We compute spectra of symmetric random matrices describing graphs with general modular structure and arbitrary inter-and intra-module degree distributions, subject only to the constraint of finite mean connectivities. We also evaluate spectra of a certain class of small-world matrices generated from random graphs by introducing short-cuts via additional random connectivity components. Both adjacency matrices and the associated graph Laplacians are investigated. For the Laplacians, we find Lifshitz type singular behaviour of the spectral density in a localised region of small |λ| values. In the case of modular networks, we can identify contributions local densities of state from individual modules. For small-world networks, we find that the introduction of short cuts can lead to the creation of satellite bands outside the central band of extended states, exhibiting only localised states in the band-gaps. Results for the ensemble in the thermodynamic limit are in excellent agreement with those obtained via a cavity approach for large finite single instances, and with direct diagonalisation results.
The problem of vertex coloring in random graphs is studied using methods of statistical physics and probability. Our analytical results are compared to those obtained by exact enumeration and Monte-Carlo simulations. We critically discuss the merits and shortcomings of the various methods, and interpret the results obtained. We present an exact analytical expression for the 2-coloring problem as well as general replica symmetric approximated solutions for the thermodynamics of the graph coloring problem with p colors and K-body edges. 05.50.+q, 75.10.Nr, 02.60.Pn I. INTRODUCTIONMethods of statistical physics have recently been applied to a variety of complex optimization problems in a broad range of areas, from computational complexity [1,2] to the study of error correcting codes [3] and cryptography [4,5].Graph coloring is one of the basic Non-deterministically Polynomial (NP) problems. The task is to assign one of p-colors to each node, in a randomly connected set of vertices, such that no edge will have the same colors assigned to both ends. The feasibility of finding such a solution clearly depends on the level and nature of connectivity in the graph and the number of colors. The very existence of a solution is in the class of NP-complete problems [6]. An extension of the problem to the case of hyper-edges comprising more than two vertices is also of practical significance [7].Recent success in the application of statistical physics techniques to computational complexity problems, naturally lead to the belief that they may be applied to a wide range of computational complexity tasks, among them is the graph coloring problem.In this paper we map the graph coloring problem, with p-colors, onto the anti-ferromagnetic p-spin Potts model [8]; this facilitates the use of established methods of statistical physics for gaining insight into the dependence of graph colorability on the nature and level of its connectivity, and the phase transitions that take place. The suggested framework comes with its own limitations; we critically discuss what can, and cannot be calculated via the methods of statistical mechanics.The statistical physics approach is based on the introduction of a Hamiltonian or cost-function, and the calculation of the typical free energy in the large system limit. From the free energy one can obtain the typical ground state energy, which in turn allows one to make predictions on the graph colorability. A non-zero ground state energy indicates that, under the given conditions, random graphs are typically not colorable. Our theoretical results are restricted to the replica symmetric (RS) approximation (see [10,11]), and are, for the 2-color problem (which is solvable in linear time) in perfect agreement with those obtained by numerical methods; for the 3-color problem the results are only in qualitative agreement with those obtained by numerical methods. The theoretical results can be systematically improved by using replica symmetry breaking (RSB) approximations, although our current results do n...
We introduce models of heterogeneous systems with finite connectivity defined on random graphs to capture finite-coordination effects on the low-temperature behavior of finite dimensional systems. Our models use a description in terms of small deviations of particle coordinates from a set of reference positions, particularly appropriate for the description of low-temperature phenomena. A Born-von-Karman type expansion with random coefficients is used to model effects of frozen heterogeneities. The key quantity appearing in the theoretical description is a full distribution of effective single-site potentials which needs to be determined self-consistently. If microscopic interactions are harmonic, the effective single-site potentials turn out to be harmonic as well, and the distribution of these single-site potentials is equivalent to a distribution of localization lengths used earlier in the description of chemical gels. For structural glasses characterized by frustration and anharmonicities in the microscopic interactions, the distribution of single-site potentials involves anharmonicities of all orders, and both single-well and double well potentials are observed, the latter with a broad spectrum of barrier heights. The appearance of glassy phases at low temperatures is marked by the appearance of asymmetries in the distribution of single-site potentials, as previously observed for fully connected systems. Double-well potentials with a broad spectrum of barrier heights and asymmetries would give rise to the well known universal glassy low temperature anomalies when quantum effects are taken into account.
Abstract. We present a theoretical method for a direct evaluation of the average and reliability error exponents in low-density parity-check error-correcting codes using methods of statistical physics. Results for the binary symmetric channel (BSC) are presented for codes of both finite and infinite connectivity.
We propose a novel method for the determination of the effective interaction potential between the amino acids of a protein. The strategy is based on the combination of a new optimization procedure and a geometrical argument, which also uncovers the shortcomings of any optimization scheme. The strategy can be applied on any data set of native structures such as those available from the Protein Data Bank (PDB). In this work, however, we explain and test our approach on simple lattice models, where the true interactions are known a priori. Excellent agreement is obtained between the extracted and the true potentials even for modest numbers of protein structures in the PDB. Comparisons with other methods are also discussed.
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