Temporally-Evolving Generalised Networks and Their Reproducing Kernels

This paper considers time-evolving generalised network, intended as networks where (i) the edges connecting the nodes are nonlinear, (ii) stochastic processes are continuously indexed over both vertices and edges and (iii) the topology is allowed to change over time, that is: vertices and edges can disappear at subsequent time instants and edges may change in shape and length.Topological structures satisfying (i) and (ii) are usually represented through special classes of graphs, termed graphs with Euclidean edges.We build a rigorous mathematical framework for time-evolving networks.We consider both cases of linear and circular time, where, for the latter, the generalised network exhibits a periodic structure.Our findings allow to illustrate pros and cons of each setting.Our approach allows to build proper semi-distances for the temporallyevolving topological structures of the networks.Generalised networks become semi-distance spaces whenever equipped with semi-distances.Our final effort is then devoted to guiding the reader through the appropriate choice of classes of functions that allow to build random fields on the time-evolving networks, via their kernels, that are composed with the temporally-evolving semi-distances topological structure.