Soft Control of Swarms

Analytical Approach

Guillaume Sartoretti

∗

and Max-Olivier Hongler

STI/IMT/LPM,

Ecole Polytechnique F´ed´erale de Lausanne (EPFL), CH-1015, Lausanne, Switzerland

Keywords:

Rank-based Brownian Motions, Heterogeneous Autonomous Agents, Soft-control of Swarms, Exact Analytic

Solvable Models, Super-diffusive Shill Agent, Flocking.

Abstract:

We analytically study the collective dynamics of mutually interacting heterogeneous agents evolving in a

random environment. Our formal framework consists of a collection of N scalar drifted Brownian motions

(BM) diffusing on R. The mutual interactions are introduced via a ranked-based, real-time mechanism always

endowing the laggard (i.e the agent with the leftmost position) with an extra positive drift. The extra drift

generates a net tendency for any agents not to remain the laggard of the society. For well chosen individual

and extra laggard’s drifts, the agents organize with time to ﬂock towards a tight and stable travelling spatial

pattern. For a population of (N −1) identical agents and an atypical fellow (called hereafter the shill), we are

able to analytically discuss the dynamics. In particular we exhibit how a single turbulent shill, stylized here

by a ballistic diffusion process, can destroy the cohesion of a swarm. Conversely, we also analytically show

how a single shill is able to safely pilot a whole swarm to avoid an obstacle, via interactions with its fellows.

A series of simulations experiments comfort our analytic ﬁndings.

1 INTRODUCTION

The capability of a collection of interacting stochas-

tic agents to exhibit an emergent collective behaviour

(i.e ﬂocking behaviour) even in random environments

stimulates a strong research activity devoted to both

experimental and theoretical modeling approaches.

For suitable range of mutual interactions, ﬂocking

(phase) transitions are observed, namely the self-

organized capability to create ﬁnite and persistent

spatio-temporal patterns (Banner A. D. and Karatzas,

2005; Yates C. A. and Sumpter, 2009; Chatterjee

and Pal, 2010; Dosseti, 2012; Pal and Pitman, 2008;

Ichiba T. and Fernhold, 2011; Bialek W. and Walczak,

2012).

Agents societies can be composed of either dy-

namically homogeneous or heterogeneous individu-

als requiring for each case drastically different ap-

proaches. For large and homogeneous population

of agents, the classical statistical mechanics concepts

and in particular the mean-ﬁeld description (MF) di-

rectly offers an appropriate tool to analytically discuss

the global dynamics. In the MF description, one basi-

cally assumes that the behaviour of the global society

∗

Supported by the Swiss National Funds for Scientiﬁc

Research

can be characterized by the dynamics of a single rep-

resentative agent which feels its fellows’ interactions

via an effective external ﬁeld. In most circumstances

however, homogeneity fails, and therefore a growing

attention is now paid to heterogeneous populations.

Heterogeneity occurs typically when one or several

masked agents, that we shall from now on refer to

as shills, exhibit leaders or troublemakers behaviours.

These shills are not recognized by the regular agents

which see them as ordinary fellows, leaving the inter-

action rules between them unchanged. The presence

of shills can strongly alter the ultimate evolution of

the whole society and it is the central goal here to an-

alytically study this problematic.

Our approach involves assemblies of interacting

stochastic agents in which a single shill exhibits a dif-

ferent individual behaviour. All agents, including the

shill, interact with their fellows with ﬁxed given rules.

The paradigmatic vision of this situations has been

currently explored in ethology where one fake indi-

vidual is introduced among schools of ﬁshes, cock-

roaches, newborn chicken, etc. The shill is able to

ultimately pilot the whole population (Gribovskiy A.

and Mondada, 2010; Faria J. J. and Krause, 2010;

Vaughan R. and Cameron, 2000). This basic mech-

anism also referred as the soft control of a popula-

tion (Han and Wang, 2010; Wang X. and Han, 2011;

147

Sartoretti G. and Hongler M..

Soft Control of Swarms - Analytical Approach.

DOI: 10.5220/0004176301470153

In Proceedings of the 5th International Conference on Agents and Artiﬁcial Intelligence (ICAART-2013), pages 147-153

ISBN: 978-989-8565-38-9

 2013 SCITEPRESS (Science and Technology Publications, Lda.)

Wang Y. and Liu, 2009; Wang and Guo, 2008) is the

core of our present paper. As heterogeneous societies

preclude MF approaches, rather few analytical results

are yet available. Recent mathematical models of het-

erogeneous rank-based interacting Brownian mo-

tions (RBM), introduced in ﬁnance (Banner A. D.

and Karatzas, 2005; Chatterjee and Pal, 2010; Pal and

Pitman, 2008; Ichiba T. and Fernhold, 2011), will be

used in the sequel to analytically approach the soft

control problematic. We shall analytically show how

a troublemaker can break the cohesion of an ini-

tially tight swarm and how a single agent can efﬁ-

ciently incitate its fellows to achieve a preassigned

task.

Our paper is organized as follows: we ﬁrst recall in

section 2 relevant properties of RBM, also known in

economy as the Atlas models. In section 3, we use

the RBM’s framework to show how a troublemaker

is able to smash an initially tight swarm. Our indi-

vidual dynamics are one-dimensional diffusion pro-

cesses with piecewise constant drifts except for the

shill, which will beassumed to be drivenby a ballistic

process with quadraticvariance in time. The ballis-

tic noise is itself generated by a simple non-Gaussian

diffusive stochastic process with nonlinear drift. In

section 4, we address the dual soft control problem in

which a shill is used to steer the whole collection of

RBM’s towards slots that are drilled through an ob-

stacle board.

2 RANK-BASED BROWNIAN

MOTIONS - ATLAS MODEL

Our approach makes extensive use of recent results

(Ichiba T. and Fernhold, 2011), that we now brieﬂy

summarize. Let us consider a collection of N inter-

acting agents diffusing according to the class of pro-

cesses:

(t) =

∑

k=1

(i)

{Y (t)}+ γ

+ γ

dt+σ

(t),

(0) = y

, t ∈R

, (1)

where Y (t) = (Y

,··· ,Y

) ∈ R

and dW

(t) are

N independent standard White Gaussian Noise pro-

cesses (WGN) processes. The indicator function 1

in Eq.(1) effectively generates mutual interactions.

The effective, time-dependent drift component g

en-

tering into the drifts (g

+ γ

+ γ) of the N Brown-

ian motions on R, is rank-based. Namely, it is in-

stantaneously adjusted according to the position (i.e.

the rank) occupied by each agent with respect to

the remaining (N − 1) fellows. The constant drift

components γ

are name-based, i.e. they are deﬁ-

nitely assigned to each individual agent (γ

is time-

independent). Finally, a constant drift component γ

can be added, which is common to all agents. Ac-

cordingly Eq.(1) describes a collection of N diffusion

processes having piecewise deterministic drifts. The

somehow simpler situation obtained when γ

≡ 0 ∀i

has been thoroughly studied (Pal and Pitman, 2008).

In the sequel and without loss of generality, we sys-

tematically choose γ to be the (average) barycentric

speed of the swarm. This is achieved, provided one

has:

∑

k=1

+ γ

] = 0. (2)

For future use, we introduce the following notations

and deﬁnitions:

1. We write

Y (t) =

∑

k=1

(t),

Y ∈ R, (3)

for the barycenter position. The set Σ

stands for

the set of all the permutations of {1, 2, ...,N}. It

is proven (Ichiba T. and Fernhold, 2011) that, for

almost every initial conditions and when the set of

constraints

∑

k=1



+ γ

p(l)



< 0 (4)

are fulﬁlled for all possible permutations p =

(p(1),..., p(N)) ∈ Σ

, then the N deviations pro-

cesses:

(t) = [Y

(t) −

Y (t)] (5)

converge to stationary probability measures. Note

that Eq.(4) yields therefore a set of (N −1) ×

|Σ

| = N! (N −1) constraints to be veriﬁed.

2. When all constraints in Eq.(4) are fulﬁlled, the dy-

namics given by Eq.(1) then converges to a tight

swarm described by a stationary multi-variate

process characterizing the (N −1)-gap processes

(t) = [Y

i+1

(t) −Y

(t)]. (6)

The associated probability density ψ(z), z ∈R

N−1

can be written as a sum-of-product-of-exponential

form:

ψ(z) =

∑

q∈Σ

N−1

∏

k=1

−1

p,k

−1

∑

p∈Σ

exp(−hλ

,zi)

(7)

with the parameters explicitly given by



p,k



N−1

k=1

p,k

−4

∑

l=1



+ γ

p(l)



+ σ

k+1

. (8)

Remark. It is worth observing that the diffusion co-

efﬁcients σ

do not enter into the set of tightness con-

straints given in Eq.(4). However, the σ

’s do enter

into the parameters Eq.(7) characterizing the station-

ary probability measure.

ICAART2013-InternationalConferenceonAgentsandArtificialIntelligence

148

3 HOW DOES A

SUPER-DIFFUSIVE FELLOW

SMASH TIGHTNESS

We ﬁrst investigate how a single ”turbulent” fellow

can destroy the cohesion of a tight swarm. To this

aim, we use the RBM formalism introduced in section

(2). We consider a conﬁguration involving (N −1)

identical mutually interacting RBM’s, referred from

now on as the regular agents, interacting with a sin-

gle fellow, the shill, say agent number one, which is

itself driven by a ballistic diffusion process (remem-

ber from Eq.(1) that the (N −1) regular agents are

drivenby independent WGN’s). The ballistic process,

to be introduced below, exhibits a variance ∝ t + βt

with a ballistic parameter β ≥ 0 a constant. For

β = 0, the shill simply behaves as a regular fellow

and therefore, in this β = 0 limit, our global dynamics

reduces to a standard version of Eq.(1). We set the

speciﬁc parameters: g

= (N −1)g and g

= −g for

1 ≤ k ≤ N −1. In addition, we shall further assume

that:

∑

i=1

= 0,

ii) max

1≤i≤N

< g.

The couple of constraints i) and ii) imply that the

Eq.(4) is fulﬁlled. Indeed, we directly verify that :

−g+ γ

p(1)

< 0, −2g+ γ

p(1)

+ γ

p(2)

< 0, ···

−(N −1)g+

N−1

∑

k=1

p(k)

< 0. (9)

Hence, for β = 0, our collection of agents behaves

as a tight swarm enjoying a stationary joint proba-

bility law for the inter-distance between successive

agents.

Let us now focus on β > 0. Now, the shill Y

(t) be-

haves as a turbulent fellow which interact with the

remaining (N −1) regular fellows. Speciﬁcally the

dynamics of Y

(t) is chosen to be described by:

(t) =

∑

k=1

(i)

{Y (t)}+ γ

+ γ

dt + dZ(t) ,

(0) = y

t ∈ R

, (10)

where the process dZ(t) is a super-diffusive bal-

listic noise generator introduced in (Hongler M. O.

and Blanchard, 2006; Hongler M. O. and Rodriguez,

2012):

dZ(t) = {βtanh[βZ(t)]}dt + dW (t), Z(t = 0) = 0.

(11)

The non-Gaussian Markov diffusion process Z(t)

given in Eq.(11) is fully characterized by its transition

probability density P

(z,t | 0) which simply reads:

(z,t | 0) =



+ P

−



(z,t | 0)

−

√

2πt

(x±βt)

(12)

with average hZ(t)i = 0 and variance hZ

(t)i = t +

The simple form given in Eq.(12) suggests the exis-

tence of an alternative representation for the ballistic

noise Z(t). Indeed, writing BM

±β

(t) for the Brown-

ian motions with ±β constant drifts, we observe that

the transition probability for the process Z(t) can be

rewritten as Z(t) = B BM

±β

(t) where B is a symmet-

ric Bernoulli r.v., taking the values ±1 with equal

probability 1/2. Hence, one realisation of the Z(t)

consists ﬁrst in choosing with probability 1/2 one

among the couple processes BM

±β

(t), and then fol-

low the realisation of the selected process (see ex-

ample 2 in (Rogers and Pitman, 1981) and (Hongler

M. O. and Blanchard, 2006; Hongler M. O. and Ro-

driguez, 2012)).

We now come back to the dynamics jointly involv-

ing a turbulent fellow given by Eq.(10) and (N −1)

regular agents described by Eq.(1). We then view the

(N −1) regular agents as being inﬁltrated by a the

shill Y

(t).

Now we have to investigate the values of β enabling

the swarm to remain tight (i.e. if a stationary prob-

ability measure for the intervals between successive

agents exists). In view of the representation given in

Eq.(12), for each realisation of the noise source Z(t),

we effectively deal with a standard RBM model with

a re-normalized name-based drift of Y

(t), namely

7→ γ

±β depending on the outcome of B . Accord-

ingly, to infer on the tightness of the swarm, we sim-

ply have to separately examine Eq.(4) for the couple

of outcomes ±β.

Realisation +β. Let us deﬁne γ =

, γ

N−1

and γ

= −γ = −

for 2 ≤ i ≤N. The constraints

Eq.(6) are required for the swarm to be tight. This

yields, for l = 1 and p = (1,2,3...,N), to the most

critical constraint:

+ γ

p(1)

< 0 ⇒ −g+

N −1

β < 0

⇔ β <

N −1

g. (13)

Provided that Eq.(13) holds, the swarm remains

tight when the Z

noise induces a +β extra drift.

Realisation −β. The same reasonning applied to

the −β case yields:

γ = −

= −

N −1

β γ

(2 ≤ i ≤ N) .

(14)

SoftControlofSwarms-AnalyticalApproach

149

The critical constraint arises when l = 1 and γ



,...,

,−

N−1



with:

−g+

< 0 ⇔ β < N ·g. (15)

This implies that for β < N ·g, the swarm remains

tight when the Z

noise induces a −β extra drift.

The previous considerations can be summarized by

observing that when the constraints Eq.(13) are ful-

ﬁlled, so are also those given in Eq.(15). This

suggests to distinguish an alternative “semi”-tight

regime which arises when the constraints Eq.(15)

alone are fulﬁlled. In this “semi”-tight case, the

swarm exhibits a tight conﬁguration when the −β re-

alisation is achieved and is nottight for the other alter-

native +β. This intrinsic asymmetry can be easily un-

derstood as our RBM dynamics only pushes the lag-

gard towards the others. Hence, regarding the tight-

ness, the shill is obviously less inﬂuential when the

−β extra drift is realised.

3.1 Spatial Dispersion of the Agents

The extra ±β-drifts due to the super-diffusive process

driving the shill dynamics obviously affect the spatial

dispersion of the swarm. As all noise sources (includ-

ing the dZ(t) process) have zero average, the barycen-

ter of the whole population will remain unchanged by

the presence of dZ(t). However, due to the presence

of the shill, an initial single-modal cluster of agents

will, as time increases, be split into two separately

evolving population subgroups. According to the ±β

realisation taken by dZ(t), the overall effect of the

shill will steer the swarm either towards the negative

or positive direction; this generates the formation of

two distinct clusters. Each cluster has an individual

barycentricvelocity: γ =

±β

; this guaranteesthat

the average barycenter remains unchanged.

Speciﬁcally, whenever the shill’s drift assumes the

value −β, the shill has an overall propensity to stay at

the rear of the swarm. Accordingly, all regular agents

will, with high probability, be endowed with the rank-

based drift −g. Hence the regular agents possess a

clear tendency to be driven toward the negative direc-

tion on R. Conversely, in presence of the +β reali-

sation, the shill is very likely to belong to the group

of leaders. This imposes to the remaining (N −1)

regular fellows to equally share, with alternations, the

rank-baseddrift of (N −1)g which drives the laggard.

Therefore, with the +β realisation, the whole popu-

lation is driven towards the positive direction of R.

We now can isolate three regimes depending on the

strength β.

a) β < β

N−1

g. In this regime, the shill is able

to steer the whole population in one of the two

possible directions while remaining itself attached

to the swarm. In other words, a stationary proba-

bility measure exists for the distance between the

agents (i.e. all agents evolve in a single ﬂock).

Note however that the presence of the shill breaks

an initially uni-modal spatial repartition into a bi-

modal repartition.

For three agents (i.e. one shill and two regular

fellows), the resulting spatial repartition obtained

by simulation is shown in Figure (1). In Figures

(1) and (2), the shill’s position is represented in

black, the others being displayed in grey.

Figure 1: Probability of spatial repartition of the agents at

end

= 10, numerical computation over 10

runs, with N =

3, g = 1, β = 1.1 < β

b) β

≤β < g·N. Here the strength of the ballistic

noise precludes to reach a global stationary state

for the inter-distance processes between the N

agents. Indeed, the remormalized drift associated

with the +β realisation of dZ(t) (i.e.

= γ

+ β),

violates the constraint Eq.(13). However, for the

−β realisation, the constraint for

= γ

−β is ful-

ﬁlled, implying that the shill remains ﬂocked with

its (N−1) remaining fellows. An experimentwith

β chosen in this range, shows the spatial reparti-

tion in Figure (2) (left).

Figure 2: Left: End position distribution at t

end

= 10, with

N = 3, g = 1, β = β

= 1.5.

Right panel: Probability density of the position at t

end

= 10

for N = 3, g = 1, β = 4 > N ·g. Both numerical computa-

tions include 10

runs.

c) Finally, for β > g·N, the shill becomes highly

turbulent and the tightness constraints are never

ICAART2013-InternationalConferenceonAgentsandArtificialIntelligence

150

fulﬁlled. The shill escapes from the ﬂock and the

resulting spatial repartition is typically shown in

Figure (2) (right).

For all choices of the ballistic strength β, the numeri-

cal results intimately match the analytical predictions.

In particular, the shill escapes from the ﬂock when the

critical ballistic strength β = β

is reached (for the +β

realisation, or when β = N ·g for the −β realisation

of dZ(t). In these non-stationary regimes, the shill

quits the (N −1)- regular tight swarm with velocity

V (β) ∝ β.

3.2 Distance between Successive Agents

Complementary to the tightness constraints, let us

now brieﬂy discuss the stationary probabilitymeasure

which characterizes the distances between successive

agents. To this aim, we shall choose β to ensure the

existence of a global stationary regime (i.e. ensures

tightness), namely 0 ≤β < β

N−1

Note however that using the results derived in

(Ichiba T. and Fernhold, 2011), heterogeneous dif-

fusion constants can also be analytically discussed.

The distances separating consecutive agents deﬁned

in Eq.(6) can be explicitly computed by using the re-

sults summarized in section (2). For agent Y

(t) and

in view of the extra β-drift induced by the shill bal-

listic driving, we are now led to deﬁne a couple of

name-based drifts vectors as:

= ±



N−1

β −

··· −



. (16)

The fulﬁllment of the tightness constraints given in

Eq.(4) ensures that the λ

p,k

from Eq. (8) are strictly

positive (∀p,k). Hence, it results a couple of sta-

tionary probability densities (one for each realisation

+β and −β), characterizing the inter-distance process

Ξ(t) (see Eq.(6)):

P {z ≤Ξ(t) ≤ z+ dz — ±β is realised}= P

±β

(z)dz





∑

p∈Σ

N−1

∏

k=1

p,k

−1





−1

∑

p∈Σ

exp



−





dz.

(17)

The complete stationary probability density then

reads:

P (z) =



+β

(z) + P

−β

(z)



. (18)

4 USING A SHILL TO GUIDE

THE COLLECTIVE CROSSING

OF OBSTACLES

The shill Y

(t) of section (3) has so far been viewed

as a mere trouble maker. However, for speciﬁc tasks

to be achieved, a shill’s presence might become truly

beneﬁcial. Indeed, the presence of a shill can softly

control a swarm towards a preassigned target.

To concretely illustrate this alternative view point, let

us, once again, consider a collection of (N −1) iden-

tical drifted Brownian (regular) agents and one shill,

say Y

(t), all diffusing on R. From now on, we shall

represent the evolution on R

with the time being

identiﬁed with the x-axis, and positions with the y-

axis. At time t = 0, all N agents start at location

(0) = 0, 1 ≤ i ≤ N. We may think of agents running

on the x-axis with constant unit speed, as we therefore

identify the x-axis with time. At the x-location T (see

Figure 3), we introduce a solid wall (i.e. the obstacle)

in which two-slots are drilled. The global objective

for the N agents is to try to avoid the fatal collisions

with the wall by traveling trough one of the couple of

slots. The two slots have width W and are symmetri-

cally drilled at the ordinates ±D, see Figure (3) for a

sketch of the conﬁguration.

Figure 3: Initial conﬁguration for the soft control problem.

The agents start their diffusion at (t

) = (0,0).

In absence of shill, one has an homogeneous swarm

(all agents are dynamically identical) and the N-

swarm proceeds according to the rule deﬁned in

Eq.(1). As a consequence, the wall will be hit with

high probability as, at time T, the probability den-

sity of the positions will be given by a collection of

N centered Gaussians (we basically have N Brow-

nian motions with constant drifts having their mean

barycenter located at zero). As seen in section (3), the

presence of a super-diffusive shill Y

(t) may strongly

modify this nominal β = 0 picture. IndeedY

(t), with

suitable parameters β and g is able to steer the whole

swarm with high preference to one of the slots. This

basic and simple idea can be made fully quantita-

tive as discussed below. In our simulations, we set

β = 0 once the slots are crossed. Therefore, the global

swarm continues its nominal path with a tight conﬁg-

uration after the obstacle.

4.1 Optimal Barycentric Driving for

Suitable Choice of β

First, we select the Y

(t) parameters to ensure global

SoftControlofSwarms-AnalyticalApproach

151

tightness. This is achieved by deﬁning β < β

N−1

g. For a single realisation of the Z(t) noise, we

already know that the average barycentric y-speed of

the swarm is

±β

. Hence, at time T, thegroup barycen-

ter reaches the y-position located at

±βT

. The center

of the slots being located at ±



D+



, we therefore

naturally require:

±βT

= ±



D+



⇔ β

opt



D+



·N

(19)

The choice β

opt

, doesnot yet ensurethat the shillY

(t)

itself remains attached to the ﬂock. This second re-

quirement can be achieved provided one has:



D+



·N

= β

opt

< β

N −1

⇔ g >



D+



·(N −1)

=: g

. (20)

This choice of the couple parameters β

opt

and g > g

now jointly ensures that i) the barycenter is steered

towards one of the slots centers and ii) Y

(t) remains

tightly attached to the swarm. The overall swarm’s

y-dispersion is itself dependent on the diffusion con-

stants σ

, which were here taken as (σ

:= σ ∀i).

Figure 4: Left panel: N = 10 agents, W = 20 and D = 10.

Right panel: N = 20 agents, W = 20 and D = 50.

4.2 Adjusting the Spatial Dispersion

Resolution for a Given Drift g

When the drift g is ﬁxed, the ballistic component β

cannot always be chosen to simultaneously ensure

tightness and the collective drive into one of the slots.

Keeping the barycentric driving β = β

opt

, as deﬁned

in Eq.(19), three different scenarii are now possible:

a) β = β

opt

< β

. In this case, the shill nicely

steers the group towards one of the slots while

staying attached to the ﬂock, as we already estab-

lished in section 4.1.

b) β = β

opt

≥ β

. Here, Y

(t) escapes from the

(N − 1)-ﬂock with a constant drifting velocity.

Two sub-cases have to be distinguished, i) drift

β−g resulting when +β is realised by dZ(t) and

conversely ii) drift (N −1)g−β for the alternative

−β case. In both cases, the regular agents evolve

with an average drift γ

ave

which reads:

γ =

(

(β−g)·1+(N−1)·γ

ave

for +β

−

((N−1)g−β)+(N−1)γ

ave

for −β

⇔ γ

ave



N−1

−g

(21)

c) β = β

opt

≥β

< N ·g. Here, the shill remainsat-

tached to the ﬂock for the −β realisation of dZ(t),

but escapes otherwise. This then leads to a mix of

cases a) and b) depending on whether +β or −β

is realised.

The computation of γ

ave

from case b) shows that with

g ﬁxed such that β

opt

≥ β

, the shill escapes from the

ﬂock (hence, no stationary probability measure ex-

ists). The remaining (N −1) regular fellows evolve

with average speed γ

ave

(which is β-independent).

Whatever the values taken by β, the shill is never

able to drive the swarm through one of the slots, the

swarm’s speed being only g-dependent if the shill is

not attached to the swarm.

So far, only the swarm’s directions has been con-

sidered. Obviously, the dispersion is also a determi-

nant feature for efﬁcient slots crossings. Here, not

only g but the ratio

will be determinant. Clearly

for small g values, the swarm dispersion will exhibit

a clear tendency to exceed the slots widths, altering

therefore the overall efﬁciency.

4.3 Multi-slots Conﬁgurations

So far, we did consider the capability of a shill to steer

the swarm through a couple of slots. For multi-slots

conﬁgurations, the shill construction used before nat-

urally suggests to deﬁne more general shills to steer

swarms in many different directions. This is achiev-

able by replacing the ballistic noise driving the shill

with more complex stochastic processes. Doing so

however, the shill dynamics cannot be anymore rep-

resented by a simple diffusive stochastic differential

equation like Eq.(11). As an illustration, consider a

three symmetrical slots conﬁguration for which one

is naturally driven to introduce the following noise

source:

dZ(t) =

(

βtanh(βZ(t))dt + dW (t) with prob.

dW (t) with prob.

(22)

The shill dynamics is taken as before, namely with

probability

it steers the swarm towards the positive

or negative slots with ballistic parameter β and with

probability

, it behaves as a regular agent driving the

swarm on a centered path, see Figure (5).

For arbitrary number of slots and conﬁgurations, one

can generalize our construction by suitably adjusting

the properties of the noise source driving the shill.

ICAART2013-InternationalConferenceonAgentsandArtificialIntelligence

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Figure 5: Trajectories for β and g optimal, with N

= 3

doors, under different conditions for the position/width of

the doors.

5 CONCLUSIONS

It is a true challenge, to analytically discuss the swarm

dynamics of heterogeneous interacting agents. Due to

heterogeneity, ordinary analytical tools like the mean-

ﬁeld approach are to be ruled out a priori. Hence be-

sides simulation experiments, very little hope remains

for rigorous theoretical results. We think particularly

to modeling approaches relying on statistical mechan-

ics and phase transitions to explain the emergence of

self-organized spatio-temporal patterns (i.e ﬂocking).

Howeveras ourpaper intends to show, facing inhomo-

geneous swarms problems, complete hope for analyt-

ical results should not be abandoned. The theoretical

analysis, when achievable, deﬁnitely offers sources of

inspiration for new and unexpected research axis.

In our present contribution, we barely scratched

the wealth of analytical possibilities. Indeed, swarm

heterogeneity has numerous origins, affecting the in-

dividual drifts and/or the variances, modeling the sen-

sitivity of each agent to the external random environ-

ment. In parallel, heterogeneity can either be sys-

tematic, thus implying that each agent behaves dif-

ferently, or can be limited to one or only a few indi-

viduals who exhibit singular behaviors. In the latter

conﬁguration, the emergent swarm dynamics can be

affected, sometimes even strongly, by the exotic be-

havior of this (or these) individual(s). The inﬂuence

of the exotic fellow(s) can hence be viewed as a soft

control mechanism, either harmful or beneﬁcial. The

exotic insiders, acting as leaders (or as shills in econ-

omy) are not detected to be singular by the other fel-

lows, offering the (politically frightening possibility!)

to drive large swarms towards global goals known

only to the manipulators. A formal analytical ap-

proach (complemented with simulations) to this gen-

eral problematic is a truly fascinating challenge.

Further works include analysis of multiple shills

inﬂuence, leading to the separation of the initial

swarm into multiple ﬂocks, one shill soft-controlling

each ﬂock. Generalisation in two or three dimensions

would also provide more realistic applications.

REFERENCES

Banner A. D., R. F. and Karatzas, I. (2005). Atlas models

of equity markets. Ann. Appl. Prob. 15, 2296-2330.

Bialek W., Cavagna A., G. I. M. T. S. E. M. V. and Walczak,

A. M. (2012). Statistical mechanics for natural ﬂocks

of birds. PNAS 13, vol. 109, 4786-4791.

Chatterjee, S. and Pal, S. (2010). A phase transition

behaviour for brownian motions interacting through

their ranks. Prob. Theory Relat. Fields 147, 123-159.

Dosseti, V. (2012). Cohesive motion in one-dimensionnal

ﬂocking. Physica A: Math. Theor. 45.

Faria J. J., Dyer J. R. G., . . . and Krause, J. (2010). A novel

method for investigating the collective behaviour of

ﬁsh: Introducing ’roboﬁsh’. Behavioral Ecology and

Sociobiology 8, vol. 64, 1211-1218.

Gribovskiy A., Halloy J., D. J. L. H. B. and Mondada,

F. (2010). Towards mixed societies of chickens and

robots. IEEE/RSJ 2010 International Conference on

Intelligent Robots and Systems, Conference Proceed-

ings pp. 4722-4728.

Han, J. and Wang, L. (2010). New strategy of the shill:

Consistent moving. Proceedings of the 29th Chinese

Control Conference, pp. 4459-4464.

Hongler M. O., Filliger R., P. B. and Rodriguez, J. (2012).

”Contemporary Topics in Mathematics and Statis-

tics with Applications” in ”On Stochastic Processes

Driven By Ballistic Noise Sources”. Ed. M. R. Ad-

hikari.

Hongler M. O., R. F. and Blanchard, P. (2006). Solu-

ble models for dynamics driven by a super-diffusive

noise. Physica A 370, 301-355.

Ichiba T., Papathanakos V., B. A. I. K. and Fernhold, R.

(2011). Hybrid atlas model. Ann. Appl. Prob. 21(2)

609-644.

Pal, S. and Pitman, J. (2008). One dimensional brownian

particles with rank-dependent drifts. Ann. Appl. Prob.

18 2179-2207.

Rogers, L. C. G. and Pitman, J. W. (1981). Markov func-

tions. Ann. of Prob. 9(4), 573-582.

Vaughan R., Sumpter N., H. J. A. F. and Cameron, S.

(2000). Experiments in automatic ﬂock control.

Robotics and Autonomous Systems 1, vol. 31, 109-

117.

Wang, L. and Guo, L. (2008). Robust consensus and soft

control of multi-agent systems with noises. Journal of

Systems Science and Complexity 3, vol. 21, 406-415.

Wang X., J. H. and Han, H. (2011). Special agents can

promote cooperation in the population. PLoS ONE

12, vol. 6.

Wang Y., X.-Z. M. and Liu, J.-P. (2009). Soft control for

swarm systems with simple attraction and repulsion

functions. Proceedings of the 2nd International Con-

ference on Intelligent Networks and Intelligent Sys-

tems, pp. 482-485.

Yates C. A., Erban R., . . . and Sumpter, D. J. T. (2009).

Inherent noise can facilitate coherence in collective

swarm motion. PNAS 14, vol. 106, 5464-5469.

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