COMMUNITY DETECTION IN STOCHASTIC BLOCK MODELS VIA SPECTRAL METHODS

Laurent Massoulié (MSR-Inria Joint Centre, Inria)

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

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

Courant-Fisher minimax theorem

For Hermitian matrix :

Controlling perturbation of eigen-elements of Hermitian matrices

Where : observed; : signal; : perturbation

Perturbation Lemmaorder ,

1) (Weyl)

2) Let . IfThen for all i and for any normed eigenvector , there exists s.t.

Application to SBM

Spectrum of : spectrum of scaled by , hence

If there exist block-specific vectors such that for all

Clustering of spectral representatives follows ( eg take )

Another application of Courant-Fisher minimax theorem

Cauchy interlacing theorem:For Hermitian matrix , where : orthogonal projection on -dimensional space, then for all

Where : eigenvalues of viewed as operator on -dimensional space

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

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

Result à la Furedi-Komlos

centered independent with Then whp in probability as

e.g. in Gaussian case , :

Supremum of over unit sphere = typical value

A toy version illustrating the « trace method »

For centered independent with

For all , whp

Implies if for some fixed , A first sufficient condition for consistency of basic method

Another tool: Matrix Bernstein inequality [Tropp’10&’14]

For dimensional independent Hermitian matrices such that:, almost surely,Note and Then for all :

Hence:

Application

Show that in SBM, with high probability,

Deduce consistency of spectral method in SBM for signal strength

Key lemmas

Lemma 1 (« Master inequality »)

Lemma 2 (consequence of Lieb’s theorem):For independent Hermitian matrices , Then

Lemma 3: For hermitian such that and a.s.,

Result follows by proper choice of …

spectral separation properties “à la Ramanujan”

s-regular graph Ramanujan if

[Lubotzky-Phillips-Sarnak’88]

[Friedman’08]: random s-regular graph verifies whp

[Feige-Ofek’05]: for Erdős-Rényi graph and , then whp Also: . Result carries over to SBM

Optimality of Ramanujan graphs

Alon-Boppana theorem:

For s-regular graph with diameter Then