📝 SummarXiv

Abstract

Networks are structures that encode relationships between pairs of elements or nodes. However, there is no imposed connection between these relationships, i.e., the relationship between two nodes can be independent of every other one in the network, and need not be defined for every possible pair of nodes. This is not true for metric spaces, where the triangle inequality imposes conditions that must be satisfied by triads of distances in the space, and these distances must be defined for every pair of nodes. In this paper, we study how to project networks into q-metric spaces, a generalization of metric spaces that encompasses a larger class of structured representations. In order to do this, we encode as axioms two intuitively desirable properties of the mentioned projections. We show that there is only one way of projecting networks onto q-metric spaces satisfying these axioms. Moreover, for the special case of (regular) metric spaces, this method boils down to computing the shortest path between every node and, for the case of ultrametric spaces, it coincides with single linkage hierarchical clustering. Furthermore, we show that the projection method satisfies two properties of practical relevance: optimality, which enables its utilization for the efficient estimation of combinatorial optimization problems, and nestedness, which entails consistency of the structure induced when projecting onto different q-metric spaces. Finally, we illustrate how metric projections can be used to efficiently search networks aided by metric trees.