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