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# Limit Sets of Generalized, Multi-Threshold Networks

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## Abstract

Standard two-state (Boolean) threshold networks have been used to model a broad range of social and biological systems. In this paper, we generalize this class of systems to arbitrary finite sets with nonsymmetric thresholds. For this new class of systems, we derive sufficient conditions on the threshold parameters to ensure that the limit sets are fixed points. In contrast to the standard Boolean threshold networks, this broader class can have long periodic orbits and here we identify bifurcation points of these systems. Our focus is mainly on asynchronous systems, but we also discuss synchronous systems. The extension we introduce is directly motivated by applications in the social sciences. However, we also expect that our results will be useful for modeling biological phenomena where a finer level of expression than 0/1 or on/off is needed.

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