Rotem Brand | Searcharxiv

arXiv2026

On the most reliable graphs with fixed redundancy

The all-terminal reliability of a graph $G$ is the probability that $G$ remains connected when each edge fails independently with probability $p$. For fixed $n$ and $m$, the uniformly most reliable problem asks which graph with $n$ vertices and $m$ edges maximizes reliability for all $p \in [0,1]$. Although such graphs do not always exist, optimal graphs in the regime $p \to 0$ always do and are determined by the structure of their minimal cut sets. We establish a structural characterization of graphs that are most reliable near $p=0$. Our results partially resolve a conjecture of Bourel et al., showing that, under suitable conditions, regular graphs with maximal girth are optimal. Extending this analysis to graphs with fixed redundancy $r=m-(n-1)$ and sufficiently large $n$, we show that the most reliable graphs are obtained by subdividing the most reliable cubic graphs with $2(r-1)$ vertices. The general conjecture remains open. Unlike previous results, which resolved only small redundancy cases or very dense regimes, our approach yields a substantial extension of the known range. We determine the unique cubic candidates for uniformly most reliable graphs for all redundancy levels $m-n \le 19$, and prove the non-existence of uniformly most reliable graphs for several infinite families with fixed redundancy and asymptotically large $n$. These results significantly enlarge both the candidate class and the range of provable non-existence.

math.CO↗

arXiv2023

Cost-effective network robustness

Modern society relies heavily on infrastructure networks, from communication to power systems, making their reliability under failure and disruption critical. Robustness typically requires redundancy, providing alternative paths that ensure continuous service, yet redundancy is costly and often limited. Real-world networks therefore operate under a fundamental trade-off between resource investment and reliability. Here we derive theoretical bounds governing this trade-off and identify the network architectures that minimize expected downtime under constraints of cost, component durability, and spatial embedding. We show that, in the relevant limit, optimal configurations consist of a 3-connected structural core linked by intermediate chains of uniform weight. This reveals a previously unrecognized family of spatial networks that achieves near-optimal robustness with orders-of-magnitude fewer resources. Distinct from commonly studied network models, these architectures expose general structural principles underlying the efficient organization of reliable infrastructure networks.

cs.DM↗