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doi: 10.1038/srep08665.

Understanding the influence of all nodes in a network

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Understanding the influence of all nodes in a network

Glenn Lawyer. Sci Rep..

Abstract

Centrality measures such as the degree, k-shell, or eigenvalue centrality can identify a network's most influential nodes, but are rarely usefully accurate in quantifying the spreading power of the vast majority of nodes which are not highly influential. The spreading power of all network nodes is better explained by considering, from a continuous-time epidemiological perspective, the distribution of the force of infection each node generates. The resulting metric, the expected force, accurately quantifies node spreading power under all primary epidemiological models across a wide range of archetypical human contact networks. When node power is low, influence is a function of neighbor degree. As power increases, a node's own degree becomes more important. The strength of this relationship is modulated by network structure, being more pronounced in narrow, dense networks typical of social networking and weakening in broader, looser association networks such as the Internet. The expected force can be computed independently for individual nodes, making it applicable for networks whose adjacency matrix is dynamic, not well specified, or overwhelmingly large.

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Figures

Figure 1
Figure 1. Deriving the expected force from the possible outcomes of two transmissions.
This network will be in one of eight possible states after two transmissions from the the seed node (red). Two states are illustrated, where the seed has transmitted to the two orange nodes along the solid black edges. Each state has an associated number of (dashed orange) edges to susceptible nodes (blue), the cluster degree. States containing two neighbors of the seed (panel a) can form in two ways or, if they are part of a triangle, four ways. The eight network states associated with the pictured seed node arrise from thirteen possible transmission clusters. The expected force of a seed node is the entropy of the distribution of the (normalized) cluster degree over all (here 13) possible transmission clusters.
Figure 2
Figure 2. Correlation of spreading power metrics to epidemic outcomes on simulated networks.
Violin plots show the distribution of observed correlation values for each spreading process outcome in each network family. The expected force and ExFM (orange shades) are consistently strong, with mean correlations greater than 0.85 and small variance. The other measures (k-shell, eigenvalue centrality, and accessibility, blue-green shades) show both lower mean values and higher variance, as seen in the position and vertical spread of their violins. Each violin summarizes correlations computed on 100 simulated networks. Spreading processes (x axis) are suffixed to indicate simulations in continuous (-C) or discrete (-D) time. The epidemic outcome for SI processes is the time until half the network is infected. For SIS and SIR processes it is the probability that an epidemic is observed.
Figure 3
Figure 3. Correlation of spreading power metrics to epidemic outcomes on real networks.
Point and error bar plots show the observed correlation and 95% confidence interval between each measure and spreading process outcome on the 24 real networks. The expected force and ExFM (orange shades) show strong performance, consistently outperforming the other metrics (k-shell, eigenvalue centrality, and accessibility when computed, blue-green shades). The epidemic outcome for SI processes is the time until half the network is infected. For SIS and SIR processes it is the probability that an epidemic is observed. The suffix “-D” indicates spreading processes simulated in discrete time. Individual panels are given as separate (larger) figures in Supplementary Figures 1-6.
Figure 4
Figure 4. Spreading power is a factor of a node's first and second order degree.
Plotting expected force (x-axis) versus node degree (orange), the sum of the degree of all neighbors (blue), and the sum of the degree of all neighbors at distance 2 (green) shows that for nodes with low ExF, the neighbor's degree has strong correlation to ExF, while for nodes with high ExF their own degree is more closely correlated. The result is accentuated in denser collaboration networks in comparison to more diffuse Pareto networks. Correlation between ExF and neighbor degree is 0.94 ± 0.01 in collaboration networks, and drops to 0.84 ± 0.02 in Pareto networks (mean taken over 50 networks; See Supplementary Table 3 for the correlations over all network structures).
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References

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