=Paper= {{Paper |id=Vol-2215/paper5 |storemode=property |title=A Kinetic Model of the Dynamics of Compromise in Large Multi-Agent System |pdfUrl=https://ceur-ws.org/Vol-2215/paper_5.pdf |volume=Vol-2215 |authors=Federico Bergenti,Stefania Monica |dblpUrl=https://dblp.org/rec/conf/woa/BergentiM18 }} ==A Kinetic Model of the Dynamics of Compromise in Large Multi-Agent System == https://ceur-ws.org/Vol-2215/paper_5.pdf

A Kinetic Model of the Dynamics of
Compromise in Large Multi-Agent Systems
Federico Bergenti, Stefania Monica
Dipartimento di Scienze Matematiche, Fisiche e Informatiche
Università degli Studi di Parma, 43124 Parma, Italy
Email: {federico.bergenti,stefania.monica}@unipr.it

Abstract—Compromise is one of the primary phenomena that In this paper, the long-time asymptotic behaviour of the
govern the dynamics of the opinion in multi-agent systems. collective dynamics of the opinion in multi-agent systems is
In this paper, compromise is isolated from other phenomena, studied using the very general approach proposed by math-
and it is studied using a statistical framework designed to
investigate collective properties of large multi-agent systems. The ematical kinetic theories (e.g., [8] and referenced literature),
proposed framework is completed with the details needed to which are intended to investigate the collective properties of
model compromise, and differential problems which describe the groups of interacting peers. The prototypical example of a
dynamics of the opinion under suitable hypotheses are presented. mathematical kinetic theory is the classic kinetic theory of
Long-time asymptotic solutions of obtained differential problems gases, which studies collective properties of gases, like tem-
are discussed to confirm that compromise makes multi-agent
systems tend to reach consensus. It is proved that compromise perature and pressure, starting from the details of interactions
makes all agents tend to share the same opinion, and that among molecules (or atoms, for noble gases). When studied
the value of the asymptotic opinion can be expressed in terms gases are made of different types of molecules, classic kinetic
of the characteristics of the multi-agent system and of the theory of gases is normally generalized to the kinetic theory of
initial distribution of the opinion. Obtained analytic results are gas mixtures, which is another mathematical kinetic theory that
confirmed by independent simulations in an illustrative case.
accounts for gases with molecules with different properties.
A rather obvious parallelism between the molecules of a gas
I. I NTRODUCTION and the agents of a multi-agent system can be drawn to adopt
generalisations of the kinetic theory of gases to study collective
The study of the multiple aspects of opinion formation in properties of multi-agent systems. This has already been done,
multi-agent systems is an important research topic that finds e.g., in [9], where the similarity between the distribution of
applications in various fields, e.g., control theory, robotics, bi- wealth in a simple economy and the density of molecules in
ology, sociology, and artificial intelligence. Usually, the study a gas is studied, or in [10], where simple models of opinion
of opinion formation in multi-agent systems assumes that each dynamics are studied. Note that besides the general framework
agent has an opinion on a given topic, and that the opinion can of mathematical kinetic theories, few results from the kinetic
be expressed in terms of a value in a suitable range. Agents theory of gases can be adapted to other contexts because the
interact by exchanging messages on a discussed topic, and details of the interaction rules which model collisions among
interactions make agents change their opinions. Interactions molecules in gases are significantly different from those of
are typically described by suitable interaction rules [1] that the interaction rules that model cooperation and competition
model how each interaction changes the opinions of involved among agents in multi-agent systems.
agents. Interaction rules take into account the sociological The general framework of mathematical kinetic theories is
phenomena that describe how agents form their opinions, and used in this paper to study the characteristics of compro-
they model the dynamics of the opinion of each agent. When mise [10], which is one of the most important phenomena that
the number of agents in the considered multi-agent system is govern the dynamics of the opinion in multi-agent systems.
large, the study of the dynamics of the opinion of each agent The major contribution of this study is to generalise results
is not feasible, and the analysis of collective properties of the from the literature (e.g., [1], [11]) by allowing each agent to
opinion of the multi-agent system as a whole is preferred. Such have a specific propensity to change its opinion because of
collective properties are typically investigated using statistical interactions. Such a generalisation is important because it is
approaches that account for the dynamics of the opinion in meant to model agents that act with some degree of autonomy.
terms of the long-time asymptotic dynamics of aggregate val- This paper is organized as follows. Section II completes the
ues, like the average opinion and the variance of the opinion. generic framework of mathematical kinetic theories with the
The literature already proposes various approaches to study the details needed to model compromise, and it provides results
dynamics of collective properties of the opinion, which include on the dynamics of compromise. Section III reports results of
those based on thermodynamics [2], on Bayesian networks [3], illustrative simulations which confirm the behaviors predicted
on gossip protocols [3], on flocking models [4], on graph by the proposed model of compromise. Finally, Section IV
Laplacians [5], and on cellular programming [6], [7]. concludes the paper and outlines future developments.

27
II. A K INECTIC M ODEL OF C OMPROMISE where v ∗ and w∗ are the opinions of agents s and r, respec-
tively, after the interaction. Observe that if n is the number
The study of the dynamics of the opinion normally considers of agents in the multi-agent system, the considered model
a number of sociological phenomena that can be used to model involves n2 parameters {γs,r }ns,r=1 , where γs,r measures the
the behaviours of agents (e.g., [11] and referenced literature). propensity of a generic agent s to change its opinion in favor
Among considered phenomena, some of the most extensively of the opinion of another agent r. Note that, according to
studied are: (1), if γs,r is nearly 0, agent s is not inclined to change its
• Compromise: the tendency of agents to move their opin- opinions towards that of agent r. For this reason, values of
ions towards those of agents they interact with, trying to γs,r close to 0 characterise skeptical agents. At the opposite,
reach consensus [10]; if γs,r ' 1/2, then v ∗ ' 1/2(v + w), and such a γs,r can be
• Diffusion: the phenomenon according to which the opin- used to characterize agents that get easily convinced.
ion of each agent can be influenced by the social con- The choice of parameters {γs,r }ns,r=1 is crucial to determine
text [12]; the characteristics of the model, and it deserves further dis-
• Homophily: the process according to which agents inter-
cussions. First, note that post-interaction opinions v ∗ and w∗
act only with those with similar opinions [13]; still belong to interval I where opinions are defined because
• Negative influence: the idea according to which agents |v ∗ | ≤ (1 − γs,r + γs,r ) max{|v|, |w|}
evaluate their peers, and they only interact with those |w∗ | ≤ (1 − γr,s + γr,s ) max{|v|, |w|},
with positive scores [14];
• Opinion noise: the process according to which a random and, since max{|v|, |w|} ≤ 1, it can be readily concluded that
additive variable may lead to arbitrary opinion changes |v ∗ | ≤ 1 and |w∗ | ≤ 1. Then, from (1) it can be derived that
with small probability [15]; and the difference of the opinions of two interacting agents after
• Striving for uniqueness: the phenomenon based on the an interaction is
idea that agents want to distinguish themselves from v ∗ − w∗ = εrs (v − w), (2)
others and, hence, they decide to change their opinions
if too many agents share the same opinion [16]. where
εrs = 1 − (γr,s + γs,r ). (3)
Models based on mathematical kinetic theories have already
been proposed in the literature to study all mentioned phe- Observe that the model aims at describing compromise, which
nomena analytically (e.g., [1], [11], [17]–[28], and referenced is the idea that the opinions of two interacting agents get
literature). The major contribution of this paper with respect closer after an interaction, and from (2), such a requirement
to existing literature regards the possibility to consider agents corresponds to
with different propensity to change their opinions because of |v ∗ − w∗ | = |εrs ||v − w| < |v − w| (4)
interactions with other agents. The paper studies a kinetic
framework that associates agents with some level of autonomy for values of εrs such that |εrs | ≤ 1. This condition can be
by allowing each agent to have a specific propensity to change written in terms of the parameters of the model as
its opinion because of interactions. Compromise is isolated 0 < γs,r + γr,s < 2, (5)
from all other phenomena and its dynamics is studied quanti-
tatively using the proposed framework. Note that the analytic and it is certainly satisfied under the assumption that
model described in this section can be enriched to incorporate 0 < γs,r < 1. (6)
all mentioned phenomena by adding specific contributions to
adopted interaction rules, but such a generalization is not Under this assumption, it is reasonable to expect that, after
discussed here. a sufficiently large number of interactions, all agents would
end up with the same opinion. In addition, a complete model
Starting from the pioneering work of De Groot [29], a key
of compromise requires that if two agents with different
ingredient of opinion models has been the idea of updating the
opinions interact, they tend to preserve their opinions. This
opinions of agents after an interaction by properly weighting
can be modelled by introducing the assumption that the post-
the pre-interaction opinions of all interacting agents. Here,
interaction opinion of an agent is normally closer to its pre-
only binary interactions are considered, and an agent s with
interaction opinion than to the pre-interaction opinion of the
opinion v ∈ I is supposed to interact with another agent
agent it interacts with, which corresponds to the following
r with opinion w ∈ I, where I = [−1, 1] without loss
conditions
of generality. For binary interactions, the post-interaction
opinions of the two interacting agents are assumed to depend |v ∗ − v| < |v ∗ − w| and |w∗ − w| < |w∗ − v|. (7)
on the respective pre-interaction opinions as described by the
Observe that conditions (6) are not sufficient to guarantee
following interaction rules
that inequalities (7) are satisfied. However, by imposing the
( additional condition
v ∗ = v − γs,r (v − w)
(1) 1
w∗ = w − γr,s (w − v), 0 < γs,r < (8)
2

28
it can be easily proved that the post-interaction opinion of an asymptotically. Note that some of the major properties of
agent is normally closer to its pre-interaction opinion than to matrix C are similar to those derived in [4] using a related,
the pre-interaction opinion of the agent it interacts with but significantly different, model.
In order to properly analyse relevant properties of C, some
|v ∗ − v| = γs,r |v − w| < |v ∗ − w|
(9) classic results valid for complex matrices are needed. First,
|w∗ − w| = γr,s |w − v| < |w∗ − v|. recall that a generic m × m complex matrix B is said to be
In summary, in order to properly model compromise, from diagonally dominant if the following inequality holds for all
now on all parameters {γs,r }ns,r=1 are assumed to be defined in its elements {bs,r }m
s,r=1
the interval (0, 1/2). Such a choice of parameters can be used m
X
to study relevant collective properties of multi-agent systems |bs,s | ≥ |bs,r |. (12)
where only compromise is relevant to opinion formation, and r=1
r6=s
where adopted interaction rules are (1).
The interest now is on investigating the temporal evolution Then, also recall that a generic m × m complex matrix B is
of the opinion of a generic agent s. This can be done using the said to be strictly diagonally dominant if the following strict
general results of mathematical kinetic theory, as described, inequality holds for all its elements {bs,r }m
s,r=1
e.g., in [1]. In particular, using the adopted interaction rules, m
X
the weak form of the Boltzmann equation [1] relative to agent |bs,s | > |bs,r |. (13)
s and test function φ(v) = v can be written as r=1
r6=s
n n
d X X
Two classic results on complex matrices, i.e., the Gershgorin
us (t) = γs,r βs,r ur (t) − us (t) γs,r βs,r , (10)
dt r=1 r=1 circle theorem (e.g., [30]) and the Levy-Desplanques theorem
r6=s r6=s (e.g., [30]), are sufficient to prove the following propositions,
where us (t) is the opinion of agent s at time t ≥ 0, ur (t) is which characterise matrix C.
the opinion of another agent r at the same time t, and βs,r Proposition 1. Matrix C is singular.
is the probability that agent s interacts with agent r per unit
time, with βs,s = 0 by definition. Proof. The singularity of matrix C follows from the fact that
Equation (10) is for a generic agent s, and when considering the sum of the elements in each row is zero. Each diagonal
n agents, it represents a single equation of a homogeneous element of C is defined as the opposite of the sum of the
system of first-order linear differential equations. Such a kind remaining elements on the same row. Actually, denoting as ds
of system can be solved in closed form, and its solution is the s−th column of matrix C, the following equality holds
generally expressed using matrix notation. Let u be the vector n
X
of size n whose s−th component is us , then the system of ds = 0 (14)
equations whose s−th equation is (10), can be written in s=1
matrix notation as where 0 is a vector of length n with all elements equal to 0.
d Therefore, it can then be concluded that det(C) = 0.
u(t) = C u(t) (11)
dt Proposition 2. Matrix C is diagonally dominant, but not
where C is the n × n matrix of the coefficients of the system, strictly diagonally dominant, and the following hold
and it has the following explicit expression n
X
 n
X
 |cs,s | = |cs,r | (15)
− γ1,r β1,r γ1,2 β1,2 ... γ1,n β1,n  r=1
 r=1  r6=s
 
 r6=1  Proof. Observe that the non-diagonal elements of C are
 n 
positive. Adopted assumptions ensures that the parameters
X
 γ2,1 β2,1 − γ2,r β2,r . . . γ2,n β2,n 
 

r=1
 {γs,r }ns,r=1 are positive, and therefore
C=  r6=2
.

 .. .. .. ..  |cs,r | = cs,r for s 6= r. (16)
.
 
 . . . 

 Xn 
 At the opposite, it is easy to show that diagonal elements of
 γn,1 βn,1 f
 γn,2 βn,2 ... − γn,r βn,r 
 C are negative. Therefore, the following, which corresponds
r=1 to (15), hold
r6=n
n
X n
X
The rest of this section is devoted to the study of the properties |cs,s | = −cs,s = γs,r βs,r = |cs,r | (17)
of C which are needed to study the long-time asymptotic r=1 r=1
dynamics of the solutions of (11). Actually, presented prop- r6=s r6=s

erties of C are sufficient to prove that the opinions of all where the last equality follows from the definition of C and
agents tend to the same value, and that consensus is reached from (16).

29
Proposition 3. The rank of matrix C is n − 1. Proof. In order to prove the proposition, the solution of the
system of first-order linear differential equations (11) must
Proof. Since, according to Proposition 1, matrix C is singular,
be studied. Observe that the entire system depends on the
it is not full rank, and the following inequality holds
n × n matrix C. Denoting as λ0 = 0 and {λh }kh=1 the
rank(C) < n. (18) (k + 1) eigenvalues of C, and as {ηh }kh=0 their corresponding
multiplicities, with η0 = 1 for Proposition 4, the solutions of
In order to show that rank(C) = n − 1, the principal minor
(11) can be written as
of order n − 1 is proved to be full rank. First, let us show
that the principal minor of order n − 1 is strictly diagonally k
X
dominant. This is easily proved by the following inequality us (t) = eλh t Ps(h) (t) (23)
n n−1 n−1 h=0
X X X
|cs,s | = |cs,r | > γs,r βs,r = |cs,r |, (19) (h)
r=1 r=1 r=1
where Ps (t) are polynomials of degree ηh − 1
r6=s r6=s r6=s
h −1
ηX
where the first equality follows from Proposition 2, the in- Ps(h) (t) =
(h)
aj,s tj . (24)
equality follows from the fact that in the second sum the (pos- j=0
itive) term relative to r = n is omitted, and the last equality
follows from (16). Note that the Levy-Desplanques theorem Since, according to Proposition 4, λ0 is an eigenvalue with
states that a matrix which is strictly diagonally dominant is multiplicity η0 = 1, the degree of the n polynomials
(0)
also full rank, which proves the proposition because it can be {Ps (t)}ns=1 is η0 − 1 = 0, so that for all s
concluded that the principal minor of order n − 1 is full rank
and, therefore, that the rank of C is n − 1. Ps(0) (t) = a0,s . (25)

Proposition 4. One of the eigenvalues of matrix C is 0, and Moreover, according to classic results on systems of linear
it has multiplicity 1. differential equations, {a0,s }ns=1 are proportional to the com-
ponents of an eigenvector relative to the eigenvalue λ0 = 0.
Proof. The fact that 0 is an eigenvalue of C follows from the
In other words
fact that, as observed in Proposition 1, C is singular. The fact
that the multiplicity of eigenvalue 0 is 1 follows from the fact |
a0,1 a0,2 . . . a0,n = l0 e0 , (26)
that, as shown in Proposition 3, the rank of C is n − 1.
Proposition 5. The real parts of the eigenvalues of matrix C where e0 is a vector of length n which satisfies
are not positive, and they are 0 only for eigenvalue λ0 = 0.
C e0 = 0. (27)
Proof. Consider the Gershgorin disks relative to matrix C,
which are defined as disks in the complex plane with the Recalling the explicit expression of matrix C, it is evident that
following structure vector 1, whose n components are all equal to 1, satisfies (27)
  and it is therefore an eigenvector of C relative to λ0

 n n


 
e0 = 1. (28)
X X
Ks = z ∈ C : |z + γs,r βs,r | ≤ γs,r βs,r . (20)
 
r=1 r=1
From (26) it can then be concluded that {a0,s }ns=1 are all equal

 

r6=s r6=s

Since the elements of C are real, each disk Ks is centered on a0,s = l0 . (29)
the real axis at n
X
−ρs = γs,r βs,r , (21) According to this results, the solutions us (t) computed in (23)
r=1 can be written as
r6=s
k
and its radius is exactly ρs , so that all disks intersect the
X
us (t) = l0 + eλh t Ps(h) (t). (30)
immaginary axis only at the origin. Therefore, if {λs }ns=1 h=1
are the eigenvalues of matrix C, from (20), the following
inequalities hold From (30) it can be concluded that for all solutions

−2ρs ≤ Re(λs ) ≤ 0, (22) lim us (t) = l0 , (31)
t→∞
which proves the proposition and it also provides a lower
since all the addends in the sum in (23) converge to zero
bound on the real parts of eigenvalues.
because all eigenvalues {λh }kh=1 have negative real part for
Proposition 6. The long-time asymptotic opinions of all Proposition 5. Therefore, all agents would eventually end up
agents are equal to a real value which depends on initial with the same opinion, whose value l0 depends on initial opin-
opinions, on values {γr,s }nr,s=1 , and on values {βs,r }nr,s=1 . ions, on values {γr,s }nr,s=1 , and on values {βs,r }nr,s=1 .

30
0.25

compromise is considered relevant. Finally, the paper showed
0.2
simulations ran in an illustrative case by directly implementing
0.15 chosen interaction rules to confirm analytic results.
0.1
The proposed analytic framework can be used to study
0.05
all sociological phenomena that are normally associated
with opinion formation, e.g., compromise, diffusion, and ho-
0

mophily. This paper considered only compromise in order to
-0.05

derive analytic results intended to characterise the dynamics of
-0.1
compromise independently from other phenomena. Actually,
-0.15 the fact that compromise makes the opinion of all agents tend
-0.2
to a single value is not surprising, but the dynamics of such
-0.25
an asymptotic behaviour was not so obvious. Compromise
6.15E+03 7.40E+03 8.65E+03 9.90E+03 1.12E+04 1.24E+04 1.37E+04 1.49E+04 1.62E+04 1.74E+04

makes a multi-agent system reach consensus exponentially
Fig. 1. Plots of minimum (blue), maximum (red), and expected (gray) values fast, and the proposed model allows estimating the rate at
of the opinion for the studied scenario as a function of time. which the consensus value is approached on the basis of
the propensity of each agent to change its opinion because
of interactions, and on other relevant characteristics of the
III. V ERIFICATION BY S IMULATION multi-agent system. Obtained analytic results are confirmed by
This section shows illustrative simulations concerning the independent simulations performed by directly implementing
dynamics of the opinion in a specific scenario, which is adopted interaction rules.
used to verify the expected long-time asymptotic behaviour Methodologically, the major advantage that is expected from
modelled in previous section. A multi-agent system made of the adoption of a kinetic approach to the study of the dynamics
n = 103 agents is considered, and simulations implementing of the opinion is that mathematical kinetic theories are inher-
studied interaction rules are performed by iteratively selecting ently analytic and they can provide analytic descriptions of
a random agent and by making selected agent interact with the collective properties of the opinion. Such a characteristic
another randomly chosen agent. The opinions of agents are of mathematical kinetic theories ensures that obtained results
initialised to random values uniformly distributed in interval can be used not only as descriptive tools capable of explaining
I = [−1, 1], and their specific propensity to change opinion observations, but that they can also be used as prescriptive
because of interactions, i.e., parameters {γs,r }ns,r=1 , are fixed tools to govern the dynamics of studied multi-agent systems.
to random values uniformly distributed in interval ( 14 , 12 ). The As a prescriptive tool, the proposed approach can support the
distribution of the opinion after the execution of simulations design of multi-agent systems with desired properties because
is compared with the expected long-time asymptotic opinion analytic results can be used to identify the actual values of
l0 , which is computed as discussed in previous section. The specific parameters to have the multi-agent system behave
coherence of the results of simulations with expected value l0 as intended. In addition, as a descriptive tool, the analytic
is evaluated in terms of approach that is developed in this paper can be used as an
alternative to simulation. The validity of results of simulations
ǔ(t) = min us (t) and û(t) = max us (t), (32) depends on how much selected simulations are representative
1≤s≤n 1≤s≤n
of studied multi-agent systems. On the contrary, the validity of
and all simulations are performed until (û(t) − ǔ(t)) ≤ 10−3 . analytic results is clearly identified by the assumptions adopted
Note that presented simulations do not use analytic results to derive them, and such assumptions can also be studied in
from previous section, rather they are direct implementation order to be possibly generalised.
of considered interaction rules and they are intended only to Planned developments of the presented work involve four
validate analytic results. generalizations. First, the deterministic parameters {γs,r }ns,r=1
Figure 1 shows the dynamics of ǔ(t) and û(t), and it also which characterize the propensity of agents to change their
shows the expected long-time asymptotic value of the opinion opinions because of interactions could be replaced by random
l0 . Note that the opinions of agent converge to the expected variables with suitable distributions. Second, the topology of
value after 18.6 × 103 interactions, with each agent involved the multi-agent system could be taken into account and the
in less than 60 interactions. hypothesis that each agent can freely interact with any other
agent could be dropped. Third, more complex interaction
IV. C ONCLUSIONS rules could be considered to take into account how various
This paper presented analytic results that characterise com- phenomena that contribute to opinion formation interact in real
promise, which is one of the major phenomena used to situations, and how they jointly contribute to the dynamics of
describe opinion formation in multi-agent systems. The paper the opinion. Fourth, note that the ideas behind the discussed
used the general framework of mathematical kinetic theories framework are not limited to the study of opinion dynamics,
to model compromise and to derive results on the long- and the proposed approach could be applied to describe other
time asymptotic behaviour of multi-agent systems where only collective properties of large multi-agent systems.

31
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