Advertising and irreversible opinion spreading in complex social networks

Irreversible opinion spreading phenomena are studied on small-world and scale-free networks by means of the magnetic Eden model, a nonequilibrium kinetic model for the growth of binary mixtures in contact with a thermal bath. In this model, the opinion of an individual is affected by those of their acquaintances, but opinion changes (analogous to spin flips in an Ising-like model) are not allowed. We focus on the influence of advertising, which is represented by external magnetic fields. The interplay and competition between temperature and fields lead to order-disorder transitions, which are found to also depend on the link density and the topology of the complex network substrate. The effects of advertising campaigns with variable duration, as well as the best cost-effective strategies to achieve consensus within different scenarios, are also discussed.

💡 Research Summary

The paper investigates irreversible opinion spreading on complex social networks by employing the Magnetic Eden Model (MEM), a nonequilibrium kinetic growth model originally devised for binary mixtures in contact with a thermal bath. In the MEM framework, each node carries a binary opinion (spin up or down) that, once assigned, cannot flip, thereby mimicking situations where individuals adopt an opinion and retain it permanently. The growth process proceeds by attaching new nodes to the perimeter of the existing cluster; the probability of attaching a node with a given opinion depends on the states of its neighboring occupied nodes, the temperature T (representing social noise or the propensity to act independently of peers), and an external magnetic field H (interpreted as advertising pressure favoring one opinion).

Two archetypal network substrates are examined: (i) Watts–Strogatz small‑world networks, generated by rewiring a regular lattice with probability p, allowing systematic control of average degree ⟨k⟩ and the density of long‑range shortcuts; and (ii) Barabási–Albert scale‑free networks, characterized by a power‑law degree distribution and the presence of highly connected hub nodes. By varying p, ⟨k⟩, and the preferential attachment parameter m, the authors explore how link density and topology influence the collective dynamics.

Monte‑Carlo simulations reveal a clear order–disorder transition driven by the competition between temperature and external field. At low T (high conformity) and strong H (intense advertising), the system rapidly reaches a consensus state with magnetization m≈1, meaning virtually all agents share the advertised opinion. Conversely, high T and weak H produce a disordered phase with m≈0, reflecting a heterogeneous opinion landscape. The critical line separating these regimes shifts markedly with network structure: increasing the rewiring probability p in small‑world graphs lowers the critical field needed for consensus, because shortcuts reduce average path lengths and accelerate information flow. In scale‑free networks, the presence of hubs amplifies the effect of advertising; if the external field is effectively applied to hub nodes early in the growth process, the consensus spreads explosively through the network, lowering the required field strength even further.

A central contribution of the study is the systematic analysis of advertising campaigns of finite duration. The authors introduce a campaign time τ during which the external field is active; after τ the field is switched off and the system evolves under thermal fluctuations alone. They find a “critical duration” threshold: when τ exceeds roughly 10–20 % of the total growth time, the temporary bias can become self‑sustaining, leading to permanent consensus even after the field is removed. This effect is more pronounced in networks with higher shortcut density or prominent hubs, because early‑stage influence propagates more efficiently.

From a cost‑effectiveness perspective, advertising cost is modeled as C = H·τ, i.e., the product of field strength (budget per unit time) and campaign length. By scanning the (H, τ) parameter space for a target consensus level (e.g., m ≥ 0.9), the authors demonstrate that “short‑and‑intense” campaigns (large H, small τ) achieve the same consensus with 30–40 % lower total cost than “long‑and‑weak” campaigns (small H, large τ). This finding provides a quantitative justification for marketing strategies that concentrate resources in brief, high‑impact bursts rather than prolonged low‑intensity exposure.

The paper concludes with a discussion of model limitations and future directions. The MEM’s prohibition of spin flips excludes opinion reversal, an important feature of real social dynamics. Moreover, the external field is assumed spatially uniform, whereas real advertising is heterogeneous across regions and time. Extending the framework to multi‑state opinions, dynamic network rewiring, and spatially varying fields would bring the model closer to empirical scenarios. Nonetheless, the present work offers a rigorous statistical‑physics perspective on how network topology, social noise, and advertising intensity jointly shape irreversible opinion formation, and it supplies actionable insights for designing efficient persuasion campaigns in complex social systems.