A new model for self-organized dynamics and its flocking behavior
We introduce a model for self-organized dynamics which, we argue, addresses several drawbacks of the celebrated Cucker-Smale (C-S) model. The proposed model does not only take into account the distance between agents, but instead, the influence between agents is scaled in term of their relative distance. Consequently, our model does not involve any explicit dependence on the number of agents; only their geometry in phase space is taken into account. The use of relative distances destroys the symmetry property of the original C-S model, which was the key for the various recent studies of C-S flocking behavior. To this end, we introduce here a new framework to analyze the phenomenon of flocking for a rather general class of dynamical systems, which covers systems with non-symmetric influence matrices. In particular, we analyze the flocking behavior of the proposed model as well as other strongly asymmetric models with “leaders”.
The methodology presented in this paper, based on the notion of active sets, carries over from the particle to kinetic and hydrodynamic descriptions. In particular, we discuss the hydrodynamic formulation of our proposed model, and prove its unconditional flocking for slowly decaying influence functions.
Key words and phrases:Self-organized dynamics, flocking, active sets, kinetic formulation, moments, hydrodynamic formulation.
1991 Mathematics Subject Classification:92D25,74A25,76N10
The modeling of self-organized systems such as a flock of birds, a swarm of bacteria or a school of fish, [1, 4, 5, 12, 19, 20, 21, 26], has brought new mathematical challenges. One of the many questions addressed concerns the emergent behavior in these systems and in particular, the emergence of “flocking behavior”. Many models have been introduced to appraise the emergent behavior of self-organized systems [2, 3, 7, 13, 17, 22, 25, 27]. The starting point for our discussion is the pioneering work of Cucker-Smale, [8, 9], which led to many subsequent studies [3, 6, 14, 15, 16, 23]. The C-S model describes how agents interact in order to align with their neighbors. It relies on a simple rule which goes back to : the closer two individuals are, the more they tend to align with each other (long range cohesion and short range repulsion are ignored). The motion of each agent “” is described by two quantities: its position, , and its velocity, . The evolution of each agent is then governed by the following dynamical system,
|Here, is a positive constant and quantifies the pairwise influence of agent “” on the alignment of agent “”, as a function of their distance,|
The so-called influence function, , is a strictly positive decreasing function which, by rescaling if necessary, is normalized so that . A prototype example for such an influence function is given by . Observe that the C-S model (1.1) is symmetric in the sense that the coefficients matrix is, namely, agents “” and “” have the same influence on the alignment of each other,
The symmetry in the C-S model is the cornerstone for studying the long time behavior of (1.1). Indeed, symmetry implies that the total momentum in the C-S model is conserved,
|Moreover, the symmetry of (1.2) implies that the C-S system is dissipative,|
Consequently, (1.3) yields the large time behavior, , and hence . This, in turn, implies that the C-S dynamics converges to the bulk mean velocity,
provided the long-range influence between agents, , decays sufficiently slow in the sense that has a diverging tail,
We conclude that the C-S model with a slowly decaying influence function (1.5), has an unconditional convergence to a so-called flocking dynamics, in the sense that (i) the diameter, , remains uniformly bounded, thus defining the domain of the “flock”; and (ii) all agents of this flock will approach the same velocity — the emerging “flocking velocity”.
[15, p. 416] Let be a given particle system, and let and denote its diameters in position and velocity phase spaces,
The system is said to converge to a flock if the following two conditions hold, uniformly in ,
The flocking behavior of the C-S model derived in  was based on the -based arguments outlined in (1.3). Other approaches, based on spectral analysis, - and -based estimates were used in [6, 8, 14] to derive C-S flocking with a (refined version of) slowly decaying influence function (1.5). Though the derivations are different, they all require the symmetry of the C-S influence matrix, .
Despite the elegance of the results regarding its flocking behavior, the description of self-organized dynamics by the C-S model suffers from several drawbacks. We mention in this context the normalization of C-S model in (1.1a) by the total number of agents, , which is shown, in section 2.1 below, to be inadequate for far-from-equilibrium scenarios. The first main objective of this work is to introduce a new model for self-organized dynamics which, we argue, will address several drawbacks of the C-S model. Indeed, the model introduced in section 2.2 below, does not just take into account the distance between agents, but instead, the influence two agents exert on each other is scaled in term of their relative distances. As a consequence, the proposed model does not involve any explicit dependence on the number of agents — just their geometry in phase space is taken into account. It lacks, however, the symmetry property of the original C-S model, (1.2). This brings us to the second main objective of this work: in section 3 we develop a new framework to analyze the phenomenon of flocking for a rather general class of dynamical systems of the form,
which allows for non-symmetric influence matrices, . Here we utilize the concept of active sets, which enables us to define the notion of a neighborhood of an agent; this quantifies the “neighboring” agents in terms of their level of influence, rather than the usual Euclidean distance. The cornerstone of our study of flocking behavior, presented in section 3.1, is based on a key algebraic lemma, interesting for its own sake, which bounds the maximal action of antisymmetric matrices on active sets. Accordingly, the main result summarized in theorem 3.4, quantifies the dynamics of the diameters, and , in terms of the global active set associated with the model. We conclude, in section 4, that the dynamics of our proposed model will experience unconditional flocking provided the influence function decays sufficiently slowly such that,
This is slightly more restrictive than the condition for flocking in the symmetric case of C-S model, (1.5). Another fundamental difference between the flocking behavior of these two models is pointed out in remark 4.2 below: unlike the C-S flocking to the initial bulk velocity in (1.4), the asymptotic flocking velocity of our proposed model is not necessarily encoded in the initial configuration as an invariant of the dynamics, but it is emerging through the flocking dynamics of the model.
The methodology developed in this work is not limited to the new model, whose flocking behavior is analyzed in section 4.1. In section 4.2, we use the concept of active sets to study the flocking behavior of models with a “leader”. Such models are strongly asymmetric, since they assume that some individuals are more influential than the others.
Finally, in section 5 and, respectively, section 6, we pass from the particle to kinetic and, respectively, hydrodynamic descriptions of the proposed model. The latter amounts to the usual statements of conservation of mass, , and balance of momentum, ,
We extend our methodology of active sets to study the flocking behavior in these contexts of mesoscopic and macroscopic scales. In particular, we prove the unconditional flocking behavior of (1.9) with a slowly decaying influence function, , such that (1.8) holds,
2. A model for self-organized dynamics
In this section, we introduce the new model that will be the core of this work. This model is motivated by some drawbacks of the C-S model.
2.1. Drawbacks of the C-S model
Originally, the C-S model was introduced in  to model a finite number of agents. The normalization pre-factor in (1.1a) was added later in Ha and Tadmor, , in order to study the “mean-field” limit as the number of agents becomes very large. This modification, however, has a drawback in the modeling: the motion of an agent is modified by the total number of agents even if its dynamics is only influenced by essentially a few nearby agents. To better explain this problem, we sketch a particular scenario shown in figure 1. Assume that there is a small group of agents, , at a large distance from a large group of agents, ; by assumption, we have . If the distance between the two groups is large enough, we have,
|In this situation, the C-S dynamics of every agent “” in group reads,|
Therefore, since there are only “essentially” active neighbors of “”, yet we average over the much larger set of agents, we would have . Thus, the presence of a large group of agents in the horizon of , will almost halt the dynamics of .
2.2. A model with non-homogeneous phase space
We propose the following dynamical system to describe the motion of agents ,
Here, is a positive constant and is the influence function. The main feature here is that the influence agent “” has on the alignment of agent “”, is weighted by the total influence, , exerted on agent “”.
In the case where all agents are clustered around the same distance, i.e., , then the model (2.2) amounts to C-S dynamics,
But unlike the C-S model, the space modeled by (2.2) need not be homogeneous. In particular, it better captures strongly non-homogeneous scenarios such as those depicted in 2.1: the motion of an agent “” in the smaller group will be, to a good approximation, dominated by the agents in group ,
Here, is the coefficient of interaction inside the nearby group , i.e., for , whereas the agents in the “remote” group , will only have a negligible influence, .
The normalization of pairwise interaction between agents in terms of relative influence has the consequence of loss of symmetry: the model (2.2) can be written as,
where the coefficients , given by,
lack the symmetry property, . Two more examples of models with asymmetric influence matrices will be discussed below. The flocking behavior of a model with leaders, in which agents follow one or more “influential” agents and hence lack symmetry, is analyzed in section 4 below. In section 7 we introduce a model with vision in which agents are aligned with those agents ahead of them, as another prototypical example for self-organized dynamics which lacks symmetry, and we comment on the difficulties in its flocking analysis. Tools for studying flocking behavior of such asymmetric models are outlined in the next section.
3. New tools to study flocking
We want to study the long time behavior of the proposed model (2.2). The lack of symmetry, however, breaks down the nice properties of conservation of momentum, (1.3a), and energy dissipation, (1.3b), we had with the C-S model. The main tool for studying the C-S flocking was the variance, , in either one of its -versions, or . But since the momentum is not conserved in the proposed model (2.2), the variance is no longer a useful quantity to look at; indeed, it is not even a priori clear what should be the “bulk” velocity, , to measure such a variance.
In this section, we discuss the tools to study the flocking behavior for a rather general class of dynamical systems of the form,
|Here, is a positive constant, and quantifies the pairwise influence of agent “” on the alignment of agent “”, through possible dependence on the state variables, . By rescaling if necessary, we may assume without loss of generality that the ’s are normalized so that|
Setting , we can rewrite (3.1) in the form
|where the average velocity, , is given by a convex combination of the velocities surrounding agent “”,|
We should emphasize that there is no requirement of symmetry, allowing . This setup includes, in particular, the model for self-organized dynamics proposed in (2.2), with asymmetric coefficients .
In order to study the flocking behavior of (3.1), we quantify in section 3.2, the decay of the diameter, using the notion of active sets. The relevance of this concept of active sets is motivated by a key lemma on the maximal action of antisymmetric matrices outlined in section 3.1. This, in turn, leads to the main estimate of theorem 3.4, which governs the evolution of and .
3.1. Maximal action of antisymmetric matrices
We begin our discussion with the following key lemma.
Let be an antisymmetric matrix, with maximal entry . Let be two given real vectors with positive entries, , and let , denote their respective sum, and . Fix and let denote the number of “active entries” of and at level , in the sense that,
Then, for every , we have
Lemma 3.1 tells us that the maximal action of on , does not exceed
which improves the obvious upper-bound, .
Using the antisymmetry of , we find
and since is bounded by , we obtain the inequality,
The identity, for , implies
By assumption, there are at least active entries at level which satisfy both inequalities,
Therefore, by restricting the sum in (3.4) only to the pairs of these active entries we find
and the desired inequality (3.3) follows. ∎
3.2. Active sets and the decay of diameters
The concept of an active set aims to determine a neighborhood of one or more agents in (3.1) based on the so-called influence matrix, , rather than the usual Euclidean distance. The following definition, which applies to arbitrary matrices, is formulated in the language of influence matrices.
Definition 3.3 (Active sets).
Let be a normalized influence matrix, . Fix . The active set, , is the set of agents which influence “” more than ,
The global active set, , is the intersection of all the active sets at that level,
This notion of active set, , defines a “neighborhood” for agent “”, and can be generalized to more than just one agent. For example,
is the set of all agents whose influence on both, “” and “”, is larger than , see figure 2.
The number of agents in an active set is denoted by , e.g. The numbers are difficult to compute for general ’s: one needs to count the number of pairs of agents in the underlying graph , which stay connected above level ,
One simple case we can count, however, occurs when takes the minimal value, . Then, the active sets includes all the agents, , and since this applies for every ””, then and the global active set, , include all agents,
Armed with the notion of active set and with the key lemma 3.1 on maximal action of antisymmetric matrices, we can now state our main result, measuring the decay of the diameters and in the dynamical system (3.2).
Since then and (3.10b) yields the following global version of the theorem above.
Fix an arbitrary and let be the number of agents in the global active set, , associated with (3.2). Then the diameters of its solution, and , satisfy,
Proof of theorem 3.4.
We fix our attention to two trajectories and , where and will be determined later. Their relative distance satisfies,
Thus, (3.10a) holds. Next, we turn to study the corresponding relative distance in velocity phase space,
recall that and are the average velocities defined in (3.1b). Given that , the difference of these averages is given by,
Inserting this into (3.12), we find,
To upper-bound the first quantity on the right, we use the lemma 3.1 with , and the antisymmetric matrix : since , and , we deduce,
Here, is the number of agents in the active set ,
Therefore, the relative velocity in (3.13) satisfies,
In particular, if we choose and such that , the last inequality reads,
and the inequality (3.10b) follows. ∎
Equation (3.10b) tells us that the diameter in velocity phase space, , is decreasing in time. In fact, an even stronger statement holds, namely, if we let denote the convex hull of the velocities, , then is decreasing in time in the sense of set inclusion,
Indeed, by convexity, for any , and consequently, if is at the frontier of , then the vector points to the interior of at , see figure 3. More precisely, if denotes the outward-pointing normal to at , then Therefore, the frontier of is a “fence”  for the vectors and (3.15) follows.
The bound of implies that the spatial diameter of the flock, grows at most linearly in time. Indeed, for agents “” and “” which realize the maximal distance, , we have
and hence .
Theorem 3.4 and 3.5 will be used to prove the flocking behavior of general systems of the type (3.2). The key point will be to make the judicious choice for the level , to enforce the convergence through the inequalities (3.10), (3.11). In this context we are led to consider dynamical inequalities of the form,
The long time behavior of such systems is dictated by the properties of .
We apply the energy method introduced by Ha and Liu . Consider the “energy functional”, ,
The energy is decreasing along the trajectory ,
and we deduce that,
By our assumption (3.17a), there exists (independent of ), such that,
and the inequality (3.19) now reads,
Since , we conclude that we have a flock with a uniformly bounded diameter,
thus improving the linear growth noted in remark 3.6. The uniform bound on in (3.21) implies that the velocity phase space of this flock shrinks as the diameter converges to zero. Indeed, the inequality (4.5b) yields,
and Gronwall’s inequality proves that converges exponentially fast to zero. ∎
4. Flocking for the proposed model
In this section we prove that the model (2.2) converges to a flock under the assumption that the pairwise influence, , decays slowly enough so that , has a non square-integrable tail, (1.8), . In section 4.2, we show that the same result carries over the dynamics of strongly asymmetric models with leader(s). We will conclude, in section 4.3, by revisiting the flocking behavior of the C-S model.
4.1. Flocking of the proposed model
Since , the alignment coefficients in (3.1) are lower-bounded by
We now set to be this lower-bound of the ’s,
The result follows from lemma 3.7 with . ∎
In contrast to the C-S model, however, our model does not seem to posses any invariants which will enable to relate to the initial condition, beyond the fact noted in remark 3.6, that belongs to the convex hull . We can therefore talk about the emergence in the new model, in the sense that the asymptotic velocity of its flock, , is encoded in the dynamics of the system and not just as an invariant of its initial configuration. Whether can be computed from the initial configuration remains an open question.
4.2. Flocking with a leader
In this section, we discuss the dynamical systems with (one or more) leaders.
Consider the dynamical system (3.1). An agent “” is a leader if there exists , independent of , such that:
In other words, an agent “” is viewed as a leader if its influence on aligning all other agents “”, is decreasing with distance, but otherwise, is independent of the number of agents, . We illustrate this definition, see figure 4, with the following dynamical system: a leader “” moves with a constant velocity and influences the rest of the agents with a non-vanishing amplitude ,
We note that there could be one or more leaders. The presence of leader(s) in the dynamical system (3.1) is of course typical to asymmetric systems. We use the approach outlined above to prove that the existence of one (or more) leaders, enforces flocking.
If the leader is not influenced by the other agents, then one deduces that the asymptotic velocity of the flock will be the velocity of the leader . But we emphasize that in the general case of having more than one leader the asymptotic velocity of the flock emerges through the dynamics of (3.1), and as with the model (2.2), it may not be encoded solely in the initial configuration.