Network localization governs social contagion dynamics with macro-level reinforcement

TL;DR

This work addresses how macro-level social reinforcement interacts with local contagion on networks, by introducing a linear feedback SIR-like model where transmissibility grows as $\beta'(t)=\min(1,\beta+\alpha\frac{R(t-1)}{N})$. Using simulations and dynamic message passing, it shows a phase diagram with a stable critical point $\beta_c$ and a reinforcement threshold $\alpha_c$ that triggers a mixed-order transition, characterized by an abrupt outbreak jump with lingering criticality. A key contribution is the localization-based metric $\mathcal{L}$, derived from non-backtracking centrality, which predicts $\alpha_c$ via $\alpha_c = \lambda \mathcal{L}^{\eta}$ (empirically $\lambda\approx2.63$, $\eta\approx1$), and links topology to diffusion efficiency through $\beta_c = \frac{1}{\langle k\rangle^3 (\alpha_c/\lambda)^{2/\eta} + \langle k\rangle -1}$. The framework reveals a fundamental trade-off: networks that localize weak contagions tend to slow diffusion but enable broader spread under suitable reinforcement, challenging the notion that stronger local connectivity always facilitates contagion. These results offer a structural lens to optimize or mitigate diffusion by tuning network localization and macro-level feedback.

Abstract

The spread of ideas, behaviors, and technologies generally depends on feedback mechanisms operating across multiple scales. Previous studies have extensively examined pairwise transmission and local reinforcement. However, the role of macro-level social influence -- where widespread adoption enhances further adoption -- remains understudied. Here, we focus on a contagion process that incorporates both pairwise interactions and macro-level reinforcement. We show that the contagion undergoes a shift from continuous to mixed-order transition as macro-level influence exceeds a reinforcement threshold. Simulations on various real-world networks indicate that network localization governs the contagion outcomes by determining the critical point and the reinforcement threshold. Building on this insight, we develop a structural metric linking network localization to contagion dynamics, revealing a key trade-off: networks that facilitate weak contagion tend to experience slower diffusion and lower adoption rates, while networks that suppress weak contagions enable faster and more widespread adoption. These findings challenge the conventional belief that stronger local connectivity uniformly promotes contagion.

Figures (4)

• Figure 1: Definition and behavior of the proposed model. (a) The diagram depicts the contagion process within a simplified network consisting of eight nodes. The transmissibility, $\beta'(t)$, of the SIR-like model depends on both intrinsic attractiveness, $\beta$, and macro-level reinforcement, $\alpha\frac{R(t-1)}{N}$. At time $t$, an adopter tries to convert two susceptible neighbors, with the macro-level influence contribution to transmissibility being $\alpha \frac{3}{8}$, since there are three recovered nodes in the system. At time $t+1$, the level of feedback intensity rises to $\alpha \frac{4}{8}$ due to the addition of one more recovered node. (b) The final proportion of recovered nodes, $R(\infty)/N$, is depicted as a function of $\beta$ for three different values of $\alpha$ across three real-world networks (Advogato, Enron-Large, and CondMat networks). Dots represent simulation results, while solid lines denote theoretical predictions derived from the dynamic message passing (DMP) method. As the level of feedback intensity increases, the contagion exhibits a finite jump in the outbreak size, as indicated by the gray backgrounds. Additional details are provided in the main text. (c) Phase diagrams for the same networks as in (b) are divided by a vertical line at the critical point $\beta_c$, separating the "vanished" and "prevalent" contagion states. In the former state, the contagion decays, while in the latter state, it spreads widely. Solid lines ($\alpha<\alpha_c$) indicate continuous transitions, while dashed curves ($\alpha>\alpha_c$) indicate transitions that develop a finite jump in outbreak size. The white dot marks the value of $\alpha_c$ at which this qualitative change in transition behavior occurs.
• Figure 2: Critical point ($\beta_c$) and reinforcement threshold ($\alpha_c$) for real-world networks. (a) Comparison between simulated and predicted values of $\beta_c$, based on Eq. (\ref{['eq:eqs23']}). (b) Comparison between simulated and predicted values of $\alpha_c$, derived from Eq. (\ref{['eq:alpha_l']}). Each dot represents a real-world network, and the solid diagonal lines ($y=x$) indicate where predicted values match simulated results. The mean squared logarithmic errors (MSLE) between simulations and predictions are annotated in each plot.
• Figure 3: Effect of network localization on the contagion process. We simulate the proposed model with the parameters $\alpha=1$ and $\beta=\beta_c$ (the theoretical critical value) on two synthetic networks, shown in (a) and (b), respectively. The network in (b) is generated by randomly rewiring the edges of the real-world network in (a), while preserving its degree sequence. Node sizes correspond to non-backtracking centrality, with gray nodes indicating the final susceptible states and blue nodes indicating the final recovered states; the initial seed is marked in orange. The histograms display the distribution of node non-backtracking centrality indices, with each bar's color reflecting the proportion of recovered nodes within the respective bin. The figure also includes annotations for the localization metric, $\mathcal{L}$, for both networks. For the critical thresholds, we observe $\beta_c=0.104$, and $\alpha_c=1.17$ for network (a), and $\beta_c =0.183, \alpha_c = 0.58$ for network (b).
• Figure 4: Relationship between the critical point and the reinforcement threshold. Values of $\beta_{c}$ and $\alpha_{c}$ are obtained through numerical simulations. Each marker represents a network, with its position indicating the corresponding intrinsic threshold and reinforcement threshold. Solid lines illustrate the relationship between $\beta_c$ and $\alpha_c$ as described by Eq. (\ref{['eq:beta_alpha_relation']}) for the specified average degree. (a) Equation (\ref{['eq:beta_alpha_relation']}) is plotted with $\langle k \rangle = 2$. (b) Networks are categorized by average degree, with lines representing different $\langle k \rangle$ values. A logarithmic scale is used to highlight finer details. (c) Adjusting network assortativity yields configurations with varying $\beta_c$ and $\alpha_c$ values for four selected real-world networks (InterdisPhysics, Delicious, Advogato, Fb-pages-artist). Arrows indicate the directions of these changes. (d) Similar to (c), but applied to three scale-free networks generated by the configuration model.