Evolution of opinions on social networks in the presence of competing committed groups

Public opinion is often affected by the presence of committed groups of individuals dedicated to competing points of view. Using a model of pairwise social influence, we study how the presence of such groups within social networks affects the outcome and the speed of evolution of the overall opinion on the network. Earlier work indicated that a single committed group within a dense social network can cause the entire network to quickly adopt the group’s opinion (in times scaling logarithmically with the network size), so long as the committed group constitutes more than about 10% of the population (with the findings being qualitatively similar for sparse networks as well). Here we study the more general case of opinion evolution when two groups committed to distinct, competing opinions $A$ and $B$, and constituting fractions $p_A$ and $p_B$ of the total population respectively, are present in the network. We show for stylized social networks (including Erd\H{o}s-R'enyi random graphs and Barab'asi-Albert scale-free networks) that the phase diagram of this system in parameter space $(p_A,p_B)$ consists of two regions, one where two stable steady-states coexist, and the remaining where only a single stable steady-state exists. These two regions are separated by two fold-bifurcation (spinodal) lines which meet tangentially and terminate at a cusp (critical point). We provide further insights to the phase diagram and to the nature of the underlying phase transitions by investigating the model on infinite (mean-field limit), finite complete graphs and finite sparse networks. For the latter case, we also derive the scaling exponent associated with the exponential growth of switching times as a function of the distance from the critical point.

💡 Research Summary

This paper investigates how the presence of two competing, committed groups influences opinion dynamics on social networks. Building on earlier work that showed a single committed minority (≈10 % of the population) can rapidly drive a dense network to consensus, the authors extend the model to include two immutable factions, each supporting a distinct opinion (A or B) with fractions p_A and p_B of the total population. The underlying interaction rule is a binary‑agreement variant of the Naming Game: each node holds either opinion A, opinion B, or both (state AB). At each discrete time step a random speaker‑listener pair is selected; the speaker attempts to transmit one of its opinions (chosen uniformly if it holds both). If the listener already possesses that opinion, both parties retain only that opinion; otherwise the listener adds it to its repertoire. Committed nodes never change their opinion but otherwise behave identically to ordinary nodes.

The authors first derive mean‑field equations for the densities of uncommitted nodes in states A, B, and AB, together with the constraints imposed by the committed fractions. In the limit of infinite network size (equivalently, a complete graph), the dynamics reduce to a pair of coupled ordinary differential equations (Eq. 1). When only one committed group is present (p_A>0, p_B=0), the system reproduces the known first‑order (spinodal) transition at a critical committed fraction p_c≈0.1: below p_c the system possesses an absorbing consensus state and a metastable mixed state separated by a saddle point; above p_c the metastable state disappears and consensus is reached in logarithmic time.

When both p_A and p_B are non‑zero, the authors explore the (p_A, p_B) parameter plane numerically by integrating the mean‑field equations from two opposite initial conditions (all‑A‑dominant and all‑B‑dominant). They discover two distinct regions separated by two fold‑bifurcation (spinodal) lines that meet tangentially at a cusp (critical point). Inside the “bistable” region (region I) the phase space contains two stable fixed points (one A‑dominant, one B‑dominant) separated by an unstable saddle. Outside (region II) only a single stable fixed point exists, corresponding to a global consensus on one opinion. The cusp occurs at p_A = p_B = p_c, where the two spinodal lines coalesce; this is a second‑order (continuous) critical point analogous to equilibrium critical phenomena.

To validate the mean‑field predictions, extensive Monte‑Carlo simulations are performed on finite complete graphs and on sparse networks (Erdős‑Rényi and Barabási‑Albert). For each (p_A, p_B) pair, the authors run ten realizations from each of the two opposite initial conditions, measuring the order parameter m = (n_B − n_A)/(1 − p_A − p_B), its variance X_N, and the Binder cumulant U_N = 1 − ⟨m⁴⟩/(3⟨m²⟩²). Along the diagonal line p_B = p_A (c = 1) the system exhibits a continuous transition: the distribution of m changes from a symmetric double‑delta (two coexisting states) to a single Gaussian centered at zero as the cusp is crossed, and U_N drops from 2/3 to 0. Along off‑diagonal lines (e.g., c = 0.5) a discontinuous transition is observed: the magnetization jumps abruptly, and U_N shows a characteristic dip at the spinodal. The simulation results converge to the mean‑field bifurcation curves as N increases, confirming the analytical phase diagram.

Beyond the static phase structure, the paper examines the scaling of switching (or consensus) times τ on finite sparse networks. Near the critical point, τ grows exponentially with the inverse distance Δ = |p_A − p_c| (or |p_B − p_c|): τ ∝ exp(α Δ^−ν). The exponent ν depends on the network topology, reflecting how heterogeneity in degree distribution influences the barrier between the two metastable basins. This result implies that even a modest increase in the size of a competing committed minority can dramatically accelerate opinion reversal, whereas when both groups are near the critical size the system becomes extremely sluggish.

In summary, the authors provide a comprehensive theoretical and computational framework for opinion dynamics with two competing committed groups. They map out a full phase diagram featuring both first‑order (spinodal) and second‑order (cusp) transitions, validate it across mean‑field, complete‑graph, and sparse‑graph settings, and derive scaling laws for transition times. The work deepens our understanding of how polarized, unwavering factions can shape collective opinion, offering quantitative guidance for scenarios ranging from political campaigns to public‑health messaging in complex social networks.