We present a complete analytical solution of a system of Potts spins on a random k-regular graph in both the canonical and microcanonical ensembles, using the Large Deviation Cavity Method (LDCM). The solution is shown to be composed of three different branches, resulting in a non-concave...

We discuss various ensembles of homogeneous complex networks and a Monte-Carlo method of generating graphs from these ensembles. The method is quite general and can be applied to simulate micro-canonical, canonical or grand-canonical ensembles for systems with various statistical weights. It can...

A foam is a space-filling cellular pattern, that can be decomposed into successive layers or strata. Each layer contains all cells at the same topological distance to an origin (cell, cluster of cells, or basal layer). The disorder of the underlying structure imposes a characteristic roughening...

The random graph of Erdos and Renyi is one of the oldest and best studied models of a network, and possesses the considerable advantage of being exactly solvable for many of its average properties. However, as a model of real-world networks such as the Internet, social networks or biological...

The technique of approximating the mean path of Markov chains by differential equations has proved to be a useful tool in analyzing the performance of heuristics on random graph instances. However, only a small family of algorithms can currently be analyzed by this method, due to the need to...

Recent work has demonstrated that many social networks, and indeed many networks of other types also, have broad distributions of vertex degree. Here we show that this has a substantial impact on the shape of ego-centered networks, i.e., sets of network vertices that are within a given distance...

Clusters of parallel spins in the square-lattice Ising model are defined as groups of traders acting together on the Cont-Bouchaud stock market model. As in the random percolation case, we see a crossover from a power-law behaviour to a more Gaussian distribution for the market fluctuations.

We study the small-world network model, which mimics the transition between regular-lattice and random-lattice behavior in social networks of increasing size. We contend that the model displays a normal continuous phase transition with a divergent correlation length as the degree of randomness...

We present a simple model of a stock market where a random communication structure between agents gives rise to a heavy tails in the distribution of stock price variations in the form of an exponentially truncated power-law, similar to distributions observed in recent empirical studies of high...