Dynamical phase transitions in graph cellular automata

Freya Behrens, Barbora Hudcová, and Lenka Zdeborová

Phys. Rev. E 109, 044312 – Published 18 April 2024

Abstract

Discrete dynamical systems can exhibit complex behavior from the iterative application of straightforward local rules. A famous class of examples comes from cellular automata whose global dynamics are notoriously challenging to analyze. To address this, we relax the regular connectivity grid of cellular automata to a random graph, which gives the class of graph cellular automata. Using the dynamical cavity method and its backtracking version, we show that this relaxation allows us to derive asymptotically exact analytical results on the global dynamics of these systems on sparse random graphs. Concretely, we showcase the results on a specific subclass of graph cellular automata with “conforming nonconformist” update rules, which exhibit dynamics akin to opinion formation. Such rules update a node's state according to the majority within their own neighborhood. In cases where the majority leads only by a small margin over the minority, nodes may exhibit nonconformist behavior. Instead of following the majority, they either maintain their own state, switch it, or follow the minority. For configurations with different initial biases towards one state we identify sharp dynamical phase transitions in terms of the convergence speed and attractor types. From the perspective of opinion dynamics this answers when consensus will emerge and when two opinions coexist almost indefinitely.

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Received 25 October 2023
Accepted 22 March 2024

DOI:https://doi.org/10.1103/PhysRevE.109.044312

Physics Subject Headings (PhySH)

Dynamical phase transitions

Statistical Physics & Thermodynamics

Authors & Affiliations

Freya Behrens ¹, Barbora Hudcová ^2,3, and Lenka Zdeborová¹

¹Statistical Physics Of Computation Laboratory, École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland
²Algebra Department, Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic
³Czech Institute of Informatics, Robotics and Cybernetics, Czech Technical University, Prague, Czech Republic

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Vol. 109, Iss. 4 — April 2024

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Figure 1
A phase transition diagram for a particular instance of a 5-regular GCA with a conforming anticonformist rule 001011. An illustration of the system's two phases that depend on the density (i.e., the average number of black-colored nodes) in the initial configuration. The phases are illustrated by space-time diagrams for a system of size $n = 1000$ nodes, though only a window of 75 nodes is shown. (Left) Rapid phase: fast convergence to the all-0 attractor. (Right) Chaotic phase: apparent randomness in the nodes state, convergence takes longer. (Middle) In the large system limit, when $n \to \infty$ , there is a dynamical phase transition. At a particular initial density value $ρ_{init}$ , the typical behavior of the system abruptly switches from the rapid to the chaotic phase. For each $ρ_{init}$ and each system size $n$ we sampled 1024 initial configurations with the given $ρ_{init}$ and computed how often the system enters a chaotic phase. For practical purposes, we conclude the system is in a chaotic phase if it does not converge within $100 * {log}_{2} (n)$ time-steps. The resulting frequency exhibits a sharp phase transition between 0.217 and 0.218, where the solid red line is our prediction from the DCM and the shaded red area comes from an empirical approximation. This transition separates the behavior on the left and the right.
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Figure 2
Example of a GCA and its dynamics. (Left) Defining the GCA from a 3-regular graph $G$ , state set $S$ , and local rule $f$ following the majority. (Middle) A configuration $x$ (0 is white and 1 is black) and the GCA's space-time diagram starting from $x$ . (Right) For the majority GCA defined on the left, we show a part of its configuration graph.
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Figure 3
Complete configuration graph of a majority GCA. Every node is a unique configuration of the system. The edges show how the dynamics evolve from one configuration into another. The different colors distinguish configurations that eventually evolve into different types of attractors. The orange color marks configurations leading to cyclic attractors of size 2 marked red, the blue configurations converge to point attractors in cyan. This is the absolute majority rule, the GCA with code 0011 on a 3-regular graph with $n = 12$ nodes.
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Figure 4
Two examples for CNC dynamics. We show two examples for conforming nonconformist dynamics on small random regular graphs with $n = 100$ nodes. We sample initial configurations for a fixed initial density $ρ$ and show how they evolve (light gray lines). Some samples are highlighted in color and their corresponding space-time diagrams are shown. (Left) The anticonformist GCA 001011 is started from $ρ = 0.2$ , in this case, the time axis is broken for visualization purposes, as for some samples the time to an attractor is extremely long. We highlight three different behaviours: Two samples converge the the configuration with only 0s (blue), but one takes a very long time to reach it (light green) and one converges faster to a configuration with only 1s (dark green). (Right) The volatile independent GCA $00 - 11$ if started from $ρ = 0.75$ . Here the orange solid line represents the dynamics of a sample where eventually all nodes change their state in a length-2 limit cycle. We call such nodes rattling (ratl.) The dashed line shows a sample which is partial rattling, i.e., some nodes are stable in the attractor, but some are rattling. Finally, the black line is a sample which ends up in the attractor with only 1s.
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Figure 5
Numerical experiments for four types of CNC rules for $d = 3, 4, 5$ . For rules 0011, $00 + 11$ , and $00 - 11$ we sampled 1024 graphs for every $n \in {10^{3}, 10^{4}, 10^{5}, 10^{6}}$ and every initial density $ρ_{init} = \frac{k}{100}, k \in {0, 1, ..., 100}$ . For 001011, due to the exponential explosion of the transient length in $\sim (0.2, 0.8)$ , we used $n \in {50, 100, 150, 200}$ . (First and second row) Histograms of the properties of attractors: their density and the fraction of rattlers. With the exception of GCA 001011, they were computed for $n = 10^{3}$ with a binning on the $y$ axis for both $ρ_{attr}$ and $α$ with 101 bins. For the GCA 001011 graphs of size $n = 200$ and 51 bins on the $y$ axis were used. (Third row) Average transient length $p$ for 0011, $00 + 11$ , and $00 - 11$ ; median transient length for 001011. We observe behavior consistent with either exponential or logarithmic growth of the transient lengths as a function of the system size. (Last row) Diagram showing the dynamical phases corresponding to each of the attractor and transient type. Transitions between the phases correspond to peaks in the transient length or a change in the scaling of the transient regime. We use the color schemes from Table 1 when the convergence to the attractor is rapid. The light green color denotes a slow convergence. Note that the same data for the rules $00 + 11$ and $00 - 11$ were already used in Ref. [19] to illustrate the results that the BDCM can obtain.
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Figure 6
Transient growth, chaotic phase classification and relaxation time for the anticonformist GCA 001011. (Left) For $ρ_{init} < 0.5$ we display the transient growth for graphs of size $n \in {50, 100, 150, 200}$ , generated as in Fig. 5. The resolution of $ρ_{init}$ is limited by $n = 50$ , a stepsize of 0.02. A transition between an exponential (straight line in the log-linear plot) and a much slower transient growth between $ρ_{init} = 0.2$ and 0.22 is clearly visible. (Middle) Empirical phase transition for the onset of a chaotic phase, which in this case is defined as the attractor taking more (chaotic) or less (homogeneous stable) than $log (n) * 100$ time steps to reach an attractor. The resolution of $ρ_{init}$ is 0.001 and narrows the interval of the dynamical phase transition down to [0.2165,0.2185], the interval for $n = 10^{6}$ between which no samples out of 1024 exhibit a behavior that is not consistent with their phase. (Right) The relaxation time describes the number of time steps required until either the chaotic regime or an attractor is reached. We empirically conclude the system is in the chaotic regime if the densities of 100 consecutive configurations remain in the interval $(0.5 - \frac{3}{\sqrt{n}}, 0.5 + \frac{3}{\sqrt{n}})$ . For all values of $n$ the maximal length of the two largest $ρ_{init}$ we observe are $ρ_{init} = 0.21$ and 0.22.
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Figure 7
DCM prediction of dynamical phase transition for the anticonformist GCA 001011. (Left) The prediction of the DCM for the density $ρ_{p}$ after $p$ steps, for different initial configurations $ρ_{init}$ . (Middle) Zoom into the region of the phase transition, with data for $p = 7$ added. (Right) Table of the crossover points between the different lines. The curves in the middle zoom were fitted with a linear regression and then the intersection was computed. Extrapolating $p \to \infty$ gives a transition at $ρ_{init} = 0.2168 \pm 0.0001$ (see Appendix Fig. 17).
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Figure 8
BDCM prediction for the absolute majority GCA 0011. (Left) Entropy of the basin of attraction for the homogeneous and partially rattling attractors, for increasing path lengths $p = 1, 2, 3$ . (Middle) The activity in the limit cycle for fixed points for $(p / c = 2)$ backtracking attractors. The dashed line shows the range of $ρ_{init}$ for which our numerics did not find any fixed points. (Right) The table shows the values of the smallest $ρ_{init}^{*, p} > 0.5$ for which $α (ρ_{init}^{*, p}) = 0.0$ together with the normalized entropy at the corresponding given $ρ_{init}$ . It shows the extrapolation $p \to \infty$ and compares it with the numerical results and related work (see Appendix pp3).
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Figure 9
BDCM prediction for the stubborn independent rule $00 + 11$ . (Left) Entropy of the basin of attraction for the all-1 and mixed stable attractors respectively, for increasing steps into the basin of attraction $p = 1, 2, 3$ . The dynamical phase transition is marked in red. (Middle) The density of 1s in the attractor, $ρ_{attr}$ , as a function of $ρ_{init}$ for attractors with $c = 1$ . (Right) The transitions is the first $ρ_{init}$ for which the $ρ_{attr} = 1$ . These values are recorded in the table together with the entropy of the basin of attraction at that point. The extrapolation agrees well with the empirical estimate of the transition from the maximal slowing down (see Appendix pp3).
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Figure 10
Dynamical phase transition for the volatile independent GCA $00 - 11$ . Comparison of the analytical and empirical prediction of a dynamical phase transition for the volatile independent GCA $00 - 11$ . We examine the transition between the all and partially rattling 2-cycles. (Left) Analytical prediction of the entropy for each $ρ_{init}$ for the two different types of attractors for increasing transient lengths $p$ . The intersection of the two entropy marks the phase transition for a given $p$ and is marked in red. Because the computed entropy is not close enough to the maximal entropy, as shown by the grey line, the approximation of the transition is not very conclusive and extrapolating the four data points would result in very high uncertainty. (Middle) Zoom in on the average transient length around the phase transition from Fig. 5. (Right) Probability of obtaining a smaller than $o (n)$ fraction of rattlers, i.e., the fraction of nodes in the attractor. To determine a reasonable threshold for having a constant $o (n)$ fraction of rattlers, when $n$ is finite, we analysed the scaling of the rattler fraction as a function of $n$ , which resulted in an attractor having no more than 0.07% of nonrattlers to be classified as a partially rattling attractor (Appendix pp3). While the thresholds agree roughly, the accuracy is worse than for the GCAs discussed previously.
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Figure 11
Phase diagram scheme for GCAs with rules of type 01, $0 - 1$ , and $0 + 1$ . While for all GCAs the homogeneous stable all-white and all-black phases are at the end of the $ρ_{init}$ spectrum, the intermediate behavior is qualitatively different. We argue that for each rule type, when increasing $ρ_{init}$ from 0 to 1, the phases occurring always obey the order illustrated in the diagram, though, for degenerate cases, some of the phases might be missing (e.g., the constant 0 GCA only has the homogeneous all-0 phase).
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Figure 12
Weak agreement region with scaling (very roughly) in $θ \sim \sqrt{d}$ . Increasing the degree $d$ for the stubborn/volatile independent GCAs and the anticonformist GCA, while scaling the weak agreement region (very roughly) as $\sqrt{d}$ . Samples were obtained as described in Fig. 5.
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Figure 13
Weak agreement region with $θ \in {0, 1}$ . Increasing the degree $d$ for the absolute majority and stubborn/volatile independent GCAs, while keeping the weak agreement region constant. Eventually, all GCAs behave like the absolute majority GCA. Samples were obtained as described in Fig. 5.
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Figure 14
Length of limit cycles for the anticonformist GCA (code 001011). We show the length of the limit cycles for the data collected for Fig. 5, i.e., 4-regular graphs and different initial densities. (Left) Probability that a sample has a limit cycles length $c \leq 2$ . Since we only sample few such long limit cycles, we combine the data for the different initializations $ρ_{init}$ on the (right). The dashed grey line shows the minimal resolution we are limited to due to our sample size, which was 1024 for each of the 100 different initial densities.
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Figure 15
Illustration of chaotic phase for anticonformist GCA 001011: evolution of distances of two close-by trajectories. For GCA 001011 we randomly generated an initial configuration with length $n = 10 000$ , and density $ρ_{init} = 0.5$ and a close-by configuration with $ε \cdot n$ different bits; $ε = 0.01$ . While both configurations' trajectories remain in the chaotic regime, we measure their average Hamming distance and plot the probabilities for 100 such experiments averaged.
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Figure 16
Histogram of the density $ρ$ for transients with a chaotic behavior (after relaxation and before convergence to an attractor) for the anticonformist GCA 001011. We sample 1024 graphs of size $n = 100$ with the dynamics of the rule 001011 started at $ρ_{init} = 0.2$ . The first 100 time steps after the start of the dynamics and last 100 time steps before reaching the attractor are removed for every sampled trajectory. Since the convergence is fast in the phase where an attractor is reached rapidly (see Fig. 5), such a cut-off effectively removes all transients that converge rapidly (less than 200 time steps long). The histogram shows the density of the remaining trajectory which exhibits a chaotic behavior. The red lines mark the phase transition measured empirically for rule 001011 between the phase of rapid convergence to the homogeneous state (left and right side) and the chaotic phase (between the red lines). Note that during this time almost all samples lie within the regime of the chaotic phase.
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Figure 17
Anticonformist GCA 001011—Extrapolation. Extrapolation of the crossover points from Fig. 7 to $p \to \infty$ . The plot shows the distance to the critical $ρ_{init}^{c}$ , the possible location of phase transition points. Under the assumption of an exponentially fast convergence in $p$ , the best fit seems to lie at roughly $ρ_{init}^{c} = 0.2168 \pm 0.0001$ .
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Figure 18
Absolute majority rule 0011—Extrapolation. Extrapolation of the theoretical BDCM and empirical experiments to $n \to \infty$ and $p \to \infty$ , respectively. (Left) Scaling of the distance to different values of extrapolated $ρ_{init}^{*, \infty}$ for data for $p = 1, . . ., 5$ for the BDCM. Assuming exponentially fast convergence in $p$ , a $ρ_{init}^{*, \infty} \sim 0.7875 \pm 0.005$ is reasonable. (Right) The location $ρ_{init}$ of the slowest average convergence for experiments over 4,096 samples of random regular graphs and initial configurations for a given graph size $n$ are recorded, and then extrapolated to $n \to \infty$ . With this we estimate that in the large system limit, the transition happens at $\sim 0.785 \pm 0.005$ .
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Figure 19
Stubborn independent GCA $00 + 11$ —Extrapolation. As in Fig. 18 we show different extrapolations of the BDCM's predictions for dynamical phase transitions for fixed $p$ and extrapolate them to $p \to \infty$ , concluding that convergence within $\sim 0.73 \pm 0.005$ . The extrapolation for the empirics is taken from Ref. [19].
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Figure 20
Scaling of the nonrattling nodes for different $ρ_{init}$ . For some $ρ_{init}$ the nonrattling nodes scales logarithmic in $o (n)$ , for others they are a constant fraction in $Θ (n)$ . The red value marks the intermediate threshold we selected to classify the all-rattling and partially rattling phases in Fig. 10 (right), which had 0.7% of stable nodes.
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