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Networks are ubiquitous throughout science and engineering. A number ofmethods, including some from our own group, have explored how one goes aboutcomputing or predicting the dynamics of networks given information aboutinternal models of individual nodes and network connectivity, possibly withadditional information provided by statistical or descriptive metrics thatcharacterize the network. But what can be inferred about network dynamics whenthere is no knowledge or information about the internal model or dynamics ofparticipating nodes? Here, we explore how connected subsets of nodescompetitively interact in order to activate a common downstream node theyconnect into. We achieve this by assuming a simple set of rules borrowed fromneurophysiology. The model we develop reflects a local process from whichglobal network dynamics emerges. We call this model a competitive refractorydynanics model. It is derived from a consideration of spatial and temporalsummation in biological neurons, whereby summating post synaptic potentials(PSPs) along the dendritic tree contribute towards the membrane potential atthe initial segment reaching a threshold potential. We first show how the'winning node' or set of 'winning' nodes that achieve activation of adownstream node is computable by the model. We then derive a formal definitionof optimized network signaling within our framework. We define a ratio betweenthe signaling latencies on the edges of the network and the internal time ittakes individual nodes to process incoming signals. We show that an optimalratio is one where the speed of information propagation between connected nodesdoes not exceed the internal dynamic time scale of the nodes. We then show howwe can use these results to arrive at a unique interpretation for theprevalence of the small world network topology in natural and engineeredsystems.

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