Efficiency of informational transfer in regular and complex networks

We analyze the process of informational exchange through complex networks by measuring network efficiencies. Aiming to study non-clustered systems, we propose a modification of this measure on the local level. We apply this method to an extension of the class of small-worlds that includes {\it declustered} networks, and show that they are locally quite efficient, although their clustering coefficient is practically zero. Unweighted systems with small-world and scale-free topologies are shown to be both globally and locally efficient. Our method is also applied to characterize weighted networks. In particular we examine the properties of underground transportation systems of Madrid and Barcelona and reinterpret the results obtained for the Boston subway network.Comment: 10 pages and 9 figure

Degree-dependent intervertex separation in complex networks

We study the mean length $\ell(k)$ of the shortest paths between a vertex of degree $k$ and other vertices in growing networks, where correlations are essential. In a number of deterministic scale-free networks we observe a power-law correction to a logarithmic dependence, $\ell(k) = A\ln [N/k^{(\gamma-1)/2}] - C k^{\gamma-1}/N + ...$ in a wide range of network sizes. Here $N$ is the number of vertices in the network, $\gamma$ is the degree distribution exponent, and the coefficients $A$ and $C$ depend on a network. We compare this law with a corresponding $\ell(k)$ dependence obtained for random scale-free networks growing through the preferential attachment mechanism. In stochastic and deterministic growing trees with an exponential degree distribution, we observe a linear dependence on degree, $\ell(k) \cong A\ln N - C k$. We compare our findings for growing networks with those for uncorrelated graphs.Comment: 8 pages, 3 figure

Efficiency of Scale-Free Networks: Error and Attack Tolerance

The concept of network efficiency, recently proposed to characterize the properties of small-world networks, is here used to study the effects of errors and attacks on scale-free networks. Two different kinds of scale-free networks, i.e. networks with power law P(k), are considered: 1) scale-free networks with no local clustering produced by the Barabasi-Albert model and 2) scale-free networks with high clustering properties as in the model by Klemm and Eguiluz, and their properties are compared to the properties of random graphs (exponential graphs). By using as mathematical measures the global and the local efficiency we investigate the effects of errors and attacks both on the global and the local properties of the network. We show that the global efficiency is a better measure than the characteristic path length to describe the response of complex networks to external factors. We find that, at variance with random graphs, scale-free networks display, both on a global and on a local scale, a high degree of error tolerance and an extreme vulnerability to attacks. In fact, the global and the local efficiency are unaffected by the failure of some randomly chosen nodes, though they are extremely sensititive to the removal of the few nodes which play a crucial role in maintaining the network's connectivity.Comment: 23 pages, 10 figure

Small-World File-Sharing Communities

Web caches, content distribution networks, peer-to-peer file sharing networks, distributed file systems, and data grids all have in common that they involve a community of users who generate requests for shared data. In each case, overall system performance can be improved significantly if we can first identify and then exploit interesting structure within a community's access patterns. To this end, we propose a novel perspective on file sharing based on the study of the relationships that form among users based on the files in which they are interested. We propose a new structure that captures common user interests in data--the data-sharing graph-- and justify its utility with studies on three data-distribution systems: a high-energy physics collaboration, the Web, and the Kazaa peer-to-peer network. We find small-world patterns in the data-sharing graphs of all three communities. We analyze these graphs and propose some probable causes for these emergent small-world patterns. The significance of small-world patterns is twofold: it provides a rigorous support to intuition and, perhaps most importantly, it suggests ways to design mechanisms that exploit these naturally emerging patterns

Statistical Analysis of Airport Network of China

Through the study of airport network of China (ANC), composed of 128 airports (nodes) and 1165 flights (edges), we show the topological structure of ANC conveys two characteristics of small worlds, a short average path length (2.067) and a high degree of clustering (0.733). The cumulative degree distributions of both directed and undirected ANC obey two-regime power laws with different exponents, i.e., the so-called Double Pareto Law. In-degrees and out-degrees of each airport have positive correlations, whereas the undirected degrees of adjacent airports have significant linear anticorrelations. It is demonstrated both weekly and daily cumulative distributions of flight weights (frequencies) of ANC have power-law tails. Besides, the weight of any given flight is proportional to the degrees of both airports at the two ends of that flight. It is also shown the diameter of each sub-cluster (consisting of an airport and all those airports to which it is linked) is inversely proportional to its density of connectivity. Efficiency of ANC and of its sub-clusters are measured through a simple definition. In terms of that, the efficiency of ANC's sub-clusters increases as the density of connectivity does. ANC is found to have an efficiency of 0.484.Comment: 6 Pages, 5 figure