Temporally-Evolving Generalised Networks and their Reproducing Kernels

TL;DR

This paper introduces a framework for modeling and analyzing temporally-evolving generalized networks with stochastic processes, focusing on constructing semi-metrics and reproducing kernels for dynamic, complex topologies.

Contribution

It develops methods to handle changing network topologies over time and constructs semi-metrics and kernels suited for these evolving structures, advancing the analysis of dynamic networks.

Findings

Networks with Euclidean edges can be extended to include temporal evolution.

Proper semi-metrics can be constructed for time-varying topologies.

Reproducing kernels can be built for dynamic network structures.

Abstract

This paper considers generalised network, intended as networks where (a) the edges connecting the nodes are nonlinear, and (b) stochastic processes are continuously indexed over both vertices and edges. Such topological structures are normally represented through special classes of graphs, termed graphs with Euclidean edges. We build generalised networks in which topology changes over time instants. That is, vertices and edges can disappear at subsequent time instants and edges may change in shape and length. We consider both cases of linear or circular time. For the second case, the generalised network exhibits a periodic structure. Our findings allow to illustrate pros and cons of each setting. Generalised networks become semi-metric spaces whenever equipped with a proper semi-metric. Our approach allows to build proper semi-metrics for the temporally-evolving topological structures of the networks. Our final effort is then devoted to guiding the reader through appropriate choice of classes of functions that allow to build proper reproducing kernels when composed with the temporally-evolving semi-metrics topological structures.