Higher-order modeling of face-to-face interactions

Technical Summary: Higher-order modeling of face-to-face interactions

Problem Statement
Face-to-face interactions constitute the most fundamental form of human social contact. While high-resolution data has revealed universal spatio-temporal features of these interactions—such as bursty activity, heterogeneity in behavior, and the absence of a characteristic contact duration scale—existing mathematical models are insufficient to capture their full complexity. Current agent-based models primarily rely on dyadic (pairwise) relationships or temporal networks focusing on links and connected components. These approaches fail to characterize interactions within larger groups, treating many-to-many interactions merely as collections of one-to-one contacts. Furthermore, existing higher-order network approaches often lack spatial mobility, while current mobile agent models fail to capture the spatio-temporal evolution and specific dynamics of group formation and dissolution.

Methodology: The Group Attractiveness Model (GAM)
To address these limitations, the authors propose the Group Attractiveness Model (GAM), an agent-based framework where mobile agents interact by forming groups of varying sizes. The model operates in a square environment with periodic boundary conditions and includes the following key mechanisms:

• Agent Attractiveness: Each agent $i$ possesses an intrinsic attractiveness value $a_i$, sampled from a uniform distribution $[0,1]$, representing the likelihood of others interacting with them.
• Group Attractiveness: The attractiveness of a group $g$, denoted $a_g$, is defined as the product of the attractiveness of its members: $a_g = \prod_{j \in g} a_j$. This formulation implies that group attractiveness scales nonlinearly with size; larger groups are, on average, less attractive than smaller ones.
• Interaction Dynamics: At each time step, an active agent considers groups within a distance $d$. The probability $p_i(t)$ of interacting with a neighboring group is proportional to the mean attractiveness of those groups.
  • If an agent chooses to interact, it joins the group (increasing group size by one) or rejoins its existing group.
  • If an agent chooses not to interact (with probability $1 - p_i(t)$), it moves in a random direction.
  • A group persists in time only if all its members decide to interact with one another.
• Activity States: Agents can be active or inactive. Inactive agents do not move or interact but can become active with a certain probability, modeling individuals entering or leaving a social context.
• Homophily Extension: The model is extended to include homophilic preferences by associating agents with attributes (e.g., gender). Interaction probabilities are modulated by homophily matrices ($H^{(2)}, H^{(3)}$, etc.), which dictate the likelihood of an agent joining a group based on the attributes of existing members.

The model was tested against thirteen high-resolution datasets (including SocioPatterns, Utah experiments, and the Copenhagen Network Study). Group interactions were reconstructed from dyadic contact data by identifying time steps where all possible dyads among $s$ individuals were present.

Key Results
The authors compared the GAM against a baseline (individual) Attractiveness Model (AM) which only accounts for dyadic relationships. The results demonstrate:

1. Group Size Distribution: The GAM successfully reproduces the empirical distribution of unique groups of different sizes across various social contexts. In contrast, the AM significantly overestimates the number of larger groups.
2. Correlation of Group Counts: The model accurately captures the high Pearson correlation between the number of groups of size two and size three that individuals participate in. The AM systematically underestimates this correlation, failing to recognize that groups are not merely collections of pairs.
3. Hierarchical Burstiness: Empirical data shows that contact duration distributions are broad-tailed and hierarchically organized, with smaller groups exhibiting broader distributions (lasting longer) than larger groups. The GAM reproduces this hierarchical structure. The AM fails here, predicting that larger groups are more stable than smaller ones.
4. Higher-Order Homophily: When extended to include gender attributes, the GAM captures complex mixing patterns at the group level. For instance, in the "HS11" dataset, the model reveals that while men exhibit homophily in pairwise interactions, women exhibit stronger homophily in groups of three. The GAM outperforms the Social-Attractiveness Model (SAM), which relies solely on pairwise homophily to infer group mixing, by correctly predicting the fraction of unique triads in different gender configurations.

Significance and Claims
The paper claims that higher-order interactions are essential for accurately describing human face-to-face contacts. The primary contribution is the demonstration that modeling group dynamics through a "group attractiveness" mechanism allows for the reproduction of statistical properties (distribution, correlation, persistence) and structural features (hierarchical burstiness, higher-order homophily) that dyadic models cannot replicate.

The authors position the GAM as a tool to investigate how microscopic group dynamics influence macroscopic social phenomena, such as epidemic spreading, social contagion, and the emergence of segregation. They acknowledge limitations, noting that the current model is Markovian (lacking memory effects), assumes a single mechanism for joining/leaving groups, and does not distinguish between different types of social groups or contexts. However, the work provides a foundational framework for moving beyond pairwise approaches to understand the collective behavior of complex social systems.