Emergent phenomena in collective dynamics are observed in a wide range of biological systems and across different scales—from cells to bacteria, from insects to fish, from humans to other mammals. Accordingly, it has been a topic of scientific interest in a wide range of disciplines, including biology, ecology, physics, mathematics and computer science [1]. In this context, one is concerned with 'active particles' which consist of living agents (and likewise, certain types of mechanical agents) equipped with senses and sensors with which they probe the environment. These are responsible for small-scale pairwise interactions. The phenomenon of emergence is observed when a crowd of agents, driven by those small-scale interactions, is self-organized into large-scale formations: ants form colonies, insects swarm, birds fly in flocks, mobile networks coordinate a rendezvous or create traffic jams, human opinions evolve into political parties and so on. Thus, with no apparent central control or a built-in bias in the dynamics, the following questions arise: where does this unity from within come from and what is behind the seemingly spontaneous self-organization?

Let denote the amplitude of pairwise interaction of agents positioned at xi and xj . Recent studies of collective dynamics have identified different classes of interaction kernels that play a decisive role in governing the different features of their emergent behavior [2–4]. These include metric kernels depending on the metric distance between agents [5, 6]

Then there are topologically based kernels depending on how crowded is the region enclosed between agents positioned at xi and xj , rather than their metric distance [7, 8]

Further, we distinguish between the class of long-range heavy-tailed kernels , expressed in terms of their radial envelope [9, 10]

and singular-headed kernels [11–13], versus short range, compactly supported kernels [14] . Our primary interest is in self-organization that is independent of external forces/stimuli; for a mathematical analysis of the latter see [15], for example.

One can classify three main types of pairwise interactions that govern the emergent phenomena observed in biological systems, namely attraction, repulsion and alignment [16–18]. The first two main features are attraction, which acts as a cohesion towards the average position of neighboring agents, and repulsion, which steers to avoid collisions. These are familiar from particle dynamics. A typical first-order attraction–repulsion dynamics can be expressed by

Here, the agent positioned at xi interacts with its neighbors at , of size . Thus, with the pre-factor normalization in (4), it can be interpreted as a local environmental averaging of positions. Short-range versus long-range kernels translate into local versus global neighborhoods. Attraction and repulsion are dictated by the positive, respectively negative, parts of . The balance between attraction and repulsion is responsible for the phenomenon of aggregation, where a crowd of agents is self-organized into one or more large-scale stationary clusters with an observable geometric configuration. Different kernels lead to a great variety of different limiting configurations. These are observed in cell biology, with tissue formation (mediated by cell-to-cell recognition and cell adhesion) being the prototypical example [19]; cell aggregation also plays a fundamental role in cellular differentiation [20], proliferation [21, 22] and viability [22, 23]. We mention on passing the important role played by aggregation in cellular viability, for example when it is utilized in biofilms as a survival mechanism for bacterial cells and for cellular adhesion in chemo- and radio-resistance [24–27]. Aggregates of cells also commonly coordinate their movement to collectively migrate; prominent biological processes displaying this behavior are wound healing and cancer invasion [28], as well as chemotaxis and phototaxis [29, 30]. Aggregation is of course not limited to cells; thus, for example, many species of insects (e.g. monarch butterflies overwintering) and animals form complex social structures for a diverse set of evolutionary reasons [31].

A third main feature in emergent dynamics is driven alignment—the steering towards the average heading of neighboring agents. A typical second-order alignment dynamics can be expressed by

Here, τ > 0 is a fixed scaling parameter and pi stands for the velocity of the agent positioned at [5, 6], , or its orientation [14, 32, 33], . In a typical case of long-range interactions in a crowd of N agents, , one can adjust to short- and long-range interactions, replacing [34]. The alignment encoded in (5) describes environmental averaging of velocities/orientations. Alignment may be either local or global, depending on the heavy-tailed scale of the interaction kernel. Alignment governs the emergent phenomena of flocking or swarming, found in animal populations [35], in which agents attempt to align their heading and/or speed in a large-scale coordinated movement. Schools of fish [36–38], flocks of birds [39–42] and herds of animals [43] are some of the most well-known examples. We mention in passing that the evolutionary roles played by flocking are diverse and species dependent: examples include reproductive efficiency, predation avoidance and route learning in migration [44–46]. Flocking can manifest itself via synchronization, in which pairwise interactions between agents are coordinated in time into large-scale crowd oscillations. Well-known examples include the frequency of flashing of firefly lights [47], the 'chorusing' behavior of some species of crickets [48] and the firing of neurons [49]. Flocking occurs in behavioral contexts as well, with consensus building being an emergent phenomenon in opinion dynamics [50]. It is realized on many different scales, from populations of cells to populations of humans [51].

The full complexity of self-organization observed in biological systems is realized when combining attraction, repulsion and alignment. This was originally advocated in the pioneering work of Reynolds [16] for realistic simulation of boids—birds-like objects. Reynolds' model remains one of the most commonly utilized methods of describing collective motion, with extensions proposed to incorporate the effect of pheromone signaling [52] and obstacle avoidance [53], as well as a motivation for development of particle swarm optimization [54]. The incorporation of social hierarchy via leadership has also been explicitly incorporated into Reynolds' rules for boids using an additional steering force that allows an agent to change the course of the flock based on the agent's position with respect to the flock [55]. We note that although most boid models are presented as discrete velocity update rules, they typically can be translated to either deterministic or discrete second-order systems (see section 3.2).

A systematic framework for combining attraction, repulsion and alignment mechanisms is offered by anticipation dynamics induced by a radial potential U, and acting at the 'anticipated positions' [56] (here we make the simplification of long-range interactions )

Expanding at the small 'anticipated time' , one finds

Here, attraction and repulsion are dictated by , and alignment is dictated by the Hessian , with a scalar leading-order term . Thus, for example, a standard U-shaped potential-based anticipation dictates three-zone dynamics in three concentric regions, ranging from interior repulsion (), through intermediate alignment where and surrounded with exterior attraction (). Such three-zone dynamics is encountered in many models for flocking and swarming. For example, many species of insects exhibit swarming behavior in which their motion is self-organized into approximately concentric trajectories, known as milling, or vortex formation [57]. This enables the insects to carry out specific tasks in the form of collective intelligence. Examples of swarming include the marching of locust nymphs [58, 59] and lane formation and obstacle avoidance in army ants [33]. Milling is most commonly associated with fish populations during schooling and mating rituals [32, 60]. It also occurs in cell clusters [61, 62], and less frequently in ants during extreme conditions [63].

Finally, we note that although it is not a focus of the present work understanding collective motion for biological crowds has numerous applications in the engineering sciences. Examples include mobile sensing networks and the utilization of cooperative uncrewed aerial vehicles [64–68].

Certain forms of emergent behavior can be classified as possessing degrees of social hierarchy, where individual agents conform to distinct roles. As with all emergent behavior, hierarchy can arise across a vast range of scales, from small groups of cells (e.g. in cell migration [69, 70]), to colonies of insects [71], to extraordinarily complex systems in vertebrates [72]. A well-known example occurring in bacterial motion is that of fingering, which serves as a primary motivation for the mathematical model introduced in this work. Fingering is a motility pattern that is often observed in cell cultures and is characterized by cellular populations, which initially undergo essentially random and independent motion, forming structured 'finger-like' protrusions from their initial homogeneous state [73–75]. These protrusions indicate the emergence of social hierarchy via 'leader-type' cells at the leading edge of the protrusions; the remaining cells 'follow' in the paths determined by leading cells, often in very straight lines [75]. Fingering is most closely associated with populations exposed to optical gradients (phototaxis), but is also observed in wound healing, where cellular communication is determined primarily via chemical (chemotaxis) and mechanical signaling [76–79]. The formation of leaders/followers is also observed in other biological systems, such as in trail formation and cooperative transport in groups of ants [33, 80–83] and the marching swarms of locusts as mentioned above [58, 59]. Many biological mechanisms exist by which leader/follower hierarchy emerges, including pheromone signaling [81], slime formation [75] and mechanical pressure [79], although many scientific questions remain [84, 85].

It is the goal of this work to present a minimal mathematical model that describes the emergence of a social hierarchy of leaders and followers via pairwise interactions; for a visualization of typical simulations exhibiting line formation see figure 1. Our proposed model can be understood from a simple phenomenological perspective: rather than metric-based interaction, , we propose projected–based interactions

where the agent positioned at xi interacts with the traces of neighboring agents in the forward-looking cone (figure 5 shows a geometric illustration of the projection). This leads to the spontaneous formation of leaders and followers, defined with respect to their relative positions in a linear aggregate. Observe that the interactions in (8) are not symmetric; further, they are not Galilean invariant. Accordingly, there is a need to shift the fixed origin and trace the dynamics relative to the center of mass, .

Figure 1. Trajectory plot of the second-order system described in section 3.2 with the initial configuration shown in figure 2. The coloring of the agents is described in the caption of figure 2, and the gray trailing lines indicate the path of an agent's trajectory. Note that the agents furthest away may not become 'leaders'.

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Figure 2. Initial configuration of positions/velocities of a second-order system as described in section 3.2. We simulate N = 100 agents (blue squares), with 10 agents (red diamonds) chosen at the initial time as the furthest from the center of mass (cyan star) at .

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Such interactions can be readily understood in many of the applications described above, such as the sensing of pheromone trails left by neighboring ants and slime model deposits in bacterial cultures. Although inspired by fingering in phototaxis and chemotaxis, the model assumes no external forcing, so that the emergence of lines is intrinsic to the interactions of the agents alone. Furthermore, the model is sufficiently generic to describe a wide variety of phenomena, including spatial positions and velocity, but also emotions, frequencies, headings, opinions, etc as described previously.

The remainder of the paper is organized as follows. We provide a brief discussion of the mathematical models of collective motion and chemotaxis/phototaxis in section 2. In section 3, we provide a detailed description of the modeling framework, with details of the first- and second-order systems provided in sections 3.1 and 3.2, respectively. The formation of lines in first-order systems is summarized in theorem 1 in section 4; Numerical results are provided in section 5, and concluding remarks are provided in section 6.

In this section we restrict our attention to alignment dynamics, suppressing the additional roles of attraction and repulsion. We begin with a brief overview of two alignment models: we refer to [17, 18] for a thorough discussion on the biological phenomena and to [4] for a recent mathematically rigorous discussion of alignment models. The first alignment model originates from the 1995 work of Vicsek et al [14], in which self-propelled particle systems go through local averaging of velocity orientations. Indeed, many physical and biological systems utilize one form or another of environmental averaging [86–90]. A second velocity alignment model was introduced in 2007 by Cucker and Smale [5, 6]. The model presented in this manuscript is directly inspired by the Cucker–Smale (CS) model, so we describe it in detail here. The system consists of N identical interacting agents, each identified by its position xi and velocity vi in , for . Their dynamics is governed by

with pairwise interactions driven by . The scalar communication kernel, φ, quantifies the dynamic influence of agent j on agent i. In the original CS model, the authors advocate a class of long-range decreasing metric kernels

with constants . We previously discussed other classes of singular kernels that emphasize nearby agents over those farther away [11–13, 91, 92], , and the class of short-range kernels, . Metric kernels reflect, by definition, symmetric interactions, , and we notice the tacit assumption that communication decays with distance. Motivated by the original CS model, the general framework of alignment based on pairwise interactions has inspired considerable work, including the hydrodynamic description of its large crowd limit [2, 9, 93–96], incorporation of collision avoidance [97], steering [98] and stochasticity [99]. The large-time behavior of CS alignment dynamics (9) should lead the crowd to aggregate into a finite-size cluster, , which in turn leads to flocking . However, left without attraction/repulsion, dynamics driven solely by alignment does not support the emergence of any preferred spatial configuration.

As mentioned in section 1, the goal of this work is to provide a minimal mathematical model that exhibits the emergence of a simple form of social hierarchy through pairwise interactions. The model is a direct analog of CS alignment and is inspired by the biological phenomenon of fingering in chemotaxis and photoaxis. It is advocated as a simple alignment mechanism by which a priori identical agents evolve to form fingering structures with internal hierarchy. It should be emphasized that here, no attempt was made to model the external environment, which is of course necessary to accurately describe an externally signaled process such as phototaxis/chemotaxis; instead, we limit ourselves to cellular inter-communication mechanisms which, we claim, are an essential part of the more complicated processes. In this sense, this work is complimentary to theoretical and experimental work studying social hierarchy as well as chemotaxis/phototaxis. For example, many works formulate interacting agent systems similar to the Vicsek model [100], which may include an internal excitation variable to model phototaxis both deterministically [101] and stochastically [102–104]. Slime deposition [105] is also a common mechanism used to describe fingering, with agent-based [30, 106] and continuum partial differential equations [75] being proposed. Similar approaches exist in describing chemotaxis, including modeling fingering as a free boundary value problem [107], and extensions to the classical chemotaxis equations introduced by Keller and Segel [108–110]. Hierarchy and leadership have been investigated in the CS model [111] as well as in network graphs with switching topologies [112]. Leadership arising via external signaling was introduced and analyzed in [113], moreover leadership in cells due to feedback in speed and curvature can be formed [114–116], which we note may be particularly relevant for phototaxis and chemotaxis.

Motivated by the discussion in section 1, we propose both first- and second-order models that describe the emergence of hierarchical structure in interacting agent systems for active particles, which we term generally 'line alignment models'. For both systems, we consider a total of N interacting agents. Each agent is assigned a position , and, in the case of second-order models, agents are assigned an additional velocity . We utilize the projected position as a way to realize the tendency of agents 'to look ahead'. In order to avoid the discussion of absolute origin, we also use the center of mass position of the whole system as the reference. We believe that this assumption is physically reasonable, as groups of bacteria/cells/animals should not utilize a global coordinate system with specified fixed origin but rather measure positions with respect to their local environment, for example the center of mass of their flock, school or other social structural unit. Coordinate systems in local environments may be species dependent; for example, bacteria undergoing phototaxis may measure their position relative to a dominant light source [117] while humans at a concert may measure their positions with respect to the main stage. In an isotropic environment, a 'natural' coordinate system is the center of mass reference frame. That is, we assume that the interacting agents measure their positions relative to the agent-system itself. For example, we consider the relative positions and defined with respect to the center of mass of the system

Here xi and xj denote the positions of the agents with respect to an arbitrary origin . This is visualized in figure 3. We note that when interactions occur through symmetric differences of positions, as in the CS and Vicsek models, absolute versus relative positions result in identical dynamical systems so that the distinction is irrelevant. However, when considering non-symmetric interactions that arise via projected distances, as in (8), the resulting systems possess distinct vector fields. Of course, certain species may indeed have global coordinate systems, such as in the mass migration of some species of birds [118].