Structural diversity in social contagion

Abstract

The concept of contagion has steadily expanded from its original grounding in epidemic disease to describe a vast array of processes that spread across networks, notably social phenomena such as fads, political opinions, the adoption of new technologies, and financial decisions. Traditional models of social contagion have been based on physical analogies with biological contagion, in which the probability that an individual is affected by the contagion grows monotonically with the size of his or her "contact neighborhood"--the number of affected individuals with whom he or she is in contact. Whereas this contact neighborhood hypothesis has formed the underpinning of essentially all current models, it has been challenging to evaluate it due to the difficulty in obtaining detailed data on individual network neighborhoods during the course of a large-scale contagion process. Here we study this question by analyzing the growth of Facebook, a rare example of a social process with genuinely global adoption. We find that the probability of contagion is tightly controlled by the number of connected components in an individual's contact neighborhood, rather than by the actual size of the neighborhood. Surprisingly, once this "structural diversity" is controlled for, the size of the contact neighborhood is in fact generally a negative predictor of contagion. More broadly, our analysis shows how data at the size and resolution of the Facebook network make possible the identification of subtle structural signals that go undetected at smaller scales yet hold pivotal predictive roles for the outcomes of social processes.

Fig. 1.

Contact neighborhoods during recruitment. (A) An illustration of a small friendship neighborhood and a highlighted contact neighborhood consisting of four nodes and three components. (B–D) The relative conversion rates for two-node, three-node, and four-node contact neighborhood graphs. Shading indicates differences in component count. For five-node neighborhoods, see Fig. S1. Invitation conversion rates are reported on a relative scale, where 1.0 signifies the conversion rate of one-node neighborhoods. Error bars represent confidence intervals and implicitly reveal the relative frequency of the different topologies.

Fig. 2.

Recruitment contact neighborhoods and component structure. (A) Conversion as a function of edge count neighborhoods with one connected component (1 CC) with four to six nodes, where variations in edge count predict no meaningful difference in conversion. (B) Conversion as a function of neighborhood size, separated by CC count. When component count is controlled for, size is a negative indicator of conversion. (C) Conversion as a function of tie strength in two-node neighborhoods, measured by photo co-tags, a negative indicator of predicted conversion. Recruitment conversion rates are reported on a relative scale, where 1.0 signifies the conversion rate of one-node neighborhoods. Error bars represent confidence intervals.

Fig. 3.

Inviter position during recruitment. Shown is recruitment conversion as a function of neighborhood graph topology and inviter position in neighborhoods of size 4. The position of the inviter within the neighborhood graph is described exactly (up to symmetries) by node degree. Shading indicates differences in component count. Recruitment conversion rates are reported on a relative scale, where 1.0 signifies the conversion rate of one-node neighborhoods. Error bars represent 95% confidence intervals.

Fig. 4.

Engagement and structural diversity for 50-node friendship neighborhoods. (A) Illustration of the connected components in a friendship neighborhood, delineating connected components and components of size . (B) Illustration of the k-core and the k-brace, delineating the connected components of the 2-core and the 1-brace. (C) Engagement as a function of connected component count. (D) Engagement as a function of the number of components of size , for , with connected component (CC) count shown for comparison. (E) Engagement as a function of k-core component count for , with CC count shown for comparison. (F) Engagement as a function of k-brace component count for , with CC count shown for comparison. Engagement rates are reported on a relative scale, where 1.0 signifies the average conversion rate of all 50-node neighborhoods. All error bars are 95% confidence intervals. For other neighborhood sizes, see Fig. S4.

Fig. 5.

Engagement as a function of edge density. For five different neighborhood sizes, , we see that when component count is not accounted for, an internal engagement optimum is observed, showing the combined forces of focused context and structural heterogeneity. Engagement rates are reported on a relative scale, where 1.0 signifies the average conversion rate of all 50-node neighborhoods. All error bars are 95% confidence intervals.