Emergent Directedness in Social Contagion

TL;DR

This work tackles the unpredictability of complex contagion diffusion in social networks by introducing a causal diffusion framework that simulates multiple potential diffusion paths and computes edges and nodes with maximal causal impact. It defines Causal Tie Importance and Causal Node Importance to trace activation sequences and reveals emergent directedness: even in undirected networks, complex contagions often diffuse along predominantly one-way paths, with directionality increasing as contagion complexity grows. The findings overturn conventional centrality intuitions, showing, for example, that the periphery can become diffusion cores and that moderate-strength ties can dominate diffusion, while tie-range and triadic-closure dynamics shape bridge symmetry. These insights have practical implications for network design and interventions, highlighting how weak ties, bridge formation, and reinforcement requirements influence real-world diffusion of norms, technologies, and behaviors.

Abstract

An enduring challenge in contagion theory is that the pathways contagions follow through social networks exhibit emergent complexities that are difficult to predict using network structure. Here, we address this challenge by developing a causal modeling framework that (i) simulates the possible network pathways that emerge as contagions spread and (ii) identifies which edges and nodes are most impactful on diffusion across these possible pathways. This yields a surprising discovery. If people require exposure to multiple peers to adopt a contagion (a.k.a., 'complex contagions'), the pathways that emerge often only work in one direction. In fact, the more complex a contagion is, the more asymmetric its paths become. This emergent directedness problematizes canonical theories of how networks mediate contagion. Weak ties spanning network regions - widely thought to facilitate mutual influence and integration - prove to privilege the spread contagions from one community to the other. Emergent directedness also disproportionately channels complex contagions from the network periphery to the core, inverting standard centrality models. We demonstrate two practical applications. We show that emergent directedness accounts for unexplained nonlinearity in the effects of tie strength in a recent study of job diffusion over LinkedIn. Lastly, we show that network evolution is biased toward growing directed paths, but that cultural factors (e.g., triadic closure) can curtail this bias, with strategic implications for network building and behavioral interventions.

Figures (6)

• Figure 1: An Example of Emergent Directedness in Complex Contagion. All panels depict complex contagions with a threshold $T = 2$. (a) A symmetric bridge between the red and green community where contagion can spread in both directions. (b) An asymmetric bridge where it is possible to spread from the red to the green neighborhood but not in the reverse direction. (c) An exemplary undirected network before analysis. (d) The same network visualized with edge opacity scaled by our measure of Causal Tie Importance, revealing emergent asymmetries and directed pathways along which complex contagions are most likely to flow, even in this structurally undirected system. This visualization demonstrates how the interaction between network topology and complex contagion dynamics induces emergent directional biases.
• Figure 2: Effect of threshold values on symmetry. The boxplots display the distribution of the symmetry measure $\Xi_s(\text{RCS})$, where higher values indicate more symmetric spreading dynamics. Panel (a) shows results for absolute thresholds $T_i$ on Watts--Strogatz graphs ($n = 200$, $k = 8$), while panels (b)–(d) report results for relative thresholds $\theta_i$ on Watts--Strogatz (b), AddHealth (c), and Banerjee et al. (d) networks. For (a) and (b), 1000 Watts–Strogatz graphs were generated using 100 evenly spaced $\beta$ values in $[0, 1]$, with 10 graphs per $\beta$. For each giant component on each network, we define a simulation scenario based on its specific graph structure and a given threshold value. For each scenario, $100N$ independent and randomly chosen clustered seed sets comprised of $5\%$ of all nodes are generated to compute the symmetry measure, where $N$ denotes the number of nodes in the graph. Across all settings, the boxplots reveal a consistent pattern: increasing the threshold systematically reduces symmetry, indicating a shift toward more directional and asymmetric contagion pathways.
• Figure 3: Structural Dependencies Between Tie Range and Causal Tie Importance Asymmetry Across Graph Types. (a--b) Schematic representations of Tie Range, defined as the second-shortest path between two adjacent nodes. In (a), the red tie spans a range of 2, while in (b), it spans a range of 4. (c--e) Heatmaps of Tie Importance asymmetry ($\Delta$) as a function of Tie Range and the maximum importance of each tie, across (c) 1000 generated Watts--Strogatz graphs ($n = 400$, $k = 8$) using 100 evenly spaced $\beta$ values in $[0, 1]$, with 10 graphs per $\beta$, (d) AddHealth, and (e) Banerjee et al. networks with threshold $T = 2$. For each giant component on each network, we define a simulation scenario based on its specific graph structure and a given threshold value $\theta_i$. Each scenario was repeated 10 times, $10N$ independent and randomly chosen clustered seed sets comprised of $5\%$ of all nodes are generated to compute the maximum importance of each tie, where $N$ denotes the number of nodes in each graph. The x-axis denotes the maximum CPC value across both directions of each tie. The y-axis shows the Tie Range. The color spectrum indicates the strength of Tie Importance Delta, which captures the level of emergent directedness, with higher deltas (warmer colors) indicating more directedness (i.e., more asymmetric flow favoring one direction along a tie).
• Figure 4: Tie importance in complex contagion spreading and job mobility. (A) Mean tie importances for weak, medium, and strong ties under a complex contagion model across three network datasets: the Addhealth and Banerjee et al. (2012) empirical networks, and scale-free network with tunable clustering holme2002growing ($n=1000$) generated for degrees $k \in {2, 3, 4, 5}$ and clustering probabilities $p$ linearly spaced in $[0, 1]$, matching the number of AddHealth graph instances. For each giant component on each network, we define a simulation scenario based on its specific graph structure and a given threshold value $\theta_i \in [0.1, 0.15, 0.2, 0.25, 0.3, 0.35]$. For each scenario, $5N$ independent and randomly chosen clustered seed sets comprised of $2\%$ of all nodes are generated to compute the Tie Importance values of each tie, where $N$ denotes the number of nodes in the graph. Data points show the partial effect of tie type estimated by an OLS regression that includes fixed effects by dataset, network, and threshold. (B) Empirical LinkedIn job mobility data showing the effect of weak, medium, and strong ties (defined by mutual friend terciles) on job transmissions, reproduced from rajkumar2022causal based on values extracted from the original figure, for visual comparison. Both results, using the same tie strength definition, highlight the disproportionate impact of medium-strength ties in diffusion. Error bars indicate 95% confidence intervals.
• Figure 5: Global Realignments of Node Importance and Influence Flow Induced by Complex Contagions. (a) Visualization of the shift in degree-normalized node importance from the network core to the periphery as the spreading mechanism transitions from simple ($T=1$; T is used in place of $\theta$ for absolute thresholds) to complex contagions with increasing relative thresholds ($\theta = 0.05, 0.1, 0.2$) on a scale-free network with tunable clustering ($n = 1000$, $m = 4$, $p = 0.4$) holme2002growing. Node color indicates degree-normalized importance, with red representing high importance and blue representing low importance. As $\theta$ increases, influence moves away from central hubs toward peripheral nodes. (b) Boxplot of the correlation $\rho(\Delta S, \Delta k)$ on the AddHealth network, quantifying how tie importance asymmetry ($\Delta S$) relates to degree differences ($\Delta k$) across varying $\theta$. Negative correlations reflect core-to-periphery spreading, while positive values indicate a reversal toward periphery-to-core flow. For each threshold configuration and graph (including panel (a) and each graph in the AddHealth dataset for (b)), $100N$ independent clustered seed sets covering $5\%$ of nodes were used. Results for panel (b) include only simulation scenarios achieving full network activation.