Finite-horizon competition of branching random walks on trees and sparse random graphs: Size-biased genealogy and annealed simulation diagnostics

We study a finite-horizon first-arrival competition generated by two independent branching random walks on regular tree exhaustions and sparse Erdős–Rényi graphs. The paper focuses on a variance-sensitive mechanism that is invisible at the level of first moments. If two offspring laws have the same mean but different second factorial moments $ \eta = \mathbb E[K(K-1)] $, then their many-to-one occupation fields agree, while their second moments and the genealogies selected by size-biased sampling can differ. We make this distinction explicit through a size-biased spine calculation: Under the associated size-biased measure, the expected number of non-spine sibling subtrees per spine generation is $ \eta/m $. For any target set fixed before the branching realization, we further derive an exact finite-time $ \eta $-response theorem: Changing $ \eta $ by $ \Delta\eta $ changes the target second moment by $ \Delta\eta\, \Gamma_T(A) $, where $ \Gamma_T(A) $ is an explicit nonnegative random-walk kernel, and changes the terminal-size-weighted survivor target occupation by $ (\Delta\eta/m)P^T(a, A)\sum_{\ell = 0}^{T-1}m^\ell $. These identities quantify both intermittency and size-biased enrichment without identifying the spine law with ordinary survival conditioning. A finite-time Paley–Zygmund bound records how unconditional hitting, second moments, and survival probabilities interact. The computational protocol is designed accordingly. The Erdős–Rényi experiment is annealed over independent graph instances, red and blue start from an exchangeable pair of neighbours of a high-degree root, and graph-level cluster bootstrap and paired graph-level comparisons are used for sparse-graph summaries. The numerical results are reported as finite-size diagnostics, not as an infinite-volume coexistence theorem. They support the following picture: Fixed-mean high variance is not uniformly advantageous, but after survival filtering its surviving genealogies can have a larger observed contribution near the outer frontier.