The prevalence of small world networks explained by modeling the competing dynamics of local signaling events in geometric networks (1510.08729v5)

Abstract: Networks are ubiquitous throughout science and engineering. A number of methods, including some from our own group, have explored how one goes about computing or predicting the dynamics of networks given information about internal models of individual nodes and network connectivity, possibly with additional information provided by statistical or descriptive metrics that characterize the network. But what can be inferred about network dynamics when there is no knowledge or information about the internal model or dynamics of participating nodes? Here, we explore how connected subsets of nodes competitively interact in order to activate a common downstream node they connect into. We achieve this by assuming a simple set of rules borrowed from neurophysiology. The model we develop reflects a local process from which global network dynamics emerges. We call this model a competitive refractory dynanics model. It is derived from a consideration of spatial and temporal summation in biological neurons, whereby summating post synaptic potentials (PSPs) along the dendritic tree contribute towards the membrane potential at the initial segment reaching a threshold potential. We first show how the 'winning node' or set of 'winning' nodes that achieve activation of a downstream node is computable by the model. We then derive a formal definition of optimized network signaling within our framework. We define a ratio between the signaling latencies on the edges of the network and the internal time it takes individual nodes to process incoming signals. We show that an optimal ratio is one where the speed of information propagation between connected nodes does not exceed the internal dynamic time scale of the nodes. We then show how we can use these results to arrive at a unique interpretation for the prevalence of the small world network topology in natural and engineered systems.