Entropy of microcanonical finite-graph ensembles

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 th...

We consider the set of all graphs on n labeled vertices with prescribed degrees D=(d_1, ..., d_n). For a wide class of tame degree sequences D we prove a computationally efficient asymptotic formula approximating the number of graphs within a relative error which approaches 0 as n grows. As a corollary, we prove that the structure of a random graph with a given tame degree sequence D is well described by a certain maximum entropy matrix computed from D. We also establish an a...

Generalised degrees provide a natural bridge between local and global topological properties of networks. We define the generalised degree to be the number of neighbours of a node within one and two steps respectively. Tailored random graph ensembles are used to quantify and compare topological properties of networks in a systematic and precise manner, using concepts from information theory. We calculate the Shannon entropy of random graph ensembles constrained with a specifi...

We calculate explicit formulae for the Shannon entropies of several families of tailored random graph ensembles for which no such formulae were as yet available, in leading orders in the system size. These include bipartite graph ensembles with imposed (and possibly distinct) degree distributions for the two node sets, graph ensembles constrained by specified node neighbourhood distributions, and graph ensembles constrained by specified generalised degree distributions.

For a random graph subject to a topological constraint, the microcanonical ensemble requires the constraint to be met by every realisation of the graph (`hard constraint'), while the canonical ensemble requires the constraint to be met only on average (`soft constraint'). It is known that breaking of ensemble equivalence may occur when the size of the graph tends to infinity, signalled by a non-zero specific relative entropy of the two ensembles. In this paper we analyse a fo...

The Gibbs entropy of a microcanonical network ensemble is the logarithm of the number of network configurations compatible with a set of hard constraints. This quantity characterizes the level of order and randomness encoded in features of a given real network. Here we show how to relate this entropy to large deviations of conjugated canonical ensembles. We derive exact expression for this correspondence using the cavity methods for some hard constraints.

We study the tailoring of structured random graph ensembles to real networks, with the objective of generating precise and practical mathematical tools for quantifying and comparing network topologies macroscopically, beyond the level of degree statistics. Our family of ensembles can produce graphs with any prescribed degree distribution and any degree-degree correlation function, its control parameters can be calculated fully analytically, and as a result we can calculate (a...

We generate new mathematical tools with which to quantify the macroscopic topological structure of large directed networks. This is achieved via a statistical mechanical analysis of constrained maximum entropy ensembles of directed random graphs with prescribed joint distributions for in- and outdegrees and prescribed degree-degree correlation functions. We calculate exact and explicit formulae for the leading orders in the system size of the Shannon entropies and complexitie...

It is generally believed that, in the thermodynamic limit, the microcanonical description as a function of energy coincides with the canonical description as a function of temperature. However, various examples of systems for which the microcanonical and canonical ensembles are not equivalent have been identified. A complete theory of this intriguing phenomenon is still missing. Here we show that ensemble nonequivalence can manifest itself also in random graphs with topologic...

We present a statistical mechanics approach for the description of complex networks. We first define an energy and an entropy associated to a degree distribution which have a geometrical interpretation. Next we evaluate the distribution which extremize the free energy of the network. We find two important limiting cases: a scale-free degree distribution and a finite-scale degree distribution. The size of the space of allowed simple networks given these distribution is evaluat...