arXiv:cond-mat/0305102v3 [cond-mat.stat-mech] 28 May 2004 EPJ manuscript No. (will be inserted by the editor) Analytical results for the Sznajd model of opinion formation Frantiˇ sek Slanina 1a and Hynek Laviˇ cka 2 1 Institute of Physics, Academy of Sciences of the Czech Republic, Na Slovance 2, CZ-18221 Praha, Czech Republic 2 Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Bˇ rehov´a 7, CZ-11519 Praha 1, Czech Republic Abstract. The Sznajd model, which describes opinion formation and social influence, is treated analytically on a complete graph. We prove the existence of the phase transition in the original formulation of the model, while for the Ochrombel modification we find smooth behaviour without transition. We calculate the average time to reach the stationary state as well as the exponential tail of its probability distribution. An analytical argument for the observed 1/n dependence in the distribution of votes in Brazilian elections is provided. PACS. 89.65.-s Social and economic systems – 05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion – 02.50.-r Probability theory, stochastic processes, and statistics 1 Introduction There is significant convergence between statistical physics and mathematical sociology in approaches to their respec- tive fields [1]. Ising model, the single most studied statis- tical physics model, finds its numerous applications in so- ciophysics simulations. Conversely, sociologically inspired models pose new challenges to statistical physics. We be- lieve this is the case of the Sznajd model we are studying here. The model of K. Sznajd-Weron and J. Sznajd [2] was designed to explain certain features of opinion dynamics. The slogan “United we stand, divided we fall” lead to simple dynamics, in which individuals placed on a lattice (one-dimensional in the first version) can choose between two opinions (political parties, products etc.) and in each update step a pair of neighbours sharing common opinion persuade their neighbours to join their opinion. Therefore, it was noted that contrary to the Ising or voter [3] models, information does not flow from the neighbourhood to the selected spin, but conversely, it flows out from the selected cluster to its neighbours. The model initiated a surge of immediate interest [4,5, 6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25] and the results of numerical simulations can be briefly summarised as follows. The results do not depend much on the spatial dimensionality or on the type of the neigh- bourhood selected [11]. In the case of q choices of opinion, the system has q obvious homogeneous stationary (absorb- ing) states, where all individuals choose the same opinion. There is no way to go out of the homogeneous state, so it is a e-mail: slanina@fzu.cz an attractor of the dynamics. This is reminiscent of a zero- temperature dynamics, which in Ising model leads to rich behaviour [26]. However, in the Sznajd model, the pos- sible metastable states, like the “antiferromagnetic” con- figuration have negligible probability to occur, unless we introduce explicitly also an “antiferromagnetic” dynamic rule as it was used in the very first formulation [2]. The case q = 2 was studied mostly, denoting the opin- ions by Ising variables +1 and −1. The probability of hit- ting the stationary state of all +1 (or, complementary, all −1) was studied, depending on the initial fraction p of the individuals choosing +1. Sharp transition was ob- served at value p =0.5 [11]; for p> 0.5 the probability to reach eventually the state of all opinions +1 is close to one, while for p< 0.5 it is negligible, which can be inter- preted as a dynamical phase transition. The distribution of times needed to reach the stationary state was mea- sured, revealing a peak followed by relatively fast decay. This means that the average hitting time is a well-defined quantity [11]. It was also found in one and two-dimensional lattices that the fraction of individuals who never changed opin- ion decays as a power with time, similarly to Ising model. While the exponent in one dimension agrees with the Ising case, the two-dimensional Sznajd model gives different ex- ponent than Ising model, indicating different dynamical universality class [13]. Also the waiting time between two subsequent opinion changes is distributed according to a power-law [2]. Among other studies, let us mention the influence of advertising effects [18,19] and price formation [20]. Long- range interactions were studied in [21].