W, 11other

1490R, 247D, 19other

Year

Sen

ator

10 20 30 40 50 60 70 80 90 100 110CTMEMANHRI VTDE NJNY PAIL INMI OHWI IAKSMNMONENDSDVA ALAR FLGA LAMSNCSC TXKYMDOK TNWVAZCO IDMTNVNMUTWYCAORWAAK HI

Congress #

A

B

Fig. 3. Multislice community detection of U.S. Senate roll call vote similarities (23) withw = 0.5 couplingof 110 slices (i.e., the number of 2-year Congresses from 1789 to 2008) across time. (A) Colors indicateassignments to nine communities of the 1884 unique senators (sorted vertically and connected acrossCongresses by dashed lines) in each Congress in which they appear. The dark blue and red communitiescorrespond closely to the modern Democratic and Republican parties, respectively. Horizontal barsindicate the historical period of each community, with accompanying text enumerating nominal partyaffiliations of the single-slice nodes (each representing a senator in a Congress): PA, pro-administration;AA, anti-administration; F, Federalist; DR, Democratic-Republican; W, Whig; AJ, anti-Jackson; A, Adams; J,Jackson; D, Democratic; R, Republican. Vertical gray bars indicate Congresses in which three communitiesappeared simultaneously. (B) The same assignments according to state affiliations.

www.sciencemag.org SCIENCE VOL 328 14 MAY 2010 877

REPORTS

on

July

6, 2

012

ww

w.s

cien

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ag.o

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P. J. Mucha & M. A. Porter [2010], Chaos 20(4):041108

Much more on neuronal networks on Friday (Kaiser, Buldú)

Experiments on learning of a simple motor task

– fMRI data: network from correlated time series

– Examine role of modularity in human learning by identifying dynamic changes in modular organization over multiple time scales

– Main result: “flexibility”, as measured by allegiance of nodes to communities over temporal layers, in one session predicts amount of learning in subsequent session

• D. S. Bassett, N. F. Wymbs, M. P. Rombach, MAP, P. J. Mucha, & S. T. Grafton [2013], PLoS Comput. Bio.9(9):1003171

• Flexible nodes are consistently in a “periphery” as computed for static networks encompassing given time windows

• Nodes that are not flexible (call them “stiff”) are consistently in a structural core in these static networks

• Uses our methodology for computing core–periphery structure.– M. P. Rombach, MAP, J. H. Fowler, & P. J.

Mucha [2014], SIAM J. App. Math.74(1):167–190.

Temporal core–periphery organization ≈ Geometrical core–periphery organization!(the latter is a density-based core using network structure in individual layers)

– L. G. S. Jeub, M. W. Mahoney, P. J. Mucha, & MAP [2015], arXiv:1510.05185

– Extension to multilayer networks: allow spreading dynamics along both intralayer and interlayer edges

ZU064-05-FPR local˙multiplex15 1 September 2015 20:47

A local perspective on community structure in multilayer networks 3

et al., 2015; Kuncheva & Montana, 2015). For our purposes, we define

p

ia(t +1) = Âj,b

P

jbia p

jb (t) , (1)

where p

jb (t) is the probability for a random walker to be at node j in layer b at time t andP

jbia is the probability for a random walker to transition from node j in layer b to node i in

layer a in a time step. We also want the random walk to be ergodic, so that it has a well-defined stationary distribution p

ia(•). We then use the stationary distribution to define theconductance (Jerrum & Sinclair, 1988) of a set of state nodes

1S as

f(S) =Â

(i,a)2S

Â( j,b )/2S

P

iajb p

ia(•)

Â(i,a)2S

p

ia(•), (2)

which we use as a quality measure for local communities. Once we select an appropri-ate random walk (or other Markov process2), we can define the associated personalizedPageRank (PPR) score of state node (i,a) as the solution to the equation

PPR(g)ia = g Â

j,bP

jbia PPR(g)

jb +(1� g)sia , (3)

where s is a probability distribution that determines the seed nodes for the method (Gleich,2014). We then approximate the solution to equation (3) locally (Andersen et al., 2006) tofind local communities.

Given a random walk on a multilayer network, one can analyze communities in multi-layer networks using the same methods as for single-layer networks. See Jeub et al. (2015)for a detailed discussion of a few different methods and their application to several classesof networks (which exhibit rather different types of behavior with respect to the dynam-ics of diffusion processes). Our code for identifying local communities and visualizingnetworks is available from https://github.com/LJeub.

In the present article, we illustrate some features that one can encounter as a consequenceof the particular structure of multiplex networks. As examples, we use two different randomwalks (see Fig. 1) on a transportation network (see Fig. 2) and a social network (see Fig. 3).We highlight some key properties of these two networks in Table 1. An interesting aspectof multilayer networks is that one could use either physical nodes or state nodes as seedsto sample local communities. We compare the results of these two sampling proceduresin Figs. 2 and 3. For directed networks (e.g., the network in Fig. 3), random walks arenot necessarily ergodic and hence might not have a unique stationary distribution. We useunrecorded edge teleportation (Lambiotte & Rosvall, 2012; De Domenico et al., 2015) to

1 Following (De Domenico et al., 2015), we use the term state node to refer to a node-layer tupleand the term physical node to refer to the collection of all state nodes that represent the same node.

2 More generally, it would also be both fruitful and interesting to develop local community-detection methods using dynamical processes that are not Markovian. A good start would be touse our approach through suitable adaptations of other processes that have been used to examinecommunity structure in networks. Examples include Kuramoto phase oscillators (Arenas et al.,2006); epidemic spreading processes (Ghosh et al., 2014); and higher-order Markovian processes,such as those that have been employed in the study of “memory networks” (Rosvall et al., 2014).




p

ia(t +1) = Âj,b

P

jbia p

jb (t) , (1)

where p





1S as

f(S) =Â

(i,a)2S

Â( j,b )/2S

P

iajb p

ia(•)

Â(i,a)2S

p

ia(•), (2)


PPR(g)ia = g Â

j,bP

jbia PPR(g)

jb +(1� g)sia , (3)









p

ia(t +1) = Âj,b

P

jbia p

jb (t) , (1)

where p





1S as

f(S) =Â

(i,a)2S

Â( j,b )/2S

P

iajb p

ia(•)

Â(i,a)2S

p

ia(•), (2)


PPR(g)ia = g Â

j,bP

jbia PPR(g)

jb +(1� g)sia , (3)








100 101 102 10310�3

10�2

10�1

100

number of state nodes

cond

ucta

nce

w = 0.1 w = 1 w = 10

(a) Classical random walk

100 101 102 10310�3

10�2

10�1

100

number of state nodes

cond

ucta

nce

r = 0.01 r = 0.1 r = 1

(b) Relaxed random walk

Wizz Air

Ryanair

(c) Best community with 173 state nodes forw = 0.1 (physical node as seed)

SunExpressPanagra Airways

Turkish Airlines

(d) Best community with 169 state nodes forr = 0.1 (state node as seed)

Fig. 2: European Airline Network: Multiplex transportation network with 37 layers. Eachlayer includes the flights for a single airline (Cardillo et al., 2013). Panels (a) and (b)show (mostly downward-sloping) network community profiles (NCPs) for this network,where we plot the quality (as measured by conductance) of the best community of eachsize (as measured by the number of state nodes that are a member of the community).Observe that sampling using physical nodes (the thin curves) and sampling using statenodes leads to similar results. Panels (c) and (d) illustrate some of the communities thatwe obtain. We shade the state nodes in a community from dark red to light grey basedon their rank within the community. The large arrows point to the seed nodes. For smalllayer-jumping probability r in the relaxed random walk and small interlayer edge weight win the classical random walk, the best communities tend to consist of sets of similar typesof airlines (e.g., they fly to the same airport, are low-cost airlines, or share some otherfeature). The prominent dips in the NCPs in panel (b) for communities that consist of twostate nodes are the result of a spurious connection in the data set that creates a bottleneckfor the relaxed random walk. Even for r = 1, the relaxed walk still predominantly identifiesthis type of community. By contrast, for large values of w , the classical random walk resultsin relatively geographically localized communities.

Mesoscale structures can give fascinating insights about networks, but be careful about

how you apply these tools and ideas.

– Numerous different types: communities, core–periphery structures, roles and positions, block models (arbitrary block structures), etc.

– Not just communities!– Community structure (to examine assortative structures) is the most popular

and best-studied type of mesoscale structure, but it’s far from the only one, and there is no reason to think it is the most important one.

– Our focal question: If we examine a given type of structure (or given types of structures), what can we learn about a network?

– A different question: How does one infer the statistically most likely block structure? If we want to study “large-scale” structure in networks broadly, what should we be looking for?– Statistical inference and model selection (see presentation by Tiago Peixoto)

– Multilayer modularity maximization, community-detection method of Jeub et al., code for visualization and analysis of multilayer networks, and other methods available at http://www.plexmath.eu/?page_id=327

Coming Next

Tiago Peixoto’s presentation: statistically principled approaches

Mesoscale Structures in Networks

Science

Transcript of Mesoscale Structures in Networks