Opinion formation on adaptive networks with intensive average degree

PHYSICAL REVIEW E 82, 066104 共2010兲

Opinion formation on adaptive networks with intensive average degree B. Schmittmann and Abhishek Mukhopadhyay Department of Physics, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061-0435, USA 共Received 15 August 2010; published 2 December 2010兲 We study the evolution of binary opinions on a simple adaptive network of N nodes. At each time step, a randomly selected node updates its state 共“opinion”兲 according to the majority opinion of the nodes that it is ˜ 兲 if they connect nodes with equal linked to; subsequently, all links are reassigned with probability ˜p 共q 共opposite兲 opinions. In contrast to earlier work, we ensure that the average connectivity 共“degree”兲 of each node is independent of the system size 共“intensive”兲, by choosing ˜p and ˜q to be of O共1 / N兲. Using simulations and analytic arguments, we determine the final steady states and the relaxation into these states for different system sizes. We find two absorbing states, characterized by perfect consensus, and one metastable state, characterized by a population split evenly between the two opinions. The relaxation time of this state grows exponentially with the number of nodes, N. A second metastable state, found in the earlier studies, is no longer observed. DOI: 10.1103/PhysRevE.82.066104

PACS number共s兲: 89.75.Hc, 02.50.Le, 05.70.Ln, 89.65.Ef

I. INTRODUCTION

Statistical physics establishes the bridge from microscopic characteristics to macroscopic behaviors, for systems containing a large number of interacting components. Using both analytical and numerical tools, it has contributed greatly to our understanding of various complex systems. Its conceptual and methodological approaches are very general, allowing for a diverse range of applications including biology, economics and sociology. In this paper, we are motivated by the statistical physics of a sociological problem, namely, opinion formation 关1兴. Among the most popular models in this field are voter 关2–4兴 and majority rule 共or threshold兲关5,6兴 models which have been considered on regular lattices as well as on networks. Even if many social details are neglected, such approaches can identify essential and/or universal features of very complex systems and guide more sophisticated research. More elaborate versions of opinion formation models, such as the Axelrod 关7–9兴 or Sznajd 关10,11兴 models, have also been investigated, constituting the first steps toward more realistic studies. A detailed recent review can be found in 关1兴. In the following, we investigate a threshold model on a coevolving network, reflecting both, changes in opinion and the restructuring of social interactions. Specifically, we consider a social network with individuals or agents holding two different kinds of opinions or party affiliations. Individuals are represented as nodes which can exist in two states 共opinions or “republicans” and “democrats”兲. These states can reflect political, religious, or ethical views, or denote different languages. Interactions 共e.g., conversations, professional contacts, or friendships兲 will be represented by links between pairs of individuals. The presence or absence of these links depends on whether the individuals in question share the same state 共opinion, political party, language, etc.兲 or not, and the connections of an individual drive his or her changes in opinion. In other words, the network structure is adaptive: nodes and links coevolve. Studies of coevolving networks have become more frequent over the past few years. Typically, a pair of connected 1539-3755/2010/82共6兲/066104共7兲

nodes is selected and a decision is made whether to retain their connection or whether to rewire it to another node, selected according to a separate criterion 关12–20兴. In a few instances, links are simply deleted so that the network becomes less and less connected 关21兴. In other studies, links are created or destroyed, based on the state of the participating nodes 关22,23兴. Two recent reviews can be found in 关24,25兴. Three key factors will be incorporated into our study. 共i兲 With advances in information and transportation technology, social interactions are no longer confined to a neighborhood or locality so that geographical closeness does not play a strong role. Therefore, a network topology without an underlying notion of proximity or spatial structure is more suitable for our problem than a regular lattice. 共ii兲 The connections or relationships 共links兲 between individuals 共nodes兲 can change over time. Increased mobility and new modes of communication make it very simple for an agent to break old connections and establish new ones. 共iii兲 Finally, no matter how outgoing an individual is, his or her number of connections is necessarily finite, i.e., it does not grow with the size of the community or network to which the individual belongs. In an earlier study by Benczik et al. 关22,23兴, the dynamics of opinion formation in a simple model incorporating only 共i兲 and 共ii兲 was studied. Using both analytical methods and numerical simulations, it was shown that large societies or groups eventually settle into four different phases in a twodimensional phase space spanned by the initial number of individuals holding a certain opinion and their likelihood to connect to others holding the same or different opinion. The four phases are complete consensus in which all individuals adopt the same opinion or support the same party, an evenly divided state 共gridlock兲 in which half of the population holds one opinion and the other half holds the other, and a state in which the system remembers the initial distribution of opinions 共status quo兲. Upon closer analysis, however, one notes that the two consensus states are absorbing states so that the other two phases are just metastable. Both evolve into the absorbing states, on time scales which grow exponentially with the system size. Thus, for smaller societies, complete consensus emerges on observable time scales.

066104-1

©2010 The American Physical Society


B. SCHMITTMANN AND ABHISHEK MUKHOPADHYAY

While intriguing in its analytic simplicity, this earlier study is unrealistic in one respect: the typical degree 共average number of connections兲 of each individual increases with the size of their community, i.e., it is extensive in the number of nodes, N. In the earlier model, someone moving from a small town of, say, 104 residents to a large city with a population of 106 people would suddenly expand her social network by a factor of 100. Here, we will modify the model so as to introduce an average degree that remains intensive in N, reflecting the limited number of connections. Our major findings are as follows: by definition of the model, the two “consensus” states remain absorbing and therefore control the long-time behavior of all finite systems. The gridlock state remains metastable, with a relaxation time that grows exponentially with the number of nodes. The most dramatic change, compared to the earlier study, involves the status quo state: here, we find that it is no longer long lived; instead, it decays exponentially fast to one of the absorbing states. This paper is organized as follows: we begin with a definition of our model, followed by a discussion of the underlying master equation and its analytic predictions for steadystate properties. These are then put into the context of our simulation results. Dynamic quantities, such as the relaxation into the steady state, are investigated numerically. We conclude with a summary and outlook. II. MODEL

Our model consists of N nodes 共individuals兲, each of which carries a spin variable 共opinion兲 ␴ j = ⫾ 1, with j = 1 , 2 , . . . , N. All pairs of spins are either connected by a link or not. Nodes and links coevolve, according to the following dynamics. In each time step, we first update a randomly selected spin, based on a simple majority rule: if it is connected to more positive than negative spins, its state becomes positive and negative otherwise. In the case of a tie, the spin remains unchanged. This update rule is identical to that of the majority-vote model 关26兴, with zero noise parameter; it is also equivalent to Glauber dynamics 关27兴 at zero temperature. Next, we reassign all links: pairs of spins in the same ˜ 兲. This 共different兲 state are connected with a probability ˜p 共q stochastic reassignment of all links constitutes a massive simplification since it eliminates all history dependence, allowing us to treat the dynamics as a Markov process. In the earlier study 关22,23兴, ˜p and ˜q were fixed parameters of the model, independent of the number of nodes. Since the number of pairs grows as N共N − 1兲 / 2, that implies that the typical degree of a given spin increases with N. Here, we rescale ˜p and ˜q with N in such a way that the average degree remains independent of N: ˜p ⬅ ␩ p / N and ˜q ⬅ ␩q / N are the probabilities of connecting a pair of equal or opposite spins, respectively. Here, the 共positive兲 parameter ␩ controls the average degree, while p and q control the preference for linking similar or opposite opinions. For simplicity, we consider the special point p + q = 1 共as well as 0 ⱕ p , q ⱕ 1兲 but our results do not depend significantly on this constraint. Any other choice for the value of the sum simply leads to a scale factor which can be absorbed in ␩. The network is adaptive since the

dynamic evolution couples nodes and links to one another. We study the system both analytically and by numerical simulations. Our focus is the long-time behavior of the system: what states does the system approach, in the long-time limit, and how does it approach them? To answer these questions, we investigate the probability P共M , t兲 to find the system with M positive spins at time t as well as the first moment of this distribution, i.e., the average fraction of positive spins at time t, N

1 ¯ 共t兲 ⬅ 兺 MP共M,t兲. m N M=0

共1兲

This density-also referred to as the “popularity”-takes values in the interval 关0,1兴. The extremes correspond to completely ¯ = 0.5 ordered states 共all spins positive or negative兲 while m characterizes the completely disordered 共gridlocked兲 state. III. MASTER EQUATION AND THE PHASE DIAGRAM

Before turning to simulation data, we first summarize a few analytical observations. Since the number, M, of positive spins changes by one in each time step, the dynamics can be written as a birth-death process. Also, the random reassignment of the links after each update in conjunction with the absence of any spatial structure implies that the birth and death rates at time t depend only on M at that time, leading to a Markov process. The corresponding master equation can be written as

⳵t P共M,t兲 = bM−1 P共M − 1,t兲 + dM+1 P共M + 1,t兲 − 关bM + d M 兴P共M,t兲,

共2兲

where the birth 共death兲 rate bM 共d M 兲 is the rate of flipping a negative 共positive兲 spin given M positive spins. In the language of Monte Carlo simulations, the time scale here is set by a single update attempt. Following 关22,23兴, the birth and death rates are easily determined. Focusing on dM , we first ask for the probability, M / N, of choosing a positive spin out of M positive spins. Second, we write the probability that this spin is connected to exactly k of the remaining M − 1 positive spins, given by the binomial distribution BM−1,␩p/N共k兲 ⬅

冉冊冉冊冉冊 M−1

␩p

k

N

k

1−

␩p

M−1−k

N

共3兲

.

The probability that this spin is also connected to exactly k⬘ of the remaining N − M negative spins is given by BN−M,␩q/N共k⬘兲 =

冉冊冉冊冉冊 N−M k⬘

␩ q k⬘ N

1−

␩q N

N−M−k⬘

.

共4兲

The update of the spins follows a simple majority rule: if k⬘ ⬎ k, the selected positive spin will flip its state; otherwise, it will remain unchanged. This can be accommodated by introducing a Heavyside theta function ⌰共k⬘ − k兲. Collecting all of the above, the death rate d M can be written as

066104-2


OPINION FORMATION ON ADAPTIVE NETWORKS WITH…

(a)

(b)

(a)

(b)

FIG. 2. 共Color online兲 The birth 共solid line兲 and death 共dashed line兲 rates for a network of N = 1000 nodes at p = 0.9 and q = 0.1 for 共a兲 extensive and 共b兲 intensive average degree. For 共b兲, ␩ = 10. (d)

(c)

FIG. 1. 共Color online兲 The death 共d M , black dashed line兲 and birth 共b M , red solid line兲 rates, given by Eqs. 共5兲 and 共6兲 for different probabilities p: 共a兲 0.1, 共b兲 0.3, 共c兲 0.5, and 共d兲 0.7 for a network of N = 1000 nodes and ␩ = 10.

dM =

M N

M−1 N−M

兺兺 BM−1,␩p/N共k兲BN−M,␩q/N共k⬘兲⌰共k⬘ − k兲. k=0

共5兲

k⬘=0

Similar arguments give rise to the birth rate: N−M bM = N

N−M−1 M

兺兺 BN−M−1,␩p/N共l兲BM,␩q/N共l⬘兲⌰共l⬘ − l兲. l=0 l⬘=0

共6兲 We note, in passing, that a simpler expression can be derived for the special case q = 1 − p, following 关23兴. The expressions for the birth and death rates can be evaluated numerically. They are plotted in Figs. 1共a兲–1共d兲, as functions of M, for four different values of p, with q = 1 − p, N = 1000, and ␩ = 10. Since the dynamics is symmetric under Z2, i.e., ␴ j → −␴ j for all j, we have b M = dN−M . The phase diagram will exhibit the same Z2 symmetry 共i.e., M / N → 1 − M / N兲 and can be immediately generated from these plots, by realizing that for b M ⬎ d M , the number of positive spins, M, will increase and vice versa. Considering Fig. 1共a兲共p = 0.1兲, an initial condition M 0 / N ⬍ 0.5 共⬎0.5兲 will induce an increase 共decrease兲 in the number of positive spins, driving the system toward M / N = 0.5. When that value is reached, births and deaths balance each other, up to statistical fluctuations, and the system settles into the gridlock steady state ¯ ⬁ = 0.5. In contrast, for p = 0.5 or characterized by limt→⬁ ⬅ m 0.7 shown in Figs. 1共c兲 and 1共d兲, the situation is reversed: an initial condition M 0 / N ⬍ 0.5 共⬎0.5兲 will induce an decrease 共increase兲 in the number of positive spins, driving the system toward the consensus states M / N = 0 共M / N = 1兲. In these cases, b M and d M cross only once, at M / N = 0.5 共with the exception of the end points, M / N = 0 or 1 where bM and d M both vanish兲. For intermediate p, e.g., p = 0.3 关Fig. 1共b兲兴, a third scenario is observed. Here, the birth and death rates exhibit three crossings, leading to the following behavior. For a very small initial condition, the death rate dominates and the system is driven toward consensus at M / N = 0; similarly, if M 0 / N is close to 1, the birth rate dominates and drives the system to M / N = 1. For intermediate M 0 / N, there is a sector where the birth 共death兲 rate dominates a narrow

region to the left 共right兲 of M / N = 0.5 so that the gridlock state 共M / N = 0.5兲 is selected. Since the two consensus states are absorbing, they are the only true steady states for a system with a finite number of nodes. In Sec. IV, we will investigate the stability of the gridlock phase and show that it decays exponentially slowly in N toward one of the two absorbing states. Strictly speaking, this phase is metastable even though it can be extremely long lived. In contrast to our earlier study, where a second long-lived metastable phase, the so-called status quo, was observed for p near unity, this phase is no longer present here. To illustrate this fact, Fig. 2 shows the birth and death rates for N = 1000 and p = 0.9 共q = 1 − p兲 for the extensive and intensive models. For the extensive case, links between nodes in the same 共opposite兲 states are established with probability p; i.e., almost all links are established among individuals sharing the same opinion and hardly any among differing opinions, making any changes in opinion extremely unlikely. Figure 2共a兲 illustrates that the birth and death rates effectively vanish within a broad central region, centered on M / N = 0.5 and bounded approximately by 0.1⬍ M / N ⬍ 0.9. Any initial condition within this region will remain effectively frozen for an exponentially long time. In contrast, for our model p = 0.9 translates into a rather small probability of interaction within and across groups of opinions: the actual probability of mak˜ ing links between equal 共opposite兲 opinions is ˜p ⬅ ␩ p / N共q ⬅ ␩共1 − p兲 / N兲 which are of order 10−2 for both ˜p and ˜q, given the chosen parameters. Hence, the probabilities of making links among individuals of the same vs the opposite opinion are comparable, and the status quo does not persist. Figure 2共b兲 confirms that both the birth and the death rate remain distinctly nonzero across the full interval 0 ⬍ M / N ⬍ 1. The resulting phase diagram is shown in Fig. 3. It clearly

FIG. 3. 共Color online兲 The fraction of positive spins M / N after 150 and averaged over 10 runs, as a function of p and M 0 / N for a network of N = 1000 nodes and ␩ = 10. The black lines denote the analytic phase boundaries obtained from the birth and death rates 关Eqs. 共5兲 and 共6兲兴. The sectors are filled in by Monte Carlo simulations, to be discussed in Sec. IV.

066104-3



exhibits the three expected states—again, with the caveat that the gridlock phase is only metastable. The Z2 symmetry 共i.e., M / N → 1 − M / N兲 is clearly displayed. In the following, we describe two complementary approaches to probe the dynamics of the system. The first is based on a Kramers-Moyal 关28兴 expansion of the master equation and illustrates that the initial relaxation of the system, into both the stable and metastable states, is exponential. We find an analytic expression for the relaxation rate which will be compared to simulation data below. This naive perturbative analysis does not capture the eventual decay of the metastable state into the absorbing states, and so we turn to a second approach: following 关23兴, we compute the slowest nonzero eigenvalue of the Liouville operator which controls the true long-time behavior of the model. Noting that M only changes by at most 1 in each update, which is typically a small change on the scale of N, we can perform a Kramers-Moyal expansion 关28兴 of the master equation. Rewriting b M and d M as two functions of m ⬅ M / N and defining ⑀ ⬅ 1 / N, we take the limit N → ⬁ while keeping m 苸关0 , 1兴 finite. The master equation 共2兲 then becomes

⳵P = 关d共m + ⑀兲P共m + ⑀,t兲 − d共m兲P共m,t兲兴 ⳵t

After reaching the metastable state, the system relaxes extremely slowly to one of the absorbing states. This can be captured from a formal solution of the master equation. For details, we refer the reader to 关23兴. Adopting matrix notation, we write Eq. 共2兲 as d兩P共t兲典 = − Lˆ 兩P共t兲典, dt

where 兩P共t兲典 is the N + 1 dimensional column vector with entries P共M , t兲, M = 0 , 1 , . . . , N. Lˆ is the time evolution 共Liouville兲 operator, given by

Lˆ =

共7兲

共m − m⬁兲P共m,t兲dm

共8兲兩00典 =

and its equation of motion can be derived easily from Eq. 共7兲:

冕冕

共m − m⬁兲

0

=−⑀

⳵ 关共d − b兲P兴 dm ⳵m

共d − b兲Pdm ⬅ ⑀具d共m兲 − b共m兲典, 共9兲

where we have used the fact that both the birth and death rate ¯ 共t兲 vanish at the ends of the interval 关0,1兴. For large times, m approaches m⬁. Since d共m兲 − b共m兲 is analytic around m⬁, we can expand it in a Taylor series. For the average, this implies ¯ 共t兲 − m⬁兴 ⳵⳵m 关d共m兲 − b共m兲兴兩m=m . This form 具d共m兲 − b共m兲典 = 关m ⬁ can be integrated easily, resulting in

再

冎

⳵ ¯ 共t兲 − m⬁ ⬀ exp − ⑀t 关d共m兲 − b共m兲兴兩m=m . m ⬁ ⳵m

− dN 0

− bN−1

共10兲

Noting that ⑀ = 1 / N, we recognize that ⑀t = t MC, and so ␬共p , ␩ , N兲 ⬅ ⳵⳵m 关d共m兲 − b共m兲兴兩m=m⬁ is our prediction for the characteristic decay rate which will be compared to our Monte Carlo data in the next section.

冣

.

共12兲

共13兲

冢冣冢冣 1

0

⳵ ¯ 共t兲 − m⬁兴 = ⑀ 关m ⳵t

¯

where ␭␮ denotes the eigenvalues, associated with right and left eigenvectors 兩␮R典 and 具␮L兩, and 兩P共0兲典 is the initial state. There are two vanishing eigenvalues, ␭0 = ␭N = 0, associated with the two eigenkets 兩00典 and 兩0N典 which describe the two absorbing states:

1

1

− b1

␮

¯ 共t兲, we have therefore For the average, m

冕

− d1

兩P共t兲典 = 兺 e−␭␮t兩␮R典具␮L兩P共0兲典,

⳵ 兵关d共m兲 − b共m兲兴P共m,t兲其. ⳵m

¯ 共t兲 − m⬁ = m

冢

0

− b0 共b1 + d1兲

The Z2 symmetry of our model is reflected in dM = bN−M , and b0 = dN = 0 identifies the absorbing states. The formal solution of the master equation can be written as

− 关− b共m − ⑀兲P共m − ⑀,t兲 + b共m兲P共m,t兲兴 =⑀

共11兲

0

]

0

0

and 兩0N典 =

0

]

1

Projecting Eq. 共13兲, for an initial state with M 0 positive spins, onto the space of zero eigenvalues allows us to compute the probability with which a given initial condition reaches one of the absorbing states, e.g., the one in which all spins are positive: P共N , ⬁兲. For details, we refer the reader to 关23兴, specifically Eq. 共30兲. To investigate the approach to the absorbing states, we compute the first nonzero eigenvalue ␭1 numerically, for systems with relatively small N. Our results for both of these quantities will be discussed in the next section, where we turn to the Monte Carlo simulations. IV. SIMULATION RESULTS

In this section, we report Monte Carlo data for our model, using a random sequential update scheme. All initial configurations are random, i.e., M 0 randomly selected nodes are assigned positive spin and the remainder negative. First, we randomly select a node i. In the most naive scheme, we would then consider all pairs, 共i , j兲, with j ⫽ i, and the associated spins 共␴i , ␴ j兲, and decide whether to establish links between these pairs, according to the following rule: if ␴i

066104-4



(b)

(a) (a)

(b)

(c)

(c)

(d)

FIG. 4. 共Color online兲 The fraction of positive spins M / N after 150 MCS averaged over 10 runs, as a function of p and M 0 / N for a network of 1000 nodes for the 共a兲 extensive and intensive case with 共b兲 ␩ = 5, 共c兲 ␩ = 10, and 共d兲 ␩ = 20.

= ␴ j, the two nodes are linked with probability ˜p; if ␴i ⫽ ␴ j, they are linked with probability ˜q. Obviously, the probabilities for not making a link are 1 − ˜p and 1 − ˜q, respectively. Once all choices are made, ␴i is updated following a simple majority rule: ␴i共t + 1兲 = sgn 兺 j⫽i ␴ j共t兲 if 兺 j⫽i␴ j共t兲 ⫽ 0; otherwise, ␴i remains unchanged. A much more efficient algorithm implements the majority rule directly by computing 关29兴 the probability that ␴i is connected to more or fewer positive than negative spins, according to Eqs. 共5兲 and 共6兲. This completes one update attempt. Before the next update, all links are removed, and the next node is selected at random. One Monte Carlo step 共MCS兲 corresponds to N update attempts; we define Monte Carlo time as tMC = t / N, where t counts the number of update attempts. We performed simulations for a range of system sizes N = 100, 500, and 1000; connectivities ␩ = 5, 10, and 20; and multiple p values between 0 and 1 共with q = 1 − p兲. Other parameters are specified in the figure captions, as needed. Our reference system, for which most of the data were obtained, is characterized by N = 1000 and ␩ = 10. The phase diagram for our reference system is shown in Fig. 3. We observe good agreement with the theoretically predicted phases and phase boundaries. Due to statistical fluctuations, the boundaries of the simulated phase diagram are of course not perfectly sharp, and due to the finite length of the simulation, a very weak remnant of the “status quo”

(a)

FIG. 6. 共Color online兲 The fraction of positive spins M / N after 共a兲 10 MCS, 共b兲 30 MCS, and 共c兲 100 MCS, averaged over 20 runs, as a function of p and M 0 / N for N = 1000 nodes and ␩ = 10.

phase can still be observed for p very close to 1. Figure 4 illustrates the dependence on the connectivity ␩. In the first panel, we show the phase diagram of the extensive model, with all four phases. For the intensive case, the phase boundaries depend weakly on ␩ but the qualitative behaviors are very similar. The system size N plays two key roles: first, for larger system sizes, statistical fluctuations are less pronounced, leading to sharper phase boundaries 共Fig. 5兲. Second, the remnants of the gridlock state decay more slowly for larger N so that they are more visible in larger systems. Figure 6 illustrates how these remnants decay with Monte Carlo time. We note that, compared to the relaxation of this state in the extensive system which takes of order eN MCS, the times shown here 共100 MCS or less兲 are extremely short. In the remainder of this section, we will investigate the relaxation into the stable and metastable phases, and out of the metastable phase, in more detail. We focus on the aver¯ 共t兲, defined in Eq. 共1兲, as a age density of positive spins, m key indicator. Starting from an arbitrary initial condition, this density relaxes exponentially quickly to the appropriate stable or ¯ 共t兲 − m ¯ ⬁ ⬃ e−␬t, where t is measured in metastable value: m MCS and ␬共p , ␩ , N兲 is a characteristic relaxation rate. In Fig. ¯ 共t兲 for four typical parameter 7 we show the behavior of m sets: 共m0 = 0.1, p = 0.4兲 and 共m0 = 0.4, p = 0.1兲, for N = 1000 and ␩ = 10 and 20. Averages were taken over 200 realiza¯ 共t兲 relaxes toward the absorbing tions. For m0 = 0.1, p = 0.4, m ¯ ⬁ = 0 while approaching the metastable state m ¯⬁ state at m = 1 / 2 for m0 = 0.4, p = 0.1. The relaxation rate ␬ clearly depends on p, m0, and ␩, but only weakly on N. The measured

(b)

(a)

(c)

FIG. 5. 共Color online兲 Simulation results for the fraction of positive spins M / N after 150 MCS averaged over 10 runs, as a function of p and M 0 / N for a network of 共a兲 N = 100, 共b兲 N = 500, and 共c兲 N = 1000 nodes and ␩ = 10.

(b)

FIG. 7. 共Color online兲 The time evolution of the fraction of positive nodes 共M / N兲 for N = 1000 with 共a兲 ␩ = 10 and 共b兲 ␩ = 20 starting with M 0 / N = 0.1 and 0.4. The red diamonds are the simulation results 共averaged over 20 runs兲 and the black dashed curves are theoretical predictions based on Eq. 共10兲.

066104-5



(b)

(a) (b)

(a)

FIG. 8. 共Color online兲 The probability P共N , ⬁兲, to reach the absorbing state with N positive spins at infinite times plotted against the initial density of the positive spins for system size N = 1000, three different values of ␩ and 共a兲 p = 0.25 and 共b兲 p = 0.7. P共N , ⬁兲 = 0 implies extinction of the positive population, while P共N , ⬁兲 = 1 represents a purely positive population. P共N , ⬁兲 is calculated from Eq. 共30兲 in 关23兴.

behaviors agree well with the results from the theoretical prediction, Eq. 共10兲. On a much longer time scale, the metastable state relaxes to either one of the stable states. Figure 8 shows our results for P共N , ⬁兲 for two typical situations, namely, p = 0.25 and p = 0.70. The figure shows analytic results, based on the exact projection of Eq. 共13兲 onto the nullspace of the operator Lˆ 共see 关23兴兲. For p = 0.70, the relaxation behavior is deterministic: for M 0 ⬍ N / 2, all initial configurations end in the absorbing state M = 0 reflected in P共N , ⬁兲 = 0, and for M 0 ⬎ N / 2, all fall into the absorbing state M = N reflected via P共N , ⬁兲 = 1. For p = 0.25, such deterministic behavior persists only at the edges of the M 0 range. In the central region, P共N , ⬁兲 = 0.5, indicating that the system chooses to fall into either one of the two absorbing states with equal probability. Not surprisingly, the width of this central region is identical to the range of the metastable state in the phase plot, shown in Fig. 3. In Fig. 9, we illustrate the long-time behavior of ¯ 共t兲, for p = 0.2 and m0 = 0.4, and a range of N and ␩. The m plots show very clearly that the system relaxes extremely slowly into the absorbing state, with a relaxation rate ␭共p , ␩ , N兲 that depends very strongly on the system size, N. In Fig. 10, we plot these rates versus N, and the dependence is manifestly exponential, i.e., ␭共p , ␩ , N兲 ⬃ exp关−a共p , ␩兲N兴.

(b)

(a)

(c)

FIG. 9. 共Color online兲 Relaxation of the disordered metastable ¯ = 0.5 to the completely ordered absorbing state m ¯ = 0 for state with m networks of different size 共N = 20, 30, 40, and 50兲 and for different values of ␩: 共a兲 ␩ = 5, 共b兲 ␩ = 10, and 共c兲 ␩ = 20. The simulation data 共solid red lines兲 are averaged over 104 runs. The dashed black lines are exponential fits. Other simulation parameters are p = 0.2 and M 0 / N = 0.4. Note the log scale on the vertical axis.

(c)

FIG. 10. 共Color online兲 The first nonzero eigenvalues 共red dots兲 ␭1 of the operator L and 共fitted兲 slopes from Fig. 9 共black crosses兲 vs system size N, for different values of ␩: 共a兲 ␩ = 5, 共b兲 ␩ = 10, and ¯ 0 = 0.4 and p = 0.2. Due to the 共c兲 ␩ = 20. Other parameters are m excellent agreement, it is difficult to distinguish dots and crosses. The dashed lines are exponential fits. In the logarithmic-normal plot shown, the slopes are 0.135, 0.155, and 0.161 for ␩ = 5, 10, and 20, respectively.

The measured values for ␭ are compared to the first nonzero eigenvalue of the time evolution operator, computed numerically from Eq. 共12兲, and the quantitative agreement is found to be excellent. V. CONCLUSIONS

In this paper, we investigated the evolution of opinions on an adaptive network, subject to a dynamics in which nodes and links coevolve. Each node can exist in two states, reflecting two opinions, and is updated according to the majority opinion of connected nodes. Each link is then reassigned, with probabilities depending on the states of the associated nodes. In contrast to an earlier study 关22,23兴, we choose these probabilities to be of O共1 / N兲 to ensure that the average degree of each node remains intensive in N as the size of the network increases. The motivation for this choice arises from real social networks: when individuals move from a small town to a large city, the number of their social contacts is not expected to increase with the size of the city. Instead, we introduce a parameter, ␩, which controls the average degree. Specifically, we connect two equal 共different兲 nodes with a ˜ ⬅ ␩q / N兲, where p and q are now paprobability ˜p ⬅ ␩ p / N共q rameters of O共1兲. In general, larger values of ␩ lead to network behaviors which approximate the properties of the extensive model more closely. The key difference between the models with intensive and extensive average degree is observed in the phase diagram. Both models exhibit two absorbing states, corresponding to ¯ = 1 or m ¯ = 0兲; also, both models exhibit perfect consensus 共m ¯ = 1 / 2兲. a metastable state with evenly divided opinions 共m However, the second metastable state 共referred to as status ¯ = M 0 / N兲 is no longer observed in the intensive model. quo, m The source of this behavior is easily identified: in the extensive model, the status quo phase persists when the probability for linking equal nodes is close to unity, massively reducing the likelihood of any changes of opinion. In contrast, in the intensive case this probability is always scaled by N and remains 共relatively兲 small so that changes of opinion are not excluded.

066104-6



Other key results are as follows. First, the phase boundaries are only weakly dependent on the system size: larger values of N just serve to reduce statistical noise. Second, the relaxation time from the metastable into the stable absorbing states increases exponentially with N, i.e., smaller communities reach consensus much faster than larger ones. In larger societies, a diversity of different opinions can persist for extremely long times. A study of different languages in the Solomon islands may provide an example for this behavior 关30兴. Clearly, this simple model neglects many aspects of a real social system, such as spatial and age structures, a spectrum of opinions, memory, and history, spontaneous changes in

opinions, the influence of mass media, etc., to name just a few characteristics. However, the compelling feature of our naive model resides in its mathematical simplicity which allows us to extract many of its properties analytically. In future work, we plan to incorporate greater complexity, supported by the mathematical foundations established so far.

We thank Royce Zia and Izabella Benczik for useful discussions and K. Holsinger for the binomial generator. We gratefully acknowledge partial support from the National Science Foundation, under Grant No. DMR-0705152.

关1兴 C. Castellano, S. Fortunato, and V. Loreto, Rev. Mod. Phys. 81, 591 共2009兲. 关2兴 P. Clifford and A. Sudbury, Biometrika 60, 581 共1973兲. 关3兴 R. Holley and T. M. Liggett, Ann. Probab. 3, 643 共1975兲. 关4兴 T. M. Liggett, Stochastic Interacting Systems: Contact, Voter, and Exclusion Processes 共Springer, New York, 1999兲. 关5兴 S. Galam, Eur. Phys. J. B 25, 403 共2002兲. 关6兴 P. L. Krapivsky and S. Redner, Phys. Rev. Lett. 90, 238701 共2003兲. 关7兴 C. Castellano, M. Marsili, and A. Vespignani, Phys. Rev. Lett. 85, 3536 共2000兲. 关8兴 K. Klemm, V. M. Eguiluz, R. Toral, and M. San Miguel, Phys. Rev. E 67, 045101共R兲共2003兲. 关9兴 F. Vazquez and S. Redner, EPL 78, 18002 共2007兲. 关10兴 K. Sznajd-Weron and J. Sznajd, Int. J. Mod. Phys. C 11, 1157 共2000兲. 关11兴 R. Dickman and R. Vidigal, J. Phys. A 35, 1147 共2002兲. 关12兴 F. Vazquez, V. M. Eguíluz, and M. S. Miguel, Phys. Rev. Lett. 100, 108702 共2008兲. 关13兴 C. Nardini, B. Kozma, and A. Barrat, Phys. Rev. Lett. 100, 158701 共2008兲. 关14兴 F. Fu and L. Wang, Phys. Rev. E 78, 016104 共2008兲. 关15兴 M. Liu and K. E. Bassler, Phys. Rev. E 74, 041910 共2006兲. 关16兴 P. Holme and M. E. J. Newman, Phys. Rev. E 74, 056108 共2006兲.

关17兴 L. B. Shaw and I. B. Schwartz, Phys. Rev. E 77, 066101 共2008兲. 关18兴 L. B. Shaw and I. B. Schwartz, Phys. Rev. E 81, 046120 共2010兲. 关19兴 S. Mandra, S. Fortunato, and C. Castellano, Phys. Rev. E 80, 056105 共2009兲. 关20兴 R. Lambiotte and J. C. Gonzalez-Avella, e-print arXiv:0906.3389. 关21兴 S. Gil and D. Zanette, Phys. Lett. A 356, 89 共2006兲. 关22兴 I. J. Benczik, S. Z. Benczik, B. Schmittmann, and R. K. P. Zia, EPL 82, 48006 共2008兲. 关23兴 I. J. Benczik, S. Z. Benczik, B. Schmittmann, and R. K. P. Zia, Phys. Rev. E 79, 046104 共2009兲. 关24兴 T. Gross and B. Blasius, J. R. Soc., Interface 5, 259 共2008兲. 关25兴 Adaptive Networks. Theory, Models, and Applications, edited by T. Gross and H. Sayama 共Springer, Dordrecht, 2009兲. 关26兴 T. M. Liggett, Interacting Particle Systems 共Springer, New York, 1985兲. 关27兴 R. J. Glauber, J. Math. Phys. 4, 294 共1963兲. 关28兴 See, e.g., N. G. Van Kampen, Stochastic Processes in Physics and Chemistry 共North-Holland, Amsterdam, 1981兲. 关29兴 The binomial generator used the BTPEC algorithm described in V. Kachitvichyanukul and B. W. Schmeiser, Commun. ACM 31, 216 共1988兲. 关30兴 J. Terrel, Fieldiana Anthropology 68, 1 共1977兲.

ACKNOWLEDGMENTS

066104-7