Abstract
This paper contributes to the theoretical analysis of the qualitative behavior of two types of Boolean networks: biased and cooperative ones. A Boolean network is biased if at least a specified fraction of its regulatory functions returns one Boolean value more often than the other and is cooperative if there are no negative interactions between the variables. We prove nontrivial upper bounds on the maximum length of periodic orbits in such networks under the assumption that the maximum number of inputs and outputs per node is a fixed constant r. For the case of n-dimensional networks with r = 2 in which only AND and OR are allowed, we find an upper bound of 10 n/4 , which is asymptotically optimal in view of previously published counterexamples. The theoretical results are supplemented by simulations of the generic behavior of cooperative networks which indicate that for large indegrees, trajectories tend to converge rapidly towards a steady state or a small periodic orbit. The latter starkly contrasts with the behavior of random arbitrary Boolean networks. MSC: 05A15; 06A07; 34C12; 39A33; 94C10; 92C42
Highlights
Mathematical models of biological problems have become increasingly sophisticated with the advent of new quantitative techniques
In Section we briefly explore the analogue for cooperative Boolean networks
6 Generic behavior of cooperative networks While our work so far has investigated the most extreme possible behavior, we focus on the average behavior of cooperative Boolean networks
Summary
Mathematical models of biological problems have become increasingly sophisticated with the advent of new quantitative techniques. The above simulations indicate that for sufficiently large r, random cooperative networks tend to have highly ordered dynamics, and that the amount of chaos, at least in terms of the lengths of observed periodic orbits and the percentage of eventually frozen nodes, is very low.
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