Outer-totalistic cellular automata on graphs

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.

The formalism-Within the CA framework, the discrete (binary) state x i ∈ Σ = {0, 1} of a node i at time t + 1 solely depends on its own state and the states of its d neighboring nodes at time t. All cells are updated synchronously by the same, time-independent rule f : Σ d+1 → Σ. To implement CA on a directed or undirected graph G, we have to account for different neighborhood sizes d i due to the heterogeneous connectivity and thus, in general, to allow for individual rules f i . Our strategy instead is to impose constraints on the rule space, motivated by simple physical requirements, in order to obtain a set of discrete rules, implementable on arbitrary topologies:

• Homogeneity f i = f ∀ i, i.e. the same rule applies to all nodes in the graph.

• Isotropy f = f (x i , ρ i ), i.e. rules may not depend on the order of neighboring states and are thus functions of the density of neighboring states, ρ i (t) =

Here, G is represented by the adjacency matrix A: If a link connects node j to node i, A ij = 1, and we call j an input node of i. The number of all input nodes is called the in-degree of node i, d i = j A ij .

• Functional simplicity, i.e. the rule f is a piecewise constant function of the density ρ i .

Elementary Cellular Automata-The simplest CA, termed elementary CA (ECA) [2], are defined on a onedimensional grid with minimal neighborhood size, d = 2, and a binary state space, Σ = {0, 1}. The 2 3 = 8 different neighborhood configurations x i-1 , x i , x i+1 result in 2 2 3 = 256 possible rules. In this set, 2 6 = 64 rules fulfill the conditions mentioned above and depend only on the state x i (0 or 1) and on the density ρ i of neighboring states (0, 1/2, or 1). These 64 rules are called outer-totalistic [2] and are now parametrized with the rule parameter set (α, β, γ):

We distinguish the following cases for the rule parameters α, β, γ: The state x i (t + 1) may be 0 or 1 independently of the state x i (t) itself, or it may remain unchanged (+) or be flipped (-), α, β, γ ∈ {0, 1, +, -}. The frequently used majority rule [5,6,7,8], for example, where a node i is mapped onto 0/1 if the density ρ i is below/above 0.5, and stays in its state otherwise, is described in our formalism by (α, β, γ) = (0, +, 1). For α, β, γ ∈ {0, 1}, the corresponding CA rules are called totalistic [2], since x i (t+1) depends exclusively on the density ρ i of the input states. Only these rules have strict RBN rule equivalents (see Table I). Aside from the initial system state x(0) := (x 1 , x 2 , . . . , x N ) at t = 0, the patterns of rule (0, 0, 0) and rule (1, 1, 1) are perfectly symmetric under the action of the operator T : ξ → 1 -ξ, ξ ∈ {0, 1}. The operator T exchanges all 0s and 1s in an array of elements, which can be both a pattern consisting of 0’s and 1’s or a set of rule parameters. Note that the elements {+, -} remain unaffected under the action of T . Generally, the symmetric rule to (α, β, γ) is rule T (γ, β, α). The patterns emerging from the action of a rule onto an initial state, written as (α, β, γ) • x(0), are identical to the inverted patterns emerging from the inverted initial state T x(0) due to T (γ, β, α):

Explicitly, the symmetric rule to (0, 1, +), corresponding to the ECA with rule number 218 [2], is (+, 0, 1) with ECA rule number 164 (see Table I for more examples). Some rules, like the majority rule (0, +, 1), are self-symmetric. After elimination of all symmetric counterparts, 34 different ECA rules remain. (1,+,–) 10 9

(1,0,+) 13 3

(1,0,–

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