Holme and Newman (2006) Evolving voter model. Extreme Cases. Community sizes N =3200, M =6400, γ =10. Our model: continuous time. Finite size scaling

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1 Evolving voter model Rick Durrett 1, James Gleeson 5, Alun Lloyd 4, Peter Mucha 3, Bill Shi 3, David Sivakoff 1, Josh Socolar 2,andChris Varghese 2 1. Duke Math, 2. Duke Physics, 3. UNC Math, 4. NC State Math, 5. MACSI, U of Limerick, Ireland A result of the SAMSI program on complex networks PNAS 19 (212) Holme and Newman (26) TheybeginwithanetworkofN nodes and M edges, where each node x has an opinion ξ(x) fromasetofg possible opinions and the number of people per opinion γ N = N/G stays bounded as N gets large. On each step a vertex x is picked at random. If its degree d(x) =, nothing happens. If d(x) >, (i) with probability 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 ) a random neighbor y of x is selected and the opinion of x is set to ξ(y). Eventually there are no edges that connect different opinions and the system freezes at time τ N. Durrett (Duke) MBI 4/2/212 1/34 Durrett (Duke) MBI 4/2/212 2/34 Extreme Cases Community sizes N =32, M =64, γ =1. When =1only rewiring steps occur, so once all of the M edges have been touched the graph has been disconnected into G components, each of which is small. By results for the coupon collector s problem, τ N M log M updates. When φ =this is a voter model on a static graph. Ifweusean Erdös-Renyi random graph in which each vertex has average degree λ>1 then there is a giant component with a positive fraction of the vertices and a large number of small components with size O(log N). The giant component will reach consensus after τ N = O(N 2 ) updates, so the end result is one opinion with a large number of followers while all of the other populations are small. Durrett (Duke) MBI 4/2/212 3/34 Durrett (Duke) MBI 4/2/212 4/34 Finite size scaling Our model: continuous time Events happen on each oriented edge (x, y) at times of a rate one Poisson process. (Isothermal voter model.) N 2 updates time N. If the voters at the two ends of the edge agree then we do nothing. If they disagree, then with probability 1 the voter at x adopts the opinion of the voter at y. With probability, x breaks its connection to y and makes a new connection to a voter chosen at random: (i) from all of the vertices in the graph rewire to random, (ii) from those that share its opinion rewire to same. Opinions {, 1}. Initial state product measure with density u. Durrett (Duke) MBI 4/2/212 5/34 Durrett (Duke) MBI 4/2/212 6/34

2 Rewire to random with u =1/2 Rewire to random: A universal curve? Number in minority Fraction in minority state initial fraction =.5 initial fraction =.1 initial fraction =.25 initial fraction = Figure: Erdos-Renyi, λ =4,N = 1, Figure: Critical value c (u) depends on u, butwhen< c (u) minority fraction agrees with curve for u =1/2. Durrett (Duke) MBI 4/2/212 7/34 Durrett (Duke) MBI 4/2/212 8/34 Rewire to same: discontinuous transition The simulation that showed us the answer.5 N=1, p=.4, u=.5, =.3, Rewire to Random initial fraction =.5 initial fraction =.1 initial fraction =.25 initial fraction =.5 minority fraction Fraction in Minority State N actual time Figure: N = 1, λ =4,u =1/2, Initial N 1 = 1. Durrett (Duke) MBI 4/2/212 9/34 Durrett (Duke) MBI 4/2/212 1 / 34 Holley and Liggett (1975) Cox and Greven (199) Consider the voter model on the d-dimensional integer lattice Z d 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. In d 2, the system approaches complete consensus. That is if x y then P(ξ t (x) ξ t (y)). In d 3ifwestartfromξ p product measure with density p, i.e., ξp (x) are independent and equal to 1 with probability then ξt p converges in distribution to a limit ν p, which is a stationary distribution for the voter model. The voter model on the torus in d 3attimeNt then it locally looks like ν θ(t) where the density changes according to the Wright-Fisher diffusion: dθ t = β d 2θ t (1 θ t )db t The quantity under the square root is the fraction of discordant edges under ν θ(t). There is a one parameter family of quasi-stationary distributions, and the parameter changes according to a diffusion. quasi-stationary since in the finite voter model all s and all 1 s are absorbing. Durrett (Duke) MBI 4/2/ / 34 Durrett (Duke) MBI 4/2/ / 34

3 N 1 versus N 1, =.5 Graph fission for =.65 Figure: Process comes quickly to the arch then diffuses along it, splitting into two when it reaches the end. Durrett (Duke) MBI 4/2/ / 34 Durrett (Duke) MBI 4/2/ / 34 Finite dim. distr. on a random graph N 1 versus N 1, =.5 A definition from the theory of graph limits of Lovasz et al. N ijk is the number of homomorphisms from the labeled graph i j k a b c into our labeled graph (G,ξ). When i =,j =1,k =everytripleis counted twice but this seems like the natural definition. Durrett (Duke) MBI 4/2/ / 34 Durrett (Duke) MBI 4/2/ / 34 N ijk are polynomials? Evolution Equations Bill Shi s simulations for λ =4, =.5 N 1 = 3.42x x.38 N 11 = 13.53x x x.3 N 1 =13.54x x x 1.77 N 11 = 1.14x x x +.8 N 1 =1.15x x x 1.2 N 11 (x) =N 1 (1 x), N 11 (x) =N 1 (1 x) dn 1 = (2 )N 1 +(1 )[N 1 N 1 + N 11 N 11 ] dt 1 dn 11 =(1 (1 u))n 1 +(1 )[N 11 N 11 ] 2 dt 1 dn =(1 u)n 1 +(1 )[N 1 N 1 ] 2 dt Of course N 11 +2N 1 + N = M, the number of edges. N ijk = d(y)(d(y) 1) y d dt ijk ijk N ijk = 2[N 11 + N 1 + N 1 + N 11 ]+4N 1 M N Durrett (Duke) MBI 4/2/ / 34 Durrett (Duke) MBI 4/2/ / 34

4 One equation short Arches for rewire to random When λ =4, =.5 from equations from simulation N 11 /N 1 (2a 3 +2b 3 ) 2b 3 u u N 1 /N 1 2a 3 +2b 3 u u N 11 /N 1 (2a 3 +2b 3 +1) (2b 3 1)u u N 1 /N 1 (2a 3 +2)+(2b 3 1)u u From (d/dt) ijk N ijk = we get 2λ(1 ) =4a b 3 Equations and simulation agree if 2a 3 =2.73 and 2b 3 = Durrett (Duke) MBI 4/2/ / 34 Durrett (Duke) MBI 4/2/212 2 / 34 Arches for rewire to same Why do the arches behave differently in the two versions of the model? Figure: Note that constant term, explaining discontinuous distribution. Durrett (Duke) MBI 4/2/ / 34 Durrett (Duke) MBI 4/2/ / 34 Extensions Degree distribution Poisson? N = 1 5 =.6 u = γ = We get the same result if we start with a random 4-regular graph OR if we designate un vertices as 1 and (1 u)n as and connect an i node to a j node with probability p ij /N. Inthesecondcasebychoosingthep ij correctly we can achieve any possible value of N 1 /N and N 1 /M where M = λn/2. For these initial conditions we quickly move to the arch of quasistationary distributions. p k simulation Poisson(4) k Durrett (Duke) MBI 4/2/ / 34 Durrett (Duke) MBI 4/2/ / 34

5 Poisson iff fraction of neighbors is constant Fraction of degree k nodes =1 N=1 5, =.6, u=.5, γ =.17, Average over 1 trials q k /p k.14 γ k Durrett (Duke) MBI 4/2/ / 34 Durrett (Duke) MBI 4/2/ / 34 Open Problems Prove quasi-stationary distributions exist when is small. When = we know the quasi-stationary distributions for the voter model, so it is natural to try a perturbation argument. However when we consider (G,ξ)forthevotermodeltheG does not change. Singular perturbation problem. Work with Jonathan Mattingly and David Sivakoff. First step: understand lim which is systemwith =. Answer: rewire, voter model equilibrates, rewire again. Harder second step: There is a unique stationary distribution on the space of graphs with N vertices and M edges. Conjecture. Intherewiretorandommodelif< c (1/2) and v() < u 1/2 then starting from product measure with a density u of 1 s, the evolving voter model converges rapidly to a quasi-stationary distribution ν,u. At time tn the evolving voter model looks locally like ν,θ(t) where the density changes according to a generalized Wright-Fisher diffusion process dθ t = (1 )[c θ t (1 θ t ) b ]db t until θ t reaches v() or 1 v(). Rewire to same is similar but b =. What happens with more than two initial types? Durrett (Duke) MBI 4/2/ / 34 Durrett (Duke) MBI 4/2/ / 34 Arch is c (1 i u2 i )/2 b for same c, b? c and b as a function of β = /(1 ) Generator L = V + βr, Voter, Rewire 5.2 # discordant edges c() y =.51*x /1 b() y =.26*x / Figure: Conjecture. c = a +2bβ, b = bβ Linear in β, not a power series. Durrett (Duke) MBI 4/2/ / 34 Durrett (Duke) MBI 4/2/212 3 / 34

6 Infinitely many phase transitions Higher order statistics Suppose c = a +2bβ, b = bβ, β = 1 If there are k opinions the smallest value of 1 i u2 i is 1 1/k N = c 2 (1 1/k) b This is when β k = a(k 1)/2b. (needs2b and b). a =1.3, b =.25 k =2,β 2 =2.6, 2 = β 2 /(1 + β 2 )=.72 k =3,β 3 =5.2, 3 =.84 k is the critical value for starting with k types. If we start with a large number types then for < 2 we may end up with two or more types at the end, but if 2 << 3 we will always end up with three or more, etc. = y = N 1 /N versus x = N 1 /N.1 y =8.3256x x x y =8.3574x x x y =8.696x x x y =8.9222x x x y =9.9584x x x y = x x x y = x x x ax(1 x) 2 + bx(1 x) =ax 3 (2a + b)x 2 +(a + b)x a = c λ p(x y z) b = c λ p(xz y) c λ = x d x(d x 1)/N, p coalesce probs averaged over triples Durrett (Duke) MBI 4/2/ / 34 Durrett (Duke) MBI 4/2/ / 34 N 1 Coefficients Quadratic: Constant Term Exactly solvable model?.5 data 1 quadratic y =.18*x 2.75*x.51 Graph statistics N 1, N 1, N 1, etc. are polynomials in u and β Unfortunately there does not seem to be a dual for the evolving voter model N 1 N 1 =(1+βu)N 1 Durrett (Duke) MBI 4/2/ / 34 Durrett (Duke) MBI 4/2/ / 34