  We investigate the eigenvalue density in ensembles of large sparse Bernoulli
random matrices. We demonstrate that the fraction of linear subgraphs just
below the percolation threshold is about 95\% of all finite subgraphs, and the
distribution of linear chains is purely exponential. We analyze in detail the
spectral density of ensembles of linear subgraphs, discuss its ultrametric
nature and show that near the spectrum boundary, the tail of the spectral
density exhibits a Lifshitz singularity typical for Anderson localization. We
also discuss an intriguing connection of the spectral density to the Dedekind
$\eta$-function. We conjecture that ultrametricity is inherit to complex
systems with extremal sparse statistics and argue that a number-theoretic
ultrametricity emerges in any rare-event statistics.
