PHYSICAL REVIEW E 84, 066101 (2011) Strategy of competition between two groups based on an inflexible contrarian opinion model Qian Li, 1,* Lidia A. Braunstein, 2,1 Shlomo Havlin, 3 and H. Eugene Stanley 1 1 Department of Physics and Center for Polymer Studies, Boston University, Boston, Massachusetts 02215, USA 2 Instituto de Investigaciones F´ ısicas de Mar del Plata (IFIMAR), Departamento de F´ ısica, Facultad de Ciencias Exactas y Naturales, Universidad Nacional de Mar del Plata-CONICET, Funes 3350, 7600 Mar del Plata, Argentina 3 Department of Physics, Bar Ilan University, Ramat Gan, Israel (Received 17 August 2011; revised manuscript received 26 October 2011; published 1 December 2011) We introduce an inflexible contrarian opinion (ICO) model in which a fraction p of inflexible contrarians within a group holds a strong opinion opposite to the opinion held by the rest of the group. At the initial stage, stable clusters of two opinions, A and B, exist. Then we introduce inflexible contrarians which hold a strong B opinion into the opinion A group. Through their interactions, the inflexible contrarians are able to decrease the size of the largest A opinion cluster and even destroy it. We see this kind of method in operation, e.g., when companies send free new products to potential customers in order to convince them to adopt their products and influence others to buy them. We study the ICO model, using two different strategies, on both Erd¨ os-R´ enyi and scale-free networks. In strategy I, the inflexible contrarians are positioned at random. In strategy II, the inflexible contrarians are chosen to be the highest-degree nodes. We find that for both strategies the size of the largest A cluster decreases to 0 as p increases as in a phase transition. At a critical threshold value, p c , the system undergoes a second-order phase transition that belongs to the same universality class of mean-field percolation. We find that even for an Erd¨ os-R´ enyi type model, where the degrees of the nodes are not so distinct, strategy II is significantly more effective in reducing the size of the largest A opinion cluster and, at very small values of p, the largest A opinion cluster is destroyed. DOI: 10.1103/PhysRevE.84.066101 PACS number(s): 89.75.Hc, 89.65.−s, 64.60.−i, 89.75.Da I. INTRODUCTION Competition between two groups or among a larger number of groups is ubiquitous in business and politics: the decades- long battle between the Mac and the PC in the computer industry, between Procter & Gamble and Unilever in the personal products industry, among all major international and local banks in the financial market, and among politicians and interest groups in the world of governance. All competitors want to increase the number of their supporters and thus increase their chances of success. In gathering supporters, competitors put much effort into persuading skeptics and those opponents who may actually be potential supporters. This kind of activity is normally modeled as a dynamic process on a complex network in which the nodes are the agents and the links are the interactions between agents. The goal of these models is to understand how an initially disordered configuration can become an ordered configuration through the interaction between agents. In the context of social science, order means agreement and disorder means disagreement [1,2]. Most of these models—e.g., the Sznajd model [3], the voter model [4,5], the majority rule model [6,7], and the social impact model [8,9]—are based on two-state spin systems which tend to reduce the variability of the initial state and lead to a consensus state in which all the agents share the same opinion. However this consensus state is not very realistic, since in many real competitions there are always at least two groups that coexist at the same time. Recently a nonconsensus opinion (NCO) model [10] was developed, where two opinions A and B compete and reach * liqian@bu.edu a nonconsensus stable state. At each time step each node adopts the opinion of the majority in its “neighborhood,” which consists of its nearest neighbors and itself. When there is a tie, the node does not change its state. Considering also the node’s own opinion leads to the nonconsensus state. The dynamics are such that a steady state in which opinions A and B coexist is quickly reached. It was conjectured, and supported by intensive simulations [10], that the NCO model in complex networks belongs to the same universality class as percolation [10–12]. The concepts of inflexible agents and contrarian agents were introduced by Galam et al. in their recent work on opinion models [13–16]. However, till now, no one has explored the opinion model with “inflexible contrarians.” Here we test how competition strategies are affected when “inflexible contrarians” are introduced. Inflexible contrarians are agents who hold a strong opinion that is opposite to the opinion held by the rest of the group [13,14]. And the inflexible here means that once the contrarians are chosen, they will not change their opinions under any circumstances [15,16]. We develop a spin-type inflexible contrarian opinion (ICO) model in which inflexible contrarian agents are introduced into the steady state of the NCO model. The goal of the inflexible contrarians is to change the opinions of the current supporters of the rival group [17]. We see this strategy in operation, for example, when companies send free new products to potential customers in order to convince them to adopt the products and encourage their friends to do the same. We can observe it also in political campaigns when candidates “bribe” voters by offering favors. The questions we ask in our model are as follows. Do these free products and bribes work and how? Who are the best individuals to chose as inflexible contrarians in order to make the most impact on opinion change. 066101-1 1539-3755/2011/84(6)/066101(8) ©2011 American Physical Society