Strategy of Competition between Two Groups based on a Contrarian Opinion Model

We introduce a contrarian opinion (CO) model in which a fraction p of 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 contrarians which hold a strong B opinion into the opinion A group. Through their interactions, the 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 the product and influence others. We study the CO model, using two different strategies, on both ER and scale-free networks. In strategy I, the contrarians are positioned at random. In strategy II, the contrarians are chosen to be the highest degrees nodes. We find that for both strategies the size of the largest A cluster decreases to zero 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 ER type model, where the degrees of the nodes are not so distinct, strategy II is significantly more effctive in reducing the size of the largest A opinion cluster and, at very small values of p, the largest A opinion cluster is destroyed.

💡 Research Summary

The paper introduces a “contrarian opinion” (CO) model that extends the previously studied non‑consensus (NCO) opinion dynamics on complex networks. In the NCO model, agents hold one of two binary opinions (A or B) and at each synchronous update a node adopts the majority opinion among its neighbors plus itself; this leads to a stable coexistence of A‑ and B‑clusters above a critical initial fraction f_c of opinion A. The authors take the steady state of the NCO model as the initial condition for the CO model. They then convert a fraction p of the agents that originally hold opinion A into “contrarians”: agents that permanently hold the opposite opinion B and are immune to influence, yet they can influence their neighbors. After this conversion the NCO dynamics are run again until a new steady state is reached.

Two network topologies are examined: Erdős–Rényi (ER) random graphs with Poisson degree distribution and scale‑free (SF) graphs with a power‑law degree distribution (exponent λ≈3.5). For each topology two strategies for selecting the contrarians are considered: (I) random selection of a fraction p of A‑agents, and (II) targeted selection of the highest‑degree A‑agents (the hubs).

The main observable is S₁, the relative size of the largest A‑cluster after the second NCO relaxation. As p increases, S₁ shrinks continuously and vanishes at a critical contrarian fraction p_c, indicating a second‑order phase transition. The control parameter can be either the initial A‑fraction f (as in the original NCO model) or the contrarian fraction p; both lead to the same critical behavior. The transition is characterized by the usual percolation signatures: the size of the second‑largest cluster S₂ peaks at p_c, the derivative dS₁/dp shows a sharp jump that becomes more pronounced with system size, and the cluster‑size distribution at criticality follows n_s ∝ s^‑τ with τ≈2.5. The susceptibility exponent γ≈1 is also measured, confirming that the CO model belongs to the mean‑field percolation universality class.

Strategy II (targeted) is consistently more efficient than strategy I (random). In ER networks, the critical contrarian fraction drops from p_c≈0.20 (random) to p_c≈0.12 (targeted) for typical initial A‑densities; the average degree of A‑nodes ⟨k⟩ falls below the percolation threshold (⟨k⟩=1) at a lower p in the targeted case, leading to earlier fragmentation of the A‑component. In SF networks the effect is amplified because hubs dominate the connectivity. Targeting hubs can destroy the giant A‑cluster with a contrarian fraction as low as p≈0.05, whereas random placement requires p≈0.15–0.20. The authors demonstrate that high‑degree nodes are overwhelmingly located in the largest A‑cluster before contrarian insertion, so targeting them maximizes the number of affected edges.

The paper also explores the role of majority versus minority status. When the group employing contrarians is the majority, its average degree is already high, making it harder to reduce ⟨k⟩ below the percolation threshold; nevertheless, targeted contrarians still outperform random ones. When the contrarians belong to the minority, a very small p suffices to fragment the majority’s giant component, highlighting the disproportionate power of a few well‑placed influencers.

Overall, the study provides a clear theoretical framework for “influence‑by‑contrarians” strategies. It shows that inserting a modest fraction of stubborn agents with the opposite opinion can trigger a percolation‑type collapse of the opponent’s opinion cluster. Targeting high‑degree nodes dramatically reduces the required fraction, especially in heterogeneous (scale‑free) networks that resemble many real social systems. These insights have practical implications for marketing (free product seeding), political campaigning (targeted persuasion), and any scenario where a minority seeks to destabilize a dominant opinion through strategic placement of unwavering advocates.