We consider a simplified model of a social network in which individuals have one of two opinions (called 0 and 1) and their opinions and the network connections coevolve. Edges are picked at random. If the two connected individuals hold different opinions then, with probability 1 - α, one imitates the opinion of the other; otherwise (i.e., with probability α), the link between them is broken and one of them makes a new connection to an individual chosen at random (i) from those with the same opinion or (ii) from the network as a whole. The evolution of the system stops when there are no longer any discordant edges connecting individuals with different opinions. Letting ρ be the fraction of voters holding the minority opinion after the evolution stops, we are interested in how ρ depends on α and the initial fraction u of voters with opinion 1. In case (i), there is a critical value αc which does not depend on u, with ρ ≈ u for α > αc and ρ ≈ 0 for α < αc. In case (ii), the transition point αc(u) depends on the initial density u. For α > αc(u), ρ ≈ u, but for α < αc(u), we have ρ(α,u) = ρ(α,1/2). Using simulations and approximate calculations, we explain why these two nearly identical models have such dramatically different phase transitions.
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In recent years, a variety of research efforts from different disciplines have combined with established studies in social network analysis and random graph models to fundamentally change the way we think about networks. Significant attention has focused on the implications of dynamics in establishing network structure, including preferential attachment, rewiring, and other mechanisms (1–5). At the same time, the impact of structural properties on dynamics on those networks has been studied, (6), including the spread of epidemics (7–10), opinions (11–13), information cascades (14–16), and evolutionary games (17, 18). Of course, in many real-world networks the evolution of the edges in the network is tied to the states of the vertices and vice versa. Networks that exhibit such a feedback are called adaptive or coevolutionary networks (19, 20). As in the case of static networks, significant attention has been paid to evolutionary games (21–24) and to the spread of epidemics (25–29) and opinions (30–35), including the polarization of a network of opinions into two groups (36, 37). In this paper, we examine two closely related variants of a simple, abstract model for coevolution of a network and the opinions of its members.
Holme–Newman Model
Our starting point is the model of Holme and Newman (38–41). They begin with a network of N vertices and M edges, where each vertex x has an opinion ξ(x) from a set of G possible opinions and the number of people per opinion γN = N/G stays bounded as N gets large. On each step of the process, a vertex x is picked at random. If its degree d(x) = 0, nothing happens. For d(x) > 0, (i) with probability α an edge attached to vertex x is selected and the other end of that edge is moved to a vertex chosen at random from those with opinion ξ(x); (ii) otherwise (i.e., with probability 1 - α) a random neighbor y of x is selected and we set ξ(x) = ξ(y). This process continues until there are no longer any edges connecting individuals with different opinions.
When α = 1, only rewiring steps occur, so once all of the M edges have been touched, the graph has been disconnected into G components, each consisting of individuals who share the same opinion. Because none of the opinions have changed, the components are small (i.e., their sizes are Poisson with mean γN). By classical results for the coupon collector’s problem, this requires approximately M log M updates (see, e.g., ref. 42, p. 57).
In contrast, for α = 0, this system reduces to the voter model on a static graph. If we suppose that the initial graph is an Erdös–Rényi random graph in which each vertex has average degree λ > 1, then (see, e.g., ref. 12, chap. 2) there is a “giant component” that contains a positive fraction, μN, of the vertices and the second largest component is small having only O(log N) vertices; i.e., when N is large, the size will be approximately Cλ log N, where Cλ is a constant that depends on λ. The voter model on the giant component will reach consensus in O(N2) steps (see, e.g., ref. 12, sect. 6.9), so the end result is that one opinion has μN followers while all of the other groups are small.
Using simulation and finite size scaling, Holme and Newman showed that there is a critical value αc so that for α > αc all of the opinions have a small number of followers at the end of the process, whereas for α < αc “a giant community of like-minded individuals forms” (38). When the average degree λ = 2M/N = 4 and the number of individuals per opinion γN → 10, this transition occurs at αc ≈ 0.46.
Our Model and Simulation Results
The “rewire-to-same” model we study differs from that of Holme and Newman in two ways: (i) we consider two opinions (called 0 and 1) instead of a number proportional to the size of the graph; and (ii) on each step, we pick a discordant edge (x,y) at random rather than a vertex, avoiding the problem of picking vertices with degree zero or vertices that agree with all of their neighbors. With probability 1 - α, the voter at x adopts the opinion of the voter at y. Otherwise (i.e., with probability α), x breaks its connection to y and makes a new connection to a voter chosen at random from those that share its opinion. The process continues until there are no edges connecting voters that disagree.
Despite the differences in implementation, this rewire-to-same model has a phase transition similar to that of Holme and Newman. In particular, the final fraction ρ of voters with the minority opinion undergoes a discontinuous transition at a value αc that does not depend on the initial density. Fig. 1 shows results of simulations for the rewire-to-same model starting from an initial graph that is Erdös–Rényi with N = 100,000 vertices and average degree λ = 4. Opinions are initially assigned randomly with the probability of opinion 1 given by u = 0.5, 0.25, 0.1, and 0.05. The figure shows the final fraction ρ of voters with the minority opinion from five realizations for each u. For α > αc ≈ 0.43, we observe ρ ≈ u and for α < αc, ρ ≈ 0.
Fig. 1.
Simulation results for rewire-to-same model, starting from Erdös–Rényi graphs with N = 100,000 nodes and average degree λ = 4.
We also study a “rewire-to-random” variant of this model that differs from the rewire-to-same model in only one way: x makes its new connection to a voter chosen at random from all of the vertices in the graph. This single difference leads to fundamentally different model outcomes, as seen in Fig. 2, showing simulation results for the rewire-to-random model on initially Erdös–Rényi graphs with N = 100,000 nodes and average degree λ = 4 for u = 0.5, 0.25, 0.1, and 0.05. When u = 0.5, the fraction in the minority is constant at 0.5 over [αc(0.5),1] and then decreases continuously to a value near zero as α decreases to zero.
Fig. 2.
Simulation results for the rewire-to-random model, starting from Erdös–Rényi graphs with N = 100,000 nodes and average degree λ = 4.
The behavior of our models for α > αc is easy to understand. As in the case of the Holme and Newman model, we expect consensus to be reached in O(N log N) steps when α = 1 and in O(N2) steps when α = 0. We define the boundary between the fast and slow consensus regimes to be the value of α, where the average number of steps needed to reach consensus is N3/2 (any power between one and two would give the same results when N → ∞). When an edge is chosen between voters with different opinions, then a rewiring event does not change the number of ones, whereas a voting event will increase and decrease the number of ones with equal probability; i.e., the number of ones is a random walk that on each step stays constant with probability α. The central limit theorem implies that when consensus is reached in O(N3/2) steps, the typical change in the number of ones from the initial configuration is O(N3/4). Hence, when the initial fractions of ones is u ≤ 1/2, the final fraction ρ with the minority opinion will be approximately equal to u.
Turning to the curves in Fig. 2 for u = 0.25, 0.1, and 0.05, we see that each initial density u has a critical value αc(u), so that for α > αc(u), we have ρ(α,u) = u, whereas for α < αc(u), we have ρ(α,u) = ρ(α,0.5). Because all of the ρ(α,u) agree with ρ(α,0.5) when they are < u, we call the graph of ρ(α,0.5) on [0,αc(0.5)] the universal curve. The main goal of this paper is to explain this phenomenon.
Quasi-Stationary Distributions
Let Ni be the number of vertices in state i. Our first clue to the reason for a universal curve in the rewire-to-random model came from Fig. 3, which shows the change over time of the fraction of vertices with the minority opinion min{N1,N0}/N and the number of edges connecting vertices with opposite opinions, N10, for a simulation in which the initial density of ones is u = 1/2, α = 0.3, the number of nodes is N = 1,000, and we start with an Erdös–Rényi graph with average degree λ = 4. In the visualization of these results and the theoretical discussions that follow, the model is considered in continuous time with each edge subject to change at times of a rate one Poisson process. The sequence of states visited by the model is the same in discrete or continuous time, but tM updates correspond to continuous time t. Hence, in the slow consensus regime, O(N2) updates becomes time O(N).
Fig. 3.
Fraction of nodes with the minority opinion (min{N0,N1}/N) and the number of discordant edges N10 versus time, for a simulation of N = 1,000 nodes, u = 0.5, and α = 0.3.
There are M ≈ 2,000 edges in this graph simulated in Fig. 3, so the initial number of 1-0 edges is approximately 1,000, but the curve drops very quickly to a value near 600, and then begins to change more slowly. The second key observation is that the number of 0-1 edges and the fraction with the minority opinion min{N1,N0}/N appear to be strongly correlated. The initial transient and the reason for the correlation will be seen more clearly in Fig. 4.
Fig. 4.
Plot of N10/M versus N1/N when α = 0.5 in the rewire-to-random case. Five simulations starting from u = 0.2, 0.35, 0.5, 0.65, and 0.8 are plotted in different colors. These results are from graphs with N = 10,000 vertices and plotted every 1,000 steps.
To explain the key insight derived from this simulation, we recall results for the voter model on the d-dimensional integer lattice , in which each vertex decides to change its opinion at rate 1, and when it does, it adopts the opinion of one of its 2d nearest neighbors chosen at random. Let ξt(x) be the opinion of the voter at x at time t. Holley and Ligget (43) and Liggett (44) proved the following result.
Theorem.
In d ≤ 2, the voter model approaches complete consensus; that is, if x ≠ y then P[ξt(x) ≠ ξt(y)] → 0. In d≥3, if the voter model starts from product measure with density p [i.e., are independent and equal to one with probability p], then converges in distribution to a limit νp, which is a stationary distribution for the voter model.
Simulations of the voter model are done on a finite set, typically the torus . In this setting, the behavior of the voter model is “trivial” because it is a finite Markov chain with two absorbing states, all ones and all zeros. As the next result shows (see Cox and Greven, ref. 45), the voter model has interesting behavior along the road to absorption.
Theorem.
If the voter model on the torus in d≥3 starts from product measure with density p, then at time Nt it looks locally like νθ(t) where the density θt changes according to the Wright–Fisher diffusion process
and βd is the probability that two random walks starting from neighboring sites never hit.
In the next section we will describe conjectures for the evolving voter model that are analogues of the last theorem. To prepare for stating our conjectures, note that (i) although the voter model on the torus does not have a nontrivial stationary distribution, it does have a one parameter family of “quasi-stationary distributions” that look locally like νp, and (ii) the quantity under the square root in the Wright–Fisher diffusion is, by results of Holley and Liggett (43), the expected value of N10/M under νθ(t).
Conjectures
Our next goal is to use simulation results to formulate the analogues of the Cox and Greven result (45) for our two evolving voter models, beginning with the more interesting rewire-to-random case. Fig. 4 shows results from simulations of the system with α = 0.5. The initial graph is Erdös–Rényi with N = 10,000 vertices and average degree λ = 4. Observations of the pair (N1/N,N10/M) are plotted every 1,000 steps starting from densities u = 0.2, 0.35, 0.5, 0.65, and 0.8. The plotted points converge quickly to a curve that is approximately (fitting to a parabola) 1.707x(1 - x) - 0.1867 and then diffuse along the curve until they hit the axis near 0.125 or 0.875. Thus the final fraction with the minority opinion ρ ≈ 0.125, a value that agrees with the universal curve in Fig. 2 at α = 0.5.
The fact that, after the initial transient, N10/M is a function of N1/N supports the conjecture that the evolving voter model has a one parameter family of quasi-stationary distributions, for if this is true, then the values of all of the graph statistics can be computed from N1/N. To further test this conjecture, we examined the joint distribution of the opinions at three sites. Let Nijk be the number of oriented triples x-y-z of adjacent sites having states i, j, k, respectively. Note for example, in the 010 case, this will count all such triples twice, but this is the approach taken in the theory of limits of dense graphs (46), where the general statistic is the number of homomorphisms of some small graph (labeled by ones and zeros in our case) into the random graph being studied.
Fig. 5 shows a plot of N010/N versus N1/N. After an initial transient, the observed values stay close to a curve that is well approximated by a cubic. Simulations of the other Nijk show similar behavior. Because the numbers of 010 triples must vanish when the number of 1-0 edges do, the fitted cubic shares two roots with the quadratic approximating the graph of N10/M versus N1/N. This quadratic curve (see again Fig. 4 for α = 0.5) is fundamental to our understanding of the observed system behavior, and we hereafter refer to it as the “arch.”
Fig. 5.
Plot of N010/N versus N1/N when α = 0.5 in the rewire-to-random case. All simulations start at u = 0.5 because multiple runs from one starting point are enough to explore all of the arch. These results are from graphs with N = 100,000 vertices and plotted every 10,000 steps.
The phenomena just described for α = 0.5 also hold for other values of α. Fig. 6 shows the arches that correspond to α = 0.1,0.2,…,0.7. Numerical results show that the curves are well approximated by cαu(1 - u) - bα. Let (v(α),1 - v(α)) be the “support interval” where the arch has positive values. Simulations show that if u < v(α), then the simulated curve rapidly goes almost straight down and hits the axis where N10 = 0.
Fig. 6.
Observed arches for the rewire-to-random model. The specified parabolas are fits to simulation data with N = 10,000, λ = 4.
Conjecture 1.
In the rewire-to-random model, if α < αc(1/2) and v(α) < u ≤ 1/2, then starting from product measure with density u of ones, the evolving voter model converges rapidly to a quasi-stationary distribution να,u. At time tN, the evolving voter model looks locally like να,θ(t) where the density changes according to a generalized Wright–Fisher diffusion process
until θt reaches v(α) or 1 - v(α).
Here the quantity under the square root is (1 - α)N10/M with (1 - α) equal to the fraction of steps that are voter steps because rewiring steps do not change the number of ones.
If Conjecture 1 is true, then the universal curve in Fig. 2 has ρ(α,0.5) = v(α) for α < αc(0.5). When α is close to αc(0.5), v(α) ≈ 1/2, so when the evolving voter model hits N10 = 0 both opinions are held by large groups, and the graph splits into two giant connected components (that is, their size is proportional to N for large graphs). Fig. 7 visualizes a network shortly before this “graph fission” for α = 0.65.
Fig. 7.
Visualization of the rewire-to-random model soon before fission occurs, for N = 500 nodes, average degree λ = 4, and α = 0.65. Colors correspond to the two opinions.
Though the nature of the phase transition looks different in the rewire-to-same model, the underlying picture is the same. Fig. 8 shows arches computed from simulations for the rewire-to-same model that correspond to the ones in Fig. 6 for the rewire-to-random model. However, now all the arches have the same support interval, (0,1), and the formulas in that figure show that the curves are well approximated by cαu(1 - u) for different values of cα.
Fig. 8.
Observed arches for rewire-to-same model. The specified parabolas are fits to simulation data with N = 10,000, λ = 4.
In the rewire-to-same case, Vazquez et al. (40) noticed that the fraction of active links N10/M plotted versus the fraction of ones converged rapidly to an arch and then diffused along it (see ref. 40, figure 4). However, they did not formulate the following:
Conjecture 2.
In the rewire-to-same model, the behavior is as described in Conjecture 1 but now bα = 0, so αc is independent of the initial density u, and for α < αc, ρ ≈ 0.
Approximate Calculations
Our next question is why is there a difference in the behavior of the support intervals of the arches in the two models? Consider first the rewire-to-random model. By considering all of the possible changes, one can write differential equations for the graph statistics. For example,
To be mathematically precise, for any T < ∞ this holds for t ≤ T in the limit as the number of vertices tends to ∞. Thus these equations describe the initial behavior (i.e., as the evolving voter model converges to the arch of quasi-stationary distributions) and their equilibria as a function of u describe the arch itself.
Note that the derivatives of the two-vertex quantities Nij involve two or three vertices. Derivatives of three-vertex quantities involve up to four vertices. To be able to solve the equations, we need some approximation to close the equations. In physics and ecology (see ref. 47 and references therein), it is common to use the pair approximation (PA), which in essence assumes the equilibrium state is a Markov chain: N100 = N10N00/N0. A little algebra then shows (see SI Materials for more details about the results in this section) that, for u∈(0,1/2],
[1]
When λ = 4, αc(1/2) = 6/7 and αc(u) → 3/4 as u → 0, so the predicted fraction ρ with the minority opinion at the end rises from zero to one-half on [3/4,6/7]. Fig. 9 compares the predicted value of ρ with simulations in Fig. 2 and shows the agreement is very poor.
Fig. 9.
Predictions of the ending minority fraction ρ(α) in rewire-to-random case from pair approximation (dashed line) and approximate master equation (solid line), compared with simulation (x).
One gets much better estimates of ρ from the approximate master equation (AME) framework introduced in ref. 48 to analyze binary state dynamics such as epidemic models. Here we follow the approach in refs. 49. Let [] be the number of nodes at time t that are in state 0 (susceptible) [in state 1 (infected)], have degree k, and have m neighbors in state 1.
To describe the logic behind the AME, let x be a vertex and let y be one of its neighboring vertices. Three types of things can happen: (i) rewiring may break the connection between x and y or bring a new edge to connect to x; (ii) x or y may influence the other by a voting step; or (iii) the opinion of y may be changed by imitating one of its neighbors z ≠ x.
Exact equations can be written for the first two types of events in terms of and , but the third type requires an approximation. For example, if x and y are both in state 0, we postulate that y changes from zero to one at rate N001/N00, the expected number of 1 neighbors of a 0-0 edge. This reasoning is similar to the PA, but now we do not suppose that N001/N00 = N01/N0, which, if numerical results are accurate, approximates the ratio of two cubic by a quadratic over the density u. Instead, we use identities such as
to compute the evolution of N001/N00 and other similar terms. As shown in Fig. 10, the AME gives a better approximation of the final minority fraction ρ than the PA. More importantly, the AME gives the correct qualitative behavior: the predicted ρ(α) > 0 for all α > 0 and tends to zero as α → 0.
Fig. 10.
Arches computed by approximate master equation (solid lines) versus simulation (dashes) for rewire-to-random model with α = 0.4, 0.5, 0.6, 0.7. The curves decrease as α increases.
One can repeat the analysis described above for the rewire-to-same model. Using the PA, we conclude that αc = (λ - 1)/λ and the arches N10/N = u(1 - u)[λ - 1/(1 - α)] always span (0,1). This qualitative behavior agrees with Fig. 8, but the PA estimate of αc = 3/4 when λ = 4 drastically overestimates the value αc ≈ 0.43 that comes from simulation (see Fig. 1). Again, one can numerically solve differential equations to employ the AME. The computed arches span (0,1) but the numerical predictions of ρ and the estimate of αc are more accurate. See SI Materials for details of the application of the PA and AME to both models.
Discussion
We have considered a model in which the opinions of individuals and network structure coevolve. Based on a combination of simulation and approximate calculations we conclude that (i) there is a discontinuous transition in the rewire-to-same model, similar to that in Holme and Newman (38), which occurs at an αc independent of the initial fraction u of ones; and (ii) there is a continuous transition in the rewire-to-random model at the critical value αc(u) that depends on u, and the curves for the final fraction ρ(α,u) of voters in the minority agree with ρ(α,1/2) for α < αc(u).
Thus, a small change in the dynamics of the model results in a large change in the qualitative behavior and in a manner that we find counterintuitive. One would think that the rewire-to-same dynamic would result in a more rapid division of the population into two noninteracting groups with different opinions. The critical value for the amount of rewiring αc needed to produce rapid disconnection is smaller in the rewire-to-same case than αc(1/2) for the rewire-to-random. Moreover, in the rewire-to-same case, the size of the minority opinion shrinks to almost zero for α < αc, whereas in the rewire-to-random case, the group fissions into two, leaving a significant minority group.
Calculations based on the approximate master equation reproduce the qualitative behavior of the phase transition. However, it would be nice to derive results directly from the exact differential equations and in a way that gives some insight into the mechanisms underlying the differences between the two models.
Acknowledgments.
The authors thank Raissa D’Souza, Eric Kolaczyk, Tom Liggett, and Mason Porter for their many helpful suggestions. This work began during the 2010–2011 program on Complex Networks at the Statistical and Applied Mathematical Sciences Institute. This work was partially supported by National Science Foundation Grants DMS-1005470 (to R.D.) and DMS-0645369 (to P.J.M.), by Science Foundation Ireland Grants 06/IN.1/I366, and Mathematics Applications Consortium for Science and Industry 06/MI/005 (to J.P.G.), and by the Research and Policy for Infectious Disease Dynamics program at National Institutes of Health (A.L.L.).
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Cover image: Connected to the Nile’s floodwaters, the oasis pictured here was developed after 280 BCE by Ptolemy II, son of one of Alexander’s generals. Cultivation and productivity in the region were expanded through a regulator dam that controlled the influx of Nile water, while an outer ring of master canals helped secure irrigation for new lands. The article by Karl W. Butzer, part of the Critical Perspectives on Historical Collapse Special Feature, reports that a succession of plague, reduced Nile floods, and mismanagement, led to the collapse of the upstream canals, depopulation, and widespread abandonment, illustrating the ecological dereliction of a landscape that did not fully recover until about 1900 CE. See the Perspective by Karl W. Butzer on pages 3632–3639. Image courtesy of Karl W. Butzer and John Oswald (University of Texas at Austin, Austin, TX).
The authors thank Raissa D’Souza, Eric Kolaczyk, Tom Liggett, and Mason Porter for their many helpful suggestions. This work began during the 2010–2011 program on Complex Networks at the Statistical and Applied Mathematical Sciences Institute. This work was partially supported by National Science Foundation Grants DMS-1005470 (to R.D.) and DMS-0645369 (to P.J.M.), by Science Foundation Ireland Grants 06/IN.1/I366, and Mathematics Applications Consortium for Science and Industry 06/MI/005 (to J.P.G.), and by the Research and Policy for Infectious Disease Dynamics program at National Institutes of Health (A.L.L.).
To whom correspondence should be addressed. E-mail: [email protected].
Contributed by Richard T. Durrett, January 13, 2012 (sent for review October 26, 2011)
Author contributions: R.D., J.P.G., A.L.L., P.J.M., F.S., D.S., J.E.S.S., and C.V. performed research; and R.D. and P.J.M. wrote the paper.
Competing Interests
The authors declare no conflict of interest.
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R. Durrett, J.P. Gleeson, A.L. Lloyd, P.J. Mucha, F. Shi, D. Sivakoff, J.E.S. Socolar, C. Varghese, Graph fission in an evolving voter model, Proc. Natl. Acad. Sci. U.S.A.
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, in which each vertex decides to change its opinion at rate 1, and when it does, it adopts the opinion of one of its 2d nearest neighbors chosen at random. Let ξt(x) be the opinion of the voter at x at time t. Holley and Ligget (43) and Liggett (44) proved the following result.
are independent and equal to one with probability p], then 
converges in distribution to a limit νp, which is a stationary distribution for the voter model.
. In this setting, the behavior of the voter model is “trivial” because it is a finite Markov chain with two absorbing states, all ones and all zeros. As the next result shows (see Cox and Greven, ref. 45), the voter model has interesting behavior along the road to absorption.
and βd is the probability that two random walks starting from neighboring sites never hit.
until θt reaches v(α) or 1 - v(α).
To be mathematically precise, for any T < ∞ this holds for t ≤ T in the limit as the number of vertices tends to ∞. Thus these equations describe the initial behavior (i.e., as the evolving voter model converges to the arch of quasi-stationary distributions) and their equilibria as a function of u describe the arch itself.
When λ = 4, αc(1/2) = 6/7 and αc(u) → 3/4 as u → 0, so the predicted fraction ρ with the minority opinion at the end rises from zero to one-half on [3/4,6/7]. Fig. 9 compares the predicted value of ρ with simulations in Fig. 2 and shows the agreement is very poor.
[
] be the number of nodes at time t that are in state 0 (susceptible) [in state 1 (infected)], have degree k, and have m neighbors in state 1.
and 
, but the third type requires an approximation. For example, if x and y are both in state 0, we postulate that y changes from zero to one at rate N001/N00, the expected number of 1 neighbors of a 0-0 edge. This reasoning is similar to the PA, but now we do not suppose that N001/N00 = N01/N0, which, if numerical results are accurate, approximates the ratio of two cubic by a quadratic over the density u. Instead, we use identities such as
to compute the evolution of N001/N00 and other similar terms. As shown in Fig. 10, the AME gives a better approximation of the final minority fraction ρ than the PA. More importantly, the AME gives the correct qualitative behavior: the predicted ρ(α) > 0 for all α > 0 and tends to zero as α → 0.
