Growing Networks with Enhanced Resilience to Perturbation

The need to build a link between the structure of a complex network and the dynamical properties of the corresponding complex system (comprised of multiple low dimensional systems) has recently become apparent. Several attempts to tackle this problem have been made and all focus on either the controllability or synchronisability of the network --- usually analyzed by way of the master stability function, or the graph Laplacian. We take a different approach. Using the basic to...

We show how scale-free degree distributions can emerge naturally from growing networks by using random walks for selecting vertices for attachment. This result holds for several variants of the walk algorithm and for a wide range of parameters. The growth mechanism is based on using local graph information only, so this is a process of self-organisation. The standard mean-field equations are an excellent approximation for network growth using these rules. We discuss the effec...

Complex evolving systems such as the biosphere, ecosystems and societies exhibit sudden collapses, for reasons that are only partially understood. Here we study this phenomenon using a mathematical model of a system that evolves under Darwinian selection and exhibits the spontaneous growth, stasis and collapse of its structure. We find that the typical lifetime of the system increases sharply with the diversity of its components or species. We also find that the prime reason ...

We present a novel type of weighted scale-free network model, in which the weight grows independently of the attachment of new nodes. The evolution of this network is thus determined not only by the preferential attachment of new nodes to existing nodes but also by self-growing weight of existing links based on a simple weight-driven rule. This model is analytically tractable, so that the various statistical properties, such as the distribution of weight, can be derived. Fina...

This article addresses the degree distribution of subnetworks, namely the number of links between the nodes in each subnetwork and the remainder of the structure (cond-mat/0408076). The transformation from a subnetwork-partitioned model to a standard weighted network, as well as its inverse, are formalized. Such concepts are then considered in order to obtain scale free subnetworks through design or through a dynamics of node exchange. While the former approach allows the imm...

We propose a possible relation between complex networks and gravity. Our guide in our proposal is the power-law distribution of the node degree in network theory and the information approach to gravity. The established bridge may allow us to carry geometric mathematical structures, which are considered in gravitational theories, to probabilistic aspects studied in the framework of complex networks and vice versa.

We develop a simple theoretical framework for the evolution of weighted networks that is consistent with a number of stylized features of real-world data. In our framework, the Barabasi-Albert model of network evolution is extended by assuming that link weights evolve according to a geometric Brownian motion. Our model is verified by means of simulations and real world trade data. We show that the model correctly predicts the intensity and growth distribution of links, the si...

Many biological, ecological and economic systems are best described by weighted networks, as the nodes interact with each other with varying strength. However, most network models studied so far are binary, the link strength being either 0 or 1. In this paper we introduce and investigate the scaling properties of a class of models which assign weights to the links as the network evolves. The combined numerical and analytical approach indicates that asymptotically the total we...

In this paper we investigate networks whose evolution is governed by the interaction of a random assembly process and an optimization process. In the first process, new nodes are added one at a time and form connections to randomly selected old nodes. In between node additions, the network is rewired to minimize its pathlength. For timescales, at which neither the assembly nor the optimization processes are dominant, we find a rich variety of complex networks with power law t...

A new mechanism leading to scale-free networks is proposed in this letter. It is shown that in many cases of interest, the connectivity power-law behavior is neither related to dynamical properties nor to preferential attachment. Instead, we show that without increasing the number of vertices in time and without applying the so called {\it ``rich-get-richer''} condition we obtain networks whose statistical properties are scale-free. Assigning a quenched fitness value $x_i$ to...