The Bounded Confidence Model Of Opinion Dynamics

# The Bounded Confidence Model Of Opinion Dynamics

GÓMEZ-SERRANO, JAVIER    GRAHAM, CARL    LE BOUDEC, JEAN-YVES
###### Abstract

The bounded confidence model of opinion dynamics, introduced by Deffuant et al, is a stochastic model for the evolution of continuous-valued opinions within a finite group of peers. We prove that, as time goes to infinity, the opinions evolve globally into a random set of clusters too far apart to interact, and thereafter all opinions in every cluster converge to their barycenter. We then prove a mean-field limit result, propagation of chaos: as the number of peers goes to infinity in adequately started systems and time is rescaled accordingly, the opinion processes converge to i.i.d. nonlinear Markov (or McKean-Vlasov) processes; the limit opinion processes evolves as if under the influence of opinions drawn from its own instantaneous law, which are the unique solution of a nonlinear integro-differential equation of Kac type. This implies that the (random) empirical distribution processes converges to this (deterministic) solution. We then prove that, as time goes to infinity, this solution converges to a law concentrated on isolated opinions too far apart to interact, and identify sufficient conditions for the limit not to depend on the initial condition, and to be concentrated at a single opinion. Finally, we prove that if the equation has an initial condition with a density, then its solution has a density at all times, develop a numerical scheme for the corresponding functional equation, and show numerically that bifurcations may occur.

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Keywords: Social networks; reputation; opinion; mean-field limit; propagation of chaos; nonlinear integro-differential equation; kinetic equation; numerical experiments.

MSC2010: 91D30,60K35,45G10,37M99

## 1 Introduction

Some models about opinion dynamics (or belief or gossip propagation, etc.) are based on binary values, and often lead to attractors that display uniformity of opinions. These models are not valid for scenarios such as the social network of truck drivers interested in the quality of food of a highway restaurant or the critics’ ratings about the new opening movies, for which it is required to have a continuous spectrum of opinions, as is also the case in politics when people are positioned on a scale going from extreme left-wing to right-wing opinions.

The bounded confidence model introduced by Deffuant et al. is a popular model for such scenarios. Peers have -valued opinions; repeatedly in discrete steps, two peers are sampled, and if their opinions differ by at most a deviation threshold then both move closer, in barycentric fashion governed by a confidence factor. These parameters are the same for all peers, and the system is in binary mean-field interaction. The model has been studied and generalized, notably to other interaction graphs than the fully-connected one, to vector-valued opinions, and to peer-dependant deviation thresholds.

Reputation systems have lately emerged due to the necessity to measure trust about users while doing transactions over the internet; popular examples can be found in e-Bay or Bizrate. Some models for trust evolution and the potential effects of groups of liars attacking the system can be seen as a generalization of the bounded confidence model, in particular when there are no liars nor direct observations and the system evolves only by interaction between the peers. The “Rendez-vous” model used by Blondel et al has qualitative resemblance to the model used in this paper; like ours, it converges to a finite number of clusters in finite time for the finite case. However, the interaction model is different, and our techniques (based on convexity and conservation of mean, see Proposition 3.2) do not seem to apply to this model.

The mean-field approximation method for large interacting systems has a very long history. Its heuristic and rigorous use started in statistical physics and entered many other fields, notably communication networks, TCP connections, robot swarms, transportation systems, and online reputation systems in which is particularly appealing since the number of users may be very large (over 400 million for Facebook).

This paper provides some rigorous proofs of old and new results on the Deffuant et al. model, which has been studied intensively, but essentially by heuristic arguments and simulations. Notably, justifying the validity of the mean-field approach is not a simple matter, and classical methods do not apply, as seen below.

We prove that as time increases to infinity, opinions eventually group after some random finite time into a constant number of clusters, which are separated by more than the deviation threshold, and cannot influence one another. Thereafter, all opinions within every such cluster converge to their barycenter. The limit distribution of opinions is thus of a degenerate form, in which there are only a small number of fixed opinions which differ too much to influence each other, called a “partial consensus”; when it is constituted of one single opinion, it is called a “total consensus”. Note that the limit distribution is itself random, i.e. different sample runs of the same model with same initial conditions always converge, but perhaps to different limiting distributions of opinions.

We then prove a mean-field limit result, called “propagation of chaos” in statistical mechanics: if the number of peers goes to infinity, the systems are adequately started, and time is rescaled accordingly, then the processes of the opinions converge in law to i.i.d. processes. Each of these is a so-called nonlinear Markov (or McKean-Vlasov) process, corresponding to an opinion evolving under the influence of opinions drawn independently from the marginal law of the opinion process itself, at a rate which is the limit of that at which a given peer in the finite system encounters its peers. Moreover, these marginals are the unique solution of an adequately started nonlinear integro-differential equation.

This implies a law of large numbers: the empirical measures of the interacting processes converge to the law of the nonlinear Markov process. Such process level results imply results for the marginal laws, but they are much stronger: limits are derived for functionals of the sample paths, such as hitting times or extrema. In particular, a functional law of large numbers holds for the marginal processes of these empirical measures, with limit the solution of the integro-differential equation.

The probabilistic structure of this limit equation is similar to that of kinetic equations such as the cutoff spatially-homogeneous Boltzmann or Kac equations, classically used in statistical mechanics to describe the limit of certain particle systems with binary interaction. Under quite general assumptions, satisfied here, it has long been known that it is well-posed in the space of probability laws, and that if the initial law has a density, then the solution has a density at all times satisfying a functional formulation of this equation.

###### Remark 1.1

There are two main difficulties in the propagation of chaos proof:

1. the interaction is binary mean-field, since two opinions change simultaneously,

2. the indicator functions related to the deviation threshold are discontinuous.

A system in which only one opinion would change at a time would be in simple mean-field interaction, and one could write equations for the opinions in almost closed form, which could be passed to the limit in various classical ways. This cannot be done for binary interaction, in which there is much more feedback between peers; moreover, this would require continuous coefficients. See Section 4.2 for details.

Such difficulties have been solved before. In order to adapt results obtained for a class of interacting systems inspired by communication network models using stochastic coupling techniques, which can be applied to various Boltzmann and Kac models, we introduce an intermediate auxiliary system, a continuous-time variant of the discrete-time model of Deffuant et al. interacting at Poisson instants, which itself constitutes a relevant opinion model.

For this auxiliary system, we prove propagation of chaos, in total variation norm with estimates on any finite time interval. We then control the distance between this auxiliary system and the Deffuant et al. model, and prove propagation of chaos for a weaker topology, but still at the process level and allowing discontinuous measurable dependence on the -values taken by the opinions.

The method can be easily generalized, for instance to vector-valued opinions, or to randomized interactions with a joint law governing whether one or both peers change opinion and by how much; for instance, choosing uniformly at random one peer to change opinion and leaving the other fixed would lead to a simple mean-field interacting model, and the limit model would be slowed down by a factor two.

To the best of our knowledge, this is the first rigorous mean-field limit result for this model. Similar integro-differential equations were used without formal justification before, and appear to be incorrect by a factor 2 (perhaps by disregarding that two peers change opinion at once), which illustrates the interest of deriving the macroscopic equation from a microscopic description, as we do here.

We thank a referee to have brought to our attention the preprint Como-Fagnani. It contains results for the marginal laws of a continuous-time variant of the model with two major simplifications: the interaction is simple mean-field (only one opinion changes at a time), and the indicator functions are replaced by Lipschitz-continuous functions; this removes difficulties (1.1) and (1.1) in Remark 1.1, to which its techniques do not apply. We have overcome these difficulties in the precise model of Deffuant et al., and have given much stronger results, for process laws in total variation norm and not for marginal laws in weak topologies.

One expects that the long-time behavior for the mean-field limit should be highly related to the behavior for an large number of peers of the long-time limit of the finite model. This heuristic inversion of long-time and large-number limits can be sometimes rigorously justified, for instance by a compactness-uniqueness method, but here the limit nonlinear integro-differential equation may have multiple equilibria, and formal proof would constitute a formidable task.

We prove that the long-time behavior of the solution of the limit integro-differential equation is similar to that of the model with finitely many peers: it converges to a partial consensus constituted of a small number of fixed opinions which differ too much to influence each other.

We then develop a numerical method for the limit equation, and use it to explore the properties of the model. We observe phase transitions with respect to the number of limit opinions, while varying the deviation threshold for some fixed initial condition. We model the scenario of a company fusion, dividing the workers into an “undecided” group and two “extremist” factions, and obtain that having 20% of the workers “undecided” is enough to achieve consensus between all.

Last, we establish a bound on the deviation threshold, allowing to determine if there is total consensus or not, under the assumption of symmetric initial conditions.

In the sequel, Section 2 describes the finite model, and Section 3 studies some of its long time properties. Section 4 rigorously derives the mean-field limit, Section 5 studies some of its long time properties, and Section 6 is devoted to numerical results. The appendix contains some probabilistic complements in Section A, the details of the algorithm in Section B and all proofs in Section C.

## 2 Interacting system model, and reduced descriptions

The model for interacting peers introduced by Deffuant et al. is as follows. The random variable (r.v.) with values in denotes the reputation record kept at peer at time , representing its opinion (or belief, etc.) about a given subject, the same for all peers. The discrete-time process of the states taken by the system of peers is

 XN=(XN(k),k∈N),XN(k)=(XNi(k))1≤i≤N,

and evolves in function of the deviation threshold and the confidence factor . At each instant , two peers and are selected uniformly at random without replacement, and:

• if then , the two peers’ opinions being too different for mutual influence,

• if then the values of peers and are updated to

 {XNi(k+1)=wXNi(k)+(1−w)XNj(k),XNj(k+1)=wXNj(k)+(1−w)XNi(k),

and the values of the other peers do not change at time , the two peers having sufficiently close opinions to influence each other.

Small values of and large values of mean that the peers trust very much their own opinions in comparison to the new information given by the other interacting peer. The extreme excluded values or correspond to peers never changing opinion, and to peers switching opinions if close enough. For , two close-enough peers would both end up with the average of their opinions.

A reduced, or macroscopic, description of the system is given by the empirical measure , and by its marginal process also called the occupancy process,

 ΛN=1NN∑i=1δXNi=1NN∑i=1δ(XNi(k),k∈N),MN(k)=1NN∑i=1δXNi(k).

The random measure has samples in , the space of probability measures on ; its projection , which carries much less information, has sample paths in , the space of sequences of probability measures on . For measurable and ,

 ⟨g,ΛN⟩=1NN∑i=1g(XNi),⟨h,MN(k)⟩=1NN∑i=1h(XNi(k)).

We will also re-scale time as , and consider in particular the rescaled occupancy process given by , which in Section 4 will be shown to converge to a deterministic process .

## 3 Long-time behavior of the finite N model

We consider a fixed finite number of peers and let time go to infinity. We prove that the distribution of peer opinions converges almost surely (a.s.) to a random distribution . Note that the limiting distribution depends on chance as well as on the initial condition. We prove that is a combination of at most Dirac measures at points separated by at least . A key observation here is that if is any convex function then is non-increasing in . Dittmer and Krause obtained similar results, but for a deterministic model.

###### Definition 3.1

We say that is a partial consensus with components if with , for , and . Necessarily and . If , i.e., if is a Dirac measure, we say that is a total consensus.

If is a partial consensus, then peers are grouped in a number of components too far apart to interact, and within one component all peers have the same value. Thus, a partial consensus is an absorbing state for , and Theorem 3.9 below will show that converges a.s., as , to one such state.

### 3.1 Convexity and Moments

We start with results about convexity and moments, which are needed to establish the convergence result and are also of independent interest.

###### Proposition 3.2

For any convex function , any and in ,

 h(wx+(1−w)y)+h(wy+(1−w)x)−h(x)−h(y)≤0

with equality when is strictly convex possible only if or or .

The following corollary is immediate from the interaction structure of the model.

###### Corollary 3.3

If is a convex function, then is a non-increasing function of along any sample path.

Applying this to , and yields that in any sample path, the first moment is constant and other moments are non-increasing with time.

###### Corollary 3.4

For and , let denote the -th moment of , and let be the standard deviation given by . Then:

1. The mean is stationary in , i.e., for all .

2. The moments and the standard deviation are non-increasing in : if then and .

Moreover, stationarity of moments is equivalent to reaching partial consensus:

###### Proposition 3.5

If is a partial consensus, then for all and . Conversely, if for some there exists a (random) instant such that for all , then for all and is a partial consensus, almost surely.

### 3.2 Almost Sure Convergence to Partial Consensus

###### Definition 3.6

We say that two peers and are connected at time if their values and satisfy . We say that is a cluster at time if it is a maximal connected component.

In other words, a cluster is a maximal set of peers such that every peer can pass the deviation test with one neighbour in the cluster. The set of clusters at time is a random partition of the set of peers. The following proposition states that a cluster can either split or stay constant, but cannot grow.

###### Proposition 3.7

Let be the set of clusters at time . Then either or where is a partition of , for some .

The number of clusters is thus non decreasing, and since it is bounded by it must be constant after some time, yielding the following:

###### Corollary 3.8

There exists a random time , a.s. finite, such that

 CN(k)=CN(KN)fork≥KN.

Finally, we prove that the occupancy measure converges to a partial consensus (see Appendix A. Probabilistic, Topological and Measurability issues for the usual weak topology on ):

###### Theorem 3.9

As goes to infinity, converges almost surely, for the weak topology on , to a random probability , which is a partial consensus with components, where is the final number of clusters.

Theorem 3.9 notably implies that there is convergence to total consensus if and only if . The probability of convergence to total consensus is not necessarily or , but:

1. If the diameter of is less than (i.e., ) then (obvious);

2. If there is more than 1 cluster in then (see Proposition 3.7).

## 4 Mean-field limit results when N goes to infinity

This section is devoted to a rigorous statement for the following heuristic statistical mechanics limit picture: all peers act independently, as if each were influenced by an infinite supply of independent statistically similar peers, of which the instantaneous laws solve a nonlinear equation obtained by consistency from this feedback.

In statistical mechanics and probability theory, such convergence to an i.i.d. system is called chaoticity, and the fact that chaoticity at time implies chaoticity at further times is called propagation of chaos.

Some other probabilistic definitions and facts are recalled in Appendix A. Probabilistic, Topological and Measurability issues.

We introduce an intermediate auxiliary system, in which the peer meet at the instants of a Poisson process, which itself constitutes a relevant opinion model. We adapt results in Graham-Méléard to prove that the sample paths of the auxiliary system are well approximated by the limit system. We eventually control the distance between the auxiliary system and the model of Deffuant et al.

### 4.1 Mean-field regime, rescaled and auxiliary systems

The number of peers is typically large, and we let it go to infinity. At each time-step two peers are possibly updated, and the empirical measures have jumps of order , hence time must be rescaled by a factor . This is a mean-field limit, in which time is usually rescaled by physical considerations (such as “the peers meet in proportion to their numbers”). It is also related to fluid limits.

For a Polish space , let denote the space of probability measures on , with the Borel -field, and denote the Skorohod space of paths from to which are right-continuous with left-hand limits, with the product -field.

A non-trivial continuous-time limit process is expected for the rescaled system

 ˜XN=(˜XNi)1≤i≤N,˜XN=(˜XN(t),t∈R+)=(XN(⌊Nt⌋),t∈R+), (4.1)

with sample paths in . The corresponding empirical measure and the process constituted of its marginal laws are given by

 ˜ΛN=1NN∑i=1δ˜XNi=1NN∑i=1δ(˜XNi(t),t∈R+),˜MN(t)=1NN∑i=1δ˜XNi(t), (4.2)

respectively with samples in and sample paths in .

An auxiliary (rescaled) system is obtained by randomizing the jump instants of the original model by waiting i.i.d. exponential r.v. of mean between selections, instead of deterministic durations. A convenient construction using a Poisson process of intensity  is that, with sample spaces as above,

 ˆXN=(ˆXNi)1≤i≤N, ˆXN=(ˆXN(t),t∈R+)=(XN(A(Nt)),t∈R+), (4.3) ˆΛN=1NN∑i=1δˆXNi, ˆMN=(ˆMN(t),t∈R+),ˆMN(t)=1NN∑i=1δˆXNi(t).

If for are given by and the jump instants of , then

 ˜XN(t)=ˆXN(t′)=XN(k),kN≤t

Note that and , but that the relationship between and and is more involved.

The process is a pure-jump Markov process with rate bounded by , at which two peers are chosen uniformly at random without replacement, say and at time , and:

• if then ,

• if then only the values of peers and change to

 ⎧⎨⎩ˆXNi(t)=wˆXNi(t−)+(1−w)ˆXNj(t−),ˆXNj(t)=wXNj(t−)+(1−w)ˆXNi(t−).
###### Remark 4.1

Each of the unordered pairs of peers is thus chosen at rate , and then both peers undergo a simultaneous jump in their values if these are close enough. Each peer is thus affected at rate .

The generator of acts on (the Banach space of essentially bounded measurable functions on ) as

 ANf((xn)1≤n≤N)=2N−1∑1≤i

where is obtained from by replacing and with and and leaving the other coordinates fixed. Its operator norm is bounded by , and the law of the corresponding Markov process is well-defined in terms of the law of .

For , this generator acts on which depend only on the -th coordinate, of the form for some , as

 2N−1∑1≤j≤N:j≠k[h(wxk+(1−w)xj)−h(xk)]1{|xk−xj|≤Δ}:=A(1N−1∑1≤j≤N:j≠kδxj(dy))h(xk) (4.6)

where the generators act on as

 A(μ)h(x)=2⟨[h(wx+(1−w)y)−h(x)]1{|x−y|≤Δ},μ(dy)⟩,μ∈P[0,1]. (4.7)

Heuristically, if the converge in law to i.i.d. r.v. of law , then the are expected to converge in law to i.i.d. processes of law , the law of a time-inhomogeneous Markov process with initial law and generator at time , where is the instantaneous law of this same process, the marginal of . Such a process is called a nonlinear Markov process, or a McKean-Vlasov process. Considering the forward Kolmogorov equation for this Markov process, should satisfy the following weak (or distributional-sense) formulation of a nonlinear integro-differential equation.

###### Definition 4.1 (Problem 1)

We say that with is solution to Problem 1 with initial value if and

 ⟨h,m(t)⟩−⟨h,m(0)⟩=∫t0⟨A(m(s))h,m(s)⟩ds :=∫t02⟨[h(wx+(1−w)y)−h(x)]1{|x−y|≤Δ},m(s)(dy)m(s)(dx)⟩ds (4.8)

for all test functions ; this can be written more symmetrically as

 ⟨h,m(t)⟩−⟨h,m(0)⟩ =∫t0⟨[h(wx+(1−w)y)+h(wy+(1−w)x) (4.9)

The distance in total variation norm of and in is given by

 |μ−μ′|=sup∥ϕ∥∞≤1⟨ϕ,μ−μ′⟩=2sup{μ(A)−μ′(A):measurable A⊂S}. (4.10)
###### Theorem 4.2

Consider the generators given by (4.7), and in .

1. There is a unique solution to Problem 1 starting at . For the total variation norm on , is continuous, and is continuous for uniform convergence on bounded time sets.

2. There is a unique law on for an inhomogeneous Markov process with generator at time and initial law . Its marginal is given by .

###### Remark 4.2

Such nonlinear Markov processes and equations are well-known to probabilists. The equations 1 have same probabilistic structure as the weak forms (2.1), (2.2), (2.4) (with ) of the spatially homogeneous version (without -dependence) of the Boltzmann equation (1.1) in Graham-Méléard, the weak form (1.7) of the (cutoff) Kac equation (1.1)-(1.2) in Desvillettes et al., the nonlinear Kolmogorov equation (2.7) in Graham, and the kinetic equation (9.4.4) in Graham. The weak formulation involves explicitly the generator of the underlying Markovian dynamics and allows to understand it more directly. The functional formulation (for probability density functions) of this integro-differential equation involves an adjoint expression of this generator, and will be seen in Section 6.

### 4.2 Difficulties for classical mean-field limit proofs

The system exhibits simultaneous jumps in two coordinates, and is in binary mean-field interaction in statistical mechanics terminology.

A system in which only one opinion would change at a time would be in simple mean-field interaction; the generator in (4.5) would be replaced by a simpler expression, which could be written as a sum over of terms acting only on the -th coordinate in terms of the value and of . Consequently, the empirical measures would satisfy an equation in almost closed form, which could be exploited in various ways to prove convergence to a limit satisfying the closed nonlinear equation in which the empirical distribution is replaced by the law itself.

A binary mean-field interacting system is much more complex, since there is much more feedback between peers. It is impossible to relate it in a simple way to an independent system, in which the coordinates cannot jump simultaneously. Because of that, the coupling methods introduced by Sznitman, see also Méléard and Graham-Robert, cannot be adapted here. Moreover, these use contraction techniques, and the metric used is too weak for the indicator functions.

Elaborate compactness-uniqueness methods are also used for proofs, see Sznitman, and also Méléard, Graham-Méléard Section 4, and Graham, but require weak topologies for compactness criteria, and continuity properties in order to pass to the limit; hence, the indicator functions prevent using them here.

###### Remark 4.3

The indicator functions require quite strong topologies. For instance, if and , then there exists with support not intersecting and converging weakly to , and starting there and have at least two clusters and support outside . There exists also with support inside and converging weakly to , and and have one cluster and support inside for any , and will be a total consensus after some random time.

### 4.3 Rigorous mean-field limit results for the auxiliary system

Systems of this type were studied in Graham-Méléard, see also Ref. ?. The first paper studied a class of not necessarily Markovian multitype interacting systems, as a model for communication networks. The second studied Monte-Carlo methods for a class of Boltzmann models, and in particular expressed some notions and results of the first in this framework. Their results yield the following.

For and and laws and on , let denote the distance in variation norm (4.10) of the restrictions of and on . When clear, the processes will be restricted to without further mention.

###### Theorem 4.3

Consider the auxiliary system (4.3) for . If the are i.i.d. of law , then there is propagation of chaos. More precisely, let and be as in Theorem 4.2 for , and .

1. For ,

 |L(ˆXN1,…,ˆXNk)−L(ˆXN1)⊗⋯⊗L(ˆXNk)|T≤2k(k−1)2T+4T2N−1,

and

 ∣∣∣1NN∑i=1L(ˆXNi)−Q∣∣∣T≤|L(ˆXNi)−Q|T≤6exp(2T)−1N+1.
2. For any such that ,

 E[⟨ϕ,ˆΛN−1NN∑i=1L(ˆXNi)⟩2]≤4+8T+16T2N.

Moreover

 ˆΛNin probab.−−−−−−−→N→∞Q,ˆMNin probab.−−−−−−−→N→∞m,

respectively for the weak topology on with the Skorohod topology on , and for the topology of uniform convergence on bounded time intervals on with the weak topology on .

The assumption that the initial conditions are i.i.d. can be appropriately relaxed, as in Theorem 1.4 in Graham-Mélééard.

These very strong results are obtained for a relevant opinion model given by the auxiliary (rescaled) system, and are of independent interest. In the next section we will derive from them some weaker results for the original discrete-time model.

###### Remark 4.4

The convergence result for is equivalent to convergence in law to (Ethier-Kurtz, Corollary 3.3.3). The convergence result for implies convergence in law to for test functions which are continuous, bounded, and measurable for the product -field (Ref. ?, Theorem 3.10.2). Separability issues restrict these results, see Appendix A. Probabilistic, Topological and Measurability issues; in fact, convergence of holds for any convergence induced by a denumerable set of bounded measurable functions.

### 4.4 From the auxiliary to the rescaled system

For , let denote the Skorohod metric on given by (3.5.21) in Ethier-Kurtz for the atomic metric on (which induces the topology of all subsets of , for which any function is continuous). Note that is measurable with respect to the usual Borel -field on .

A time-change is an increasing homeomorphism of , i.e., a continuous function from to which is null at the origin and strictly increasing to infinity. Two paths are close for if there is a time-change close to the identity such that the time-change of one path is equal to the other path.

Eq. (4.4) is the key to obtain the following quite general result showing that the rescaled system is very close to the the auxiliary system , up to a well-controlled (random) time-change.

###### Theorem 4.4

Consider the rescaled system (4.1) and the auxiliary system (4.3) for . Then in probability.

This result and Theorem 4.3 now yield the main mean-field convergence result.

###### Theorem 4.5

Consider the rescaled system (4.1) for . If the are i.i.d. of law , then there is propagation of chaos. More precisely, let and be as in Theorem 4.2 for .

1. For ,

 limN→∞L(˜XN1,…,˜XNk)=Q⊗k,

for the weak topology on induced by test functions which are either uniformly continuous for the Skorohod metric , bounded, and measurable for the usual product -field (for the usual Borel -field on ), or continuous for the usual Skorohod topology (for the usual metric on ) and bounded.

2. For the usual topology of ,

 ˜ΛNin probab.−−−−−−−→N→∞Q,˜MNin probab.−−−−−−−→N→∞m,

respectively for the weak topology on with the Skorohod topology on , and for the topology of uniform convergence on bounded time intervals on with the weak topology on .

The assumption that the initial conditions are i.i.d. may again be relaxed. For the second result, see again Remark 4.4.

## 5 Infinite N Model

We now study the mean-field limit obtained in Section 5 when goes to infinity. As for the finite model, we find that there is convergence to a partial consensus as goes to infinity. The limit may depend on the initial conditions, as might the random limit when is finite. We are able to say more, and notably find tractable sufficient conditions for the limit to be a total consensus.

### 5.1 Convexity and Moments

Applying Proposition 3.2 to the equivalent definition of Problem 1 given by (4.1) yields the following:

###### Corollary 5.1

Let be a solution of Problem 1. If is convex, then is a non-increasing function of . Moreover, for and , let denote the -th moment of , and its standard deviation (i.e., ). Then:

1. The mean is stationary: for all .

2. The moments are non-increasing in : if then .

3. The standard deviation is also a non-increasing function of .

Furthermore, we have some bounds.

###### Proposition 5.2

For all , we have .

Note that Corollary 5.1 and Proposition 5.2 generalize results of Ref ?, which established similar results for the case . However, the bound in Proposition 5.2 is different, as the equation considered in Ref. ? misses a factor 2.

### 5.2 Convergence to Partial Consensus

It is immediate that a partial consensus is a stationary point for Problem 1, i.e., if is solution of Problem 1 with initial value a partial consensus , then for all . Conversely, we show, in Theorem 5.5 below, that any trajectory converges to a partial consensus.

It is useful to consider the essential sup and inf of , defined as follows.

###### Definition 5.3

For , let and .

Note that if and , then the support of is included in , i.e., for any measurable , .

###### Proposition 5.4

Let be solution of Problem 1. Then [resp. ] is a non-increasing [resp. non-decreasing] function of .

See Definition 3.1 for partial and total consensus.

###### Theorem 5.5

Let be a solution of Problem 1. As goes to infinity, converges, for the weak topology on , to some which is a partial consensus for every , i.e., of the form with , for , and .

Note that the limit may depend on the initial condition , and may or may not be a total consensus (as shown in the next section). We are in particular interested in finding initial conditions that guarantee that is a total consensus. The following is an immediate consequence of Proposition 5.4.

###### Corollary 5.6

If the diameter of is less than , i.e., if , then is a total consensus.

Note that the converse is not true: if the diameter of is larger or equal than , there may be convergence to total consensus (see next section for an example).

### 5.3 Convergence to Total Consensus

We find sufficient criteria for guaranteeing some upper bounds on the number of components of , in particular, we find some sufficient conditions for convergence to total consensus. Although the bounds are suboptimal, to the best of our knowledge, they are the first of their kind. The bounds are based on Corollary 5.1.

First define, for and , the set of partial consensus with components and mean , i.e., iff there is some sequence with , some sequence for with and such that and .

Second, for any convex, continuous , let be the set of strict lower bounds of the image by the mapping of , i.e., iff for any consensus with components and mean , it holds that . If is empty, let .

Note that is necessarily an interval with lower bound . The following proposition states that is non decreasing with .

###### Proposition 5.7

For any and and convex continuous , it holds that .

Combining Proposition 5.7 with Corollary 5.1, we obtain:

###### Theorem 5.8

Let be the solution of Problem 1 with initial condition , and be the number of components of the limiting partial consensus . Assume that, for some , some convex continuous , and some , we have , where is the mean of .

Under these assumptions, if then .

Here is an example of use of the theorem, for and .