July 15, 2019
We present an analytical approach for describing spectrally constrained maximum entropy ensembles of finitely connected regular loopy graphs, valid in the regime of weak loop-loop interactions. We derive an expression for the leading two orders of the expected eigenvalue spectrum, through the use of infinitely many replica indices taking imaginary values. We apply the method to models in which the spectral constraint reduces to a soft constraint on the number of triangles, which exhibit `shattering' transitions to phases with extensively many disconnected cliques, to models with controlled numbers of triangles and squares, and to models where the spectral constraint reduces to a count of the number of adjacency matrix eigenvalues in a given interval. Our predictions are supported by MCMC simulations based on edge swaps with nontrivial acceptance probabilities.
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January 29, 2016
I report on the development of a novel statistical mechanical formalism for the analysis of random graphs with many short loops, and processes on such graphs. The graphs are defined via maximum entropy ensembles, in which both the degrees (via hard constraints) and the adjacency matrix spectrum (via a soft constraint) are prescribed. The sum over graphs can be done analytically, using a replica formalism with complex replica dimensions. All known results for tree-like graphs ...
August 25, 2020
We analyze maximum entropy random graph ensembles with constrained degrees, drawn from arbitrary degree distributions, and a tuneable number of 3-loops (triangles). We find that such ensembles generally exhibit two transitions, a clustering and a shattering transition, separating three distinct regimes. At the clustering transition, the graphs change from typically having only isolated loops to forming loop clusters. At the shattering transition the graphs break up into exten...
October 19, 2009
The spectral density of random graphs with topological constraints is analysed using the replica method. We consider graph ensembles featuring generalised degree-degree correlations, as well as those with a community structure. In each case an exact solution is found for the spectral density in the form of consistency equations depending on the statistical properties of the graph ensemble in question. We highlight the effect of these topological constraints on the resulting s...
October 12, 2013
The asymptotic behaviour of dynamical processes in networks can be expressed as a function of spectral properties of the corresponding adjacency and Laplacian matrices. Although many theoretical results are known for the spectra of traditional configuration models, networks generated through these models fail to describe many topological features of real-world networks, in particular non-null values of the clustering coefficient. Here we study effects of cycles of order three...
June 7, 2012
We present a general method for obtaining the spectra of large graphs with short cycles using ideas from statistical mechanics of disordered systems. This approach leads to an algorithm that determines the spectra of graphs up to a high accuracy. In particular, for (un)directed regular graphs with cycles of arbitrary length we derive exact and simple equations for the resolvent of the associated adjacency matrix. Solving these equations we obtain analytical formulas for the s...
March 11, 2014
We develop a thorough analytical study of the $O(1/N)$ correction to the spectrum of regular random graphs with $N \rightarrow \infty$ nodes. The finite size fluctuations of the resolvent are given in terms of a weighted series over the contributions coming from loops of all possible lengths, from which we obtain the isolated eigenvalue as well as an analytical expression for the $O(1/N)$ correction to the continuous part of the spectrum. The comparison between this analytica...
July 20, 2011
We derive exact equations that determine the spectra of undirected and directed sparsely connected regular graphs containing loops of arbitrary length. The implications of our results to the structural and dynamical properties of networks are discussed by showing how loops influence the size of the spectral gap and the propensity for synchronization. Analytical formulas for the spectrum are obtained for specific length of the loops.
April 10, 2014
Networks observed in the real world often have many short loops. This violates the tree-like assumption that underpins the majority of random graph models and most of the methods used for their analysis. In this paper we sketch possible research routes to be explored in order to make progress on networks with many short loops, involving old and new random graph models and ideas for novel mathematical methods. We do not present conclusive solutions of problems, but aim to enco...
August 6, 2012
We study random graphs with arbitrary distributions of expected degree and derive expressions for the spectra of their adjacency and modularity matrices. We give a complete prescription for calculating the spectra that is exact in the limit of large network size and large vertex degrees. We also study the effect on the spectra of hubs in the network, vertices of unusually high degree, and show that these produce isolated eigenvalues outside the main spectral band, akin to imp...
April 10, 2008
We derive the spectral properties of adjacency matrix of complex networks and of their Laplacian by the replica method combined with a dynamical population algorithm. By assuming the order parameter to be a product of Gaussian distributions, the present theory provides a solution for the non linear integral equations for the spectra density in random matrix theory of the spectra of sparse random matrices making a step forward with respect to the effective medium approximation...