Metric structures on networks and applications

It is a common belief that biological networks are determined by their network motifs. Empirical results show that simulated networks with similar motif structure also reflect similarity across other structural measures. In this work we tackle this question from a theoretical perspective and announce our results proposing a family of invariants, the motif sets, as a natural modelization of “bags of motifs”. We prove that these motif sets precisely characterize networks which are weakly isomorphic, in the sense that the networks are at 0-distance with respect to a certain pseudometric on the space of all networks. This notion differs from the usual notion of (strong) isomorphism. We further explore a notion of skeleton of a network and establish that this construction mediates between strong and weak isomorphism, in the sense that two networks are weakly isomorphic if and only if there exists a strong isomorphism between their respective skeleta.