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We analyse a model of fixed in-degree Random Boolean Networks in which thefraction of input-receiving nodes is controlled by a parameter gamma. Weinvestigate analytically and numerically the dynamics of graphs under aparallel XOR updating scheme. This scheme is interesting because it isaccessible analytically and its phenomenology is at the same time undercontrol, and as rich as the one of general Boolean networks. Biologically, itis justified on abstract grounds by the fact that all existing interactionsplay a dynamical role. We give analytical formulas for the dynamics on generalgraphs, showing that with a XOR-type evolution rule, dynamic features aredirect consequences of the topological feedback structure, in analogy with therole of relevant components in Kauffman networks. Considering graphs with fixedin-degree, we characterize analytically and numerically the feedback regionsusing graph decimation algorithms (Leaf Removal). With varying gamma, thisgraph ensemble shows a phase transition that separates a tree-like graph regionfrom one in which feedback components emerge. Networks near the transitionpoint have feedback components made of disjoint loops, in which each node hasexactly one incoming and one outgoing link. Using this fact we provideanalytical estimates of the maximum period starting from topologicalconsiderations.

Feedback topology and XOR-dynamics in Boolean networks with varying input structure