Stochastic Process-based Method for Degree-Degree Correlation of Evolving Networks

We introduce a minimalistic model based on dynamic node deletion and node duplication with heterodimerisation. The model is intended to capture the essential features of the evolution of protein interaction networks. We derive an exact two-step rate equation to describe the evolution of the degree distribution. We present results for the case of a fixed-size network. The results are based on the exact numerical solution to the rate equation which are consistent with Monte Car...

Many networks contain correlations and often conventional analysis is incapable of incorporating this often essential feature. In arXiv:0708.2176, we introduced the link-space formalism for analysing degree-degree correlations in evolving networks. In this extended version, we provide additional mathematical details and supplementary material. We explore some of the common oversights when these correlations are not taken into account, highlighting the importance of the form...

Introduced recently, the concept of hierarchical degree allows a more complete characterization of the topological context of a node in a complex network than the traditional node degree. This article presents analytical characterization and studies of the density of hierarchical degrees in random and scale free networks. The obtained results allowed the identification of a hierarchy-dependent power law for the degrees of nodes in random complex networks, with Poisson density...

Dynamical processes on complex networks such as information propagation, innovation diffusion, cascading failures or epidemic spreading are highly affected by their underlying topologies as characterized by, for instance, degree-degree correlations. Here, we introduce the concept of copulas in order to artificially generate random networks with an arbitrary degree distribution and a rich a priori degree-degree correlation (or `association') structure. The accuracy of the prop...

In many complex systems, states and interaction structure coevolve towards a dynamic equilibrium. For the adaptive contact process, we obtain approximate expressions for the degree distributions that characterize the interaction network in such active steady states. These distributions are shown to agree quantitatively with simulations except when rewiring is much faster than state update, and used to predict and to explain general properties of steady-state topologies. The m...

We review the recent fast progress in statistical physics of evolving networks. Interest has focused mainly on the structural properties of random complex networks in communications, biology, social sciences and economics. A number of giant artificial networks of such a kind came into existence recently. This opens a wide field for the study of their topology, evolution, and complex processes occurring in them. Such networks possess a rich set of scaling properties. A number ...

We introduce a simple one-parameter network growth algorithm which is able to reproduce a wide variety of realistic network structures but without having to invoke any global information about node degrees such as preferential-attachment probabilities. Scale-free networks arise at the transition point between quasi-random and quasi-ordered networks. We provide a detailed formalism which accurately describes the entire network range, including this critical point. Our formalis...

A key problem in the study and design of complex systems is the apparent disconnection between the microscopic and the macroscopic. It is not straightforward to identify the local interactions that give rise to an observed global phenomenon, nor is it simple to design a system that will exhibit some desired global property using only local knowledge. Here we propose a methodology that allows for the identification of local interactions that give rise to a desired global prope...

We study the evolution of networks when the creation and decay of links are based on the position of nodes in the network measured by their centrality. We show that the same network dynamics arises under various centrality measures, and solve analytically the network evolution. During the complete evolution, the network is characterized by nestedness: the neighbourhood of a node is contained in the neighbourhood of the nodes with larger degree. We find a discontinuous transit...

We propose generalizations of a number of standard network models, including the classic random graph, the configuration model, and the stochastic block model, to the case of time-varying networks. We assume that the presence and absence of edges are governed by continuous-time Markov processes with rate parameters that can depend on properties of the nodes. In addition to computing equilibrium properties of these models, we demonstrate their use in data analysis and statisti...