# Robustness of Regular Graphs Based on Natural Connectivity

## TL;DR

This paper investigates the robustness of regular graphs using natural connectivity, revealing that regular ring lattices are more robust than random regular graphs, with natural connectivity independent of network size and increasing monotonically.

## Contribution

It provides an analytical reformulation of natural connectivity for regular ring lattices and compares robustness between regular ring lattices and random regular graphs.

## Key findings

- Natural connectivity of regular ring lattices is size-independent.
- Natural connectivity increases monotonically with network parameters.
- Regular ring lattices are more robust than random regular graphs.

## Abstract

It has been recently proposed that the natural connectivity can be used to characterize efficiently the robustness of complex networks. The natural connectivity quantifies the redundancy of alternative routes in the network by evaluating the weighted number of closed walks of all lengths and can be seen as an average eigenvalue obtained from the graph spectrum. In this paper, we explore both analytically and numerically the natural connectivity of regular ring lattices and regular random graphs obtained through degree-preserving random rewirings from regular ring lattices. We reformulate the natural connectivity of regular ring lattices in terms of generalized Bessel functions and show that the natural connectivity of regular ring lattices is independent of network size and increases with monotonically. We also show that random regular graphs have lower natural connectivity, and are thus less robust, than regular ring lattices.

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Source: https://tomesphere.com/paper/0912.2144