Higher-order modeling of face-to-face interactions

TL;DR

This work proposes a model in which agents interact with each other by forming groups of different sizes, each has a degree of social attractiveness, based on which neighboring agents decide whether to join, and captures homophilic patterns at the level of higher-order interactions, going beyond standard pairwise approaches.

Abstract

The most fundamental social interactions among humans occur face to face. Their features have been extensively studied in recent years, owing to the availability of high-resolution data on individuals' proximity. Mathematical models based on mobile agents have been crucial to understand the spatio-temporal organization of face-to-face interactions. However, these models focus on dyadic relationships only, failing to characterize interactions in larger groups of individuals. Here, we propose a model in which agents interact with each other by forming groups of different sizes. Each group has a degree of social attractiveness, based on which neighboring agents decide whether to join. Our framework reproduces different properties of groups in face-to-face interactions, including their distribution, the correlation in their number, and their persistence in time, which cannot be replicated by dyadic models. Furthermore, it captures homophilic patterns at the level of higher-order interactions, going beyond standard pairwise approaches. Our work sheds light on the higher-order mechanisms at the heart of human face-to-face interactions, paving the way for further investigation of how group dynamics at a microscopic scale affects social phenomena at a macroscopic scale.

Figures (5)

• Figure 1: Schematic illustration of the Group Attractiveness Model. At each time step $t$, each active agent $i$ (blue) considers the groups lying within a radius $d$ from it, and interacts with all of them with a probability $p_i(t)$ that depends on the mean attractiveness of the neighboring groups. The agent interacts only with the group members within its scope and ignores inactive nodes (gray). With the complementary probability $1 - p_i(t)$, the agent moves away in a random direction, with a step of length $v$.
• Figure 2: The GAM reproduces the empirical group statistics. Panels a to f report the distribution of groups of different sizes in a given social system (black circles), as well as the predictions of the Group Attractiveness Model (blue squares) and the Attractiveness Model starnini2013modeling (red diamonds). Markers represent the average number of groups generated by the models over 100 simulations, while error bars indicate the standard deviation. Whereas the Attractiveness Model largely overestimates the number of large groups in the data, the Group Attractiveness Model is able to correctly predict the group statistics.
• Figure 3: The GAM reproduces the correlation between the number of groups of sizes two and three. Correlation in the number of groups of size two and three in various social contexts (black circles) are compared to the predictions of the Group Attractiveness Model (blue squares) and the Attractiveness Model (red diamonds). Markers represent the average correlation over 100 simulations, while error bars indicate the standard deviation. The Attractiveness Model systematically underestimates correlations in the number of groups, while the Group Attractiveness Model better reproduces empirical values of correlation.
• Figure 4: Hierarchical organization of higher-order burstiness. We show the distributions of contact duration for groups of different sizes in the "HS11" dataset (panel a), as well as those predicted by the Group Attractiveness Model (panel b) and the Attractiveness Model (panel c). The Attractiveness Model predicts large groups to be more stable than small ones. Instead, the Group Attractiveness Model correctly reproduces the hierarchical organization of the distributions observed in the data, as groups with less individuals remain in contact for longer than groups with more individuals.
• Figure 5: Higher-order homophily in face-to-face interactions. Panel a shows a schematic of the Group Attractiveness Model with homophily. At each time step $t$, an active agent $i$ (blue) considers the groups within its scope, and decides with a probability $p_i(t)$ based on the mean attractiveness of the neighboring groups (see \ref{['eq:eq2']}); if it stays, the agent chooses the group(s) to which it connects based on the homophily matrices $H^{(2)}$, $H^{(3)}$, and so on, which depend on its own attributes (shapes and colors) and those of the group member(s). Panels b and c display the homophily matrices $H^{(2)}$ and $H^{(3)}$, modulating the formation of groups of two and three individuals, respectively, obtained for the interactions in the "HS11" dataset. Men prefer to interact with other men at the level of pairwise interactions, while women are more homophilic in groups of three individuals. Panel d shows the fraction of unique groups of size three in the different gender configurations present in the "HS11" dataset (black bars), together with those predicted by the Group Attractiveness Model (blue bars), and by the Social-Attractiveness Model oliveira2022group (yellow bars). The value of the bars represents the average fraction over 100 simulations, while the error bars indicate the standard deviation. The Social-Attractiveness Model overestimates the tendency of men to interact with their same gender, while the Group Attractiveness Model is in good agreement with the empirical mixing patterns.