Belief Propagation-guided Decimation Research Articles

The Suppression of Epidemic Spreading Through Minimum Dominating Set

COVID-19 has infected millions of people, with deaths in more than 200 countries. It is therefore essential to understand the dynamic characteristics of the outbreak and to design effective strategies to restrain the large-scale spread of the epidemic. In this paper, we present a novel framework to depress the epidemic spreading, by leveraging the decentralized dissemination of information. The framework is equivalent to finding a special minimum dominating set for a duplex network which is a general dominating set for one layer and a connected dominating set for another layer. Using the spin glass and message passing theory, we present a belief-propagation-guided decimation (BPD) algorithm to construct the special minimum dominating set. As a consequence, we could immediately recognize the epidemic as soon as it appeared, and rapidly immunize the whole network at minimum cost.

Dismantling and Vertex Cover of Network Through Message Passing

The dismantling problem and minimum vertex cover (MVC) problem of network are two fundamental NP-hard problems where the former aims to find a minimal subset of nodes whose removal leaves the network broken in small components of sub-extensive size and the latter is also to find a minimal vertex set which contains at least one incident vertex of every edge. Both of them have a wide range of applications related to circuit design, communication, network security, transportation, epidemic control, molecular biology and economics. In this brief, we first propose a novel belief-propagation equation for the spin glass model of the MVC problem. A belief-propagation-guided decimation (BPD) algorithm is then presented which could construct approximate optimal vertex cover of network. In addition, based on the relationship between the network dismantling problem and the MVC problem, we propose a simple and fast algorithm formed by the MVC and a general node reinsertion strategy, for solving the network dismantling problem. The effectiveness of the proposed algorithms is finally verified by comparing with other well-known algorithms on a number of real world networks.

Efficient Network Dismantling via Node Explosive Percolation**Supported by the 2016 Basic Operational Outlays for the Research Activities of Centric University, Civil Aviation University of China under Grant No. 3122016L010, and the National Natural Science Foundation of China under Grant No. 11705279, and also supported by the Zhejiang Provincial Natural Science Foundation of China under Grant No. LR16A050001 and the National

The network dismantling problem asks the minimum separate node set of a graph whose removal will break the graph into connected components with the size not larger than the one percentage of the original graph. This problem has attracted much attention recently and a lot of algorithms have been proposed. However, most of the network dismantling algorithms mainly focus on which nodes are included in the minimum separate set but overlook how to order them for removal, which will lead to low general efficiency during the dismantling process. In this paper, we reformulate the network dismantling problem by taking the order of nodes’ removal into consideration. An efficient dismantling sequence will break the network quickly during the dismantling processes. We take the belief-propagation guided decimation (BPD) dismantling algorithm, a state-of-the-art algorithm, as an example, and employ the node explosive percolation (NEP) algorithm to reorder the early part of the dismantling sequence given by the BPD. The proposed method is denoted as the NEP-BPD algorithm (NBA) here. The numerical results on Erdös-Rényi graphs, random-regular graphs, scale-free graphs, and some real networks show the high general efficiency of NBA during the entire dismantling process. In addition, numerical computations on random graph ensembles with the size from 210 to 219 exhibit that the NBA is in the same complexity class with the BPD algorithm. It is clear that the NEP method we used to improve the general efficiency could also be applied to other dismantling algorithms, such as Min-Sum algorithm, equal graph partitioning algorithm and so on.

Encoding of Spatially Coupled LDGM Codes for Lossy Source Compression

It has been shown that a class of spatially coupled low-density generator-matrix (SC-LDGM) code ensembles displays distortion saturation for the lossy binary symmetric source coding problem with the belief propagation guided decimation (BPGD) algorithm, i.e., the BPGD distortion approaches the optimal expected distortion of the underlying ensemble asymptotically in code length. We investigate the distortion performance of a practical class of protograph-based SC-LDGM code ensembles and demonstrate distortion saturation numerically. Moreover, taking advantage of the convolutional structure of the SC-LDGM codes, we propose an efficient windowed encoding (WE) algorithm with two decimation techniques for lowering the WE complexity that maintain distortion performance close to the rate-distortion bound.

Identifying optimal targets of network attack by belief propagation

For a network formed by nodes and undirected links between pairs of nodes, the network optimal attack problem aims at deleting a minimum number of target nodes to break the network down into many small components. This problem is intrinsically related to the feedback vertex set problem that was successfully tackled by spin-glass theory and an associated belief propagation-guided decimation (BPD) algorithm [Zhou, Eur. Phys. J. B 86, 455 (2013)EPJBFY1434-602810.1140/epjb/e2013-40690-1]. In the present work we apply the BPD algorithm (which has approximately linear time complexity) to the network optimal attack problem and demonstrate that it has much better performance than a recently proposed collective information algorithm [Morone and Makse, Nature 524, 65 (2015)NATUAS0028-083610.1038/nature14604] for different types of random networks and real-world network instances. The BPD-guided attack scheme often induces an abrupt collapse of the whole network, which may make it very difficult to defend.

A spin glass approach to the directed feedback vertex set problem

A directed graph (digraph) is formed by vertices and arcs (directed edges) from one vertex to another. A feedback vertex set (FVS) is a set of vertices that contains at least one vertex of every directed cycle in this digraph. The directed feedback vertex set problem aims at constructing a FVS of minimum cardinality. This is a fundamental cycle-constrained hard combinatorial optimization problem with wide practical applications. In this paper we construct a spin glass model for the directed FVS problem by converting the global cycle constraints into local arc constraints, and study this model through the replica-symmetric (RS) mean field theory of statistical physics. We then implement a belief propagation-guided decimation (BPD) algorithm for single digraph instances. The BPD algorithm slightly outperforms the simulated annealing algorithm on large random graph instances. The RS mean field results and algorithmic results can be further improved by working on a more restrictive (and more difficult) spin glass model.

Solving the undirected feedback vertex set problem by local search

An undirected graph consists of a set of vertices and a set of undirected edges between vertices. Such a graph may contain an abundant number of cycles, in which case a feedback vertex set (FVS) is a set of vertices intersecting with each of these cycles. Constructing a FVS of cardinality approaching the global minimum value is an optimization problem in the nondeterministic polynomial-complete complexity class, and therefore it might be extremely difficult for some large graph instances. In this paper we develop a simulated annealing local search algorithm for the undirected FVS problem by adapting the heuristic procedure of Galinier et al. [P. Galinier, E. Lemamou, M.W. Bouzidi, J. Heuristics 19, 797 (2013)], which worked for the directed FVS problem. By defining an order for the vertices outside the FVS, we replace the global cycle constraints by a set of local vertex constraints on this order. Under these local constraints the cardinality of the focal FVS is then gradually reduced by the simulated annealing dynamical process. We test this heuristic algorithm on large instances of Erdös-Rényi random graph and regular random graph, and find that this algorithm is comparable in performance to the belief propagation-guided decimation algorithm.

Spin glass approach to the feedback vertex set problem

A feedback vertex set (FVS) of an undirected graph is a set of vertices that contains at least one vertex of each cycle of the graph. The feedback vertex set problem consists of constructing a FVS of size less than a certain given value. This combinatorial optimization problem has many practical applications, but it is in the nondeterministic polynomial-complete class of worst-case computational complexity. In this paper we define a spin glass model for the FVS problem and then study this model on the ensemble of finite-connectivity random graphs. In our model the global cycle constraints are represented through the local constraints on all the edges of the graph, and they are then treated by distributed message-passing procedures such as belief propagation. Our belief propagation-guided decimation algorithm can construct nearly optimal feedback vertex sets for single random graph instances and regular lattices. We also design a spin glass model for the FVS problem on a directed graph. Our work will be very useful for identifying the set of vertices that contribute most significantly to the dynamical complexity of a large networked system.