Human crowds as social networks: Collective dynamics of consensus and polarization

Abstract

A ubiquitous type of collective behavior and decision-making is the coordinated motion of bird flocks, fish schools, and human crowds. Collective decisions to move in the same direction, turn right or left, or split into subgroups arise in a self-organized fashion from local interactions between individuals, without central plans or designated leaders. Strikingly similar phenomena of consensus (collective motion), clustering (subgroup formation), and bipolarization (extreme groups) are also observed in social networks. As we developed models of crowd dynamics and analyzed crowd networks, we found ourselves going down the same path as models of opinion dynamics in social networks. In this paper, we draw out the parallels between human crowds and social networks. We show that models of crowd dynamics and opinion dynamics have a similar mathematical form and generate analogous phenomena in multi-agent simulations. We suggest that they can be unified by a common collective dynamics, which may extend to other psychological collectives. Models of collective dynamics thus offer a means to account for collective behavior and collective decisions without appealing to a priori mental structures.

A shared aim of post-Cartesian psychology is to purge our theories of ‘homunculi’, internal replicas of the very phenomena we are trying to explain. We seek to account for the organization in behavior without subtly assuming it a priori, that is, without attributing it to prior organization in the form of plans, priors, internal models, or other preformed structures in our heads. The most promising way forward, we believe, is to look to principles of self-organization in complex systems. In physical and biological systems, local interactions among many components can lead to the formation of global spatial and temporal patterns (Camazine et al., 2001; Haken, 2004; Schneider & Sagan, 2005). In psychology, such pattern formation is observed at sub-personal (neural), personal (agent-environment), and inter-personal (social) scales (Frank et al., 2009; Grossberg, 2013; Kelso, 1995; Riley et al., 2011; Warren, 2006). Collective behavior offers relatively uncharted territory in which to extend principles of self-organization at the social scale (Goldstone & Gureckis, 2009).

A paradigmatic example is the collective motion of bird flocks, fish schools, and human crowds (Couzin & Krause, 2003; Sumpter, 2010). Collective motion is thought to emerge from local interactions between neighbors in the absence of central plans or designated leaders, although the precise nature of the interactions varies across species. Such ‘flocking’ behavior also exhibits a basic form of collective decision-making (Couzin et al., 2005; Dyer et al., 2009; Leonard et al., 2012; Sridhar et al., 2021), such as a group decision to move in the same direction, to collectively turn left or right, to split into two groups, or to fragment into multiple subgroups.

Strikingly analogous phenomena of consensus (collective motion), clustering (fragmentation), and bipolarization (splitting into two extreme groups) are also manifested in social networks. As we pursued experiments on pedestrian behavior and developed models of crowd dynamics, we discovered that we were going down a path very similar to modelers of opinion dynamics (Flache et al., 2017; Mason et al., 2007). Indeed, models of consensus, clustering, and bipolarization created independently in these domains have an analogous mathematical form. As we analyzed our crowd data as visual influence networks, we realized they had properties very similar to social influence networks (Baek et al., 2021). The purpose of this paper is to draw out the parallels between human crowds and social networks, which suggest they may share an underlying collective dynamics (see also Couzin et al., 2011).

Modeling flocks, schools, and crowds

In ‘macroscopic’ or continuum models of collective motion the aggregate behavior of large crowds is treated as similar to a fluid or a granular material, greatly simplifying the behavior of individual agents. In contrast, ‘microscopic’ or agent-based models describe the local behaviors that generate collective motion (Giardina, 2008; Warren, 2018). The ultimate goal of many theorists is to formally derive macro-level phenomena from micro-level behavior, inspired by the success of statistical mechanics in deriving thermodynamic laws from the kinetic theory of particle motion (Cristiani et al., 2014; Degond et al., 2013; Flierl et al., 1999). Here we focus on microscopic models.

Microscopic models aim to characterize (a) the rules of engagement that govern how an individual responds to a neighbor, and (b) the neighborhood of interaction over which the rules operate and the influences of multiple neighbors are combined. A popular ‘zonal’ model (Reynolds, 1987) posited three local rules operating in concentric zones around each agent. The agent’s acceleration was determined by (i) repulsion from neighbors in a near zone, to avoid collisions, (ii) alignment or velocity matching with neighbors in an intermediate zone, to generate common motion, and (iii) attraction to neighbors in a far zone, to ensure group cohesion. The strength of influence decreased with neighbor distance, and was combined by a weighted average of neighbors in each zone. A cottage industry of physicists and mathematicians began to explore the space of microscopic models in the 1990s, proposing models based solely on alignment (Cucker & Smale, 2007; Czirók et al., 1997; Vicsek et al., 1995), on attraction and repulsion (Helbing & Molnár, 1995; Helbing et al., 2001; Romanczuk et al., 2009), or some combination (Couzin et al., 2002; Grégoire et al., 2003).

Rather than attempting to explain biological behavior, this physical approach sought to derive minimal models having ‘universality’. Something similar occurred in the development of social network models, as agent-based modelers explored different classes of rules, relatively unconstrained by psychological data (Castellano et al., 2009; Flache et al., 2017). Building on what has been learned from these generic models, we advocate for model development that is driven bottom-up by experimental data about the local interactions that yield emergent collective behavior (Warren, 2018) (see Calovi et al., 2018; Gautrais et al., 2012, for a similar approach in fish). We thus attempted to derive a pedestrian model from experiments on individual locomotor behavior (Fajen et al., 2003; Warren & Fajen, 2008) and pedestrian interactions (Dachner & Warren, 2014; Rio et al., 2014), to investigate the resulting crowd dynamics in multi-agent simulations, and to test the model against pedestrian and crowd data. This iteration of experimentation and model construction at the local level, and simulation and analysis of group data at the global level, constitutes what Sumpter, et al. (2012) call the ‘modeling cycle’ for collective behavior. In what follows, we briefly describe our models of collective motion (consensus), fragmentation (clustering), and splitting into two groups (bipolarization), drawing explicit parallels with social influence networks as we go.

Our initial experiments on pairs of participants showed that velocity matching (alignment) captured pedestrian following quite well (Dachner & Warren, 2014). To build a bridge to collective motion, we then studied a participant walking with a crowd in virtual reality (VR), and perturbed the walking direction (heading) or speed of a subset of virtual neighbors (Rio et al., 2018). The results showed that pedestrians adopt the mean heading and mean speed of their neighbors, and that neighbor influence decreases exponentially with distance. This led to a weighted averaging model of collective motion based on alignment, which predicted individual trajectories in real crowd data (Rio et al., 2018), generated collective motion in multi-agent simulations (Warren & Dachner, 2018), and generalized surprisingly well to new experimental scenarios (Wirth & Warren, 2021).

Like most previous models, however, our model was omniscient: the model assumed the positions and velocities of all neighbors were simply given. In contrast, real pedestrians are visually coupled: locomotion is controlled by visual information, and near neighbors often visually occlude far neighbors (Bastien & Romanczuk, 2020; Moussaïd et al., 2011; Ondrej et al., 2010; Strandburg-Peshkin et al., 2013). Our second model was thus derived from experiments on the visual control of following, in which we used VR to manipulate the optical information (Bai & Warren, 2019; Dachner & Warren, 2017, 2019). In the resulting visual model of collective motion (Dachner et al., 2022), each pedestrian cancels the optical motions (expansion and angular velocity) of their neighbors, weighted by the neighbor’s visibility (0–1). This visual model fits the data even better than the omniscient model, generates collective motion in simulation, and explains the human neighborhood of interaction (Wirth et al., 2022). For simplicity, below we report simulations of Rio, et al’s (2018) omniscient model, which closely approximates human data.

Crowds as social networks

Flocks, schools, and crowds are social groups, and the local interactions are often described as social interactions (Couzin et al., 2005; Miller et al., 2013). Even though they are usually modeled as particles moving in a continuous space, they can also be represented as networks or graphs, in which nodes correspond to individuals, edges to connections between them, and edge weights represent the influence of one individual on the motion of another. This seems to be directly analogous to social networks, in which nodes correspond to individuals, edges to communication links, and edge weights represent social influence. But what type of social network is a crowd, exactly?

Spatial, locally connected networks.

Crowds are spatial networks, for the influence between neighbors is determined by their positions in physical space and decreases with distance. Further, because pedestrians visually influence each other, links in the network are limited by visual occlusion (Dachner & Warren, 2019; Poel et al., 2021). Taken together, this results in networks that are locally connected. By contrast, in many social networks such as gossip networks and social media, influence does not depend on spatial position or distance (but see Feliciani et al., 2017). Thus, we represent a crowd network by embedding the graph in a metric space and plotting nodes in their relative spatial positions. A microscopic model can be directly converted into a weighted graph: formally, a graph is a matrix of weights between pairs of nodes, and influence weights between pairs of neighbors can be computed from the model, given their current positions (Cucker & Smale, 2007).

Directed networks.

In social networks, the influence between two individuals may be bidirectional (mutual) or unidirectional (directed); the latter are represented in directed graphs. Opinion models often assume a bidirectional influence, such that an interaction mutually affects the opinions of both individuals, whereas persuasion models often assume a unidirectional influence, in which a source changes the belief of a target (Mason et al., 2007).

In crowd networks, influence is predominantly unidirectional. This follows first from the fact that the human field of view is approximately 180° across, so neighbors ahead of each pedestrian are usually visible while those behind are not. Closed loops are thus rare in a crowd. But even in species with a nearly 360° field of view, like pigeons, the influence in a flock flows from front to back, as each bird follows those ahead (Nagy et al., 2010). We observed the same pattern in human ‘swarms’, groups of participants instructed to walk around a large hall, veering left and right but staying together as a group. When we measured the time-delays between participant trajectories, we found that influence also flows from front to back (Rio et al., 2018). Thus, whereas an opinion network might be represented by an undirected graph with loops, a crowd network is best represented by a directed acyclic graph, which has no loops.

Dynamic networks.

Crowd networks are also dynamic, for they evolve in time as pedestrians move about. Not only do edge weights vary as neighbor positions change, but the topology (connectivity) of the network changes due to dynamic occlusion and a limited field of view. Whereas most social network and flocking models are concerned with the dynamics on a fixed network, visual models reveal that the dynamics of the network are crucial, as neighbors are constantly going in and out of view. The network dynamics cannot be predicted from the current topology, but must be computed from the motions of pedestrians in space. Thus, crowd networks exhibit coevolution of network topology and individual states (Gross & Blasius, 2008; Leonard et al., 2012; Zschaler, 2012): the evolution of the topology depends on the motions of individuals, which in turn depend on the network topology.

Visual influence networks

We can thus conclude that crowd networks are dynamic, directed visual influence networks. This places strong constraints on crowd dynamics. Co-author Kei Yoshida has been working on the reconstruction of visual influence networks from human swarm data (Yoshida & Warren, 2022), with the goals of estimating the influential positions in a crowd, the information transfer through the network, and the local interactions predicted by the visual model.

After reviewing various information theoretic approaches to network reconstruction (Pilkiewicz et al., 2020), we settled on a correlational method known as time-dependent delayed correlation (TDDC) (Alderisio, Fiore, & di Bernardo, 2017; Giuggioli et al., 2015; Nagy et al., 2010). When one individual $i$ turns and another $j$ follows, there is a time delay between their changes in heading. A positive delay indicates that $i$ is leading $j$, and its proportional duration indicates the strength of $i$’s influence on $j$. As proof of concept, we used this method to reconstruct the influence network in four walking participants (Lombardi et al., 2020). To obtain visual influence networks, the links between $i$ and $j$ are then pruned. First, we remove time delays shorter than 300 ms, because they are likely to be coincidental given locomotor response times on the order of a half-second. Second, we delete links and adjust weights based on neighbor visibility. Figure 1a presents a sequence of 1s visual influence networks reconstructed from a 10s segment of swarm data, revealing the dynamic nature of crowd networks as edges appear, disappear, and reverse direction over a few seconds.

Figure 1.

Visual influence networks. (a) A sequence of 1s networks reconstructed from a 10s segment of human swarm data $(\mathrm{N}=16)$. Note that network topology changes as the group turns to the left. Nodes (numbered participants) are plotted in their relative spatial positions (m), with the mean heading direction upward; edges represent influence from visible neighbors, weighted by percentage of time leading (thickness). (b) Spatial heat map of leadership in human swarms $(\mathrm{N}=10,16,20)$. The region of strongest influence (dark blue) is at the front of the crowd, the region of weakest influence (dark red) is at the rear. Temperature represents the mean normalized net leadership ranking (see text) in each 0.5 × 0.5 m cell for 27 min of data. The crowd’s center of mass is plotted at the origin with the mean heading direction upward.

To estimate the influential positions in a crowd, we developed a ‘net leadership’ measure that subtracts the total weight of incoming edges from the total weight of outgoing edges for each node. The nodes are then ranked by net leadership, the range is normalized by the number of nodes (on a scale of 0–1), and the means of 27 min of data are plotted a spatial heat map (Figure 1b). The results clearly indicate that the most influential region in a crowd is near the front, the least influential is at the rear, and influence flows from front to back.

The results thus far point to several conclusions. First, visual influence networks are dynamic: we find that the network topology changes on the order of seconds, thanks to dynamic occlusion in a moving crowd. Second, the most influential positions in a crowd are near the front, which has practical implications for steering crowds to safety in emergency situations. Third, as the crowd turns, leadership passes from one set of individuals to another. Consequently, ‘leadership’ is largely emergent, depending more on position in the crowd than on personal qualities.

These findings lead us to expect that, if aspiring leaders wish to steer or split a crowd, they should move to the front of the group. We are currently conducting field experiments in which we place four confederates at various positions in a crowd of 20 participants, and record crowd motion using a camera drone. This paradigm allows us to test whether covert ‘leaders’ in the front row are indeed most effective in turning and splitting an uninformed crowd. We are also testing explicit leaders (confederates carrying pennants) to manipulate the network topology and test predictions of centralized leadership (Alderisio, Fiore, Salesse, et al., 2017; Sueur et al., 2012).

Modeling consensus, clustering, and bipolarization

We are now in a position to compare the development of models of crowd dynamics and opinion dynamics in social networks. We rely on an analogy between opinions and headings, in which holding a particular opinion is equivalent to heading in a particular direction. Several caveats are in order. First, although heading has one dimension (angle), opinions can have multiple dimensions. Nevertheless, the modeling literature often operationalizes them as a value on one scale, which represents a position on an issue, an attitude toward a statement, or a belief about a state of affairs. Second, opinions are usually represented on a linear interval [0, 1], whereas heading direction is a circular variable [0°, 360°]. This has consequences for modeling extremism and bipolarization. Third, as noted above, influence is typically mutual in basic opinion networks, whereas it is directed in crowd networks. Fourth, crowd networks are locally connected with heterogeneous weights that depend on distance, while basic opinion networks tend to be completely connected with homogeneous weights. Finally, in discussing collective decisions, we focus on the dynamic interactions between individuals that yield a common outcome for a group, postponing consideration of the subtleties of the individual decision-making process. We heavily rely on the distillation of opinion dynamics models provided by Flache et al. (2017).

Consensus.

The self-organizing phenomenon that all models seek to explain is consensus: the collective decision to move in the same direction or adopt the same opinion, from initially random states (for example, Couzin et al., 2002; Cucker & Smale, 2007; Vicsek et al., 1995). In Rio, et al.’s (2018) collective motion model, a pedestrian turns (with angular acceleration ${\ddot{\varphi }}_p$) until the mean difference between its current heading $\left({\varphi }_p\right)$ and that of each neighbor $\left({\varphi }_j\right)$ is brought to zero, so the pedestrian matches the weighted average of its neighborhood:

$${\ddot{\varphi }}_p=-\frac{k}{n}\sum _{j=1}^n{w}_{j}sin\left({\varphi }_p-{\varphi }_j\right)$$ (1a)

The coupling strength $k=3.15$ determines the rate of the heading response, and $n$ is the number of neighbors within a 5m radius and a 180° field of view. The coupling weight $\left(w_j\right)$ of each neighbor decays exponentially (from 1 to 0) with distance $\left(d_j)\right.$:

$$w_j=\frac{a}{e^{\omega d_j}+a}$$ (1b)

The decay rate $(\omega =1.3)$ and the constant $(a=9.2)$ were fit to human swarm data, yielding a neighborhood with a soft radius at 4–5m. Note that Equation la has the same form as the Kuramoto model of synchronization in coupled oscillators, another paradigm of self-organization (Kelso, 2021; Strogatz, 2000), with a pedestrian’s heading standing in for an oscillator’s phase (Escaff & Delpiano, 2020; Leonard et al., 2012).

To visualize the dynamics of the model, we plot its phase portrait, the pedestrian’s angular acceleration $\left({\ddot{\varphi }}_p\right)$ as a function of the current heading difference $\left({\varphi }_p-{\varphi }_j\right)$, in Figure 2a The negative sine function creates an attractor (negative slope) at a heading difference of 0° and peak acceleration at heading differences of ±90°. Thus, if $p$’s heading is to the right of $j$’s heading (positive), $p$ accelerates leftward (negative), and vice versa. Because heading is a circular variable, the sine function smoothly wraps around to form a repeller (positive slope) at 180°. The sine function thus defines a basin of attraction 360° wide, so for any initial heading difference, $p$ eventually aligns with $j$. With multiple neighbors $p$ aligns with the weighted mean heading in the neighborhood.

Figure 2.

Dynamics of weighted averaging models: Consensus. (a) Phase portrait of the collective motion model, with an attractor (negative slope) at a heading difference of 0° and a repeller (positive slope) at a heading difference of 180°. (b) Classical opinion dynamics model, with attractor at an opinion difference of zero. (c) Multi-agent simulation of collective motion model (Eq. 1) with 100 agents, a uniform distribution of initial headings (range 360°), initial spacing 0.5 times the neighborhood radius, randomly jittered by <0.125 of the radius; 800 time steps, $\mathrm{\Delta }\mathrm{t}=100\mathrm{m}\mathrm{s}$. (d) Multi-agent simulation of the classical model (Eq. 2) with 100 agents, $\mu =0.1$, and a uniform distribution of initial opinions (range 0–1) (reproduced from Flache, et al., 2017).

Classical models of opinion dynamics in social networks similarly sought to produce consensus (Abelson, 1964; DeGroot, 1974; French, 1956), predicated on the assumption of assimilative influence: the difference between two opinions is always reduced. Specifically, the change in $i$’s opinion $\left(\mathrm{\Delta }{o}_i\right)$ is proportional to the mean difference between its current opinion $\left(o_i\right)$ and that of each neighbor $j\left(o_j\right)$, weighted by $j’s$ social influence $\left(w_{ij}\right)$:

$$\mathrm{\Delta }{o}_i=-\mu \sum _{j=1}^n{w}_{ij}\left(o_i-o_j\right)$$ (2)

The gain $(0<\mu <1)$ determines the rate of convergence of opinions, and the weights typically sum to one. The dynamics of the opinion model is illustrated in Figure 2b, which plots the change in $i$’s opinion as a function of the current opinion difference $\left(o_i-o_j\right)$. This linear function defines an attractor at a difference of 0, always resulting in compromise. With multiple neighbors, $i$ adopts the weighted mean opinion of all neighbors, which always reduces the difference in opinions. In multi-agent simulations, the classical opinion model invariably converges to a consensus that lies within the initial range (see Figure 2d).

To explore the collective motion model (Eq. 1), co-author Ben Falandays conducted multi-agent simulations of 100 agents moving on a torus (rather than a plane) because it allows us to identify stable solutions, analogous to a social network continuing to interact until opinions stabilize. We varied the initial range of headings (30° to 360°) with a uniform distribution, and the initial spacing between agents (from 0.25 to 1.0 times the neighborhood radius), and we measured the proportion of 100 runs with a bimodal distribution of final headings (bimodality coefficient > 0.55) (Pfister et al., 2013). A sample run (Figure 2c) illustrates convergence to a common heading direction, and simulation results confirm that final headings are overwhelmingly unimodal in all conditions. In other words, the model converges to consensus.

The parallel is clear: weighted averaging models of collective motion and opinion formation reduce the difference between individuals, leading to consensus. Consequently, however, this class of models fails to account for other qualitative behavior. If individuals always reach consensus, how can diversity emerge and persist (clustering)? If initial states are always averaged, how can extremes arise (bipolarization)?

Clustering.

Our original collective motion model presented related puzzles for crowd behavior. When two groups walk directly toward each other in ‘counterflow’ (at 180°), they spontaneously form alternating lanes moving in opposite directions (Mullick et al., 2022). According to the collective motion model, however, pedestrians are repelled from heading differences of 180°, so they should make U-turns and follow the opposing group. By the same token, if a crowd splits in two, with one group turning left and the other turning right, pedestrians behind them should average the two headings and continue walking straight ahead, losing both groups.

Co-author Trent Wirth investigated the latter question experimentally by asking participants to walk with a virtual crowd while wearing a VR headset. After a few seconds, half of the avatars turned to the left and the other half turned an equal angle to the right (angular differences of 20° to 120°). Preliminary results show that participants average the two groups up to an angular difference of 60°, but at that point they started to follow one group or the other on different trials (Warren & Wirth, 2022). This ‘decision’ can be understood as a bifurcation from unimodal to bimodal stable heading directions, at a critical angle of 60°.

Both the bifurcation and counterflow phenomena might be explained if pedestrians ignored neighbors with headings that diverge significantly from their own. Co-author Brian Free thus derived a new bounded attraction model in which the sine function of Eq. 1a is replaced with a double-sigmoid function, dsig:

$${\bar{\varphi }}_p=-\frac{k}{n}\sum _{j=1}^n{w}_{j}dsig\left({\varphi }_p-{\varphi }_j\right)$$ (3)

The dynamics of the bounded attraction model appears in Figure 3a. When dsig is fit to the splitting crowd data, the attractor at 0° is flanked by acceleration peaks at ±31° with a basin of attraction 180° wide, surrounded by a neutrally stable region with virtually zero response. A pedestrian is thus strongly attracted to headings within 31° of their own, but is not influenced by heading differences greater than 90°. The model reproduces the bifurcation in final heading at 60°.

Figure 3.

Dynamics of bounded averaging models: Clustering. (a) Bounded attraction model of crowd dynamics, fit to human bifurcation data. Attractor at a heading difference of 0°, strongest attraction between peaks (±31.4°), basin of attraction 180° wide, surrounded by a flat neutral region. (b) Bounded confidence model of opinion dynamics. Attractor at an opinion difference of 0, confidence threshold at ±0.5, surrounded by zero response region. (c) Multi-agent simulation of bounded attraction model with 180° basin produces 2 clusters (initial heading range 360°). (d) Same with 120° basin produces 3 clusters. (e) Multi-agent simulation of bounded confidence model, yielding 3 clusters (reproduced from Flache, et al, 2017).

To explore the effect of bounded attraction on collective behavior, we assigned the dsig function to each agent in multi-agent simulations, and varied the width of the basin of attraction. The model broke the grip of consensus and generated clusters, as illustrated in Figure 3c,d. In other words, a group of agents with the empirical dsig function exhibited a collective decision to fragment, or split into subgroups.

Opinion modelers faced the same dilemma: if averaging produces consensus, how do opinion clusters form? This led to bounded confidence models, predicated on the concept of similarity bias, in which an individual is influenced by others with similar opinions, but not by those with whom they disagree (Deffuant et al., 2000; Hegselmann & Krause, 2002; Urbig et al., 2008). The similarity cutoff is determined by a ‘confidence threshold’ beyond which an individual loses confidence in their neighbor’s opinion and ignores them. In this model, the change in $i$’s opinion $\left(\mathrm{\Delta }{o}_i\right)$ is proportional to the current difference with $j’s$ opinion $\left(o_i-o_j\right)$ only up to the confidence threshold $\left(ϵ\right)$, at which point $j’s$ influence goes to zero:

$$\mathrm{\Delta }{o}_i=-\sum _{j=1}^n{w}_{ij}\left(o_i-o_j\right)$$ (4a)

$$w_{ij}=\left\{\begin{array}{c}\mu ,\text{if}\left|o_i-o_j\right|\le ϵ\\ 0,\text{otherwise}\end{array}\right.$$ (4b)

The gain $(0<\mu <0.5)$ determines the rate of opinion convergence.

The dynamics of the bounded confidence model is illustrated in Figure 3b. Similar to the classical model, it defines an attractor at an opinion difference of 0, but the linear function is truncated at the confidence threshold, creating a smaller basin of attraction ($-ϵ$ to $+ϵ$) surrounded by a flat region that elicits no response. This is conceptually similar to our dsig function (Figure 3a). With a large confidence threshold, initially diverse opinions converge to consensus, but with a smaller confidence threshold, stable clusters of opinions form and persist (Figure 3e).

We investigated the bounded attraction model of crowd dynamics (Equation 3) in more detail using multi-agent simulations. Intuitively, one might expect that the basin of attraction would ‘tile’ the range of possible final headings (360°), such that a 180° basin would produce two clusters 180° apart (Figure 3d), a 120° basin would generate three clusters 120° apart (Figure 3e), and so on. We thus varied the basin width (30° to 180°) and the initial heading range (30° to 360°), and estimated the number of clusters in final heading using the gap statistic together with k-means clustering (Tibshirani et al., 2001). The ‘tiling’ hypothesis predicts that the number of clusters should be equal to 360°/basin width, and that clusters should be 1 basin width apart.

The resulting phase diagram (Figure 4a) reveals that the mean number of clusters increases as the basin of attraction shrinks, as expected. To break consensus, the initial heading range must exceed the basin width (red boundary), and must be quite large (>270°) before the expected number of clusters (2, 3, 4, 6, 8, 12) tends to appear. When we set k to the expected number and computed the distance between adjacent clusters, the mean spacing was very close to 1 basin width (mean $\mathrm{\Delta }={3.1}^{\circ }$) with low variability (mean $\mathrm{S}\mathrm{D}={30.4}^{\circ }$), consistent with the tiling hypothesis.

Figure 4.

Phase diagrams of the mean number of clusters for (a) the bounded attraction model, and (b) the attraction-repulsion model. The number of clusters (color temperature) increases with the initial range of headings (deg), and decreases with wider basins of attraction (deg). Multi-agent simulations of 100 agents, with an initial spacing of 0.5 of the neighborhood radius (4m), and initial positions randomly jittered (within 0.125 of the radius). Number of clusters in final heading was estimated using the gap statistic and k-means clustering (range 1–15) to determine the mean optimal value of k across 100 runs per cell. Each run lasted 80s (800 time steps, $\mathrm{\Delta }\mathrm{t}=100\mathrm{m}\mathrm{s}$, synchronous updating).

Bipolarization.

Bipolarization refers to the formation of two stable groups at opposite extremes, which may diverge beyond the initial range of values. To drive groups to extreme states, one could combine a repulsive influence with the attractive influence of the previous models. That is, instead of ignoring neighbors with significantly different headings from one’s own, one would be repelled from them.¹

Ben Falandays formulated an attraction-repulsion model by replacing the dsig function in Eq. 3 with a new attrep function, based on a sum of Gaussians:

$${\ddot{\varphi }}_p=-\frac{k}{n}\sum _{j=1}^n{w}_{j}attrep\left({\varphi }_p-{\varphi }_j\right)$$ (5)

The dynamics of the model is illustrated in Figure 5a. The attractor at 0° is again flanked by two acceleration peaks, but they are now bordered by two repellers. The basin of attraction is thus surrounded by a repulsion region that acts to amplify the heading difference until it asymptotes at 180°. This added repulsion might drive bipolarization – for example, stabilizing opposing lanes at 180° in counterflow. Sample time series demonstrate that the model can generate two extreme groups 180° apart (Figure 5c) as well as multiple clusters (Figure 5d), depending on the basin width.

Figure 5.

Dynamics of attraction-repulsion models: Clustering. (a) Attraction-repulsion model of crowd dynamics. Attractor at 0° (negative slope), flanked by repellers (positive slopes) that define a basin of attraction, with neutral stability at 180°. (b) Social repulsion model of opinion dynamics. Linear function changes sign beyond the confidence threshold. (c) Multi-agent simulation of attraction-repulsion model with a 180° basin produces bipolarization, and (d) a 120° basin produces 3 stable groups (initial heading range 360° for both). (e) Social repulsion model drives opinions to both ends of the range. (f) Bipolarization driven by extremists in the bounded attraction model, (g) the attraction-repulsion model (initial range 240°, initial spacing 0.5 radius for both), and (h) the bounded confidence model of opinion dynamics. (Panels e and h reproduced from Flache, et al. (2017).

Analogously, the problem of bipolarization in opinion dynamics led modelers to formulate social repulsion models, based on a process of social differentiation in addition to the previous assimilative influence (Jager & Amblard, 2005; Macy et al., 2003; Salzarulo, 2006). Instead of ignoring neighbors with whom one disagrees, one’s opinion shifts away from theirs. Specificallly, the change in $i$’s opinion $\left(\mathrm{\Delta }{o}_i\right)$ is proportional to the current difference in opinion $\left(o_i-o_j\right)$ up to a critical level (0.5), at which point $j’s$ influence becomes negative:

$$\mathrm{\Delta }{o}_i=-\sum _{j=1}^n{w}_{ij}\left(o_i-o_j\right)$$ (6a)

$$w_{ij}=\mu \left(1-2\left|o_i-o_j\right|\right)$$ (6b)

The dynamics of the social repulsion model is illustrated in Figure 5b. Instead of setting the influence weight to zero at the confidence threshold, it crosses zero and keeps growing. Thus, if $i$’s current opinion is very different from $j’s$ (>0.5), the difference will continue to increase until it reaches the end of the interval [0, 1], whereupon it is typically truncated. Social repulsion models can thus produce bipolarization (Figure 5e). However, this runaway process is not intrinsically stable, whereas the attractor-repeller model of crowd dynamics generates stable solutions 180° apart (Figure 5c).

To further investigate the attraction-repulsion model of crowd dynamics (Equation 5), we varied the basin of attraction and the initial heading range in multi-agent simulations. The phase diagram appears in Figure 4b. The number of clusters again increases as the basin of attraction shrinks, and is close to the expected number when the initial heading range is 360° (right column). Setting $k$ equal to the expected number, we find that the mean spacing between adjacent clusters is again very close to 1 basin width (mean $\mathrm{\Delta }={3.9}^{\circ }$) with low variability (mean $\mathrm{S}\mathrm{D}={27.0}^{\circ }$) – but surprisingly, no better than the bounded attraction model. However, sample time series indicate a more rapid convergence to the final clusters (shorter ‘relaxation time’) in the attraction-repulsion model (Figure 5d) than the bounded attraction model (Figure 4d). Thus, although the repulsive influence does not yield more evenly spaced final clusters, it does produce more rapid differentiation.

Most importantly, we observe spontaneous bipolarization in both models. This is evident in Figures 3c and 5c, where 180° basins of attraction generate two groups 180° apart, and an initial heading range >180° reliably leads to bipolarization (Figure 4). Thus, bipolarization does not require a repulsive influence, as has also been reported for opinion models (Mäs & Flache, 2013). This is particularly important given the lack of experimental evidence for a repulsive social influence (Takács et al., 2016). Even with repulsion, it very difficult to obtain biploarization outside the initial range of heading values, suggesting that it is not a robust effect.

Extremists can effectively drive bipolarization in both classes of models as well. We ran a series of multi-agent simulations in which 20% of the agents were extremists, randomly positioned in the crowd and uninfluenced by other agents, who had 180° basin widths. On each run, an equal number of extremists turned to head in opposite directions (±90°). The extremists produce strong bipolarization (bimodality coefficient > 0.55 on over 90% of runs) for both models, but only if the initial heading range exceeds 180° and the initial spacing is less than the neighborhood radius. Sample time series illustrate that the bounded attraction model yields bipolarization (Figure 5f) much like the bounded confidence model of opinion dynamics (Figure 5h) (Hegselmann & Krause, 2015). Adding a repulsive influence primarily serves to drive groups apart more rapidly (Figure 5g), but they are very rarely driven outside the initial range. Thus, bipolarization is only observed if some agents are initially near both extreme states.

In sum, the bounded attraction model and the attraction-repulsion model behave quite similarly: they both form clusters consistent with the tiling principle, produce spontaneous bipolarization, and bipolarization driven by extremists. The main contribution of a repulsive influence appears to be a more rapid differentiation of clusters. More complex models of polarization dynamics have recently expanded to include multiple opinion dimensions and influences of identity such as partisan sorting (homophily), affective polarization (xenophobia), and political elites (see Levin et al., 2021, a special issue of PNAS). Related effects of subgroup identity on crowd behavior have recently been reported as well (Templeton et al., 2018, 2019; Willcoxon & Warren, 2022), suggesting that the analogy could be pursued further.

Conclusions

In this article we have pointed out striking parallels between the domains of crowd dynamics in physical space and opinion dynamics in social networks. First, the local interactions between neighbors in a flock or crowd can be represented as a visual influence network, analogous to a social influence network. Second, the basic models of collective motion and opinion dynamics have a similar mathematical form, and yield analogous pattern formation in simulations. These models have become more complex, but the present observations suggest that crowd dynamics, opinion dynamics, and other kinds of collective decision-making might be unified by an underlying collective dynamics.

We believe that the common collective dynamics can be understood in terms of attractor/repeller dynamics, with its rich bestiary of feedbacks, bifurcations, tipping points, hysteresis, multistability, and irreversibility (Macy et al., 2021). Its Rosetta Stone is the phase portrait, which displays the attractor layout that governs the qualitative behavior of the system. Specifically, consensus results from a large basin of attraction that spans the range of possible states. Clustering results from a smaller basin of attraction surrounded by a neutrally stable region, and the number of clusters depends on the ratio of state range to basin width. Bipolarization results from a basin of attraction equal to half the state range, and the addition of a repulsive surround produces more rapid differentiation of clusters. Perhaps most importantly, collective decisions can be understood in terms of bifurcations in the system’s dynamics, which act to create and destroy attractors as conditions change.

The promise of understanding the behavior of psychological collectives in terms of collective dynamics is not merely mathematical formalization. The scientific contribution lies in its potential to explain where global behavioral patterns come from, as consequences of the dynamics of local interactions among individuals. Collective social and psychological phenomena are inherently system-level phenomena, emergent properties (Bar-Yam, 2004) of the system’s self-organizing dynamics. The discipline of psychology is considered the study of the individual mind and behavior, and seeks explanations in terms of internal structures and processes. But trying to explain collective behavior by appealing to internal models or blueprints not only revivifies the homunculus, it misses a larger opportunity. The formation of global patterns of behavior may be explained by the collective dynamics – and may even shed light on the origins of individual behavior. Models of collective dynamics thus offer the field the opportunity to account for collective behavior and collective decisions without resorting to either ghosts or machines.