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Register a new userThe entropy of randomized network ensembles
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Ginestra Bianconi
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Randomized network ensembles are the null models of real networks and are extensivelly used to compare a real system to a null hypothesis. In this paper we study network ensembles with the same degree distribution, the same degree-correlations or the same community structure of any given real network. We characterize these randomized network ensembles by their entropy, i.e. the normalized logarithm of the total number of networks which are part of these ensembles. We estimate the entropy of randomized ensembles starting from a large set of real directed and undirected networks. We propose entropy as an indicator to assess the role of each structural feature in a given real network.We observe that the ensembles with fixed scale-free degree distribution have smaller entropy than the ensembles with homogeneous degree distribution indicating a higher level of order in scale-free networks.
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In this paper we generalize the concept of random networks to describe networks with non trivial features by a statistical mechanics approach. This framework is able to describe ensembles of undirected, directed as well as weighted networks. These networks might have not trivial community structure or, in the case of networks embedded in a given space, non trivial distance dependence of the link probability. These ensembles are characterized by their entropy which evaluate the cardinality of networks in the ensemble. The general framework we present in this paper is able to describe microcanonical ensemble of networks as well as canonical or hidden variables network ensemble with significant implication for the formulation of network constructing algorithms. Moreover in the paper we define and and characterize in particular the structural entropy, i.e. the entropy of the ensembles of undirected uncorrelated simple networks with given degree sequence. We discuss the apparent paradox that scale-free degree distribution are characterized by having small structural entropy but are so widely encountered in natural, social and technological complex systems. We give the proof that while scale-free networks ensembles have small structural entropy, they also correspond to the most likely degree distribution with the corresponding value of the structural entropy.
Ground state entropy of the network source location problem is evaluated at both the replica symmetric level and one-step replica symmetry breaking level using the entropic cavity method. The regime that is a focus of this study, is closely related to the vertex cover problem with randomly quenched covered nodes. The resulting entropic message passing inspired decimation and reinforcement algorithms are used to identify the optimal location of sources in single instances of transportation networks. The conventional belief propagation without taking the entropic effect into account is also compared. We find that in the glassy phase the entropic message passing inspired decimation yields a lower ground state energy compared to the belief propagation without taking the entropic effect. Using the extremal optimization algorithm, we study the ground state energy and the fraction of frozen hubs, and extend the algorithm to collect statistics of the entropy. The theoretical results are compared with the extremal optimization results.
Theory of Random Matrix Ensembles have proven to be a useful tool in the study of the statistical distribution of energy or transmission levels of a wide variety of physical systems. We give an overview of certain q-generalizations of the Random Matrix Ensembles, which were first introduced in connection with the statistical description of disordered quantum conductors.
The fidelity susceptibility measures sensitivity of eigenstates to a change of an external parameter. It has been fruitfully used to pin down quantum phase transitions when applied to ground states (with extensions to thermal states). Here we propose to use the fidelity susceptibility as a useful dimensionless measure for complex quantum systems. We find analytically the fidelity susceptibility distributions for Gaussian orthogonal and unitary universality classes for arbitrary system size. The results are verified by a comparison with numerical data.
We present a simple, perturbative approach for calculating spectral densities for random matrix ensembles in the thermodynamic limit we call the Perturbative Resolvent Method (PRM). The PRM is based on constructing a linear system of equations and calculating how the solutions to these equation change in response to a small perturbation using the zero-temperature cavity method. We illustrate the power of the method by providing simple analytic derivations of the Wigner Semi-circle Law for symmetric matrices, the Marchenko-Pastur Law for Wishart matrices, the spectral density for a product Wishart matrix composed of two square matrices, and the Circle and elliptic laws for real random matrices.
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