Given a complex biological or social network, how many clusters should it be decomposed into? We define the distance di,j from node i to node j as the average number of steps a Brownian particle takes to reach j from i. Node j is a global attractor of i if di,j ≤ d i,k for any k of the graph; it is a local attractor of i, if j ∈ Ei (the set of nearest-neighbors of i) and di,j ≤ d i,l for any l ∈ Ei. Based on the intuition that each node should have a high probability to be in the same community as its global (local) attractor on the global (local) scale, we present a simple method to uncover a network's community structure. This method is applied to several real networks and some discussion on its possible extensions is made.PACS numbers: 89.75.-k,89.20.-a,87.10.+e A complex networked system, such as an organism's metabolic network and genetic interaction network, is composed of a large number of interacting agents. The complexity of such systems originates partly from the heterogeneity in their interaction patterns, aspects of which include the small-world [1] and the scale-free properties [2,3] observed in many social, biological, and technological networks [4,5,6]. Given this high degree of complexity, it is necessary to divide a network into different subgroups to facilitate the understanding of the relationships among different components [7,8].A complex network could be represented by a graph. Each component of the network is mapped to a vertex (node), and the interaction between two components is signified by an edge between the two corresponding nodes, whose weight is related to the interaction strength. The challenge is to dissect this graph based on its connection pattern. We know that to partition a graph into two equally sized subgroups such that the number of edges in between reaches the absolute minimum is already a NP-complete problem, a solution is not guaranteed to be found easily; however it is still a well-defined question. On the other hand, the question "How many subgroups should a graph be divided into and how?" is ill-posed, as we do not have an objective function to optimize; and we have to rely on heuristic reasoning to proceed.If we are interested in identifying just one community that is associated with a specified node, the maximum flow method [9] turns out to be efficient. Recently, it is applied to identifying communities of Internet webpages [10]. An community thus uncovered is usually very small; and for this method to work well one needs a priori knowledge of the network to select the source and sink nodes properly. Another elegant method is based on the concept of edge betweenness [11]. The degree of betweenness of a edge is defined as the total number of shortest paths between pair of nodes which pass through it. By removing recursively the current edge with the highest degree of betweenness, one expects the connectivity of the network to decrease the most efficiently and minimal cutting operations is needed to separate the network into subgroups [7]. This idea of Girvan and...

Abstract. 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.

How humans make decisions in non-cooperative strategic interactions is a big question. For the fundamental Rock-Paper-Scissors (RPS) model game system, classic Nash equilibrium (NE) theory predicts that players randomize completely their action choices to avoid being exploited, while evolutionary game theory of bounded rationality in general predicts persistent cyclic motions, especially in finite populations. However as empirical studies have been relatively sparse, it is still a controversial issue as to which theoretical framework is more appropriate to describe decision-making of human subjects. Here we observe population-level persistent cyclic motions in a laboratory experiment of the discrete-time iterated RPS game under the traditional random pairwise-matching protocol. This collective behavior contradicts with the NE theory but is quantitatively explained, without any adjustable parameter, by a microscopic model of win-lose-tie conditional response. Theoretical calculations suggest that if all players adopt the same optimized conditional response strategy, their accumulated payoff will be much higher than the reference value of the NE mixed strategy. Our work demonstrates the feasibility of understanding human competition behaviors from the angle of non-equilibrium statistical physics.