Fig 1.
Colour intensities and edge widths are chosen to reflect relative magnitude of betweenness measures. Yellow and thick edges have high betweenness scores, dark grey and thin edges have low betweenness scores. (A) The SP betweenness. It fails to identify the upper and lower “bridge” as important global bottleneck edges. (B) The CF betweenness. (C) The LF betweenness with λ = 1. It is almost identical to the CF betweenness. (D) The LF betweenness with λ = 2/5. It completely ignores the global bottleneck “bridges” and turns attention to edges within each block.
Fig 2.
Characteristics of the original networks.
(A) Network community profiles (NCPs, see Methods for a brief introduction). The NCPs have been computed using the Local Graph Clustering API [39, 40] based on the original paper [24]. The markers in the NCPs in Fig 2A correspond to tightly-knit clusters of nodes that we used to initialize the epidemic models for Fig 2C. (B) Degree distributions. (C) Predicted epidemic curves without intervention.
Fig 3.
Simulation results for Facebook County.
(A) Initially infected cluster of counties. (B) Predicted epidemic curves at 25% coverage level, i.e., intervention is applied to 25% of all edges, along with the original epidemic curve without intervention (NI). The abbreviations in the legend are introduced at the beginning of Results. LF(a), where a ∈ {1/2, 1/10, 1/50}, represents LF intervention with parameter λ = a. (C) Predicted epidemic peaks (peak prevalence) over a range of intervention coverage levels. (D) Predicted epidemic sizes (total infection) over a range of intervention coverage levels. Observe that EG intervention does not reduce peak prevalence at all. This is likely due to the epidemic curve has two modes and EG mostly targets on edges connected to the counties where the first peak happens, e.g., Fig 3B shows that EG significantly reduces the first epidemic peak but has almost no effect on the second. Fig 3A uses Plotly Python Open Source Graphing Library [41] with base map data from Natural Earth @ naturalearthdata.com. Direct link to base map data: https://www.naturalearthdata.com/http//www.naturalearthdata.com/download/110m/cultural/ne_110m_admin_1_states_provinces.zip.
Fig 4.
Distribution of target edges reflected by county-level colours.
A county is coloured red if intervention is applied to most edges incident to it, a county is coloured dark blue if intervention is applied to very few edges incident to it. (A) SP intervention. (B) CF intervention. (C) LF intervention, λ = 1/50. The plots are made with base map data from Natural Earth @ naturalearthdata.com. Direct link to base map data: https://www.naturalearthdata.com/http//www.naturalearthdata.com/download/110m/cultural/ne_110m_admin_1_states_provinces.zip.
Fig 5.
Characteristics of the modified Facebook County networks due to targeted interventions using different edge-betweenness measures.
(A) NCPs (see Methods for a brief introduction). (B) Distribution of small-size clusters (having less than 100 nodes) by conductance. (C) Percentage of targeted out-link edges from the initial cluster of infection. Fig 5A and 5B show that the modified network due to LF intervention has many more well-defined (i.e., low conductance) small clusters, which can be effective at slowing down or stopping the spread of disease among tightly-knit small communities. For example, Fig 5C shows that only the top 5% edges based on LF betweenness already include all the out-link edges from the initial cluster of infection in Fig 3A.
Fig 6.
Simulation results for Wi-Fi Montreal.
(A) Predicted epidemic curves at 25% coverage level. (B) Predicted epidemic peaks over a range of intervention coverage levels. (C) Predicted epidemic sizes over a range of intervention coverage levels.
Fig 7.
Characteristics of the modified Wi-Fi Montreal networks due to targeted interventions using different edge-betweenness measures.
Locality bias of LF betweenness for Wi-Fi Montreal is illustrated by the dramatic difference in Fig 7C: When the network has too many singleton nodes, LF intervention isolates those nodes. (A) NCPs (see Methods for a brief introduction). (B) Distribution of clusters by conductance. (C) Isolation of singleton nodes.
Fig 8.
Simulation results for both sub-sampled and full Portland networks.
For the full Portland network we used λ = 1/1000 for scalability. (A) Predicted epidemic peaks, sub-sampled network. (B) Predicted epidemic sizes, sub-sampled network. (C) Predicted epidemic peaks, full network. (D) Predicted epidemic sizes, full network.
Fig 9.
Characteristics of the modified Portland networks due to targeted interventions.
LF intervention causes the networks to contain significantly more nodes having low degrees, and thus greatly reducing the probability of disease spread over these nodes. (A) NCPs, sub-sampled network. (B) Degree distributions, sub-sampled network. (C) NCPs, full network. (D) Degree distributions, full network.
Fig 10.
Simulation results under alternative model parametrization.
The transmission rate parameter is calibrated so that 55% of the population would be affected without intervention. We initialized both population-based and individual-based models using random initialization, where a few randomly selected nodes are labelled as infectious at time 0. The plots show for each dataset the final epidemic sizes under different intervention strategies and various percentage coverages. We average over 50 runs for random initialization. (A) Results for Facebook County. LF for λ ∈ {1/10, 1/50} still leads to the most reduction in epidemic sizes when the coverage level is less than 30%. When targeting more than 30% edges, uniform intervention (UI), which uniformly reduces the transmission rate over all edges, becomes more effective for Facebook County. This is because the transmission rate parameter is getting close to a threshold under which a pandemic would not emerge, which is seen in the 0% final size under UI at 50% coverage level. (B) Results for Wi-Fi Montreal. LF interventions still lead to the most reduction in the final sizes. (C) Results for Port. Sub. CF and LF with λ = 1/2 have the best performance overall, while LF with λ = 1/10 is the most effective when targeting less than 15% edges. (D) Results for Portland. We used λ = 1/1000 in order to scalably compute LF. We see that LF has better performance when targeting no more than 40% of all edges.
Fig 11.
Simulation results for node immunization on sub-sampled and full Portland networks.
The results are obtained from agent-based SEIR model with random initializations and are averaged over 50 trials. (A) Predicted epidemic peaks, sub-sampled network. (B) Predicted epidemic sizes, sub-sampled network. (C) Predicted epidemic peaks, full network. (D) Predicted epidemic sizes, full network.
Fig 12.
Computation time of betweenness measures.
All computations are carried out on a personal laptop with 32GB RAM and 2.9 GHz 6-Core Intel Core i9. We used NetworkX [55] for computing SP and CF. We implemented LF computation in Julia. Note that the computation time scales linearly with λ. Moreover, since the range of λ values shown in the figures includes the values we used for our experiments, one may obtain accurate estimates on the computation times of all betweenness measures we used for the experiments. (A) Computation times for Facebook County. (B) Computation times for Port. Sub. (C) Computation times for Wi-Fi Montreal. CF is omitted for Wi-Fi Montreal because it takes too long to finish.