  We analyse a model of fixed in-degree Random Boolean Networks in which the
fraction of input-receiving nodes is controlled by a parameter gamma. We
investigate analytically and numerically the dynamics of graphs under a
parallel XOR updating scheme. This scheme is interesting because it is
accessible analytically and its phenomenology is at the same time under
control, and as rich as the one of general Boolean networks. Biologically, it
is justified on abstract grounds by the fact that all existing interactions
play a dynamical role. We give analytical formulas for the dynamics on general
graphs, showing that with a XOR-type evolution rule, dynamic features are
direct consequences of the topological feedback structure, in analogy with the
role of relevant components in Kauffman networks. Considering graphs with fixed
in-degree, we characterize analytically and numerically the feedback regions
using graph decimation algorithms (Leaf Removal). With varying gamma, this
graph ensemble shows a phase transition that separates a tree-like graph region
from one in which feedback components emerge. Networks near the transition
point have feedback components made of disjoint loops, in which each node has
exactly one incoming and one outgoing link. Using this fact we provide
analytical estimates of the maximum period starting from topological
considerations.
