A decomposition theorem for balanced measures

TL;DR

The paperAddressess the structure of balanced measures on finite graphs by proving a decomposition theorem: every balanced measure is a convex combination of compatible basic balanced measures. It introduces a one-sided extrapolation lemma and a poset on the pair $(\mathrm{spt}\mu,M_\mu)$ to identify extremal basic measures and generate all balanced measures from them. It also analyzes the combinatorial size of the basic-measure family, establishing exponential bounds and sharpness, and studies graph-join constructions to relate balanced measures across joins. The results reveal a rich polyhedral picture: balanced measures form a finite union of polytopes with basic measures as vertices, and they show how any graph can embed as an induced subgraph in the compatibility graph of basic measures. These insights provide a practical framework for enumerating and generating balanced measures on graphs, with connections to energy maximization and potential theory on networks.

Abstract

Let $G = (V,E)$ be a connected graph. A probability measure $μ$ on $V$ is called "balanced" if it has the following property: if $T_μ(v)$ denotes the "earth mover's" cost of transporting all the mass of $μ$ from all over the graph to the vertex $v$, then $T_μ$ attains its global maximum at each point in the support of $μ$. We prove a decomposition result that characterizes balanced measures as convex combinations of suitable "extremal" balanced measures that we call "basic." An upper bound on the number of basic balanced measures on $G$ follows, and an example shows that this estimate is essentially sharp.

Figures (7)

• Figure 1: A balanced measure on the $n$-vertex path graph $P_n$. While a graph may admit many balanced measures, the equidistributed probability measure supported on the endpoints is in fact the unique balanced measure on the path graph.
• Figure 2: The supports of two balanced measures on the $44$-vertex Grinberg graph.
• Figure 3: Left: The support of a balanced measure on the Dürer graph. Right: A decomposition of this measure as a convex combination of three basic balanced measures, each of which is equidistributed on its support.
• Figure 4: The $C_8$ graph equipped with the measures $\mu$, $\nu$, and $\rho$ of Equation \ref{['eq:basic-measures-C4']}.
• Figure 5: The bipartite graph $G = \bar{K_3} + \bar{K_3}$ on three vertices. The uniform measure depicted above is a basic balanced measure that can be decomposed into the two basic balanced measures depicted below it. Note that this does not contradict Corollary \ref{['cor:weak-minimality']}.

Theorems & Definitions (33)

• Theorem 1.1: Steinerberger steinerberger2023sums
• Definition 1.1
• Definition 1.2
• Theorem 1.2: Decomposition of Balanced Measures
• Proposition 1.3
• Lemma 2.1
• Lemma 2.2
• Corollary 2.3
• Corollary 2.4
• Lemma 2.5