Exact dynamics of Boolean networks with connectivity one
We study boolean dynamics on the simplest class of network topologies: those in which each node
has a single input (K = 1). Despite their simplicity, they exhibit highly intricate bahaviour. We
give the exact solution for the size and number of attractors on a loop and multiple loops of the same
size. By expressing the dynamics of a network as a composition of the dynamics of its modules, we
give a detailed solution to the critical K = 1 Kauﬀman model, and show that the minimum number
of attractors scales as 2n−√2n log2 √2n , where n is the number of nodes in loops.
Fink, T; Ahnert, S; Bar On, R; Harer, J