Fig 1.
The heatmap centrality algorithm pseudocode.
Fig 2.
The rankings of nodes in the example network.
The top two-ranked nodes in the example network G with 15 nodes and 19 edges with respect to the (A) degree, (B) eigenvector, (C) closeness, (D) betweenness, and (E) heatmap centrality measures are colored orange, while the remaining 13 nodes are colored grey.
Table 1.
The ranking of all 15 nodes in the example network G with respect to the degree (CD), eigenvector (CE), closeness (CC), betweenness (CB), and heatmap (CHM) centrality measures.
Table 2.
The Spearman-rank correlation for the betweenness (CB) and heatmap (CHM) centrality measures is calculated based upon the difference (di) in the final ranking with respect to the rankings of CB and CHM for each node (vi) in network G.
Table 3.
The Kendall-rank correlation for the betweenness (CB) and heatmap (CHM) centrality measures is calculated based upon both the total number of concordant (NC) and discordant (ND) pairs observed in network G.
Table 4.
The value of represents the number of edges added to each new node in each time step such that the network of size N is of the desired density d.
In the experiment, a total of 30 scale-free networks of various size and density are simulated.
Table 5.
The mean and standard deviation of the average degree <CD> for each network size and density calculated from the 100 simulated networks.
Table 6.
The mean and standard deviation of the clustering coefficient <cc> for each network size and density calculated from the 100 simulated networks.
Table 7.
The mean and standard deviation of the diameter for each network size and density calculated from the 100 simulated networks.
Fig 3.
The degree distribution of the four real-world networks.
The degree distribution of each real-world network exhibits a power law distribution P(k)~k−γ, where γ>1. The R2 calculated from the linear regression analysis on log(P(k))~ −γlog(k) is provided for each network as a measure of the goodness of fit for the power law model on the degree distribution. The closer R2 is to 1, the higher the degree of fit to the power-law distribution.
Table 8.
The basic structural features of the four real-world scale-free networks.
Fig 4.
The CPU time for the centrality measures.
The CPU time (in seconds) of the both the (A) betweenness and (B) heatmap centrality measures required to calculate the value of each node in the scale-free networks of size N and density d averaged over 100 iterations. The standard deviation of the CPU times at each point is included.
Fig 5.
The Spearman-rank correlation coefficient ρ.
The value of ρ for the rankings with respect to the (A) heatmap and degree, (B) heatmap and eigenvector, (C) heatmap and closeness, and (D) heatmap and betweenness centrality measures applied to each simulated scale-free network of size N and density d. The standard deviation of ρ at each point is included.
Fig 6.
The Kendall-rank correlation coefficient τ.
The value of τ for the rankings with respect to the (A) heatmap and degree, (B) heatmap and eigenvector, (C) heatmap and closeness, and (D) heatmap and betweenness centrality measures applied to each simulated scale-free network of size N and density d. The standard deviation of τ at each point is included.
Table 9.
The top-10 ranked nodes of the Email network with respect to the degree (CD), eigenvector (CE), closeness (CC), betweenness (CB), and heatmap (CHM) centrality measures.
The nodes in bold are identified by all five measures.
Table 10.
The top-10 ranked nodes of the Polblogs network with respect to the degree (CD), eigenvector (CE), closeness (CC), betweenness (CB), and heatmap (CHM) centrality measures.
The nodes in bold are identified by all five measures.
Table 11.
The top-10 ranked nodes of the USFlights network with respect to the degree (CD), eigenvector (CE), closeness (CC), betweenness (CB), and heatmap (CHM) centrality measures.
The nodes in bold are identified by all five measures.
Table 12.
The top-10 ranked nodes of the Facebook network with respect to the degree (CD), eigenvector (CE), closeness (CC), betweenness (CB), and heatmap (CHM) centrality measures.
The nodes in bold are identified by all five measures.
Fig 7.
The spreading capability F(t) in the Email network.
A comparison of F(t) of the top-10 nodes in the Email network between the heatmap centrality and the (A) degree, (B) eigenvector, (C) closeness, and (D) betweenness centrality measures. The standard deviation of F(t) at each point is included.
Fig 8.
The spreading capability F(t) in the Polblogs network.
A comparison of F(t) of the top-10 nodes in the Polblogs network between the heatmap centrality and the (A) degree, (B) eigenvector, (C) closeness, and (D) betweenness centrality measures. The standard deviation of F(t) at each point is included.
Fig 9.
The spreading capability F(t) in the USFlights network.
A comparison of F(t) of the top-10 nodes in the USFlights network between the heatmap centrality and the (A) degree, (B) eigenvector, (C) closeness, and (D) betweenness centrality measures. The standard deviation of F(t) at each point is included.
Fig 10.
The spreading capability F(t) in the Facebook network.
A comparison of F(t) of the top-10 nodes in the Facebook network between the heatmap centrality and the (A) degree, (B) eigenvector, (C) closeness, and (D) betweenness centrality measures. The standard deviation of F(t) at each point is included.
Table 13.
The Spearman-rank correlation ρ of the rankings of the heatmap (CHM) centrality with respect to the rankings of the degree (CD), eigenvector (CE), closeness (CC), and betweenness (CB) centrality measures, respectively, for each of the four real-world networks.
For each network, the value of ρ(CB,CHM) is the largest.
Table 14.
The Kendall-rank correlation τ of the rankings of the heatmap (CHM) centrality with respect to the rankings of the degree (CD), eigenvector (CE), closeness (CC), and betweenness (CB) centrality measures, respectively, for each of the four real-world networks.
For each network, the value of τ(CB,CHM) is the largest.