We study the computational complexity of counting the fixed point configurations (FPs), the predecessor configurations and the ancestor configurations in certain classes of graph or network automata viewed as discrete dynamical systems. Some early results of this investigation are presented in two recent ECCC reports [39, 40]. In particular, it is proven in [40] that both exact and approximate counting of FPs in the two closely related classes of Boolean network automata, called Sequential and Synchronous Dynamical Systems (SDSs and SyDSs, respectively), are computationally intractable problems when each node is required to update according to a monotone Boolean function. In the present paper, we further strengthen those results by showing that the intractability of exact enumeration of FPs of a monotone Boolean SDS or SyDS still holds even when (i) the monotone update rules are restricted to linear threshold functions, and (ii) the underlying graph is uniformly sparse. By uniform sparseness we mean that every node in the graph has its degree bounded by for a small value of constant. In particular, we prove that exactly enumerating FPs in such SDSs and SyDSs remains #P-complete even when no node degree exceeds. Among other consequences, we show that this result also implies intractability of determining the exact memory capacity of discrete Hopfield networks with uniformly sparse and nonnegative integer weight matrices.

Abstract. Modeling, designing and analyzing large scale multi-agent systems (MAS) with anywhere from tens of thousands to millions of autonomous agents will require mathematical and computational theories and models substantially different from those underlying the study of small- to medium-scale MAS made of only dozens, or perhaps hundreds, of agents. In this paper, we study certain aspects of the global behavior of large ensembles of simple reactive agents. We do so by analyzing the collective dynamics of several related models of discrete complex systems based on cellular automata. We survey our recent results on dynamical properties of the complex systems of interest, and discuss some useful ways forward in modeling and analysis of large-scale MAS via appropriately modified versions of the classical cellular automata. 1

Abstract — We are interested in simple cellular automata (CA) and their computational and dynamical properties. In our past and ongoing work, we have been investigating (i) asymptotic dynamics of various types of CA and (ii) different communication models for CA. In this paper, we specifically focus on the convergence properties of a very simple kind of totalistic CA, namely, those defined on one-dimensional arrays where each cell or node updates according to the Boolean Majority function: the new state of a cell becomes 1 if and only if a simple majority of its inputs are currently in state 1, and it becomes 0 otherwise. We have observed in our prior work that such CA tend to have relatively simple asymptotic dynamics: a short transient chain followed by convergence to a “fixed point”. We now provide solid statistical evidence for these conjectures, based on our recent extensive computer simulations of Majority 1-D CA. In particular, we study the convergence properties of such CA for two communication models: one is the classical, parallel CA model with perfectly synchronous cell updates, and the other are CA whose cells update sequentially, one at a time; we consider two variants of such sequential update regimes. We simulate CA whose sizes range up to 1,000 cells, and demonstrate very fast (in particular, sublinear), and very slowly decreasing with an increase in the total number of cells, speeds of convergence. Finally, we draw conclusions based on our extensive simulations and outline some interesting questions to be considered in the future work.

This technical report addresses a particular approach to modeling and analysis of the behavior of large-scale multi-agent systems. A broad variety of multi-agent systems are modeled as appropriate variants of cellular and network automata. Several fundamental properties of the collective dynamics of those cellular and network automata are then formally analyzed. Various loosely coupled large-scale distributed information systems are of an increasing interest in a variety of areas of computer science and its applications – areas as diverse as team robotics, intelligent transportation systems, open distributed software environments, disaster response management, distributed databases and information retrieval, and computational theories of language evolution. A popular paradigm for abstracting such distributed infrastructures is that of multi-agent systems (MAS) made of typically a large number of autonomous agents that locally interact with each other. This report is an attempt at a cellular and network automata based mathematical and computational theory of such MAS. The