Asymptotic behavior of two-terminal series-parallel networks

Abstract

This paper discusses the enumeration of two-terminal series-parallel networks, i.e. the number of electrical networks built with n identical elements connected in series or parallel with two-terminal nodes. They frequently occur in applied probability theory as a model for real networks. The number of networks grows asymptotically like R^n/n^alpha, as for some models of statistical physics like self-avoiding walks, lattice animals, meanders, etc. By using a exact recurrence relation, the entropy is numerically estimated at R = 3.5608393095389433(1), and we show that the sub-leading universal exponent alpha is 3/2.