The ultrametric backbone is the union of all minimum spanning forests

TL;DR

It is shown that the ultrametric backbone is the union of minimum spanning forests in undirected graphs and provides a new generalization of minimum spanning trees to directed graphs that, unlike minimum equivalent graphs and minimum spanning arborescences, preserves all max−min shortest paths and De Morgan's law consistency.

Abstract

Minimum spanning trees and forests are powerful sparsification techniques that remove cycles from weighted graphs to minimize total edge weight while preserving node connectivity. They have applications in computer science, network science, and graph theory. Despite their utility and ubiquity, they have several limitations, including that they are only defined for undirected networks, they significantly alter dynamics on networks, and they do not generally preserve important network features such as shortest distances, shortest path distribution, and community structure. In contrast, distance backbones, which are subgraphs formed by all edges that obey a generalized triangle inequality, are well defined in both directed and undirected graphs and preserve those and other important network features. The backbone of a graph is defined with respect to a specified path-length operator that aggregates weights along a path to define its length, thereby associating a cost to indirect connections. The backbone is the union of all shortest paths between each pair of nodes according to the specified operator. One such operator, the max function, computes the length of a path as the largest weight of the edges that compose it (a weakest link criterion). It is the only operator that yields an algebraic structure for computing shortest paths that is consistent with De Morgan's laws. Applying this operator yields the ultrametric backbone of a graph in that (semi-triangular) edges whose weights are larger than the length of an indirect path connecting the same nodes (i.e., those that break the generalized triangle inequality based on max as a path-length operator) are removed. We show that the ultrametric backbone is the union of all minimum spanning forests in undirected graphs and provides a new generalization of minimum spanning trees to directed graphs.

Figures (1)

• Figure 1: The ultrametric backbone is distinct from unions of MST analogs in directed graphs. (a) An example distance graph with thicker edges corresponding to smaller distance weights. (b) The ultrametric backbone is shown with edge weights omitted for visual clarity. Edge $d_{2,4}$ is removed, as indicated by the red dashes; it is redundant for $\max-\min$ shortest paths because it breaks the $\max-\min$ transitivity. (c) A minimum equivalent graph is shown, which in this example is unique. Note that it is distinct from the ultrametric backbone and does not preserve the shortest $\max-\min$ path from $x_2$ to $x_4$. (d) Five (in this case, unique) minimum spanning arborescences with the root node filled in with black are shown. The red dashed line indicates an edge, $d_{2,3}$, that is not in any minimum spanning arborescence, but is in the ultrametric backbone and required for $\max-\min$ shortest paths (its weight increases from $5$ to $6$). The blue edge, $d_{2,4}$, is present in the union of these five graphs, but is redundant for $\max-\min$ shortest paths and therefore is not in the ultrametric backbone.

Theorems & Definitions (19)

• Definition 1.1: ultrametric backbone
• Lemma 2.1: Cycle Property of MSTs
• proof : Proof of Lemma \ref{['res:cycle_property']}
• Lemma 2.2
• proof : Proof of Lemma \ref{['res:U_has_all_T']}
• Lemma 2.3
• proof : Proof of Lemma \ref{['res:no_extra_edges_in_U']}
• Theorem 2.4
• Corollary 2.4.1
• Remark 2.5