Efficient Behavior of Small-World Networks

We introduce the concept of efficiency of a network, measuring how efficiently it exchanges information. By using this simple measure small-world networks are seen as systems that are both globally and locally efficient. This allows to give a clear physical meaning to the concept of small-world, and also to perform a precise quantitative a nalysis of both weighted and unweighted networks. We study neural networks and man-made communication and transportation systems and we show that the underlying general principle of their construction is in fact a small-world principle of high efficiency.Comment: 1 figure, 2 tables. Revised version. Accepted for publication in Phys. Rev. Let

Hierarchy measure for complex networks

Nature, technology and society are full of complexity arising from the intricate web of the interactions among the units of the related systems (e.g., proteins, computers, people). Consequently, one of the most successful recent approaches to capturing the fundamental features of the structure and dynamics of complex systems has been the investigation of the networks associated with the above units (nodes) together with their relations (edges). Most complex systems have an inherently hierarchical organization and, correspondingly, the networks behind them also exhibit hierarchical features. Indeed, several papers have been devoted to describing this essential aspect of networks, however, without resulting in a widely accepted, converging concept concerning the quantitative characterization of the level of their hierarchy. Here we develop an approach and propose a quantity (measure) which is simple enough to be widely applicable, reveals a number of universal features of the organization of real-world networks and, as we demonstrate, is capable of capturing the essential features of the structure and the degree of hierarchy in a complex network. The measure we introduce is based on a generalization of the m-reach centrality, which we first extend to directed/partially directed graphs. Then, we define the global reaching centrality (GRC), which is the difference between the maximum and the average value of the generalized reach centralities over the network. We investigate the behavior of the GRC considering both a synthetic model with an adjustable level of hierarchy and real networks. Results for real networks show that our hierarchy measure is related to the controllability of the given system. We also propose a visualization procedure for large complex networks that can be used to obtain an overall qualitative picture about the nature of their hierarchical structure.Comment: 29 pages, 9 figures, 4 table

Effects of different connectivity patterns in a model of cortical circuits

The final publication is available at Springer via http://dx.doi.org/10.1007/3-540-44868-3_11Proceedings of 7th International Work-Conference on Artificial and Natural Neural Networks, IWANN 2003 Maó, Menorca, Spain, June 3–6, 2003, Part ICortical circuits are usually modeled as a network of excitatory and inhibitory neurons with a completely regular or a random connectivity pattern. However, neuroanatomy of the macaque and the cat cortex shows that cortical neurons are organized into densely linked groups that are sparsely and reciprocally interconnected. Interesting properties arise in the average activity of an ensemble of cortical neurons when the topology of the network itself is an intrinsic parameter of the model that can vary with a given set of rules. In this work we show that both the temporal activity and the encoded rhythms in an ensemble of cortical neurons depend on the topology of the network.We thank the Ministerio de Ciencia y Tecnolog a (BFI 2000-015). (PP) and (CA) are partially supported by BFM2002-02359. (PP) and (CA) also receive a partial support by POCTI/MAT/40706/2001. (ES) receive a partial support by (TIC 2002-572-C02-02)