Abstract

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.

Information

Preprint No.SS-2024-0345
Manuscript IDSS-2024-0345
Complete AuthorsTobia Filosi, Claudio Agostinelli, Emilio Porcu
Corresponding AuthorsEmilio Porcu
Emailsgeorgepolya01@gmail.com

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Acknowledgments

The authors are grateful to Havard Rue for insightful discussions during the preparation

of this manuscript. This project has also been sustained by the prompt help of Valeria

Simoncini and Valter Moretti.

Claudio Agostinelli was partially funded by BaC INF-ACT S4 - BEHAVE-MOD

PE00000007 PNRR M4C2 Inv. 1.3 - NextGenerationEU, CUP: I83C22001810007 and

by the PRIN funding scheme of the Italian Ministry of University and Research (Grant

No. P2022N5ZNP).

Supplementary Materials

. However, we stress that the semi-distance and the resulting

covariance functions remain the same for x = 0 (thus L⋆= L), as shown in Proposition

Supplementary materials are available for download.