Interpolation method and scaling limits in sparse random graphs
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Interpolation method and scaling limits in sparse random graphs
David Gamarnik
MIT
Workshop on Counting, Inference and Optimization on Graphs
November, 2011
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• 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
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Erdos-Renyi graph G(N,c)
N nodes,
M=cN edges chosen u.a.r. from N2 possibilities
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K=3
Erdos-Renyi hypergraph G(N,c)
N nodes,
M=cN (K-hyper) edges chosen u.a.r. from NK possibilities
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MAX-CUT
Note:
Conjecture: the following limit exists
Goal: find the limit.
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Partition function on cuts
Note:
Conjecture: the following limit exists
Goal: find this limit.
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General model: Markov Random Field (Forney graph)
Spin assignments
Random i.i.d. potentials
Example: Max-Cut Example: Independent Set
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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]
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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.
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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]
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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.
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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 .
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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)
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• t=0 (no red edges) : G(N,c)
Interpolation between G(N,c) and G(N1,c) + G(N2,c)
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• t=cN (no blue edges) : G(N1, c) + G(N2, c)
Interpolation between G(N,c) and G(N1, c) + G(N2, c)
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Claim:
As a result the sequence of optimal values is nearly super-additive
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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
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Proof sketch. MAX-CUT
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Proof sketch. MAX-CUT
Convexity of f(x)=x2 implies
QED
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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
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“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
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This covers MAX-CUT, Coloring problems and Independent Set problems:
MAX-CUT
Coloring
Independent set
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• 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.
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Thank you