Voter Model with Arbitrary Degree Dependence: Clout, Confidence and Irreversibility
In this paper, we consider the voter model with popularity bias. The influence of each node on its neighbors depends on its degree. We find the consensus probabilities and expected consensus times for each of the states. We also find the fixation probability, which is the probability that a single node whose state differs from every other node imposes its state on the entire system. In addition, we find the expected fixation time. Then two extensions to the model are proposed and the motivations behind them are discussed. The first one is confidence, where in addition to the states of neighbors, nodes take their own state into account at each update. We repeat the calculations for the augmented model and investigate the effects of adding confidence to the model. The second proposed extension is irreversibility, where one of the states is given the property that once nodes adopt it, they cannot switch back. The dynamics of densities, fixation times and consensus times are obtained.
To study social systems, one can focus on collective large-scale social phenomena (macro behavior) or on the behavior of individuals and their interactions (micro behavior). Macro social behaviors are simultaneously consequences and determinants of micro behaviors. For example, culture is a product of many individual actions, and in turn affects the action of each individual. The recent upsurge in network science has led to a panoply of models of opinion dynamics. These models are agent-based and seek to unify the micro-macro duality [1, 2, 3, 4, 5, 6]. Contributions and applications also exist in diverse strands of research such as economics [7, 8], biology [9, 10] and physics [11, 12, 13].
In models of opinion dynamics, each node has a ‘state’. This state can be continuous [14, 15, 16, 17], or discrete [18, 19, 20]. One of the most studied models is the voter model, where states can take only two possible values. The voter model was first introduced in  (hereafter, we call the version of the voter model in  the ‘pure’ voter model). It is a simple stochastic process which models dynamics of dissensus and consensus between agents (hereafter, nodes). One rationale behind this model is the fact that people are evidently influenced by their peers when they make decisions , and that their observations and interactions with others can affect their behavior remarkably .
In the voter model, nodes are endowed with dichotomous states, typically denoted by . This simplified representation is applicable to situations where there are two choices to take, e.g., seeing or not seeing a film, choosing between two major political stances, and whether or not adopt a new product or technology. Time increments are discrete. At each timestep, one node is randomly selected to update its state. A fraction of the neighbors of this node adopt state and another fraction adopt state at that instant. These fractions constitute two probabilities, and the node adopts one of the states with respective probabilities.
The voter model has been studied on lattices in different dimensions  and arbitrary heterogeneous networks [24, 19]. Typical lines of inquiry include the probability that the system will reach eventual unanimity on either of the states, and the expected time it takes to reach those states. This model has been also generalized to include three states . In [20, 26, 27], the existence of stubborn nodes are considered. These nodes never alter their states. In , nodes have inertia. The longer a node stays with the same state, the less likely it gets for that node to change it. In , time-dependent transition rates are considered through a noise-reduction scheme. In [30, 31], nodes have heterogeneous conviction (or persuasion). In [32, 33, 34] an ‘adaptive’ network is envisaged, where links whose incident nodes are of opposing states are rewired so that the new link has two agreeing nodes at its ends. In , a popularity bias is incorporated into the model. At each timestep, a link is selected and then with probabilities that depend on the degrees of incident nodes to that link, one of them imposes its state on the other one. The information that is considered to be known about the topology of the underlying network varies among models. For example, in [19, 20, 5] the adjacency matrix is assumed to be known completely, whereas in [24, 35, 36] only some statistics of the network are known, such as the degree distribution, in addition to the assumption that the network is connected (see  for a thorough discussion).
In this paper, we consider that each node assigns ‘weights’ to the state of its neighbors (equivalent to [35, 37]). If a neighbor has degree , the weight its state will have is denoted by . This extension could be used to emulate the idea that in social settings, the nodes with higher degrees are more ‘central’ and cast more influence on the decision making of their peers. We find the probability of consensus on either state, expected time to reach unanimity, and the expected time to reach consensus conditional on the final state. Also, we find the ‘fixation probability’, which is the probability that a system in which all nodes are unanimous except one anti-conformist node will eventually reach consensus on the state adopted by that single node.
The pure voter model lacks an important element present in realistic social settings, which is confidence. In this paper, we endow the nodes with confidence. This means that each node, in addition to the state of its neighbors, accounts for its own (weighted) state as well. This extension is proposed to remedy a peculiarity of the voter model. Consider a star graph, where the central node has state and degree . Let all the peripheral nodes have state . In the pure voter model, the probabilities of reaching consensus on either or for this system are both equal to . This is at odds with the intuitive expectation from social interactions. In social networks, opinion leaders are typically characterized by such centrality. Such central nodes usually have ‘clout’, and influence others’ opinions heavily. In the degree-dependent voter model without confidence, this problem persists, because all neighbors of the central node have the same degree, and weighting them as a function of degree will give identical weights. However, as shall be described, if we endow nodes with confidence, so that they account for their own opinions as well, this problem is resolved and the central node will have a higher chance of imposing its state, compared to the peripheral nodes.
Another scenario that the pure voter model is not applicable to is evoked in marketing applications. Let nodes with state represent those who have seen a film, and nodes with state represent those who have not. In this case, the voter model must be modified, since a node with state cannot ‘un-see’ a film and flip back to . To emulate this setting, we propose an ‘irreversible’ version of the voter model, where nodes with state cannot flip back to (note that the case with no degree-dependence is akin to the SI model of disease epidemics ). We initiate the system with a certain fraction of nodes adopting state at the outset (which in realistic cases is typically small, because most of the population have not seen the film), and study the dynamics of the system. We find the time required for the system to reach unanimity, and the time it takes for a single node with state to impose it on the entire network.
Ii The Model
Let the nodes be connected on the graph . We do not assume that is known. We assume that is connected and that the degree distribution is known. Let us denote the state of node at time by . States are binary, so that . States are updated asynchronously. We take the time unit to be equal to , where is the number of nodes. Consequently, at each timestep, on average all nodes are selected once to update their states. At each timestep, node calculates probabilities to adopt each of the two states at the next timestep. Denote the probability that node will adopt and at time by and , respectively. Let denote the set of neighbors of node . Also let and denote the set of neighbors of with state and respectively. For any node , let denote its degree, i.e., the number of its neighbors. The update scheme is modeled as follows:
Note that if is a constant, then these probabilities become the fraction of nodes adopting each of the states, which is the case for the pure voter model .
We assume that the underlying graph is connected. We analyse the model within the framework used in [24, 35], which is observed to perform remarkably well for scale-free graphs with negligible degree correlations [36, 24, 37].
Throughout, the total number of nodes is denoted by . We denote the fraction of all nodes that are adopting the state at time by . Number of nodes with degree is denoted by , and the fraction of nodes with degree is denoted by . So denotes the degree distribution of the graph. The fraction of nodes with degree whose state at time is is denoted by .
Denote the expected value of the state of node at time by . This quantity can be obtained by the weighted average of the neighbors of node in the following way
Let denote the network average of the quantity , that is, . For example, denotes the average degree of the graph. Now we find the expected value of the numerator and the denominator of (2), assuming that the degree-degree correlations of the underlying graph is negligible, which is true for random graphs realized, for example, in [38, 39, 40], and works for a broad range of heterogeneous networks . For the denominator, we have
For the numerator, we have
Let us define the following weighted density
So (4) can be rewritten as follows
Let us multiply both sides by and then sum over all . On the right hand side we will have the sum , which is equal to . On the left hand side we will have the sum . This sum can be expressed equivalently as follows
So after multiplying by the factor and summing over all , Equation (7) transforms into the following
This implies that , so that the quantity is conserved.
Ii-a Consensus Probability
The conservation of leads us to the probability of eventual consensus on each of the states. Let us define the consensus probabilities
At the value of is the same as that at . When the system is at state , then the value of is equal to . When is zero, then the value of is equal to zero. From conservation of , we have:
For the probability of eventual consensus on state we obtain
As (8) indicates, we can express in terms of the initial conditions as follows
Plugging this expression into (12), we find the following alternative form of the eventual consensus probability
Similarly, the eventual consensus probability on state is obtained
In Fig. 1, theoretical predictions are compared to simulation results. The underlying graph is one obtained by the preferential attachment scheme proposed in , which yields graphs with power-law degree distributions with exponent . We used , which means that in the sequential construction of the graph, each new node attaches to existing nodes with a mechanism described in . A network of size is used for simulations. The special case of is considered as an example.
Ii-B Time to Consensus
In this section we will first focus on the expected time to reach unanimity. We denote this time by . In the next section we will turn our attention to the expected time to reach consensus on state and that to reach state . We denote these two quantities by and , respectively.
We begin by introducing the increment and decrement probabilities for nodes of degree . The increment probability is the probability that a node of degree flips its state from to , and the decrement probability is defined vice versa. When these events occur, the fraction of nodes of degree with state changes by . We define
Let us represent the state of the system by a vector , where encodes the densities of sub-populations of nodes with different degrees. Let denote the unit vector along the -th dimension. We denote the expected time to reach unanimity by . If a node of degree flips its state, then will vary by . Let us denote this change by .
The following recurrence relation holds for the expected unanimity time:
This equation relates for densities at two successive time steps; the time-dependence is omitted in the expression to improve readability. The first sum on the right hand side accounts for the case that an increment or decrement occurs. The second sum refers to the case where no change occurs.
Now let us consider the Taylor expansion of (17) up to second order. After some algebraic steps, we obtain
We can simplify this equation further. Using the chain rule and rearranging the terms, we arrive at the following equation
To continue, we need the increment and decrement probabilities. For the increment probability, we have
The factor is the portion of all nodes who have degree and state . The factor that multiplies this fraction is the probability that a node whose state is switches to state . Similarly, for the decrement probability we have
For the first term on the left hand side, note that the following is true
So (22) simplifies to
To simplify this equation further, note that the expected change in is . So we have
Integrating this equation yields
This means that the deviation of from decays exponentially fast in time. This happens for all values of . After this rapid change, the dynamics of the system enters a more slowly-varying phase [24, 35].
Using this fact, we can approximate (24) by confining the range of time to the second phase of the dynamics.
So the differential equation for the unanimity time (24) transforms into the following
Let us define the new variables
In terms of the new variables, (27) transforms into the following
In order to solve this second-order differential equation we require two boundary conditions. Note that when all nodes are at state , i.e., when is equal to unity, then will be zero. Also when all nodes have state , which means that is zero, then will be equal to zero. Thus the two boundary conditions are . Integrating (29) twice and applying these boundary conditions, we obtain
Replacing by its explicit expression given in (28), we obtain
Ii-C Time to Reach Consensus Conditional on the Final State
First let us focus on the expected time to reach consensus on the state , which we denote by . Since this time is conditional upon the eventual consensus being on state , we have to make adjustments to (17). The recurrence equation becomes:
Let us define
Then (32) reduces to the following equation
We proceed by restricting the time domain within the second phase of the dynamics similar to the previous stage. Thus (34) simplifies to the following
Integrating this equation twice and dividing by , we obtain
where and are integration constants. The first boundary condition is , which gives . So we obtain
Note that symmetry of the dynamics readily determines the following result
To determine note that the following holds
Plugging in the results obtained in (38) and (39) into the left hand side and using the expression for in (30) on the right hand side, we obtain . After replacing the explicit expression for given in (28), we arrive at the conditional consensus times
Theoretical predictions are presented along with simulation results in Fig. 3.
Ii-D Example: Linear Clout
Now let us focus on a simple case of that can encode popularity bias, namely . Popularity bias means that when node is updating its state, the higher the degree of a neighbor is, the more influence that neighbor will have on the state to be adopted by node . Linear popularity bias is referred to as linear clout hereafter
As an example, let us investigate the dependence of the expected time to reach unanimity for the special case of graphs with power-law degree distribution whose degree-degree correlations are negligible (through the recipe articulated in [38, 39, 40], for example). For these networks, we have
From (31) we see that the quantities and are required to study the behavior of expected unanimity time. For a network with power-law degree distribution, first we estimate the maximum degree, denoted by , through a heuristic method used in [24, 42, 40]. For we have:
This gives . We focus on the scale-free family of graphs introduced in  where . In this case, we have . So for the numerator of (31) we have . Similarly, for the denominator of (31) we obtain . Inserting these limits into (31), we arrive at
Ii-E Fixation Probability
Now let us focus on the fixation probability, namely the probability of the system reaching consensus on state , for the initial condition of a single node at state and all other nodes at state . In social contexts, the fixation probability quantifies the likelihood of the emergence of a leader or the takeover of a minority. In linguistics, it quantifies the probability that a singular way of pronouncing a word overspreads the population .
We refer to the single deviant node as the mutant. Suppose that the degree of this node is . This means that . Let denote the fixation probability for a mutant with degree . Then (12) gives the probability of eventual consensus over as follows
To proceed, we consider the case of linear clout over the uncorrelated networks with power-law degree distribution . We focus on the case networks grown by preferential attachment as described in , for which we have . For the denominator of (45), similar to Section II-D, we have . Assume that the mutant has a small degree, namely . Then for the fixation probability we obtain.
This probability indicates that for large system sizes, the chance for the system to fixate on the state imposed by a single mutant are scant. However, as can be seen from Fig. 5, this probability is less than that of the pure voter model, for which .
Iii The Role of Confidence
As discussed in the introduction, in the pure voter model and also the generalization we considered above, nodes do not account for their own state when they are updating their states. One drawback that stems from this characteristic is the failure to incorporate the model with confidence. In social settings, nodes have different levels of confidence and persuasiveness, which gives rise to role structures such as hierarchy of influence [44, 45]. To be able to model those phenomena, the model of opinion dynamics must incorporate confidence. Here we discuss a generalization to the voter model in order to address this issue.
As an example, consider a star graph where the central node has degree . The central node has state and those on the periphery have state . Suppose nodes do not have confidence, so that each node updates its state according to (1). Consider the case of for expository brevity. In  it is contended that the system reaches consensus over or with equal probability . Now let us add confidence to nodes, which means that they also account for their own states. Now if the central node is chosen to update its state, then the probability of not changing state is , higher than that of switching, whereas the peripheral nodes adopt the state of the central node with certainty if they are chosen to update state. This elevates the chance of the central node to impose its state over the system above , which is closer to the intuitive expectation that central nodes have higher influence.
We augment the model by assuming that in the update process, each agent also accounts for its own state. In the pure voter model, this would not change the dynamics significantly, but if there is degree-dependent weighting, then nodes with large degrees will assign a large weight to their own state, which models hierarchy in social influence.
We begin by modifying (2) to account for self-influence as follows:
Multiplying both sides by the denominator of the right hand side, and then multiplying both sides by , we get
Now define (the definition for is repeated here for convenience of reference)
Now let us sum up (49) over all and then divide both sides of the equation by . We get
This can be simplified to
So the quantity is conserved throughout the dynamics. This immediately leads us to the consensus probability over each of the states.
Iii-a Consensus Probability
Note that when all nodes have state , then takes the value . Using the conservation of , we find that the consensus probability over state satisfies
So we obtain
We can use the explicit form for the numerator and obtain the following expression for the consensus probability as a function of initial conditions
Instead of sum over all degrees, we can express this result as a sum over all individual nodes. The result is
Similarly, for consensus over we have
Iii-B Time to Consensus
In this section, we will take the similar steps undertaken in Section II-B to obtain the expected time to reach unanimity.
Let us define two distinct sums:
For the increment probability, we have to make adjustments to (20) as follows
Similarly, for the decrement probability we have
From the increment and decrement probabilities, we obtain
Inserting this expression in and also using the fact that , we obtain
By substituting by its explicit expression given in (50), we can rewrite the numerator of the second factor of the summand as follows
Using this expression, (63) can be equivalently expressed as follows
Using the chain rule, we have
Note that the summand is anti-symmetric with respect to the exchange of the indices and . So summing over all values of and yields zero. So we have .
Returning to (58), we have the following simplified differential equation for the expected time to reach unanimity
Employing the explicit expression for as given in (50), the numerator can be simplified as follows
So (70) becomes
Plugging this into (69), we get
To proceed, we confine ourselves to the case of linear clout, namely . In this case, from (50) we can see that and are the same, so is conserved. We can use the chain rule to transfer the differentiation to . Then (73) transforms into
Following the identical steps that led to (31), we obtain
This is longer than the consensus time given by (31), due to the extra term in the numerator that stems from the confidence of nodes.
Theoretical prediction and simulation results are presented in Fig. 7. The expected consensus time for the absence of confidence is also depicted, and it can be seen that confidence slows down the convergence of the system towards unanimity, as predicted by (76).
For the expected time to consensus conditional on the final state, similar steps that led to (41) apply here, and we arrive at
Theoretical predictions for the conditional consensus times are presented along with simulation results in Fig. 8.
Iv Irreversible Dynamics
As discussed in the introduction, the voter model is not applicable to marketing problems where once a node sees a film (correspondingly, adopts ), it cannot go back, i.e., it cannot un-see it. Here we consider an irreversible version of the voter model. The probability of an increment in the population of adopters of state with degree is
So for the evolution of densities we have
Taking the time derivative of and using (79), we obtain
This can be simplified to yield the following differential equation
This can be equivalently expressed as follows
which can be integrated to give
Rearranging the terms, we can find as a function of time
Theoretical prediction and simulation results for the evolution of are presented in Fig. 9 for the case of for . As can be seen from the figure, higher values of converge faster towards . This signifies the role of high-degree nodes in spreading the influence throughout the system.
To solve this equation, let us temporarily use the following definition for brevity
Then we can rewrite (85) as follows
This is a linear first order differential equation. Multiplying both sides by the integration factor in order to make both sides equal to , we find that