Properties and Expressivity of Linear Geometric Centralities

2025-06-24

AI Summary

This paper investigates the expressive power and node-discrimination capability of Linear Geometric Centrality (LGC), a family of centrality measures based on shortest-path distances. Addressing the core question—“What is the maximum number of distinct node rankings that LGC can induce?”—we propose a unified modeling framework: all LGC variants are formalized as linear transformations of distance-count vectors. Leveraging axiomatic reasoning, Farkas’ Lemma, and linear programming duality, we characterize the theoretical limits of LGC’s ranking capacity. Our main contributions are threefold: (1) the first tight upper bound on the number of distinct node rankings inducible by LGC; (2) necessary and sufficient conditions—and an efficient decision criterion—for whether a given node ranking is realizable by some LGC; and (3) a formal characterization of LGC’s robustness foundation in adversarial graph analysis. These results provide principled theoretical grounding for position encoding and centrality design in graph neural networks.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

Centrality indices are used to rank the nodes of a graph by importance: this is a common need in many concrete situations (social networks, citation networks, web graphs, for instance) and it was discussed many times in sociology, psychology, mathematics and computer science, giving rise to a whole zoo of definitions of centrality. Although they differ widely in nature, many centrality measures are based on shortest-path distances: such centralities are often referred to as geometric. Geometric centralities can use the shortest-path-length information in many different ways, but most of the existing geometric centralities can be defined as a linear transformation of the distance-count vector (that is, the vector containing, for every index t, the number of nodes at distance t). In this paper we study this class of centralities, that we call linear (geometric) centralities, in their full generality. In particular, we look at them in the light of the axiomatic approach, and we study their expressivity: we show to what extent linear centralities can be used to distinguish between nodes in a graph, and how many different rankings of nodes can be induced by linear centralities on a given graph. The latter problem (which has a number of possible applications, especially in an adversarial setting) is solved by means of a linear programming formulation, which is based on Farkas' lemma, and is interesting in its own right.

Problem

• Study linear geometric centralities in graphs
• Analyze expressivity of linear centralities for node distinction
• Determine possible node rankings using linear programming

Innovation

• Linear transformation of distance-count vectors
• Axiomatic approach for centrality analysis
• Linear programming based on Farkas' lemma