Outer-totalistic cellular automata on graphs

Abstract

We present an intuitive formalism for implementing cellular automata on arbitrary topologies. By that means, we identify a symmetry operation in the class of elementary cellular automata. Moreover, we determine the subset of topologically sensitive elementary cellular automata and find that the overall number of complex patterns decreases under increasing neighborhood size in regular graphs. As exemplary applications, we apply the formalism to complex networks and compare the potential of scale-free graphs and metabolic networks to generate complex dynamics.Introduction-Cellular automata (CA) on graphs in principle provide the possibility to monitor systematic changes of dynamics under variation of network topology. In practice, however, unambiguously studying the relation between topology and dynamics with CA is conceptually difficult, since changes in topology inevitably induce changes in the rule space. Proposed by von Neumann [1] as a model system for biological selfreproduction, a surge of research activity from the 80's onwards [2] established them as the standard tool of complex systems theory and spatio-temporal pattern formation [3] on regular grids. Another discrete and binary modeling approach for complex biological systems are random Boolean networks (RBNs), introduced by Kauffman [4]. While the CA framework introduces one rule for all regularly ordered cells with bi-directional links, the original RBNs consist of randomly and directionally linked nodes with individual rules. Here, we present a formalism that generalizes CA to arbitrary architectures. It allows (i) the establishment of a general correspondence between CA and isotropic RBNs and (ii) the comparison of the potential of different topologies to generate complex dynamics. As applications we examine the topological sensitivity of elementary CA, monitor the number of complex rules of CA under increasing neighborhood size, and compare the dynamic potential of scale-free graphs and representations of metabolism as substrate graphs.

Cellular Automata on Graphs: Topological Properties of ER Graphs Evolved towards Low-Entropy Dynamics

“…CA have been used in a vast number of investigations to explore the emergence of complex patterns from simple dynamic rules. Originally defined on regular lattices [27], they have also been studied on more complex topologies [15,16,36] and in noisy environments [32,37]. It should be noted that due to the diverse neighborhood sizes (compared to a regular 1D or 2D lattice) and the lack of ordering of neighbors, only a very small set of rules (from classical CA) can be plausibly transferred to general graphs.…”

“…The framework we employ is that of cellular automata (CA) on graphs [15,16], together with the notion of simulated evolution for enhancing a specific dynamic function.…”

“…The principal goal of discussing CA on graphs [15] is to explore the relationship between network architecture and dynamics from the perspective of pattern formation (and, more specifically, the Wolfram classes [27,28], as a way of characterizing observed dynamic behaviors). In a series of numerical studies we have employed this framework and related systems to analyze discrete dynamical processes on graphs [16,[32][33][34]. Furthermore, the concept of probing real networks with binary dynamics has been formulated for assessing the regularizing capacity of networks and, conversely, its ability to display complex dynamics [16].…”

“…In a series of numerical studies we have employed this framework and related systems to analyze discrete dynamical processes on graphs [16,[32][33][34]. Furthermore, the concept of probing real networks with binary dynamics has been formulated for assessing the regularizing capacity of networks and, conversely, its ability to display complex dynamics [16]. With such "dynamic probes" we found that complex (high-entropy) dynamics are systematically reduced on metabolic networks compared to randomized networks with identical degree sequences.…”

“…With such "dynamic probes" we found that complex (high-entropy) dynamics are systematically reduced on metabolic networks compared to randomized networks with identical degree sequences. Already small topological modifications substantially enhance the capacity of a network to host complex dynamic behavior and thus reduce its regularizing potential [16,35].…”