Fig 1.
Sketch of the models of contagion considered.
In all sketches, black nodes represent infectious hosts, empty nodes are susceptible, and colored nodes represent the hosts that can be contaminated by the infectious ones. Left: Simple contagion on weighted graphs. Contagion events occur along the network edges, with probability per unit time given by β multiplied by the weight Wij of the edge (i, j) between a susceptible and an infected node. Center: Simplicial model on weighted hypergraphs. Contagions can take place both along network edges (rate βWij) and if a susceptible node i is part of a group (i, j, k) with j and k both infectious (rate , with
the weight of the hyperedge (i, j, k)). Right: Threshold model on weighted graphs. A susceptible node becomes infected when the sum of the weights of its connections with infected nodes, divided by the total weight of its connections, exceeds a threshold θ.
Fig 2.
Toy network illustrating the asymmetry of the infection pattern and its dissimilarity with the adjacency matrix. The upper sketch shows the weighted adjacency matrix (links’ width proportional to their weights, nodes’ size proportional to their weighted degree). The lower sketch represents the infection pattern for a simple SIR contagion with R0 = 2 (averaged over 500 simulations). For each connection only the direction with higher probability of infection is shown and the arrows’ width is proportional to the probability. The nodes are colored according to their spreader index.
Fig 3.
A: Cosine similarity between the infection patterns of different models of simple contagion, simulated with the same R0 = 2.5 (see Methods for the description of the models). B: Cosine similarity between the infection patterns obtained at varying R0 for the SIR model of simple contagion. C: Cosine similarity between infection patterns at varying R0 for the SIR model, with infection patterns computed only using runs with final attack rate between 0.75 and 0.85. D: Same as C but using runs with final attack rate between 0.5 and 0.6. E: Same as C but using runs with final attack rate lower than 0.2. The results in panel C have been obtained by comparing, for each value of R0, infection patterns obtained by averaging over 1000 simulations with final attack rate a in the chosen range. For panels D and E the number of simulations to average on has been increased to 10000 and 50000, respectively. Indeed, smaller values of a mean that less nodes and links are involved in each run, so that one needs to average over more runs to compute the infection probability for each link.
Fig 4.
Comparison, for the SIR model, between a reference R0 = 4 and five testing parameter values (R0 from 1.5 to 3.5). Each curve in the upper panel represents the similarity in time between the temporal infection pattern Cref(t) of the reference and the infection pattern of each testing parameter. Cref(t) is computed by averaging, over 1000 numerical simulations of the SIR model at R0 = 4, the contagion events occurring until t.
is instead obtained by averaging all contagion events of 1000 numerical simulations of the SIR model at R0. The middle panel shows as a black curve the temporal evolution of the non-zero mode of the distributions of attack rates of the reference spread, also computed over all 1000 simulations at R0 = 4 and at each time. The colored dots show, for each R0 ∈ {1.5, 2, 2.5, 3, 3.5}, the value of the non-zero mode of the final attack rate distribution, computed over 1000 simulations at each R0. The corresponding attack rate distributions are shown in the smaller panels below.
Fig 5.
A: Cosine similarity between infection patterns at varying different combinations of β| and βΔ. B: Number of contagions taking place via first and second order simplices in the simulations of the previous panel. C1 is the infection pattern matrix obtained considering only infections taking place via pairwise links and C2 is the analogous for triads infections, with C1 + C2 = C. In the plot we report the sum of all elements of the matrices ∑ij(C1)ij and ∑ij(C2)ij, which give the respective fractions of contagion events of each type. C: Cosine similarity between infection patterns at varying different combinations of β and βΔ, when computing the infection patterns using only simulations with attack rate between 0.6 and 0.7. D: Number of contagions taking place via first and second order simplices in the simulations of the previous panel. E: Cosine similarity between infection patterns of simplicial contagion (for the same range of values of β| and βΔ) and simple contagion (for different values of R0).
Fig 6.
A: Cosine similarity between infection patterns at varying θ. B: Cosine similarity between infection patterns of threshold contagion (for different values of θ) and simple contagion (for different values of R0).
Table 1.
Simple contagion model parameters.