Fluctuation versus fixation in the one-dimensional constrained voter model

Fluctuation versus fixation in the one-dimensional constrained voter model

Abstract

The constrained voter model describes the dynamics of opinions in a population of individuals located on a connected graph. Each agent is characterized by her opinion, where the set of opinions is represented by a finite sequence of consecutive integers, and each pair of neighbors, as defined by the edge set of the graph, interact at a constant rate. The dynamics depends on two parameters: the number of opinions denoted by and a so-called confidence threshold denoted by . If the opinion distance between two interacting agents exceeds the confidence threshold then nothing happens, otherwise one of the two agents mimics the other one just as in the classical voter model. Our main result shows that the one-dimensional system starting from any product measures with a positive density of each opinion fluctuates and clusters if and only if . Sufficient conditions for fixation in one dimension when the initial distribution is uniform and lower bounds for the probability of consensus for the process on finite connected graphs are also proved.

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Constrained voter model \runauthorN. Lanchier and S. Scarlatos \addressSchool of Mathematical and Statistical Sciences
Arizona State University
Tempe, AZ 85287, USA. \addressDepartment of Mathematics
University of Patras
Patras 26500, Greece.

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Interacting particle systems, constrained voter model, fluctuation, fixation.

1 Introduction

The constrained voter model has been originally introduced in [9] to understand the opinion dynamics in a spatially structured population of leftists, centrists and rightists. As in the popular voter model [3, 5], the individuals are located on the vertex set of a graph and interact through the edges of the graph at a constant rate. However, in contrast with the classical voter model where, upon interaction, an individual adopts the opinion of her neighbor, it is now assumed that this imitation rule is suppressed when a leftist and a rightist interact. In particular, the model includes a social factor called homophily that prevents agents who disagree too much to interact.

Model description – This paper is concerned with a natural generalization of the previous version of the constrained voter model that includes an arbitrary finite number of opinions and a so-called confidence threshold . Having a connected graph representing the network of interactions, the state at time is a spatial configuration

Each individual looks at each of her neighbors at rate one that she imitates if and only if the opinion distance between the two neighbors is at most equal to the confidence threshold. Formally, the dynamics of the system is described by the Markov generator

where configuration is obtained from by setting

and where means that the two vertices are connected by an edge. Note that the basic voter model and the original version of the constrained voter model including the three opinions leftist, centrist and rightist can be recovered from our general model as follows:

The main question about the constrained voter model is whether the system fluctuates and evolves to a global consensus or fixates in a highly fragmented configuration. To define this dichotomy rigorously, we say that fluctuation occurs whenever

(1)

and that fixation occurs if there exists a configuration such that

(2)

In other words, fixation means that the opinion of each individual is only updated a finite number of times, therefore fluctuation (1) and fixation (2) exclude each other. We define convergence to a global consensus mathematically as a clustering of the system, i.e.,

(3)

Note that, whenever , the process reduces to the basic voter model with instead of two different opinions for which the long-term behavior of the process is well known: the system on lattices fluctuates while the system on finite connected graphs fixates to a configuration in which all the individuals share the same opinion. In particular, the main objective of this paper is to study fluctuation and fixation in the nontrivial case when .

Figure 1: Two typical realizations of the constrained voter model on the torus with 600 vertices for two different pairs of parameters. Time goes down from time 0 to time 3000. The black lines represent the boundaries between the different domains, that is the edges that connect individuals with different opinions.

Main results – Whether the system fluctuates or fixates depends not only on the two parameters but also on the initial distribution. In particular, we point out that, throughout the paper, it will be assumed that the initial distribution is the product measure with constant densities. To avoid trivialities, we also assume that the initial density of each of the opinions is positive:

(4)

For the constrained voter model on the one-dimensional torus with vertices, the mean-field analysis in [9] suggests that, in the presence of three opinions and when the threshold is equal to one, the average domain length at equilibrium is

(5)

when the initial density of centrists is small and is large. Vázquez et al. [9] also showed that these predictions agree with their numerical simulations from which they conclude that, when the initial density of centrists is small, the system fixates with high probability in a frozen mixture of leftists and rightists. In contrast, it is conjectured in [1] based on an idea in [7] that the infinite system fluctuates and clusters whenever , which includes the threshold one model with three opinions introduced in [9]. To explain this apparent disagreement, we first observe that, regardless of the parameters, the system on finite graphs always fixate and there is a positive probability that the final configuration consists of a highly fragmented configuration, thus showing that spatial simulations of the necessarily finite system are not symptomatic of the behavior of its infinite counterpart. Our first theorem shows that the conjecture in [1] is indeed correct.

Theorem 1

– Assume (4) and . Then,

  1. The process on fluctuates (1) and clusters (3).

  2. The probability of consensus on any finite connected graph satisfies

The intuition behind the proof is that, whenever , there is a nonempty set of opinions which are within the confidence threshold of any other opinions. This simple observation implies the existence of a coupling between the constrained and basic voter models, which is the key to proving fluctuation. The proof of clustering is more difficult. It heavily relies on the fact that the system fluctuates but also on an analysis of the interfaces of the process through a coupling with a certain system of charged particles. In contrast, our lower bound for the probability of consensus on finite connected graphs relies on techniques from martingale theory. Note that this lower bound is in fact equal to the initial density of individuals who are in the confidence threshold of any other individuals in the system. Returning to the relationship between finite and infinite systems, we point out that the simulation pictures of Figure 1, which show two typical realizations of the process on the torus under the assumptions of the theorem, suggest fixation of the infinite counterpart in a highly fragmented configuration, in contradiction with the first part of our theorem, showing again the difficulty to interpret spatial simulations. Note also that, for the system on the one-dimensional torus with vertices, the average domain length at equilibrium is bounded from below by

which, together with the second part of the theorem, proves that the average domain length scales like the population size when and that (5) does not hold. While our fluctuation-clustering result holds regardless of the initial densities provided they are all positive, whether fixation occurs or not seems to be very sensitive to the initial distribution. Also, to state our fixation results and avoid messy calculations later, we strengthen condition (4) and assume that

(6)

The next theorem looks at the fixation regime in three different contexts.

Theorem 2

– Assume (6). Then, the process on fixates (2) in the following cases:

  1. and is small enough.

  2. is large, and where

  3. and and .

The first part of the theorem is the converse of the first part of Theorem 1, thus showing that the condition is critical in the sense that

  • when , the one-dimensional constrained voter model fluctuates when starting from any nondegenerate distributions (4) whereas

  • when , the one-dimensional constrained voter model can fixate even when starting from a nondegenerate distribution (4).

The last two parts of the theorem specialize in two particular cases. The first one looks at uniform initial distributions in which all the opinions are equally likely. For simplicity, our statement focuses on the fixation region when the parameters are large but our proof is not limited to large parameters and implies more generally that the system fixates for all pairs of parameters corresponding to the set of white dots in the phase diagram of Figure 2 for the one-dimensional system with up to twenty opinions. Note that the picture suggests that the process starting from a uniform initial distribution fixates whenever even for a small number of opinions. The second particular case returns to the slightly more general initial distributions (6) but focuses on the threshold one model with four opinions for which fixation is proved when is only slightly less than one over the number of opinions = 0.25. This last result suggests that the constrained voter model with four opinions and threshold one fixates when starting from the uniform product measure, although the calculations become too tedious to indeed obtain fixation when starting from this distribution.

2468104681012141618202121416182000votermodelTHM 1THM 2.aTHM 2.bTHM 2.c
Figure 2: Phase diagram of the one-dimensional constrained voter model in the plane along with a summary of our theorems. The black dots correspond to the set of parameters for which fluctuation and clustering are proved whereas the white dots correspond to the set of parameters for which fixation is proved.

Structure of the paper – The rest of the article is devoted to the proof of both theorems. Even though our proof of fluctuation-clustering and fixation differ significantly, a common technique we introduce to study these two aspects for the one-dimensional process is a coupling with a certain system of charged particles that keeps track of the discrepancies along the edges of the graph rather than the actual opinion at each vertex. In contrast, our approach to analyze the process on finite connected graphs is to look at the opinion at each vertex and use, among other things, the optimal stopping theorem for martingales. The coupling with the system of charged particles is introduced in section 2 and then used in section 3 to prove Theorem 1. The proof of Theorem 2 is more complicated and carried out in the last five sections 48. In addition to the coupling with the system of charged particles introduced in the next section, the proof relies on a characterization of fixation based on so-called active paths proved in section 4 and large deviation estimates for the number of changeovers in a sequence of independent coin flips proved in section 5.

2 Coupling with a system of charged particles

To study the one-dimensional system, it is convenient to construct the process from a graphical representation and to introduce a coupling between the process and a certain system of charged particles that keeps track of the discrepancies along the edges of the lattice rather than the opinion at each vertex. This system of charged particles can also be constructed from the same graphical representation. Since the constrained voter model on general finite graphs will be studied using other techniques, we only define the graphical representation for the process on , which consists of the following collection of independent Poisson processes:

  • for each , we let be a rate one Poisson process,

  • we denote by its th arrival time.

This collection of independent Poisson processes is then turned into a percolation structure by drawing an arrow at time and, given a configuration of the one-dimensional system at time , we say that this arrow is active if and only if

The configuration at time is then obtained by setting

(7)

An argument due to Harris [4] implies that the constrained voter model starting from any configuration can indeed be constructed using this percolation structure and rule (7). From the collection of active arrows, we construct active paths as in percolation theory. More precisely, we say that there is an active path from to , and write , whenever there exist

such that the following two conditions hold:

  1. For , there is an active arrow from to at time .

  2. For , there is no active arrow that points at .

Note that conditions 1 and 2 above imply that

Moreover, because of the definition of active arrows, the opinion of vertex at time originates and is therefore equal to the initial opinion of vertex so we call vertex the ancestor of vertex at time . One of the key ingredients to studying the one-dimensional system is to look at the following process defined on the edges: identifying each edge with its midpoint, we set

and think of edge as being

  • empty whenever ,

  • occupied by a pile of particles with positive charge whenever ,

  • occupied by a pile of particles with negative charge whenever .

The dynamics of the constrained voter model induces evolution rules which are again Markov on this system of charged particles. Assume that there is an arrow at time and

indicating in particular that there is a pile of particles with positive charge at . Then, we have the following alternative:

425425124142545444
Figure 3: Schematic illustration of the coupling between the constrained voter model and the system of charged particles along with their evolution rules. In our example, the threshold which makes piles of three or more particles blockades with frozen particles and piles of two or less particles live edges with active particles.
  • There is no particle at edge or equivalently in which case the individuals at vertices and already agree so nothing happens.

  • There is a pile of particles at edge in which case and disagree too much to interact so nothing happens.

  • There is a pile of particles at in which case

    In particular, there is no more particles at edge and a pile of particles all with the common charge at edge .

Similar evolution rules are obtained by exchanging the direction of the interaction or by assuming that we have from which we can deduce the following description:

  • piles with more than particles cannot move therefore we call such piles blockades and the particles they contain frozen particles.

  • piles with at most particles jump one step to the left or one step to the right at the same rate one therefore we call the particles they contain active particles.

  • when a pile with positive/negative particles jumps onto a pile with negative/positive particles, positive and negative particles annihilate by pair which results in a smaller pile of particles all with the same charge.

We refer to Figure 3 for an illustration of these evolution rules. Note that whether an arrow is active or not can also be characterized from the state of the edge process:

In particular, active arrows correspond to active piles of particles.

3 Proof of Theorem 1

The key ingredient to proving fluctuation of the one-dimensional system and estimating the probability of consensus on finite connected graphs is to partition the opinion set into two sets that we shall call the set of centrist opinions and the set of extremist opinions:

Note that the assumption implies that the set of centrist opinions is nonempty. Note also that both sets are characterized by the properties

(8)

as shown in Figure 4 which gives a schematic illustration of the partition. Fluctuation is proved in the next lemma using this partition and relying on a coupling with the voter model.

Lemma 3

– The process on fluctuates whenever and .

Proof.

It follows from (8) that centrist agents are within the confidence threshold of every other individual. In particular, for each pair we have the transition rates

(9)

and similarly

(10)

Now, we introduce the process

Since for all the transition rates are constant over all according to (9), we have the following local transition rate for this new process:

Using (10) in place of (9) and some obvious symmetry, we also have

This shows that the spin system reduces to the voter model. In particular, the lemma directly follows from the fact that the one-dimensional voter model itself, when starting with a positive density of each type, fluctuates, a result proved based on duality in [7], pp 868–869. ∎

12threshold threshold centrist extremist extremist
Figure 4: Partition of the opinion set.
Lemma 4

– The process on clusters whenever and .

Proof.

The proof strongly relies on the coupling with the voter model in the proof of the previous lemma. To begin with, we define the function

which, in view of translation invariance of the initial configuration and the evolution rules, does not depend on the choice of . Note that, since the system of charged particles coupled with the process involves deaths of particles but no births, the function is nonincreasing in time, therefore it has a limit: as . Now, on the event that an edge is occupied by a pile of at least one particle at a given time , we have the following alternative:

  • is a blockade. In this case, since the centrist agents are within the confidence threshold of all the other agents, we must have

    But since the voter model fluctuates,

    In particular, at least of one of the frozen particles at is killed eventually.

  • is a live edge. In this case, since one-dimensional symmetric random walks are recurrent, the active pile of particles at eventually intersects another pile of particles, and we have the following alternative:

    • The two intersecting piles of particles have opposite charge, which results in the simultaneous death of at least two particles.

    • The two intersecting piles have the same charge and merge to form a blockade in which case we are back to the previous case: since the voter model fluctuates, at least one of the frozen particles in this blockade is killed eventually.

    • The two intersecting piles have the same charge and merge to form a larger active pile in which case the pile keeps moving until, after a finite number of collisions, we are back to one of the previous two possibilities: at least two active particles annihilate or there is creation of a blockade with at least one particle that is killed eventually.

In either case, as long as there are particles, there are also annihilating events indicating that the density of particles is strictly decreasing as long as it is positive. In particular, the density of particles decreases to zero so there is extinction of both the active and frozen particles:

In particular, for all with , we have

which proves clustering. ∎



The second part of the theorem, which gives a lower bound for the probability of consensus of the process on finite connected graphs, relies on very different techniques, namely techniques related to martingale theory following an idea from [6], section 3. However, the partition of the opinion set into centrist opinions and extremist opinions is again a key to the proof.

Lemma 5

– For the process on any finite connected graph,

Proof.

The first step is to prove that the process that keeps track of the number of supporters of any given opinion is a martingale. Then, applying the martingale convergence theorem and optimal stopping theorem, we obtain a lower bound for the probability of extinction of the extremist agents, which is also a lower bound for the probability of consensus. For , we set

and we observe that

(11)

Letting denote the natural filtration of the process, we also have

indicating that the process is a martingale with respect to the natural filtration of the constrained voter model. This, together with (11), implies that also is a martingale. It is also bounded because of the finiteness of the graph therefore, according to the martingale convergence theorem, there is almost sure convergence to a certain random variable:

and we claim that can only take two values:

(12)

To prove our claim, we note that, invoking again the finiteness of the graph, the process gets trapped in an absorbing state after an almost surely stopping time so we have

Assuming by contradiction that gives an absorbing state with at least one centrist agent and at least one extremist agent. Since the graph is connected, this implies the existence of an edge such that

but then we have and

showing that is not an absorbing state, in contradiction with the definition of time . This proves that our claim (12) is true. Now, applying the optimal stopping theorem to the bounded martingale and the almost surely finite stopping time and using (12), we obtain

from which it follows that

(13)

To conclude, we observe that, on the event that , all the opinions present in the system after the hitting time are within distance of each other therefore the process evolves according to a voter model after that time. Since the only absorbing states of the voter model on finite connected graphs are the configurations in which all the agents share the same opinion, we deduce that the system converges to a consensus. This, together with (13), implies that

This completes the proof of the lemma. ∎

4 Sufficient condition for fixation

The main objective of this section is to prove a sufficient condition for fixation of the constrained voter model based on certain properties of the active paths.

Lemma 6

– For all , let

Then, the constrained voter model fixates whenever

(14)
Proof.

This is similar to the proof of Lemma 2 in [2] and Lemma 4 in [8]. To begin with, we define recursively a sequence of stopping times by setting

In other words, the th stopping time is the th time the individual at the origin changes her opinion. Now, we define the following random variables and collection of events:

The assumption (14) together with reflection symmetry implies that the event occurs almost surely for some positive integer , which implies that

Since is the event that the individual at the origin changes her opinion infinitely often, in view of the previous inequality, in order to establish fixation, it suffices to prove that

(15)

To prove equations (15), we let

be the set of descendants of at time which, due to one-dimensional nearest neighbor interactions, is necessarily an interval and its cardinality, respectively. Now, since each interaction between two individuals is equally likely to affect the opinion of each of these two individuals, the number of descendants of any given site is a martingale whose expected value is constantly equal to one. In particular, the martingale convergence theorem implies that

therefore the number of descendants of converges to a finite value. Since in addition the number of descendants is an integer-valued process,

which further implies that, with probability one,

(16)

Finally, we note that, on the event , the last time the individual at the origin changes her opinion is at most equal to the largest of the stopping times for therefore

according to (16). This proves (15) and the lemma. ∎

5 Large deviation estimates

In order to find later a good upper bound for the probability in (14) and deduce a sufficient condition for fixation of the process, the next step is to prove large deviation estimates for the number of piles with  particles with a given charge in a large interval. More precisely, the main objective of this section is to prove that for all and all the probability that

decays exponentially with . Note that, even though the initial opinions are chosen independently, the states at different edges are not independent. For instance, a pile of particles with a positive charge is more likely to be surrounded by negative particles. In particular, the result does not simply follow from large deviation estimates for the binomial distribution. The main ingredient is to first show large deviation estimates for the number of so-called changeovers in a sequence of independent coin flips. Consider an infinite sequence of independent coin flips such that

where is the outcome: heads or tails, at time . We say that a changeover occurs whenever two consecutive coin flips result in two different outcomes. The expected value of the number of changeovers before time can be easily computed by observing that

and by using the linearity of the expected value:

Then, we have the following large deviation estimates for the number of changeovers.

Lemma 7

– For all , there exists such that

Proof.

To begin with, we let be the time to the th changeover and notice that, since all the outcomes between two consecutive changeovers are identical, the sequence of coin flips up to this stopping time can be decomposed into strings with an alternation of strings with only heads and strings with only tails followed by one more coin flip. In addition, since the coin flips are independent, the length distribution of each string is

and lengths are independent. In particular, is equal in distribution to the sum of independent geometric random variables with parameters and , namely, we have

(17)

Now, using that, for all ,

and large deviation estimates for the binomial distribution implies that

(18)

for a suitable constant and all large. Similarly,

(19)

for a suitable and all large. Combining (17)–(19), we deduce that

Taking and observing that , we deduce

for a suitable and all large. In particular, for all sufficiently large,

for suitable constants and and all sufficiently large. Using the previous two inequalities and the fact that the event that the number of changeovers is equal to is also the event that the time to the th changeover is less than but the time to the next changeover is more than , we conclude that

for all sufficiently large. This completes the proof. ∎



Now, we say that an edge is of type if it connects an individual with initial opinion on the left to an individual with initial opinion on the right, and let

denote the number of edges of type in the interval . Using the large deviation estimates for the number of changeovers established in the previous lemma, we can now deduce large deviation estimates for the number of edges of each type.

Lemma 8

– For all , there exists such that

Proof.

For any given , the number of edges and with has the same distribution as the number of changeovers in a sequence of independent coin flips of a coin that lands on heads with probability . In particular, applying Lemma 7 with gives

(20)

for all sufficiently large. In addition, since each preceding a changeover is independently followed by any of the remaining opinions,