Closeness Centralization Measure for Two-mode Data of Prescribed Sizes

TL;DR

This paper proves a conjecture that the maximum closeness centralization in bipartite graphs with fixed partition sizes is achieved by a specific rooted tree structure, confirming a theoretical optimal network configuration.

Contribution

It confirms a conjecture by Everett, Sinclair, and Dankelmann, establishing the extremal structure for maximizing closeness centralization in bipartite graphs with fixed sizes.

Findings

Maximum closeness is achieved by a rooted tree of depth 2.

Extremal configuration has a root with neighbors having nearly equal children.

The result applies to bipartite graphs with fixed bipartition sizes.

Abstract

We confirm a conjecture by Everett, Sinclair, and Dankelmann~[Some Centrality results new and old, J. Math. Sociology 28 (2004), 215--227] regarding the problem of maximizing closeness centralization in two-mode data, where the number of data of each type is fixed. Intuitively, our result states that among all networks obtainable via two-mode data, the largest closeness is achieved by simply locally maximizing the closeness of a node. Mathematically, our study concerns bipartite graphs with fixed size bipartitions, and we show that the extremal configuration is a rooted tree of depth~$2$, where neighbors of the root have an equal or almost equal number of children.