Exponential Generalised Network Descriptors

TL;DR

This paper introduces exponential generalizations of network descriptors that account for decreasing communication with distance, providing new insights into network structure and extremal properties.

Contribution

It presents a novel exponential weighting scheme for network descriptors, extending previous concepts to incorporate distance-dependent communication decay.

Findings

Derived extremal values of the new descriptors

Analyzed how these descriptors vary with the decay parameter

Provided theoretical insights into network structure under the new metrics

Abstract

In communication networks theory the concepts of networkness and network surplus have recently been defined. Together with transmission and betweenness centrality, they were based on the assumption of equal communication between vertices. Generalised versions of these four descriptors were presented, taking into account that communication between vertices $u$ and $v$ is decreasing as the distance between them is increasing. Therefore, we weight the quantity of communication by $\lambda^{d(u,v)}$ where $\lambda \in \left\langle0,1 \right\rangle$. Extremal values of these descriptors are analysed.