Trajectory Networks and Their Topological Changes Induced by   Geographical Infiltration
  In this article we investigate the topological changes undergone by trajectory networks as a consequence of progressive geographical infiltration. Trajectory networks, a type of knitted network, are obtained by establishing paths between geographically distributed nodes while following an associated vector field. For instance, the nodes could correspond to neurons 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. Therefore trajectory networks are natural models of a large number of geographical structures. 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. The way in which progressive geographical infiltrations affect trajectory networks is investigated in terms of the degree, clustering coefficient, size of the largest component and the lengths of the existing chains measured along the infiltrations. It is shown 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 shown not to be related to the percolation of the network also implied by the infiltrations. 
Luciano da Fontoura Costa