Nucleation during phase transitions in random networks

Dynamical processes on complex networks, ranging from biological, technological and social systems, show phase transitions between distinct global states of the system. Often, such transitions rely upon the interplay between the structure and dynamics that takes place on it, such that weak connectivity, either sparse network or frail interactions, might lead to global activity collapse, while strong connectivity leads to high activity. Here, we show that controlling dynamics ...

The Ising model in clustered scale-free networks has been studied by Monte Carlo simulations. These networks are characterized by a degree distribution of the form P(k) ~ k^(-gamma) for large k. Clustering is introduced in the networks by inserting triangles, i.e., triads of connected nodes. The transition from a ferromagnetic (FM) to a paramagnetic (PM) phase has been studied as a function of the exponent gamma and the triangle density. For gamma > 3 our results are in line ...

Within the conventional statistical physics framework, we study critical phenomena in a class of configuration network models with hidden variables controlling links between pairs of nodes. We find analytical expressions for the average node degree, the expected number of edges, and the Landau and Helmholtz free energies, as a function of the temperature and number of nodes. We show that the network's temperature is a parameter that controls the average node degree in the who...

During the past two decades, percolation has long served as a basic paradigm for network resilience, community formation and so on in complex systems. While the percolation transition is known as one of the most robust continuous transitions, the percolation transitions occurring in complex systems are often of different types such as discontinuous, hybrid, and infinite-order phase transitions. Thus, percolation has received considerable attention in network science community...

The internal organization of complex networks often has striking consequences on either their response to external perturbations or on their dynamical properties. In addition to small-world and scale-free properties, clustering is the most common topological characteristic observed in many real networked systems. In this paper, we report an extensive numerical study on the effects of clustering on the structural properties of complex networks. Strong clustering in heterogeneo...

A free zero-range process (FRZP) is a simple stochastic process describing the dynamics of a gas of particles hopping between neighboring nodes of a network. We discuss three different cases of increasing complexity: (a) FZRP on a rigid geometry where the network is fixed during the process, (b) FZRP on a random graph chosen from a given ensemble of networks, (c) FZRP on a dynamical network whose topology continuously changes during the process in a way which depends on the c...

A model for growing networks is introduced, having as a main ingredient that new nodes are attached to the network through one existing node and then explore the network through the links of the visited nodes. From exact calculations of two limiting cases and numerical simulations the phase diagram of the model is obtained. In the stationary limit, large network sizes, a phase transition from a network with finite average connectivity to a network with a power law distributio...

An important characteristic of flocks of birds, school of fish, and many similar assemblies of self-propelled particles is the emergence of states of collective order in which the particles move in the same direction. When noise is added into the system, the onset of such collective order occurs through a dynamical phase transition controlled by the noise intensity. While originally thought to be continuous, the phase transition has been claimed to be discontinuous on the bas...

In a previous paper, we defined a version of the percolation triangle condition that is suitable for the analysis of bond percolation on a finite connected transitive graph, and showed that this triangle condition implies that the percolation phase transition has many features in common with the phase transition on the complete graph. In this paper, we use a new and simplified approach to the lace expansion to prove quite generally that for finite graphs that are tori the tri...

In this paper we study the phase transitions of different types of Random Boolean networks. These differ in their updating scheme: synchronous, semi-synchronous, or asynchronous, and deterministic or non-deterministic. It has been shown that the statistical properties of Random Boolean networks change considerable according to the updating scheme. We study with computer simulations sensitivity to initial conditions as a measure of order/chaos. We find that independently of th...