Phase Transitions in Possible Dynamics of Cellular and Graph Automata Models of Sparsely Interconnected Multi-Agent Systems

We are interested in Communicating Finite State Machine (CFSM) based models of large-scale multi-agent systems and their emerging behavior. CFSM-based models are suitable for studying large ensembles of simple reactive agents, and collective dynamics of such ensembles. In this paper, we focus on the asymptotic dynamics of a class of the classical (finite) Cellular Automata (CA) and more general Network or Graph Automata (GA). We restrict our attention to CA and GA with Boolean-valued linear threshold functions as the node update rules, inspired by the well-known classical models of biological neurons. The linear threshold update rules on which we focus the most are the Boolean-valued functions that do not allow negative weights. We fully characterize the configuration spaces of such simple threshold CA, with a focus on the Majority update rule that results in the most interesting dynamics among the CA in this class. In particular, we show the combinatorics behind determining the total number of fixed point configurations for simple threshold CA. Even when the combinatorics is non-trivial, such as in the case of Majority update rule, the counting problems of interest are computationally tractable. We then discuss a stark contrast with respect to intractability of counting for the related classes of Boolean graph automata with the same restrictions on the node update rules. The GA with proven complex dynamics have only slightly more complex structures when it comes to (i) the underlying interconnection topologies (``cellular spaces'') and (ii) the diversity of the node update rules, i.e., whether all the nodes use the same update rule (as in the classical CA), or just two different rules from the given class are allowed.