1. COMMUNITY DETECTION IN STOCHASTIC BLOCK MODELS VIA SPECTRAL METHODS Laurent Massoulié (MSR-Inria Joint Centre, Inria)

2. Outline – remainder of the course  Control of eigen-elements’ perturbation  Courant-Fisher min-max theorem  Weyl’s inequalities  Bounding spectral norm of random noise matrices  Trace method  Matrix Bernstein inequalities  Alon-Boppana theorem re. Ramanujan property  The tree reconstruction problem  Branching number of a tree & Threshold for reconstruction  From tree reconstruction to SBM reconstruction  Proof elements for modified spectral methods  Matrix expansion formula  « Local analysis »: quasi-deterministic growth

3. Outline – remainder of the course  Control of eigen-elements’ perturbation  Courant-Fisher min-max theorem  Weyl’s inequalities  Bounding spectral norm of random noise matrices  Trace method  Matrix Bernstein inequalities  Alon-Boppana theorem re. Ramanujan property  The tree reconstruction problem  Branching number of a tree & Threshold for reconstruction  From tree reconstruction to SBM reconstruction  Proof elements for modified spectral methods  Matrix expansion formula  « Local analysis »: quasi-deterministic growth

4. Detection by modified spectral method i j i j

5. Spectral separation “à la Ramanujan” a/2a/2b/2b/2 a/2a/2 b/2b/2

6. Illustrations for n=200

7. Reconstruction from 2 nd eigenvector i + + +- -

9. Proof Strategy I

10. Proof Strategy II

11. Proof elements 1) matrix expansion  Expected adjacency matrix  Centered simple path adjacency matrix  Expansion: “small” terms

12. “Smallness” of matrix coefficients

13.  Sum over circuits made of concatenation of 2k simple paths of length l  Encode circuit as repetition of 3 distinct phases  Phase 1: walk on tree of node discoveries  Phase 2: sequence of new node discoveries  Phase 3: edge towards already discovered node

14. Proof elements 2) Quasi-deterministic growth of node neighborhoods i + + +- -

15. Weak Ramanujan property  Previous results combined give Use spectral radius bounds Use bounds from quasi-deterministic growth

16. “Spectral redemption” and the non- backtracking matrix e f e f

17. a/2a/2b/2b/2 a/2a/2 b/2b/2

18. Main result [Bordenave-Lelarge-M’15]

19. Illustration for 2-community symmetric Stochastic block model

21. Proof elements Low-rank

22. Proof elements

23. Outlook

24. Thanks!