Consensus and disagreement: the role of quantized behaviours in opinion dynamics
This paper deals with continuous-time opinion dynamics that feature the interplay of continuous opinions and discrete behaviours. In our model, the opinion of one individual is only influenced by the behaviours of fellow individuals. The key technical difficulty in the study of these dynamics is that the right-hand sides of the equations are discontinuous and thus their solutions must be intended in some generalized sense: in our analysis, we consider both Carathéodory and Krasowskii solutions. We first prove existence and completeness of Carathéodory solutions from every initial condition and we highlight a pathological behavior of Carathéodory solutions, which can converge to points that are not (Carathéodory) equilibria. Notably, such points can be arbitrarily far from consensus and indeed simulations show that convergence to non-consensus configurations is very common. In order to cope with these pathological attractors, we then study Krasowskii solutions. We give an estimate of the asymptotic distance of all Krasowskii solutions from consensus and we prove its tightness via an example: this estimate is quadratic in the number of agents, implying that quantization can drastically destroy consensus. However, we are able to prove convergence to consensus in some special cases, namely when the communication among the individuals is described by either a complete or a complete bipartite graph.
Keywords. Opinion dynamics, quantized consensus, disagreement, discontinuous differential equations.
A fundamental assumption in opinion dynamics is that one’s opinion is attracted by other’s opinions ([17, 7]). If the opinion of individual is described by a variable , then this assumption leads to describe the evolution of the opinions by a set of differential equations:
where and means that the opinion of the individual is influenced by the opinion of the individual . As long as there is at least one individual who can influence (albeit indirectly) all others, then it can be proved that consensus is asymptotically achieved, namely there exists such that as for all . However, experience suggests that in reality consensus is not always achieved, but disagreement persists: clearly, the opinion dynamics model should include some additional feature.
In this paper, we postulate that the individuals can not directly perceive the private opinions of the others, but only observe their displayed behaviours. Indeed, in some situations individuals may not be able to express their opinions precisely, but only through a limited number of behaviours or actions: let us think of consumers’ choices, electoral votes, or stereotyped interactions in online social media. Even though we are assuming that the opinions are real-valued, behaviours are better described as elements of a finite or discrete set: hence, the behaviour of an individual shall be a suitable quantization of his/her opinion. If we choose for simplicity to represent behaviours as integers, equations (1) can be replaced by
where the map is defined as . This new model is a very simple modification of (1), which has no pretension to describe social dynamics precisely, but has the aim to make the role of quantization evident. The feature that distinguishes it from (1) is that consensus is not achieved in general. From the point of view of opinion dynamics, this observation allows us to explain the persistence of disagreement as an effect of the limited number of behaviours allowed.
The aim of this paper is to study the effect of quantization of others’ states in a consensus model. This is done by undertaking a rigorous mathematical analysis of (2). The key technical difficulty in the study of this model is that the right-hand sides of the equations are discontinuous and their solutions must be intended in some generalized sense. As the first contribution, we prove existence and completeness of Carathéodory solutions from every initial condition. This is a relevant fact because most discontinuous systems, including well-known models of opinion dynamics, do not have complete Carathéodory solutions : even proving existence for most initial conditions can be a daunting task [4, 3].
As the second contribution, we highlight a pathological behavior of Carathéodory solutions, which can converge to points that are not (Carathéodory) equilibria: actually, these points can be arbitrarily far from consensus. The presence of these pathological attractors leads us to consider also Krasowskii solutions, which are trivially seen to exist and be complete. Indeed, Krasowskii solutions can not converge to points that are not (Krasowskii) equilibria. Another advantage is that, as Carathéodory solutions are particular Krasowskii solutions, results that hold for Krasowskii solutions are more general.
As the third contribution, we derive an estimate of the asymptotic distance of Krasowskii solutions from consensus and we prove by means of an example that it can not be substantially improved in general. This estimate is quadratically increasing in the number of agents and its tightness implies that quantization can drastically destroy consensus. Actually, simulations show that most graphs imply convergence to non-consensus configurations. However, consensus can happen in some special cases. Indeed, as the fourth contribution, we prove convergence to consensus when communication among the individuals is described either by a complete graph or by a complete bipartite graph.
Relations with the literature.
This paper relates to various bodies of work in control theory and in mathematical sociology. In the last ten years, control theorist have widely studied quantized versions of consensus algorithms. Since giving a complete overview would be impossible here, we just mention some papers whose approaches are particularly close to ours. First of all, in  the authors consider a discrete-time version of (2): their analysis is limited to observing that the algorithm may not converge to consensus and then abandoned. The poor perfomance of the dynamics in terms of approaching consensus explains the absence111A preliminary and incomplete study on the topics of this article was published in the Proceedings of the European Control Conference as , where only all-to-all communication was considered. of known results about (2). Instead, papers like [8, 16, 14, 18, 21] have considered other possible quantizations of (1): in [8, 16] all states are quantized in the right-hand side, while in [14, 18] distances between couples of states are seen through the quantizer. In these papers convergence to consensus is proved under appropriate but generally mild assumptions .
In the opinion dynamics literature, different explanations of persistence of disagreement have been proposed: among them, we recall in  the persisting influence of the initial opinions; in  the bounded confidence of the individuals; in [23, 1] the presence of stubborn individuals; and in  the occurrence of antagonistic interactions. Here, we support the claim that quantization can be a source of disagreement. The fact that others’ opinion are materialized by means of discrete behaviours is a natural observation, which has been made by social scientists and socio-physicists and has been addressed in a few models including [24, 22, 11] and [17, Chapter 10]. Discrete behaviours are the outcome of limited verbalization capabilities in  or represent actions taken by the indiduals (CODA models) in . These papers feature dynamics that may not reach consensus and that involve quantization together with other, possibly nonlinear, effects. In comparison, our proposed model (2) can be understood as an effort to single out the effects of quantization only. We consider the case of multiple quantization levels, but two-level quantization as in  can be easily obtained as a particular case.
Outline of the paper.
In Section 2 we present some alternative forms of (2), introduce Carathéodory and Krasowskii solutions and prove some of their basic properties. Section 3 is devoted to equilibria, providing the relevant definitions and examples. In Section 4 we prove the asymptotic estimate of the distance of solutions from consensus and show that it is sharp. In Section 5 we study the special cases of all-to-all and bipartite all-to-all communication and prove that in these cases consensus is achieved. Finally, Section 6 presents some simulations and Section 7 concludes the paper.
2 Fundamental properties of the dynamics
We begin this section by rewriting equations (2) in some alternative forms that allow us to see the model from different points of view.
First of all, we can think that communication among individuals is described by a weighted directed graph whose vertices are individuals. The numbers are the entries of the weighted adjaciency matrix of the graph, namely if the agents and are linked, if they are not linked and . We can then observe that
If we denote by the weighted degree of the -th vertex, by , and by , then we can define the Laplacian matrix of the graph and write
where for the second equality we have used and , together with the fact that . System (3) can be seen as the classical consensus system perturbed by others’ states quantization errors. This interpretation will become useful in Section 4. Clearly, we can also write (3) as
which prompts a connection with the theory of neural networks.
Remark 1 (Neural networks).
By adding a constant term in (4) we get
which is the typical system used in order to describe neural networks. In this context, individuals represent neurons and the function is called activation function. In the basic models is assumed to be smooth and increasing, but the literature has also considered weaker assumptions. In particular, in  can be discontinuous but it must be increasing and bounded, and the matrix is assumed invertible. A major difference between our model and the one in  is that we have multiple equilibria (Section 3), whereas  assumes a single equilibrium point.
When we look at system (4), we observe that for every the vector is constant on each set
This fact makes it evident that the system is affine if restricted to each set . Moreover its right-hand side is discontinuous on the set , where is the boundary of . In general, existence of solutions of equations with discontinuous right-hand side is not guaranteed. For this reason different types of solutions have been introduced in the literature (see ). The notion of solution nearest to the classical one is that of Carathéodory solution. The main problem in using Carathéodory solutions is that, often, they do not exist. Here we prove that Carathéodory solutions do exists and are complete. Nevertheless the discontinuity causes the strange phenomenon of solutions converging to points which are not equilibria. In order to better understand this phenomenon we also study Krasowskii solutions: such points are in fact Krasowskii equilibria.
We now formally introduce Carathéodory and Krasowskii solutions and study their basic properties. We refer the reader to [12, 19] for overviews on generalized solutions of discontinuous differential equations. System (2) can be cast in the general form
provided is the vector field given by . Let be an interval of the form . An absolutely continuous function is a Carathéodory solution of (6) if it satisfies (6) for almost all or, equivalently, if it is a solution of the integral equation
An absolutely continuous function is a Krasowskii solution of (6) if for almost every , it satisfies
We recall that in general any Carathéodory solution is also a Krasowskii solution and we observe that, for the specific dynamics (6), Krasowskii solutions coincide with Filippov solutions that are often adopted for discontinuous systems. The following result states the basic properties of the solutions of (6).
Theorem 1 (Properties of solutions).
(i-C) First of all, we remark that the right-hand side of (2) is continuous at any point in the interior of for any , then local solutions with initial conditions in do exist. Then, we consider initial conditions in . For any we denote by the subset of of the indices such that for some and by the cardinality of .
We first consider initial conditions such that and , i.e. for some and for any and any . Let us denote
We have that (the -th vector of the canonical basis)
If , then there is a solution starting at which satisfies the equations and stays in in an interval of the form for some . If , then there is a solution starting at which satisfies the equation and stays in in an interval of the form for some . Note in particular that if , then the vector field is tangent to not only in but also in a neighbourhood of intersecated with the discontinuity surface. In fact there exists a neighbourhood of such that for all one has for all . Moreover if one gets .
We now consider initial conditions such that . The vector field has limit values at corresponding to the sectors defined by the inequalities and , . We describe these sectors by means of vectors . Let if and if . We define if and if . Let . In the following we denote . We want to prove that there exists such that for all . This means that the vector field points inside the sector at .
Preliminarily, note that the th component of can be written as
Now, we start by considering the sector such that for all . If for all , we have finished. Otherwise, there exists such that . Assume without loss of generality that , i.e. . Then for all we have . In fact if then also .
We then examine only those such that . In particular the next we consider, which we call , is such that and for all other . If for all , then we have finished. Otherwise, there exists such that . Assume that such , i.e. . Then for all we have . We can then restrict our attention to those such that , and so forth. By proceeding in this way, in step at most we find the sector with the desired property.
As already mentioned, the meaning of the condition for all is that the vector field is directed inside . If points strictly inside , i.e. for all , then there is a solution that enters the sector. If instead some components of are null, is tangent to the boundary of , and, more precisely, to , where is the subset of such that if . Thanks to (8) there exists a neighbourhood of such that for all and for all , i.e. remains tangent to so that there exists a solution of (2), and such that and for almost every .
(i-K) Existence of Krasowskii solutions follows from the fact that the function is locally bounded.
(ii-C) Let be a Carathéodory solution and let be any index in such that . For , . Let . If , one has for all and then . This implies that is lower bounded by . The previous calculation also shows that the function is nondecreasing. Analogously, if is any index such that , , we get that is upper bounded by , and the fuction is nonincreasing.
(ii-K) Let now be a Krasowskii solution be defined as in (ii-C). As for Carathéodory solutions, if , one has for all and then while if , may be negative and then is lower bounded by and is upper bounded by .
(iii-C-K) Both Carathéodory and Krasowskii solutions can be continued up to thanks to local existence and boundedness of solutions. ∎
Remark 2 (Monotonicity and limit of minimum and maximum quantization level for Carathéodory solutions).
Many consensus-seeking dynamics enjoy the property that the smallest (largest) component is nondecreasing (nonincreasing). This fact is not true for system (2). Instead, we have shown in the proof of Theorem 1 that, for Carathéodory solutions, the smallest and the largest quantization levels and are nondecreasing and nonincreasing, respectively. Since they are bounded and take values in , it follows that they are definitively constant. For any Carathéodory solution of (2) there exist and such that for any and . This monotonicity of the extremal quantization levels does not hold for Krasowskii solutions, as can be seen from Example 4 in the next section.
Remark 3 (Infinite versus finite quantization levels).
The function takes values in , and this means that we a priori admit infinite types of behaviours. Nevertheless a consequence of the previous remark is that the number of quantization levels actually assumed in the evolution of the system is finite, once the initial condition is fixed. In particular, if the initial values of all the components are in the interval , the quantizer only takes the values and : in this way we get the case of binary behaviours.
The following example shows that in general uniqueness of solutions is not guaranteed.
Example 1 (Multiple Carathéodory solutions).
Consider the dynamics
with initial condition . There are two solutions issuing from this point, whose trajectories are the line segments joining the initial condition with the points and ; see Figure 2.
3 Equilibria and lack of consensus
We call a Carathéodory equilibrium of (6) if the function is a Carathédodory solution of (6), and we call a Krasowskii equilibrium of (6) if the function is a Krasowskii solution of (6). Carathéodory equilibria are found by looking for solutions of the equation , whereas Krasowskii equilibria are points such that . If we denote by the set of Carathéodory equilibria of (6) and by the set of Krasowskii equilibria, it is evident that . As an example of a Krasowskii equilibrium that is not a Carathéodory equilibrium we can take the point in Example 1. Actually points of the form are always Krasowskii equilibria of (2).
Let us call consensus point a point such that for all . Clearly, integer consensus points like with are Carathéodory equilibria of (2) because . Such a point belongs to the interior of and, consequently, is locally asymptotically stable. However, Carathéodory equilibria are not necessarily consensus points and may belong to the discontinuity set.
Example 2 (Non-consensus Carathéodory equilibrium).
Consider the system
The point is a Carathéodory equilibrium point which lies on the boundary of . The set is contained in the attraction region of , but is not Lyapunov stable.
This example also shows that solutions can converge to points that are not consensus points.
For smooth systems, if a (classical) solution converges to a point, then such point is a (classical) equilibrium. This property does not hold true for Carathéodory solutions of systems with discontinuous right-hand side, and, in particular, in the case of systems of the form (4). This motivates the following definition of extended equilibrium. Let and the vector field whose components are
Clearly coincides with on the set . We call extended equilibrium of (2) a point such that there exists such that and . We denote by the set of extended equilibria of (2). It is evident that . The following example shows that these inclusions are strict in general.
Example 3 (4-path graph: equilibria).
Let us consider (2) on a 4-node line graph:
The point is an extended equilibrium, because and . However, can not be a Carathéodory equilibrium, because .
The point is a Krasowskii equilibrium, as it can be computed . However, can not be an extended equilibrium, because can not be equal to any quantizer value, so that the first component of the vector field can not be zero in any neighbourhood of the point.
This example shows that extended equilibria include points, like , that are not consensus points. Interestingly, there exist Carathéodory solutions that asymptotically converge to : it is enough to take the solution issuing from an initial condition in . In spite of being attractive, is not a Carathéodory equilibrium and actually Carathéodory solutions originating from converge to . This pathological behaviour might appear surprising, but is allowed by the discontinuity of the vector field.
In the previous example one can find a Krasowskii solution that is not a Carathéodory solution: its slides on the discontinuity set and connects two equilibria.
Example 4 (4-path graph: Krasowskii trajectories).
Let us consider again the dynamics (3) on the 4-node path graph. Consider the parametrized segment for , which interpolates between the Krasowskii equilibria and . For every it holds that
Then there is a Krasowskii solution such that
for any there exists such that ,
Note that can not be a Carathéodory solution as .
This example also shows that the minimum quantizer level assumed along Krasowskii solutions may be decreasing (see Remark 2). In fact, if we compute for the Krasowskii solution showed above, then whereas if .
3.1 Path graph: equilibria
We now show, by means of an example, that equilibria can be significantly far from consensus: in the case the communication graph is a path, such distance can be proportional to the square of number of individuals.
In the case the graph is a path, equations (2) read
If is an extended equilibrium, then
Furthermore, there exists an extended equilibrium such that
Let us characterize extended equilibria of (3.1). Let . If is such that then
Then, if and only if the following inequalities are satisfied
Since , from the first inequality it follows that and from the last one it follows that . Moreover the inequalities with can be written in the form
From the left inequality we deduce that
and from the right inequality
These two inequalities imply that
that can be written as
Assuming without loss of generality that , we can deduce that
Let us distinguish the cases is even and is odd. If is even, , and . We have
If is odd, , and . We have
The first statement then follows from the fact .
On the other hand, if we select such that and
we do obtain an extended equilibrium that achieves the above bounds. ∎
This example shows that the system, provided is large enough, has extended equilibria that are arbitrarily far from the consensus. Figure 3 shows convergence to a non-consensus state.
4 Quantization as disturbance
A classical approach to obtain (conservative) results about quantized systems is to see the quantized dynamics as the perturbation of a “nominal” system, where the perturbation is due to the quantization error . This idea is also useful for our problem. In this section we get an asymptotic estimate of the distance of solutions from consensus. In order to prove it we need to assume the graph to be weight balanced. Below, we recall a few concepts of graph theory needed to make use of this assumption.
We have already defined . Let . The (directed weighted) graph with adjacency matrix is said to be weight-balanced if for all . Note that and if and only if the graph is weight-balanced. Given an edge , we shall refer to and to as the tail and the head of the edge, respectively. A path is an ordered list of edges such that the head of each edge is equal to the tail of the following one. The graph is said to be
strongly connected if for any there is an path from to ;
connected if there exists one node such that for any there is path from to ;
weakly connected if for each pair of nodes , one can construct a path which connects and by possibly reverting the direction of some edges.
These three notions of connectedness coincide when the graph is symmetric, that is when if and only if . On the contrary, weakly connected weight-balanced graphs are strongly connected [13, Proposition 2]. We recall the following result, which can be derived from [5, Theorem 1.37] and [13, Formula (1) and Section 2.2].
Let be the Laplacian matrix of a weight-balanced and weakly connected graph. Then:
The matrix is positive semi-definite.
Denoted by the smallest non-zero eigenvalue of ,
for all .
Theorem 4 (Convergence to a set).
Assume that the graph with adjacency matrix is weight balanced and weakly connected. If is any Carathéodory or Krasowskii solution of (2) and
then as .
We prove the statement for Krasowskii solutions, as Carathéodory solutions are a special case. Let . Then Consider the function . We have that