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COMPLEX NETWORKS: THEORY, METHODS, AND APPLICATIONS (2ND EDITION)

May 16-20, 2016

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- 1. Mesoscale Structures in Networks Lake Como School on Complex Networks, 2016 Expander?
- 2. – I do not expect to have time to cover all of my slides!
- 3. – 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.
- 4. Studying mesoscale network structures goes very far beyond studying only community structure!
- 5. – 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
- 6. Puck Rombach CENSORED!
- 7. – 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)
- 8. – 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
- 9. – Other structures besides assortative structures: different types of block sructures – Bipartite, core–peripherystructure, 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 networkstructure 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
- 10. 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.
- 11. – 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.
- 12. – 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
- 13. Traud et al., SIAM Review, 2011
- 14. – “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?
- 15. – 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!
- 16. – Network Scientists with Karate Trophies: http://networkkarate.tumblr.com
- 17. • Popular approach: Use a “modularity” quality function • GOAL: Assign nodes to communities to maximize Q. (Use some computational heuristic.)
- 18. • 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)
- 19. • 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
- 20. – 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.
- 21. – 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”
- 22. – Full networks (single university, largest connected component)
- 23. 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
- 24. 1. Z-scores for Rand, AdjustedRand,Fowlkes- Mallows, & gamma indices are provably identical 2. Analytical formulas exist for the above indices (need permutation tests for Jaccard andMinkowski)
- 25. Legends gives disk size as a function of maximum distance d between the 6 different partitions Full – We visualize social organization using barycentric coordinates. – Center aroundYear 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), andUC 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
- 26. Female Networks Male Networks
- 27. – 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
- 28. – 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/LocalCommunit ies THINK LOCALLY, ACT LOCALLY: DETECTION OF . . . PHYSICAL REVIEW E 91, 012821 (2015) 100 101 102 103 104 10−3 10−2 10−1 100 size conductance CA-GrQc FB-Johns55 US-Senate (a) NCP 100 101 102 103 104 10−3 10−2 10−1 100 101 102 size conductanceratio CA-GrQc FB-Johns55 US-Senate (b) CRP (c) CA-GrQc (d) FB-Johns55 0 0.5 1 (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, and US-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 Senates that have at least one senator in common with those in the communities for US-SENATE). We ﬁnd good small communities but no good large communities in CA-GRQC, some weak large-scale structure in FB-JOHNS55 that does not create substantial bottlenecks for the random-walk dynamics, and signatures of low-dimensional structure (i.e., good large communities but no good small communities) for US-SENATE. The low-dimensional structure in US-SENATE results from the multilayer structure that encapsulates the network’s temporal properties. [The dashed line in panel (b) indicates a conductance ratio of 1.] reason for the downward-sloping shape is that US-SENATE and structure using the MOVCUT (see Appendix C) and EGONET
- 29. – 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 details on 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 one end 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 of edges that are attached to nodes in the set. The volume vol(S,S) between a set S and its complement S has a natural interpretation as the “surface area” of the “boundary” between S and S. In this study, a set S is a hypothesized community. Informally, the conductance of a set S of nodes is the 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 the isoperimetric structure of G. Somewhat more formally, the conductance of a set of nodes S ⊂ V is vol(S,S) To gain insight into how to understand an NCP and what it reveals about network structure, consider Fig. 2. In Fig. 2(a), we illustrate three possible ways that an NCP can behave. In each case, we use conductance as a measure of community quality. The three cases are the following ones. (1) Upward-sloping NCP. In this case, small communities are “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 to be 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 of a network. An NCP for very large communities, which we do not show in ﬁgures as a result of this choice, roughly mirrors that for small communities, as the complement of a good small community is a good large community because of the inherent symmetry in conductance [see Eq. (3)]. In Fig. 2(b), we show an NCP of a LIVEJOURNAL network from Ref. [25]. It demonstrates an empirical fact about a large variety of large social and information networks: There exist good small conductance-based communities, but there do not exist any good large conductance-based communities JEUB, BALACHANDRAN, PORTER, MUCHA, AND MAHONEY PHYSICAL REVIEW E 91, 012821 (2015) we then discuss our extensions of such ideas. For more details on 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 one end 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 of edges that are attached to nodes in the set. The volume vol(S,S) between a set S and its complement S has a natural interpretation as the “surface area” of the “boundary” between S and S. In this study, a set S is a hypothesized community. Informally, the conductance of a set S of nodes is the 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 the isoperimetric structure of G. Somewhat more formally, the conductance of a set of nodes S ⊂ V is φ(S) = vol(S,S) min (vol(S),vol(S)) . (3) Thus, smaller values of conductance correspond to better communities. The conductance of a graph G is the minimum conductance of any subset of nodes: φ(G) = min S⊂V φ(S). (4) Computing the conductance φ(G) of an arbitrary graph is an intractable problem (in the sense that the associated decision problem is NP-hard [69]), but this quantity can be approximated by the second-smallest eigenvalue λ2 of the To gain insight into how to understand an NCP and what it reveals about network structure, consider Fig. 2. In Fig. 2(a), we illustrate three possible ways that an NCP can behave. In each case, we use conductance as a measure of community quality. The three cases are the following ones. (1) Upward-sloping NCP. In this case, small communities are “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 to be 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 of a network. An NCP for very large communities, which we do not show in ﬁgures as a result of this choice, roughly mirrors that for small communities, as the complement of a good small community is a good large community because of the inherent symmetry in conductance [see Eq. (3)]. In Fig. 2(b), we show an NCP of a LIVEJOURNAL network from Ref. [25]. It demonstrates an empirical fact about a large variety of large social and information networks: There exist good small conductance-based communities, but there do not exist any good large conductance-based communities in many such networks. (See Refs. [24–26,37,67,68] for more empirical evidence that large social and information networks tend not to have large communities with low conductances.) On the contrary, Fig. 2(c) illustrates a small toy network—a so-called “caveman network”—formed from several small cliques connected by rewiring one edge from each clique to create a ring [70]. As illustrated by the downward-sloping NCP in Fig. 2(d), this network possesses good conductance-based communities, and large communities are better than small ones. One obtains a similar downward-sloping NCP for the Zachary Karate Club network [59] as well as for many other Network Community Profile (NCP)
- 30. – Upper right plot from two slides ago: ratio of conductance to internal conductance – Smaller ratio è better communities 00 101 102 103 size ipartite structure he idealized example Karate Club network. om a block model with -R´enyi graph. (d) NCP rtite block model. e only connected via s than the periphery. phery structure tends Figs. 1(b) and 3(b). ply to all networks riphery structure. If d (though still much ger observes good, ike expanders from om walkers, so they les of such networks model that we used (b) [61]. mogeneous expander s tend to have poor Appendix A for a not have any charac- NCP of a bipartite in the network. For o types of nodes are P [see Fig. 3(d)] has r. tent of NCPs tness properties of priori, as an NCP is ever, the qualitative upward-sloping, or of nodes and edges, ocessing decisions, –26]. For example, y small communities behave in a roughly similar manner to conductance-based NCPs, whereas measures that capture only one of the two criteria exhibit qualitatively different behavior (typically for rather trivial reasons) [26]. Although the basic NCP that we have been discussing yields 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 number or density of communities with different community quality scores. (This contributes to the robustness properties of NCPs with respect to perturbations of a network.) Accordingly, the communities that are revealed by an NCP need not be representative of the majority of communities in a network. However, the extremal features that are revealed by an NCP have important system-level implications for the behavior of dynamical processes on a network: They are responsible for the most severe bottlenecks for associated dynamical processes on networks [72]. Another property that is not revealed by an NCP is the internal structure of communities. Recall from Eq. (3) that the conductance of a community measures how well (relative to 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 several disjoint pieces. Recent work has addressed how spectral-based approximations to optimizing conductance also approximately optimize measures of internal connectivity [73]. We augment the information from basic NCPs with some additional computations. To obtain an indication of a community’s internal structure, we compute the internal conductance of the communities that form an NCP. The internal 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 in the community S. The internal conductance is equal to the conductance of the best partition into two communities of the network G|S viewed as a graph in isolation. Because a good community should be well separated from the remainder of a network and also relatively well connected internally, we expect good communities to have low conductance but high internal conductance. We thus compute the conductance ratio (S) = φ(S) φin(S) (7) 012821-7 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 of edges that are attached to nodes in the set. The volume vol(S,S) between a set S and its complement S has a natural interpretation as the “surface area” of the “boundary” between S and S. In this study, a set S is a hypothesized community. Informally, the conductance of a set S of nodes is the 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 the isoperimetric structure of G. Somewhat more formally, the conductance of a set of nodes S ⊂ V is φ(S) = vol(S,S) min (vol(S),vol(S)) . (3) Thus, smaller values of conductance correspond to better communities. The conductance of a graph G is the minimum conductance of any subset of nodes: φ(G) = min S⊂V φ(S). (4) Computing the conductance φ(G) of an arbitrary graph is an intractable problem (in the sense that the associated decision problem is NP-hard [69]), but this quantity can be approximated by the second-smallest eigenvalue λ2 of the normalized Laplacian [67,68]. If the “surface area to volume” (i.e., isoperimetric) inter- pretation captures the notion of a good community as a set of nodes that is connected more densely internally than with the remainder of a network, then computing the solution to Eq. (4) leads to the “best” (in this sense) community of any size in the network. are “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 to be 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 of a network. An NCP for very large communities, which we do not show in ﬁgures as a result of this choice, roughly mirrors that for small communities, as the complement of a good small community is a good large community because of the inherent symmetry in conductance [see Eq. (3)]. In Fig. 2(b), we show an NCP of a LIVEJOURNAL network from Ref. [25]. It demonstrates an empirical fact about a large variety of large social and information networks: There exist good small conductance-based communities, but there do not exist any good large conductance-based communities in many such networks. (See Refs. [24–26,37,67,68] for more empirical evidence that large social and information networks tend not to have large communities with low conductances.) On the contrary, Fig. 2(c) illustrates a small toy network—a so-called “caveman network”—formed from several small cliques connected by rewiring one edge from each clique to create a ring [70]. As illustrated by the downward-sloping NCP in Fig. 2(d), this network possesses good conductance-based communities, and large communities are better than small ones. One obtains a similar downward-sloping NCP for the Zachary Karate Club network [59] as well as for many other networks for which there exist meaningful visualizations [25]. The wide use of networks that have interpretable visualizations (such as the Zachary Karate Club and planted-partition models [71] with balanced communities) to help develop and evaluate methods for community detection and other procedures can lead to a strong selection bias when evaluating the quality of those methods. the most severe bottlenecks for associated dynamical processes on networks [72]. Another property that is not revealed by an NCP is the internal structure of communities. Recall from Eq. (3) that the conductance of a community measures how well (relative to 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 several disjoint pieces. Recent work has addressed how spectral-based approximations to optimizing conductance also approximately optimize measures of internal connectivity [73]. We augment the information from basic NCPs with some additional computations. To obtain an indication of a community’s internal structure, we compute the internal conductance of the communities that form an NCP. The internal 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 in the community S. The internal conductance is equal to the conductance of the best partition into two communities of the network G|S viewed as a graph in isolation. Because a good community should be well separated from the remainder of a network and also relatively well connected internally, we expect good communities to have low conductance but high internal conductance. We thus compute the conductance ratio (S) = φ(S) φin(S) (7) 21-7 Conductance Ratio Profile (CRP)
- 31. K LOCALLY, ACT LOCALLY: DETECTION OF . . . PHYSICAL REVIEW E 91, 012821 (2015) low-dimensional space. Spectral clustering or other ing methods often ﬁnd meaningful communities in such rks, and one can often readily construct meaningful and etable visualizations of network structure. Core-periphery structure. In Fig. 1(b), we illustrate se in which α11 ≫ α12 ≫ α22. This is an example network with a density-based “core-periphery” struc- 24,25,62–64]. There is a core set of nodes that are ely well connected both among themselves and to a set pheral nodes that interact very little among themselves. Expander or complete graph. In Fig. 1(c), we illustrate se in which α11 ≈ α12 ≈ α22. This corresponds to a rk with little or no discernible structure. For example, = α12 = α22 = 1, then the graph is a clique (i.e., the ete graph). Alternatively, if the graph is a constant- expander, then α11 ≈ α12 ≈ α22 ≪ 1. As discussed pendix A, constant-degree expanders yield the metric that embed least well in low-dimensional Euclidean . In terms of the idealized block model in Fig. 1, they ke complete graphs, and partitioning them would not network structure that one should expect to construe as ngful. Informally, they are largely unstructured when d at large size scales. Bipartite structure. In Fig. 1(d), we illustrate the case ch α12 ≫ α11 ≈ α22. This corresponds to a bipartite or bipartite graph. Such networks arise, e.g., when there (a) Three possible NCPs (b) Realistic NCP from [25] (c) A caveman network 100 101 102 103 10−4 10−3 10−2 10−1 100 size conductance (d) NCP of caveman network FIG. 2. (Color online) Illustration of network community pro- ﬁles (NCPs) of conductance versus community size. (a) Stylized versions 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 illustrates the characteristic upward-sloping NCP that is typical for many large empirical social and information networks [25]. (c) A toy “caveman
- 32. – 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. there exists an ϵ > 0 such that h(Gt ) ϵ mally, a given graph G is an expander if its rge. Ref. [114], one can view expanders from tary viewpoints. From a combinatorial ers are graphs that are highly connected e has to sever many edges to disconnect a nder graph. From a geometric perspective, ifﬁculty implies that every set of nodes has ry relative to its size. From a probabilistic ders are graphs for which the natural ss converges to its limiting distribution as Finally, from an algebraic perspective, hs in which the ﬁrst nontrivial eigenvalue erator is bounded away from 0. (Because d-regular graphs, note that this statement mbinatorial Laplacian and the normalized tion, constant-degree (i.e., d-regular, for d) expanders are the metric spaces that d strong sense [114]) embed least well in aces (such as those discussed informally in se interpretations imply that smaller values pond more closely to the intuitive notion ies (whereas larger values of expansion nition, to better expanders). ies between Eq. (A2) and Eq. (A3), which and Eq. (3) and Eq. (4), which deﬁne equations make it clear that the difference and conductance simply amounts to a he size (or volume) of sets of nodes and the y (or surface area) between a set of nodes t. This difference is inconsequential for owever, because of the deep connections nd rapidly mixing random walks, the latter densely connected nodes) [9,32,53,115–118] as well as for ﬁnding sets of nodes that are related to each other in other ways [48,54,115,119,120]. In this paper, we build on the idea that random walks and related diffusion-based dynamics, as well as other types of local dynamics (e.g., ones, like geodesic hops, that depend on ideas based on egocentric networks), should get “trapped” in good communities. We examine three dynamical methods for community identiﬁcation. 1. Dynamics type 1: Local diffusions (the “ACLCUT” method) In this procedure, we consider a random walk that starts at a given seed node s and runs for some small number of steps. We take advantage of the idea that if a random walk starts inside a good community and takes only a small number of steps, then it 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 solution ⃗pr(α,⃗s) of the equation ⃗pr(α,⃗s) = αD−1 A ⃗pr(α,⃗s) + (1 − α)⃗s, (B1) where 1 − α is a “teleportation” probability and ⃗s is a seed vector. From the perspective of random walks, evolution occurs either by the walker moving to a neighbor of the current node or by the walker “teleporting” to a random node (e.g., determined uniformly at random as in the usual PageRank procedure, or to a random node that is biased towards ⃗s in the PPR procedure). The PPR vector ⃗pr(α,⃗s) represents the stationary distribution of this random walk. In general, teleportation results in a bias to the random walk, and one usually tries to minimize such a bias when detecting communities. (See Ref. [123] for clever ways to choose ⃗s with this goal in mind.)
- 33. – 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/LocalCommunit ies THINK LOCALLY, ACT LOCALLY: DETECTION OF . . . PHYSICAL REVIEW E 91, 012821 (2015) 100 101 102 103 104 10−3 10−2 10−1 100 size conductance CA-GrQc FB-Johns55 US-Senate (a) NCP 100 101 102 103 104 10−3 10−2 10−1 100 101 102 size conductanceratio CA-GrQc FB-Johns55 US-Senate (b) CRP (c) CA-GrQc (d) FB-Johns55 0 0.5 1 (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, and US-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 Senates that have at least one senator in common with those in the communities for US-SENATE). We ﬁnd good small communities but no good large communities in CA-GRQC, some weak large-scale structure in FB-JOHNS55 that does not create substantial bottlenecks for the random-walk dynamics, and signatures of low-dimensional structure (i.e., good large communities but no good small communities) for US-SENATE. The low-dimensional structure in US-SENATE results from the multilayer structure that encapsulates the network’s temporal properties. [The dashed line in panel (b) indicates a conductance ratio of 1.] reason for the downward-sloping shape is that US-SENATE and structure using the MOVCUT (see Appendix C) and EGONET
- 34. – 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. 🤔
- 35. – Studying community structure can be very insightful—I spend time doing this, after all!—but one has to use such tools carefully. 😉
- 36. Another important type of mesoscale structure, which is becoming increasingly popular to study.
- 37. – P. Csermely, A. London, L.-Y. Wu, & B. Uzzi [2013], “Structure and Dynamics of Core–Periphery Networks”, Journal of Complex Networks 1:93–123. – 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.
- 38. Core–Periphery StructureCommunity Structure
- 39. 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 peripheralbanks, transportation networks, etc. ì Borgatti–Everett (1999): ì Discrete notions: simpler one is to compare networkto an ideal block model consisting of a fully connected core and a periphery with no internal connections but fully connected to the core ì 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 graph ì Da Silva et al (2008): Defined a core coefficient using closeness centrality and a measure of shortest paths ì Leskovec et al. (2009): Onions and whiskers ì Leskovec and collaborators (2013): Core regions from overlap of communities
- 40. – 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)
- 41. New York & Erie Railroad, diagram from about 160 years ago The London Underground (“The Tube”)
- 42. – 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.
- 43. 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
- 44. 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 shuffled ì 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
- 45. Maximize where Cij = 1 if i or j is in the core and Cij = 0 otherwise. Find the best fit to a core–peripheryblockmodel. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0
- 46. Find the best fit to a core–peripheryblockmodel. 1 1 1 1 a a a a 1 1 1 1 a a a a 1 1 1 1 a a a a 1 1 1 1 a a a a a a a a 0 0 0 0 a a a a 0 0 0 0 a a a a 0 0 0 0 a 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.
- 47. 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.
- 48. v Interpolates between continuous and discrete notions of core-periphery structure v 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:
- 49. ì We obtain a core score for each node by averaging results over different values of α and β: ì Z = normalizationconstant; ensures that max[CS(i)] = 1 ì It would be interesting to develop more sophisticated procedures for sampling values of α and β.
- 50. α = 0.1 β = 0.5 α = 0.9 β = 0.5 α = 0.5 β = 0.9 α = 0.5 β = 0.1
- 51. Picture courtesy of Aaron Clauset
- 52. Tube data available from my website.
- 53. – 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.
- 54. 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)
- 55. – 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).
- 56. – 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. E 89:032810
- 57. 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
- 58. Data available at https://sites.google.com/site/lshlj82/
- 59. S. H. Lee & P. Holme, Phys. Rev. Lett. 108, 128701 (2012) Data (100 cites) available at https://sites.google.com/site/lshlj82/
- 60. 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.a bstract]; 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)
- 61. Given a network, can we assign “roles” (i.e., colors) to nodes to identify their type? (Not based on density of connections!)
- 62. – 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
- 63. – 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 andpositions are taken or adapted from slides by Tom Snijders(5/2/2012): http://www.stats.ox.ac.uk/~s nijders/Equivalences.pdf
- 64. 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”.)
- 65. – 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
- 66. Tom Snijders
- 67. Tom Snijders
- 68. – 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 Homomorphisms on Networks of Relations”, Social Networks 5:193–234 A coloring is a regular equivalence if two nodes of the same color also have neighbors of the same color. Tom Snijders
- 69. – 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.
- 70. ➞ – 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.
- 71. – 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
- 72. – 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
- 73. Some illustrative examples and basic ideas for examining community structure in more general types of networks.
- 74. – 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
- 75. – 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.
- 76. – 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’s lecture and review article). – Useful to develop and consider null models that incorporate spatial information.
- 77. – 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
- 78. – 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
- 79. 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 network Ø Communities obtained from optimization of modularity (with “physical null model”) match well with empirical granular force networks in both laboratory and computational experiments matrix W is oen called a “weight matrix.” To obtain force chains from W, we want to determ particles for which strong inter-particle forces occ densely connected sets of particles. We can obtain a this problem via “community detection”,34,35,44 in whi sets of densely connected nodes called “modules” o nities.” A popular way to identify communities in a by maximizing a quality function known as modu respect to the assignment of particles to sets called nities.” Modularity Q is dened as Q ¼ X i;j Â Wij À gPij Ã d À ci; cj Á ; where node i is assigned to community ci, node j is a community cj, the Kronecker delta d(ci, cj) ¼ 1 if and cj, the quantity g is a resolution parameter, and expected weight of the edge that connects node i j under a specied null model. One can use the maximum value of modularity the quality of a partition of a force network into sets that are more densely interconnected by strong f expected under a given null model. The resolution g provides a means of probing the organization of in forces across a range of spatial resolutions. To pro intuition, we note that a perfectly hexagonal packing uniform forces should still possess a single comm small values of g and should consist of a collection particle (i.e., singleton) communities for large valu intermediate values of g, we expect maximizing mo yield a roughly homogeneous assignment of par communities of some size (i.e., number of particles) and the total number of particles. (The exact size d the value of g.) The strongly inhomogeneous c assignments that we observe in the laboratory and packings (see Section IV) are a direct consequen Publishedon23February2015.DownloadedbyCaliforniaInstituteofTechnologyon21/04/
- 80. we observe a maximum of guniform at g ¼ 0.9 (for g ˛ {0.1, 0.3,., 2.1}) in high-pressure packings (5.9 Â 10À3 E) and at g ¼ 1.5 for low-pressure packings (2.7 Â 10À4 E). In the numerical packings, we observe a maximum of guniform at g ¼ 1.1 for all pressures. In comparison to our observations in the main text from employ- ing the size-weighted systemic gap factor g, we nd that the optimal value of g is larger when we instead employ guniform (compare Fig. 10 to Fig. 5 and 7). We also observe that the curves of the systemic gap factor versus resolution parameter exhibit larger variation for the uniformly-weighted gap factor than for the size-weighted gap factor. Optimal value of the resolution parameter The large variation in the maximum of guniform over packings and pressures makes it diﬃcult to choose an optimal resolu- tion-parameter value. We choose to take gopt ¼ 1.1 because (1) it corresponds to the maximum of guniform in the numerical packings and (2) it corresponds to the mean of the maximum of guniform in the laboratory packings. To facilitate the comparison of optimal values of g from the two weighting schemes, we denote 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 resolution parameter The force chains that we identify for the optimal value for the uniformly-weighted gap factor (at ^guniform ¼ 1.1) diﬀer from those that we identied in the main text for the optimal value of the size-weighted gap factor (at ^g ¼ 0.9). We show our comparison in Fig. 11. For both laboratory and numerical Fig. 11 In both (A) (frictional) laboratory and (B) (frictionless) numerical packings, we identify larger and more branched force chains at the optimal resolution determined by (left; g ¼ 0.9) the size-weighted gap factor g, and we identify smaller and less branched force chains at the optimal resolution determined by (right; g ¼ 1.1) the uniformly- weighted gap factor guniform. These observations are consistent across all pressure values, but they are especially evident at high pressures in Paper Soft Matter bruary2015.DownloadedbyCaliforniaInstituteofTechnologyon21/04/201522:40:58. View Article Online – For larger pressures, we obtain larger and more branched force chains in both the (frictional) laboratory packings described earlier and in (frictionless) numerical packings
- 81. – 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 modelbased on gravity model – Found French vs. Flemish communities in mobile phone networkin 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
- 82. – Example: importance based on node strength (weighted degree):
- 83. – 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).
- 84. 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
- 85. – M. Kivelä et al., “Multilayer Networks”, JCN, 2014
- 86. – 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
- 87. • 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 6 6 6 6 6 6 6 6 6 6 6 6 4 0 1 1 ! 0 0 0 0 0 1 0 0 0 ! 0 0 0 0 1 0 0 0 0 ! 0 0 0 ! 0 0 0 1 1 ! 0 0 0 ! 0 1 0 1 0 ! 0 0 0 ! 1 1 0 0 0 ! 0 0 0 ! 0 0 0 1 0 0 0 0 0 ! 0 1 0 1 0 0 0 0 0 ! 0 1 0 3 7 7 7 7 7 7 7 7 7 7 7 7 5 Fig. 3.1. Example of (left) a multilayer network with unweighted intra-layer connections (solid lines) and uniformly weighted inter-layer connections (dashed curves) and (right) its corresponding adjacency matrix. (The adjacency matrix that corresponds to a multilayer network is sometimes called 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: ˆr(C, t) = N|T | X i,j=1 ✓ ⇡i ⇥ ij + t⇤ii(Mij ij) ⇤ ⇡i⇢i|j ◆ (ci, cj) , (3.1)
- 88. • Find communities algorithmically by optimizing “multislice modularity” – 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].
- 89. • 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
- 90. munities under Laplacian dynamics (13), which we have generalized to recover the null models for bipartite, directed, and signed networks (14). First, we obtained the resolution-parameter generaliza- standard null model for directed networks (16, 17) (again with a resolution parameter) by generaliz- ing the Laplacian dynamics to include motion along different kinds of connections—in this case, the differe lution par for signed We ap models fo existing q an additio between s by adjace interslice r to itself attention (Aijs = Ajis incorpora couplings single-slic each node and acros multislice time Lapla p˙is respects interslice probabilit ∑jrkjr, w terms of t slice s con tureallow for intra- 1800 1820 1840 1860 1880 1900 1920 1940 1960 1980 2000 40PA, 24F, 8AA 151DR, 30AA, 14PA, 5F 141F, 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 Senator NM UT WY CA OR WA AK HI A B
- 91. P. J. Mucha & M. A. Porter [2010], Chaos 20(4):041108
- 92. Much more on neuronalnetworkson Friday (Kaiser, Buldú)
- 93. Experiments on learning of a simple motor task
- 94. – 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
- 95. • 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.
- 96. Temporal core–periphery organization ≈ Geometrical core–periphery organization! (the latter is a density-based core using network structure in individual layers)
- 97. – 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 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 pia (t +1) = Â j,b P jb ia pjb (t), (1) where pjb (t) is the probability for a random walker to be at node j in layer b at time t and P jb ia 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 pia (•). We then use the stationary distribution to deﬁne the conductance (Jerrum & Sinclair, 1988) of a set of state nodes1 S as f(S) = Â (i,a)2S Â ( j,b)/2S Pia jb pia (•) Â pia (•) , (2) 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 pia (t +1) = Â j,b P jb ia pjb (t), (1) where pjb (t) is the probability for a random walker to be at node j in layer b at time t and P jb ia 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 pia (•). We then use the stationary distribution to deﬁne the conductance (Jerrum & Sinclair, 1988) of a set of state nodes1 S as f(S) = Â (i,a)2S Â ( j,b)/2S Pia jb pia (•) Â (i,a)2S pia (•) , (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 personalized PageRank (PPR) score of state node (i,a) as the solution to the equation PPR(g)ia = g ÂP jb ia PPR(g)jb +(1 g)sia , (3) et al., 2015; Kuncheva & Montana, 2015). For our purposes, we deﬁne pia (t +1) = Â j,b P jb ia pjb (t), where pjb (t) is the probability for a random walker to be at node j in layer b at ti P jb ia is the probability for a random walker to transition from node j in layer b to layer a in a time step. We also want the random walk to be ergodic, so that it ha deﬁned stationary distribution pia (•). We then use the stationary distribution to d conductance (Jerrum & Sinclair, 1988) of a set of state nodes1 S as f(S) = Â (i,a)2S Â (j,b)/2S Pia jb pia (•) Â (i,a)2S pia (•) , which we use as a quality measure for local communities. Once we select an ate random walk (or other Markov process2), we can deﬁne the associated pers PageRank (PPR) score of state node (i,a) as the solution to the equation PPR(g)ia = g Â j,b P jb ia PPR(g)jb +(1 g)sia , where s is a probability distribution that determines the seed nodes for the method 2014). We then approximate the solution to equation (3) locally (Andersen et al., ﬁnd local communities. Given a random walk on a multilayer network, one can analyze communities layer networks using the same methods as for single-layer networks. See Jeub et a for a detailed discussion of a few different methods and their application to severa of networks (which exhibit rather different types of behavior with respect to the ics of diffusion processes). Our code for identifying local communities and vi networks is available from https://github.com/LJeub. In the present article, we illustrate some features that one can encounter as a con
- 98. ZU064-05-FPR local˙multiplex15 1 September 2015 20:47 A local perspective on community structure in multilayer networks 5 100 101 102 103 10 3 10 2 10 1 100 number of state nodes conductance w = 0.1 w = 1 w = 10 (a) Classical random walk 100 101 102 103 10 3 10 2 10 1 100 number of state nodes conductance r = 0.01 r = 0.1 r = 1 (b) Relaxed random walk Wizz Air Ryanair (c) Best community with 173 state nodes for w = 0.1 (physical node as seed) SunExpress Panagra Airways Turkish Airlines (d) Best community with 169 state nodes for r = 0.1 (state node as seed) Fig. 2: European Airline Network: Multiplex transportation network with 37 layers. Each layer 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 each size (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 state nodes leads to similar results. Panels (c) and (d) illustrate some of the communities that we obtain. We shade the state nodes in a community from dark red to light grey based on their rank within the community. The large arrows point to the seed nodes. For small layer-jumping probability r in the relaxed random walk and small interlayer edge weight w
- 99. Mesoscale structures can give fascinating insights about networks, but be careful about how you apply these tools and ideas.
- 100. – 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)
- 101. – 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
- 102. Coming Next Tiago Peixoto’s presentation: statistically principled approaches

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