Dynamical Processes on Complex Networks by Alain Barrat

By Alain Barrat

The provision of enormous info units have allowed researchers to discover complicated houses similar to huge scale fluctuations and heterogeneities in lots of networks that have bring about the breakdown of ordinary theoretical frameworks and versions. until eventually lately those platforms have been regarded as haphazard units of issues and connections. fresh advances have generated a full of life learn 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, energy grids and the net, might be represented as huge advanced networks. This new and up to date account provides a accomplished clarification of those results.

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6. If there is a small number of important connections around a node, the quantity Y2 is of order 1/m with m of order unity. In contrast, if all the connections are of the same order, Y2 is small and of order 1/k where k is the node’s degree. Y2 (i) = j∈V (i) wi j si 2 . 6). A similar finding is encoded in the local entropy, defined for nodes of degree larger than 2 as wi j wi j 1 ln . 34) f (i) = − ln ki si si j∈V (i) This quantity goes from 0 if the strength of i is fully concentrated on one link to the maximal value 1 for homogeneous weights: it can thus be used as an alternative or complement to the disparity Y2 to investigate the local heterogeneity of the weights.

24) Punc (k | k) = k k Nk where considering that P(k) = Nk /N , finally yields Punc (k | k) = 1 k P(k ). 25) This expression states that even in an uncorrelated network, the probability that any edge points to a node of a given degree k is not uniform but proportional to 5 Operatively, the maximally random network can be thought of as the stationary ensemble of networks visited by a process that, at any time step, randomly selects a couple of links of the original network and exchanges two of their ending points (automatically preserving the degree distribution).

In this chapter we review the basic topological and dynamical features that characterize real-world networks and we attempt to categorize networks into a few broad classes according to their observed statistical properties. In particular, self-organized dynamical evolution and the emergence of the small-world and scale-free properties of many networks are discussed as prominent concepts which have led to a paradigm shift in which the dynamics of networks have become a central issue in their characterization as well as in their modeling (which will be discussed in the next chapter).

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