Multifractal Network Generator

We introduce a new approach to constructing networks with realistic features. Our method, in spite of its conceptual simplicity (it has only two parameters) is capable of generating a wide variety of network types with prescribed statistical properties, e.g., with degree- or clustering coefficient distributions of various, very different forms. In turn, these graphs can be used to test hypotheses, or, as models of actual data. The method is based on a mapping between suitably chosen singular measures defined on the unit square and sparse infinite networks. Such a mapping has the great potential of allowing for graph theoretical results for a variety of network topologies. The main idea of our approach is to go to the infinite limit of the singular measure and the size of the corresponding graph simultaneously. A very unique feature of this construction is that the complexity of the generated network is increasing with the size. We present analytic expressions derived from the parameters of the -- to be iterated-- initial generating measure for such major characteristics of graphs as their degree, clustering coefficient and assortativity coefficient distributions. The optimal parameters of the generating measure are determined from a simple simulated annealing process. Thus, the present work provides a tool for researchers from a variety of fields (such as biology, computer science, biology, or complex systems) enabling them to create a versatile model of their network data.Comment: Preprint. Final version appeared in PNAS

Dependence of ground state energy of classical n-vector spins on n

We study the ground state energy E_G(n) of N classical n-vector spins with the hamiltonian H = - \sum_{i>j} J_ij S_i.S_j where S_i and S_j are n-vectors and the coupling constants J_ij are arbitrary. We prove that E_G(n) is independent of n for all n > n_{max}(N) = floor((sqrt(8N+1)-1) / 2) . We show that this bound is the best possible. We also derive an upper bound for E_G(m) in terms of E_G(n), for m<n. We obtain an upper bound on the frustration in the system, as measured by F(n), which is defined to be (\sum_{i>j} |J_ij| + E_G(n)) / (\sum_{i>j} |J_ij|). We describe a procedure for constructing a set of J_ij's such that an arbitrary given state, {S_i}, is the ground state.Comment: 6 pages, 2 figures, submitted to Physical Review

Approximating the monomer-dimer constants through matrix permanent

The monomer-dimer model is fundamental in statistical mechanics. However, it is #P-complete in computation, even for two dimensional problems. A formulation in matrix permanent for the partition function of the monomer-dimer model is proposed in this paper, by transforming the number of all matchings of a bipartite graph into the number of perfect matchings of an extended bipartite graph, which can be given by a matrix permanent. Sequential importance sampling algorithm is applied to compute the permanents. For two-dimensional lattice with periodic condition, we obtain $0.6627\pm0.0002$, where the exact value is $h_2=0.662798972834$. For three-dimensional lattice with periodic condition, our numerical result is $0.7847\pm0.0014$, {which agrees with the best known bound $0.7653 \leq h_3 \leq 0.7862$.}Comment: 6 pages, 2 figure

Stressed backbone and elasticity of random central-force systems

We use a new algorithm to find the stress-carrying backbone of ``generic'' site-diluted triangular lattices of up to 10^6 sites. Generic lattices can be made by randomly displacing the sites of a regular lattice. The percolation threshold is Pc=0.6975 +/- 0.0003, the correlation length exponent \nu =1.16 +/- 0.03 and the fractal dimension of the backbone Db=1.78 +/- 0.02. The number of ``critical bonds'' (if you remove them rigidity is lost) on the backbone scales as L^{x}, with x=0.85 +/- 0.05. The Young's modulus is also calculated.Comment: 5 pages, 5 figures, uses epsfi

Stochastic interacting particle systems out of equilibrium

This paper provides an introduction to some stochastic models of lattice gases out of equilibrium and a discussion of results of various kinds obtained in recent years. Although these models are different in their microscopic features, a unified picture is emerging at the macroscopic level, applicable, in our view, to real phenomena where diffusion is the dominating physical mechanism. We rely mainly on an approach developed by the authors based on the study of dynamical large fluctuations in stationary states of open systems. The outcome of this approach is a theory connecting the non equilibrium thermodynamics to the transport coefficients via a variational principle. This leads ultimately to a functional derivative equation of Hamilton-Jacobi type for the non equilibrium free energy in which local thermodynamic variables are the independent arguments. In the first part of the paper we give a detailed introduction to the microscopic dynamics considered, while the second part, devoted to the macroscopic properties, illustrates many consequences of the Hamilton-Jacobi equation. In both parts several novelties are included.Comment: 36 page

The complexity of the Pk partition problem and related problems in bipartite graphs

International audienceIn this paper, we continue the investigation made in [MT05] about the approximability of Pk partition problems, but focusing here on their complexity. Precisely, we aim at designing the frontier between polynomial and NP-complete versions of the Pk partition problem in bipartite graphs, according to both the constant k and the maximum degree of the input graph. We actually extend the obtained results to more general classes of problems, namely, the minimum k-path partition problem and the maximum Pk packing problem. Moreover, we propose some simple approximation algorithms for those problems

The number of matchings in random graphs

We study matchings on sparse random graphs by means of the cavity method. We first show how the method reproduces several known results about maximum and perfect matchings in regular and Erdos-Renyi random graphs. Our main new result is the computation of the entropy, i.e. the leading order of the logarithm of the number of solutions, of matchings with a given size. We derive both an algorithm to compute this entropy for an arbitrary graph with a girth that diverges in the large size limit, and an analytic result for the entropy in regular and Erdos-Renyi random graph ensembles.Comment: 17 pages, 6 figures, to be published in Journal of Statistical Mechanic

The Role of Kemeny's Constant in Properties of Markov Chains

In a finite state irreducible Markov chain with stationary probabilities \pi_i and mean first passage times m_(ij) (mean recurrence time when i = j) it was first shown by Kemeny and Snell (1960) that \sum_j \pi_j m_(ij) is a constant K, not depending on i. This constant has since become known as Kemeny's constant. A variety of techniques for finding expressions and various bounds for K are derived. The main interpretation focuses on its role as the expected time to mixing in a Markov chain. Various applications are considered including perturbation results, mixing on directed graphs and its relation to the Kirchhoff index of regular graphs.Comment: 13 page