Fig 1.
First, we extract the community structure if it is not known, then we compute the community-aware centrality measures and sort the nodes from most influential to least. Afterwards, we take a top fraction of nodes denoted as fo to be infected/activated. The size of fo varies according to a predefined budget, which we set from 1% of the network’s size to 50% in our study. Then, these selected nodes will initiate the dynamics under the four conceptually different models: the SI, SIR, IC and LT models. When the models reach their stable state, we compare their output. Every model has a different evaluation measure to evaluate the output. In the SI model, the average number of iterations needed to infect 50% of the network is computed: the lower the number of iterations, the more effective the centrality measure. In the SIR and IC models, the relative outbreak and activation size denoted as ΔR and ΔA, respectively are computed. This value quantifies the difference between the number of nodes recovered or activated based on a given community-aware centrality measure and a reference measure which is the classical degree centrality in our case: the higher ΔR and ΔA are, the better the performance of the community-aware centrality measure. Finally, in the LT model, the evaluation measure is the total number of activated nodes normalized by the size of the network: the higher Ar is, the better.
Table 1.
A summary of the studies of community-aware centrality measures. SIR means Susceptible-Infected-Recovered model and LT refers to Linear Threshold model. The character ‘-’ refers to “not applicable”. indicates that the goal is to minimize diffusion and
indicates that the goal is to maximize diffusion.
Fig 2.
Comparing the diffusion models under study.
λ is the infection rate, ψ is the recovery rate, mv is the total number of active neighbors node v possesses, ξv is node the threshold of node v, Pu,v is the likelihood of node u activating node v, and ξu,v is the threshold of edge (u, v).
Fig 3.
Behavior of the community-aware centrality measures under various dynamic models in synthetic networks while varying the mixing parameter (μ).
The first, second, third, and fourth rows indicate the results of the (A) SI model, (B) SIR model, (C) IC model, and (D) LT model.
Fig 4.
Behavior of the community-aware centrality measures under various dynamic models in synthetic networks while varying the community size distribution exponent (θ).
The first, second, third, and fourth rows indicate the results of the (A) SI model, (B) SIR model, (C) IC model, and (D) LT model.
Fig 5.
Behavior of the community-aware centrality measures under various dynamic models in synthetic networks while varying the degree distribution exponent (γ).
The first, second, third, and fourth rows indicate the results of the (A) SI model, (B) SIR model, (C) IC model, and (D) LT model.
Fig 6.
Behavior of the community-aware centrality measures under various dynamic models in real-world networks with varying community structure strengths.
The first, second, third, and fourth rows indicate the results of the (A) SI model, (B) SIR model, (C) IC model, and (D) LT model.
Fig 7.
Comparing the position of the top nodes in the Kegg Metabolic network (μ = 0.466).
The top nodes are chosen at a low budget availability (fo = 1%), medium budget availability (fo = 25%), and high budget availability (fo = 40%). The bigger nodes in the left, middle, and right figures are the top nodes ranked by Comm Centrality (αComm), K-shell with Community (αks), and Modularity Vitality targeting hubs (), respectively.
Fig 8.
Comparing the position of the top nodes in the Facebook Politician Pages network (μ = 0.111).
The top nodes are chosen at a low budget availability (fo = 1%), medium budget availability (fo = 25%), and high budget availability (fo = 40%). The bigger nodes in the left, middle, and right figures are the top nodes ranked by Comm Centrality (αComm), Modularity Vitality targeting hubs (), and Modularity Vitality targeting bridges (
), respectively.
Fig 9.
Comparing the position of the top nodes in the Hamsterster and Facebook Politician networks.
The top nodes are chosen at a low budget availability (fo = 1%) and medium budget availability (fo = 25%). The bigger nodes in the left, middle, and right figures are the top nodes ranked by Map Equation Centrality (αMapEq), Community Hub-Bridge (αCHB), and Comm Centrality (αComm), respectively.
Fig 10.
Comparing the position of the top nodes in the Facebook Politician Pages and Ego Facebook networks.
The top nodes are chosen at a medium budget availability (fo = 25%) and high budget availability (fo = 40%). The bigger nodes in the left, middle, and right figures are the top nodes ranked by Modularity Vitality targeting hubs and bridges (|αMV|), Community Hub-Bridge (αCHB), and Comm Centrality (αComm), respectively.
Fig 11.
Comparing the position of the top nodes in the Ego Facebook networks.
The top nodes are chosen at a high budget availability (fo = 40%). The bigger nodes in the left, middle, and right figures are the top nodes ranked by Modularity Vitality targeting hubs and bridges (|αMV|), hubs only (), and bridges only (
), respectively.
Table 2.
Summary of the best performing community-aware centrality measures.
Fig 12.
Comparing the trends of the various dynamic models in Hamsterster with its communities identified by Infomap and Louvain.
The first, second, third, and fourth rows indicate the results of the (A) SI model, (B) SIR model, (C) IC model, (D) LT model.
Fig 13.
Comparing the position of the top nodes in the Hamsterster network having its communities identified by Infomap and Louvain.
The top nodes are chosen at a low budget availability (fo = 1%) and medium budget availability (fo = 25%). The bigger nodes in the left, middle, and right figures are the top nodes ranked by Comm Centrality (αComm), Community Hub-Bridge (αCHB), and Modularity Vitality targeting hubs (), respectively.
Fig 14.
Histograms of the community size distribution of the Hamsterster network.
Communities are identified by Infomap and Louvain.