Complex Webs: Dynamics of chance

In this chapter we explore web dynamics using a master equation in which the rate of change of probability is determined by the probability flux into and out of the state of interest. The master equation captures the interactions among large numbers of elements each with the same internal dynamics; in this case the internal dynamics consist of the switching of a node between two states. A two-state node may be viewed as the possible choices of an individual, say whether or not that person will vote for a particular candidate in an election. This is one of the simplest dynamical webs which has been shown mathematically to result in synchronization under certain well-defined conditions. We focus on the intermittent fluctuations emerging from a phase-transition process that achieves synchronized behavior for the strength of the interaction exceeding a critical value. This model provides a first step towards proving that these intermittent fluctuations, rather than being a nuisance, are important channels of information transmission allowing communication within and between different complex webs. The crucial power-law index μ discussed earlier and there inserted for mathematical convenience is here determined by the web dynamics. This observation on the inverse power-law index leads us to define the network efficiency in a form that might not coincide with earlier definitions proposed through the observation of the network topology. Both the discrete and the continuous master equation are discussed for modeling web dynamics. One of the most important aspects of the analysis concerns the perturbation of one complex network by another and the transfer of information between complex clusters. A cluster is a network with a uniformity of opinion due to a phase transition to a given state. This modeling strategy is not new, but in fact dates back to Yule [71], who used the master-equation approach, before the approach had been introduced, to obtain an inverse power-law distribution. We investigate whether the cluster’s opinion is robust or whether it can be easily changed by perturbing the way members of the cluster interact with one another.