Equilibrium Ensemble Approach to Disordered Systems I: General Theory, Exact Results

An outline of Morita's equilibrium ensemble approach to disordered systems is given, and hitherto unnoticed relations to other, more conventional approaches in the theory of disordered systems are pointed out. It is demonstrated to constitute a generalization of the idea of grand ensembles and to be intimately related also to conventional low--concentration expansions as well as to perturbation expansions about ordered reference systems. Moreover, we draw attention to the variational content of the equilibrium ensemble formulation. A number of exact results are presented, among them general solutions for site-- and bond-- diluted systems in one dimension, both for uncorrelated, and for correlated disorder.Comment: 24 pages, Late

Contagion in an interacting economy

We investigate the credit risk model defined in Hatchett & K\"{u}hn under more general assumptions, in particular using a general degree distribution for sparse graphs. Expanding upon earlier results, we show that the model is exactly solvable in the $N\rightarrow \infty$ limit and demonstrate that the exact solution is described by the message-passing approach outlined by Karrer and Newman, generalized to include heterogeneous agents and couplings. We provide comparisons with simulations of graph ensembles with power-law degree distributions.Comment: 21 pages, 6 figure

Gibbs vs. Non-Gibbs in the Equilibrium Ensemble Approach to Disordered Systems

We describe the salient ideas of the equilibrium ensemble approach to disordered systems, paying due attention to the appearance of non-Gibbsian measures. A canonical scheme of approximations – constrained annealing – is described and characterised in terms of a Gibbs ’ variational principle for the free energy functional. It provides a family of increasing exact lower bounds of the quenched free energy of disordered systems, and is shown to avoid the use of non-Gibbsian measures. The connection between the equilibrium ensemble approach and conventional lowconcentration expansions or perturbation expansions about ordered reference systems is also explained. Finally applications of the scheme to a number of disordered Ising models and to protein folding are briefly reviewed. AMS subject classification: 82B05 Classical equilibrium statistical mechanics 82B44 Disordered systems Key words: Morita method, disordered systems, variational bounds, non-Gibbsiannes

Spectral properties of the trap model on sparse networks

One of the simplest models for the slow relaxation and aging of glasses is the trap model by Bouchaud and others, which represents a system as a point in configuration-space hopping between local energy minima. The time evolution depends on the transition rates and the network of allowed jumps between the minima. We consider the case of sparse configuration-space connectivity given by a random graph, and study the spectral properties of the resulting master operator. We develop a general approach using the cavity method that gives access to the density of states in large systems, as well as localisation properties of the eigenvectors, which are important for the dynamics. We illustrate how, for a system with sparse connectivity and finite temperature, the density of states and the average inverse participation ratio have attributes that arise from a non-trivial combination of the corresponding mean field (fully connected) and random walk (infinite temperature) limits. In particular, we find a range of eigenvalues for which the density of states is of mean-field form but localisation properties are not, and speculate that the corresponding eigenvectors may be concentrated on extensively many clusters of network sites.Comment: 41 pages, 15 figure

Distance distribution in configuration model networks

We present analytical results for the distribution of shortest path lengths between random pairs of nodes in configuration model networks. The results, which are based on recursion equations, are shown to be in good agreement with numerical simulations for networks with degenerate, binomial and power-law degree distributions. The mean, mode and variance of the distribution of shortest path lengths are also evaluated. These results provide expressions for central measures and dispersion measures of the distribution of shortest path lengths in terms of moments of the degree distribution, illuminating the connection between the two distributions.Comment: 28 pages, 7 figures. Accepted for publication in Phys. Rev.

Statistical analysis of articulation points in configuration model networks

An articulation point (AP) in a network is a node whose deletion would split the network component on which it resides into two or more components. APs are vulnerable spots that play an important role in network collapse processes, which may result from node failures, attacks or epidemics. Therefore, the abundance and properties of APs affect the resilience of the network to these collapse scenarios. We present analytical results for the statistical properties of APs in configuration model networks. In order to quantify their abundance, we calculate the probability $P(i \in {\rm AP})$, that a random node, i, in a configuration model network with P(K=k), is an AP. We also obtain the conditional probability $P(i \in {\rm AP}|k)$ that a random node of degree k is an AP, and find that high degree nodes are more likely to be APs than low degree nodes. Using Bayes' theorem, we obtain the conditional degree distribution, $P(K=k|{\rm AP})$, over the set of APs and compare it to P(K=k). We propose a new centrality measure based on APs: each node can be characterized by its articulation rank, r, which is the number of components that would be added to the network upon deletion of that node. For nodes which are not APs the articulation rank is $r=0$, while for APs $r \ge 1$. We obtain a closed form expression for the distribution of articulation ranks, P(R=r). Configuration model networks often exhibit a coexistence between a giant component and finite components. To examine the distinct properties of APs on the giant and on the finite components, we calculate the probabilities presented above separately for the giant and the finite components. We apply these results to ensembles of configuration model networks with a Poisson, exponential and power-law degree distributions. The implications of these results are discussed in the context of common attack scenarios and network dismantling processes.Comment: 53 pages, 16 figures. arXiv admin note: text overlap with arXiv:1804.0333