Development of Nodal Dispersion Indices for Networks

Introduction

We consider a simple undirected graph G=(V,E), where V is the set of n vertices and E is the set of m edges. Our goal is to develop an index that quantifies the dispersion of a specified subset of vertices S⊆V, providing insight into their spatial arrangement within the graph. Below, we outline the key motivations for such an index.

• Expanding the Graph Theory Toolkit: We aim to complement existing graph-theoretic metrics such as degree centrality, clustering coefficients, and graph density by focusing on the spatial relationships within subsets of vertices. The dispersion index introduces a new dimension to the study of graph properties, particularly in scenarios involving localized subsets.

• Generality Across Graph Types: Our approach is designed to apply universally to various graph types, including spatial graphs (e.g., road networks), social graphs, and abstract graphs. This versatility ensures that the index will be broadly relevant across multiple domains and disciplines.

• Characterizing Subset Distributions: We propose the dispersion index as a means of analyzing the spatial arrangement of subset S within G. This metric will enable us to understand the separation and distribution of key vertices, providing novel insights into structural properties of the graph.

• Supporting Optimization Problems: The dispersion index can serve as an objective function in optimization problems, such as placing critical resources (e.g., sensors, evacuation centers) to maximize coverage or minimize redundancy. We see this as an important contribution to the intersection of graph theory and operations research.

• Assessing Network Robustness: Dispersion often correlates with resilience in network structures. By evaluating how dispersed critical nodes are, we can study network robustness and identify vulnerabilities. This application is particularly relevant in fields like disaster preparedness and network reliability.

• Benchmarking and Evaluation: We propose the dispersion index as a tool for comparing different graph configurations. This will allow us to benchmark algorithms and validate models, particularly in scenarios where the spatial distribution of a subset S is a critical factor.

• Computational Perspectives: From an algorithmic standpoint, developing efficient methods to compute the dispersion index is a meaningful challenge. We aim to address scalability and computational complexity, making the metric suitable for both small and large graphs.

• Bridging Theory and Practice: Although rooted in graph theory, the dispersion index has practical applications in transportation, communication networks, and disaster management. By emphasizing both theoretical rigor and real-world applicability, we hope to foster interdisciplinary collaboration.

Formalization

We are currently exploring mathematical formulations of the dispersion index. Initial approaches include:

• Distance-based metrics (e.g., average or maximum pairwise shortest-path distances among nodes in S).

• Structural metrics incorporating connectivity patterns.

• Robustness measures under perturbation or removal of edges.