The distance backbone of complex networks

Abstract

Redundancy needs more precise characterization as it is a major factor in the evolution and robustness of networks of multivariate interactions. We investigate the complexity of such interactions by inferring a connection transitivity that includes all possible measures of path length for weighted graphs. The result, without breaking the graph into smaller components, is a distance backbone subgraph sufficient to compute all shortest paths. This is important for understanding the dynamics of spread and communication phenomena in real-world networks. The general methodology we formally derive yields a principled graph reduction technique and provides a finer characterization of the triangular geometry of all edges—those that contribute to shortest paths and those that do not but are involved in other network phenomena. We demonstrate that the distance backbone is very small in large networks across domains ranging from air traffic to the human brain connectome, revealing that network robustness to attacks and failures seems to stem from surprisingly vast amounts of redundancy.

Contact networks have small metric backbones that maintain community structure and are primary transmission subgraphs

“…where the resulting distance weights are symmetrical and inversely proportional to the intensity of social interaction, with d ij = +OE for individuals x i and x j who have no contact, d ij = 0 when they are simultaneously active and always in contact, and d ii = 0. The metric backbone [13] of a distance graph D(X) is defined as its invariant subgraph B(X) in the computation of the graph's metric closure D T (X). The metric closure is the graph obtained after computing the shortest paths between all pairs of nodes of the distance graph and replacing the original distance edges d ij with the length of the shortest path between x i and x j .…”

“…1 No shortest path is a ected by the metric backbone. In particular, even the distance between nodes directly linked by a removed edge is not a ected as the shortest path between these nodes is then an indirect path via other nodes (edges not on the backbone break the triangle inequality; see [13] for details).…”

“…In contrast, the metric backbone is a parameter-free, principled method to obtain a subgraph that fully preserves the shortest paths of the original graph 1 [13]. It is typically a subgraph much smaller than the original network across domains ranging from topical spaces of large document corpora [25, 26, 27] to the brain connectome and functional networks [13, 28, 29, 30].…”

“…In contrast, the metric backbone is a parameter-free, principled method to obtain a subgraph that fully preserves the shortest paths of the original graph 1 [13]. It is typically a subgraph much smaller than the original network across domains ranging from topical spaces of large document corpora [25, 26, 27] to the brain connectome and functional networks [13, 28, 29, 30]. For instance, the metric backbone of a knowledge graph of more than 3 million concepts extracted from Wikipedia and used for automated fact-checking [31] contains only 2% of the original edges but is sufficient to compute all shortest-paths of the original graph [13].…”

“…Additionally, distance backbones such as the metric backbone are parameter-free, so we compare its community structure to threshold and random graphs obtained by removing exactly the same number of edges as removed in forming the metric backbone ( ‡(D).|d i>j |) in a parameter-free manner. A comparison and discussion of the advantages of distance backbones in regard to alternative backbone subgraphs is already available[13].…”