A note on the consensus time of mean-field majority-rule dynamics

 

Damián H. Zanette¹*

*E-mail: zanette@cab.cnea.gov.ar

Consejo Nacional de Investigaciones Científicas y Técnicas, Centro Atómico Bariloche and Instituto Bal-seiro, 8400 San Carlos de Bariloche, Río Negro, Argentina.

 

In this work, it is pointed out that in the mean-field versión of majority-rule opinión dynamics, the dependence of the consensus time on the population size exhibits two regimes. This is determined by the size distribution of the groups that, at each evolution step, gather to reach agreement. When the group size distribution has a finite mean valué, the previously known logarithmic dependence on the population size holds. On the other hand, when the mean group size diverges, the consensus time and the population size are related through a power law. Numerical simulations valídate this semi-quantitative analytical prediction.

 

Much attention has been recently paid, in the context of statistical physics, to models of social processes where ordered states emerge spon-taneously out of disordered initial conditions (ho-mogeneity from heterogeneity dominance from di-versity, consensus from disagreement, etc.) [1]. Not unexpectedly many of them are adaptations of well-known models for coarsening in interacting spin systems, whose dynamical rules are reinter-preted in the framework of social-like phenomena. The voter model [2, 3] and the majority rule model [4, 5] are paradigmatic examples. In the latter, consensus in a large population is reached by ac-cumulative agreement events, each of them involv-ing just a group of agents. The present note is aimed at briefly revisiting previous results on the time needed to reach consensus in majority-rule dynamics, stressing the role of the size distribution of the involved groups. It is found that the growth of the consensus time with the population size shows distinct behaviors depending on whether the mean valué of the group size distribution is ñnite or not. Consider a population of N agents where, at any given time, each agent has one of two possible opin-ions, labeled +1 and -1. At each evolution step, a group of G agents (G odd) is selected from the population, and all of them adopt the opinión of the majority. Namely, if i is one of the agents in the selected group, its opinión s¿ changes as

where the sum runs over the agents in the group. Of course, only the agents, not the majority effectively change their opinión. In the mean-ñeld versión of this model, the G agents selected at each step are drawn at random from the entire population.

It is not difñcult to realize that the mean-ñeld majority-rule (MFMR) dynamics is equivalent to a random walk under the action of a forcé ñeld. For a ñnite-size population, this random walk is moreover subject to absorbing boundary conditions. Think, for instance, of the number N₊ of agents with opinión +1. As time elapses, N₊ changes randomly, with transition probabilities that depend on N₊ itself, until it reaches one of the extreme valúes,

N₊ = O or N. At this point, all the agents have the same opinión, the population has reached full consensus, and the dynamics freezes.

In view of this overall behavior, a relevant quan-tity to characterize MFMR dynamics in ñnite pop-ulations is the consensus time, i.e. the time needed to reach full consensus from a given initial condi-tion. In particular, one is interested in determining how the consensus time depends on the population size N. The exact solution for three-agent groups (O = 3) [5] shows that the average number of steps needed to reach consensus, S_c, depends on N as

for large N. The proportionality factor depends in turn on the initial unbalance between the two opinions all over the population. The analogy of MFMR dynamics with random walks suggests that this result should also hold for other valúes of the group size G, as long as G is smaller than N. This can be easily veriñed by solving a rate equation for the evolution of N₊ [1]. Numerical results and semi-quantitative arguments [6] show that Eq. (2) is still valid if, instead of being constant, the valué of G is uniformly distributed over a ñnite interval.

What would happen, however, if, at each step, G is drawn from a probability distribution pa that al-lows for valúes larger than the population size? If, at a given step, the chosen group size G is equal to or largen than N, full consensus will be instantly at-tained and the evolution will cease. In the random-walk analogy, this step would correspond to a single long jump taking the walker to one of the bound-aries. Is it possible that, for certain forms of the distribution pa, these single large-G events could dominate the attainment of consensus? If it is so, how is the ./V-dependence of the consensus time modiñed?

To give an answer to these questions, assume that G is drawn from a distribution which, for large G, decays as

with 7 > 1. Tuning the exponent 7 of this power-law distribution, large valúes of G may become suf-ñciently frequent as to control consensus dynamics.

The probability that at the S-ih step the selected group size is G > N, while in all preceding steps G < N, reads

where the last relation holds for large N when pa veriñes Eq. (3).

Compare now Eqs. (2) and (5). For 7 > 2 (re-spectively, 7 < 2) and asymptotically large population sizes, one has S_w 3> S_c (respectively, S_w <C S_c). This suggests that above the critical exponent 7_cr¡_t = 2, the attainment of consensus will be driven by the asymptotic random-walk features that lead to Eq. (2). For smaller exponents, on the other hand, consensus will be reached by the occur-rence of a large-G event, in which all the population is entrained at a single evolution step. Note that 7_cr¡_t stands at the boundary between the domain for which the mean group size is ñnite (7 > 7_cr¡t) and the domain where it diverges (7 < 7_cr¡t)-

In order to validate this analysis, numerical sim-ulations of MFMR dynamics have been performed for population sizes ranging from 10² to 10⁵. The probability distribution for the group size G has been introduced as follows. First, deñne G = 2g-\-l. Choosing g = 1, 2, 3,... ensures that the group size is odd and G > 3. Then, take for g the probability distribution

where £(z, a) is the generalized Riemann (or Hur-witz [7]) zeta function. In the numerical simula-tions, both opinions were equally represented in the initial condition. The total number of steps needed to reach full consensus, S, was recorded and aver-aged over series of 10² to 10⁶ realizations (depend-ing on the population size N).

Figure 1: Numerical results for the number of steps needed to reach consensus, S, normalized by the population size N, as a function of N, for three valúes of the exponent 7. The straight dotted lines emphasize the validity of Eq. (2) for 7 = 2.5 and 3. For 7 = 2 the line is horizontal, suggesting S oc N.

The two upper data sets in Fig. 1 show the ratio S/N for two valúes of the exponent 7 > 7_crit. Since the horizontal scale is logarithmic, a linear depen-dence in this graph corresponds to the proportion-ality given by Eq. (2). Dotted straight lines illus-trate this dependence. For these valúes of 7, there-fore, the relation between the consensus time and the population size coincides with that of the case of constant G. For the lowest data set, which corresponds to 7 = 7_crit, the relation ceases to hold. The horizontal dotted line suggests that now S oc N, as predicted for 7 = 2 by Eq. (5).

The log-log plot of Fig. 2 shows the number of steps to full consensus as a function of the population size for three exponents 7 < 7_crit- The dotted straight line has unitary slope, representing the proportionality between S and N for 7 = 2. For lower exponents, the full curves are the graphic rep-resentation of S_w as given by Eq. (5). The excellent

Figure 2: Number of steps needed to reach consensus as a function of the population size, for three valúes of the exponent 7. The slope of the straight dotted line equals one. Full curves correspond to the function S_w given in Eq. (7).

agreement between S_w and the numerical results for S demonstrates that, for these valúes of 7, the consensus time in actual realizations of the MFMR process is in fact dominated by large-G events.

Figure 3: Fraction of realizations where consensus is attained through a large-G event as a function of the population size, for several valúes of the exponent 7.

A further characterization of the two regimes of consensus attainment is given by the fraction of realizations where consensus is reached through a large-G event. This is shown in Fig. 3 as a function of the population size. For 7 < 7_cr¡t, consensus is the result of a step involving the whole population in practically all realizations. As N grows, the fre-quency of such realizations increases as well. The opposite behavior is observed for 7 > 7_cr¡t. For the critical exponent, meanwhile, the fraction of large-G realizations is practically independent of N, and fluctuates slightly around 0.57.

In summary it has been shown here that in majorityrule opinión dynamics, the dependence of the consensus time on the population size exhibits two distinct regimes. If the size distribution of the groups of agents selected at each evolution step de-cays fast enough, one reobtains the logarithmic analytical result for constant group sizes. If, on the other hand, the distribution of group sizes decays slowly as a power law with a sufñciently small exponent, the dependence of the consensus time on the population size is also given by a power law. The two regimes are related to two different mechanisms of consensus attainment: in the second case, in particular, consensus is reached during events which involve the whole population at a single evolution step. The logarithmic regime occurs when the mean group size is ñnite, while in the power-law regime the mean valué of the distribution of group sizes diverges. In connection with the random-walk analogy of majority-rule dynamics, this is reminiscent of the contrasting features of standard and anomalous diffusion [8].

 

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