Spectral Clustering for Dynamic Block Models

Transcription

1 Spectral Clustering for Dynamic Block Models Sharmodeep Bhattacharyya Department of Statistics Oregon State University January 23, 2017 Research Computing Seminar, OSU, Corvallis (Joint work with Shirshendu Chatterjee, City College, CUNY) Sharmodeep Bhattacharyya (Oregon State) Dynamic Spectral Clustering January 23, / 53

2 Outline 1 Introduction and Motivation 2 Community Detection in Networks Community Detection Algorithms Community Detection Algorithms: Spectral Methods 3 Feature and Models of Networks Dynamic Network Models Spectral Clustering Methods Theoretical Results 4 Resullts Simulation Resullts Real Networks: Neuroscience Example 5 Summary Sharmodeep Bhattacharyya (Oregon State) Dynamic Spectral Clustering January 23, / 53

3 Introduction and Motivation Networks Networks Nodes Edges Social Network People Friendship/kinship Biological Network Gene/Protein Interaction Citation Networks Papers citation Sharmodeep Bhattacharyya (Oregon State) Dynamic Spectral Clustering January 23, / 53

4 Introduction and Motivation Network Data G = (V, E): undirected graph and V = {v 1,, v n } arbitrarily labeled vertices. Adjacency matrices (Symmetric), [A ij ] n i,j=1 numerically represent network data: 1 if node i links to node j, A ij = 0 otherwise. Sharmodeep Bhattacharyya (Oregon State) Dynamic Spectral Clustering January 23, / 53

5 Introduction and Motivation Drosophila protein interactions Guruharsha et al., A protein complex network of Drosophila melanogaster, Cell, 147: , 2011 Experimentally measured and scored protein interactions 1612 nodes; 10,421 edges (edge density ˆρ = ) Sharmodeep Bhattacharyya (Oregon State) Dynamic Spectral Clustering January 23, / 53

6 Introduction and Motivation Political blogs Understanding political patterns Adamic, Lada A., and Natalie Glance. "The political blogosphere and the 2004 US election: divided they blog." Proceedings of the 3rd international workshop on Link discovery. ACM, Sharmodeep Bhattacharyya (Oregon State) Dynamic Spectral Clustering January 23, / 53

7 Introduction and Motivation Online Social Network Sharmodeep Bhattacharyya (Oregon State) Dynamic Spectral Clustering January 23, / 53

8 Introduction and Motivation Dynamic/Time-varying Networks Figure: Dynamic Network Examples Sharmodeep Bhattacharyya (Oregon State) Dynamic Spectral Clustering January 23, / 53

9 Introduction and Motivation A Motivating Example: Electro-Corticograph Array Data for Speech Figure: a. MRI reconstruction of a single subject brain with vsmc electrodes (dots), colored according to distance from the Sylvian fissure (black and red are the most dorsal and ventral positions, respectively). b. Expanded view of vsmc anatomy: cs, central sulcus; PoCG, post-central gyrus; PrCG, pre-central gyrus; Sf, Sylvian fissure. c - e.top, vocal tract schematics for three consonants (/b/, /d/, /g/), produced by occlusion at the lips, tongue tip and tongue body, respectively (red arrow). Middle, spectrograms of spoken consonant-vowel syllables (Bouchard et.al., Nature, 2013). Sharmodeep Bhattacharyya (Oregon State) Dynamic Spectral Clustering January 23, / 53

10 Introduction and Motivation Other Examples of Network Data Biological Networks: Biochemical pathway networks Gene transcription networks Epidemiological Networks Social Networks: Academic networks such as collaboration and citation networks Networks arising from text-mining Technological Networks: Internet Cell-phone tower and telephone exchange networks Airport and Transport Networks Sharmodeep Bhattacharyya (Oregon State) Dynamic Spectral Clustering January 23, / 53

11 Introduction and Motivation Two Main Classes of Problems for Networks (I) Formation of networks given information on vertices as data. (II) Inference on networks given complete network with node and edge structure as data. Sharmodeep Bhattacharyya (Oregon State) Dynamic Spectral Clustering January 23, / 53

12 Introduction and Motivation Two Main Classes of Problems for Networks (I) Formation of networks given information on vertices. (II) Inference on networks given a complete network with node and edge structure. Sharmodeep Bhattacharyya (Oregon State) Dynamic Spectral Clustering January 23, / 53

13 Introduction and Motivation Commonly Questions Asked Community Detection. Link Prediction. Covariate or Latent Variable Estimation. Sampling of nodes and subgraphs. Dynamic network inference and information exchange in networks. Most of these questions can be answered by performing inference on network models. Sharmodeep Bhattacharyya (Oregon State) Dynamic Spectral Clustering January 23, / 53

14 Introduction and Motivation Commonly Questions Asked Community Detection Link Prediction Covariate or Latent Variable Estimation Sampling of nodes and subgraphs Information exchange Most of these questions can be answered by performing inference on network models. Sharmodeep Bhattacharyya (Oregon State) Dynamic Spectral Clustering January 23, / 53

15 Introduction and Motivation Community in Networks Physical Topological Definition How to Find Topological Nodes within a community has more edges Community detection algorithms among themselves than with nodes proposed by Statisticians/ outside community in average Computer Scientists/ Mathematicians Physical Nodes or Edges within community Verified by Scientists have some shared property Sharmodeep Bhattacharyya (Oregon State) Dynamic Spectral Clustering January 23, / 53

16 Community Detection in Networks Community Detection Algorithms Outline 1 Introduction and Motivation 2 Community Detection in Networks Community Detection Algorithms Community Detection Algorithms: Spectral Methods 3 Feature and Models of Networks Dynamic Network Models Spectral Clustering Methods Theoretical Results 4 Resullts Simulation Resullts Real Networks: Neuroscience Example 5 Summary Sharmodeep Bhattacharyya (Oregon State) Dynamic Spectral Clustering January 23, / 53

17 Community Detection in Networks Community Detection Algorithms Community Detection Algorithms Popular algorithms for community detection are - 1 Modularity maximizing methods. (Newman and Girvan (2006)) 2 Spectral clustering based methods. (McSherry (2001)) 3 Likelihood and its approximation maximization (a) Profile Likelihood Maximization (Bickel and Chen (2009)). (b) Variational Likelihood Maximization. (Celisse et. al. (2011)) (c) Pseudo-likelihood Maximization (Chen et. al. (2012)). Sharmodeep Bhattacharyya (Oregon State) Dynamic Spectral Clustering January 23, / 53

18 Community Detection in Networks Community Detection Algorithms: Spectral Methods Outline 1 Introduction and Motivation 2 Community Detection in Networks Community Detection Algorithms Community Detection Algorithms: Spectral Methods 3 Feature and Models of Networks Dynamic Network Models Spectral Clustering Methods Theoretical Results 4 Resullts Simulation Resullts Real Networks: Neuroscience Example 5 Summary Sharmodeep Bhattacharyya (Oregon State) Dynamic Spectral Clustering January 23, / 53

19 Community Detection in Networks Community Detection Algorithms: Spectral Methods General Spectral Clustering Algorithm Sharmodeep Bhattacharyya (Oregon State) Dynamic Spectral Clustering January 23, / 53

20 Community Detection in Networks Community Detection Algorithms: Spectral Methods Well-known Examples of M n For community identification in network, there are some well-known operators M n. Adjacency matrix M n = A (Sussman et.al (2012)) Normalized Laplacian matrices M n = L rw n = D 1 L n and L sym n = D 1/2 L n D 1/2 with L n = D A n (Rohe. et.al. (2011)). These operators although perform well in regime (a) fail to perform well in both regime (b) and (c) described previously. Sharmodeep Bhattacharyya (Oregon State) Dynamic Spectral Clustering January 23, / 53

21 Community Detection in Networks Community Detection Algorithms: Spectral Methods M n for Sparse Networks For community identification in sparse networks, there are some regularized variations of L n or A n. Adjacency matrix A τ = A + τ11 T, where 1 is a vector of 1 s of length n. (Amini et.al (2012)) Laplacian matrix L τ n = (D + τi ) 1/2 A(D + τi ) 1/2 (Chaudhuri. et.al. (2012)). Trimmed adjacency matrix A τ, where, high-degree nodes are trimmed (Coja-Oghlan (2010)). Theoretical performance of first two regularized operators for sparse networks is under investigation. Sharmodeep Bhattacharyya (Oregon State) Dynamic Spectral Clustering January 23, / 53

22 Feature and Models of Networks Dynamic Models Outline 1 Introduction and Motivation 2 Community Detection in Networks Community Detection Algorithms Community Detection Algorithms: Spectral Methods 3 Feature and Models of Networks Dynamic Network Models Spectral Clustering Methods Theoretical Results 4 Resullts Simulation Resullts Real Networks: Neuroscience Example 5 Summary Sharmodeep Bhattacharyya (Oregon State) Dynamic Spectral Clustering January 23, / 53

23 Feature and Models of Networks Dynamic Models A Motivating Example: Electro-Corticograph Array Data for Speech Figure: a. MRI reconstruction of a single subject brain with vsmc electrodes (dots), colored according to distance from the Sylvian fissure (black and red are the most dorsal and ventral positions, respectively). b. Expanded view of vsmc anatomy: cs, central sulcus; PoCG, post-central gyrus; PrCG, pre-central gyrus; Sf, Sylvian fissure. c - e.top, vocal tract schematics for three consonants (/b/, /d/, /g/), produced by occlusion at the lips, tongue tip and tongue body, respectively (red arrow). Middle, spectrograms of spoken consonant-vowel syllables (Bouchard et.al., Nature, 2013). Sharmodeep Bhattacharyya (Oregon State) Dynamic Spectral Clustering January 23, / 53

24 Feature and Models of Networks Dynamic Models Dynamic Network Models: A Myopic Review Dynamic time-evolving formation of networks: Barabasi and Albert (1999) and a large literature. Extension of static models of network: Latent space models, Sarkar and Moore (2005), Sewell and Chen (2014). Mixed membership block models, Xing et.al. (2010), Ho et.al. (2011). Random dot-product models, Tang et.al. (2013). Stochastic block models, Xu et.al. (2013), Ghasemian et.al. (2015). Graphon models, Crane (2015). Bayesian models: Ho et.al. (2011), Durante et.al. (2014). Sharmodeep Bhattacharyya (Oregon State) Dynamic Spectral Clustering January 23, / 53

25 Feature and Models of Networks Dynamic Models Nonparametric Latent Variable Models Derived from representation of exchangeable random infinite array by Aldous and Hoover (1983). NP Model Define P({A ij } n i,j=1 ) conditionally given latent variables {ξ i} n i=1 associated with vertices {v i } n i=1 respectively. (Bickel & Chen (2009), Bollobás et.al. (2007), Hoff et.al. (2002)). ξ 1,..., ξ n iid U(0, 1) Pr(A ij = 1 ξ i = u, ξ j = v) = h n (u, v) = ρ n w(u, v), Sharmodeep Bhattacharyya (Oregon State) Dynamic Spectral Clustering January 23, / 53

26 Feature and Models of Networks Dynamic Models Nonparametric Latent Variable Models Derived from representation of exchangeable random infinite array by Aldous and Hoover (1983). NP Model Define P({A ij } n i,j=1 ) conditionally given latent variables {ξ i} n i=1 associated with vertices {v i } n i=1 respectively. (Bickel & Chen (2009), Bollobás et.al. (2007), Hoff et.al. (2002)). ξ 1,..., ξ n iid U(0, 1) Pr(A ij = 1 ξ i = u, ξ j = v) = h n (u, v) = ρ n w(u, v), w(u, v) is the conditional latent variable density given A ij = 1. Define λ n nρ n as the expected degree parameter and P = [P ij ] n i,j = [ρ nw(ξ i, ξ j )] n i,j. h n : not uniquely defined. h n ( ϕ(u), ϕ(v) ), with measure-preserving ϕ, gives same model. Sharmodeep Bhattacharyya (Oregon State) Dynamic Spectral Clustering January 23, / 53

27 Feature and Models of Networks Dynamic Models Stochastic Block Model (Holland, Laskey and Leinhardt 1983) A K-block stochastic block model with parameters (π, P) is defined as follows. Consider latent variable corresponding to vertices as z = (z 1, z 2,..., z n ) with z 1,..., z n iid Multinomial(1; (π1,..., π K )) Pr(A ij = 1 z i, z j ) = P zi z j, where P = [P ab ] is a K K symmetric matrix for undirected networks. Sharmodeep Bhattacharyya (Oregon State) Dynamic Spectral Clustering January 23, / 53

28 Feature and Models of Networks Dynamic Models Dynamic Nonparametric Latent Variable Models Now we try to introduce a time component to the exchangeable model. The most general version of the model becomes ξi t (ξi t 1 = u) ( P A (t) ij = 1 ξi t = u, ξj t = v, A (t 1) ) ij = z ξ 0 i iid U(0, 1) (1) iid F (u) (2) = h n (u, v, z, t) = ρ n w(u, v, z, t) (3) where, F is an univariate distribution and 0 h n 1 and 0 t T is the time variable. Random re-wiring mechanism: h n depends on both t and z (Harry Crane, 2015). Evolving Communities: h n depends on (u, v) only, F non-trivial (Ghasemian et.al., 2015). Sharmodeep Bhattacharyya (Oregon State) Dynamic Spectral Clustering January 23, / 53

29 Feature and Models of Networks Dynamic Models Dynamic Stochastic Block Model (DSBM) Specialize to Dynamic Stochastic Block Model with parameters (π, B) and latent variables z, z 1,..., z n iid Mult(1; (π1,..., π K )), (4) ( P A (t) ij = 1 z i, z j ) = B z (t) i z j. (5) where, B t = [Bab t ] are K K symmetric matrix for undirected networks for each time step t and 0 t T is the time variable. Sharmodeep Bhattacharyya (Oregon State) Dynamic Spectral Clustering January 23, / 53

30 Feature and Models of Networks Dynamic Models Dynamic Degree Corrected Block Model (DDCBM) Specialize to Dynamic Degree Corrected Block Model with parameters (π, B, ψ) and latent variables z, z 1,..., z n iid Mult(1; (π1,..., π K )), (6) ( P A (t) ij = 1 z i, z j, ψ) = ψ i ψ j B z (t) i z j. (7) where, B t = [Bab t ] are K K symmetric matrix for undirected networks for each time step t and 0 t T is the time variable. Sharmodeep Bhattacharyya (Oregon State) Dynamic Spectral Clustering January 23, / 53

31 Feature and Models of Networks Spectral Methods Outline 1 Introduction and Motivation 2 Community Detection in Networks Community Detection Algorithms Community Detection Algorithms: Spectral Methods 3 Feature and Models of Networks Dynamic Network Models Spectral Clustering Methods Theoretical Results 4 Resullts Simulation Resullts Real Networks: Neuroscience Example 5 Summary Sharmodeep Bhattacharyya (Oregon State) Dynamic Spectral Clustering January 23, / 53

32 Feature and Models of Networks Spectral Methods Dynamic Spectral Clustering Algorithms Sharmodeep Bhattacharyya (Oregon State) Dynamic Spectral Clustering January 23, / 53

33 Feature and Models of Networks Theory Outline 1 Introduction and Motivation 2 Community Detection in Networks Community Detection Algorithms Community Detection Algorithms: Spectral Methods 3 Feature and Models of Networks Dynamic Network Models Spectral Clustering Methods Theoretical Results 4 Resullts Simulation Resullts Real Networks: Neuroscience Example 5 Summary Sharmodeep Bhattacharyya (Oregon State) Dynamic Spectral Clustering January 23, / 53

34 Feature and Models of Networks Theory First Method: In Detail In the first method, we sum the adjacency matrices to obtain T A = A (t). t=1 We obtain leading K eigenvectors of A corresponding to its largest eigenvalues. Suppose Ûn K contains those eigenvectors as columns. Then we use (1 + ɛ) approximate k-means clustering algorithm to obtain Ẑ M n,k and ˆΘ R K K such that Ẑ ˆΘ Û 2 F (1 + ɛ) min ZΘ Û 2 Z M n K,Θ R K K F. Ẑ is the estimate of Z = (z 1,..., z n ) from this method. Sharmodeep Bhattacharyya (Oregon State) Dynamic Spectral Clustering January 23, / 53

35 Feature and Models of Networks Theory First Method: Consistency of Ẑ Adjacency matrices, A generated from the DSBM with n nodes and K communities with parameters (π, {B (t) } T t=1 ), γ n be the smallest non-zero singular value of P, d := max k,l [K],t [T ] B (t) k,l n be the maximum expected degree of a node at any time. Sharmodeep Bhattacharyya (Oregon State) Dynamic Spectral Clustering January 23, / 53

36 Feature and Models of Networks Theory First Method: Consistency of Ẑ Theorem Let A is generated from DSBM. Suppose γ n is large enough so that K γ 2 n max{td, log 2 n/td} = o(1). For any ɛ, c > 0, there is a constant C = C(ɛ, c) > 0 such that if Ẑ is the estimate of Z as described in Algorithm 1, and if f i, i [K] is the fraction of nodes belonging to C i which are misclassified in Ẑ, then K f i C K γn 2 max{td, log 2 n/td} i=1 with probability at least 1 n c. Sharmodeep Bhattacharyya (Oregon State) Dynamic Spectral Clustering January 23, / 53

37 Feature and Models of Networks Theory First Method: Consistency of Ẑ Corollary In the special case of Theorem when (i) the minimum eigenvalue of n d B(t) is positive and uniformly bounded away from zero for all t [T ], (ii) the community sizes are balanced, i.e. n max /n min = O(1), then consistency holds for Ẑ if either Td log(n) and K = o(td), or (log(n)) 2/3 << Td < log(n) and K = o((td) 3 /(log(n)) 2 ). Sharmodeep Bhattacharyya (Oregon State) Dynamic Spectral Clustering January 23, / 53

38 Feature and Models of Networks Theory First Method: Consistency of Ẑ Corollary In the special case of Theorem when (i) the minimum eigenvalue of n d B(t) is positive and uniformly bounded away from zero for all t [T ], (ii) the community sizes are balanced, i.e. n max /n min = O(1), then consistency holds for Ẑ if either Td log(n) and K = o(td), or (log(n)) 2/3 << Td < log(n) and K = o((td) 3 /(log(n)) 2 ). Sharmodeep Bhattacharyya (Oregon State) Dynamic Spectral Clustering January 23, / 53

39 Feature and Models of Networks Theory Algorithm 2: In Detail In the second method, we sum the squares of the adjacency matrices to obtain, A [2] and then subtract its diagonal to obtain, A [2], T A [2] Ä := A (t) ä 2 A[2] T Ä, := A (t) ä 2. t=1 t=1 We obtain leading K eigenvectors of A [2] corresponding to its largest eigenvalues. Suppose Ŭ R n K contains those eigenvectors as columns. Then we use (1 + ɛ) approximate K-means clustering algorithm to obtain Z M n,k and Θ R K K such that Z Θ Ŭ 2 F (1 + ɛ) min Z M n K,Θ R K K ZΘ Ŭ 2 F. Z is the estimate of Z = (z 1,..., z n ) from this method. Sharmodeep Bhattacharyya (Oregon State) Dynamic Spectral Clustering January 23, / 53

40 Feature and Models of Networks Theory Consistency of Z In order to prove consistency of Z, we need some notations and observations. Let T B [2] Ä := B (t) ä 2 t=1 T P [2] Ä := P (t) ä 2 T Ä = Z 1 B (t) ä 2 1 Z T (8) t=1 t=1 The main assumption about the connection probabilities that we need is At least one B (t), t [T ], must be nonsingular. (9) Sharmodeep Bhattacharyya (Oregon State) Dynamic Spectral Clustering January 23, / 53

41 Feature and Models of Networks Theory More Notations and Conditions for Consistency of Z A is generated from DSBM with n nodes and K communities and the parameters (aπ, {B (t) } T t=1 ). γ n be the smallest non-zero singular value of P [2] d := max k,l [K],t [T ] B (t) k,l n be the maximum expected degree of a node at any time. Sharmodeep Bhattacharyya (Oregon State) Dynamic Spectral Clustering January 23, / 53

42 Feature and Models of Networks Theory Second Method: Consistency of Z Theorem Let A is generated from DSBM satisfying assumption (9). Suppose γ n is large enough so that K (Td 3 (1 T 1 d 1 log n + log 10 n) = o(1). For any ɛ, c > 0, there γn 2 is a constant C = C(ɛ, c) > 0 such that if Z is the estimate of Z as described in Algorithm 2, and if f i, i [K] is the fraction of nodes belonging to C i which are misclassified in Z, then K i=1 f i CK Td 3 (1 T 1 d 1 log n) + (Td 2 log 2 (n) log 10 (n)) (Td 2 log 12 (n)) γ 2 n with probability at least 1 4n c. Sharmodeep Bhattacharyya (Oregon State) Dynamic Spectral Clustering January 23, / 53

43 Feature and Models of Networks Theory Second Method: Consistency of Z Corollary In the special case of Theorem when (i) the number of nonsingular matrices among { n d B(t) : t [T ]} (whose singular values are bounded away from 0 uniformly) grows faster than max{d 2 log 5 n,» T /d}, and (ii) the community sizes are balanced, i.e. n max /n min = O(1), then consistency holds for Z. Sharmodeep Bhattacharyya (Oregon State) Dynamic Spectral Clustering January 23, / 53

44 Feature and Models of Networks Theory Algorithm 3: Spherical Spectral Clustering In the third method, obtain the sum of the squared adjacency matrices without its diagonal, A [2] := T t=1 Ä A (t) ä 2. Obtain Ŭ R n K consisting of the leading K eigenvectors of corresponding to its largest absolute eigenvalues. A [2] Let n + be the number of nonzero rows of Ŭ. Obtain Ŭ + R n + K consisting of the normalized nonzero rows of Ŭ, i.e. Ŭ + i, = Ŭ i, / Ŭi, for i such that 2 > 0. 2 Ŭi, Use (1 + ɛ) approximate K-median clustering algorithm on the row vectors of Ŭ + to obtain Z + M n +,K and X R K K. Extend Z + to obtain Z by (arbitrarily) adding n n + many canonical unit row vectors at the end, such as, Z i = (1, 0,..., 0) for i such that = 0. 2 Z is the estimate of Z. Ŭi, Sharmodeep Bhattacharyya (Oregon State) Dynamic Spectral Clustering January 23, / 53

45 Results Simulation Outline 1 Introduction and Motivation 2 Community Detection in Networks Community Detection Algorithms Community Detection Algorithms: Spectral Methods 3 Feature and Models of Networks Dynamic Network Models Spectral Clustering Methods Theoretical Results 4 Resullts Simulation Resullts Real Networks: Neuroscience Example 5 Summary Sharmodeep Bhattacharyya (Oregon State) Dynamic Spectral Clustering January 23, / 53

46 Results Simulation Simulation Results: DSBM (a) (b) Figure: (a) For Sparse network λ n = 3 (b) Dense network, λ n = 8. Sharmodeep Bhattacharyya (Oregon State) Dynamic Spectral Clustering January 23, / 53

47 Results Simulation Simulation Results: DSBM (a) Figure: Dense network, λ n = 10, with B nearly singular. Sharmodeep Bhattacharyya (Oregon State) Dynamic Spectral Clustering January 23, / 53

48 Results Simulation Simulation Results: DDCBM (a) (b) Figure: Dense: (a) B nearly singular (b) B non-singular. Sharmodeep Bhattacharyya (Oregon State) Dynamic Spectral Clustering January 23, / 53

49 Results Real Networks Outline 1 Introduction and Motivation 2 Community Detection in Networks Community Detection Algorithms Community Detection Algorithms: Spectral Methods 3 Feature and Models of Networks Dynamic Network Models Spectral Clustering Methods Theoretical Results 4 Resullts Simulation Resullts Real Networks: Neuroscience Example 5 Summary Sharmodeep Bhattacharyya (Oregon State) Dynamic Spectral Clustering January 23, / 53

50 Results Real Networks Neuroscience ECoG Example Figure: Clustering of the network correctly identifies the lip region (upper right hand part of the vsmc) involved in the production of /b/, which engages the lips. (a): Location of Electrode Clusters based on BolBO-based graph Estimation (b): Organization of articulator representations in the vsmc (black: larynx; red: lips; blue: tongue; green: jaw). (c): Estimated graph of electrodes. harmodeep Bhattacharyya (Oregon State) Dynamic Spectral Clustering January 23, / 53

51 Conclusion Outline 1 Introduction and Motivation 2 Community Detection in Networks Community Detection Algorithms Community Detection Algorithms: Spectral Methods 3 Feature and Models of Networks Dynamic Network Models Spectral Clustering Methods Theoretical Results 4 Resullts Simulation Resullts Real Networks: Neuroscience Example 5 Summary Sharmodeep Bhattacharyya (Oregon State) Dynamic Spectral Clustering January 23, / 53

52 Conclusion Summary and Future Works Summary We consider two methods of spectral clustering for dynamic SBM. We give theoretical justifications of each method. Works in Progress Extension of more general dynamic SBM. Extension of dynamic models. Sharmodeep Bhattacharyya (Oregon State) Dynamic Spectral Clustering January 23, / 53

53 Conclusion Future Problems in Networks Methodological Detection of dynamic communities. Detection of communities in presence of covariates. Comparison of networks and communities for multiple networks. Theoretical Condition for community recovery for general K and connectivity matrix. Condition for community recovery for dynamic networks. Condition for community recovery for networks with covariate information. Sharmodeep Bhattacharyya (Oregon State) Dynamic Spectral Clustering January 23, / 53