1. DM GROUP MEETING PRESENTATION 10-07-2015

2. PLAN Eigenvector-based Centrality Measures For Temporal Networks by D Taylor et.al. Uncovering the Small Community Structure in Large Networks: A Local Spectral Approach WWW 2015, by Y Li et.al.

3. P ART 1 Eigenvector-based Centrality Measures For Temporal Networks by D Taylor et.al.

4. M OTIVATION Many existing eigenvector-based centralities quantify importance of each node based on eigenscore for static graphs Generalize eigenvector based centralities to temporal networks Average graphs do not give correct answers. (Eg. Random walks)

5. P ROBLEM D EFINITION Given G(V,E) for T timestamps Find time-averaged centralities of each node without neglecting the temporal properties of the graph Find how the centralities of the nodes change over time

6. E XAMPLE

7. APPROACH Change the given N*N*T tensor to NT*NT Supra-centrality matrix Independent of choice of centrality matrix Adjacency Hub and Authorities (AA T, A T A) Any other centrality characterized by dominant eigenvector Couple nearest neighbor temporal layers using inter- layer edges of weight w.

8. N OTATIONS

9. NAÏVE APPROACH Kronecker product of T*T matrix and N*N matrix.

10. NAÏVE APPROACH Now, Apply standard eigen-vector based centrality formulation to supra-adjacency matrix A treating it like static adjacency matrix. Doesn’t distinguish between intra-layer and inter-layer edges. Problem : - Spectral and Connectedness properties do not carry over to supra-adjacency matrix naturally.

11. I NTER - LAYER COUPLING OF CENTRALITY MATRICES To preserve the role of inter-layer edges, couple matrices that define eigenvector based centrality within each layer, i.e. define some matrix M as function of adjacency matrix A Example hub :- AA T Let M (t) denote centrality matrix for layer t Define e = 1/w where w is inter-layer coupling weight

12. I NTER - LAYER COUPLING OF CENTRALITY MATRICES Entries of λ max gives the centrality measure for node layer pair (i,t).

13. E XAMPLE MNC:- Marginal Node Centrality MLC:- Marginal Layer Centrality

14. E XAMPLE

15. S INGULAR P ERTURBATION IN STRONG C OUPLING LIMIT When e = 0, inter-layer connectivity is completely elimitnated Network reduces to N connected components No longer irreducible, Perron-Forbenius theorem does not hold Singulairy

16. S INGULAR P ERTURBATION IN STRONG C OUPLING LIMIT Examine time averaged centrality for the limit e tends to zero from right. Result: Where X is N*N matrix and α i is the time- averaged node centrality of node I

17. Similarly, first order-mover (m i ) score is given by:

18. Calulating α i and m i requires solving size N linear systems Computationally efficient than NT*NT when T>>1.

19. CASE-STUDY Doctoral degree exchange in the mathematics genealogy project Nodes : - universities Directed edge from i -> j at time t, if a PhD from university j who graduated in year t, later advised at least one student in university I Consider academic reputation based on Authority with respect to HITS centrality

20. CASE STUDY

22. P ART 2 Uncovering the Small Community Structure in Large Networks: A Local Spectral Approach WWW 2015, by Y Li et.al.

23. M OTIVATION We find communities of size hundreds in networks of billion nodes Taking entire graph into account is not practical Focus on microscopic structure to discover small communities Reduction in computational cost

24. A PPROACHES FOR COMMUNITY DETECTION Globally based community finding algorithms Find communities by optimizing objective functions Random Walk based detection algorithms Among divergent approaches, reveal communities that bear closest resemblance to the ground truth communities Seed set expansion based approaches Finds communities based on the set of seed nodes fed as input

25. P ROBLEM STATEMENT Given a network G = (V,E) and set of members S in the target community C, where |C|<<|V| and |S|<<|C|, find remaining members if C in time functional to the size of the community

26. ALGORITHM LEMON(Local expansion via minimum norm) Vertices around seed members are more likely to be in the community, therefore start random walks for few steps Instead of considering single probability vector, consider few dimensions of vector after short random walks, use it as approximate invariant subspace. Look for minimum zero norm vector in the span of invariant subspace such that seed members are in support Want to find rows in invariant subspace that are roughly in the same direction as the seed

27. ALGORITHM Step 1: Generate local spectra Let Be the normalized adjacency matrix Consider random walk from exemplary vertices in S Let p o denote initial probability vector Consider the span of l-dimensional probability vectors of successive random walks Initial invariant subspace is given by orthonormal basis of the span P 0,l, denoted by V 0,l. Use to calculate l-dimensional orthonormal basis after k steps of random walk

28. ALGORITHM Step 2: Solve the following linear programming problem: Where e is 1 vector, x and y are unknown The element of y indicate likelihood of vertex being in community Choose top |C| elements in y

29. ALGORITHM Step 3:Reseeding Add chosen top elements of y to S and denote new seed by S’ Repeat 1 and 2 until stop criteria is met Stop criteria based on: |C| If |C| is not known: Adding irrelevant nodes would cause conductance to increase Estimate |C|based on the first minimum conductance encountered

30. S EEDING METHOD High degree seeding Low degree seeding Triangle seeding Random seeding High-inward ratio seeding

31. P ERFORMANCE AGAINST LOCALIZED ALGORITHMS

32. P ERFORMANCE AGAINST GLOBAL ALGORITHMS