Answer to the Commentary on “A note on the consensus time of mean-field majority-rule dynamics”

 

Dami´an H. Zanette¹

*E-mail: zanette@cab.cnea.gov.ar
Consejo Nacional de Investigaciones Científicas y Técnicas, Centro Atómico Bariloche and Instituto Balseiro, 8400 San Carlos de Bariloche, Río Negro, Argentina.

 

In his Commentary [1], H. Fort begins by point-ing out two aspects of the majority-rule (MR) model that, as presented in the main paper [2], may need some clariñcation.

It was suggested in the paper that the equiva-lence of the mean-ñeld MR dynamics and a ran-dom walk can be formulated in terms of the evo-lution of N+, the number of agents with opinión +1. Speciñcally in order to write down a master equation for the process, one should ñnd the tran-sition probability that, in a single evolution step, N₊ changes to any other N^. It can be readily seen that such event requires that the size G of the group of agents chosen to reach consensus at that step satisñes G > 2|A|, with A = N'₊ - N₊. If A is positive (respectively negative) the number of agents with opinión -1 (respectively +1) in the group must be exactly equal to |A|. Summing the probabilities for all the possible valúes of G yields the transition probability for a given A.

Of course, one could cut off and renormalize the probability distribution for the group size, pa, in such a way that G can be at most equal to the population size N. The two regimes in the size dependence of the consensus time would certainly still exist -although explicitly working out analytical results such as Eq. (7) of the main paper may become trickier. However, it is somehow artificial to conceive that, in a social process driven by events which involve agent groups, probabilities depend on the total population size. It sounds more natural to just allow any group size and, in the case that G ≥ N, admit that the population falls into an absorbing, frozen state. Also, from an opera-tional viewpoint, a cut-off probability distribution with G ≤ N would be difficult to implement if the population size varies with time.

The Commentary also addresses a series of gen-eralizations of the MR dynamics -some of which have already been considered in the literature- that certainly add realism to the model. Hetero-geneity among agents is crucial to more realisti-cally approach any population-based complex sys-tem. In the context of the MR model, inflexible and unsettled agents as well as “contrarians” have been considered in a series of papers by S. Galam and coworkers (and recently reviewed by Castellano et al. [3]). Also, we have discussed the ef-fects of several forms of heterogeneity on synchro-nization dynamics [4, 5], which bear close simi-larities with opinion formation. Overall, the ex-pected consequence of heterogeneity is that full consensus is never reached, but a dynamical state - where the degree of consensus fluctuates with time around a well-defined average- is asymptotically approached.

Spatially-distributed populations, where agent groups are localized in space, have been taken into account since the first formulations of the MR dy-namics [6, 7]. It has been shown that dimensionality has nontrivial consequences in the attainment of consensus. For a fixed group size G, the consensus time grows with the population size N as a power whose exponent depends on the dimension [3]. In one-dimensional arrays, the final state is sensibly dependent on the initial condition, while in higher dimensions, the dynamics coincides with diffusive coarsening.

Either random or deterministic time-dependent effects, such as stochastic fluctuations in the rules that govern single consensus events or population dynamics, are certainly worth considering. One may ask, for instance, whether sufficiently frequent “births” or “arrivals” of dissenters are able to over-come consensus attainment. To my knowledge, population dynamics has not been addressed in the context of MR model. On the other hand, random-ness in each consensus event has already been taken into account [3].

Finally, global opinion-formation factors such as mass media, advertising, and propaganda play the role of external fields in the spin-like dynamics of the MR model, as also discussed by S. Galam [3].

 

[1] H Fort, Commentary on “A note on the con-sensus time of mean-field majority-rule dy-namics”, Pap. Phys. 1, 010003 (2009).

[2] D H Zanette, A note on the consensus time of mean-field majority-rule dynamics, Pap. Phys. 1, 010002 (2009).

[3] C Castellano, S Fortunato, V Loreto, Statis-tical physics of social dynamics, Rev. Mod. Phys. 81, 591 (2009).

[4] G H Paissan, D H Zanette, Synchronization of phase oscillators with heterogeneous coupling: A solvable case, Physica D 237, 818 (2008).

[5] D H Zanette, Interplay of noise and coupling in heterogeneous ensembles of phase oscilla-tors, Eur. Phys. J. B 69, 269 (2009).

[6] P L Krapivsky, S Redner, Dynamics of major-ity rule in two-state interacting spin systems, Phys. Rev. Lett. 90, 238701 (2003).

[7] C J Tessone, R Toral, P Amengual, H S Wio, M San Miguel, Neighborhood models of minor-ity opinion spreading, Eur. Phys. J. B 39, 535 (2004).