Agent swarms cooperation and coordination under stringent communications constraint Paul Kinsler Department of Electronic Electrical Engineering University of Bath Bath BA2 7AY United Kingdom

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Agent swarms: cooperation and coordination under stringent communications constraint

Paul Kinsler∗

Department of Electronic & Electrical Engineering University of Bath, Bath, BA2 7AY, United Kingdom

Sean Holman†

Department of Mathematics, University of Manchester, Manchester, M13 9PL, United Kingdom

Andrew Elliott‡

School of Mathematics and Statistics, University of Glasgow, Glasgow, G12 8QQ, United Kingdom

Cathryn N. Mitchell§

Department of Electronic and Electrical Engineering University of Bath, Bath, BA2 7AY, United Kingdom

R. Eddie Wilson

Department of Engineering Mathematics, University of Bristol, Bristol, BS8 1TW, United Kingdom

(Dated: Friday 7th April, 2023)

Here we consider the communications tactics appropriate for a group of agents that need to “swarm” together

in a highly adversarial environment. Speciﬁcally, whilst they need to cooperate by exchanging information with

each other about their location and their plans; at the same time they also need to keep such communications

to an absolute minimum. This might be due to a need for stealth, or otherwise be relevant to situations where

communications are signiﬁcantly restricted. Complicating this process is that we assume each agent has (a) no

means of passively locating others, (b) it must rely on being updated by reception of appropriate messages; and

if no such update messages arrive, (c) then their own beliefs about other agents will gradually become out of

date and increasingly inaccurate. Here we use a geometry-free multi-agent model that is capable of allowing for

message-based information transfer between agents with different intrinsic connectivities, as would be present

in a spatial arrangement of agents. We present agent-centric performance metrics that require only minimal

assumptions, and show how simulated outcome distributions, risks, and connectivities depend on the ratio of

information gain to loss. We also show that checking for too-long round-trip-times can be an effective minimal-

information ﬁlter for determining which agents to no longer target with messages.

I. INTRODUCTION

For more than a decade, the availability and range of appli-

cations for Unmanned Aerial Vehicles (UAVs) or “drones” has

greatly increased. They now span areas such as logistics, agri-

culture, remote sensing, communication, security, and defense

[1–6]. In particular, UAVs appear to be useful for tasks which

are too dangerous, expensive, or innaccessible by manned ve-

hicles. They also offer the advantage of being able to use

a swarm of smaller and less expensive drones in place of a

single UAV. In addition, the operating capabilities may differ

across individual elements for the swarm. Groups of drones

operating cooperatively have also been proposed for surveil-

lance [2, 7, 8], search and rescue [9–11], and military missions

[12, 13]. Using a swarm has the advantage of being robust

against losses of individual drones. Further, since smaller and

more agile individual swarm constituents can be difﬁcult to

detect and attack, they can have a natural advantage for appli-

cations in which the environment is contested.

Combining a group of drones into a “swarm" is a difﬁcult

engineering problem with potential to draw from a wide range

of disciplines [7, 11, 12, 14]. The advantage is that it acts and

can be directed as a single entity while minimising commu-

nication and remaining robust to both physical and cyber at-

tacks. Swarm robotics and swarm engineering are emerging

ﬁelds [14–17] which study automated decision making and

∗https://orcid.org/0000-0001-5744-8146; Dr.Paul.Kinsler@physics.org

†https://orcid.org/0000-0001-8050-2585

‡https://orcid.org/0000-0002-4536-5244

§https://orcid.org/0000-0003-1964-8723

control for large groups of robots using only local communi-

cation between nearby swarm members. Much work in that

ﬁeld considers larger swarms (≈102−105) than of interest

for ﬂying drone swarms (≈10 −102). There are special chal-

lenges inherent in the creation and real-time maintenance of

non-centralised ad hoc networks [11, 18, 19] for groups of

UAVs. Note that these are variously termed FANETs (Flying

Ad Hoc Networks) [1] and UAANETs (UAV Ad Hoc Net-

works) [20]; these have been studied in recent years and vari-

ous architectures and protocols have been proposed [1, 3, 20].

A number of groups have developed in silico test-beds for

FANETs [21–23] with some moving on to in robotico realisa-

tions, and there has been recent interest in local algorithms for

adaptive FANETs as well as consensus [8, 22, 24]. However,

work with speciﬁc application to search and rescue [9–11],

rarely considers network resilience to external attacks or the

need to minimise risk of detection, although there has been

research on autonomous swarms with covert leaders [25].

In this work we focus on scenarios in which the drones

(hereafter agents) must act in the extreme limit of minimal

information sharing. This is most easily represented as situ-

ations where communications must be restricted in order to

maintain stealth. This is the opposite of typical scenarios,

where information sharing and other communications are con-

sidered as “free”. In such a typical scenario, each agent almost

automatically has an excellent knowledge as to the state of

the swarm, or at least some coordinating agent or supervisor

has such information with which to efﬁciently direct swarm

operations. It should be noted that this “cooperation under

communications constraint” is a different problem to control

under communications constraint [26, 27].

In the minimal information cases we consider in this paper

arXiv:2210.01163v2 [cs.MA] 6 Apr 2023

we should no longer talk of what an agent “knows”, since that

unhelpfully suggests a likely misleading degree of accuracy;

but instead what it “believes”, since that implies an appropri-

ate degree of doubt and uncertainty. Although up to date and

accurate information might be received in any new message

just received, since communications will be sparse, we need

to keep in mind that this will nevertheless become less reli-

able as time passes. What this means in practice is that many

of the typical treatments of swarm activities as given above

(e.g. [1, 3–7, 11, 15, 17]) become secondary problems to the

fundamental issue [28]: i.e. how well does each agent know

where the others are, so that it might send a message, and what

tactics should it use to maximise the accuracy of its beliefs,

whilst minimising its exposure to adversarial action? A game

theory [29, 30] problem arises here because an agent gains

no new information by sending a message, but such transmis-

sions only expose it to more risk. Instead, remaining silent

might allow agents to risklessly accumulate information from

the transmissions of others – except that if all agents do this,

the swarm cannot behave coherently. The situation is a com-

parable to an inverse tragedy of the commons [31].

In our scenario, agents must both cooperate and coordinate.

Here “cooperation” refers to the necessity that all agents must

cooperate by all sending location messages, because without

this, agents will end up with inaccurate information, and so

be unable to target messages successfully, so that swarm con-

nectivity must then fail. Further, “coordination” refers to the

fact that a connected swarm can only be formed using multi-

hop message paths (as per Sec. 4.3) if other agents cooperate

by forwarding such messages. Without this cooperation, any

message must be sent directly, requiring accurate information

about all other agents. In such a direct-signalling paradigm,

not only would an increased transmission power be needed to

reach the longer ranges, increasing risk, but any blocked mes-

saged path would be fatal for connectivity.

Our contribution here is that we construct a mathemat-

ical multi-agent information model that can be specialized

to both continuum communications and discrete communi-

cations models. The discrete communications model is then

used to design a stochastic simulation code, which we used to

benchmark and test a minimal approach that optimises com-

munications whilst minimising risk, whilst using only mini-

mal assumptions. In particular, we focus in on the underlying

basics of the “to transmit ... or not to transmit?” problem

without the many additional complications of as movement in

space, speciﬁc communications physics, or how to build op-

timal communications networks within the swarm. As well

as considering what properties of the model an agent might

be permitted to use when taking action, we present some per-

formance and risk metrics that help evaluate the performance-

under-constraint scenario, and also propose a round-trip tim-

ing test that is based solely on data an agent is aware of, and

which can be used to deprecate poor links.

In what follows we present our model and its concepts in

Sec. II. This is followed by a continuum communication im-

plementation in Sec. III, where an agent might act to modify

its transmission priorities on the basis of the rate of informa-

tion arrival from other agents. This rate-equation description

can be used to generate indicative steady-state answers, and

allow some preliminary conclusions. To assist with judge-

ments about performance, we deﬁne some agent and swarm

metrics in Sec. IV. Then we describe our implementation of

a discrete communications approach in Sec. V, where we use

a Monte-Carlo approach to produce a more sophisticated un-

derstanding of the distribution of possible outcomes. Here, the

informational basis on which an agent might act is no longer

a rate, but instead the timings (and time-delays) of messages

from other agents, and so in Sec. VI we show and explain

some results using this approach. After a discussion in Sec.

VII, we conclude in Sec. VIII.

II. MULTI-AGENT MODEL

We consider a swarm of Nagents that intend to cooperate

and send messages about their activities to each other. Our

model state is intended to mimic the behavior of a spatially

distributed swarm of agents, but without requiring a detailed

spatial model and all the additional complications that would

entail. As such it contains only a minimal set of features, and

is primarily intended to facilitate an initial understanding of

how our novel scenario might be handled. The model contains

four types of information, intended to represent as simply as

possible the accuracy of an agents beliefs, the degradation of

that accuracy as time passes, the environment’s effect on sig-

nalling efﬁciency, and agent messaging choices. Thus:

First,: each agent ahas an information store about all other

agents j, which we summarize using values Φa

j. If

this store contains recent and reliable information, we

would expect it to result in a high probability of mes-

saging success, but if the information is outdated or oth-

erwise unreliable, the probability would instead be low.

Thus these accuracies are represented as probabilities,

using real numbers Φa

j∈[0,1]; with zero representing

entirely inaccurate beliefs and 1 representing perfectly

accurate beliefs. Since agent ais presumably perfectly

informed about itself, Φa

a=1 should always hold.

Second,: we allow for the possibility that an agent’s beliefs

about others slowly become out of date and degraded.

We model this by assuming that all the Φa

j(iff j6=a) de-

cay exponentially as determined by some loss parame-

ter γ. However, we assume there is also a minimum

“ﬁnd by chance” probability Φmsuch that Φa

j≥Φmfor

any aand all j.

Third,: there is an agent-to-agent transmission efﬁciency,

which represents environmental constraints that might

hinder communications between agents. This agent-

to-agent transmission efﬁciency Lab ∈[0,1]enables the

representation of a wide range of networks, including

networks based on spatial positions and signal models,

as well as those with Lab generated randomly according

to some algorithm, e.g. abstract Erdos-Renyi (ER) net-

works [32]. However, at this early stage we do not spec-

ify how the efﬁciencies Lab might have been generated,

and can even allow Lab 6=Lba, i.e. the transmission efﬁ-

ciency from ato bmay be different to the transmission

efﬁciency from bto a. An important feature is that we

do not assume any agent ahas any information about

either Lba or Lab.

Fourth,: since an agent might transmit information at differ-

ent rates towards different targets, we also specify its

set of transmission rates αa

Φa

αa

Φb

αb

Φc

αc

Lba '0

Lab '0

Lac '1

Lca '1

Lbc '1

FIG. 1. Diagram of a simple swarm with three agent-drones a,b,

and c; where the link a↔bis blocked so that Lab and Lba are near

zero, whereas the a↔cand b↔clinks are free of obstruction. This

means we would expect the values Φa

band Φb

ato be small (since the

beliefs, being poorly updated, will become ever more inaccurate),

whereas the unobstructed messaging along links a↔cand b↔c

should mean that it is possible to maintain Φa

c,Φc

a,Φb

c, and Φc

bat

values near 1.

These deﬁnitions mean that while the probability of mes-

sage transmission from an agent ato another bis straighfor-

wardly given by the product of Φa

band Lab, the actual rate of

information arrival is αa

iΦa

bLab. This model is broadly con-

sistent with an implict assumption that transmisions are sent

directionally and need to be aimed, being sent from ato b

with an accuracy Φa

bacross a link with efﬁciency Lab. Also,

in this model, messages are only ever received by the intended

recipient. A simple depiction with just three agents and one

blocked (inefﬁcient) link is given in ﬁg.1.

A. Index convention: subscruipts and superscripts

To enable easier interpretation of the model parameters and

values, we use an index convention where each indexing let-

ter, and its positioning as a super- or sub-script, implies ex-

tra meaning. If we are referring to some speciﬁc agent we

use one of {a,b,c}, where a,b,c∈ {1,2, ...,N}; but if refer-

ring to a range of other agents will use one of {i,j,k}, where

i,j,k={1,2,...,N}. Further, a superscript denotes that the

quantity is a property of that superscripted agent, but for a

subscript, there is no such implication. That is, the value Φa

is a property of afor any j, but for none of those j(if a6=j) is

it a property. Thus Φa

bis a number that is a property of agent a,

and Φa

iis a collection of numbers that is a property of agent a.

However, the collection of Φi

j(or even Φi

a) is global informa-

tion. This is because iand jeach encompasses many agents,

so that Φi

jis not a property of any single agent in the swarm.

Other characters used as sub- or superscripts will indicate not

agents but special cases or particular values of e.g. Φor L.

We do not use any implied summation convention.

B. Abstractions are not knowledge

When using models of the type proposed here, it is impor-

tant to note that an agent property (e.g. Φa

i) is deﬁned within

the model as being attributable to an agent a, and may affect

the outcomes of a’s actions. However, even though the model

of agent acontains a collection of properties, this does not

mean that the agent decision making can necessarily make use

of each and every agent property. In particular, here we have

that Φa

iis a representation or abstraction of agent a’s beliefs

about the spatial location of agent i, thus telling the model

how efﬁciently agent acan target that agent i. This is why the

model will use it when calculating either what fraction of the

information contained in the messages sent actually arrives,

as in the continuum communications model; or alternatively

whether or not a whole message is received, as in the discrete

communications model.

Despite this, there is no reason why any actual agent awill

be aware of and be able to use the value of Φa

iin decision

making. For example, the model could specify that an agent a

has an accuracy Φa

b=0.50 when messaging b. However, the

agent amight not be aware that that is the accuracy, so that

it cannot use its value of 0.50 when decision making, e.g. by

using it in a formula or algorithm. This is because an actual

agent will instead only be cognisant of some speciﬁc “basket”

of data – containing entries such as position estimates, likely

errors, future plans for movement, and so on – which need not

be reducible in an algorithmic way to the model’s substitute,

i.e. the abstraction Φa

i’s particular value. That is, any actual

agent amight only be aware of a basket of speciﬁc details,

but not how to synthesise the abstract Φa

jfrom those details.

Indeed, when we use this model, we do not know this synthe-

sising process either, nor anything about the basket contents.

As discussed above, and as indcated in ﬁg. 2, we have that

Φa

iis a property of the agent amodel, but that agent ais not

aware of Φa

j. In contrast, an agent should be aware of its

choices or settings for transmission rates αa

i, so these would

be usable in decision making. However, whether parameters

such as γor Φmare agent properties that the agent is aware

of, agent properties that are unknown, or even parameters en-

tirely unrelated to the agent description, will depend on how

we envisage the model of agent loss and message transmis-

sion. Nevertheless, since an agent ais unaware of the agent

property Φa

j, it seems reasonable to decide likewise that γis

also (at best) only an (unknown) agent property. However, if

γwere (e.g.) dependent on the environment, it might not even

be considered an agent property.

This distinction between the model’s abstractions and an

agent’s actual awareness or beliefs – whatever they might be

– means that to implement a generalisable communications

tactic we must avoid reliance on our model’s abstractions, and

instead use only quantities that an agent is aware of, can mea-

sure, or believes.

C. Terminology

In this work we will often refer to a “link”, meaning the po-

tential for communication between two agents aand b. How-

ever, here “link” is just a short and convenient word for that

potential, and it does not imply that such a communication

is guaranteed to be easy – or even possible – in any partic-

ular case. When discussing links, we will also use three ad-

jectives – efﬁcent, accurate, reliable – to describe them, and

these three adjectives have speciﬁc meanings which we will

now deﬁne. However, this terminology is only intended to

make general discussion clearer by specifying preferred ad-

jectives, rather than as any unique mathematical speciﬁcation;

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Agentswarms:cooperationandcoordinationunderstringentcommunicationsconstraintPaulKinslerDepartmentofElectronic&ElectricalEngineeringUniversityofBath,Bath,BA27AY,UnitedKingdomSeanHolmanDepartmentofMathematics,UniversityofManchester,Manchester,M139PL,UnitedKingdomAndrewElliottSchoolofMathematicsandS...

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Agent swarms cooperation and coordination under stringent communications constraint Paul Kinsler Department of Electronic Electrical Engineering University of Bath Bath BA2 7AY United Kingdom

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