1. Interpolation method and scaling limits in sparse random graphs
David Gamarnik
MIT
Workshop on
Counting, Inference and Optimization on Graphs
November, 2011

2. 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

3. Erdos-Renyi graph G(N,c)
N nodes,
M=cN edges chosen u.a.r. from N2 possibilities

4. Erdos-Renyi hypergraph G(N,c)
N nodes,
M=cN (K-hyper) edges chosen u.a.r. from NK possibilities
K=3

5. MAX-CUT Note: Conjecture: the following limit exists
Goal: find the limit.

6. Partition function on cuts
Note:
Conjecture: the following limit exists
Goal: find this limit.

7. General model: Markov Random Field (Forney graph)
Spin assignments
Random i.i.d. potentials
Example: Max-Cut
Example: Independent Set

8. Ground state (optimal value)
Partition function:
Conjecture: the following limit exists
Equivalently, the sequence of random graphs is right-converging
Borgs, Chayes, Kahn & Lovasz [10]

9. 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.

10. 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]

11. 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.

12. 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 .

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

14. Interpolation between G(N,c) and G(N1,c) + G(N2,c)
t=0 (no red edges) : G(N,c)

15. Interpolation between G(N,c) and G(N1, c) + G(N2, c)
t=cN (no blue edges) : G(N1, c) + G(N2, c)

16. Claim:
As a result the sequence of optimal values is nearly super-additive

17. Claim: for every graph G0 ,
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
Observation:

18. Proof sketch. MAX-CUT

19. Proof sketch. MAX-CUT
Convexity of f(x)=x2 implies
QED

20. For what general model the interpolation method works?
Random i.i.d. potentials
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:

21. “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

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

23. Actual value of limits.
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.

24. Thank you