We study the satisfiability of randomly generated formulas formed by M clauses of exactly K literals over N Boolean variables. For a given value of N the problem is known to be most difficult when α = M/N is close to the experimental threshold αc separating the region where almost all formulas are SAT from the region where all formulas are UNSAT. Recent results from a statistical physics analysis suggest that the difficulty is related to the existence of a clustering phenomenon of the solutions when α is close to (but smaller than) αc. We introduce a new type of message passing algorithm which allows to find efficiently a satisfying assignment of the variables in this difficult region. This algorithm is iterative and composed of two main parts. The first is a message‐passing procedure which generalizes the usual methods like Sum‐Product or Belief Propagation: It passes messages that may be thought of as surveys over clusters of the ordinary messages. The second part uses the detailed probabilistic information obtained from the surveys in order to fix variables and simplify the problem. Eventually, the simplified problem that remains is solved by a conventional heuristic. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2005

We show that a message-passing process allows us to store in binary "material" synapses a number of random patterns which almost saturate the information theoretic bounds. We apply the learning algorithm to networks characterized by a wide range of different connection topologies and of size comparable with that of biological systems (e.g., [EQUATION: SEE TEXT]). The algorithm can be turned into an online-fault tolerant-learning protocol of potential interest in modeling aspects of synaptic plasticity and in building neuromorphic devices.

We study the network dismantling problem, which consists of determining a minimal set of vertices in which removal leaves the network broken into connected components of subextensive size. For a large class of random graphs, this problem is tightly connected to the decycling problem (the removal of vertices, leaving the graph acyclic). Exploiting this connection and recent works on epidemic spreading, we present precise predictions for the minimal size of a dismantling set in a large random graph with a prescribed (lighttailed) degree distribution. Building on the statistical mechanics perspective, we propose a three-stage Min-Sum algorithm for efficiently dismantling networks, including heavy-tailed ones for which the dismantling and decycling problems are not equivalent. We also provide additional insights into the dismantling problem, concluding that it is an intrinsically collective problem and that optimal dismantling sets cannot be viewed as a collection of individually well-performing nodes.graph fragmentation | message passing | percolation | random graphs | influence maximization A network (a graph G in the discrete mathematics language) is a set V of N entities called nodes (or vertices), along with a set E of edges connecting some pairs of nodes. In a simplified way, networks are used to describe numerous systems in very diverse fields, ranging from social sciences to information technology or biological systems (reviews are in refs. 1 and 2). Several crucial questions in the context of network studies concern the modifications of the properties of a graph when a subset S of its nodes is selected and treated in a specific way. For instance, how much does the size of the largest connected component of the graph decrease if the vertices in S (along with their adjacent edges) are removed? Do the cycles survive this removal? What is the outcome of the epidemic spreading if the vertices in S are initially contaminated, constituting the seed of the epidemic? On the contrary, what is the influence of a vaccination of nodes in S preventing them from transmitting the epidemic? It is relatively easy to answer these questions when the set S is chosen randomly, with each vertex being selected with some probability independently. Classical percolation theory is nothing but the study of the connected components of a graph in which some vertices have been removed in this way.A much more interesting case is when the set S can be chosen in some optimal way. Indeed, in all applications sketched above, it is reasonable to assign some cost to the inclusion of a vertex in S: vaccination has a socioeconomic price, incentives must be paid to customers to convince them to adopt a new product in a viral marketing campaign, and incapacitating a computer during a cyber attack requires resources. Thus, one faces a combinatorial optimization problem: the minimization of the cost of S under a constraint on its effect on the graph. These problems thus exhibit both static and dynamic features, the former referring to the combinatori...