  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.
