Self-organized criticality in aquatic robot swarm

Abstract

Self-organized criticality (SOC) provides a universal framework for understanding the emergence of scale-invariant behaviors in complex systems, yet its physical realization in programmable artificial systems remains challenging. We present an aquatic robot swarm, in which individuals interact via optical attraction and hydrodynamic repulsion, demonstrating key SOC features: avalanches with sizes and durations following power-law distributions, stability of scaling exponents under system upscaling, and self-organized evolution toward a steady state independent of system parameters. In the presence of external stimuli, the system not only maintains criticality but also spontaneously forms directed structures and exhibits adaptive behaviors such as collective pushing, showcasing emergent capabilities arising from local interactions. This work provides a controllable experimental platform for studying SOC in dynamic physical environments and provides insights for developing adaptive autonomous swarm systems that require minimal programming.

Light-wave interactions drive aquatic robotic swarms to criticality for decentralized collective behaviors.

INTRODUCTION

Self-organized criticality (SOC) is a class of dynamical phenomena widely observed in nature (1–6). In this state, systems exhibit scale invariance, where the statistical distributions of physical quantities in space and time follow power-law patterns. Typical SOC systems (e.g., sandpile models, forest fires, and earthquake activity) exhibit universal dynamical behaviors, with scaling exponents showing similarities across different systems (7, 8). Core features also include the universality of finite-size scaling and the spontaneous evolution of systems to critical states through internal interactions without requiring fine external parameter tuning (9). However, although the SOC characteristics of various natural systems have been extensively studied, constructing and analyzing SOC properties in artificial systems—expanding fundamental understanding of SOC system self-organization and attempting to leverage it for emergent collective behaviors—remains a fascinating and worthwhile endeavor.

Constructing SOC in artificial systems faces dual challenges: reproducing the emergence of critical states due to intrinsic dynamics while overcoming inherent constraints of traditional physical systems in environmental coupling and control dimensions. For instance, experimental SOC platforms based on ultracold atomic gases, while capable of exhibiting critical features through excitation and decay processes, operate in highly isolated environments that severely restrict energy or information exchange with the surroundings, and their tunable parameters are often limited (e.g., excitation density) (10, 11). On the other hand, robotic swarms—with their programmable individual behavioral rules, flexible environmental interaction mechanisms, and multiparameter independent control—have become carriers for studying diverse physical theories or emergent phenomena, such as self-organization (12–15), emergent states (16, 17), phase separation (18, 19), and emergent chirality (20). Several robot swarm systems have been reported (21, 22), such as a well-known system using approximately a thousand robot agents to achieve self-assembly behaviors such as pattern formation through local interactions (23, 24). However, existing research has largely focused on swarms operating in static, inert environments (e.g., smooth surfaces), lacking substantial physical coupling with a dynamic medium, and long-range interactions mediated through the environment remain difficult to achieve. Furthermore, while magnetically driven microrobotic swarms demonstrate precise control over collective motion and show great potential in fields such as sampling and targeted delivery—exhibiting diverse self-organized behaviors through coupling with fluid environments—their motion is entirely dependent on external global field actuation (25–27). The individuals lack autonomy and typically do not interact directly with one another, which limits the study of their intrinsic self-organizing dynamics, particularly critical phenomena.

Thus, existing platforms leave an underexplored gap along two key dimensions: individual autonomy and physical environmental coupling. This study aims to bridge this gap by proposing an aquatic robot swarm (ARS) platform. Each robot is an active unit with autonomous motion, environmental sensing, and implicit communication capabilities. Their interactions are jointly mediated by visible light signals (attraction) and hydrodynamic wave feedback (repulsion), naturally forming a decentralized, physically realized nonlinear feedback loop within the environment. Experimental and simulation results show that the system spontaneously generates power-law–distributed correlated clusters, with scaling exponents similar to natural SOC systems. By varying the system size, we further validate the universality of finite-size scaling. Moreover, the collective dynamics drive the system to a critical state that is independent of initial conditions. This critical state remains stable under external perturbations and can give rise to adaptive functionalities such as autonomous object pushing, thereby providing a distinct and controllable experimental model for studying functional SOC in dynamic physical environments.

RESULTS

Design of the ARS

Our goal was to design an artificial robot swarm system that can exhibit characteristics of SOC. The individual robots not only need to have basic capabilities such as independent operation, environmental perception, and local interaction but also require feedback loops in the system to achieve SOC characteristics such as self-organization to a steady state and nontrivial scaling behavior. Thus, we introduced implicit communication based on visible light and chose a fluid environment, aiming to use the information exchange and mechanical forces between robots as components of the SOC feedback loop.

The robot we designed is shown in Fig. 1A. The robot, with a volume of ~80 cm³, is equipped with computing, mobility, perception, and communication capabilities. The core of its unconventional movement mode lies in the three circular thermoplastic polyurethane (TPU) films (4 cm in diameter) symmetrically distributed on the sides of the robot (Fig. 1B). Vibration motors are attached to the centers of these films. By controlling the operation of the motors, the films can vibrate. The reaction force generated by the water waves from the vibrating films enables the robot to move in any direction on the water surface. In addition, this movement produces negative feedback for the aggregation behavior of other individuals. We calculated the mean squared displacement $\text{MSD}\left(t\right)=\langle {\mid \overrightarrow{r}\left(t\right)-\overrightarrow{r}\left(0\right)\mid }^2\rangle$ of the robot by averaging the trajectories of individual robots over time (Fig. 1C). On a longer time scale, the robot’s displacement is almost proportional to time, indicating ballistic motion. Fitting the MSD in the ballistic state, we found that the robot’s average operational speed is 1.7 ± 0.1 cm/s. This mode of movement not only endows the robot with flexible mobility on the water surface but also ensures that they do not lose functionality in case of collisions or congestion. The robot emits visible light through light-emitting diodes (LEDs) distributed on its sides and can perceive the direction of light signals to achieve phototaxis, thereby generating aggregation behavior. The robot detects the direction of the light source by comparing the relative voltage values of the three photoresistors installed on the sides of the robot’s body. It adjusts the input voltage values (0 to 3 V) of the three motors according to the relative angle, changes its movement direction, and thus approaches other luminous individuals (Fig. 1D and fig. S1).

Fig. 1. Design of ARS.

(A) Each robot is equipped with three vibration motors attached to three TPU films with a diameter of 3 cm to achieve movement on the water surface. The robot detects the direction of the light source through three photoresistors. The robot communicates information with other robots by controlling the brightness of the LED. (B) The TPU film is secured to the robot body with an edge and an O-ring to ensure sealing. (C) The MSD of three repeated experiments of the robot. Inset: 100-s-long trajectories of robot in three repeated experiments. Scale bar, 40 cm. (D) The azimuth of the light source $\mathrm{\omega }$ is calculated from the light intensity values $l$ obtained by three photoresistors, and phototaxis is achieved by changing the drive voltage of the vibration motors. (E) Multiple robots perform phototaxis simultaneously and exhibit collective behavior, with the robots aggregating and forming subclusters of varying sizes.

Figure 1E shows the evolution of the positions of 10 robots on the water surface. At $t$ = 1 s, the individuals are dispersed. The robots have the same light intensity, and the distance between adjacent individuals is within the recognition range of the photosensitive sensor. Because of the phototactic characteristic, a group aggregation behavior occurs. At $t$ = 30 s, a group consisting of nine individuals is formed. After the individuals aggregate, the brightness in the aggregation area increases. The initially aggregated group changes the movement direction of the surrounding individuals, causing individuals that were not close before to approach each other. However, because the water waves generated by the vibrating films of the individuals exert a repulsive force on other individuals, the aggregated individuals also disperse. At $t$ = 60 s, the group disperses into four smaller clusters. In the ARS system, the individuals achieve positive feedback through implicit communication and phototaxis, driving continuous aggregation. In contrast, water wave disturbances and random movements introduce negative feedback, inhibiting the unlimited expansion of the aggregated group. The long-range nonlinear interactions between individuals through the optical field and hydrodynamic field of the environment constitute the feedback network of SOC.

Power-law–distributed cascade behaviors

We first examined the cascade behaviors in the swarm experiments, which is one of the important characteristics of SOC systems and is also known as avalanches. Interactions between adjacent dynamic variables lead to activity bursts in the form of avalanches, which may involve the entire system. In the ARS, interactions between individuals due to phototactic attraction and feedback repulsion result in the formation of clusters of different sizes and durations. To quantify these cascades caused by aggregation in the robot swarm, we proposed the concept of “correlated cluster.” A correlated cluster is a substructure composed of adjacent individuals in the swarm that are correlated, as shown in Fig. 2A. The determination of a correlated cluster starts from the concept of neighborhood. Assuming that $D$ is a collection of individuals in a certain cluster, then the neighborhood of a point $p$ is

$$N_{\mathrm{\epsilon }}\left(p\right)=\{q\in D\mid \text{dist}(p,q)\le \mathrm{\epsilon }$$ (1)

where $\text{dist}(p,q)$ is the Euclidean distance between individuals and ε is an artificially selected distance threshold. Furthermore, for individuals $q$ and $p$, they are considered directly correlated if

$$q\in N_{\mathrm{\epsilon }}\left(p\right) \text{and angle}({\overrightarrow{v}}_p,{\overrightarrow{v}}_q)\le \mathrm{\theta }$$ (2)

where ${\overrightarrow{v}}_p$ and ${\overrightarrow{v}}_q$ are the velocities of individuals $p$ and $q$ and θ is an artificially selected angle threshold. All experimental data were processed using fixed thresholds: the distance threshold ε = 11 cm and the angular threshold θ = 60°. Direct correlation describes pairs of individuals in the cluster that are in close proximity and have similar velocity directions; this is a short-range, direct correlation relationship. However, correlations in the cluster can be transmitted; the influence of one individual can be continuously passed through other individuals, and an individual at one end of the cluster can also be correlated with an individual at the other end. Naturally, we define correlation such that if there is a chain of individuals $p_1,\dots ,p_n$, $p_1=q$, $p_n=p$, and $p_{i+1}$ and $p_i$ are directly correlated, then individual $p$ is correlated with individual $q$. This relation is transitive and symmetric. Figure 2A illustrates the correlation relationships among individuals, with multiple individuals divided into different sets based on their correlation. We define the largest set composed of pairwise-correlated individuals in the cluster as a correlated cluster, a cluster $C$ with respect to ε and θ is a non-empty subset of $D$, and $\forall p,q\in D$, if $p\in C$, and individual $p$ is related to individual $q$ with respect to ε and θ, then $q\in C$. We use Algorithm 1 to calculate the correlated clusters of a swarm at a specific moment (Supplementary Text). In addition, we can define the size of a correlated cluster as

$$S\left(C\right)=\mid C\mid$$ (3)

Fig. 2. Correlated clusters of the ARS.

(A) The correlated cluster of a swarm composed of four robots is displayed, with relevance determined by the Euclidean distance and velocity angle between robots. [The velocity vectors, and thus the angles, were calculated from position data tracked across consecutive video frames using a time difference of $\mathrm{\Delta }t$ = 5 frames (~3 s), i.e., $\overrightarrow{v}=\mathrm{\Delta }\overrightarrow{r}/\mathrm{\Delta }t$.] This illustration includes two correlated clusters of sizes 3 and 1, respectively. Arrows represent the direction of the robots’ velocities, individuals connected by red dots are directly relevant, and individuals enclosed by black dashed lines belong to the same correlated cluster. The color of the robots and the background corresponds to the size of the correlated cluster. (B) The typical composition of correlated clusters in a stable system with a scale of $N$ = 64. The color represents the size of the correlated cluster to which the robot individual belongs. Arrows are used to indicate the direction of the robot’s velocity. (C) The number of occurrences of correlated clusters of different sizes at various time throughout the experimental period. The color of the area under the curve corresponding to the size of the relevant clusters. Panel (C) uses the same colormap as panel (B). (D) The empirical probability density distribution of the sizes of correlated clusters and (E) the duration times of correlated clusters, with the red dashed line as the fitting curve. The P value indicates the goodness-of-fit of the power-law distribution, with $P$ > 0.1 suggesting a plausible fit (see Materials and Methods). (F) The average duration times of correlated clusters of different sizes, with the red dashed line as the fitting curve.

We placed 64 robots uniformly on the water surface, started them simultaneously, captured experimental images, and obtained the position changes of the robots through image processing. Figure 2B shows the typical composition of correlated clusters in the robot swarm. There are substantial fluctuations in the sizes of the formed correlated clusters in the system, with the smallest and largest correlated clusters differing by about an order of magnitude. Because the influence between individuals is a long-range action with transitivity and the feedback action between individuals is not uniform in direction, it can be seen that adjacent individuals in the swarm may not belong to the same correlated cluster, while two nonadjacent individuals may belong to the same correlated cluster. Figure 2C shows the changes in the number of correlated clusters of different sizes over time. At the beginning of the experiment, small-scale ($S$ < 4) correlated clusters mainly compose the system. As individuals aggregate through phototaxis, the number of large-scale ($S$ ≥ 4) correlated clusters increases, and after about 100 s, the number of correlated clusters of different sizes stabilizes.

Figure 2D shows the empirical probability distribution curve of the sizes of correlated clusters in a typical swarm system ($N$ = 64), based on 430 frames of swarm correlated cluster data comprising 9023 data points. The empirical probability distribution function of the correlated clusters can be described by a power law. The calculation results of correlated clusters are affected by the selection of angle and distance thresholds, but the selection of thresholds does not change the power-law characteristic of the distribution (fig. S3). The swarm system is not an infinite system, and when the size of the correlated cluster approaches the system size, the distribution deviates from the power law to some extent and shows a cutoff. We used maximum likelihood estimation (MLE) to obtain a scaling exponent of τ = 1.68(3). The scaling exponent is close to that of other complex systems speculated to exhibit SOC, such as brain neurons (28), forest fires (29), and atomic gases (10) (the scaling exponents of these systems are 1.5, 1.4, and 1.37, respectively). This result demonstrates an important characteristic of SOC, namely, power-law–distributed cascade behaviors.

In addition to spatial distribution, we also analyzed the temporal aspect. The duration of a correlated cluster $T$ is defined as the number of seconds during which the composition of individuals in the cluster remains unchanged. The durations of correlated clusters statistically recorded in the experiments ranged from close to zero to as long as about 400 s. As shown in Fig. 2E, their distribution also follows a power-law statistical pattern, with a scaling exponent of α = 1.54(2). The horizontal alignment of multiple points, particularly in the distribution tail, reflects the finite-sample statistics of rare events in the long-tailed distribution. Moreover, as shown in Fig. 2F, we also statistically analyzed the average duration $⟨T⟩$ of correlated clusters of specific sizes, revealing a power-law relationship between them, with a scaling exponent of γ = 0.74(4). Power-law distributions in both space and time indicate that the system has spatiotemporal scale invariance. Moreover, the power-law distribution of cluster lifetimes suggests that $1/f$ noise may be present. We therefore perform a spectral analysis $S\left(f\right)$ of the clusters to examine how their power spectral density varies with frequency $f$. The analysis reveals a power-law scaling in the range $f\in [0.01,1]$ Hz with exponent β = 1.26(3) (fig. S4). As a temporal “fingerprint” of SOC, $1/f$ noise is ubiquitous in critical systems (4, 7, 30).

Finite-size scaling universality

Next, we analyzed the effect of system scale changes on the scaling behavior of the system. We constructed a two-dimensional (2D), lattice-based simulation to analyze the behavior of larger-scale systems and precisely quantify the impact of system parameters. In this model, continuous space is discretized into an $L\times L$ square lattice, with each lattice representing an actual area of 7 cm by 7 cm. The robot individuals are simplified as point particles occupying a single lattice, and their movement directions are limited to the eight-neighbor discrete directions of the lattice. Each lattice site is assigned an excitation field value, and the presence of a robot individual increases the excitation field of surrounding sites, with values inversely proportional to the square of the distance. This simulates the light-emitting effect of individuals in the physical system. Meanwhile, when an individual moves, the excitation field of the lattice site in the opposite direction of its velocity decreases, simulating the water wave force generated by the vibrating film. The superposition of these two excitation fields simulates the SOC feedback loop in the physical experiment. During movement, individuals tend to migrate toward the direction of higher excitation field, but their perception is limited to the excitation field information of the eight surrounding lattice sites (see Supplementary Text for more details). In the lattice simulations, correlated clusters were identified using the same relational definitions as in the experiments but with thresholds adapted to the discrete environment: a distance threshold of 2 lattice units and an angular threshold of θ = 90°. Figure 3A shows a typical excitation field distribution map and the composition of local correlated clusters in a lattice simulation with 1000 robots, where $L$ = 230. In the simulation, similar to the physical experiment, individuals formed correlated clusters ranging in size from 1 to 70. The composition and size of each cluster continuously evolved during the simulation, but after a certain period, the number of correlated clusters of different sizes remained stable.

Fig. 3. Statistical results of correlated clusters in simulations with varying scales.

(A) A lattice simulation with a scale of 1000, where the colors in the left figure represent the magnitude of the lattice excitation field and the right side shows the composition of some correlated clusters. The color represents the size of the correlated cluster to which the robot individual belongs. Arrows are used to indicate the direction of the robot’s velocity. (B) The empirical probability density distribution of the sizes of correlated clusters and (C) the duration times of correlated clusters in simulations of different scales, with the red dashed line represents the fitted curve of the simulation data for $N$ = 1000. (D) The average duration times of correlated clusters of different sizes in simulations of different scales, with the red dashed line represents the fitted curve of the simulation data for $N$ = 1000.

We kept the swarm density ρ = 5 robots/m² constant and conducted simulations with scales of $N$ = 100, 1000, and 10,000, each lasting for 2 × 10³ frames, and statistically analyzed the distribution of correlated clusters. For the simulations with a scale of $N$ = 1000, we observed that the sizes and durations of correlated clusters still followed power-law distributions (Fig. 3, B and C), and there was also a power-law relationship between the sizes and durations of correlated clusters (Fig. 3D). The scaling exponents τ, α, and γ are 1.16(2), 2.67(1), and 0.11(1), respectively. Although these values differ from the results of the physical experiment, they remain within the same category of phenomena under power-law scaling and align with findings reported for other systems in the SOC field (29). This discrepancy stems from inherent simplifications in the model. First, the simulation maps continuous physical time to discrete steps (where one simulation time step corresponds to ~4 s in the physical system), thereby altering temporal dynamics (e.g., scaling α). Second, spatial discretization is used in the simulation: Unlike the continuous motion of physical robots, the simulation restricts individual movement to eight lattice directions, reducing directional diversity. Last, the excitation field approximation in the simulation neglects the subtle nuances of real-world light-wave coupling, and interactions such as collisions—which in the aqueous environment involve complex momentum transfer, wave scattering, and wetting-edge effects—are reduced to a simple exclusion rule. These simplifications collectively bias the statistical properties of cluster fragmentation and reorganization. Crucially, despite quantitative shifts, the simulation qualitatively reproduces the core SOC features—specifically, power-law–distributed cascade behaviors. This consistency underscores the robustness of SOC in our system across implementation scales. Despite a 100-fold difference in scale, we observed that the system’s power-law regularity persists, and the scaling exponents remain essentially fixed (with SDs of 0.0033, 0.0015, and 0.00048 for τ, α, and γ, respectively). We observed similar scale invariance in physical experiments with different scales (fig. S5). This result indicates that the SOC characteristics of the robot swarm system are not affected by changes in system scale, exhibiting the universality of finite-size scaling, which is also one of the characteristics of SOC systems. This result originates from the self-similar structure of the system, that is, the interaction patterns between individuals within subclusters are similar to the overall swarm dynamics.

Self-organization to steady state

Unlike traditional phase transition systems that require precise tuning of system parameters (such as temperature to the critical temperature $T_{\mathrm{c}}$), SOC systems can spontaneously evolve to a critical state through their internal dynamics. To verify whether the robotic swarm has this characteristic, we substantially increased the individual light intensity and swarm density in both the physical experiment and simulation shown in Fig. 4A (in the physical experiments, both the driving voltage of the light-emitting circuit and the swarm density were four times those used in the previous experiments; in the simulations, the individual light-emission intensity and the swarm density were also four times those of the earlier simulations.). Nevertheless, as shown in Fig. 4B, the size distribution of correlated clusters within the system still follows a power law. The scaling exponents obtained from the physical experiments and simulations are τ_exp = 1.35(2) and τ_sim = 1.23(1), respectively, both of which are consistent with our earlier SOC experiments and simulations reported in this study. This indicates that the system can still exhibit SOC characteristics even when parameters undergo substantial changes.

Fig. 4. Shannon entropy and correlated cluster distributions in experiments and simulations with different system parameters.

(A) Physical experiment and lattice-based simulation with increased robot luminosity and swarm density. (B) Correlated cluster distributions obtained from physical experiments and simulations. Dashed lines show the maximum likelihood estimate of the exponent. exp, experiment; sim, simulation. (C) Time evolution of Shannon entropy for correlated clusters in simulations with $N$ = 500 agents, calculated over 500 frames. Each curve is the average of three independent runs. When light intensity $I$ < 4 cd, the Shannon entropy increases to distinct steady-state values, with these values rising as $I$ increases. For $I$ ≥ 4 cd, the system entropy stabilizes near a common fixed value of 3.57 bits. (D) Correlated cluster distributions under different light intensities. For $I$ < 4 cd, the curves deviate from a power-law distribution. When $I$ ≥ 4 cd, the distributions align with a power law, with the red dashed line representing the fitting curve for the $I$ = 6 cd case. (E) Shannon entropy evolution under varying robot densities ($N$ = 500 agents, 500 frames). Each curve is the average of three independent runs. For densities ρ < 3.99 robots/m², the entropy exhibits logarithmic growth, followed by stabilization at distinct steady-state values. When ρ ≥ 3.99 robots/m², the entropy converges to a fixed value of 3.44 bits. (F) Correlated cluster distributions for different densities. For ρ < 3.99 robots/m², the curves deviate from a power-law distribution. At ρ ≥ 3.99 robots/m², the distributions follow a power law, with the red dashed line showing the fitting curve for ρ = 10.20 robots/m².

To further analyze the effects of light intensity and density, two system parameters, on the system state, we introduced the Shannon entropy as a measure. Shannon entropy $H$ is a parameter used to quantify the uncertainty or degree of disorder of information (see Materials and Methods for details). We used Shannon entropy to characterize the uncertainty of the size distribution of correlated clusters: The higher the entropy value, the greater the diversity of correlated clusters. We first studied the complete time evolution of Shannon entropy in the simulations under different light intensities $I$. The number of individuals $N$ = 500, the field size $L$ = 160, and the maximum light intensity produced by individuals ranged from 1 to 9 cd. Figure 4C shows that the time evolution of Shannon entropy exhibits substantial nonlinearity. The initial Shannon entropy was about 1.4 bits, increased logarithmically to a certain value, and then stopped growing. When $I$ < 4 cd, the system Shannon entropy approached different fixed values, which increased with increasing light intensity. When $I$ ≥ 4 cd, the curves of Shannon entropy change were similar, and all eventually approached a fixed value of 3.57 bits, which remained constant over a wide range of light intensities (with an SD of 0.12 bits). Similar distinctions were also reflected in the probability density distribution results of correlated cluster sizes, as shown in Fig. 4D. When $I$ < 4 cd, the curves deviated from a power-law distribution. When $I$ ≥ 4 cd, the curves followed a power-law distribution, with a scaling exponent of around 1.15(1).

We further analyzed the effect of swarm density $\mathrm{\rho }$, as shown in Fig. 4 (E and F). In the simulation with $N$ = 500 individuals, the system density $\mathrm{\rho }$ was varied by altering the field size. When ρ < 3.99 robots/m², the system entropy stabilized at several scattered values. In contrast, when ρ ≥ 3.99 robots/m², the entropy converged to a fixed value of 3.44 bits (SD 0.20 bits), similar to the steady-state behavior observed in the light intensity comparison simulations. This threshold also determined whether the size distribution of correlated clusters followed a power law.

The threshold behavior observed for both light intensity and density originates from whether the system can form an effective excitation-field feedback network. When either the light intensity $I$ or the density ρ falls below their respective thresholds ($I$ < 4 cd or $\mathrm{\rho }$ < 3.99 robots/m²), the excitation field exhibits insufficient spatial gradients or fragmented distribution, preventing the long-range attraction and short-range repulsion from establishing sustained nonlinear feedback. As a result, the system fails to exhibit SOC characteristics. Only when both $I$ and ρ exceed these thresholds does the excitation field have sufficient strength and spatial continuity to drive the system—via local interactions—toward SOC. This leads to the emergence of universal power-law statistics and a steady-state entropy that is independent of the system parameters. The spatial profiles of the excitation field under different parameter regimes are illustrated in fig. S7.

The distinction marked by the threshold indicates a parameter-dependent state transition process in the system. When the system parameters exceed the threshold, although the parameters (light intensity and density) differ, the stable values of Shannon entropy for correlated clusters are identical, the distribution curves follow a power law, and the scaling exponents remain the same. This shows that the system can self-organize into a steady state independent of the system parameters, demonstrating another characteristic of SOC.

The independence of SOC characteristics from perturbations

The experimental and simulation results have already demonstrated the three characteristics of SOC in the ARS. Next, we explore the changes in the collective behavior of the swarm system when external perturbations exist in the environment. To this end, we designed a passive stimulus source (Fig. 5A). This device itself does not emit visible light (except for the luminous markers used for motion tracking) and lacks autonomous mobility, but it emits a uniform infrared signal in all directions. Each robot is equipped with an infrared receiver that can adjust its own light intensity in real time based on the received local infrared signal strength, with the two being positively correlated. Although a single robot cannot directly perceive the direction of the stimulus source, after aggregation through phototaxis, the robot swarm—due to distance differences between individuals and the stimulus source and mutual occlusion—forms a brightness gradient pointing toward the stimulus source within the aggregated subclusters (Fig. 5B). Guided by this gradient, even if the movement directions of a few individuals near the stimulus source may deviate from it, most individuals within the subcluster still move toward the stimulus source. This results in a net displacement of the entire subcluster toward the stimulus source (Fig. 5C), ultimately pushing the stimulus source to move. Figure 5 (D and E) shows the experimental scenario and the spatial density distribution of robots, respectively, under conditions with 64 robots and two stimulus sources. Figure 5F presents the velocity variation curve of one stimulus source during the experiment. It can be observed that some robots gather near the stimulus source and exert a continuous push on it. This not only increases the robot density in the area around the stimulus source but also promotes the formation of larger correlated clusters nearby. Once some robot individuals begin to push the stimulus source, they exhibit highly sustained pushing behavior, and the stimulus source can be stably transported at an average speed of 0.63 ± 0.3 cm/s. This phenomenon indicates that, without explicit transport commands or environmental object location information, the robot swarm has the capability to achieve preliminary transport of environmental objects through local interactions. The aggregation response of individual robots to local stimuli leads to coordinated group behavior, which is not explicitly programmed but emerges from the interactions between the robots.

Fig. 5. ARS system under stimulus-source perturbations.

(A) Physical stimulus source fabricated via stereolithography 3D printing, integrating an infrared emitter array, battery, and voltage regulator to provide stable external infrared signals. The tracking cap is omitted for clarity. (B) The ARS moves toward the stimulus source, with the red arrows representing the direction of individual velocities. (C) Experimental setup with eight robots and one stimulus source. (D) Typical experiment with two stimulus sources and $N$ = 64 robots, initially randomly placed. Green lines show stimulus trajectories over 600 s. (E) Robot density map corresponding to (D), with color indicating probability density per unit area calculated via Gaussian kernel density estimation. (F) Speed variation of a stimulus source during pushing. (G) The empirical probability density distribution of the sizes of correlated clusters and (H) the duration times of correlated clusters in experiments with varying numbers of stimulus sources; the red dashed line represents the fitted curve for the data obtained with $N_{\text{sti}}$ = 3. Data for each condition are pooled from three independent trials, comprising over 18,000 video frames in total. (I) The average duration times of correlated clusters of different sizes in experiments introducing different numbers of stimulus sources, with the red dashed line representing the fitted curve for the data obtained with $N_{\text{sti}}$ = 3. (J) With four fixed stimulus sources on one arena edge, 32 robots perform self-organized aggregation toward them; red line traces swarm center of mass. (K) With two fixed stimulus sources (40 cm apart), 32 robots first form local clusters around them and then connect into a stable bridge structure.

Notably, the introduction of the stimulus source does not disrupt the system’s criticality. Even when different numbers of stimulus sources are introduced, throughout the experiment, the statistical results of the swarm’s correlated clusters consistently follow three sets of power-law patterns (Fig. 5, G to I). The introduction of stimulus sources causes differences in individual light intensity, which, in turn, leads to an uneven distribution of attraction between individuals. The results show that this heterogeneity only affects the spatial distribution of aggregation behavior and does not destroy the feedback network. The system maintains scale-free statistics and exhibits SOC.

To further investigate the self-organizing behavior of the swarm when stimulus sources serve as spatial reference points, we added two scenarios with fixed stimulus sources. In the first experiment, four stimulus sources were fixed at the midpoint of one side edge of the arena (Fig. 5J). Under the control of only local phototaxis rules, 32 robots gradually aggregated toward the locations of the stimulus sources and formed stable high-density regions around them, indicating that stimulus sources can act as spatial anchor points to guide the self-organized positioning of the robots. In the second experiment, two stimulus sources were fixed in the center of the arena, 40 cm apart (Fig. 5K). The robots first formed local aggregation clusters around the two stimulus sources, respectively. Over time, a connecting structure composed of robots emerged between the two clusters, eventually forming a sustained “bridge” morphology. This structure was not preprogrammed but spontaneously emerged from the robots based on local light intensity gradients and neighbor interactions, demonstrating the system’s potential for spatial structure emergence under multiple stimulus source conditions.

DISCUSSION

The fascinating aspect of SOC lies in its seemingly paradoxical nature: The system dynamically balances on the edge between stability and instability, maintaining order while also nurturing diversity. In this study, we successfully implemented the core characteristics of SOC in an artificial robot swarm system by constructing a feedback mechanism coupled with optical and hydrodynamic fields. Both experiments and simulations demonstrated that the system not only exhibits power-law–distributed cascade behaviors but also shows the universality of finite-size scaling and steady-state evolution independent of initial conditions. Unlike traditional SOC models (such as the sandpile model) that rely on short-range interactions, our system simulates the critical dynamics widely present in natural systems through long-range nonlinear feedback (optical attraction and hydrodynamic repulsion). This provides a framework for studying critical phenomena in complex systems on a controllable physical platform. The robustness of the system under disturbances and its object-transporting behavior not only offer physical validation for SOC theory but also imply a deeper commonality of critical systems: Through local nonlinear interactions, the system can achieve functional emergence without global control.

Biological systems operating at critical points also exhibit critical characteristics. For example, neural networks display neuronal activity with power-law–distributed cluster sizes to maximize information capacity (31–34), while flocks of birds maintain scale-free correlations to balance order and flexibility during flight (35–37). Our robot swarm achieves similar benefits: Scale-free correlated clusters allow rapid adaptation to stimuli while maintaining structural coherence; the stability of scaling exponents across different group sizes is analogous to the conservative scaling laws in biological systems, such as the metabolic rate scaling law that remains unchanged across organisms spanning several orders of magnitude in size (38). These similarities suggest that SOC may serve as a universal framework for designing adaptive artificial systems with biological-level resilience.

In the field of complex systems, our system can act as a controllable platform to quantitatively explore the mechanisms of critical phase transitions through programmable parameters, bridging the gap between traditional theoretical models and real systems. It provides a scalable experimental tool for the study and modeling of complex science. In the field of robotics, the bioinspired design reveals a path for the physical realization of swarm intelligence: Through the coupling of local interaction rules (phototaxis and mechanical feedback), global coordinated behavior (such as target capture and transport) can emerge. The underlying SOC state substantiates this approach by enabling robust, decentralized control without explicit programming, as evidenced by the stimulus-source experiments where the swarm spontaneously performed pushing and structured aggregation. The system’s ability to form sustained spatial patterns, such as bridges between multiple stimuli, further reveals its potential for adaptive morphological programming via SOC. This physical-field–mediated approach demonstrates that complex collective behaviors can be encoded into hardware interactions, reducing reliance on sophisticated onboard computation and explicit communication. This principle is particularly promising for enabling robust collective intelligence at very small scales, where individual robots may have severely limited computational capabilities. This work provides valuable insights for the development of autonomous swarm systems that can adapt to dynamic environments with minimal programming.

Although this study only involves the concept of correlated clusters in 2D space and examples of physical systems, the concept of correlated clusters has the potential to be extended to 3D space, such as designing underwater or aerial robot swarm systems and analyzing their self-organized critical characteristics based on correlated clusters. In addition, several unresolved issues remain in this research. The observed differences in scaling exponents between our physical experiments and lattice simulations prompt us to further reflect on the essence of “universality” in SOC. Unlike traditional equilibrium phase transitions, the universality of SOC is manifested primarily in the system’s ability to spontaneously evolve into a scale-invariant state with power-law relationships, rather than requiring exact agreement in exponent values—indeed, exponents can be influenced by microscopic details, observational definitions, and implementation specifics (29, 39). Although the platform differences between experiment and simulation inevitably introduce dynamical shifts, both consistently exhibit the core features of SOC, including power-law distributions, finite-size scaling, and self-organization toward a steady state. This consistency indicates that the emergence of criticality is fundamentally driven by the designed local nonlinear feedback mechanism (the interplay of optical attraction and hydrodynamic repulsion), independent of the specific implementation platform. The observed differences in exponent values likely reflect systematic biases introduced by modeling and measurement rather than a fundamental change in the underlying critical behavior. Future refinements to the lattice model, such as incorporating more detailed hydrodynamic interactions or stochastic elements, could help bridge the remaining gap with experimental complexity. Furthermore, the current system only introduced a single type of external perturbation (e.g., static stimulus sources), whereas in real open environments, the superposition of multimodal interferences (e.g., dynamic obstacles and competing targets) may challenge the maintenance of the system’s SOC characteristics. Subsequent research could explore the use of heterogeneous robot designs, multiphysics field coupling, or the introduction of active adaptive algorithms to expand the system’s application boundaries in complex scenarios. These directions will deepen the understanding of SOC mechanisms and advance artificial swarms toward more advanced forms of intelligence.

MATERIALS AND METHODS

Experimental design

Robot hardware design

Each robot (total mass: 40 g) integrates motion, power, control, sensing, and communication modules. The motion module consists of three symmetrically distributed TPU films (diameter: 4 cm, thickness: 0.06 mm) and a dc vibration motor (LCM1020A2945F; rated 10,000 rpm) bonded to the center of each film (Fig. 1B). The vibration motor drives the film into vibration; motor power is controlled via pulse-width modulation. The reaction force from the water surface and changes in robot posture enable locomotion. To ensure waterproofing, the motion module is secured to the robot frame using O-rings and snap-fit fasteners. The robot frame, compression ring, lid, and other components are fabricated via stereolithography 3D printing. By fine-tuning the battery position, the center of mass is located directly below the center of buoyancy, ensuring passive roll/pitch stability.

The core control system is based on the ESP32-C3-MINI-1-N4 microcontroller (operating voltage: 3.3 V, operating current: 80 mA); all electronics are connected to a custom printed circuit board (PCB; JLC Electronics). The robot features an on-board charging circuit (TP4056) and is charged via the USB port on the PCB. Power is supplied by a 3.7-V/350-mAh lithium-ion battery (ZONCELL), regulated to 3.3 V by a dual low-dropout regulator (BL9110-330BPFB). Measured continuous runtime exceeds 1 hour. The sensing module comprises three directional photoresistors (photoconversion factor γ = 0.5) and an infrared receiver; light intensity and infrared signals are quantified via voltage dividers (fig. S2). Communication and coordination among individuals are achieved via lateral LEDs (maximum intensity: 2740 mcd) for optical signaling.

Hardware design of the stimulus source

To introduce a controllable external perturbation for studying the response characteristics of SOC in heterogeneous environments, we specifically designed a passive stimulus source (Fig. 5A). The stimulus source has a total mass of ~90 g, and its housing is fabricated by stereolithography 3D printing. Its core function is to emit infrared signals with a central wavelength of 940 nm (half-intensity angle ϕ = ±17°). The signals are generated by 20 TSAL6200 infrared emitters uniformly distributed around the periphery, operating at a forward current of 100 mA. Power is supplied by a 3.7-V/1500-mAh lithium battery (ZONCELL). A dual low-dropout regulator (BL9110-330BPFB) provides a stable 3.3-V output to power the infrared-emitting circuit. A TP4056 charging management chip is integrated, supporting recharging via an onboard USB-C port. For motion tracking, four LEDs are installed on the top of the stimulus source to illuminate its green cap marker. The housing is entirely black to avoid interfering with the robots’ visible-light—based swarm interactions. All electronic components are integrated on a custom PCB (JLC Electronics). Measurements show that under continuous infrared emission with the marker LEDs illuminated, the stimulus source can operate for more than 1 hour. The housing exhibits good sealing and water-surface stability, meeting the experimental requirements for a durable and stable external signal source.

Robot software design

The robot’s software includes a software upgrade module, a start-stop control module, and a core working module. The program is written based on Arduino. The robot implements remote firmware upgrade through HTTP over-the-air. The device connects to the HTTP server through the Wi-Fi module of the main control chip to download and upgrade the firmware. The start-stop control of multiple robots is implemented on the basis of the Bluetooth module of the main control chip using iBeacon (Bluetooth Low Energy beacon) technology. After the robot completes the software update, it controls start and stop by receiving specific iBeacon signals. The realization of remote upgrade and start-stop control for multiple robots greatly improves the efficiency of large-scale robot swarm testing and experiments. The core working module mainly realizes the reception and processing of sensor data, as well as the control of modules such as motors and LEDs. In different experiments, it implements motion modes such as phototaxis and random motion and realizes feedback to light by receiving stimulus signals.

Robot tracking

Accurate swarm tracking proved challenging due to the large experimental arena, small robot dimensions, and extremely low ambient illumination. These conditions rendered conventional computer vision methods, commercial motion capture systems (limited by dark-environment incompatibility and spatial coverage constraints), and matrix-based marker solutions (requiring high-resolution imaging) ineffective for precise collective motion tracking. To address this, we affixed high-contrast bright yellow circular stickers to each robot’s top surface, with image acquisition performed by a centrally mounted 16-megapixel camera operating at 10 frames per second. Video processing used custom Python scripts implementing a three-stage workflow: (i) color-based segmentation to detect features and generate binary images (white markers against a black background), (ii) subpixel refinement via the Trackpy package for precise centroid localization of markers, and (iii) trajectory linking using predictive algorithms. Final positional data were exported as calibrated spatial coordinates, with robustness under low-light conditions validated through benchmark testing.

Experimental procedure

Before the experiment begins, the firmware and firmware number used for the experiment are set on the HTTP server, and the robots will automatically download the firmware to ensure that the movement logic followed during the experiment is consistent.

The swarm individuals follow the phototactic movement logic. By arranging the signal values obtained from the three photoresistors, the direction of the light source is determined, and then the operation of the corresponding motor is controlled. After the robots update the firmware, they will indicate this by flashing the LED. Once all robots have completed the update, place the robots in the experimental pool.

Before the experiment begins, it is necessary to ensure that the robots are evenly and randomly distributed in the venue and as far away from the edges of the venue as possible. Start recording the video in advance to ensure that the environment is dark and that there is no light interference during the experiment. Use the controller to start all robots, and after starting, the robots will operate according to the predetermined movement logic. Each experiment lasts about 15 min, and after one experiment is completed, rearrange the robots and repeat the experiment. In experiments that require the placement of stimulus sources, the stimulus sources are also randomly placed in the experimental venue.

Statistical analysis

Power-law distribution fitting and goodness-of-fit testing

This study uses the improved power-law fitting method proposed by Marshall et al. (40) to analyze the characteristics of power-law distributions in the data. This method automatically fits a discrete power-law distribution with left and right truncation using MLE, effectively addressing common issues such as finite-size effects and sampling biases in experimental data. It is combined with the goodness-of-fit testing framework proposed by Clauset to quantify the statistical significance of the fitting results. The widely used Kolmogorov-Smirnov (KS) method is adopted for the goodness-of-fit testing. The P value represents the proportion of the KS statistic in the sample distribution that is greater than the KS statistic between the empirical probability density function and the model distribution. A P value greater than 0.1 indicates that the power-law form is a suitable fit for the data. By using a sorting-based greedy search algorithm, the truncation interval is determined. For the distribution of correlated cluster sizes, these ranges are typically on the order of a single decade, primarily because the scale of the robotic swarm is less than two decades. For the distribution of correlated cluster durations, these ranges are approximately two decades. The P values for different experimental datasets are marked in the figures, confirming the validity of the power-law distribution hypothesis.

Calculate the Shannon entropy of the correlated clusters

Shannon entropy is a parameter used to measure the uncertainty or degree of disorder of information. Shannon entropy can characterize the uncertainty of the size distribution of related clusters: The higher the entropy value, the greater the diversity of the clusters, indicating that the system is in a state of disorder or highly dynamic reorganization. Conversely, a lower entropy value reflects an increase in the stability of the cluster distribution. To calculate Shannon entropy, we counted the size distribution of all related clusters in each frame and converted it into a probability distribution. For a related cluster of size $S$, the probability mass function is

$$P\left(S\right)=\frac{\text{count}\left(S\right)}{\sum \text{count}\left(S\right)}$$ (4)

The formula for calculating Shannon entropy $H$ is

$$H=-\sum P\left(S\right){\text{log}}_{2}P\left(S\right)$$ (5)

Supplementary Materials

The PDF file includes:

Supplementary Text

Figs. S1 to S8

Legends for movies S1 to S5

Other Supplementary Material for this manuscript includes the following:

Movies S1 to S5