Evolution of networks

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 of them are scale-free and show striking resilience against random breakdowns. In spite of large sizes of these networks, the distances between most their vertices are short -- a feature known as the ``small-world'' effect. We discuss how growing networks self-organize into scale-free structures and the role of the mechanism of preferential linking. We consider the topological and structural properties of evolving networks, and percolation in these networks. We present a number of models demonstrating the main features of evolving networks and discuss current approaches for their simulation and analytical study. Applications of the general results to particular networks in Nature are discussed. We demonstrate the generic connections of the network growth processes with the general problems of non-equilibrium physics, econophysics, evolutionary biology, etc.Comment: 67 pages, updated, revised, and extended version of review, submitted to Adv. Phy

Brain modularity controls the critical behavior of spontaneous activity

The human brain exhibits a complex structure made of scale-free highly connected modules loosely interconnected by weaker links to form a small-world network. These features appear in healthy patients whereas neurological diseases often modify this structure. An important open question concerns the role of brain modularity in sustaining the critical behaviour of spontaneous activity. Here we analyse the neuronal activity of a model, successful in reproducing on non-modular networks the scaling behaviour observed in experimental data, on a modular network implementing the main statistical features measured in human brain. We show that on a modular network, regardless the strength of the synaptic connections or the modular size and number, activity is never fully scale-free. Neuronal avalanches can invade different modules which results in an activity depression, hindering further avalanche propagation. Critical behaviour is solely recovered if inter-module connections are added, modifying the modular into a more random structure.Comment: 5 pages, 6 figure

Large-scale topological and dynamical properties of Internet

We study the large-scale topological and dynamical properties of real Internet maps at the autonomous system level, collected in a three years time interval. We find that the connectivity structure of the Internet presents average quantities and statistical distributions settled in a well-defined stationary state. The large-scale properties are characterized by a scale-free topology consistent with previous observations. Correlation functions and clustering coefficients exhibit a remarkable structure due to the underlying hierarchical organization of the Internet. The study of the Internet time evolution shows a growth dynamics with aging features typical of recently proposed growing network models. We compare the properties of growing network models with the present real Internet data analysis.Comment: 13 pages, 15 eps figure

Can we avoid high coupling?

It is considered good software design practice to organize source code into modules and to favour within-module connections (cohesion) over between-module connections (coupling), leading to the oft-repeated maxim "low coupling/high cohesion". Prior research into network theory and its application to software systems has found evidence that many important properties in real software systems exhibit approximately scale-free structure, including coupling; researchers have claimed that such scale-free structures are ubiquitous. This implies that high coupling must be unavoidable, statistically speaking, apparently contradicting standard ideas about software structure. We present a model that leads to the simple predictions that approximately scale-free structures ought to arise both for between-module connectivity and overall connectivity, and not as the result of poor design or optimization shortcuts. These predictions are borne out by our large-scale empirical study. Hence we conclude that high coupling is not avoidable--and that this is in fact quite reasonable