Simple spatial processes can generate heterogeneous contact distributions in face-to-face interactions

Abstract

Face-to-face interactions reveal recurring patterns, suggesting the possibility of shared underlying mechanisms. More specifically, inter-contact durations, contact durations and number of contacts per edge share similar heavy-tail distributions in many empirical settings. A common intuition is that face-to-face interactions may be influenced by spatial constraints, and that the observed complex behaviors could arise from such physical limitations. Our models explore the impact of this constraint by simulating pedestrian dynamics, and studying the generated temporal network of contacts. Previous work showed that the inter-contact duration distribution is recovered with a pedestrian dynamic as simple as the two dimensional random walk, but this approach doesn't allow to recover the distribution of the number of times a pair of individuals has been in contact. One assumption is that the number of contact between individual arises from the social relationship between them, in other words a memory of past interactions. However, we here present models that are based on solely spatial rules, by adding simple targeting mechanisms to the two-dimensional random walk. We show that these models allow to recover a broad distribution of the number of contacts, revealing the importance of two ingredients: localized phases and controlled population mixing. This suggests that the observed heterogeneity in the contact numbers within the data does not necessarily emerge from underlying social relationships between individuals, since an equivalent distribution may be reproduced using a purely spatially based model, without the need for memory mechanisms.

Figures (4)

• Figure 1: a: Definition of the temporal observables. Looking at the timeline of activation of the link between two nodes, one can define three observables: the contact durations $\tau$, the inter-contact durations $\Delta \tau$ and the number of contacts $n$. b: Conference datasets. In the conference dataset genoisCombiningSensorsSurveys2023, those three observables are power-law distributed. c: Two dimensional random walk. Distributions obtained simulating $1000$ ghost particles undergoing a two dimensional random walk as defined in masoumi2025simplecrowddynamicsgenerate. Only the inter-contact duration distribution is recovered.masoumi2025simplecrowddynamicsgenerate
• Figure 2: Description of the models.a: Particles are moving following simple pedestrian dynamics and at each time-steps a contact is defined when two particles are facing each other within a certain radius. b: A particle alternates between (i) a Random Walk (RW), where the particle is freely diffusing and (ii) a Targeting Walk (TW), where the particle's motion is biased with a drift towards a target. The transition rates between the two walks are $r$ and $s$. c: When a particle is performing a Targeting Walk, three target position choice processes are tested: (i) Resampled on arrival destination. (ii) Fixed in time destination and (iii) Constrained resampled on arrival destination.
• Figure 3: Stability of the number of contact distributions. Each row presents the result for a different targeting mechanism. The first row corresponds to resampled on arrival destinations, the second to fixed in time destinations and the last to constrained resampled on arrival destinations. Each column correspond to different values of the rates $r$ and $s$. The first column corresponds to the $RW$-dominance case ($r=0.001$ and $s=0.1$), the second column corresponds to the mixed case ($r=s=0.01$) and the last column to the $TW$-dominance case ($r=0.1$ and $s=0.001$). In each cases is plotted the distribution for different values of $n_t$, the total amount of time-steps. The dashed ine is the slope $p(n) \sim n^{-3}$ present in Fig. \ref{['confrontation_part']}.
• Figure 4: Evolution of the average degree in the aggregated network with two homogeneous target-choice mechanism. Each panel correspond to different targeting mechanisms: (i) resampled on arrival destinations and (ii) fixed in time destination (see panel (c) Fig. \ref{['fig:methods']}). Both panels show results for three different regime: the $RW$-dominance case ($r=0.001$ and $s=0.1$), the $TW$-dominance case ($r=0.1$ and $s=0.001$) and the mixed case ($r=s=0.01$). The two horizontal dashed lines show the domain of evolution of $\left<k\right>/N$, i.e. between $10^{-3}$ (corresponding to $\left<k\right> = 1$) and $1$ (corresponding to $\left<k\right> = N$). Results were obtained by averaging $10$ simulations of $n_t=10^5$ time-steps with $N=1000$ ghost particles. Errors were computed and are too low to appear on the plot.