Modeling framework unifying contact and social networks

Temporal networks of face-to-face interactions between individuals are useful proxies of the dynamics of social systems on fast timescales. Several empirical statistical properties of these networks have been shown to be robust across a large variety of contexts. To better grasp the role of various mechanisms of social interactions in the emergence of these properties, models in which schematic implementations of such mechanisms can be carried out have proven useful. Here, we put forward a framework to model temporal networks of human interactions based on the idea of a coevolution and feedback between (i) an observed network of instantaneous interactions and (ii) an underlying unobserved social bond network: Social bonds partially drive interaction opportunities and in turn are reinforced by interactions and weakened or even removed by the lack of interactions. Through this coevolution, we also integrate in the model well-known mechanisms such as triadic closure, but also the impact of shared social context and nonintentional (casual) interactions, with several tunable parameters. We then propose a method to compare the statistical properties of each version of the model with empirical face-to-face interaction data sets to determine which sets of mechanisms lead to realistic social temporal networks within this modeling framework.

Figure 1

Sketch of the dependencies between the interaction graph $G$ and the social bond graph $B$. Edges having a higher weight in social bond graph $B\left(t\right)$ are more likely to activate, i.e., to be part of interaction graph $G\left(t\right)$. The computation of $B(t+1)$ is done in two steps: First, $B$ is updated by the feedback of which edges were active in $G\left(t\right)$: unused ties decay while used ties strengthen. The output of this first step is denoted by $B(t+\frac{1}{2})$ because it refers to an intermediary step between $B\left(t\right)$ and $B(t+1)$, which is obtained from $B(t+\frac{1}{2})$ by a pruning process, consisting of removing weak unused social ties. The arrow from $G(t-1)$ to $G\left(t\right)$ is of a different nature than the arrow from $B\left(t\right)$ to $B(t+\frac{1}{2})$. The latter accounts for the inertia of the social bond graph, as $B$ can only encounter gradual change from one time to the next (implementing long-term memory). On the contrary, the arrow from $G(t-1)$ to $G\left(t\right)$ does not ensure that $G\left(t\right)$ will be similar to $G(t-1)$; it simply describes a short-term social context memory through the fact that the more two nodes share partners in $G(t-1)$, the more likely they are to be partners in $G\left(t\right)$.

Figure 2

Social mechanisms. We focus on node $i$ for two consecutive time steps $t-1$ and $t$, so we do not represent all the ties of $B$. (i) Reinforcement: $i$ interacts with $j$ at $t-1$, leading to a reinforcement of the social affinity of $i$ toward $j$: $B_{ij}\left(t\right)>B_{ij}(t-1)$. (ii) Cyclic closure: At $t-1, i$ decides to interact with a new partner. First, $i$ picks a known partner $j$ and then a known partner $l$ of $j$. The tie $B_{il}$ is created at $t$. (iii) Weakening: While $k$ is part of the egonet of $i, i$ does not interact with $k$ at $t-1$. This results in a weakening of the social affinity of $i$ toward $k$: $B_{ik}\left(t\right)<B_{ik}(t-1)$. (iv) Social context: The social affinity of $i$ toward $m$ at $t$ is temporarily increased by their common partners in $G(t-1)$. (v) Dynamic closure: Once the intentional interactions have all been drawn at $t, (m,i,l)$ is an open triangle in $G\left(t\right)$, which is likely to close because $m$ and $l$ are interacting with the same agent.

Figure 3

Dynamic triadic closure. After computation of the intentional interactions in $G\left(t\right)$, we identify its open triangles. Here $(i,k,j)$ is such an open triangle with edges $(i,k)$ and $(k,j)$ (black straight lines). Closing the triangle means either $i$ decides to interact with $j$ (probability $p_{ij}$) or vice versa (probability $p_{ji}$). $p_{ij}$ depends both on how close $i$ and $j$ are in the current social context ($c_{ij}$) and how close they are relatively to their common partner of discussion ($b_{ik}$ and $b_{kj}$): If $i$ gives a lot of attention to $k$, and $k$ a lot of attention to $j$, then it is likely that $i$ and $j$ will interact.

Figure 4

Sketch of the Hebbian-like process describing the evolution of the directed social bond weights from $B(t-1)$ to $B\left(t\right)$ due to the interactions in $G(t-1)$. Active edges are reinforced, inactive ties starting from an interacting node are weakened, and ties starting from a noninteracting node are unchanged.

Figure 5

Number of events per bursty period. The sketch represents the activation timeline of a pair of nodes $(i,j)$. A grey rectangle represents an event for $(i,j)$, i.e., a maximal period of uninterrupted interaction between $i$ and $j$. A bursty period is a maximal collection of events that follow each other in time by a delay less than a given threshold, taken here to be three time steps. We draw a green junction between two consecutive events if they belong to the same bursty period, i.e., if they are separated by $\mathrm{\Delta }t\le 3$ time steps, and a red junction if $\mathrm{\Delta }t>3$. In the example shown, we obtain two distinct bursty periods, with three events in the first and two in the second.

Figure 6

Ten most frequent (3,1)-ETN in two empirical data sets. (a) Data set “conf16” (scientific conference) and (b) data set “utah” (primary school in Utah). Although those two social contexts are very different, their ETNs are quite similar.

Figure 7

Computing the score of a model. The evaluated data sets consist of all versions of the models presented in Table 2 and the empirical data sets presented in Table 3. For each data set, a different score is computed for each observable. The higher the score of a model, the closer the distribution the associated observable is from the distributions in the empirical data sets. Said otherwise, a higher score means more realistic statistical properties for the associated observable.

Figure 8

Illustration of the scores for several observables and models. The figure displays in each panel the distribution of an observable (cc_size: size of connected components) for two data sets (the conference, conf16, and the school, utah) and three model versions: the original ADM (V14), the adjacent version with the highest average score (V9), and the version with the lowest average score (V12). The three models were optimized with respect to the utah reference. The score is computed in each case by Eq. (15): a higher score is associated with a distribution closer to the empirical one. In the top left panel, utah has a low score because the cc_size observable is the only distribution which is not similar for all empirical references.

Figure 9

Interpretation of the score of the ETN vector observable. (a)–(c) Five most frequent ETN of depth 3 at aggregation level 5 for two models tuned with respect to the empirical reference, utah. The model with the highest score (V9) has the same motifs as the reference, but not the worst version (V12). (d) ETN autosimilarity for three models (the best V9, the worst V12, and the original ADM V14) and two empirical references. The ETN autosimilarity of a given data set at aggregation level $n$ is the cosine similarity between the motifs observed in this data set at level $n$ and the motifs observed in this data set at level 1. Empirical references are highly autosimilar with respect to this measure. For models, a higher score for the ETN vector corresponds to a higher ETN autosimilarity.

Figure 10

Global ranking of models 1 to 14 according to two possible ranking strategies. In each panel, the $y$ coordinate is arbitrary: empirical data sets have been placed at $y=0$ while artificial data sets are aligned at $y=1$. The global ranking is obtained by sorting the models by increasing order of the $x$ coordinate (hence inverse global score). In each panel, the best version as well as the original ADM are highlighted as red dots. The best version in the other ranking strategy is displayed as a yellow dot. Blue vertical dashed lines indicate either the crossover or the gap between empirical and artificial data sets. (a) The $x$ coordinate is given by the model rank averaged over all observables. The best version is baseline V1, almost ex aequo with version 9. (b) The $x$ coordinate is given by the opposite of the averaged score, shifted by the maximum averaged score to take positive values. Version 9 (${\alpha }_{ij}={\beta }_{ij}=\alpha$) is here the best one. In both panels, the original ADM is ranked rather low. There is either an overlap (a) or a small gap (b) between models and empirical data sets. Thus, the class of models considered here is able to generate synthetic data sets with statistical properties close to real data sets.

Figure 11

Similarity between observables. (a) Similarity matrix obtained by computing the Kendall similarity between the associated rankings of our model versions (note that the results might be different if considering a different ensemble of models). The observables are quite independent from each other (low absolute values for the similarity for almost all observables pairs). Strong negative couplings are observed only between the point observables and some distribution observables. The ETN vector is either positively or weakly coupled to every other observable, in accordance with the conclusion of Sec. 3c: improving on motifs generally means improving on other observables. (b) The matrix is turned into a weighted network by taking as edge weight the absolute value of the Kendall similarity. A community detection algorithm based on modularity optimization detects three groups of observables (colored according to the group they belong to). The thickness of an edge is proportional to its weight and we filter out small weights for visualization purposes.

Figure 12

Score variation resulting from the change of hypothesis with respect to the baseline model. For each version, we compute the difference between its score and the score of the baseline (V1). In (a), these differences are summed over all observables of each of the groups identified in Fig. 11. The sum over group I (respectively, II and III) is called score I (respectively, II and III) and is denoted by $\mathrm{\Delta }{s}^{\text{I}}$ (respevtively, $\mathrm{\Delta }{s}^{\text{II}}$ and $\mathrm{\Delta }{s}^{\text{III}}$). (a) Score differences for each adjacent model version. Blue dots correspond to score I, red dots to scores II, and green dots to score III. The horizontal dashed line indicates the line of no changes in score. (b) Differences in score for each observable, represented as horizontal arrows. A difference is positive when the arrow points to the right and negative when it points to the left. Vertical blue dashed lines correspond to $\mathrm{\Delta }s=0$. The arrow length is proportional to the difference in score, rescaled so the longest arrow spans an arbitrary length of 0.5 unit. The edge weight and the size of connected components (cc_size) are the only observables on which a significant improvement is observed with respect to the baseline version.