Random-Priority Frontier Routing: Tight $\mathrm{\Theta }(n^c)$ Bounds Against $c$-Node Cartels

We study path diversification in trusted-node networks, where sensitive material is relayed through intermediate nodes, some of which may be compromised. Our randomized routing rule assigns each vertex an independent random priority and repeatedly expands the highest-priority vertex on the global frontier of the explored region. Let $G$ have $n$ vertices, let $s,t$ be honest endpoints, and let $C$ be a set of $c$ compromised intermediate vertices, called a cartel, whose deletion leaves $s$ and $t$ connected. For every fixed $c$ and every fixed target probability $q\in (0,1)$, we prove that $\mathrm{\Theta }(n^c)$ independent executions are sufficient in the worst case for some route to avoid $C$ with probability at least $q$.