Competing Opinions and Stubborness: Connecting Models to Data

We introduce a contagion-like model for competing opinions that includes dynamic resistance to alternative opinions. We show that this model can describe candidate vote distributions, spatial vote correlations, and slow opinion consensus with sensible parameter values. This may suggest that all these distinct behaviors, previously understood using distinct models, may be different aspects of a more unified model of human behavior introduced in this paper.

Firstly, while different aspects of opinion dynamics can be described by a wide range of quantitative models, the driving features of the global dynamics seen in empirical data remain unidentified because models with significantly different local dynamics can explain seemingly related global patterns [12,22] and because different types of models focus on different global patterns [4,12,[21][22][23].
Secondly, individuals in opinion dynamic models often reach a state of complete consensus (in which all individuals holds the same opinion) [5,11,24,25]. Depending on how model timesteps are defined, this is in disagreement with empirical observations, such as the slow dynamics of language diffusion and coexistence of multiple dialects [21] or the slow dynamics and coexistence patterns seen in the spread of religions [19].
We mitigate the previous limitations with a new model, which we call the Competing Contagions with Individual Stubbornness (CCIS) model. Our model includes features of, but is fundamentally distinct from, the Susceptible-Infectious-Susceptible (SIS) model wellknown in epidemiology (for a review, see [26]), where * keith@umd.edu † wrand@umd.edu ‡ girvan@umd.edu individuals can be "infected" (persuaded) by their neighbors and recover to a "susceptible" state (have no opinion).
The CCIS model assumes contagion-like dynamics, because rumors and opinions seem to spread virally [27][28][29][30], meaning the cumulative probability of adopting an idea increases with the number of exposures to that idea; therefore a realistic opinion model should have "viral" dynamics.
Next, unlike, for example, the influential voter model (VM), in which at each timestep an individual adopts the opinion of a randomly chosen neighbor, and similar opinion models [4,9,10,23], our model contains an unopinionated state that nodes recover to, in order to account for a stable fraction of unaffiliated voters observed in polls [31]. Also, a reduction in poll volatility has been observed before an election [32] which we might interpret as a reduction in the recovery rate as individuals increasingly need to make a stable decision before they go to the polls.
The key distinctions between our model and the traditional SIS model are: 1. Over time, individuals can become "infected" with different competing opinions or "strains". However, at any given timestep, each individual can only hold one opinion. The SIS model, in comparison, considers the spread of a single strain through the population.
2. Ideas are spread outward from individuals who have "caught" a new idea, while the SIS model assumes infections spread "inward" from a neighbor, or neutrally between two pairs of neighboring nodes.
3. Individuals become less likely to change their opinion the longer they hold it, while the SIS model has no equivalent dependence.
The CCIS model deviates from the SIS model for the following reasons: Recently, epidemic and epidemic-like models have incorporated multiple strains and multiple infections, which breaks from the traditional SIS model [33][34][35][36][37].
Similarly, it is typical in opinion dynamics research to study the interaction of multiple opinions, alike to the interaction of multiple infection strains, because rarely do opinions completely coincide (e.g. who should be president). It is therefore natural for an opinion model to contain multiple competing opinions, rather than a single homogeneous idea.
The spread of epidemics is usually assumed to be inward (i.e. an individual is picked at random, and diseases spread from infected neighbors to the ego) or neutral (a link between individuals is picked at random, and the disease similarly spreads between neighbors) [38,39]. Here, we model opinions spreading outward from the ego to its neighbors. The difference between these possibilities is subtle and not necessarily perceptible when social networks are k-regular, meaning everyone has the same number of friends. When the number of friends is highly heterogeneous, however, high degree and low degree nodes vary in the ease at which they change opinion depending on the way in which opinions spread (e.g. see [5] for variants of the VM). The outward spread hypothesis has been used to find agreement between an opinion model and election data [12], so we assume this behavior in our current model, although all the qualitative features in the current model are present in all three contagion methods (inward, outward, and neutral infection processes). Furthermore, this scenario is more plausible when we imagine that individuals actively spread ideas, instead of passively take ideas in.
Finally, we allow for individuals to increasingly resist adopting a new opinion the longer they keep their current one, by hypothesizing that individuals have "stubbornness", meaning that the probability of adopting an alternate opinion decreases the longer the current opinion is held. Evidence for this behavior is observed when studying the effects of pre-trial publicity (PTP) on juror outcomes [40]. A meta-analytical review found that a delay between the PTP exposure and the verdict, especially after one week, produced a significantly greater bias, suggesting that stubbornness towards an initial opinion increases with time, at least on the timescale of a week. Interestingly, stubbornness is not unlike the primacy effect, well studied in psychology [41,42], where the first idea someone hears can be the idea someone favors regardless of its validity. That effect, however, usually concerns ordering of choices, and not when each choice was offered.
In the following sections, we show agreement between our model and several empirical studies. Firstly, the model is shown to describe the collapse in vote distribution among candidates when votes are scaled by Q/N , where Q is the number of candidates and N , the number of voters [12,13]. Secondly, we argue that the CCIS model approaches the voter model universality class (VMUC) in certain limits [43], and therefore diffusively spreads voter preferences. The spatial vote correlations in this limit match those observed in several countries [22,44]. Lastly, we show that model parameters allow for opinion consensus to be reached at arbitrarily long times.
The remainder of the paper is structured as follows. We begin by describing past research (Section II) and details of our model and algorithm implementation (Section III), before comparing our model to empirical data (Section IV). Next, we use simulations and analytical calculations to show how our model can vary between fast and slow consensus (Section V). Finally, we conclude our paper (Section VI).

II. BACKGROUND
In this section, we review empirical data that motivates our model and recent models that, like our CCIS model, exhibit slow or non-consensus.
In recent years, Big Data has allowed researchers to observe large numbers of social interactions [12,13,22,27,30,32,45,46], leading to new insights into opinion dynamics. Here, we focus on two themes which have received recent attention: candidate vote distributions [12,13,22] and spatial vote correlations [22].
Vote-share distributions for candidates in elections show relatively universal properties. For example, two recent studies demonstrated that the vote-share distributions, when scaled by Q/N , often collapses to a universal function, based on election data from several countries across at least 50 years [12,13].
Other recent work has shown that the vote-share correlation in the US elections and the turnout rate correlation in European elections decreases as log(r), where r is the absolute distance between two voting districts [22,44], meaning correlations are very long-range. Interestingly, this is a prediction of the VM at an arbitrary, fixed time [9,10,47]. In the continuum limit of the VM, the two point correlation function in two dimensions decreases as ∼ log(r)/log(t), where t is time.
Opinion dynamics research has recently focused on how to slow or stop the consensus of opinions in models [2,4,6,14,15,23].
For example, the Non-Consensus Opinion (NCO) model assumes that individuals asynchronously update their state to be that of the majority of their neighbors plus themselves (if there is no majority, a node does not change) [4,23]. This rule, a variant of which was originally created for a cellular automaton [48], seems to converge to a frozen state of non-consensus, when each opinion initially has a roughly equal share of the population.
Several models similarly achieve non-consensus while, in addition, populations of a given opinion vary in time. For example, in a model by Galam [2], some individuals are contrarians, who oppose the majority opinion in their local neighborhood, while all others follow the local majority. Galam showed that contrarians create a non-consensus of opinions, and that there is a critical threshold in the fraction of contrarians, above which opinions are, in equilibrium, equally divided within the population. A few authors also explored the idea of adding "stubborns", which never change their opinion, but can change the opinion of their neighbors, to the Galam model [14], and to the VM [6,15]. In both cases, the equilibrium distribution is not consensus.
Lastly, we mention recent work on contagion and contagion-like models with multiple strains. A few recent studies have noted that the coexistence of two contagion strains on a single network is unstable [36,49], which is in agreement with the work presented here, although they seemed to have focused on when β 1 /δ 1 = β 2 /δ 2 , where β i and δ i are the strain-specific infection and recovery rates, respectively. Our work, in contrast, (Eq. 7) suggests that even as β 1 → β 2 = β and as δ 1 → δ 2 = δ → 0, we similarly find that two-strain dynamics is unstable because we approach the VMUC, implying finite expectation times for two strains to coexist on a finite network. Initial simulations show that the coexistence time decreases with δ and approaches a minimum, as δ → δ c , the critical point after which all strains die out exponentially fast. We have therefore taken a step towards showing that twostrain dynamics is always unstable on a single network. Theoretical (versus numerical) support for δ > 0 is still needed, however.
Methods to create non-consensus with contagion models often employ multi-layered networks where each strain, disease, or opinion exists on its own network, which is then coupled to one another [34,36,37]. Typically, these networks have distinct topologies, or, when the coupled networks are identical, certain parameter combinations are found to achieve non-consensus [37]. Alternatives include dynamics on a growing network [50], or coupled populations that are individually mixed [35].
The CCIS model is distinct from the above models in that every node holds onto their most recent opinion with increasingly stubbornness, but can return to the unopinionated state at random (see Fig. 1), and all dynamics are on a single network. Table I defines the different parameters and variables of our model: β parameterizes how quickly individuals change their opinion (i.e. their "persuasiveness"), where individuals change their opinion with a probability proportional to β. Next, µ parameterizes the speed at which individuals become stubborn: at a linear rate, individuals are decreasingly likely to change their opinion until they have kept an opinion for µ −1 timesteps, after which an individual's opinion is temporarily "frozen". Finally, δ is the rate at which individuals "recover" from an opinionated to an unopinionated state, and are thus susceptible to a new opinion again.
Our model's increasing stubbornness property was also explored in the VM with the additional rule that after a certain period, individuals could change their opinion at a fixed rate [25]. It was found that stubbornness i ncreases the speed of consensus, and for a non-trivial value of µ, it can maximize the consensus speed. We find a similar behavior in the CCIS model (Fig. 8), although our model is most alike to the VM with a freezing period, previously explored in one dimension [11], because it is easy to argue that for a period ∼ (δ −1 − µ −1 ), an individual's opinion is frozen. This model too exhibits the same qualitative behavior, suggesting it is a fairly generic property of decelerating microdynamic models. A related model, where individuals can change their opinion after a certain number of neighbors attempt to convince them [24], creates a much faster consensus on a grid graph than the VM and created curvature-driven coarsening, consistent with an effective surface tension, which the traditional VM lacks. Taken as a whole, it would seem that adding a stubbornness-like property would create exceedingly fast consensus, even though we notice that few competing opinions, e.g. opinions about operating systems or presidential candidates, achieve consensus after several years. The CCIS model can substantially increase the consensus time, however, in the limit that δ → 0. This limit is in agreement with empirical data (Fig. 2), and we claim that certain parameter combinations in this limit can produce VM-like consensus (Eq. 1). Furthermore, unlike the previous models, the CCIS model is contagion-like, which we've previously mentioned is in strong agreement with empirical data. Of course, one reason we assumed δ > 0 was to allow for a population of individuals without an opinion (i.e. are "independents"). In this paper, we will focus on when the effective number of independents is small. Opinion A density as a function of t and τ P A Fraction of individuals with opinion A at a given time

III. MODEL DETAILS
We assume in this paper that: 1. Interactions between individuals are localized on a fixed, unweighted, and undirected network topology.
2. Opinions do not evolve.
3. Some individuals begin with initial opinions and the remainder begin in the unopinionated state.
The fixed topology hypothesis is not necessarily a good approximation, because individuals tend to interact in a temporally varying network [51,52]. However, we expect the qualitative behavior to be the same in the case of slow network evolution, and the behavior in the fixed network case is analytically tractable. We believe an undirected network, however, is a good approximation for social interactions, because we expect a 2-way street between the influencer and influencee, and without a priori detailed knowledge of the real social network, we can reasonably assume the network is unweighted. In many practical instances, the assumption that opinions do not evolve works well, especially when the number of opinions possible is fixed (e.g. voting "guilty" or "not guilty" in a jury or picking among Q candidates). Again, this assumption also allows for analytic tractability, and minimizes extraneous variables.
Lastly, the assumption that individuals may have initial opinions is taken for granted in models like the VM, but is not guaranteed in many real world situations. For example, voters may start out without any knowledge of party candidates, although their understanding can increase as they interact with knowledgeable friends.
We now describe how we implement the CCIS model in an agent-based simulation. The model is run on a network of order N , where the state of each node, i, is η i ∈ {0, 1, 2, ..., Q}. Q is the total number of opinionated states, while 0 corresponds to no opinion. At t = 0, n 0 nodes are in state 0, n 1 are in state 1, ... , n Q are in state Q, such that n 0 + n 1 + ... + n Q = N .
The dynamics are as follows: 2. Count the number of opinionated individuals, N op = N − n 0 , and repeat from step 1 with time incremented by ∆t = (N op (1 + δ)) −1 .
In our model, τ is the "time a node has had its most recent opinion", µ is the stubbornness rate, β is the persuasiveness and δ parameterizes the probability of moving to the 0 state at random (see Fig. 1 for a diagram of the dynamics, and Table I).
While it may seem odd, at first, why ∆t = N −1 and the recovery parameter is not δ, the reasoning follows from the work of Boguna, Castellano, and Pastor-Satorras [39] for the SIS model. ∆t is normalized such that N op nodenode interactions take place, and δN op of the opinionated nodes recover, at time i ∆t i = 1. Basic algebra implies that, holding N op constant for each timestep ∆t = ((1 + δ)N op ) −1 . Similarly, this reasoning implies the recovery probability is δ/(1 + δ).

IV. AGREEMENT WITH DATA
We make two claims about the CCIS model's agreement with data: We can reproduce the scaled distribution of candidate votes observed previously in several countries [12,13], and the vote-shares correlations as a function of distance seen recently in presidential election data in the US [22].
Firstly, we discuss how the model reproduces the scaled distribution of candidate votes (Fig. 2). Three fitting parameters are used: degree distribution, persuasiveness (β = 0.5), and initial seeding (6% with an initial candidate preference). Individuals interact on a scale-free network with a degree distribution, p(k) ∼ k −α , where the scale-free nature of this graph is a natural construct because many social networks studied have scale-free distributions, p(k) ∼ k −α , where 2 < α < 3 [54]. It was found that α = 2.9 helps create close agreement to the distribution, although the coefficient did not strongly effect the fit to the large vote-share tail. Lastly, we let δ = 0 such that the distribution does not vary in equilibrium, when µ = 0.3.
We see, remarkably, that the model results almost lie on top of each set of data for vQ/N > 10 −1 . Of the elections modeled, we find that only Switzerland's diverges significantly from our model due to its unusual "doublehump" distribution, plausibly because votes are swayed by the local language differences (primarily French and German). We find, however, that other unusual features, such as the kink in the distribution tail of Poland's 2001 election, seem to be faithfully reproduced, suggesting it is not part of the "complexity of human nature and context-dependent factors" as initially assumed [13]. When vQ/N < 10 −1 , the model diverges from the data,  [13], where data is shifted down by decades for clarity (inset shows the original data collapse).
Here v corresponds to the number of votes, and Q and N were the same as the empirical data. Three fitting parameters were used: (1) power-law degree coefficient for each election network α = 2.9, (2) persuasiveness parameter β = 0.5, and (3) 6% initially seeded with a preference to a candidate. Two others were fixed: µ = 0.3, and δ = 0, where the latter assured non-consensus distributions in equilibrium a .
a Five elections out of 2109 were not modeled (one in Finland, 1995; one in Italy, 1958; and three in Italy, 1979) all with N < 10 2 because Q greater than 6% of the nodes. Although we could have included these with each candidate as a seed (the effect is negligible), we removed them to keep initial conditions consistent.  Figure 3: The correlation as a function of distance, with regions of 5 × 5 node squares (N = 10 6 ) after 8 × 10 3 timesteps. The top line is for a 2d grid with periodic boundary conditions while lower lines are for small-world networks built starting from a 2d grid in order of increasing fraction of edges randomly rewired. C(r) ∼ −log(r) even when a significant number of edges are re-wired, thus behavior is robust to graph "noise" symbolizing weak ties between distant neighbors [53].
which may suggest that, in this limit, a new set of dynamics comes into play. For example, candidates are who are extremely unpopular may drop out before an election, thus reducing the probability for those candidates to be on the ballot. Differences between country distributions are strongest in this limit as well, suggesting that country-dependent factors play come into play too. Our model is influenced by a previous one we call the Fortunato-Castellano (FC) model [12] which is known to approximately recreate rescaled vote distributions in Poland, Italy, and Finland. Alike to the FC model, the CCIS model considers opinions spreading outward, and individuals interacting on a scale-free network. We improve upon the previous model by adding agreement with spatial vote correlation data, slow consensus, and through simulating each election on a single social network. We describe the former two points in what follows, but for now, we focus on the final point. In the FC model, it was assumed that for every election, each of the Q candidates tries to "get out the vote" among Q disconnected social networks. Candidates however, seem to often compete for a common set of voters [55][56][57], therefore, it may be more reasonable to assume that the electoral network is connected. We can accomplish this in the CCIS model by taking δ → 0, which produces equilibrium vote distributions that seem to non-trivially collapse into a single function when votes are scaled by Q/N . Fundamentally, this is accomplished by the opinion stubbornness parameter forcing a non-consensus equilibrium, thus providing further evidence that stubbornness may exist in opinion formation.
Secondly, we show how the CCIS model creates C(r) ∼ −log(r), seen in empirical studies [22,44]. To do so, we must first argue that our model approaches the VMUC. We conjecture that the VMUC is approached in our model when [43]: where k and k 2 is the first and second moment of a network's degree distribution,respectively. The reasons are as follows, which we will expand upon in Section V: 1. δ β k 2 / k ensures the model is a 2-state system, like the VM (for a proof that this breaks down when δ β k 2 / k , see [38]).
3. δ µ is numerically found to increase the consensus time. therefore our model can reproduce the vote-share correlations as a function of distance, C(r), seen in empirical data [22,44] (see Fig. 3).
The relation above is self-consistent by setting Eq. 4 to 0 (a necessary condition for VM-like diffusive consensus), and assuming T cons , the mean time to reach consensus (Eq. 7), is 1.
Numerical studies with δ and µ > 0 have so far found that T cons log(N )/λ 1 , unlike the VM, where T cons ∼ N for a graph with bounded k 2 . If δ µ, consensus time is increased, which is equivalent to λ 1 → 0, a regime where we expect to approach diffusive consensus.
The argument that we approach the VMUC for a kregular network is as follows, given the conditions in Eq. 1: 1. After a time ∼ βkδµ −1 , a given node infects a neighbor.
(b) For a brief period µ −1 the node is susceptible.
(c) The probability of an opinion changing is always on the order of βk (d) Therefore the probability a node is susceptible times the probability a susceptible node changes its opinion is βkδµ −1 2. Nodes are updated at random. Therefore, the model acts like a VM with time rescaled by βkδµ −1 . This argument ignores the exact nature of p rec (δ) or p inf (β, µ), the time distributions to recover and change an opinion, respectively, and is therefore a general outcome of many variants of our model.
On a d-dimensional grid, where the degree of each node is 2d, the correlation as a function of time and distance in the continuous limit, is that of the VM, with time rescaled by 2βδµ −1 d [47]: where D = (2d 2 µ −1 δβ) −1 and a is the lattice spacing. The correlation approaches one exceedingly slowly for d = 2, and we find, in simulations where δ, µ → 0, that the spatial correlation can be effectively frozen in time (Fig. 3). To be consistent with previous work [22,44], we define the normalized correlation in our figures as: where P i is the fraction of voters for candidate 1 within a small region, P is the average fraction of voters for candidate 1, and σ 2 P is the variance in vote distribution across all regions. P i P j |d ij r is the 2-point correlation function between regions a distance r from each other.
We find that C(r) ∼ −log(r), even when up to 10% of edges are re-wired (Fig. 3), and that as more edges were rewired, the time-dependence on correlation falls significantly. The nearest-neighbor correlation does not change significantly for p ≤ 10 −2 , even as the mean distance falls by half. The robustness of correlation behavior as mean path length decreases is not unlike the robustness of graph clustering to rewiring seen in the Watts-Strogatz model [60].

V. APPROACHING EQUILIBRIUM
Complete consensus, where all nodes are in some state q > 0, is achieved for most parameter values, with the exception of β = 0, or δ = 0 and µ > 0. We will show that the time to consensus, however, can be arbitrarily large. But first we must first determine a realistic consensus time. It is not well defined what a timestep really is in the real world because we interact in a highly bursty manner [52,61] and the influence of a single interaction has the potential to be almost immeasurably small [30] or significant [27,29,45]. Nonetheless, if we assume adoption on the timescale of days to weeks, as has been assumed previously [2], we can search within the model's parameter space for consensus times of, for example ≥ 10 3 timesteps, given that a consensus among voters on who should be president is not achieved on the timescale of years [15].
In this section, we will begin by understanding a significant, but short-lived, transient our model creates, before we explore the consensus time in the most realistic scenario (i.e. where C(r) ∼ −log)(r)), where our model approaches the VMUC.
We begin our analytic treatment by studying the transient behavior of our model as δ → 0. We notice a transient "jump" in the fraction of individuals following a given opinion on a timescale much smaller than the time to reach consensus (see inset of Fig. 5). One way to analyze this behavior is via a mean field PDE equation, which we derive in Appendix A: In Eq. 4, ρ A is the density of individuals as a function of t (time) and τ (the time an individual has kept their most recent opinion). P A is defined as: i.e. the fraction of individuals with opinion A, while B are all contrary opinions. Finally, k is the average degree, and all other parameters are as defined previously. Cyan equation elements correspond to loss via recovery, and blue elements correspond to gain via convincing those with an alternative opinion. For simulations and analysis of the PDE, we set δ = 0, in order to observe nonconsensus equilibrium values seen in simulations (Fig.  4). We distinguish this theory from a completely distinct type of mean field theory to be discussed shortly, by calling it the PDE mean field theory (PDE MFT).
Using Appendix A and methods described in [25] for the VM with a similar stubbornness property, we find that it is unstable for everyone to have an equal vote share, at least for Q = 2. The initial condition diverges exponentially due to the largest eigenvalue being positive. If P A is the fraction of nodes with opinion A, then at initial times, |P 1 − P 2 |(t) ∼ Exp(λ 1 t), where, assuming µ 1: The derivation of this value is in Appendix A. We test this theory by showing it is effect on the timescale of the transient. A simple derivation shows that the transient time to the δ = 0 equilibrium is: where ν and c(µ) are fitting parameters. Fig. 5 demonstrates close agreement between theory and simulation. When βk = 1, this eigenvalue should agree exactly with the value cited previously [25], but we find disagreement by an overall pre-factor of 1/4 which, at least to our knowledge, may have been missed in the previous work. When δ = 0, we almost never reach consensus but in the more realistic case where δ > 0, we always reach consensus, although potentially at very long times. We argue in Section IV that we approach the VMUC as δ → 0 and µ β k , therefore, in this limit we can employ powerful theory on the VM [5] to find the scaling behavior of the mean consensus time: where, again, k 2 is the second moment of the degree distribution p(k) (see the Appendix B for a derivation of this result). To distinguish this theory from PDE MFT, we call it the VM-limit mean field theory (VML MFT). We explored the agreement with this theory by varying δ, µ, β, and k 2 , while separating the respective timescale sufficiently in order to understand the individual effects.
To explore the dependence of the VML MFT on k 2 , we set δ, µ → 0 such that δµ −1 = 1. It is easy to show that this limit is still captured by the equations, but the consensus times are found to be much faster than with δ > 0 and µ > 0, for appropriate (VM-like) values of  Figure 6: Mean consensus time, T cons , for scale-free networks (p(k) ∼ k −α ) with β = 0.5, δ = 0, and µ = 0. Inset is one example of consensus with P 1 (t) and P 2 (t). Using the VML MFT, the expected fit (solid lines) is Eq. 7. Simulations are averaged over 10 networks (30 networks for 3 × 10 4 ≤ N < 10 5 , and 20 networks for N = 1.2 × 10 5 ) with 100 trials per network.
each parameter. In a scale-free network (p(k) ∼ k −α ), k 2 diverges with network order, N : Therefore: Fig. 6 compares our simulations to the VML MFT. Although a finite size transient impedes this scaling behavior for N ≤ 10 4 , we still see agreement large enough networks. For Poisson networks, we see T cons ∼ k 2 −1 = ( k 2 − k ) −1 in the inset of Fig. 7.
Next, we focus on whether T cons ∼ β −2 , holding all other variables constant. Interestingly, agreement with theory was closest when β ∼ O(1) and k ∼ 10 − 20. The discrepancy between theory and simulation may be due to finite size effects, but future work is necessary to test this assumption.
Finally, we studied T cons by varying δ and µ. Plotting T cons versus δ (not shown), we find that T cons ∼ δ −2 for δ < β k 2 / k and µ ∼ 0.2, although this breaks down for small δ, presumably for the same reason that Eq. 7 breaks down for small β (Fig. 7). We also explored the regime where δ µ (e.g. µ = 0) and found the consensus and solid line (T cons ∼ k 2 −1 ) in the inset. varying δ. Theory (T cons ∼ µ 2 , cf. Eq. 7) seems to disagree with our simulations, at least in the parameter space observed (colored lines indicate best fit slopes, which are all around T cons ∼ µ 5 to within error bars).

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time dropped significantly when δ β k 2 / k , on networks with strongly bounded k 2 , suggesting that near the critical point for a long-lasting infection, β/δ = τ c , the consensus timescale is an order parameter that presumably drops to 0 in the thermodynamic limit. In comparison, timescales like the time to reach equilibrium blows up in this limit. Whether this can aid in determining the critical point for the SIS model for arbitrary graphs is as of yet unknown, although significant recent interest in determining the critical point for arbitrary graphs suggests this idea is worth exploring [38,39,[62][63][64].
The behavior of T cons versus µ, however, demonstrates a breakdown in our theory (Fig. 8).
Theory dictates that T cons ∼ µ 2 , for large enough µ, but instead T ∼ µ a , a ≈ 5 for the parameter values explored. Interestingly, there is also a non-trivial minimum in consensus times, as seen in similar VM-like models [11,24,25]. Lastly, there is a minimal consensus time, It should lastly be mentioned that when δ = 0, µ = 0, and βk = 1, the CCIS model is alike to the invasion process (IP) [5], where a neighbor is randomly chosen to have the same opinion as the root node [5]. In the true IP, T cons ∼ N k −1 k , but in the CCIS model, T cons ∼ N k 2 . The discrepancy is due to a fixed fraction of neighbors, 1/ k , being changed in the CCIS model, instead of exactly one in the IP. Interestingly, this implies that T cons ∼ 1/N in a complete graph. Initial simulations suggests this prediction is correct for large enough N . For moderate N , in contrast, the consensus time reaches a maximum.

VI. CONCLUSION
In conclusion, we have introduced a model of opinion spread that agrees with some current data on opinion dynamics. Furthermore, we have shown that the speed of consensus in our model can be made arbitrarily slow, in agreement with the expected long times for groups to reach consensus. Lastly, our model makes few assumptions, implying it can plausibly explain a range of behaviors mentioned in Section V.
Here, we discuss some qualitative conclusions that we can draw from the CCIS model, which future empirical investigations may be able to corroborate. Firstly, this model may qualitatively describe properties of colicin-containing bacteria. Colicins are a family of toxins produced by strains of certain bacterial species (such as Escherichia coli and Streptococcus mutans) that are active against competing strains but not their own. The stubbornness property of our model can be thought of as reducing the impact of an opposing strain on a single host, alike to colicins [65], because the build-up over time of colicins would make it less and less likely that an alternative strain can successfully take over the host. Alternative models of competing strains [34-37, 49, 50], however, assume coupled networks or a dynamic network, and furthermore, have no time-varying susceptibility. In contrast to these other models, the CCIS model suggests that colicin-containing bacteria should allow for greater genetic diversity than similar bacterial species lacking colicins, just as stubbornness in opinions allows for arbitrarily slow approaches to complete consensus. Furthermore, the relative success of a single strain (its share of the entire infected population) should follow a distribution similar to Fig. 2.
In addition, the model can be interpreted as the "viral" spread of competing products, where increasing stubbornness is instead increasing brand loyalty. With these reinterpretations, the CCIS model suggests that there is a greater number of products in the buyer network, compared to when individuals adopt a product readily, without stubbornness, or alternatively, "brand loyalty" [66,67], because the timescale for "consensus", i.e. where a single product dominates, can be extremely long. Furthermore, the brand-share distribution should exhibit a distribution similar to the one seen in Fig. 2.
Future work is also necessary to explore ways in which our model can be made more realistic. For example, it may be more appropriate in future work to add heterogeneous stubbornness, because some individuals can stubbornly refuse to relinquish an idea, while others may shift their stance like a chameleon. The most stubborn individuals can help make the political preference in a two party system near the 50/50 mark, as explored previously [6,14,15]. Expanding on previous work, we expect adding heterogeneous stubbornness to our model can slow down or stop consensus, depending on the heterogeneity, potentially creating better fits with current data.
It may also be more realistic to replace the recovery probability with a random opinion flipping probability. Adding random opinion flipping to the VM seems to have allowed for better agreement with empirical data in a recent paper [22], and, interestingly, this replacement in our model seems to create three critical points, between order (relative consensus) and chaos (no consensus). Trivially, if the flip rate is high, two opinions should be equally split. In comparison, when the flip rate is moderately low, one opinion should dominate. There seems to be a 1 st order transition between these two extremes for a high flip rate, and a continuous transition for a small but non-0 flip rate. Finally, a flip rate of 0 tends to force one opinion to dominate, unlike the behavior for an infinitesimal flip rate. Recall in our current paper, recovery always forces one opinion to dominate, regardless of how large the recovery probability is. Despite a relatively small change in the dynamics, its clear that the behavior is both interesting and drastically different.
In a similar vein, we might expect that real opinions exhibit a ladder-like process where individuals move between opinion extremes. This creates non-consensus in continuous opinion models [68] and thus may slow the consensus in a discrete model. These new dynamics may help us understand how polarization happens when individuals stubbornly hold onto an opinion.
As a final point, we only compute the theoretical consensus-time scaling in the mean field limit. More accurate results may be found with, for example, a simple pair approximation [69], or a heterogeneous pair approximation theory [70], which can also take into account degree assortativity.
Appendix A: Derivation of Eq. 5 In this appendix, we use the CCIS PDE MFT to understand the initial transient in the opinion densities (see inset in Fig. 5). This behavior has the potential to significantly effect the time it takes for our model to reach complete consensus, especially when the persuasiveness parameter (β) is of O(1) (Fig. 4).
The equation used to describe our model in the mean field is: (t) Figure 9: Schematic of the A th scalar variable in Eq. 4 as a function of time and length of time since the adoption of opinion A.
With the boundary conditions where B represents the opposing opinions and: Let us first understand the variables in our equation. The independent variable, t, is "time" and τ is the "time since an individual kept their most recent opinion". The dependent variable, ρ A (t, τ ), is the opinion density for opinion A (see Fig. 9).
The RHS describes the ability of individuals to recover (cyan term) as well as the ability to change opinions (blue term). We can interpret the boundary conditions as: 1. Normalizability 2. An increase in the infection density due to unopinionated neighbors (cyan term) and opinionated neighbors (blue term).

Initial conditions
See Fig. 10 for a visual representation. Despite the simplifications of the equation, we find good agreement with simulations (Fig. 4). We therefore have a simple equation which allows us to understand the reason for the transient.
Following [25], we set δ to 0 and let Q = 2 (creating a 2-state system), and the discretize τ to derive a set of equations that we linearize around a fixed point to determine the speed of the jump (Eq. 5 & 6). Due to the above assumptions, this is only accurate for µ 1. We define the following macroscopic variables: where τ is shorthand for ∞ τ =0 . If we let Ω(|) be the conditional probability function, andẋ ≡ d dt x, then Eq. 4 becomes (for τ > 0): and for τ = 0:  With an equivalent set of equations for ρ 2 (t, τ ) and (A4) From the above results we can sum ρ 1 (t, τ ) to find the equations for the macroscopic variables: To lowest order in µ, we also find that: I 1 (t) µ(1 − βk)P 1 (t) + βk[µP 1 (t) 2 + I 1 (t)P 1 (t) − I 1 (t)] (A6) These equations are solvable by expanding around the solution P 1 = P 2 = 1/2 and I 1 = I 2 = µ[1/(βk) − 1/2] to first order. The resulting largest eigenvalue is Eq. 5.
Appendix B: Derivation of Eq. 7 In this appendix we derive Eq. 7 using a VML MFT, which is distinct from the PDE MFT in Appendix A. Our derivation is heavily based on the derivation of consensus times for the VM and Invasive Process by Sood, Antal, and Redner [5]. We use the same conventions as in that paper, except the transition probability scaling factor (S), the degree distribution (p(k)) and associated moments, k m = k p(k)k m .
Let η(x) be the state of a node x on a network with adjacency matrix A xy and order N . Assuming 2 opinions and δ → 0, we have a two-state system. Using the conventions of Sood, Antal, and Redner [5] the opinions of the two-state system are "0" or "1" instead of "1" or "2". We stress that the 0 state is an opinionated state.
Lastly η x is the state of the system after changing a node x: η x (y) = η(y), y = x 1 − η(x), y = x (B1) The transition probability at node x is therefore: where, if we assume δ → 0: Effectively, with probability β an attempt is made to convince a neighbor but only on the order of µ −1 δ of the time is the node available to be convinced (and is otherwise stuck in its state), therefore, our model effectively scales the conviction rate by βδµ −1 .
We further assume a mean field solution, where the adjacency matrix becomes the average adjacency matrix.
Instead of individual states η(x), we can instead focus on ρ k , the density of states with degree k.
Here, x is the sum of all nodes with degree k. To clarify the below equations, we also define a variable ω: Next, we define our raising and lowering operators for ρ k , which defines the probability of increasing or decreasing ρ k by a small incriment: This is to say, the change in ρ k from a neighbor with average degree k nn = k 2 k . The raising operator is defined as: With simplification, this yields Similarly, for the lowering operator: It can be shown that, ξ 1 , the probability for all nodes to reach state one in equilibrium (also known as the "exit probability") is: And similarly, that ρ (or magnetization, if this were a spin system) is a conserved quantity. The time to consensus is: Here, the average timestep is the average number of interactions of a neighboring node with degree k nn We expand to second order in ∆ρ k and find that As was shown in in the original paper [5], this value reaches 0 for time T cons ∼ O(1) which is much smaller than the next term: Lastly, by a change in variables: Because, ρ is conserved and v k → 0, ω → ρ. Therefore, we find that this equation simplifies down to (24) in [5], where T cons ∼ N → N ef f and Thus, we have an effective scaling of our consensus time. There is a discrepancy between theory and simulation for small values of δ, β, and presumably µ, potentially because of the finite variance of p inf (β, µ) and p rec (δ), the time distributions to infect and recover, respectively, or possibly because of finite size effects. Future work could improve upon the current results by correcting for these effects. In addition, a pair approximation theory could provide correction terms to the mean field assumption [69,70].