Abstract

The field of complex networks is one of the most active areas in contemporary statistical physics. Ten years after seminal work initiated the modern study of networks, interest in the field is in fact still growing, as indicated by the ever increasing number of publications in network science. The reason for such a resounding success is most likely the simplicity and broad significance of the approach that, through graph theory, allows researchers to address a variety of different complex systems within a common framework. This special issue comprises a selection of contributions presented at the workshop 'Complex Networks: from Biology to Information Technology' held in July 2007 in Pula (Cagliari), Italy as a satellite of the general conference STATPHYS23. The contributions cover a wide range of problems that are currently among the most important questions in the area of complex networks and that are likely to stimulate future research. The issue is organised into four sections. The first two sections describe 'methods' to study the structure and the dynamics of complex networks, respectively. After this methodological part, the issue proceeds with a section on applications to biological systems. The issue closes with a section concentrating on applications to the study of social and technological networks. The first section, entitled Methods: The Structure , consists of six contributions focused on the characterisation and analysis of structural properties of complex networks: The paper Motif-based communities in complex networks by Arenas et al is a study of the occurrence of characteristic small subgraphs in complex networks. These subgraphs, known as motifs , are used to define general classes of nodes and their communities by extending the mathematical expression of the NewmanGirvan modularity. The same line of research, aimed at characterising network structure through the analysis of particular subgraphs, is explored by Bianconi and Gulbahce in Algorithm for counting large directed loops . This work proposes a belief-propagation algorithm for counting long loops in directed networks, which is then applied to networks of different sizes and loop structure. In The anatomy of a large query graph , Baeza-Yates and Tiberi show that scale invariance is present also in the structure of a graph derived from query logs. This graph is determined not only by the queries but also by the subsequent actions of the users. The graph analysed in this study is generated by more than twenty million queries and is less sparse than suggested by previous studies. A different class of networks is considered by Travençolo and da F Costa in Hierarchical spatial organisation of geographical networks . This work proposes a hierarchical extension of the polygonality index as a means to characterise geographical planar networks and, in particular, to obtain more complete information about the spatial order of the network at progressive spatial scales. The paper Border trees of complex networks by Villas Boas et al focuses instead on the statistical properties of the boundary of graphs, constituted by the vertices of degree one (the leaves of border trees). The authors study the local properties, the depth, and the number of leaves of these border trees, finding that in some real networks more than half of the nodes belong to the border trees. The last contribution to the first section is The generation of random directed networks with prescribed 1-node and 2-node degree correlations by Zamora-López et al . This study deals with the generation of random directed networks and shows that often a large number of links cannot be 'randomised' without altering the degree correlations. This permits fast generation of ensembles of maximally random networks. In the section Methods: The Dynamics , significant attention is given to the study of synchronisation processes on networks: Díaz-Guilera's contribution Dynamics towards synchronisation in hierarchical networks consists of an overview of recent studies on hierarchical networks of phase oscillators. By analysing the evolution of the synchronous dynamics, one can infer details about the underlying network topology. Thus a connection between the dynamical and topological properties of the system is established. The paper Network synchronisation: optimal and pessimal scale-free topologies by Donetti et al explores an optimisation algorithm to study the properties of optimally synchronisable unweighted networks with scale-free degree distribution. It is shown that optimisation leads to a tendency towards disassortativity while networks that are optimally 'un-synchronisable' have a highly assortative string-like structure. The paper Critical line in undirected Kauffman Boolean networksthe role of percolation by Fronczak and Fronczak demonstrates that the percolation underlying the process of damage spreading impacts the position of the critical line in random boolean networks. The critical line results from the fact that the ordered behaviour of small clusters shields the chaotic behaviour of the giant component. In Impact of the updating scheme on stationary states of networks , Radicchi et al explore an interpolation between synchronous and asynchronous updating in a one-dimensional chain of Ising spins to locate a phase transition between phases with an absorbing and a fluctuating stationary state. The properties of attractors in the yeast cell-cycle network are also shown to depend sensitively on the updating mode. As this last contribution shows, a large part of the theoretical activity in the field can be applied to the study of biological systems. The section Biological Applications brings together the following contributions: In Applying weighted network measures to microarray distance matrices , Ahnert et al present a new approach to the analysis of weighted networks, which provides a generalisation to any network measure defined on unweighted networks. The clustering coefficient constructed using this approach is used to identify a number of biologically significant genes in data sets from microarray experiments. The paper Quantifying the taxonomic diversity in real species communities by Caretta Cartozo et al reports on universal statistical properties in taxonomic trees. The results, which are obtained by sampling a large pool of species from all over the world, suggest that it is possible to quantitatively distinguish real species assemblage from random collections. In the contribution Insights into biological information processing: structural and dynamical analysis of a human protein signalling network , de la Fuente et al investigate the dynamical properties of a human protein signalling network while accounting for edge directionality and topological properties both at the local and global scale. The relationship between the node degrees and the distribution of signals through the network is characterised using degree correlation profiles. A study of a brain network is presented by de Vico Fallani et al in Persistent patterns of interconnection in time-varying cortical networks estimated from high-resolution EEG recordings in humans during a simple motor act . The authors introduce an approach based on the estimate of time-varying graph indexes that allows the capture of schemes of communication within the network. The method is applied to a set of high resolution EEG data recorded from a group of subjects performing a simple foot movement. The last section, devoted to Social and Technological Applications , includes nine contributions in the broad area of infrastructure, economic, and social systems: The paper Uncovering individual and collective human dynamics from mobile phone records by Cándia et al explores extensive phone records resolved in both time and space to study collective behaviour and the occurrence of anomalous events. At the individual level, it is shown that the distribution of time intervals between consecutive calls is heavy tailed, which agrees with results previously reported on other human activities. In Mining the inner structure of the Web graph , Donato et al present a series of measurements of the Web, which offer a better understanding of the individual components of its bow-tie structure. The scale-free properties permeate all bow-tie components although they do not exhibit self-similarity and their inner structure is quite distinct. Effects of network topology on wealth distributions , by Garlaschelli and Loffredo, shows that a networked economic system self-organises towards a stationary state whose associated wealth distribution depends crucially on the underlying interaction network. In particular, this study implies that first-order topological properties alone (such as the scale-free property) are not enough to explain the emergence of the empirically observed mixed form of the wealth distribution. In the paper Resource allocation pattern in infrastructure networks , Kim and Motter show that real communication and transportation networks tend to exhibit larger load-to-capacity ratio in nodes and links with larger capacities. This surprising pattern, which is a consequence of decentralised evolution and network traffic fluctuations, suggests that infrastructure networks have evolved to prevent local failures but not necessarily large-scale failures that can be caused

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