Numerical analysis on Cucker-Smale collective behavior models

Massimo Fornasier

Technische Universitat Munchen, Facultat fur Mathematik, Boltzmannstraße 3, D-85748 Garching

(Germany)

Francesco Vecil

Universitat de Valencia, Departament de Matematica Aplicada, calle del Doctor Moliner 50, Burjassot46100 (Spain)

Abstract

The aim of this paper is to give a contribution to the analysis of the Cucker-Smale align-ment model through numerical experiments. This work is meant as an exploration of theconnection between the discrete and the continuum systems and of the emergence of leader-ship structure through a viewing cone. First, we focus on the initial datum, and study thediscrete/continuum discrepancy by means of the Wasserstein distance. Second, we focus onthe evolution, and conjecture that more optimistic estimates than Gronwall’s exponentialdegradation might hold. In order to explore the modeling of instantaneous leadership emer-gence, which can also be extended to a continuum limit, we introduce a viewing cone, henceobtaining a partition into troops, whose shape is related to the viewing angle. Finally, wefocus on the asymptotic state for the system, trying to get an intuition about sufficient con-ditions to achieve consensus in the continuum setting, by means of phase transition diagramsin an appropriate parameter space in the discrete setting.

Keywords: collective behavior, self-organization, Cucker-Smale model, swarming2000 MSC: 70F10, 35L03, 34F05, 05C20

1. Introduction

In certain situations a large amount of interacting individuals tend to global self-or-ganization without the presence of a leader: the agents influence each others by followingmicroscopic rules, and the sum of all these contributions produces macroscopically observablepatterns [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]. An example of this is given by a flock ofbirds: none of them is the leader, or at least none of them is the leader all the time; eachbird modifies its velocity under the influence of its neighbors, by aligning its orientation andadjusting its speed to those of the others. Such situations are usually described by discrete,

Email addresses: massimo.fornasier@ma.tum.de (Massimo Fornasier), francesco.vecil@gmail.com(Francesco Vecil)

Preprint submitted to Physica D February 17, 2013

also called particle or individual-based, models: the agents are numbered from 1 to N andeach of them is assigned a position-velocity vector (x(i),v(i)) ∈ Rd × Rd, being d ∈ 1, 2, 3the dimension; moreover, the agents are endowed with any other feature that might beconsidered interesting for the modeling, like age, size, species, etc. The model includes someprinciples following which the individuals interact [1, 14, 15, 16]. For the scope of this workwe are interested in alignment models : the individuals try to mimic the orientation of thesurrounding agents. This sociological behavior belongs to a large collection of different socialforces, such as those described by the attractive/repulsive models [17, 18, 19], which representthe tendency animals have to group together, nonetheless repelling each other when theyare too close, so as to avoid collisions. All these models of so-called collective behaviorshave arisen to a role of great visibility thanks to their applicability to many fields, fromtheoretical ones, like the study of the evolution of human languages [2, 20], to applied oneslike the study of stock exchanges, the guidance of robots or other unmanned vehicles, see[21, 18, 22, 23, 24] and references therein, or the prediction of criminal behavior [13].

The starting point of this paper is the Cucker-Smale model [4]

d

dtx(i) = v(i),

d

dtv(i) =

1

N

N∑j=1

aij ·(v(j) − v(i)

), aij =

1(1 + |x(j) − x(i)|2Rd

)γ . (1)

In this alignment model the individual number i modifies its velocity vector by averagingthe direction of others, and the closer another agent is, the more influential it becomes.

A rather precise analysis of this model in terms of achieving global consensus to themean-velocity, this one being an invariant of the dynamics, has been developed, startingwith the work [4]. In [25, 26, 27] the authors address the mean-field limit to a correspondingcontinuum model:

(1) −−−→N→∞

∂f

∂t+v·∇xf−∇v ·[(v ?v U0 ?x f) f ] = 0, U0(x) =

1(1 + |x|2Rd

)γ , (2)

completed by the initial condition f(0,x,v) = f 0(x,v) and proper boundary conditionsfor the space and velocity domains. This continuum model, necessary when the number ofagents becomes very large thus making the numerical simulations unbearable, is mesoscopic[28, 29, 27, 17]: the time derivative describes the evolution of a distribution function f(t,x,v)which depends on time t ≥ 0 and the phase space (x,v) ∈ Ω × Rd, being x ∈ Ωx ⊆ Rdthe position and v ∈ Rd the velocity. The other big category of continuum description isthe hydrodynamic models [25, 27, 30, 31, 19, 32, 3, 33, 34, 35, 36, 37]: the time derivativedescribes the evolution of a distribution function ρ(t,x) which depends solely on time t ≥ 0and the position x ∈ Ωx ⊆ Rd. Of course, hydrodynamic models are coarser than themesoscopic models and can retain less information; in exchange, they are computationallycheaper. Continuum models can be derived from the individual-based ones [28, 29, 27, 17,25, 30, 31, 19, 38, 32] or be written directly [3, 33, 34, 35].

In [30] the authors show that the Wasserstein-distance, a tool used to compute howclose discrete and continuum probability measures are, between (1) and (2) grows as an

2

x(i)

v(i)

x(j)

x(k)

φ

C(i)

particle j lies inside the

i as it lies outside the viewingcone of particle i

particle k is not seen by particle

viewing angle of particle i

Figure 1: Viewing angle. In this 2D sketch, particle i can see particle j but not particle k.

exponential in time and in [39] that it converges to zero as a power of N . However, severalare the relevant aspects of such a discrete-to-continuum approximation which have not beenfully understood in the analysis of the Cucker-Smale model; in this paper we would like, bymeans of extensive numerical experiments, to address the relation between the discrete andthe continuum, and pay special attention to the emergence of a leadership structure when aviewing cone is introduced. The paper is divided into three blocks, referring to:

• The initial condition and time evolution. The mean-field approximation con-verges polynomially in N , depending on the dimension of the space, and deterioratesin time exponentially fast, due to classical Gronwall’s estimates. We shall show byextensive numerical experiments that the stability in time might actually be betterthan the expected Gronwall’s estimate.

• Leadership emergence. The drawback of the simple system (1) is that it does nottake into account the biological fact that animals cannot see all around themselves,rather they have a viewing angle, which, for the scope of this work, we assume to bea right circular cone. We want particle i to be influenced by particle j only in caseparticle j belongs to the viewing angle of particle i, i.e., in case particle j forms anangle with v(i) smaller than the aperture 2φ, as sketched in Figure 1 for a 2D case. Tothis end, we modify (1) and (2) as follows:

aij =χ[cos(x(j) − x(i),v(i)

)≥ cos(φ)

](1 + |x(j) − x(i)|2Rd

)γ , U0(x) =χ [cos (x,v) ≥ cos(φ)](

1 + |x|2Rd)γ (3)

so that they are now related not only to the relative distances between the particlesbut also to the viewing angle. We shall refer to the Cucker-Smale model (1) or (2) asthe isotropic model, and to (3) as the anisotropic model. In our notations,

cos (v′,v′′) =

d∑n=1

v′nv′′n

|v′|Rd |v′′|Rd(4)

3

is the cosine of the angle between vectors v′ and v′′ in Rd. Of course, if we set φ = π werecover the isotropic model (1). The anisotropic behavior opens doors to the emergenceof leadership structure, in the sense that we show that the model selects several groups,each of them traveling independently of the other ones, and led by an individual thatinfluences its own troop without being influenced by anyone else: we shall call suchan individual a leader. The troops can be inscribed inside cones having their leadersas vertices; we shall characterize the asymptotic states and show with numerical andtheoretical evidence that the aperture 2φ of the viewing cone and the angles at thevertices are related.

• The asymptotic state. A key role is played by the parameter γ which governs therate of communication between far distant individuals. In [26] the authors show thatcondition γ ≤ 1/2 is sufficient for the individuals to achieve consensus

v(i) −−−−→t→+∞

v :=1

N

N∑i′=1

v(i′), ∀i (5)

no matter what the initial conditions in space are. At continuum level, this propertytranslates into a delta in velocity

f(t,x,v) −−−−→t→+∞

ρ(t,x) δv(v).

In case γ > 1/2, in [4] it is proven that the discrete system (1) achieves consensus (5)provided that the initial condition is not too spread-out. As the initial datum’s supportin either space or velocity becomes larger, the system might achieve consensus or not.At continuum level, we do not dispose of any result. By studying the transition phasefrom consensus to dissent in an appropriate parameter space, at discrete level but forincreasing numbers of particles, we wish to provide a guess about the asymptotics ofthe continuum system.

This paper is organized as follows: in Section 2 we perform a numerical study on the conver-gence of the discrete model towards to the continuum one; in Section 3 we analyze the emer-gence of leadership when a viewing cone is introduced; in Section 4 we study the asymptoticstate in the regime γ > 1

2for the limit N →∞; then, in Section 5 we present our conclusions

and plans for the future.

2. The initial condition and time evolution

In [26] the authors show that the isotropic individual-based model (1) converges, asN →∞, to the continuum model (2), which we recall here:

∂f

∂t+ v ·∇xf −∇v · [(v ?v U0 ?x f) f ] = 0, U0(x) =

1

(1 + |x|2)γ.

4

The discrete and the continuum formulations are rigorously equivalent if the initial conditionis a sum of atomic measures:

f(0,x,v) =1

N

N∑j=1

δx(i)(0) ⊗ δv(i)(0)(x,v), (6)

because, in this case, the discrete and the continuum solutions are related by

f(t,x,v) =1

N

N∑j=1

δx(i)(t) ⊗ δv(i)(t)(x,v), for any t ≥ 0.

Therefore, it is only interesting to investigate the relation between the two models in casethe continuum solver is initialized with a continuous datum. From [30, 26, 25] we have atool to measure the distance between the two models: the Wasserstein-distance is defined as

W(µ, µN

):= sup

∫ϕ d

(µ− µN

)s.t. ‖ϕ‖Lip ≤ 1

, (7)

where the generic absolute continuous measure

µ(t,x) := λ ρ(t,x), λ = Lebesgue measure,

is the particle density of the continuum model in the sense of measures, and

µN(t,x) :=1

N

N∑i=1

δ(x− x(i)(t)

)is the particle distribution of the discrete model. Unfortunately, implementing inside anumerical code the computation of (7) seems almost unfeasible, hence we have to look foran alternative formulation. In general we know that the Wasserstein-distance will always belarger than the L1-norm of the difference between the probability distributions:

W(µ(t), µN(t)

)≥

∥∥∥∥∥∫⊗di=1(xmin

i ,xi)ρ (t, ξ) dξ − 1

N

N∑j=1

d∏i=1

χ]−∞,xi]

(x

(j)i (t)

)∥∥∥∥∥L1(Ωx)

but, in the 1D setting, which will be the object of our study, this inequality becomes anequality:

W(µ(t), µN(t)

)=

∥∥∥∥∥∫ x

xmin

ρ(t, ξ) dξ − 1

N

N∑i=1

χ]−∞,x]

(x(i)(t)

)∥∥∥∥∥L1(xmin,xmax)

. (8)

We shall, therefore, implement (8) inside our code. From [25, 30] we expect that

W(µ(t), µN(t)

)≤ C(t) W

(µ(0), µN(0)

), (9)

5

where C(t) is an exponential function.The plan for this section is the following: in Section 2.1 we study from the numerical

point of view the convergenceµN(0) −−−→

N→∞µ(0);

then, in Section 2.2 we study from the numerical point of view the estimation (9) andconjecture that C(t) is actually more optimistic than an exponential.

Remark 2.1 (Some implementation details). The discrete solver, third-order Total-Varia-tion-Diminishing Runge-Kutta scheme (TVD-RK-3) [40, 41], is initialized in order to ap-proximate the continuum datum, following the algorithm detailed in Appendix A.

The domain for the continuum solver, the conservative Time-Splitting (TS) FBM (FluxBalance Method, see [42] and references therein), is set large enough to avoid the presenceof mass close to the borders in our simulation times, so that the choice of the boundaryconditions at the extrema of the domain does not matter. Conversely, the atomic solver doesnot need any borders, and the particles are thus let free to move in an unbounded domain.

The continuum and the discrete models must have the same mass, hence we impose∫ xmax

xmin

∫ vmax

−vmax

f(0, x, v) dv dx = 1

because in the atomic model we have, since the beginning, implicitly assumed that all the

masses are mi ≡1

Nso that the total mass is equal to 1.

2.1. Convergence of the initial datum

We expect the Wasserstein-distance between the discrete and the continuum initial dataW(µ(0), µN(0)

)to decrease as N−δ, where δ > 0 is related to the dimension of our set-

ting. We need to perform a number of individual-based simulations sufficiently large to bestatistically meaningful; here we are using a battery of one hundred random initializationsapproximating the continuum initial datum through the algorithm detailed in Appendix A.In Figure 2 we show that

µN(0) −−−−→N→+∞

µ(0) as N−1/2,

which gives, injected into (9),

W(µ(t), µN(t)

)≤ C(t)N−1/2.

2.2. Long-time simulations

We use as reference result for the computation of the Wasserstein-difference a continuumlong-time simulation with fine meshing; moreover, we perform a number of individual-basedsimulations sufficiently large to be statistically meaningful.

6

0.01

0.1

1

8 16 32 64 128 256 512 1024 2048

W(µ

(0),

µN

(0))

(in

lo

g-s

ca

le)

points (in log-scale)

slope = -0.499046

Figure 2: Convergence of the initial datum in 1D. This picture shows how W (µ(0), µN (0)) → 0 asN−1/2. In order to get statistically meaningful results, a hundred experiments were performed for each N .The points shown in the picture are the average values, which are taken to draw the OLS straight line.

The choice of the initial datum seems crucial, because the continuum system with theatomic masses (6) exactly recovers the discrete solution. Therefore, we shall perform numer-ical experiments with different initializations:

f(0, x, v) = exp(−x2

) [exp

(−(v1− 1)2

)+ exp

(−(v1 + 1)2

)](10)

f(0, x, v) =

1 if (x, v) ∈ rectangle (−2,−3), (0,−3), (0, 1), (−2, 1)0 else

(11)

each of which must then be normalized, as pointed out in Remark 2.1. Datum (10) is closeto two Dirac masses at (0, 1) and (0,−1) and has null average velocity, while datum (11)does not try to mimic any atomic distributions and the average velocity is negative.

2.2.1. Inizialization close to Dirac masses

We use as initial datum (10). Moreover, we force the particles, after distributing themin order to mimic the continuum distribution (details about the algorithm are given inAppendix A), to have zero average velocity.

In Figure 3(a) we plot the results for different values of parameter γ and different numbersof particles, inside the range of consensus-achieving simulations (γ ≤ 0.5): the differencetends to stabilize after a certain time, due to the fact that the average velocity to whichparticles converge is zero. For γ > 0.5 the simulations are unpredictable, in the sense thatthey might achieve consensus or not, and prove a more interesting and challenging case:

7

0.002

0.004

0.006

0.008

0.01

0.012

0 0.5 1 1.5 2 2.5 3 3.5 4

W(µ

(t),

µN

(t))

time

for a fixed number of points N=32 (lin-lin scale)

γ=0.05γ=0.1γ=0.4

0.002

0.004

0.006

0.008

0.01

0.012

0 0.5 1 1.5 2 2.5 3 3.5 4

W(µ

(t),

µN

(t))

time

for a fixed value of γ=0.1 (lin-lin scale)

N=8N=16N=32N=64

(a) γ ≤ 0.5

(b) γ > 0.5

Figure 3: Convergence to the continuum model in 1D. (a): The discrete-continuum divergence, insense of the distance defined by (8), is dominated by the fast convergence of both models to their asymptotics,in the case γ ≤ 0.5 and the average velocity is 0. (b): The discrete simulation and the continuum simulationseem to diverge linearly in time in Wasserstein-distance, in the regime γ > 0.5.

our numerical experiments in Figure 3(b) suggest that the individual-based simulation andthe continuum simulation diverge linearly in time and not exponentially. Therefore, weconjecture that

C(t) = 1 +Kt, (12)

and we want to give an estimation for K depending on the parameters N and γ. Of course,the exponential growth estimation remains correct, because we have

C(t) = 1 +Kt ≤∞∑n=0

(Kt)n

n!= exp (Kt) .

Supposing now that the conjecture (12) holds, W(µ(t), µN(t)

)is then dominated by D(t) :=

8

N−1/2 +N−1/2Kt, whose slope is

D′[N, γ] =dDdt

= N−1/2K[N, γ]. (13)

The aim of the following numerical study is to get an intuition of this parameter K[N, γ], inparticular of its dependencies.In Figure 3(b) we have already pointed out that, after an initial transient, the kinetic/par-ticle divergence is linear. We have then measured the slopes D′, with different parametersN and γ, of this conjectured linear divergence, for time t ≥ 4 in our experiments. FromFigure 4(a) we evince that K does not depend on N : for any γ, we observe a N−1/2-factor,which fits the conjectured slope D′ = N−1/2K[N, γ] if K[N, γ] does actually not depend onN . Therefore, K = K[γ].Going on with this reasoning, from Figure 4(b) we empirically observe that the growth ratefor K[γ] with respect to γ is no more than linear, so K may look like

K[γ] = Kγ, K is a real constant.

2.2.2. Initialization on the rectangle

We use as initial datum (10), by mimicking the continuum distribution through thealgorithm given in Appendix A. Unlike the study in Section 2.2.1, we do not force theparticles to have the same average velocity as the continuum distribution. The results,shown in Figure 5, are very similar to those referring to the initial condition (10): thediscrepancy between discrete and continuum models seems to grow linearly in time and theN -dependency is clear.

3. Leadership emergence

3.1. The anisotropy

The Cucker-Smale model is said to be anisotropic when φ < π in (3): this means that aparticle P (i) is not influenced by all the other particles, but just by those that P (i) can see,i.e., by those that are inside its viewing angle. We define the viewing cone of particle P (i)

as (skipping time-dependency)

C(i)[φ] =x ∈ Rd such that

∣∣angle(x− x(i),v(i)

)∣∣ ≤ φ⊆ Rd, (14)

where

angle(x− x(i),v(i)

):= arccos

[ (x− x(i)

)· v(i)

|x− x(i)|Rd |v(i)|Rd

]is the angle between vector x − x(i) and vector v(i). Under these notations, particle P (i) isinfluenced by particle P (j) if and only if P (j) ∈ C(i)[φ], see Figure 1 for an example of this.

9

0.0001

0.001

0.01

8 16 32 64 128

slo

pe

N

slopes for a fixed γ

γ=.5γ=.55

γ=.6γ=.65

γ=.7γ=.75

γ=.8γ=.85

(a)

0.0005

0.001

0.0015

0.002

0.0025

0.003

0.0035

0.004

0.0045

.5 .6 .7 .8

slo

pe

γ

slopes for a fixed N

N=8N=16N=32N=64

N=128

(b)

Figure 4: Convergence to the continuum model in 1D. Estimation of the slope D′ defined by (13),i.e. the linear divergence rate observable in Figure 3(b) after some initial transient. In this picture weclearly observe a N−1/2 factor for any γ, which fits conjecture D′ = N−1/2K[N, γ] as long as K[N, γ] doesnot depend on N . Moreover, we observe a linear dependency on γ for any N . This, together with K notdepending on γ (see Figure 4(a)), suggests that K = Kγ, for K a real constant.

10

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 1 2 3 4 5 6 7 8 9 10

W(µ

(t),

µN

(t))

time

for fixed N=128

γ = 0.40γ = 0.50γ = 0.60γ = 0.70γ = 0.80γ = 0.90

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 1 2 3 4 5 6 7 8 9 10

W(µ

(t),

µN

(t))

time

for fixed γ=0.40

N = 8N = 16N = 32N = 64

N = 128

0 0.05

0.1

0.15 0.2

0.25 0.3

0.35

0.4 0.45

0.5

0 1 2 3 4 5 6 7 8 9 10

W(µ

(t),

µN

(t))

time

for fixed N=8

γ = 0.40γ = 0.50γ = 0.60γ = 0.70γ = 0.80γ = 0.90

0 0.05

0.1

0.15 0.2

0.25 0.3

0.35

0.4 0.45

0.5

0 1 2 3 4 5 6 7 8 9 10

W(µ

(t),

µN

(t))

time

for fixed γ=0.90

N = 8N = 16N = 32N = 64

N = 128

Figure 5: Convergence to the continuum model in 1D. These numerical results refer to initial condition(11), and show the time evolution of the Wasserstein distance between discrete and continuum model, fordifferent parameters of the problem. The experiments clearly suggest a liner growth, more optimistic thanGronwall’s exponential estimates.

11

3.2. Using graph theory

The set of particles is thought as a directed graph, with the particles as vertices andan arc going from particle i to particle j if and only if i is influenced be j, i.e., if andonly if P (j) ∈ C(i)[φ]. Two vertices are said to be connected if and only if there exists apath between them, i.e., if and only if a vertex is reachable from the other (obviously notnecessarily with a reciprocal visibility/reachability though); hence reachability turns intoan equivalence relation, therefore the maximum sub-graphs, called connected components,containing all the connected vertices identify a partition on the particle set. These classesof equivalence will be called troops, denoted Tp, for p = 1, . . . , Ntroops; of course, the totalnumber of troops is variable during the evolution of the system. We define the leaders Lp ofa troop Tp as those particles that are not influenced by anyone else:

Lp =P (i) ∈ Tp such that C(i)[φ] = ∅

, p = 1, . . . , Ntroops.

If card(Lp) = 1, then we can inscribe the whole troop inside a cone (a triangle for the 2Dcase) having as vertex the leader. Moreover, we introduce the following notation:

αp := the angle at the vertex of cone number p.

Remark 3.1 (leaders in the continuum setting). For the continuum case, the concept ofleader needs a proper average definition. If we introduce the map assigning the average

velocity to each point in the space domain Ωx :=d⊗

n=1

[xminn , xmax

n

]⊆ Rd:

ψ : Ωx → Rd, x 7→ ψ(x) =

∫v f(t, x,v) dv

ρ(x),

then we can define as “leaders” the portion of the x-domain containing the points x headingψ(x) without “seeing” anyone:

L :=

x ∈ ρ 6= 0 such that

∫Ωx

ρ(x)χ [cos (x− x,ψ(x)) ≥ cos(φ)] dx = 0

⊆ Ωx.

3.3. Evolution in 2D

An example of evolution for this system is given in Figure 6 for the 2D case: at thebeginning the system is initialized by placing the particles on an equilateral triangle, withall of their velocities heading upward, as shown at time 0. The topmost particle has therole of leader. Then we perturb the component of the velocities along x1 in an amountproportional to their x2-position. More in detail, we set the magnitude of the perturbationas

σ(x2) = Kmaxi

(x

(i)2

)− x2

maxi

(x

(i)2

)−min

i

(x

(i)2

) , (15)

12

where K is an arbitrary constant which we can use to adjust the intensity of the perturbation.Some particles are scattered away and are lost at the sides, either followed by some otherparticles or alone: there is kind of a peeling-effect on the sides of the initial triangle, as itis shown for time 25, time 50, time 75 and time 100. The asymptotics corresponds to eachtroop having all its particles agreeing the same velocity:

for all p = 1, . . . , Ntroops v(j) → vp ∀x(j) ∈ Tp.

This does not forbid that at the asymptotics the troops have more than one leader

card(Lp) > 1

as long as the leaders have exactly the same velocity from a certain time on (refer to Section3.5.1 for the details); notwithstanding, our simulations are constructed in such a way toinhibit that situation, and the leaders will, therefore, organize their own troops: at time 355we observe that the main troop still has several leaders; at time 365 it has split into twogroups, and finally at time 1500 we observe that there are three different groups, each onewith its own leader. It is therefore possible to inscribe the troops inside triangles having asvertices the leaders, as it is sketched in Figure 6.

3.4. Stability of triangles

As initial data for our discrete solver, we have used three different configurations, sketchedin Figure 7. We have then randomly perturbed the x2-component of the velocities as in (15)and waited for the system to reach its asymptotics, so that the angles αp stabilize toconstant values. In Figure 8, in which we plot the maximum angle

α[φ] := maxp=1,...,Ntroops

αp[φ]

against the azimuth angle φ, we see that triangles are stable configurations in the sense that

α[φ] ≤ 2φ.

Remark 3.2 (on the continuum simulations). At the state-of-the-art, we cannot perform2D continuum simulations (4D in the phase space), because they are too computationallyexpensive, as shown in Table 1. With such costs, for Nx1 = Nx2 = Nv1 = Nv2 = 64 ourmonocore implementation requires roughly one month to reach time t = 50 on a 2.13 GHzIntel Core2 Duo L9600 CPU. Therefore, we believe that performing such simulations goes outof the scope of this work: a major implementation effort is required, in terms of optimizationand CPU- or GPU-parallelization, which is left for future work.

The 3D case goes much the same as the 2D case, the only difference being that theangle is now a solid one. We have performed numerical simulations with initializations on apyramid-shaped configuration, which confirm the 2D results.

13

Figure 6: Leadership emergence in 2D. An example of partition into troops, with φ = 0.4, γ = 0.05.The arrows from i to j mean that i is influenced by j; the troops are the groups of particles connected byarcs.

14

x2

x1

(a) full triangle

x2

x1

(b) the perimeter

x2

x1

(c) only the border

Figure 7: Leadership emergence in 2D. Different initialization strategies for the solver: particles caneither (a) be a full equilateral triangle, (b) be on the sides, or (c) on the whole perimeter.

(Nx1 , Nx2 , Nv1 , Nv2)-mesh comp. time needed to reach time t = 0.1

16× 16× 16× 16 3.5 s32× 32× 32× 32 121.5 s64× 64× 64× 64 6042.6 s

Table 1: Two-dimensional continuum simulation. At the state-of-the-art, we cannot perform 2Dsimulations, which are 4D in the phase space, because the computational cost is too large. Simulations aremonocore runs performed on a 2.13 GHz Intel Core2 Duo L9600 CPU.

3.5. Theoretical support

In the following, we wish to show two important aspects of the anisotropic model: first,that each troop has only one leader in the asymptotic state, unless two or more leaders have,from a certain time on, exactly the same velocity; second, that the troops are cones with theleader as apex and its velocity as axis.

3.5.1. Only-one-leader states

The only-one-leader state is not necessary, and simulations can be constructed for whicha multiple-leader state is actually stable, where “stable” means that the followers, afterhaving their velocities perturbed, realign their directions to that of the leaders. Numericalexperiments and theoretical arguments suggest that this is possible as long as all the leadershave exactly the same velocity. In Figure 9(a) we show the initialization for a 3-particlesimulation, whose evolution is given in Figure 9(b), where the trace of the trajectories showsthat the follower, after having its x1-component of the velocity perturbed, reorientates fol-lowing the leaders’ alignment. No matter which, among the leaders, influences more thefollower, this last one will always be pushed to the same direction. Of course, this abilitydepends on the viewing angle φ and on the amount of the perturbation: simulations can beconstructed for which one or more followers are scattered because they lose visual contactwith the leaders.

The situation, for the case v(k) 6= v(`), is completely different: numerical evidence showsthat either the two-leader configuration collapses to a one-leader configuration, or the troopsplits into several troops. In Figure 10(a) we show an example of a system in which the

15

0

0.5

1

1.5

2

0 0.2 0.4 0.6 0.8 1

α

φ

full triangle

numerical αcurve α=2φ

0

0.5

1

1.5

2

0 0.2 0.4 0.6 0.8 1

α

φ

only the sides

numerical αcurve α=2φ

0

0.5

1

1.5

2

0 0.2 0.4 0.6 0.8 1

α

φ

the perimeter

numerical αcurve α=2φ

Figure 8: Leadership emergence in 2D. The angle α := maxp αp is bounded by 2φ, with φ being theangle of vision. γ is set equal to 0.05.

16

4

5

6

7

8

-0.5 0 0.5

(a) initialization

5

10

15

20

25

30

35

40

45

50

-0.5 0 0.5

(b) trace of the trajectories

Figure 9: Stability of a multiple-leader configuration. A simulation is inizialized with two leadershaving exactly the same velocity. The velocity of the rear follower is then perturbed, but it realigns to theleaders’ orientation.

leaders will never influence each other, e.g. because their directions diverge: in that case,the follower must decide which leader to follow, hence the troop is split. If, otherwise, thetrajectories of the two leaders are such that at some point one falls into the influence ofthe other, then it is likely we switch from a two-leader to a one-leader configuration. Thesituation is depicted in Figure 10(b): the “right” leader enters the viewing cone of the “left”leader and drags it towards its alignment.

We believe there is a theoretical argument that forbids two leaders inside a troop to havedifferent velocities: the asymptotics of the system corresponds to troop-by-troop constant-velocity states. We remind that the dynamics is

d

dtv(i) =

1

N

N∑j=1

χ[cos(x(j) − x(i),v(i)

)≥ cos(φ)

](1 + |x(j) − x(i)|2Rd

)γ ·(v(j) − v(i)

).

Therefore, we need for the asymptotics, for any x(i) and x(j), either

χ[cos(x(j) − x(i),v(i)

)≥ cos(φ)

]−−−−→t→+∞

0, (16)

or1(

1 + |x(j) − x(i)|2Rd)γ −−−−→

t→+∞0, (17)

orv(j) − v(i) −−−−→

t→+∞0. (18)

Condition (16) would be satisfied if no particle lay inside the viewing cone of any otherparticle, which would mean that every agent just goes on with its velocity no matter whatthe others do, thus producing dispersion, a kind of a gaseous behavior; a similar pattern is

17

0

20

40

60

80

100

120

-1 0 1 2 3 4 5 6

(a) the leaders will never influence each other

0

5

10

15

20

25

30

35

40

45

-3 -2.5 -2 -1.5 -1 -0.5 0 0.5

(b) at some point, there is interaction between theleaders

Figure 10: Stability of a multiple-leader configuration. If the orientation of the two leaders diverge,then the follower must decide which one it follows. If the orientation of the two leaders converge, then wemight end up with a one-leader configuration.

represented by condition (17), achievable either by a large γ, or by shrinking φ → 0: suchconfigurations are not difficult to obtain but are also uninteresting for the scope of this study,aiming at leadrship inside troops. The reachability condition described by (16) identifies, asalready remarked in Section 3.2, the partition into troops:

x(j) ∈ Tp, x(i) ∈ Tp′ with p 6= p′

mχ[cos(x(j) − x(i),v(i)

)≥ cos(φ)

]· χ[cos(x(j) − x(i),v(j)

)≥ cos(φ)

]= 0.

Thus, the research of asymptotics goes troop by troop, inside each of which condition (18)should be fulfilled. This means that, inside Tp, all the velocities must tend asymptoticallyto the same

v(j) → v for all x(j) ∈ Tp. (19)

Moreover, this v is necessarily that of the leader, which, by definition, is not influenced byanyone else, hence it cannot modify its orientation.

Now, let us suppose that, inside troop Tp, particles k and ` are, from a certain time t on,leaders. Therefore, by definition of leadership, their directions are not influenced by anyoneelse:

v(k)(t) = v(j) (t) , v(`)(t) = v(`) (t) , for any t ≥ t.

The only possibility for the troop’s asymptotics, i.e. for Tp to hold together, is that, from tonward, their velocities be exactly the same:

v(j) (t) = limt→+∞

v(k)(t) = v = limt→+∞

v(`)(t) = v(`) (t) =⇒ v(j) (t) = v(`) (t) , ∀t ≥ t.

This argument can of course be extended to an arbitrary number of leaders.

18

η

particle i

leader

troop

Φ

Figure 11: Leadership emergence. The shape of the troop is a right circular cone having as apex theleader Lp, as axis the direction given by v(Lp) and as aperture 2φ; we shall call the cone Cp[φ]. Supposingthat a particle lies outside Cp[φ] leads to a contradiction on the uniqueness of the leader.

3.5.2. On the estimate

We now focus on the estimate α[φ] ≤ 2φ: the analysis performed in the former sectionallows us to show, by a contradiction argument, that the shape of the troop is a right circularcone having as apex the leader Lp, as axis the direction given by v(Lp) and as aperture 2φ,as depicted in Figure 11; we shall call the cone Cp[φ].

Our hypotheses are: that in troop Tp there is solely one leader (we need this in orderto inscribe the troop inside a cone) and that we are approaching the asymptotics. Let us,moreover, suppose that particle x(i) lies outside Cp[φ] but belongs to the troop Tp.Being close to the asymptotics means (reformulating property (19)) that

∀η > 0, ∃t such that∣∣v(i) − v(Lp)

∣∣ < η ∀t ≥ t.

The real viewing cone C(i)[φ] of particle i is included inside C(i)[φ+ η]:

C(i)[φ] ⊆ C(i)[φ+ η] :=x ∈ Rd such that

∣∣angle(x− x(i),v(Lp)

)∣∣ ≤ φ+ η⊆ Rd,

as sketched in Figure 11. No matter where we place x(i), we can always take η small enoughso as to have

C(i)[φ+ η]⋂Cp[φ] = ∅,

which means that particle x(i) does not see anyone else, hence it is, together with assumptionx(i) ∈ Tp, a leader =⇒ , i.e. contradiction on the only-one-leader hypothesis.

3.6. Another observable statistical magnitude: the number of troops

The number of troops Ntroops[φ] depends on the viewing angle φ: short angles favordispersion because the particles lose visual contact with the surrounding agents easily. InFigure 12 we depict Ntroops[φ] against φ, in the 2D case, with the initialization in Figure

19

0

5

10

15

20

25

30

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

nu

mb

er

of

troop

s

Φ

K=0.5K=1

Figure 12: Number of troops. The number of troops Ntroops depends on the viewing angle φ. Theintensity of the perturbation K in (15) influences Ntroops. Here, the lines represent averages over a batteryof 100 tests. The initialization is that of Figure 7(a).

7(a), for two different intensities of the perturbation (the K parameter in equation (15)):numerical evidence supports the monotonicity of Ntroops[φ]:

φ1 ≤ φ2 =⇒ Ntroops[φ1] ≤ Ntroops[φ2], N ←−−φ→0

Ntroops[φ] −−−→φ→π

2

1.

4. The asymptotic state

From [4, 25] we know that in the discrete model the condition γ ≤ 12

is sufficient by itself

for all the particles to agree the average velocity v =1

N

N∑i=1

v(i). In [26], the extension of this

theorem to the continuum limit is proven. In case γ > 12, the discrete model might achieve

consensus or not depending on the concentration of the particles in space and velocity at theinitial time

(ΓN [x(0)],ΛN [v(0)]

), where the relative energies read

ΓN [x] :=1

2N2

∑i,j

∣∣x(i) − x(j)∣∣2 (potential), ΛN [v] :=

1

2N2

∑i,j

∣∣v(i) − v(j)∣∣2 (kinetic).

(20)

20

0

2e-06

4e-06

6e-06

8e-06

1e-05

1.2e-05

1e-16 1e-14 1e-12 1e-10 1e-08 1e-06 0.0001 0.01 1

Λ0

Γ0

32

64

128

256

Figure 13: Asymptotic state of the isotropic model. This graphic visually sketches the condition givenby Cucker and Smale (21) in [4]: consensus is ensured below the lines. This condition appears to be veryrestrictive, and more and more constraining as N → +∞.

In [4] the authors give a sufficient, non-necessary condition to ensure consensus, based onthe two-dimensional parameter space Π :=

(ΓN [x(0)],ΛN [v(0)]

):(

1

8

) 12γ−1

[(1

2γ

) 12γ−1

−(

1

2γ

) 2γ2γ−1

]︸ ︷︷ ︸

=: Cγ a constant that is only γ-dependent

1

N2

(ν2

N4

) 12γ−1

︸ ︷︷ ︸=: CN is only N -dep.

(1

ΛN [v(0)]

) 12γ−1

> 2ΓN [x(0)] +1

N2. (21)

In [4] the author give no explicit value for ν, but there is theoretical evidence that ν = N(see Appendix B); condition (21) is extremely restrictive, and it becomes more and moreconstraining as N grows larger; the situation is sketched in Figure 13, where the x-axis isplotted in log-scale because, otherwise, it could not even be possible to observe anything.Moreover, (21) does not provide hints for the continuum case, because

Cγ CN(

1

ΛN [v(0)]

) 12γ−1

> 2 ΓN [x(0)] +1

N2−−−→N→∞

0 ≥ 2 Γ[ρ1(0,x)].

Here, the continuum version of (20) would be:

Γ[ρ1] :=1

2

∫∫|x− y|2 ρ1(x)ρ1(y) dy dx, Λ[ρ2] :=

1

2

∫∫|v −w|2 ρ2(v)ρ2(w) dw dv (22)

where we have used notations ρ1(t,x) and ρ2(t,v) for the marginal densities

ρ1(t,x) :=

∫f(t,x,v) dv, ρ2(t,v) :=

∫f(t,x,v) dx.

21

The reader can check that, injecting the atomic distributions

f(t,x,v) =1

N

N∑j=1

δx(j) ⊗ δv(j)(x,v)

into (22), we recover exactly formulae (20).In the continuum case no theoretical result has been proven for γ > 1

2. In order to

have an intuition of what happens in that setting, and also in order to relax the Cucker-Smale condition (21), we shall plot, for increasing N , phase-transition diagrams: by that,we mean the threshold between consensus-achieving and dissent-achieving simulations in thetwo-dimensional Π-parameter space. By consensus-achieving simulation we mean that

∀j, v(j)(t) −−−−→t→+∞

v or, equivalently, ΛN [v(t)] −−−−→t→+∞

0. (23)

Deciding whether a simulation is achieving consensus or not is not that straightforward,because we might observe a decrease of ΛN [v(t)] during the evolution of a single run withoutbeing able to predict, even after a long observation time, whether

ΛN [v(t)] ↓ 0 or ΛN [v(t)] ↓ λ∞ 6= 0.

In Section 4.1 we shall try to give a characterization for consensus-achieving and dissent-achieving simulations, then in Section 4.2 we shall plot the phase-transition diagrams.

4.1. Asymptotic state of the isotropic model

The first problem we have, in order to construct the phase transition diagram, is to decidewhether a discrete simulation achieves consensus (23) or not. In order to do that we wishfor a better understanding of the asymptotic state.

All along the evolution of the system, the average velocity v(t) is conserved and thecenter of mass x(t) moves at the average velocity

d x(t), v(t)dt

= v,0 , x(t), v(t) :=

1

N

N∑j=1

x(j)(t),v(j)(t)

(discrete)

∫∫x,v f(t,x,v) dx dv∫∫

f(t,x,v) dv dx(continuum)

.

For the consensus-achieving simulations (γ ≤ 12, condition (21)) the distribution function is

pushed towards a Dirac-δ concentrated at the average velocity, whilst the space distributiontends to be constant after taking into account the displacement of the center of mass, i.e.the relative distances become fixed:

Λ −−−−→t→+∞

0 and Γ −−−−→t→+∞

constant.

22

(a) continuum case (b) discrete case

Figure 14: Asymptotic state of the isotropic model. The behavior of the dissent-achieving simulationsis not too different from that of the consensus-achieving ones: instead of a Dirac-δ concentrated at v = v, asort of 2D Dirac-δ appears in the phase space, relating the velocity to the position.

This translates into x(i)(t)

v(i)(t)

−−−−→t→+∞

x(i)∞ + v t

v

(discr.), f(t,x,v) −−−−→t→+∞

ρ∞(x− vt) δv(v) (cont.),

where x(i)∞ and ρ∞(x) are projections of the initial datum onto the asymptotic density.

Numerical evidence, as can be seen in Figure 14 for a 1D case, suggests that also for thedissent-achieving simulations

v(i) 9 v (discr.), f(t,x,v) 9 ρ1(t,x)δv(v) (cont.) or, equivalently, Λ 9 0

the velocities and the positions stay, in the phase space, close to a hypersurface, which tendsto a hyperplane v = constant at the asymptotic state. During the evolution, a hyperplanedoes not seem stable under the dynamics, numerical experiments suggest, and in Figure 14the particles do not actually seem to lie on a straight line, yet they do seem to try to stayon a hypersurface. Anyway, the evolution of the Ordinary Least Squares (OLS) hyperplaneapproximating the particle distribution provides very useful information, as we show in thefollowing: the OLS hyperplane is given by

v1

v2...vd

=

β1,1 β1,2 . . . β1,d

β2,1 β2,2 . . . β2,d...

......

...βd,1 βd,2 . . . βd,d

︸ ︷︷ ︸

=:A

x1

x2...xd

+

β1,0

β2,0...βd,0

.

The slopes βn,k ((n, k) ∈ 1, . . . , d× 1, . . . , d) and the intercepts βn,0 (n ∈ 1, . . . , d) are

23

obtained by minimizing, dimension by dimension, the Sum of Square Residuals (SSR)

Sn [βn,0, βn,1, . . . , βn,0] :=N∑j=1

(v(j)n −

(βn,0 +

d∑`=1

βn,` x(j)`

))2

for n = 1, . . . , d,

i.e. by letting∂Sn∂βn,k

= 0, for (n, k) ∈ 1, . . . , d × 0, . . . , d,

which gives

for n = 1, . . . , d

〈1,vn〉 = βn,0 〈1,1〉+

d∑`=1

βn,` 〈1,x`〉 ,

〈xk,vn〉 = βn,0 〈xk,1〉+d∑`=1

βn,` 〈xk,x`〉 for k = 1, . . . , d,

where we have let

〈x`,xm〉 :=N∑j=1

x(j)` x(j)

m .

We have to solve one (symmetric) linear system for each dimension:

〈1,1〉 〈1,x1〉 〈1,x2〉 . . . 〈1,xd〉〈x1,1〉 〈x1,x1〉 〈x1,x2〉 . . . 〈x1,xd〉〈x2,1〉 〈x2,x1〉 〈x2,x2〉 . . . 〈x2,xd〉

......

......

...〈xd,1〉 〈xd,x1〉 〈xd,x2〉 . . . 〈xd,xd〉

︸ ︷︷ ︸

=:M

β1,0 β2,0 . . . βd,0β1,1 β2,1 . . . βd,1β1,2 β2,2 . . . βd,2

......

......

β1,d β2,d . . . βn,d

=

〈1,v1〉 〈1,v2〉 . . . 〈1,vd〉〈x1,v1〉 〈x1,v2〉 . . . 〈x1,vd〉〈x2,v1〉 〈x2,v2〉 . . . 〈x2,vd〉

......

......

〈xd,v1〉 〈xd,v2〉 . . . 〈xd,vd〉

.

In Figure 15(a) we plot, for 1D cases, the evolution of S(t) for different values of γ. Thenumerics suggests

S(t) −−−−→t→+∞

0 as ept ⇐⇒ consensus, S(t) −−−−→t→+∞

0 as tp ⇐⇒ dissent, (24)

as stressed by the lin-log and the log-log graphics in Figure 15(a). Figure 15(b), in which wesketch in 1D the numerically-evaluated evolution of β1,1 and β1,0, shows that β1,1(t) behaveslike S(t):

β1,1(t) −−−−→t→+∞

0 as ept ⇐⇒ consensus, β1,1(t) −−−−→t→+∞

0 as tp ⇐⇒ dissent. (25)

We show in Figure 16 the decay rates for the SSR and for β1,1. We also observe that thelimit of the intercept β1,0 characterizes the consensus:

β1,0 −−−−→t→+∞

v ⇐⇒ the simulation is consensus-achieving (26)

24

(a) decay of the Sum of Square Residuals S(t)

(b) decay of the OLS’s slope β1,1(t) and convergence of the interceptβ1,0(t)

Figure 15: Asymptotic state of the isotropic model. These graphics show the convergence of theSSR S(t) −−−−−→

x→+∞0 of discrete simulations, for different values of γ, and the evolution of the slope β1,1

and of the intercept β1,0. The results suggest that the convergence for S(t) and β1,1(t) is exponential forthe consensus-achieving simulations and polynomial for the dissent-achieving ones, and that β1,0 −−−−→

t→+∞v ⇔ the simulation is consensus-achieving and β1,0 −−−−→

t→+∞0 ⇔ the simulation is dissenst-achieving. Here,

N = 16 and the initial condition is the indicatrix function of rectangle (−2,−3), (0,−3), (0, 1), (−2, 1).

25

-1.6

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

exp

on

en

tia

l d

eca

y r

ate

γ

SSRβ1,1

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0.6 0.65 0.7 0.75 0.8 0.85 0.9

po

lyn

om

ic d

eca

y r

ate

γ

SSRβ1,1

Figure 16: Asymptotic state of the isotropic model. These graphics show the decay rate for theSSR S(t) and for the slope β1,1(t). For the consensus-achieving simulations, both the SSR S(t) and theslope β1,1(t) have an exponential decay, hence the rate has to be interpreted as exp(p t). For the dissent-achieving simulations, both the SSR S(t) and the slope β1,1(t) have a polynomial decay, hence the ratehas to be interpreted as tp. Here, N = 16 and the initial condition is the indicatrix function of rectangle(−2,−3), (0,−3), (0, 1), (−2, 1).

andβ1,0 −−−−→

t→+∞0 ⇐⇒ the simulation is dissent-achieving. (27)

The multi-dimesional case, sketched in Figure 17 for the two-dimensional setting and inFigure 18 for the three-dimensional setting, show a qualitative behavior similar to the one-dimensional case: the Sidi=1 decay exponentially for the consensus-achieving simulations

and polynomially otherwise; moreover, βi,0di=1 converges to vidi=1 if and only if the systemagrees consensus, and to zero otherwise. In the following, we shall try to give an explanationto these observations.

For the consensus-achieving simulations, it is natural that

∀i, v(i)(t) ≈ v(x(i))

= A(t)x(i) + β·,0(t) −−−−→t→+∞

v =⇒(A(t),β·,0(t)

)−−−−→t→+∞

(0, v) ;

moreover, as v(i)(t) −−−−→t→+∞

v exponentially fast, A(t) and S(t) must decay exponentially fast

too. The system is approaching an asymptotic equilibrium when∑i,j

⟨v(i)(t)− v(j)(t), v(i)(t)− v(j)(t)

⟩=: F(t) −−−−→

t→+∞0,

which is equivalent to

A : W −−−−→t→+∞

0, ai,j :=1(

1 + |x(i) − x(j)|2)γ , wi,j :=

⟨v(i),v(j) − v(i)

⟩and A : W :=

∑i,j

ai,jwi,j denotes the Frobenius inner product between matrices. For a

consensus-achieving simulation, we have W → 0 exponentially fast, while A 9 0 because

26

1e-20

1e-15

1e-10

1e-05

1

100000

0 50 100 150 200 250 300 350 400

SS

R

time

γ=0.25

S1S2

0.01

0.1

1

10

100

1000

10000

100000

0.01 0.1 1 10 100

SS

R

time (log-scale)

γ=0.75

S1S2

(a) evolution of the Sum of Square Residuals

0

0.5

1

1.5

2

2.5

3

0 50 100 150 200 250 300 350 400

time

γ=0.25

β1,0average v1

β2,0average v2

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

0 50 100 150 200 250 300 350 400

time

γ=0.75

β1,0average v1

β2,0average v2

(b) evolution of the intercepts

1e-14

1e-12

1e-10

1e-08

1e-06

0.0001

0.01

1

0 50 100 150 200 250 300 350 400

time

γ=0.25

β1,1β1,2β2,1β2,2

1e-07

1e-06

1e-05

0.0001

0.001

0.01

0.1

1

0.01 0.1 1 10 100

time (log-scale)

γ=0.75

β1,1β1,2β2,1β2,2

(c) evolution of the slopes

Figure 17: Asymptotic state of the two-dimensional isotropic model. The results are qualitativelysimilar to those of the one-dimensional case: the SSR’s and the slopes show an exponential decay if and onlyif the simulation agrees consensus, and the intercepts converge to the average velocity’s components if andonly if the simulation agrees consensus (and to zero otherwise).

27

1e-16 1e-14 1e-12 1e-10 1e-08 1e-06

0.0001 0.01

1 100

10000

0 50 100 150 200 250 300 350 400

SS

R

time

γ=0.25

S1S2S3

0.01

0.1

1

10

100

1000

10000

0.01 0.1 1 10 100

SS

R

time (log-scale)

γ=0.75

S1S2S3

(a) evolution of the Sum of Square Residuals

-1.4-1.2

-1

-0.8-0.6-0.4-0.2

0

0.2 0.4 0.6

0 50 100 150 200 250 300 350 400

time

γ=0.25

β1,0avg v1

β2,0avg v2

β3,0avg v3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

0 50 100 150 200 250 300 350 400

time

γ=0.75

β1,0avg v1

β2,0avg v2

β3,0avg v3

(b) evolution of the intercepts

1e-12

1e-10

1e-08

1e-06

0.0001

0.01

1

0 50 100 150 200 250 300 350 400

time

γ=0.25

β1,1β1,2β1,3β2,1β2,2β2,3β3,1β3,2β3,3

1e-08

1e-07

1e-06

1e-05

0.0001

0.001

0.01

0.1

1

0.01 0.1 1 10 100

time (log-scale)

γ=0.75

β1,1β1,2β1,3β2,1β2,2β2,3β3,1β3,2β3,3

(c) evolution of the slopes

Figure 18: Asymptotic state of the three-dimensional isotropic model. The results are qualitativelysimilar to those of the one-dimensional case: the SSR’s and the slopes show an exponential decay if and onlyif the simulation agrees consensus, and the intercepts converge to the average velocity’s components if andonly if the simulation agrees consensus (and to zero otherwise).

28

there cannot be dispersion, i.e.∣∣x(i)(t)− x(j)(t)

∣∣ ≤ C for any t ≥ 0 and for any i and j,hence

ai,j =1(

1 + |x(i)(t)− x(j)(t)|2)γ ≥ 1

(1 + C2)γ 0.

For a dissent-achieving simulation, the situation is kind of the opposite. The velocitiesconverge v(i)(t) −−−−→

t→+∞v

(i)∞ but these limiting velocities may not coincide. The dissent

prevents W from converging to 0. As for A,

ai,j(t) =1(

1 + |x(i)(t)− x(j)(t)|2)γ ∼ 1(

1 + t2∣∣∣v(i)∞ − v(j)

∞

∣∣∣2)γ ,thus, if v

(i)∞ 6= v

(j)∞ ,

ai,j(t) −−−−→t→+∞

0 as t−2γ and wi,j(t) −−−−→t→+∞

⟨v(i)∞ ,v

(j)∞ − v(i)

∞⟩,

so the asymptotic state is reached with polynomial speed. This is related to the convergencerate of the slopes βn,k (k 6= 0). If we inject

v = Ax+ β·,0,

we obtain

F = C∑i,j

ai,j

d∑n=1

σ2n x

(i)n

(x(j)n − x(i)

n

)with σn the eigenvalues of matrix A. For the dissent-achieving case, as x(i)

n

(x(j)n − x(i)

n

)explodes as t2, the eigenvalues σn (equivalently the slopes βn,k) kill that growth by convergingat least as t−1 to zero; for the consensus-achieving case, as x(i)

n

(x(j)n − x(i)

n

)explodes only as

a polynomial and A 9 0, the eigenvalues σn (equivalently the slopes βn,k) must decreaseexponentially like the velocities; the results are confirmed in Figure 17(c) and Figure 18(c).

Remark 4.1. From numerical evidence, in the dissent-achieving simulations, only the βn,nseem to show the very expected decay rate t−1; the βn,k with n 6= k show a polynomialconvergence to zero whose rate might be different than −1. For the consensus-achievingsimulations, the behaviors of the βn,n and the βn,k (n 6= k) again seem different, even if inthis case the decay rates match better at least in the long term.

Proposition (27) may be somewhat misleading: it implies

v(t) := A(t)x+ β·,0(t) −−−−→t→+∞

0x+ 0 = 0 ⇐= the simulation is dissent-achieving.

This apparently is a contradiction, in the sense that if the particles converged to a nullvelocity, there would actually be consensus, but in fact this is not the case beacuse the

29

relative distances explode. We wish to convince the reader about that, in a simplifiedsetting: suppose d = 1, N = 2, x(1)(0) = x(2)(0) = 0 and v(1)(0) > v(2)(0). We shall denote

v(t) := v(1)(t)− v(2)(t), x(t) := x(1)(t)− x(2)(t).

The system conserves the average velocity v, therefore at any instant

v(1)(t) = v +v(t)

2, v(2)(t) = v − v(t)

2.

Saying that the relative kinetic energy ΛN is positive and monotonically decreasing is equiv-alent to saying that v(t) is positive and monotonically decreasing, therefore it possesses alimit:

v(t) ↓ v∞

= 0 if consensus

6= 0 if dissent.

As a consequence,

v(1) ↓ K(1) = v +v∞2, v(2) ↑ K(2) = v − v∞

2.

Moreover, we can estimate

x(t) ≥ v∞ t (not useful for the consensus-achieving case).

The situation is sketched in Figure 19. For a 2-particle simulation, the OLS line is trivial:it is the line through the two points

v[t](x) =v(1)(t)− v(2)(t)

x(1)(t)− x(2)(t)︸ ︷︷ ︸the slope β1,1

x+x(1)(t) v(2)(t)− x(2)(t) v(1)(t)

x(1)(t)− x(2)(t)︸ ︷︷ ︸the intercept β1,0

.

For the slope β1,1 we can estimate

0 ≤ β1,1(t) =v(t)

x(t)

−−−−→t→+∞

0 if consensus

≤ v(0)

v∞ t−−−−→t→+∞

0 if dissent

which suggests for the dissent-achieving case a decay rate of t−1, and for the consensus-achieving case an exponential γ-related decay like v(t). For the intercept β1,0 we have

β1,0(t) = v

[1− v(t) t

x(t)

]−−−−→t→+∞

v if consensus

0 if dissent

because, for the consensus-achieving case, v(t) decays exponentially, and, for the dissent-achieving case,

v(t) t

x(t)−−−−→t→+∞

1.

30

-5

-4

-3

-2

-1

0

1

2

3

-400 -300 -200 -100 0 100 200

x2

x1

traces of the two particles

particle 1

particle 2

average velocity

γ=1γ=0.9γ=0.8γ=0.7γ=0.6γ=0.5

Figure 19: Asymptotic state of the isotropic model. This graphic shows the evolution, in 1D (2D inthe phase space), of 2-particle simulations, both consensus-achieving (γ = 0.5) and unpredictable (γ ≥ 0.6).

These results are confirmed by Figure 16.In conclusion, properties (24), (25), (26) and (27) can be implemented inside the numer-

ical code to estimate whether the simulation is consensus-achieving v(j) −−−−→t→+∞

v or not, in

a clearer way than directly deciding whether ΛN [v(t)] −−−−→t→+∞

0 or not:

consensus ⇐⇒ Sn −−−−→t→+∞

0 as ept ⇐⇒ βn,n −−−−→t→+∞

0 as ept ⇐⇒ βn,0 −−−−→t→+∞

vn,

dissent ⇐⇒ Sn −−−−→t→+∞

0 as tp ⇐⇒ βn,n −−−−→t→+∞

0 as tp ⇐⇒ βn,0 −−−−→t→+∞

0.(28)

Remark 4.2 (dependency on N). In 1D, the evolution of S(t) seems to depend on therandomness of the initial datum; nevertheless, its decay rate, both in consensus-achievingand in dissent-achieving simulations, does not seem to have any dependency on the numberof points N . The slope β1,1(t) and the intercept β1,0(t) do not seem to possess any dependencyon N either. These statements are supported by numerical experiments (not shown).

Remark 4.3 (continuum case). Similar studies as for the discrete case apply, the onlydifference being the adaptation of the discussion to the continuum setting. If we let

ψ : Ωx −→ Rd, x 7→∫v f(t,x,v) dv∫f(t,x,v) dv

(29)

31

the map relating the position x to the average velocity ψ(x) at that position, the OLS is

v(x) := Ax+ β·,0; we get the same linear system as for the discrete case, i.e.

〈1, 1〉 〈1, x1〉 〈1, x2〉 . . . 〈1, xd〉〈x1, 1〉 〈x1, x1〉 〈x1, x2〉 . . . 〈x1, xd〉〈x2, 1〉 〈x2, x1〉 〈x2, x2〉 . . . 〈x2, xd〉

......

......

...〈xd, 1〉 〈xd, x1〉 〈xd, x2〉 . . . 〈xd, xd〉

β1,0 β2,0 . . . βd,0β1,1 β2,1 . . . βd,1β1,2 β2,2 . . . βd,2...

......

...β1,d β2,d . . . βn,d

=

〈1, ψ1〉 〈1, ψ2〉 . . . 〈1, ψd〉〈x1, ψ1〉 〈x1, ψ2〉 . . . 〈x1, ψd〉〈x2, ψ1〉 〈x2, ψ2〉 . . . 〈x2, ψd〉

......

......

〈xd, ψ1〉 〈xd, ψ2〉 . . . 〈xd, ψd〉

,

up to replacing the discrete summation (4.1) by an integral on the density’s support

〈ϕk, ϕ`〉 :=

∫ρ1 6=0

ϕk(ξ)ϕ`(ξ) dξ.

We leave a numerical study on the continuum setting for future work.

4.2. Phase transition

We remind that the phase transition we are referring to is between consensus-achievingand dissent-achieving simulations, in the two-dimensional Π-parameter space. By studyingthe shape and position of this threshold in the individual-based model for an increasingnumber of particles N , we wish to get an intuition of how the continuum case should behavein the Π-parameter space.

In order to do that, we fix a pair (γ0, λ0) ∈ Π by initializing the particles’ initial positions-velocities inside [−1, 1]d×N

(x(0), v(0)) ∈ [−1, 1]d×N × [−1, 1]d×N

then rescaling them

(x(0),v(0)) =

(√γ0

ΓN [x(0)]x(0),

√λ0

ΛN [v(0)]v(0)

)∈ Rd×N × Rd×N

in order to fulfill (ΓN [x(0)],ΛN [v(0)]

)= (γ0, λ0) .

We perform Na runs and count how many of them achieve consensus (28); Na sufficientlylarge is needed in order to get statistically meaningful results. We then pass to the next(γ0, λ0) ∈ Π pair. After obtaining the surface representing the probability of consensus foreach pair (γ0, λ0) in the Π-prameter space, we call “phase transition”, for given N particlesand interaction strength γ, the level curve interpolating the 50 % probability of consensus.

In Figure 20 we plot the phase-transition curve for a fixed γ = 0.95 and different numbersof particles N . We observe that the curve does not move for N ∈ 32, 64, 128; this suggeststhat it might persists even in the continuum case. This conjecture is seemingly confirmedby the fact that the transition from consensus to dissent becomes sharper as N increases,as sketched in Figure 21: this result is what we must expect if the curve really exists in thecontinuum setting because the kinetic equation has a deterministic character.

32

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0 1 2 3 4 5 6 7 8 9 10

Λ0 (

kin

etic)

Γ0 (potential)

50 % probability of success

32 particles64 particles

128 particles

Figure 20: Consensus/dissent phase transition. These graphics aim at capturing the transition curvefrom consensus to dissent in the continuum case. In order to do that, the discrete case is studied from astatistical point of view and for N →∞. The numerical experiments suggest that the transition curve existsin the discrete case and that it might persists in the continuum case. In this example, d = 1 and γ = 0.95.

33

50% consensus

0 2 4 6 8 10

Γ0

.05

.25

.45

Λ0

(a) N = 32

50% consensus

0 2 4 6 8 10

Γ0

.05

.25

.45

Λ0

(b) N = 64

50% consensus

0 2 4 6 8 10

Γ0

.05

.25

.45

Λ0

(c) N = 128

Figure 21: Consensus/dissent phase transition. These graphics aim at capturing the transition curvefrom consensus to dissent in the continuum case. In order to do that, the discrete case is studied froma statistical point of view and for N → ∞. The numerical experiments suggest that the transition fromconsensus to dissent becomes sharper as N →∞, as expected because the continuum case is deterministic.In this example, d = 1 and γ = 0.95.

34

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0 2 4 6 8 10

Λ0 (

kin

etic)

Γ0 (potential)

γ = 0.85

γ = 0.90

γ = 0.95

Figure 22: Consensus/dissent phase transition. Parameter γ governs the interaction strength betweenthe agents. As γ → +∞, dispersed states are favored because the particles lose the ability of agreeing auniform direction. In this example, d = 1 and N = 32.

The strengh of interaction between the agents is monotonically decreasing as γ → +∞,hence we expect that larger values of γ favor dispersed states. In Figure 22 we show theconsensus percentages for N = 32 and for γ ∈ 0.85, 0.90, 0.95: as expected there is adisplacement of the phase-transition curve away from the axes as γ decreases.

Remark 4.4. The simulations shown in this section are for the one-dimensional case. For-mula (21) does not have any dependency on the dimension d, and numerical evidence (forN = 32 and γ = 0.95) confirms that the phase-transition curves do not change significantlyfor d ∈ 1, 2, 3, as shown in Figure 23.

Remark 4.5 (continuum simulations). In order to control parameters Γ[ρ1] and Λ[ρ2] in thecontinuum setting, we can use as initial function

f(t = 0, x, v) = χ[−A,A] ⊗ χ[−B,B](x, v).

The reader can check that if we let

A =

(3

16

γ20

λ0

)1/6

, B =

(3

16

λ20

γ0

)1/6

then we obtainΓ[ρ1(t = 0)] = γ0, Λ[ρ2(t = 0)] = λ0.

We can thus construct initial data with the relative kinetic and potential energies that wewish. We leave numerical experiments in this setting for future work.

35

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0 1 2 3 4 5 6 7 8 9 10

Λ0 (

kin

etic)

Γ0 (potential)

1D case2D case3D case

Figure 23: Consensus/dissent phase transition. The dimension d does not seem to be influential on theposition of the phase-transition curve in the Π-parameter space. Here, N = 32 and γ = 0.95.

5. Conclusions and future plans

Numerical experiments suggest that the individual-based Cucker-Smale model tends tobe well-approximated by the mean-field continuum model with an accuracy which is deterio-rating in time only polynomially and not exponentially as expected by the rather pessimisticGronwall’s estimates. Hence, we wish to pursue an analytical refinement of such stabilityresults as suggested by the numerics. Moreover, we have given theoretical and numericalsupport to the natural tendency of the Cucker-Smale model to produce leaders wheneveran additional viewing cone condition is introduced, and we have described the shape of thetroop as a cone with the apex’s angle bounded by the viewing angle. We intend in the futureto address the numerical implementation of the corresponding continuum model, although itwill require a rather demanding 4D simulation which is beyond the efficiency of our presentcode. Finally, we have given a characterization of the consensus-achieving and the dissent-achieving simulations by means of approximating the particle distribution on a hyperplaneof the phase space; although our experiments are limited to a rather modest amount of par-ticles, they suggest that a phase transition related to kinetic and potential energy densitiesof the initial datum can be expected also for N → ∞ as indicators of possible consensusemergences when γ > 1/2. It would be of great interest to perform the numerical simulationsin the continuum case and to verify such phenomenon analytically.

36

Acknowledgments

Francesco Vecil acknowledges the MINECO project MTM2011-22741. Massimo Fornasieracknowledges the financial support provided by the START award “Sparse Approximationand Optimization in High Dimensions” no. FWF Y 432-N15 of the Fonds zur Forderungder wissenschaftlichen Forschung (Austrian Science Foundation).

Appendix A. Approximation of a continuum datum through particles

Starting from an analytic expression for the initial datum f(0, x, v), we explain a possiblestrategy to give a discrete approximation, in 1D.

• First of all, we distribute the particles in space, then in velocity. Therefore, the firstthing we need to compute is the density

ρ(0, x) =

∫Rf(0, x, v) dv,

∫ xmax

xmin

ρ(0, ξ) dξ = 1,

where we have stressed that the mass is normalized.

• The primitive of the density takes its values between 0 and 1 thanks to the massnormalization

F [ρ](x) =

∫ x

xmin

ρ(0, ξ) dξ, F [ρ](xmin

)= 0, F [ρ] (xmax) = 1.

• Take a random uniform distribution r(i)i=1,...,N ⊆ [0, 1]N and compute the counter-images through F

xi = (F [ρ])−1(r(i)), for i = 1, . . . , N.

• We now have to distribute the velocities, once all the positions xiNi=1 ⊆ [xmin, xmax]have been fixed. Actually, the strategy is the same as for the space-points, apart thatwe do not use the density ρ(0, ·) anymore but the distribution function f(0, xi, ·):

F [ω] (xi, v) =

∫ v−vmax f (0, xi, ξ) dξ∫ vmax

−vmax f (0, xi, ξ) dξ, for i = 1, . . . , N

r(i)i=1,...,N = a random uniform distribution between 0 and 1

v(i) = (F [ω])−1(xi, r(i)), for i = 1, . . . , N.

37

Appendix B. Explicit calculation of parameter ν

Parameter ν is defined as a real value such that

ν‖v‖Rd×N ≤1

2

∑i,j

∣∣v(i) − v(j)∣∣2 for any v ∈ ∆⊥, (B.1)

being ∆⊥ the orthogonal complement of the diagonal of the velocity space

∆ :=

(v,v, . . . ,v) ∈ Rd×N⊆ Rd×N .

There is an explicit characterization for ∆⊥: it can be easily checked that any v ∈ Rd×Ndecomposes in a unique way as

v = (v, v, . . . , v)︸ ︷︷ ︸v∆

+(v(1) − v,v(2) − v, . . . ,v(N) − v

)︸ ︷︷ ︸v⊥

, (v∆,v⊥) ∈ ∆×∆⊥,

which gives then the characterization v ∈ ∆⊥ ⇔ v =1

N

N∑i=1

v(i) = 0.

In [4] the authors state that ν depends only on the number of points N , and prove that

ν ≥ 1

3N, but give no explicit value. Simple calculations allow to show that parameter ν has

the explicit value ν = N : with no loss of generality

Rd×N ⊇ ∆⊥ 3 v =

(v(1),v(2), . . . ,v(n) := −

N−1∑j=1

v(j)

).

Its ‖ · ‖2Q-norm can be explicitly computed:

‖v‖2Q =1

2

N−1∑i=1

N−1∑j=1

d∑n=1

(v(i)n − v(j)n

)2+

d∑n=1

(v(i)n − v(N)

n

)2+1

2

N−1∑j=1

d∑n=1

(v(N)n − v(j)n

)2= N‖v‖2Rd×N .

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