By Alain Barrat
The provision of enormous info units have allowed researchers to discover complicated houses corresponding to huge scale fluctuations and heterogeneities in lots of networks that have bring about the breakdown of ordinary theoretical frameworks and types. till lately those platforms 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 complicated connectivity styles on dynamical phenomena. for instance, an unlimited variety of daily structures, from the mind to ecosystems, strength grids and the net, could be represented as huge advanced networks. This new and up to date account provides a entire clarification of those results.
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Extra resources for Dynamical Processes on Complex Networks
In general, topological measures do not take into account that some edges are more important than others. This can easily be understood with the simple example of a network in which the weights of all edges forming triples of interconnected vertices are extremely small. Even for a large clustering coefficient, it is clear that these triples have a minor role in the network’s dynamics and organization, and the network’s clustering properties are definitely overestimated by a simple topological analysis.
The above definitions may include a factor 1/2 to avoid counting each path twice in undirected networks. The calculation of this measure is computationally very expensive. The basic algorithm for its computation would lead to a complexity of order O(N 2 E), which is prohibitive for large networks. An efficient algorithm to compute betweenness centrality is reported by Brandes (2001) and reduces the complexity to O(N E) for unweighted networks. According to these definitions, central nodes are therefore part of more shortest paths within the network than less important nodes.
3 Mixing patterns and degree correlations As a discriminator of structural ordering of large-scale networks, the attention of the research community has initially been focused on the degree distribution, but it is clear that this function is only one of the many statistics characterizing the structural and hierarchical ordering of a network. In particular, it is likely that nodes do not connect to each other irrespective of their property or type. On the contrary, in many cases it is possible to collect empirical evidence of specific mixing patterns in networks.