Random graph generation is an important tool for studying large complex networks. Despite abundance of random graph models, constructing models with application-driven constraints is poorly understood. In order to advance state-of-the-art in this are
a, we focus on random graphs without short cycles as a stylized family of graphs, and propose the RandGraph algorithm for randomly generating them. For any constant k, when m=O(n^{1+1/[2k(k+3)]}), RandGraph generates an asymptotically uniform random graph with n vertices, m edges, and no cycle of length at most k using O(n^2m) operations. We also characterize the approximation error for finite values of n. To the best of our knowledge, this is the first polynomial-time algorithm for the problem. RandGraph works by sequentially adding $m$ edges to an empty graph with n vertices. Recently, such sequential algorithms have been successful for random sampling problems. Our main contributions to this line of research includes introducing a new approach for sequentially approximating edge-specific probabilities at each step of the algorithm, and providing a new method for analyzing such algorithms.
We consider the general problem of finding the minimum weight $bm$-matching on arbitrary graphs. We prove that, whenever the linear programming (LP) relaxation of the problem has no fractional solutions, then the belief propagation (BP) algorithm con
verges to the correct solution. We also show that when the LP relaxation has a fractional solution then the BP algorithm can be used to solve the LP relaxation. Our proof is based on the notion of graph covers and extends the analysis of (Bayati-Shah-Sharma 2005 and Huang-Jebara 2007}. These results are notable in the following regards: (1) It is one of a very small number of proofs showing correctness of BP without any constraint on the graph structure. (2) Variants of the proof work for both synchronous and asynchronous BP; it is the first proof of convergence and correctness of an asynchronous BP algorithm for a combinatorial optimization problem.
Approximate message passing algorithms proved to be extremely effective in reconstructing sparse signals from a small number of incoherent linear measurements. Extensive numerical experiments further showed that their dynamics is accurately tracked b
y a simple one-dimensional iteration termed state evolution. In this paper we provide the first rigorous foundation to state evolution. We prove that indeed it holds asymptotically in the large system limit for sensing matrices with independent and identically distributed gaussian entries. While our focus is on message passing algorithms for compressed sensing, the analysis extends beyond this setting, to a general class of algorithms on dense graphs. In this context, state evolution plays the role that density evolution has for sparse graphs. The proof technique is fundamentally different from the standard approach to density evolution, in that it copes with large number of short loops in the underlying factor graph. It relies instead on a conditioning technique recently developed by Erwin Bolthausen in the context of spin glass theory.
We establish the existence of free energy limits for several combinatorial models on Erd{o}s-R{e}nyi graph $mathbb {G}(N,lfloor cNrfloor)$ and random $r$-regular graph $mathbb {G}(N,r)$. For a variety of models, including independent sets, MAX-CUT, c
oloring and K-SAT, we prove that the free energy both at a positive and zero temperature, appropriately rescaled, converges to a limit as the size of the underlying graph diverges to infinity. In the zero temperature case, this is interpreted as the existence of the scaling limit for the corresponding combinatorial optimization problem. For example, as a special case we prove that the size of a largest independent set in these graphs, normalized by the number of nodes converges to a limit w.h.p. This resolves an open problem which was proposed by Aldous (Some open problems) as one of his six favorite open problems. It was also mentioned as an open problem in several other places: Conjecture 2.20 in Wormald [In Surveys in Combinatorics, 1999 (Canterbury) (1999) 239-298 Cambridge Univ. Press]; Bollob{a}s and Riordan [Random Structures Algorithms 39 (2011) 1-38]; Janson and Thomason [Combin. Probab. Comput. 17 (2008) 259-264] and Aldous and Steele [In Probability on Discrete Structures (2004) 1-72 Springer].
