Reachability Does Not Imply Searchability in Expanding Networks

Abstract

Routing and search respond in opposite ways to rapid network expansion. Short graph distances make a known destination easy to reach, while the neighborhood accessible within a few hops can be vastly larger than any finite inspection budget. We show that this tension imposes an algorithm-independent constraint on sparse search. If $m_q$ relevant nodes are placed without structural information among $N$ nodes and at most $M$ nodes can be inspected, any target-blind exploration protocol has success probability at most $Mm_q/N$, irrespective of correlations between successive inspections, of revisits, and of any structural preference the exploration rule may have. Geometry determines when this constraint becomes relevant: a target becomes accessible with order-one probability when the graph-distance ball volume satisfies $V(R_c)\sim N/m_q$. At this scale, any inspection mechanism achieving a fixed success probability $δ>0$ must enrich the probability of inspecting relevant nodes by at least order $N/(Mm_q)$ relative to the neutral target density. For uniformly searched candidate sets, this entails a vanishing visible fraction of the accessible region. Thus rapidly expanding networks can make rare targets geometrically close while target-blind search remains ineffective. Numerical results on Krioukov hyperbolic random graphs and two-dimensional lattices show that this separation can arise within only a few hops in the rapidly expanding case, while the lattice accessibility radius grows algebraically.