Essential metrics for Life on graphs

TL;DR

This work develops Life-like network automata (LLNAs) as a topological generalisation of outer-totalistic cellular automata on general graphs, and introduces a genotype–phenotype framework built around HW (Hamming weight), BS (Boolean sensitivity), a mean-field curve, and a Derrida plot. By linking local rule structure to global dynamics, the authors show strong correlations between genotype metrics and phenotype outcomes across network topologies, including phase-transition behaviour, and demonstrate a bottom-up approach to solving the firing squad synchronisation problem (FSSP) with high success rates. The methodology combines analytical tools (mean-field theory, Derrida analysis) with computational experiments on toroidal, small-world, and random networks, illustrating how topology and initial conditions shape state and defect evolution. The practical payoff is a principled pathway to design LLNAs with desired global behaviours, enabling efficient, interpretable engineering of synchronization tasks on complex networks.

Abstract

We present a strong theoretical foundation that frames a well-defined family of outer-totalistic network automaton models as a topological generalisation of binary outer-totalistic cellular automata, of which the Game of Life is one notable particular case. These "Life-like network automata" are quantitatively described by expressing their genotype (the mean field curve and Derrida curve) and phenotype (the evolution of the state and defect averages). After demonstrating that the genotype and phenotype are correlated, we illustrate the utility of these essential metrics by tackling the firing squad synchronisation problem in a bottom-up fashion, with results that exceed a 90% success rate.

Figures (10)

• Figure 2.1: This diagram represents the local update rule of the Game of Life, phrased in the formalism of an LLNA. The horizontal axis represents the state density of the neighbourhood. The vertical axis represents the response of the local update function, i.e. whether the central cell will be alive ($1$) or dead ($0$) in the next time step. The horizontal axis is divided into nine density intervals, representing the set $R$ whose elements are defined in Eq. \ref{['eq:r-intervals']}. The colours indicate whether the central cell is dead or alive in the current time step, which determines whether it should respond to the $B$ set or the $S$ set in Eq. \ref{['eq:local-update-rule']}. Following the notation of Eq. \ref{['eq:compact-notation']}, this rule is denoted as $\phi^9_{8,12}$.
• Figure 3.1: Local update rule diagram for $\phi^9_{72,12}$ (known as the LLCA "HighLife" eppstein2010growth) for node degrees $4$ (top) and $12$ (bottom). The vertical dashed lines indicate the possible state densities for the respective degree, and the white circles indicate the binomially distributed relative abundance of the configurations that result in this density. Top: node $v_i$ has degree $k_i=4$, such that only five densities can be taken, altogether skipping intervals $R_1, R_3$, $R_5$ and $R_7$. Bottom: for degree $k_i=12$, every density interval can be reached, but most of the configurations result in a density in $R_3$, $R_4$ or $R_5$. This results in an average behaviour that is significantly different from the degree-$4$ case (cf. Fig. \ref{['fig:mean_field_curve-derrida_curve-HW_BS_distribution-R9B72S12']}).
• Figure 3.2: Genotype parameters for rule $\phi^9_{72,12}$ on a random network with $N=900$ nodes and $3600$ edges. Left: mean-field curve for degrees $4$ (dashed black) and $12$ (dash-dotted black), and weighted (solid black) by the degree distribution. The associated HWs per degree are mapped from $\rho^0=0.5$ (dotted black): $\text{HW}_4 = 0.25$, $\text{HW}_{12}=0.37$, and $\text{HW}=0.30$. Middle: Derrida plot for average density $\rho^0=0.5$, for degrees $4$ (dashed red) and $12$ (dash-dotted red), and weighted (solid red) by the degree distribution. The associated Boolean sensitivities per degree correspond to the slope of the tangent at the origin (dotted red): $\text{BS}_4 = 2.50$, $\text{BS}_{12}=4.10$, and $\text{BS}=3.82$. Right: all degree-specific HWs (black bars) and Boolean sensitivities (red bars). The relative abundance of the degrees in the network (white circles) indicates how degree-specific values are weighted.
• Figure 4.1: The time series of state averages $\rho^t$ of an NA evolved according to rule $\phi^9_{328,52}$, with initial state density $\rho^0 = 1/4$ (full line), $\rho^0 = 2/4$ (dashed line), or $\rho^0 = 3/4$ (dash-dotted line), for three distinct network topologies. From left to right, these are the toroidal lattice (purple), small-world networks (teal), and random networks (yellow).
• Figure 4.2: Sweep of dynamical and topological values and their effect on the system's state average convergence, with colours and linestyles corresponding to the time series in Fig. \ref{['fig:state-average-evolution_three-networks_R9B328S52']}. Left: $3$ networks with $51$ linearly spaced initial state densities. Only intermediate values of the initial state density can result in an LLNA that does not go extinct ($\bar{\rho}^T=0$), with a strong cut-off for random networks. Right: $51$ networks were generated using the Watts-Strogatz algorithm with logarithmically spaced rewiring probabilities for three distinct initial state densities. For a high initial density, the system displays a phase transition between $p=0.2$ and $p=0.3$.