Detecting Tampering in a Random Hypercube

TL;DR

This paper investigates methods to detect tampering in a random hypercube graph by adding diameter paths, analyzing whether such modifications can be identified as the graph size grows large.

Contribution

It introduces two tampering scenarios involving diameter paths in a random hypercube and studies their detectability as the dimension increases.

Findings

Tampering with diameter paths can be asymptotically detectable.

Detection depends on the method of selecting the diameter path.

Results provide insights into structural vulnerabilities of hypercube graphs.

Abstract

Consider the random hypercube $H_2^n(p_n)$ obtained from the hypercube $H_2^n$ by deleting any given edge with probabilty $1-p_n$, independently of all the other edges. A diameter path in $H_2^n$ is a longest geodesic path in $H_2^n$. Consider the following two ways of tampering with the random graph $H_2^n(p_n)$: (i) choose a diameter path at random and adjoin all of its edges to $H_2^n(p_n)$; (ii) choose a diameter path at random from among those that start at $0=(0,..., 0)$, and adjoin all of its edges to $H_2^n(p_n)$. We study the question of whether these tamperings are detectable asymptotically as $n\to\infty$.