The Emergence of Lines of Hierarchy in Collective Motion of Biological Systems

James M. Greene¹, Eitan Tadmor², Ming Zhong³ ¹ Department of Mathematics, Clarkson University, Potsdam, NY, United States ² Department of Mathematics and Institute for Physical Science & Technology, University of Maryland, College Park, MD, United States ³ Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL, United States [email protected]

Abstract

The emergence of large scale structures in biological systems, and in particular the formation of lines of hierarchy, is observed in many scales, from collections of cells to groups of insects to herds of animals. Motivated by phenomena in chemotaxis and phototaxis, we present a new class of alignment models which exhibit alignment into lines. The spontaneous formation of such “fingers” can be interpreted as the emergence of leaders and followers in a system of identically interacting agents. Various numerical examples are provided, which demonstrate emergent behaviors similar to the “fingering” phenomenon observed in some phototaxis and chemotaxis experiments; this phenomenon is generally known as a challenging pattern to capture for existing models. The novel pairwise interactions provides a fundamental mechanism by which agents may form social hierarchy across a wide range of biological systems.

^†^†: Phys. Biol.

Key Words: Collective Dynamics, Emergent Behavior, Lines of Hierarchy, Active Particles

Research of ET was supported in part by ONR grant N00014-2112773. Research of MZ was supported in part by NSF grant CCF-AF-2225507. \ioptwocol

1 Introduction — emergent phenomena in biological systems

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 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 consists 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 parties and so on. Thus, with no apparent central control or a built-in bias in the dynamics, the question arises — where does this unity from within come from? what is behind the seemingly spontaneous self-organization?
Let $\phi(\mathbf{x}_{i},\mathbf{x}_{i})$ denote the amplitude of pairwise interaction of agents positioned at $\mathbf{x}_{i}$ and $\mathbf{x}_{j}$ . Recent studies of collective dynamics identified different classes of interaction kernels which play a decisive role in governing the different features of their emergent behavior, [2, 3, 4]. These include metric kernels depending on the metric distance between agents, [5, 6],

\displaystyle\phi(\mathbf{x}_{i},\mathbf{x}_{j})=\phi(|\mathbf{x}_{i}-\mathbf{x}_{j}|).

(1)

Then there are topologically-based kernels depending on how crowded is the region enclosed between agents positioned at $\mathbf{x}_{i}$ and $\mathbf{x}_{j}$ , rather than their metric distance, [7, 8],

\displaystyle\phi(\mathbf{x}_{i},\mathbf{x}_{j})=\phi\{\#k\,:\,{\mathbf{x}}_{k}\in{\mathcal{C}}(\mathbf{x}_{i},\mathbf{x}_{j})\}.

(2)

Further, we distinguish between the class of long-range heavy-tailed kernels, $\displaystyle\int_{0}^{\infty}k(r){\textnormal{d}}r=\infty$ , expressed in terms of their radial envelope, [9, 10]

\displaystyle k(r):=\min\{\phi(\mathbf{x},\mathbf{x}^{\prime})\,:\,|\mathbf{x}-\mathbf{x}^{\prime}|\leqslant r\},

(3)

and singular-headed kernels, $k(r)=r^{-\beta},\beta>0$ , [11, 12, 13] vs. short range, compactly supported kernels, [14], $k(r)\lesssim{\mathds{1}}_{r\leqslant r_{0}}$ .
Our primary interest is in self-organization which is independent of external forces/stimuli; for a mathematical analysis of the latter see e.g. [15].

1.1 Attraction, repulsion, alignment

One classifies three main types of pairwise interactions which govern the emergent phenomena observed in biological systems, namely — attraction, repulsion and alignment, [16, 17, 18]. The first two main features are attraction which acts as a cohesion towards average position of neighboring agents, while repulsion steers to avoid collisions. These are familiar from particle dynamics. A typical first order attraction-repulsion dynamics reads

\displaystyle\dot{\mathbf{x}}_{i}(t)=-\frac{1}{|\mathcal{N}_{i}(t)|}\sum_{j\in\mathcal{N}_{i}(t)}\phi(\mathbf{x}_{i},\mathbf{x}_{j})(\mathbf{x}_{i}-\mathbf{x}_{j}).

(4)

Here, the agent positioned at $\mathbf{x}_{i}$ interacts with its neighbors at $\mathcal{N}_{i}(t):=\{j:\phi(\mathbf{x}_{i}(t),\mathbf{x}_{j}(t))\neq 0\}$ , of size $|\mathcal{N}_{i}|$ . Thus, with the pre-factor normalization in (4), it can be interpreted as a local environmental averaging of positions. Short-range vs. long-range kernels translate into local vs. global neighborhoods. Attraction and repulsion are dictated by the positive, and respectively negative, parts of $\phi_{ij}=\phi(\mathbf{x}_{i},\mathbf{x}_{j})$ . The balance between attraction and repulsion is responsible tor the phenomenon of aggregation, where crowd of agents is self-organized into one or more large scale stationary clusters with observable geometric configuration. Different kernels $\phi(\cdot,\cdot)$ 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 aggregation plays in cellular viability, e.g., when it is utilized in biofilms as a survival mechanism for bacterial cells, and for cellular adhesion in chemo- and radio-resistance [24, 25, 26, 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 forming complex social structures for a diverse set of evolutionary reasons [31].
A third main feature in emergent dynamics is driven alignment — the steering towards average heading of neighboring agents. A typical second-order alignment dynamics reads

\displaystyle\dot{\mathbf{p}}_{i}(t)=-\frac{\tau}{|\mathcal{N}_{i}(t)|}\sum_{j\in\mathcal{N}_{i}(t)}\!\!\phi(\mathbf{x}_{i},\mathbf{x}_{j})(\mathbf{p}_{i}-\mathbf{p}_{j}).

(5)

Here, $\tau>0$ is a fixed scaling parameter, and $\mathbf{p}_{i}$ stands for the velocity of the agent positioned at $\mathbf{x}_{i}(t)$ , [5, 6], $\mathbf{p}_{i}(t)\mapsto\mathbf{v}_{i}(t):=\dot{\mathbf{x}}_{i}(t)$ , or its orientation, [14, 32, 33], $\mathbf{p}_{i}(t)\mapsto\boldmath{\omega}_{i}(t):=\mathbf{v}_{i}(t)/|\mathbf{v}_{i}(t)|\in{\mathbb{S}}^{d-1}$ . In a typical case of long-range interactions in a crowd of $N$ agents, $|\mathcal{N}_{i}|=N$ ; one can adjust to short- and long-range interactions, replacing $|\mathcal{N}_{i}|\mapsto\sum_{j}|\phi(\mathbf{x}_{i},\mathbf{x}_{j})|$ , [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 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, 37, 38], flocks of birds [39, 40, 41, 42], and herds of animals [43] are some of the most well-known examples. We mention in passing the evolutionary roles played by flocking are diverse and species dependent: examples include reproductive efficiency, predation avoidance, and route learning in migration, [44, 45, 46]. Flocking can manifest itself via synchronization, in which pairwise interactions of 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 neuron cells [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 which allows agents to change the course of the flock based on the agent’s position with respect to the flock [55]. We note that although most boids models are presented as discrete velocity updates 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’, $\mathbf{x}_{i}^{\tau}:=\mathbf{x}_{i}+\tau\mathbf{v}_{i}$ , [56] (here we make the simplification of long-range interactions $|\mathcal{N}_{i}|=N$ ),

\displaystyle\dot{\mathbf{v}}_{i}(t)=-\frac{1}{N}\sum_{j=1}^{N}\nabla_{i}U(|\mathbf{x}_{i}^{\tau}-\mathbf{x}_{j}^{\tau}|).

(6)

Expanding at the small “anticipated time” $t+\tau,\ \tau\ll 1$ , one finds

\displaystyle\begin{split}\dot{\mathbf{v}}_{i}(t)=&-\frac{1}{N}\sum_{j}\phi_{ij}(\mathbf{x}_{i}-\mathbf{x}_{j})\\ &+\frac{\tau}{N}\sum_{j}\Phi_{ij}(\mathbf{v}_{j}-\mathbf{v}_{i}).\end{split}

(7)

Here, attraction and repulsion are dictated by $\displaystyle\phi_{ij}:=U^{\prime}(|\mathbf{x}_{i}-\mathbf{x}_{j}|)/|\mathbf{x}_{i}-\mathbf{x}_{j}|$ , and alignment is dictated by the Hessian, $\Phi_{ij}=D^{2}U(|\mathbf{x}_{i}-\mathbf{x}_{j}|)$ , with a scalar leading order term $\psi_{ij}=U^{\prime\prime}(|\mathbf{x}_{i}-\mathbf{x}_{j}|)$ . Thus, for example, a standard U-shape potential-based anticipation dictates a 3Zone dynamics in three concentric regions, ranging from interior repulsion, $U^{\prime}<0$ , through intermediate alignment where $U^{\prime}\sim 0$ and surrounding with exterior attraction $U^{\prime}>0$ . Such 3Zone 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 of army ants [33]. Milling is most commonly associated with fish populations during schooling and mating rituals [60, 32]. It also occurs in cell clusters [61, 62], and also less frequently in ants during extreme conditions [63].

Finally we note that although 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 unmanned aerial vehicles (UAVs) [64, 65, 66, 67, 68]. who have only recently begun to develop quantitative theories of collective motion [69].

1.2 A new collective model for fingering

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 [70, 71]), to colonies of insects [72], to extraordinarily complex systems in vertebrates [73]. A well known example occurring in bacterial motion is that of fingering, which serves as a primary motivation for our mathematical model introduced in this work. Fingering is a motility pattern which is often observed in cell cultures, and is characterized by cellular populations, initially undergoing essentially random and independent motion, forming structured “finger-like” protrusions from their initial homogeneous state [74, 75, 76]. 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 [76]. 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 [77, 78, 79, 80]. The formation of leaders/followers is also observed in other biological systems, such as in trail formation and cooperative transport in groups of ants [81, 82, 33, 83, 84] and the marching swarms of locusts as mentioned above [58, 59]. Many biological mechanisms exist by which leader/follower hierarchy emerges, including pheromone signaling [82], slime formation [76], and mechanical pressure [80], although many scientific questions remain [85, 86].

It is the goal of this work to present a minimal mathematical model which describes the emergence of social hierarchy of leaders and followers via pairwise interactions; for a visualization of typical simulations exhibiting line formation, see Figure 2. Our proposed model can be understood from a simple phenomenological perspective: rather than metric-based interaction, $\phi_{ij}=\phi(|\mathbf{x}_{i}-\mathbf{x}_{j}|)$ , we propose projected-based interactions

\displaystyle\phi_{ij}=\phi(\chi_{ij}\mathbf{x}_{j}-\mathbf{x}_{i}),\quad\chi_{ij}:=\frac{\langle\mathbf{x}_{i},\mathbf{x}_{j}\rangle}{|\mathbf{x}_{j}|^{2}},

(8)

where the agent positioned at $\mathbf{x}_{i}$ interacts with the traces of neighboring agents in the forward looking cone $\mathbf{x}_{j}\in{\mathcal{N}}_{i}:=\{\beta|\mathbf{x}_{i}|/|\mathbf{x}_{j}|\leqslant\chi_{ij}\leqslant 1\}$ . For a geometric illustration of the projection, see Figure 5. This leads to the spontaneous formation of leaders and followers, defined with respect to 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 center of mass, $\mathbf{x}_{i}\mapsto\mathbf{x}_{i}-\overline{\mathbf{x}}$ .

Refer to caption — Figure 1: Initial configuration of positions/velocities of 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 away from the center of mass (cyan star) at $t_{0}=0$ .

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 mathematical models of collective motion and chemotaxis/phototaxis in Section 2. In Section 3, we provide a detailed description for the modeling framework, with details of the first- and second-order systems provided in Sections 3.1 and 3.2, respectively. Numerical results for each system are provided in Section 5.1 and Section 5.5, and concluding remarks are provided in Section 6.

2 Alignment models of collective motion and social hierarchy

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 [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 [87, 88, 89, 90, 91].
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 is identified by its position $\mathbf{x}_{i}$ and velocity $\mathbf{v}_{i}$ in $\mathbb{R}^{d}$ , for $i=1,2,\ldots,N$ . Their dynamics is governed by

\displaystyle\begin{split}\dot{\mathbf{x}_{i}}(t)&=\mathbf{v}_{i}\\ \dot{\mathbf{v}_{i}}(t)&=\frac{\tau}{N}\sum_{j=1}^{N}\phi_{ij}(t)(\mathbf{v}_{j}(t)-\mathbf{v}_{i}(t)),\end{split}

(9)

with pairwise interactions driven by $\phi_{ij}(t)=\phi(\mathbf{x}_{i}(t),\mathbf{x}_{j}(t))$ . The scalar communication kernel, $\phi$ , quantifies the dynamic influence of agent $j$ on agent $i$ . In the original CS model, the authors advocate the class of long-range, decreasing metric kernels

\displaystyle\phi_{ij}

\displaystyle=\phi(|\mathbf{x}_{i}-\mathbf{x}_{j}|),\quad\phi(r)=\frac{K}{(\alpha^{2}+r^{2})^{\beta}},

(10)

with constants $\ K,\beta>0$ . We previously discussed the other classes of singular kernels which emphasize nearby agents over those farther away [11, 12, 13, 92, 93], $\phi(r)=r^{-\beta}$ , and the class of short-range kernels, $\phi(r)={\mathds{1}}_{r\leqslant r_{0}}$ . Metric kernels reflect, by definition, symmetric interactions, $\phi_{ij}=\phi_{ji}$ , and we notice that the tacit assumption is that the communication decays with the 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, [9, 94, 95, 2, 96, 97], incorporation of collision avoidance [98], steering [99], and stochasticity [100]. The large-time behavior of CS alignment dynamics (9) should lead the crowd to aggregate into a finite-size cluster, $\max|\mathbf{x}_{i}(t)-\mathbf{x}_{j}(t)|\leqslant D$ which in turn leads to flocking $|\mathbf{x}_{i}(t)-\mathbf{v}_{j}(t)|\stackrel{{\scriptstyle t\rightarrow\infty}}{{\longrightarrow}}0$ . 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 which exhibits the emergence of a simple form of social hierarchy through pairwise interactions. The model is a direct analog of the CS alignment, and is inspired by the biological phenomena 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 empathized that there is no attempt to provide our model with external environment which is of course necessary to accurately describe an externally signaled process such as phototaxis/chemotaxis; instead, we limit ourselves to cellular communication mechanisms which, we claim, is 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 agents systems similar to the Vicsek model [101], which may include an internal excitation variable to model phototaxis both deterministically [102] and stochastically [103, 104, 105]. Slime deposition [106] is also a common mechanism used to describe fingering, with agent-based [30, 107] and continuum partial differential equation [76] proposed. Similar approaches exist in describing chemotaxis, including modeling fingering a free boundary value problem [108], and extensions to the classic chemotaxis equations introduced by Keller and Segel [109, 110, 111]. Hierarchy and leadership has been investigated in the CS model [112] as well as in network graphs with switching topologies [113]. Leadership arising via external signaling was introduced and analyzed in [114], moreover leadership in cells due to feedback in speed and curvature can be formed [115, 116, 117], which we note may be particularly relevant for phototaxis and chemotaxis.

3 Mathematical models of line alignment

Motivated by the discussion in Section 1, we propose both first- and second-order models which describe the emergence of hierarchical structure in interacting agent systems for active particles, which we term generally as “line alignment models.” For both systems, we consider a total of $N$ interacting agents. Each agent is assigned a position $\mathbf{x}_{i}\in\mathbb{R}^{d}$ , and, in the case of second order models, agents are assigned with additional velocity, $\mathbf{v}_{i}\in\mathbb{R}^{d}$ . We utilize the projected position, $\chi_{ij}\mathbf{x}_{j}$ , 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 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, e.g. 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 [118], while humans in 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 $\widetilde{\mathbf{x}}_{i}$ and $\widetilde{\mathbf{x}}_{j}$ defined with respect to the center of mass $\overline{\mathbf{x}}$ of the system:

\displaystyle\widetilde{\mathbf{x}}_{i}:=\mathbf{x}_{i}-\overline{\mathbf{x}},\quad\overline{\mathbf{x}}(t):=\frac{1}{N}\sum_{i=1}^{N}\mathbf{x}_{i}(t).

(11)

Here $\mathbf{x}_{i}$ and $\mathbf{x}_{j}$ denote the positions of the agents with respect to an arbitrary origin $0\in\mathbb{R}^{d}$ . For a visualization, see Figure 3. We note that when interactions occur through symmetric differences of positions, as in the Cucker-Smale 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 [119].

3.1 First-order model

We begin by introducing a first-order model, which governs the positions of $N$ interacting agents (cells, birds, humans, etc.). Each agent is described by its time-dependent position $\mathbf{x}_{i}(t)\in\mathbb{R}^{d}$ . Their dynamics is governed by pairwise interactions,

\displaystyle\dot{\mathbf{x}}_{i}(t)

\displaystyle=\frac{1}{|\mathcal{N}_{i}(t)|}\sum_{j\in\mathcal{N}_{i}(t)}\phi_{ij}(\chi_{ij}\widetilde{\mathbf{x}}_{j}-\widetilde{\mathbf{x}}_{i}).

(12)

Here $\phi_{ij}(t)$ quantifies the interactions depending on the projected difference

\displaystyle\phi_{ij}:=\phi(|\chi_{ij}\widetilde{\mathbf{x}}_{j}-\widetilde{\mathbf{x}}_{i}|),\quad\chi_{ij}:=\frac{\langle\widetilde{\mathbf{x}}_{i},\widetilde{\mathbf{x}}_{j}\rangle}{|\widetilde{\mathbf{x}}_{j}|^{2}}.

(13)

Note that in the case that $|\widetilde{\mathbf{x}}_{j}|=0$ (i.e. agent $j$ is located at the center of mass of the system), the projection $\chi_{ij}$ is defined as zero. The neighborhood of agent positioned at $\mathbf{x}_{i}$ is formed via a forward cone, which models the asymmetric phenomenon of “looking ahead”,

\displaystyle\mathcal{N}_{i}

\displaystyle:=\{j\,|\,\beta|\widetilde{\mathbf{x}}_{i}||\widetilde{\mathbf{x}}_{j}|\leqslant\langle\widetilde{\mathbf{x}}_{i},\widetilde{\mathbf{x}}_{j}\rangle\leqslant|\widetilde{\mathbf{x}}_{j}|^{2}\},

(14)

where $0<\beta\leqslant 1$ is a fixed constant which determines the angular size of the forward cones¹¹1For simplicity, we assume the opening of forward looking cones to be the same for all agents. As before, $\phi(\cdot)$ is a communication kernel which quantifies the dynamic influence of the traced agent $j$ on agent positioned at $\mathbf{x}_{i}$ . In this work, we limit ourselves to metric communication kernels.

To model line alignment, we specify both — the pairwise interaction of agents at distance $r_{ij}=|\chi_{ij}\mathbf{x}_{j}-\mathbf{x}_{i}|$ , as well as the spatial neighborhoods defining which agents influence the dynamics of one another. The spatial neighborhoods are necessarily non-symmetric, but rather “forward-facing”; for example, if the agent positioned at $\mathbf{x}_{i}$ is positioned ahead and “in view” of the agent positioned at $\mathbf{x}_{j}$ , then agent $j$ should be influenced by agent $i$ , but not vice versa. The notions of positioned ahead and in view of are quantified via $\mathcal{N}_{i}$ — the neighborhood of agent $i$ , i.e. the set of agents which influence the dynamics of agent $i$ .

Note that we use relative coordinates $\widetilde{\mathbf{x}}_{i}$ and $\widetilde{\mathbf{x}}_{j}$ to determine the forward cone, based at the corresponding center of mass $\overline{\mathbf{x}}$ . Each cone is defined via a central angle of $2\cos^{-1}(\beta)$ radians, which is symmetric about the $\widetilde{\mathbf{x}}_{i}$ direction; this is the left-hand inequality in (14). The right-hand inequality ensures that agents are only influenced by other agents in front of them in relative position space, so that the cone is indeed forward-facing. The latter can be understood by noting that the right-hand inequality in (14) restricts the length of the projection of $\widetilde{\mathbf{x}}_{i}$ along $\widetilde{\mathbf{x}}_{j}$ , i.e. $|\widetilde{\mathbf{x}}_{i}|\cos(\phi)\leqslant|\widetilde{\mathbf{x}}_{j}|$ where $\phi$ is the angle between $\widetilde{\mathbf{x}}_{i}$ and $\widetilde{\mathbf{x}}_{j}$ , so that we require $\widetilde{\mathbf{x}}_{j}$ to be ahead of $\widetilde{\mathbf{x}}_{i}$ in relation to $\overline{\mathbf{x}}$ . Since all positions $\widetilde{\mathbf{x}}_{i}$ are time-varying, each $\mathcal{N}_{i}$ changes in time $t$ ; a static visualization of the conic spatial region is provided in Figure 4.

Geometrically, the influence of agent $j$ on its dynamics, agent $i$ measures its difference in projected position relative to agent $j$ . Thus, the form of (12) tends to align agents along lines, as the pairwise interactions ‘aim’ to reduce the orthogonal distance between agents $i$ and $j$ . As $\phi$ is tacitly assumed to be decreasing, agents are more influenced by their nearer neighbors inside the forward-looking cone, i.e. with those that are more aligned with their current direction. A visualization is provided in Figure 5. To completely specify the dynamics, a set of initial conditions $\{\mathbf{x}_{i}(0)\}_{i=1}^{N}$ must also be prescribed. Details on initial conditions and other parameters investigated are provided in Section 5.1.

3.2 Second-order model

We can now introduce the second-order model, analogous to the first-order model detailed in Section 3.1, but in which pairwise interactions influence the acceleration of the agents; this is in contrast with (12), where agents directly regulate velocity. This framework is more closely related to a classical mechanics perspective, where the interaction terms are precisely interaction forces. The state-space is represented by the position and velocity vectors of the $N$ agents, i.e. $\{\mathbf{x}_{i},\mathbf{v}_{i}\}_{i=1}^{N}$ , where the second-order dynamics reads

\displaystyle\begin{split}\dot{\mathbf{x}}_{i}(t)&=\mathbf{v}_{i}(t)\\ \dot{\mathbf{v}}_{i}(t)&=\frac{1}{|\mathcal{N}_{i}(t)|}\sum_{j\in\mathcal{N}_{i}(t)}\Big{(}\phi(|\mathbf{r}_{ij}(t)|)\mathbf{r}_{ij}(t)\\ &\quad+\psi(|\mathbf{r}_{ij}(t)|)(\mathbf{v}_{j}(t)-\mathbf{v}_{i}(t))\Big{)}\end{split}

(15)

Here $\mathbf{r}_{ij}:=\chi_{ij}\widetilde{\mathbf{x}}_{j}-\widetilde{\mathbf{x}}_{i}$ denotes the orthogonal component of $\widetilde{\mathbf{x}}_{i}$ projected in the direction of $\widetilde{\mathbf{x}}_{j}$ , so that agents interact via relative projected directions as discussed in Section 3.1 (see equation (13)), and hence act to align “along lines”. The functions $\phi$ and $\psi$ characterize the strength of the interactions, both depend on the projected distances, $r_{ij}=|\mathbf{r}_{ij}|$ , between agent currently positioned at $\widetilde{\mathbf{x}}_{i}$ , and the agent currently positioned at $\widetilde{\mathbf{x}}_{j}$ which is traced to its backward position $\chi_{ij}\widetilde{\mathbf{x}}_{j}$ . Interactions are again local, and the net effect on the dynamics of agent $i$ is the superposition of the interaction forces from all “forward-looking” neighboring agents $j\in\mathcal{N}_{i}$ .

The pairwise interactions are determined by the two function $\phi$ and $\psi$ . We first observe that force governed by $\phi$ is identical to that of the first-order attraction/repulsion in (4). The second term, dictated by interaction kernel $\psi$ , is an extension of the Cucker-Smale velocity alignment force [5, 6]. The velocity alignment in our second-order model ensures that in equilibrium, the emergent “finger-like” lines lead to flocking, $|\mathbf{v}_{i}(t)-\mathbf{v}_{j}(t)|\stackrel{{\scriptstyle t\rightarrow\infty}}{{\longrightarrow}}0$ . Indeed, the line formation will stay stable only when all the agents are moving with the same velocity; otherwise the lines formed will not be stable.

4 From alignment to the emergence of lines

The key feature of the first- and second-order models, (12), and respectively, (15), is the emergence of geometric structure for the trails along which the crowd is aligned — specifically, we observe the large-time formation of curves turning into straight lines; consult the numerical simulations reported in Section 5 below. A detailed analysis of this phenomenon is beyond the scope of this work and will be provided in a future work. Here we quote a prototypical result. We consider the first-order line alignment model

\displaystyle\begin{split}\dot{\mathbf{x}}_{i}(t)&=\frac{\tau}{\sigma_{i}}\sum_{j\in\mathcal{N}_{i}(t)}\!\!\!\!\phi_{ij}\big{(}\chi_{ij}\mathbf{x}_{j}(t)-\mathbf{x}_{i}(t)\big{)},\\ \qquad\chi_{ij}&=\frac{\langle\mathbf{x}_{i},\mathbf{x}_{j}\rangle}{|\mathbf{x}_{j}|^{2}},\quad\sigma_{i}:=\sum_{j\in\mathcal{N}_{i}}\phi_{ij}.\end{split}

(16)

To simplify our discussion, we set the dynamics relative to a fixed origin, so that $\widetilde{\mathbf{x}}_{i}\mapsto\mathbf{x}_{i}$ . The specifics of the projected-based communication, given by $\phi_{ij}=\phi(\chi_{ij}\mathbf{x}_{j}-\mathbf{x}_{i})$ , are not essential; indeed it is remarkable that our results apply to a wide variety of communication protocols, independent of symmetry or occupying a global stencil. Here, we use an adaptive normalization of the communication protocol as in [34], replacing $N_{i}\mapsto\sigma_{i}$ , so that $\frac{1}{\sigma_{i}}\sum_{j\in\mathcal{N}_{i}}\phi_{ij}=1$ (so that the dynamics does not involve ‘counting’ the number of agents).

Theorem 1.

Consider the line alignment model (16), dictated by a decreasing kernel $\phi(r)$ , which acts inside the forward-looking cones

\mathcal{N}_{i}:=\{j\,|\,\beta|\mathbf{x}_{i}||\mathbf{x}_{j}|\leqslant\langle\mathbf{x}_{i}\mathbf{x}_{j}\rangle\leqslant|\mathbf{x}_{j}|^{2}\},\ \ \beta>0.

Then, there exist constants, $C_{0}$ depending on the initial configuration and $C_{\phi}$ depending on $\phi$ , such that the following holds,

\begin{split}\sum_{i}\sum_{j\in\mathcal{N}_{i}}\Big{(}&|\mathbf{x}_{i}(t)|^{2}\cdot|\mathbf{x}_{j}(t)|^{2}\\ &-|\langle\mathbf{x}_{i}(t),\mathbf{x}_{j}(t)\rangle|^{2}\Big{)}\leqslant C_{0}e^{-C_{\phi}\beta^{2}t}.\end{split}

Theorem 1 precisely quantifies the emergence phenomenon in the first-order model (12). Namely — the crowd forms one of more distinct straight trails, led by an agent $\mathbf{x}_{i}$ and followed by its neighbors $\{\mathbf{x}_{j},\ j\in\mathcal{N}_{i}\}$ so that $|\mathbf{x}_{i}|\cdot|\mathbf{x}_{j}|-\langle\mathbf{x}_{i},\mathbf{x}_{j}\rangle\stackrel{{\scriptstyle t\rightarrow\infty}}{{\longrightarrow}}0$ . The remarkable aspect of the line dynamics, reflected in Theorem 1, is that the ‘kinetic energy’, $\sum_{i}\sum_{j\in\mathcal{N}_{i}}|\mathbf{x}_{i}(t)|^{2}\cdot|\mathbf{x}_{j}(t)|^{2}$ is easily shown to be decreasing in time. The ‘potential energy’, however, $\sum_{i}\sum_{j\in\mathcal{N}_{i}}|\langle\mathbf{x}_{i}(t),\mathbf{x}_{j}(t)\rangle|^{2}$ , does not exhibit a time-monotone behavior and may change with the configuration. It is their difference that is decreasing in time, reflecting the emergent behavior. To our knowledge, it is the first large-time, large-crowd emergence dynamics based on local interactions (the neighborhoods $\mathcal{N}_{i}$ ’s).

5 Numerical results

In this section, we numerically investigate the models of social hierarchy presented in Section 3. Specifically, we demonstrate the emergence of lines in both the first- and second-order models, and study the effect of model parameters, including the number of agents and type of interaction kernel, on the resulting dynamics. We also show that the formation of leader agent is indeed emergent, and cannot be easily extrapolated via initial conditions alone.

5.1 First-order model line formation

We begin by simulating the first-order line alignment model(12) with parameters appearing in Table 1. We are thus simulating $N=100$ agents over a period of $50$ time units in a two spatial dimensions. Here $\mu_{0}$ denotes a probability distribution utilized for generating the initial positions, with $\mathcal{U}([0,5]^{2})$ representing the uniform distribution over $[0,5]^{2}$ . Thus, we assume that the agents are initially uniformly distributed over a square region in the plane. A fixed realization of initial positions of agents is used for all simulations in this subsection, which is provided in Figure 6. In this subsection, we assume a topological interaction kernel as discussed in Section 1:

	$\displaystyle\phi(r)$	$\displaystyle=\mathbbm{1}_{r\leqslant 1}(r)\quad\text{or}$		(17)
	$\displaystyle\phi(r)$	$\displaystyle=\frac{1}{(1+r^{2})^{0.25}},$		(18)

where $r$ denotes the projected difference between agents with respect to an agent’s forward cone (see equation (13)). Recall that $\mathbbm{1}_{r\leqslant 1}$ denotes the indicator function on the set $[0,1]$ , so that all agents with projected distance less than one unit inside of the forward cone equally influence the dynamics of the agent. All simulations, in this and other sections, are integrated using MATLAB’s built-in adaptive integrator $\text{ode}23$ for handling possible stiffness of the system.

$\mu_{0}$	$d$	$N$	$t_{0}$	$t_{f}$
$\mathcal{U}([0,5]^{2})$	$2$	$100$	$0$	$50$

Table 1: Parameters utilized to simulate the first-order model as discussed in Section 5.1.

We begin by demonstrating that the proposed first-order model asymptotically exhibits line formation. Consider Figure 7, which assumes a forward cone with central angle $\pi/3$ and interaction kernel given by (17). Blue squares in the figurerepresent agent positions $\mathbf{x}_{i}(t)$ and the cyan star represents the center of mass position of the system, i.e. $\bar{\mathbf{x}}_{i}(t)$ . Recall that the center of mass is not stationary, and that all agent measure relative coordinates with respect to $\bar{\mathbf{x}}_{i}(t)$ . In this figure, we clearly observe the formation of spatial lines which originate from center of mass of the system. This hierarchical structure emerges from the initial uniform distribution of positions in Figure 6, and hence can be thought of as a form of emergence of social hierarchy. Note that the “leaders” here correspond to the agents farthest from the center of mass of the system. The dynamics thus represent a rudimentary form of finger morphology as discussed in Section 1.2, which occurs through purely inter-agent interactions, with no reliance on external forces. It may appear that the leader’s form from those agents initially farthest from the center of mass of the system, but this is not necessarily true; see Section 5.6 for details regarding the second-order model. We also emphasize that the system has reached equilibrium, as simulating further in time (not shown) produces the same spatial pattern observed in Figure 7.

5.2 Variation of dynamics as a function of forward cone size

We investigate the dynamics of line formation as a function of the size of the forward cone. Specifically, we are interested in understanding the effect of the angular size of the cone, defined by $\beta$ (see Figure 4), on the dynamics line formation. Questions of specific interest are both the the number and density of lines formed, which thus correspond to the number of emergent leaders (equivalently, the number of “fingers” formed). Intuitively, we expect that the number of lines formed should increase as the size of the forward cone decreases, as each agent acts with a higher degree of locality. For a demonstration, see Figure 8, which utilizes the same initial conditions and parameters as in Figure 7, with the exception that a central angle corresponding to $\pi$ defines the forward cone (note that in the case, it is really a forward plane, and not a cone). We observe a similar pattern of lines compared to the smaller, and hence more local, forward cone, but with a significantly higher degree of clustering to points, so that the asymptotic behavior is more similar to points, as opposed to lines. The mechanisms that produces this behavior is due to a combination of two factors: 1) the dynamics are first-order, and hence tend to exhibit aggregation, and 2) the interaction kernel (17) assumes a uniformity of influence with respect to all neighboring agents. To see the resulting pattern formation, and the effect of varying $\beta$ , for the CS interaction kernel (18), see Section 5.4.

5.3 Variation of dynamics as a function of initial conditions

In the previous section, we studied line formation for a fixed set of initial conditions. Also of interest is the role of initial conditions on the distribution of lines. More precisely: is the resulting pattern robust or highly sensitive to the agent’s initial positions? Numerical simulations (not provided) suggest that the final configuration of lines (both number of lines, and orientations) are highly dependent on the initial conditions; resampling $\mu_{0}$ generally results in a different equilibrium distribution. We note that this is not surprising, and it is common feature of models describing collective motion. Similarly, we investigate how the behavior changes as a function of number of agents ( $N$ in Table 1). This a natural scientific question, as line formation occurs across a variety of scales; for example, the number of ants composing a trail be on the order of one hundred, while the number of bacteria generating finger morphology in phototaxis may be on the order of one thousand. Furthermore, such questions are of mathematical interest, as they may provide insight into corresponding coarse-grained macroscopic models, such as mean-field and hydrodynamic limits, described by Boltzmann-type partial differential equations. We thus investigate to what degree the social hierarchy model proposed in this work is dependent on the number of agents in the system. As an example, we repeat simulations appearing in Figures 7 and 8, with fewer agents ( $N=50$ versus $N=100$ previously; new initial conditions are provided in Figure 9); corresponding results can be found in Figures 10 and 11. Similar qualitative dynamics are apparent for the smaller system, but in general we see that the distributions of lines is quite different, even for the corresponding forward cones.

5.4 Effect of interaction function on pattern formation

In the previous sections, we assumed a topological interaction kernel given by (17), which weights all neighboring agents equally in a forward cone with a limited support. We here instead use a global interaction function, i.e. $\phi=\frac{1}{(1+r^{2})^{0.25}}$ , to demonstrate the effect of the interaction function on line formation for two different $\beta$ values. As discussed in Section 5.2, we expect a global kernel to exhibit a more pronounced response to the forward cone size with respect to angle $\beta$ . Results of simulations are provided in Figures 12 and 13. Compared to the locally supported $\phi$ (Figures 7 and 8), this system’s final distribution of lines shows considerable variation: two lines for $\beta=\pi/6$ become one for $\beta=\pi/2$ . Note that the initial configuration of position is the same as shown in Figure 6.

5.5 Second-order model simulation results

We next simulate the second-order model (15) to demonstrate the dynamics of line formation in a system where interactions affect acceleration; simulation details are provided in Table 2. Recall from Section 3.2 that we must specify two interaction kernels: $\phi$ , which governs the inter-agent interaction force of line alignment, and $\psi$ , which accounts for velocity alignment and hence stabilization. Note that $\phi$ is completely analogous to the first-order model, except it more directly corresponds to a classical Newtonian force law, as the system is second-order. We assume the same function dependence (17) for $\phi$ , and assume the classical Cucker-Smale interaction kernel for $\psi$ :

\displaystyle\psi(r)

\displaystyle=\frac{1}{(1+r^{2})^{1/2}}.

(19)

$\mu^{\mathbf{x}}_{0}$	$\mu^{\mathbf{v}}_{0}$	$d$	$N$	$t_{0}$	$t_{f}$	$\phi$	$\psi$
$[0,5]^{2}$	$\mathbb{B}^{2}$	$2$	$100$	$0$	$10$	$\mathbbm{1}_{r\leqslant 1}$	$\frac{1}{(1+r^{2})^{1/2}}$

Table 2: Parameters utilized to simulate the second-order model as discussed in Section 5.5.

Here $\mathbb{B}^{2}$ is the $2D$ unit ball centered at the origin. We simulate the models for different values of $\beta$ to investigate the effect of the various sizes of the neighborhood, as in Section 5.2 for the first-order model. All tests use the same initial condition, with the initial position $\mathbf{x}_{i}(t_{0})$ being an independent and identically distributed (i.i.d) sample from $\mu^{\mathbf{x}}_{0}$ , and the initial velocity being an i.i.d sample from $\mu^{\mathbf{v}}_{0}$ . Figure 14 provides the realization of the initial configuration of positions/velocities for all the tests of different $\beta$ values investigated in this work. The blue dots in the figures represent the position (i.e. $\mathbf{x}_{i}(t)$ ) of the agents with the yellow arrow representing the velocity (i.e. $\mathbf{v}_{i}(t)$ ), whereas the cyan dot represents the center of mass position, and $\bar{\mathbf{x}}_{i}(t)$ with the yellow arrow representing the center of mass velocity $\bar{\mathbf{v}}_{i}(t)$ . Results are provided in Figures 15-19, where we have again considered variation in forward cone size ( $\beta$ ) and number of agents ( $N=100$ versus $N=50$ ). The results are qualitatively similar to the first-order system, although the second-order system exhibits a much richer class of dynamics, as the system does not approach equilibrium configurations (this is because inter-agent forces act on accelerations and not velocities, i.e. the system is not dissipative). Note that local groups of agents initially form curved lines of hierarchy (“leaders-followers”) which eventually straighten due to the projected interaction kernel (Figure 2). It thus appears that the competition between line formation (induced by $\phi_{1}$ ) and flocking (induced by $\phi_{2}$ ) has made the emergent pattern more interesting. As shown in three of the four examples of different $\beta$ (with larger $\beta$ ), the system as a whole can produce different flocking velocities, yet social hierarchy (in this case, lines) can emerge from the initially chaotic configuration.

5.6 Initial conditions are not simple predictors of leader emergence

We note that the emergence of leaders is not a simple function of initial distance from the initial center of mass, and thus hierarchy is indeed emergent from the proposed inter-agent dynamics. Consider Figure 2, which shows a number of flocking lines (“fingers”) emerging from an initially random distribution of positions and velocities (Figure 14). In Figure 2, we have colored red the $10$ initial agents farthest from the initial center of mass. Note that some agents that were initially farthest from the center of mass become followers; for example, in the right-hand side, there is a line of agents with an initially distant agent (red) that becomes a “follower.” Similarly, we see that some agents that were initially near the center of mass become “leaders” of groups of agents. Hence, we see that the model does indeed exhibit emergence of social hierarchy.

6 Discussion and conclusions

In this work, we have presented two collective dynamic models, namely a first-order and a second-order systems, where the emergence of social hierarchy, aka line formation, is induced from the “looking ahead” tendency of agents. These two models are minimal in a sense that the “look-ahead” tendency is implemented using a projected distance together with a forward-cone neighborhood. These models show promising features of natural emergence of geometric structures for various different kinds of initial configurations. More complicated patterns can be induced by using different types of communication kernels ( $\phi$ or $\psi$ ). We have presented numerous numerical simulations, and are currently developing a mathematical theory to rigorously understand properties of the emergence of line formation in such models. .

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