Generic absorbing transition in coevolution dynamics.

Federico Vazquez,∗ Vıctor M. Eguıluz, and Maxi San MiguelIFISC, Instituto de Fısica Interdisciplinar y Sistemas Complejos (CSIC-UIB), E-07122 Palma de Mallorca, Spain

(Dated: January 12, 2008)

We study a coevolution voter model on a network that evolves according to the state of the nodes.In a single update, a link between opposite-state nodes is rewired with probability p, while withprobability 1−p one of the nodes copies its neighbor’s state. A mean-field approximation reveals anabsorbing transition from an active to a frozen phase at a critical value pc = µ−2

µ−1that only depends

on the average degree µ of the network. The approach to the final state is characterized by a timescale that diverges at the critical point as τ ∼ |pc − p|−1. The active and frozen phases correspondto a connected and a fragmented network respectively. We show that the transition in finite-sizesystems can be seen as the sudden change in the trajectory of an equivalent random walk at thecritical rewiring rate pc.

PACS numbers: 89.75.Fb, 05.40.-a, 05.65.+b, 89.75.Hc

The dynamics of collective phenomena in a system ofinteracting units depends on both, the topology of thenetwork of interactions and the interaction rule amongconnected units. The effects of these two ingredients onthe emergent phenomena in a fixed network have been ex-tensively studied. However, in many instances, both thestructure of the network and the dynamical processes onit evolve in a coupled manner [1, 2]. In particular, in thedynamics of social systems (Refs. [1, 3–5] and referencestherein), the network of interactions is not an exogenousstructure, but it evolves and adapts driven by the changesof the state of the nodes of the network. In recent modelsimplementing this type of coevolution dynamics [2, 4–12] a transition is often observed from a phase where allnodes in the network are in the same state forming asingle connected network to a phase where the networkis fragmented into disconnected components, each com-posed by nodes in the same state [13].

In this paper we address the question of how generic isthis type of transition and which is the mechanism behindit. For this purpose, we introduce a minimal model of in-teracting binary state nodes that incorporates two basicfeatures shared by many models displaying a fragmenta-tion transition: (i) two or more absorbing states in a fixedconnected network, and (ii) a rewiring rule that does notincrease the number of links between nodes in oppositestate. The state dynamics consists of nodes copying thestate of a random neighbor, while the network dynamicsresults from nodes dropping their links with opposite-state neighbors and creating new links with randomlyselected same-state nodes. This model can be thoughtas a coevolution version of the voter model [14] in whichagents may select their interacting partners according totheir states. It has the advantage of being analyticallytractable and allows a fundamental understanding of thenetwork fragmentation, explaining the transition numer-ically observed in related models [5, 8–12]. The mech-anism responsible for the transition is the competitionbetween two internal time scales, happening at a critical

value that controls the relative ratio of these scales.We consider a network with N nodes. Initially, each

node is endowed with a state +1 or −1 with the sameprobability 1/2, and it is randomly connected to exactlyµ neighbors, forming a network called degree-regular ran-dom graph. In a single time step (see Fig. 1), a node iwith state Si and one of its neighbors j with state Sj arechosen at random, then:

1. if Si = Sj nothing happens.

2. if Si 6= Sj , then with probability p, i detaches itslink to j and attaches it to a randomly chosen nodea such that Sa = Si and a is not already connectedto i; and with probability 1 − p, i adopts j’s state(Si → Si = Sj).

The rewiring probability p measures the rate at which thenetwork evolves compared to the rate at which the statesof the nodes change; the extreme values correspond to afixed network (p = 0), and to only rewiring (p = 1).

Link Dynamics. The evolution of the system can bedescribed by the densities of two different types of links:links connecting nodes with different states or active links

and links between nodes in the same state or inert links.Note that an update (either rewire or copy) only occurswhen an active link is chosen.

In Fig. 1 we describe the possible changes in the globaldensity of active links ρ and their probabilities in a singletime step, when a node of degree k is chosen. We denoteby n the number of active links connected to node i be-fore the update. With probability n/k an active link i−jis randomly selected. Then with probability p the linki − j is rewired and becomes inert (link i − a), giving alocal change of active links ∆n = −1 and a global densitychange of ∆ρ = − 2

µN , where µN/2 is the total number

of links, µ = 〈k〉 =∑

k kPk(t) is the number of links pernode or average degree and Pk(t) is the node degree dis-tribution at time t. It is worth noting that even thoughPk(t) depends on time given that the network is con-stantly evolving, µ is constant because the total number

2

j

a

i

j

a

i

k−n inert links

n active links

m = s (k−n)

j

a

i

nk

prewire

nk

(1−p) copy

n−1 active linksk−n+1 inert links

m∆ = s−2µΝ

k−n active linksn inert links

k−2n 2( )µΝ

∆ρ=

m = s (k−n+1)

∆ρ= −2µΝ

m = −s n

−2 s kµΝ∆ =m

FIG. 1: Update events and the associated changes in thedensity of active links ρ and the link magnetization m =ρ++ − ρ−− when two neighbors i and j with states Si = s

and Sj = −s are chosen (s = ±1).

of links is conserved at each time step. Numerically wefind that the dynamics leads to a narrow degree distribu-tion as expected from the random nature of the rewiring.With probability (1 − p) node i flips its state changingthe state of links around it from active to inert and vice-versa, and leading to ∆n = k − 2n and ∆ρ = 2(k−2n)

µN .Assembling these factors, the change in the average den-sity of active links in a single time step of interval dt = 1

Nis described by the master equation

dρ

dt=

∑

k

Pk

1/N

k∑

n=0

Bn,kn

k

[

(1 − p)2(k − 2n)

µN− p

2

µN

]

=∑

k

Pk2

µk

[

(1 − p)(

k〈n〉k − 2〈n2〉k)

− p〈n〉k]

,(1)

where Bn,k is the probability that n active links are con-nected to a node of degree k, and 〈n〉k and 〈n2〉k are thefirst and the second moments of Bn,k.

In this mean-field (MF) approach we consider that thestate (active or inert) of the links around a node areuncorrelated, thus we approximate the probability thata given link is active as the density of active links ρ.We also assume that all nodes are equivalent, i.e., theprobability that a link connected to a given node i withdegree ki and state Si is active is independent on ki andSi. With these assumptions Bn,k becomes the binomialdistribution being ρ the probability for the occurrenceof an independent event. Thus, the first and the secondmoments are 〈n〉k = ρk and 〈n2〉k = ρk + ρ2k(k − 1)respectively. Replacing these expressions in Eq. (1) weobtain a closed equation for the time evolution of ρ in aninfinite system, where fluctuations are neglected

dρ

dt=

2ρ

µ[(1 − p)(µ − 1)(1 − 2ρ) − 1] . (2)

Equation (2) has two stationary solutions. For p < pc,the stable solution is

ρs ≡ ξ(p) =(1 − p)(µ − 1) − 1

2(1 − p)(µ − 1), (3)

0 1p

0

0.5

pc

pc1/2

ACTIVE PHASE

FROZENPHASE

ρs

ρ+ +, ρ− −

FIG. 2: Stationary density of active links ρs and the two typesof inert links ρ++ and ρ−− vs the rewiring probability p asdescribed by the mean-field theory for a network with averagedegree µ = 4. The critical point pc separates an active froma frozen phase.

corresponding to an active steady-state with a constantfraction of active links in the system; for p > pc, thestable solution ρs = 0 corresponds to an absorbing statewhere all links are inert. Thus, the MF approach predictsan absorbing transition (Fig. 2) from an active to a frozenphase at a critical value

pc =µ − 2

µ − 1. (4)

A stationary state in the active phase, characterized byξ(p) = pc−p

2(1−p) , is composed by links continuously be-

ing rewired (evolving network) and nodes flipping theirstates. For p = 0 (original voter model), the valueξ(0) = µ−2

2(µ−1) agrees very well with the numerical values

of the voter dynamics in different random graphs [15, 16].In the frozen phase, final states correspond to a fixed net-work where connected nodes have the same state and nomore evolution is possible. The transition is continuous,with the order parameter ρs changing continuously at pc.Close and below the transition point, ρs scales as (pc−p),thus the MF critical exponent is 1.

In order to obtain an insight about the structure ofthe network in both phases we introduce ρ++ (ρ−−) asthe density of links connecting two nodes with states +1(−1). To relate ρ++ with ρ, we use the relation P (+) =P (+, +) + P (+,−), i.e, the marginal probability P (+)that a randomly selected node is positive is equal to thesum of the joint probabilities P (+, +) and P (+,−) offinding a ++ and a +− pair of nodes respectively. Weidentify P (+) with the density σ+ of + nodes, P (+, +)and P (+,−) with ρ++ and ρ/2 respectively. Combiningthese relations we arrive to

ρ++ = σ+ − ρ/2; ρ−− = σ− − ρ/2. (5)

Due to the conservation of the ensemble average of σ+

and σ− for the voter model dynamics, we have thatσ+(t) = σ−(t) = 1/2 and therefore, ρ++ = ρ−− =12 (1 − ρ). In the active phase the continuous rewiring of

3

0

0.2

0.4

0.6

0.8

1S

/N

0

0.2

0.4

0.6

0.8

1

|m|

0 0.2 0.4 0.6 0.8 1p

0

0.1

0.2

ρsurv

0 0.2 0.4 0.6 0.8 1p

0

1

2

3

τ/Ν

0 1000 2000time

0

0.1

0.2

0.3ρ

101

102

103

z

10-3

10-2

10-1

τ / N

ln(z

)

(a)

(c)

(b)

(d)

FIG. 3: (a) Average relative size of the largest network com-ponent S and (b) absolute value of the link magnetization m

vs p in the final frozen state. (c) Average convergence time τ

per system size N vs p. Inset: scaling of τ for p >∼ pc ≃ 0.38,

indicating that τ ∼ N

zln(z) with z = µ(p − pc)N . The solid

line has slope −1. (d) Stationary density of active links insurviving runs ρsurv. Inset: average time evolution of ρ. Theaverages are over 104 realizations of networks with µ = 4 andsizes N = 250 (circles), 1000 (squares) and 4000 (diamonds).

links keeps the network connected in a single component,i.e., a set of connected nodes. However, in the frozenphase only inert links are present and in the same pro-portion (ρ++ = ρ−− = 1/2), thus we expect the forma-tion of two large disconnected components with oppositestate (see Fig. 2). Therefore, the MF description revealsa fragmentation transition in the stationary structure ofthe network, which is associated with the absorbing tran-sition at pc.

Final states in a finite system. The previous MF ap-proach predicts a transition in the limit of an infinitelarge network. For any value of p, due to fluctuations, afinite size network eventually reaches an absorbing statecomposed by inert links only. We studied the structureof the network in the final state by performing numeri-cal simulations of the dynamics starting with a degree-regular random graph with connectivity µ = 4 and lettingthe system evolve until it was frozen. In Fig. 3(a) we plotthe average size of the largest network component S inthe final configuration for networks with N = 250, 1000and 4000 nodes. We observe that S is very close to N forvalues of p below a transition point pc ≃ 0.38, indicatingthat the network forms a single component [17]. Abovepc the network gets disconnected into two large compo-nents and a set of components of size much smaller thanN , giving a value of S ≃ N/2.

To compare the simulations with the MF results, wecalculated the stationary density of active links in surviv-ing runs ρsurv. As we show in Fig. 3(d), ρsurv monotoni-cally decreases with p, becoming sharper with increasing

system and indicating a transition from an active to afrozen phase as predicted by the MF theory. In the activephase, ρsurv reaches a steady value larger than zero andindependent on the system size N , while in the frozenphase ρsurv vanishes in the thermodynamic limit. Thecritical point for the active-frozen transition pc ≃ 0.38calculated from Fig. 3(d) is roughly the same as forthe fragmentation transition (Fig. 3(a)), suggesting thatthe active and frozen phases observed in infinite largesystems correspond to the connected and disconnectedphases respectively in finite systems. The MF criticalpoint pc = 2/3 calculated using Eq. (4) with µ = 4 dif-fers from the numerical value pc ≃ 0.38 (Figs. 3(a,d)) dueto correlations appearing in the rewiring process. Thesecorrelations, that are not taken into account in the ana-lytical approximation, make the first 〈n〉k and the secondmoment 〈n2〉k different from the analytical values ρk andρk + ρ2k(k − 1) respectively. These deviations have theoverall effect of decreasing the observed critical point re-spect to the theoretical one.

Approach to the absorbing states. So far, we haveshown that a finite network under the coevolving dy-namics experiments a fragmentation transition as therewiring rate is increased. We now unveil the mechanismof the fragmentation transition by studying the evolutionof the system to the frozen state.

We represent the state of the system as a point (m, ρ)in the 2 dimensional space, where the coordinates arethe link magnetization m = ρ++ − ρ−− and the den-sity of active links respectively. When a node of de-gree k connected to n active links is chosen, the possiblechanges in m and ρ and their respective probabilities arethose described in Fig. 1. In the (m, ρ) space, the sys-tem undergoes a random walk (RW) inside the triangle0 ≤ ρ + |m| ≤ 1, whose boundaries follow from the con-straint relation ρ−− + ρ++ + ρ = 1. The system reachesan absorbing configuration and stops evolving when theRW hits either one of the fixed points (−1, 0) or (1, 0)(all nodes in state − or + respectively) or a point on thefixed line ρ = 0 (frozen mixture of − and + nodes). Atthe point (−1, 0) ((1, 0)) only −− (++) links are present,the network is composed by a giant component, and thesystem is in the connected phase. Points on the line ρ = 0and close to point (0, 0) correspond to a frozen networkwith similar number of ++ and −− links arranged in twolarge + and − components (disconnected phase).

In Fig. 4 we plot trajectories of the RW in one re-alization for different values of p. For p < pc ≃ 0.38,the motion of the RW has two stages. In the first andvery short stage the RW travels along the m ≃ 0 axisfrom the starting point (m ≃ 0, ρ ≃ 1/2) to the point(m ≃ 0, ρ ≃ ξp) that corresponds to the steady stateρs = ξp in infinite large systems (see right inset of Fig.4).The value ξp agrees with the MF prediction ξ(p) fromEq. (3) only for p = 0. For p > 0, the correlations in-duced by the rewiring make the numerical plateau ξp

4

-1 -0.5 0 0.5 1m

0

0.1

0.2

0.3

0.4

0.5

ρ0 20000 40000 60000

time0

0.1

0.2

0.3

0.4

ρ

0 5000 10000 15000 20000

time

0.5

1

1.5

ρ / σ

+ σ−

4 ξ 0

ξ 0.35 = 0.11

ξ 0 = 1/3

ξ p

4 ξ 0.35

4 ξ 0.25

FIG. 4: Typical trajectories of the random walk for a networkof size N = 104 and average degree µ = 4. The upper (p =0) and lower (p = 0.35) parabolas are the trajectories forrewiring rates below the transition point p ≃ 0.38, while thequasi-vertical line is for p = 0.4. Insets: Time evolution ofthe density of active links ρ (right) and the ratio between ρ

and the product of the σ+ and σ− (left) in a single realizationfor different values of p.

smaller than ξ(p). In the second and long stage the RWdiffuses on the m direction, corresponding to the fluc-tuations of ρ down the metastable state ρs = ξp, untilit hits either point m = −1 or m = 1 (see Fig. 3(b)).We observe in Fig. 4 that the motion of the RW is notcompletely random but its trajectory fluctuates arounda curve described by

ρp(m) = ξp(1 − m2). (6)

The origin of this relation is that both ρ and m can beexpressed as functions of the density σ+. Numerical sim-ulations show that during one realization the ratio ρ

σ+σ−

fluctuates around the constant value 4 ξp (see left inset ofFig. 4). Then, using Eq. (5) we obtain ρ = 4 ξp σ+(1−σ+)and m = ρ++ − ρ−− = 2σ+ − 1, from where we arriveto Eq. (6) by eliminating σ+. For p > pc, the strongbias to the ρ = 0 line makes the RW hit a point close tothe origin (see the p = 0.4 trajectory in Fig. 4). Simula-tions show that the amplitude of the fluctuations of theRW’s trajectory around its mean value ρp(m) vanishesas N increases. Thus, starting from a fix value p < pc

and increasing N has the effect of increasing the prob-ability that the RW reaches the end points (−1, 0) or(1, 0) before it hits the line ρ = 0. As a consequence,for N finite, most realizations end in a single component,except for a small fraction, in which a giant componentand some small components appear, that vanishes as Nincreases. Eventually, in the large N limit, the RW hasthree absorbing points: either point (−1, 0) or (1, 0) (sin-gle component network) when p < pc, and point (0, 0)(two components network) when p > pc.

Convergence times. A magnitude of interest is the av-

erage time τ to reach an absorbing state. For p < pc,the m-coordinate of the walker performs a 1d symmetricrandom walk with an average jumping interval and itsprobability that scale as 1/N and ρ ∼ ξ respectively. Toreach one of the ends points m = ±1 the RW needs toattempt an average of N2/ξ steps, and given that thetime increases by 1/N in each attempt, we find thatτ ∼ N/ξ. From Eq. (2) for p >

∼ pc, ρ decays to zeroas ρ(t) ∼ −ξ e 4 ξ t/µ. The system freezes at a time τfor which ρ(τ) ∼ 1/µ N . Using the MF approximationξ(p) ∼ (pc−p) close to pc, we obtain that τ ∼ N(pc−p)−1

as p → p−c and τ ∼ (p− pc)−1 ln[µ(p − pc)N ] as p → p+

c ,thus the convergence to the final state slows down at thecritical point (see Fig.3(c)).

Summary and conclusions. In summary, the coevolu-tion mechanism on the voter model induces a fragmen-tation transition on the final frozen structure of a finitenetwork, that is a consequence of the competition be-tween the copying and the rewiring dynamics. In theconnected active phase, the system falls in a dynamicalmetastable state with a finite fraction of active links. Theslow and permanent rewiring of these links keeps the net-work evolving and connected until by a finite-size fluctu-ation the system reaches the fully ordered state (all nodesin the same state) and freezes in a single component. Inthe frozen phase, the fast rewiring dynamics quickly leadsto the fragmentation of the network into two components,before the system becomes fully ordered.

The similarity between the mean-field equation for thedensity of active links in the coevolution voter model(Eq. 2) and the one for the density of infected sites in thecontact process [19], suggests that our model could be-long to the Directed Percolation universality class. How-ever, both models seem not to be equivalent given thatour model possesses many absorbing states (any point onthe ρ = 0 line of Fig. 4), unlike the contact process wherethere is a single absorbing state characterized by the ab-sence of infected sites. Our results provide new insight inthe ongoing discussion about models with infinitely manyabsorbing states [20].

We acknowledge financial support from MEC (Spain)through project FISICOS (FIS2007-60327), CSIC(Spain) through project PIE200750I016, and the EUthrough project GABA.

∗ E-mail: federico@ifisc.uib.es

[1] T. Gross and B. Blasius, J. R. Soc. Interfacedoi:10.198/rsif.2007.1229.

[2] M.G. Zimmermann, V.M. Eguıluz, and M. San Miguel,Phys. Rev. E 69, 065102(R) (2004).

[3] G. Kossinets and D.J. Watts, Science 311, 88 (2006).[4] V.M. Eguıluz, M.G. Zimmermann, C.J. Cela-Conde, and

M. San Miguel, Am. J. Sociol. 110, 977 (2005).[5] D. Centola, J.C. Gonzalez-Avella, V.M. Eguıluz, and M.

5

San Miguel, J. of Conflict Resol. 51, 905 (2007).[6] G.C.M.A. Ehrhardt, M. Marsili, and F. Vega-Redondo,

Phys. Rev. E 74, 036106 (2006).[7] I. J. Benczik, S. Z. Benczik, B. Schmittmann, and R. K.

P. Zia, arXiv:0709.4042.[8] S. Gil and D.H. Zanette, Phys. Lett. A 356, 89 (2006).[9] P. Holme and M.E.J. Newman, Phys. Rev. E 74, 056108

(2006).[10] F. Vazquez, J. C. Gonzalez-Avella, V. M. Eguıluz, and

M. San Miguel, Phys. Rev. E 76, 046120 (2007).[11] C. Nardini, B. Kozma, and A. Barrat, arXiv:0711.1261.[12] B. Kozma and A. Barrat, arXiv:0707.4416.[13] A classical example where a social network was frag-

mented is described in W.W. Zachary, J. Anthr. Res. 13,452 (1977); an extensive description on the formation ofsocial groups can be found for example in S. Wassermanand K. Faust, Social Network Analysis (CUP, 1999).

[14] R. Holley and T. Liggett, Ann. Probab. 4, 195 (1975).

[15] K. Suchecki, V.M. Eguıluz, and M. San Miguel, Phys.Rev. E 72, 036132 (2005).

[16] C. Castellano, V. Loreto, A. Barrat, F. Cecconi, and D.Parisi, Phys. Rev. E 71, 066107 (2005).

[17] We have also calculated the critical point by means ofthe survival probability [18]. Setting as initial conditiona configuration with a few active links, the survival prob-ability at time t is calculated as the fraction of realiza-tions that have not reached an absorbing state by time t.At the critical point (pc ≃ 0.38), the survival probabilitydisplays a power law distribution.

[18] M.A. Munoz, R. Dickman, A. Vespignani, and S. Zapperi,Phys. Rev. E 59, 6175 (1999).

[19] J. Marro and R. Dickman, Nonequilibrium Phase Tran-sitions in Lattice Models (CUP, Cambridge, 1999).

[20] S.-Ch. Park and H. Park, Phys. Rev. E 76, 051123(2007).

j

a

i

j

a

i

k−n inert links

n active links

m = s (k−n)

j

a

i

nk

prewire

nk

(1−p) copy

n−1 active linksk−n+1 inert links

m∆ = s−2µΝ

k−n active linksn inert links

k−2n 2( )µΝ

∆ρ=

m = s (k−n+1)

∆ρ= −2µΝ

m = −s n

−2 s kµΝ∆ =m

Figure 1

0 1p

0

0.5

pc

pc1/2

ACTIVE PHASE

FROZENPHASE

ρs

ρ+ +, ρ− −

Figure 2

0

0.2

0.4

0.6

0.8

1

S/N

0

0.2

0.4

0.6

0.8

1

|m|

0 0.2 0.4 0.6 0.8 1p

0

0.1

0.2

ρsurv

0 0.2 0.4 0.6 0.8 1p

0

1

2

3

τ/Ν

0 1000 2000time

0

0.1

0.2

0.3ρ

101

102

103

z

10-3

10-2

10-1

τ / N

ln(z

)

(a)

(c)

(b)

(d)

Figure 3

-1 -0.5 0 0.5 1m

0

0.1

0.2

0.3

0.4

0.5

ρ0 20000 40000 60000

time0

0.1

0.2

0.3

0.4

ρ

0 5000 10000 15000 20000

time

0.5

1

1.5

ρ / σ

+ σ−

4 ξ 0

ξ 0.35 = 0.11

ξ 0 = 1/3

ξ p

4 ξ 0.35

4 ξ 0.25

Figure 4

0 2 4 6 8 10 12

k0

0.05

0.1

0.15

0.2

0.25

Pk

Poisson-4t=80t=160t=320t=640t=1280

Node degree distribution for different times.µ = 4, p = 0.1, N = 1000

Figure 5

0 0.1 0.2 0.3 0.4p

0.5

0.75

1

1.25

1.5

<n>

k / ρ

k

k=2k=4k=6

Figure 6

0 0.1 0.2 0.3 0.4p

0

0.5

1

1.5

2

2.5

3

<n2 >

k / [

ρ k

+ ρ

2 k (

k-1)

] k=2k=4k=6

Figure 7

100

101

102

103

104

time

10-3

10-2

10-1

100

S N=1000N=2000N=4000N=8000N=16000N=32000N=32000

Survival probability.p=0.382, 10

4 realizations.

slope = 1.04

Figure 8