Trajectory Networks and Their Topological Changes Induced by Geographical Infiltration

We investigate the effect of progressive geographical infiltration on the topology of trajectory networks. Trajectory networks, a type of knitted network, are obtained by establishing paths between geographically distributed nodes while following an associated vector field. These systems offer tools for modeling adaptive growth, development, and pathology of biological, transportation, or communication networks. For instance, the nodes could correspond to neurons or axonal branching points along the cortical surface and the vector field could correspond to the gradient of neurotrophic factors, or the nodes could represent towns while the vector fields would be given by economical and/or geographical gradients. The geographical infiltrations correspond to the addition of new local connections between nearby existing nodes. As such, these infiltrations could be related to several real-world processes such as contaminations, diseases, attacks, parasites, etc. Combined with a mechanism for elimination of nodes and connections, infiltration can model growth, development and adaptive plasticity in neuronal networks. The progressive geographical infiltration effect is expressed in terms of the degree, clustering coefficient, size of the largest component and the lengths of the existing chains measured along the infiltrations. We show that the maximum infiltration distance plays a critical role in the intensity of the induced topological changes. For large enough values of this parameter, the chains intrinsic to the trajectory networks undergo a collapse which is unrelated to the percolation of the network also implied by the infiltrations. (Copyright Luciano da F. Costa, 2008)