Starting from a general ansatz, we show how community detection can be interpreted as finding the ground state of an infinite range spin glass. Our approach applies to weighted and directed networks alike. It contains the at hoc introduced quality function from [1] and the modularity Q as defined by Newman and Girvan [2] as special cases. The community structure of the network is interpreted as the spin configuration that minimizes the energy of the spin glass with the spin states being the community indices. We elucidate the properties of the ground state configuration to give a concise definition of communities as cohesive subgroups in networks that is adaptive to the specific class of network under study. Further we show, how hierarchies and overlap in the community structure can be detected. Computationally effective local update rules for optimization procedures to find the ground state are given. We show how the ansatz may be used to discover the community around a given node without detecting all communities in the full network and we give benchmarks for the performance of this extension. Finally, we give expectation values for the modularity of random graphs, which can be used in the assessment of statistical significance of community structure.

We study the topology of e-mail networks with e-mail addresses as nodes and e-mails as links using data from server log files. The resulting network exhibits a scale-free link distribution and pronounced small-world behavior, as observed in other social networks. These observations imply that the spreading of e-mail viruses is greatly facilitated in real e-mail networks compared to random architectures.Comment: 4 pages RevTeX, 4 figures PostScript (extended version

A fast community detection algorithm based on a q-state Potts model is presented. Communities (groups of densely interconnected nodes that are only loosely connected to the rest of the network) are found to coincide with the domains of equal spin value in the minima of a modified Potts spin glass Hamiltonian. Comparing global and local minima of the Hamiltonian allows for the detection of overlapping ("fuzzy") communities and quantifying the association of nodes with multiple communities as well as the robustness of a community. No prior knowledge of the number of communities has to be assumed.

A Boolean network model of the cell-cycle regulatory network of fission yeast (Schizosaccharomyces Pombe) is constructed solely on the basis of the known biochemical interaction topology. Simulating the model in the computer faithfully reproduces the known activity sequence of regulatory proteins along the cell cycle of the living cell. Contrary to existing differential equation models, no parameters enter the model except the structure of the regulatory circuitry. The dynamical properties of the model indicate that the biological dynamical sequence is robustly implemented in the regulatory network, with the biological stationary state G1 corresponding to the dominant attractor in state space, and with the biological regulatory sequence being a strongly attractive trajectory. Comparing the fission yeast cell-cycle model to a similar model of the corresponding network in S. cerevisiae, a remarkable difference in circuitry, as well as dynamics is observed. While the latter operates in a strongly damped mode, driven by external excitation, the S. pombe network represents an auto-excited system with external damping.

* published as Phys. Rev. Lett. 84 (2000) 6114.We evolve network topology of an asymmetrically connected threshold network by a simple local rewiring rule: quiet nodes grow links, active nodes lose links. This leads to convergence of the average connectivity of the network towards the critical value Kc = 2 in the limit of large system size N . How this principle could generate self-organization in natural complex systems is discussed for two examples: neural networks and regulatory networks in the genome.PACS numbers: 05.65.+b, 64.60.Cn, 87.16.Yc, Networks of many interacting units occur in diverse areas as, for example, gene regulation, neural networks, food webs in ecology, species relationships in biological evolution, economic interactions, and the organization of the internet. For studying statistical mechanics properties of such complex systems, discrete dynamical networks provide a simple testbed for effects of globally interacting information transfer in network structures.One example is the threshold network with sparse asymmetric connections. Networks of this kind were first studied as diluted, non-symmetric spin glasses [1] and diluted, asymmetric neural networks [2,3]. For the study of topological questions in networks, a version with discrete connections c ij = ±1 is convenient and will be considered here. It is a subset of Boolean networks [4,5] with similar dynamical properties. Random realizations of these networks exhibit complex non-Hamiltonian dynamics including transients and limit cycles [6,7]. In particular, a phase transition is observed at a critical average connectivity K c with lengths of transients and attractors (limit cycles) diverging exponentially with system size for an average connectivity larger than K c . A theoretical analysis is limited by the non-Hamiltonian character of the asymmetric interactions, such that standard tools of statistical mechanics do not apply [1]. However, combinatorial as well as numerical methods provide a quite detailed picture about their dynamical properties and correspondence with Boolean Networks [6][7][8][9][10][11][12][13][14][15].While basic dynamical properties of interaction networks with fixed architecture have been studied with such models, the origin of specific structural properties of networks in natural systems is often unknown. For example, the observed average connectivity in a nervous structure or in a biological genome is hard to explain in a framework of networks with a static architecture. For the case of regulation networks in the genome, Kauffman postulated that gene regulatory networks may exhibit properties of dynamical networks near criticality [4,16]. However, this postulate does not provide a mechanism able to generate an average connectivity near the critical point. An interesting question is whether connectivity may be driven towards a critical point by some dynamical mechanism. In the following we will sketch such an approach in a setting of an explicit evolution of the connectivity of networks.Network models of evolving ...