interpolation method and scaling limits in sparse random graphs

Interpolation method and scaling limits in sparse random graphs

David Gamarnik

MIT

Workshop on Counting, Inference and Optimization on Graphs

November, 2011

• Structural analysis of random graphs, Erdős–Rényi 1960s

•1980s – early 1990s algorithmic/complexity problems (random K-SAT problem)

• Late 1990s – early 2000s physicist enter the picture: replica symmetry, replica symmetry breaking, cavity method (non-rigorous)

• Early 2000s, interpolation method for proving scaling limits of free energy (rigorous!)

• Goal for this work – simple combinatorial treatment of the interpolation method

Erdos-Renyi graph G(N,c)

N nodes,

M=cN edges chosen u.a.r. from N2 possibilities

K=3

Erdos-Renyi hypergraph G(N,c)

N nodes,

M=cN (K-hyper) edges chosen u.a.r. from NK possibilities

MAX-CUT

Note:

Conjecture: the following limit exists

Goal: find the limit.

Partition function on cuts

Note:

Conjecture: the following limit exists

Goal: find this limit.

General model: Markov Random Field (Forney graph)

Spin assignments

Random i.i.d. potentials

Example: Max-Cut Example: Independent Set

Conjecture: the following limit exists

Ground state (optimal value)

Partition function:

Equivalently, the sequence of random graphs is right-converging

Borgs, Chayes, Kahn & Lovasz [10]

Even more general model: continuous spins

Conjecture. (Talagrand, 2011) The following limit exists w.h.p. when is a Gaussian kernel and

General Conjecture. The limit exists w.h.p.

Existence of scaling limits

Theorem (Bayati, G, Tetali [09]). The following limits exists for Max-Cut, Independent Set, Coloring, K-SAT models.

Open problem stated in Aldous (My favorite 6 open problems) [00], Aldous and Steele [03], Wormald [99],Bollobas & Riordan [05], Janson & Thomason [08]

Notes on proof method

Guerra & Toninelli [02] Interpolation Method for Sherrington-Kirkpatrick model leading to super-additivity. Related to Slepian inequality.

Franz & Leone [03]. Sparse graphs. K-SAT.

Panchenko & Talagrand [04]. Unified approach to Franz & Leone.

Montanari [05].Coding theory.

Montanari & Abbe [10]. K-SAT counting and generalization.

Proof sketch for MAX-CUT

size of a largest independent set in G(N,c)

Claim: for every N1, N2 such that N1+N2=N

The existence of the limit

then follows by “near” superadditivity .

Interpolation between G(N,c) and G(N1,c) + G(N2,c)

Fix 0· t· cN . Generate cN-t blue edges and t red edges

Each blue edge u.a.r. connects any two of the N nodes.

Each red edge u.a.r. connects any two of the Nj nodes with prob Nj /N, j=1,2.

G(N,t)

• t=0 (no red edges) : G(N,c)

Interpolation between G(N,c) and G(N1,c) + G(N2,c)

• t=cN (no blue edges) : G(N1, c) + G(N2, c)

Interpolation between G(N,c) and G(N1, c) + G(N2, c)

Claim:

As a result the sequence of optimal values is nearly super-additive

Claim: for every graph G0 ,

Observation:

Given nodes u,v in G0 , define u» v if for every optimal cut they are on the same side. Therefore, node set can be split into equivalency classes

Proof sketch. MAX-CUT

Proof sketch. MAX-CUT

Convexity of f(x)=x2 implies

QED

Theorem. Assume existence of “soft states”. Suppose there exists large enough such that for every and every the following expected tensor product is a convex

where

Then the limit exists:

For what general model the interpolation method works?

Random i.i.d. potentials

“Special” general case.

Deterministic symmetric identical potentials

Theorem. Assume existence of “soft states”. Suppose the matrix

is negative semi-definite on

Then the limit exists

This covers MAX-CUT, Coloring problems and Independent Set problems:

MAX-CUT

Coloring

Independent set

• Replica-Symmetry and Replica-Symmetry breaking methods provide rigorous upper bounds on limits. Involves optimizing over space of functions.

• Aldous-Hoover exchangeable array approach by Panchenko (2010) gives a full answer to the problem, but this involves solving an optimization problem over space of functions with infinitely many constraints.

• Contucci, Dommers, Giardina & Starr (2010). Full answer for coloring problem in terms of minimizing over a space of infinite-dimensional distributions.

Actual value of limits.

Thank you