Graph Theory

Dynamical Processes on Complex Networks by Alain Barrat

By Alain Barrat

The supply of huge facts units have allowed researchers to discover complicated houses equivalent to huge scale fluctuations and heterogeneities in lots of networks that have result in the breakdown of ordinary theoretical frameworks and versions. until eventually lately those structures have been regarded as haphazard units of issues and connections. fresh advances have generated a full of life examine attempt in realizing the influence of advanced connectivity styles on dynamical phenomena. for instance, an enormous variety of daily structures, from the mind to ecosystems, energy grids and the web, should be represented as huge complicated networks. This new and up to date account offers a accomplished clarification of those results.

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According to these definitions, central nodes are therefore part of more shortest paths within the network than less important nodes. Moreover, the betweenness centrality of a node is often used in transport networks to provide an estimate of the traffic handled by the vertices, assuming that the number of shortest paths is a zero-th order approximation to the frequency of use of a given node. Analogously to the vertex betweenness, the betweenness centrality of edges can be calculated as the number of shortest paths among all possible vertex couples that pass through the given edge.

5 Rich-club phenomenon Several other statistical measures have been defined in the case of large-scale networks as simple proxies for their architectures and many of them are specifically devised for certain types of graphs, as seen in the density of bipartite cliques K n,m in directed graphs. Analogously, the “rich-club” phenomenon has been discussed in both social and computer sciences (de Solla Price, 1986; Wasserman and Faust, 1994; Zhou and Mondragon, 2004; Pastor-Satorras and Vespignani, 2004), and refers to the tendency of high degree nodes, the hubs of the network (the rich nodes), to be very well connected to each other, forming well-interconnected subgraphs (clubs) more easily than low degree nodes.

3 Statistical characterization of networks 11 where it is worth remarking that this measure of clustering only has a meaning for ki > 1. For ki ≤ 1 we define C(i) ≡ 0. Given the definition of ei , it is easy to check that the number of edges among the neighbors of i can be computed in terms of the adjacency matrix X as ei = 1 2 xi j x jl xli . 3, we provide an illustration of some simple examples of the clustering of vertices with a given neighborhood. The average clustering coefficient of a graph is simply given by C = 1 N C(i).

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