Table 1.
Distance graphs (D) of the social contact networks analyzed, with respective references.
|X|: number of nodes; (|di>j|): number of finite distance edges; τ(D) relative size of the metric backbone; σ(D): edge redundancy. Values of τ, and σ are shown as percentages (%). For the Exhibit networks, values of |X|, (|di>j|), τ and σ denote the mean ± standard deviation of the networks computed for each of the 69 days for which data were gathered. See Section 4.1 for additional details and a description of the networks.
Fig 1.
French high school (Fr-HS) and American high school (US-HS) contact networks.
(A-C) French high school (Fr-HS) contact network of |X| = 327 students. Colors represent the four student specializations: “MP” in blue, “PC” in green, “PSI” in orange, and “BIO” in red; lighter or darker colors separate the distinct classes within each specialization. (D-F) American high school (US-HS) contact network of |X| = 788 students, staff, and teachers. Colors represent students in blue, teachers in red, and staff in orange. Other (unspecified) individuals are shown in green. No student class metadata is available. (A, D) Original networks with node layout computed by ForceAtlas2 algorithm [47], using all proximity weights, P(X). (B, E) Metric backbone subgraphs rendered with the same node layout as in the respective networks in (A) and (D). (C, F) Metric backbone subgraphs with node layout recomputed by ForceAtlas2 using only backbone (proximity) weights. Plotted with Gephi [48].
Table 2.
SocioPatterns Fr-HS contact network and its community structure detected by the Louvain algorithm [49] for the original (proximity) network and several of its subgraphs.
Top rows (m) show the number of distinct metalabels and the number of communities detected, while bottom rows show the bidirectional modularity similarity, yAB (Eq 8 in Section 4.2). Columns show values for original graph, metric backbone subgraph, and same-size threshold and random subgraphs. Mean and standard deviation shown for 100 random subgraphs.
Fig 2.
Community structure comparison for the French high school (Fr-HS) dataset using Sankey plots.
(A) Comparison of metalabels—student attribution to their classes—with modules detected on the original graph. (B) Comparison of the modules detected on the original graph with those detected on its metric backbone subgraph. (C) Comparison of the modules detected on the original graph with those detected on the threshold subgraph of same size as the metric backbone. (D) Comparison of the modules detected on the original graph with those detected on a random subgraph of the same size as the metric backbone. Metalabel module colors assigned to student specialization and classes as in Fig 1A–1C. Modules computed using the Louvain algorithm [49].
Fig 3.
Synthetic network with connectivity generated by a Stochastic Block Model (SBM) and edge weights sampled from the SocioPatterns French Primary School (Fr-PS) contact network.
(A). Synthetic module sizes. (B). SBM generator matrix for low connectivity case. Note the hierarchical structure of modules C and D. (C). The adjacency matrix of the original graph and the metric backbone. (D). Force layout visualization of the original (proximity) graph as computed by the NetworkX python package [52]. Node positioning obtained after 100 iterations. (E). Force layout visualization of the metric backbone subgraph. Node positioning seeded from the original graph (panel C) and obtained after 100 iterations. See Sections A and 4 in S1 Text for details about synthetic network generation.
Fig 4.
Normalized time to infection using the metric backbone, threshold, or random subgraphs of the French High School (Fr-HS) network.
The horizontal axis denotes χ, a parameter to sweep the proportion of edges of the original network that are included in the subgraphs analyzed. When χ = 0% (leftmost value on axis) we have the metric backbone subgraph, or threshold and random subgraphs with the same number of edges as the metric backbone (i.e. |{bij}| = τ(D).|{dij}|, per Eq (6)). As χ increases, edges from the original network that are not on the backbone or same-size threshold and random subgraphs, are progressively added until the original network itself is reached at χ = 100% (see text for more details). (Left) Time for half of the population to be infected, t1/2, normalized by the results obtained using the entire original network, . (Right) Time for all nodes in the network to be infected, t1, normalized by the results obtained using the entire original network,
. Spreading times for every curve are averaged over nr = 10 runs of the SI model starting from a random seed node and 100 network realizations obtained via random edge removals for each value of χ. The green and blue bars, quantified against the right vertical axis in each panel, denote the fraction of networks in threshold and random baseline ensembles that are connected for a given χ. Disconnected networks are discarded to compute the spreading times. For the simulations shown, the spreading parameter was set as β = 0.9/pmax where pmax is the largest proximity weight of the original network.
Fig 5.
Distribution of semi-metric edge distortion for contact networks.
Box plots of the distributions of semi-metric distortion for semi-metric edges (sij > 1) for all contact networks in Table 1. White bars and black circles denote the median and mean values, respectively. Actual distributions are shown in SM.