The coevolving voter model with spin-dependent rewiring probability mean field approach
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1 The coevolving voter model with spin-dependent rewiring probability mean field approach Krzysztof Kułaowsi, AGH-UST, Cracow in collaboration with Joanna Toruniewsa, Krzysztof Sucheci, Janusz A. Hołyst - Warsaw TU aria Stojow, Dorota Żuchowsa-Siba, Przemysław Gawrońsi AGH-UST Summer Solstice, Gdańs, June 5-7, 018
2 Jacob L. oreno ( : first sociograms - Sing Sing prison, - a reformatory for deliquent girls, - public and private Broolyn schools.
3 within a given more or less homogeneous group members of an alien group may be introduced ( The group can assimilate, as it were, a certain number, but beyond that point assimilation is rendered difficult or impossible and the group tends to brea up along the lines of cleavage created by the alien group forming a minority group within the majority group. [J. L. oreno, Who shall survive? A new approach to the problem of human interrelations, Washington D.C., 1934]
4 outline * Homophily and social contagion * odel evolution : two inds of discrete events * ean field equation of motion after Vazquez et al., 008 * ore detailed mean field equations of motion * Analytical results vs simulations * Even more detailed mean field equations of motion * ore analytical results * Are the conclusions meaningful for minorities?
5 homophily : similarity breeds connection social contagion : connection breeds similarity Do people befriend others who are similar to them, or do they become more similar to their friends over time? movies, music homophily Exceptions: classical/jazz contagion indie/alternative anti-contagion [Lewis et al., PNAS 109 (01] EJ Newman, SIA Rev 003 (arxiv:cond-mat/
6 [F. Vazquez, V.. Eguiluz,. San iguel, PRL 100 ( ] Coevolving voter model The model system is a networ of nodes decorated with spins. Two processes compete here: - rewiring (->homophily with probability p - flips (->contagion with probability 1-p
7 Equations of motion after Vazquez et al. For the binomial distribution and n B( n / (1 then n0 d dt n0 [(1 p( 1(1 nb( n / 1] f ( Fixed points: f ( * 0, then * 1 0 (frozen phase or (1 p( 1 1 * (active phase 0 (1 p( 1 [F. Vazquez, V.. Eguiluz,. San iguel, PRL 100 ( ]
8 [F. Vazquez, V.. Eguiluz,. San iguel, PRL 100 ( ] f( * 1 =0 * is unstable, * exists and is stable, iff * 1 df ( d 0 0 df ( [(1 p ( 1 1] d 0 Hence, df ( d 0 0 for p p c 1 p is the probability of rewiring. If p is large enough, active lins disappear.
9 Parameterization: m,,, N N N N N Nm N N N hence N N (1 m N (1 N (1 - mean degree of node N N hence 1 1 m
10 ore detailed equations of motion (1 ( 1 ( (1 m p dt d m p dt dm 1 1 (1 (1 4 (1 (1 (1 (1 (1 (1 m m p p m p dt d
11 What could seem a bit strange: (1 p ( t pm( t const R p = 1 only rewiring, m = const p = 0 only flips, = const The final solution depends on the initial state.
12 Numerical results: N=510 4, #=10 3 p = 0.1, mainly flips p = 0.9, mainly rewiring
13 , m,, m, Analytical (- - - vs numerical ( results active phase: > 0 active phase: > 0 frozen phase: = 0 frozen phase: = 0
14 In silico: N= 500, p=0.1 or N=00, p=0.9
15 Even more detailed equations of motion d [ m( pb (1 (1 pb pb ] dt 1 [ m( pa (1 (1 pa pa] 1 [1 m (1 ](1 pb [1 m (1 ](1 pa (1 (1 dm dt d dt (1 m(1 pa (1 m(1 pb 1 1 (1 [1 m (1 ] pb (1 [1 (1 m(1 ] p a
16 (p a, p b for 0 =0.5, m 0 = 0, 0 = 0
17 (p a, p b for 0 =0.5, m 0 = 0.8, 0 = 0.6 p b p a (p a, p b for 0 =0.5, m 0 = 0.8, 0 = -0.6 p b p a
18 m(p a, p b for 0 =0.5, m 0 = 0, 0 = 0 m = +1 m = -1
19 contour map m(p a, p b for 0 =0.5, m 0 = 0, 0 = 0 m > +0.8 m = 0 m < - 0.8
20 m = 0 m = +1 m = -1 m(p a, p b for 0 =0.5, m 0 =- 0.8, 0 = 0.6
21 In silico again: N=100, p=0.1
22 Conclusions The model by Vazquez et al has been generalized for the case where: - the mean number of neighbors of an actor - and the probability p a,b of rewiring depend on the group (a,b. Once p a p b, the active phase vanishes. Either the groups mutually separate, or one of them is absorbed. The group a cannot be absorbed as long as p a > p b. A competition? Two bad news [J. Toruniewsa et al., PRE 96 ( KK et al., IJPC (018, in print (arxiv: ]
23 Than you
24 Equations of motion after Vazquez et al. / 1 / ( ( / / ( ( (1 0 0 N n n B P Np N n n n B P p N dt d n n [F. Vazquez, V.. Eguiluz,. San iguel, PRL 100 ( ] number of all lins prob of n active out of lins density of active lins rewire flip
25 Equations of motion after Vazquez et al. density of active lins flip active lins <-> frozen lins d dt Np N(1 P( p n0 P( B( n / n0 n B( n / 1 N / n n N / active lin -> frozen lin rewire fraction of active lins [F. Vazquez, V.. Eguiluz,. San iguel, PRL 100 ( ]
26 ab number of lins from a to b ab a -> b aa -> rewiring, probability p ab ba aa ab ba aa 1 1 flip, probability 1-p N N a b aa ab ba bb N N a b 1 1 aa ab ba bb ab ab aa ab aa aa
27 ba number of lins from b to a bb b -> a ba -> rewiring, probability p ab ba bb ab ba bb 1 1 flip, probability 1-p N N a b aa ab ba bb N N a b 1 1 aa ab ba bb bb ba ba bb ba bb
28 If Rabbit Was bigger And fatter And stronger, Or bigger Than Tigger, If Tigger was smaller, Then Tigger's bad habit Of bouncing at Rabbit Would matter No longer, If Rabbit Was taller. A. A. ilne
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