Correlation between centrality metrics and their application to the opinion model

• Degree mass is a new centrality measure that sums weighted degrees in a node's neighborhood no farther than m hops away.Source: §1 · p. 2
• The study compares centrality correlations in ER and scale-free network models and 34 real-world networks using Pearson coefficients and node-ranking similarity.Source: §1 · p. 2
• The study addresses whether understanding centrality correlations can support ranking nodes with a lower-complexity metric instead of a higher-complexity one.Source: §1 · p. 2
• In the ICO model, inflexible contrarians are selected by centrality, and strongly correlated metrics have approximately the same efficiency in reducing the largest opinion A cluster.Source: §1 · p. 2

• The networks are represented by symmetric adjacency matrices and are simple, unweighted, and without self-loops or multiple links.Source: §2 · p. 3
• The principal eigenvector x1 corresponds to the largest eigenvalue λ1, and (x1)j denotes its component at node j.Source: §2 · p. 3
• Betweenness counts shortest paths traversing a node, whereas closeness is the reciprocal of the total shortest-path hopcount to all other nodes.Source: §2 · p. 3
• K-shell decomposition identifies network core and periphery; its index predicts the infected population in SIR spreading better than degree.Source: §2 · p. 4
• Leverage compares a node's connectivity with its nearest neighbors, so a high-degree node may have low leverage when its neighbors are also highly connected.Source: §2 · p. 4
• Degree mass sums node degrees weighted by walks of length at most m, and approaches a quantity proportional to the principal eigenvector as m grows.Source: §2 · p. 4

3.1 Pearson correlation coefficients between centrality metrics

• The three strongest recurring Pearson pairings are x1 with D(2), closeness Cn with D(1), and betweenness Bn with D: correlations exceed 0.95 in both models and 0.6 in most real-world networks.Figure 1 reports the real-world comparisons across 34 networks.Source: §3.1 · p. 5, Figure 1 · p. 5

3.2 Centrality similarities M_A,B(Υ) between centrality metrics

• Centrality similarity M_A,B(Υ) measures the fraction of top-ΥN nodes shared by rankings from metrics A and B; larger values indicate greater overlap.The section evaluates this measure at Υ = 0.001, 0.01, and 0.1 in 103 realizations of each network model.Source: §3.2 · p. 6, §3.2.1 · p. 6

3.2.1 Centrality similarities in network models

• In ER and SF networks, Bn most closely matches D, Cn matches D(1) ahead of D(2) and D, and x1 has high similarity with both D(1) and D(2).In ER networks, x1’s similarity with D(2) exceeds its similarity with D(1), which in turn exceeds its similarity with D.Source: §3.2.1 · p. 6
• For small Υ, centrality similarity increases with Υ in ER networks but decreases with Υ in SF networks; near Υ = 1, it equals 1 in all networks.The reported trend is limited to small Υ.Source: §3.2.1 · p. 6

3.2.2 Centrality similarities in real-world networks

• Across 34 real-world networks, D, D(1), and D(2) are most similar to Bn, Cn, and x1, respectively, in most networks.The comparison uses Υ = 5%; the smallest network has 35 nodes, so Υ must exceed 3%.Source: §3.2.2 · p. 7, Figure 3 · p. 7

Theoretical analysis

• As m tends to infinity, the mth-order degree-mass vector is proportional to the principal eigenvector in networks with a sufficiently large spectral gap.The analysis describes the degree-mass vector as tending toward proportionality as m increases.Source: §4 · pp. 7–8
• In large sparse ER networks, the correlation of x1 with D(2) is at least that with D(1), which is at least that with D.The paper also reports using a generating-function method to analyze degree-mass expectations and variances.Source: §4 · pp. 7–8

5 Application to the inflexible contrarian opinion (ICO) model

• The application covers network models and three real-world social networks.Source: §5 · p. 8, Figure 6 · p. 10

5.1 The ICO model

• In the ICO model, a fraction po of A-opinion agents become fixed B-opinion contrarians who influence others, after which NCO update rules resume.The ICO phase threshold fc separates states with and without a giant A cluster; increasing po raises fc and reduces the largest A cluster until it disappears above p*.Source: §5.1 · pp. 8–9

5.2 Strategies of selecting inflexible contrarians using centrality metrics

• The study tests selecting contrarians by seven centrality metrics and randomly, aiming to reduce the largest steady-state A cluster and compare correlated metrics.The strategies are betweenness, degree, 1st- and 2nd-order degree mass, eigenvector component, k-shell index, leverage, and random selection.Source: §5.2 · p. 9

5.3 Comparison of inflexible contrarian selection strategies

• For leverage on ER networks, increasing po shifts fc upward and shrinks the largest A cluster; at po above p*, that cluster disappears.For the leverage strategy, p* lies between 0.3 and 0.4.Source: §5.3 · p. 9, Figure 4 · p. 9
• In ER networks, strategy efficiency ranks leverage, degree, betweenness, 1st-order degree mass, 2nd-order degree mass, k-shell index, principal eigenvector, then random selection.The same ordering is reported for ER and SF networks with N = 10^5.Source: §5.3 · pp. 9–10
• All strategies are more efficient in SF than same-sized ER networks, possibly because selected hubs influence many nearest neighbors.The authors also report that the relative change in fc is larger and p* is much smaller in SF networks.Source: §5.3 · p. 10
• In social networks, leverage, betweenness, and degree are the most efficient strategies for helping opinion B win the competition.Their similar behavior is supported by large Pearson correlation and centrality-similarity measures.Source: §5.3 · p. 10
• The principal eigenvector strategy behaves more like 2nd-order than 1st-order degree mass in the ICO model, consistent with their reported correlation ordering.The paper reports ρ(D(2), X1) ≥ ρ(D(1), X1).Source: §5.3 · p. 10

6 Conclusion

• The degree, 1st-order degree mass, and 2nd-order degree mass are strongly correlated with betweenness, closeness, and the principal eigenvector, respectively, across real-world networks and network models.The paper’s analytical result gives ρ(X1, D(2)) > ρ(X1, D(1)) > ρ(X1, D).Source: §6 · p. 11
• Degree mass D(m) is a new centrality metric whose 0th-order form is degree and whose high-order limit is proportional to the principal eigenvector x1.The conclusion also reports that network size has little influence on Pearson correlation coefficients.Source: §6 · p. 11
• Leverage is the most efficient strategy for selecting inflexible contrarians in the studied network models and real-world networks.The conclusion says strongly correlated metrics perform similarly in the ICO model.Source: §6 · p. 11
• The authors suggest low-complexity degree and leverage metrics as approximations to more complex metrics such as betweenness when locating important nodes.Examples include selecting inflexible contrarians in opinion-propagation networks and nodes for immunization in disease-transmission networks.Source: §6 · p. 11

A.1 Descriptions

• Table 1 presents descriptions of real-world networks.Source: Table 1 · p. 14

A.2 Properties of the real-world networks

• Definitions of the network properties are described in reference.The real-world network index is shown in Table 1.Source: §A.2 · p. 14, Table 2 · p. 15
• The listed properties include network size, link count, average shortest path, and clustering coefficient.N denotes the number of nodes, L the number of links, E[H] the average shortest path, and CG the clustering coefficient.Source: Table 2 · p. 15
• The table also reports degree correlation (assortativity), adjacency spectral radius, Laplacian eigenvalue measures, and effective graph resistance.The caption identifies ρD as assortativity, λ1 as the adjacency matrix's largest eigenvalue, and RG as effective graph resistance.Source: Table 2 · p. 15

B Pearson correlation coefficients between centrality metrics

• Figure 7 plots pairwise Pearson correlation coefficients against link density p in ER networks with N = 400.The figure annotations give the correlation index for each metric pair.Source: Figure 7 · p. 16
• Figure 8 plots pairwise Pearson correlation coefficients against network size N in scale-free networks with α = 2.5.The figure annotations give the correlation index for each metric pair.Source: Figure 8 · p. 16
• Table 3 reports Pearson correlation coefficients among centrality metrics in real-world networks.The correlation-index key enumerates 28 pairs, including Bn with Cn, D, x1, Ks, Ln, D(1), and D(2).Source: B Pearson correlation coefficients… · p. 15, Table 3 · p. 17

C Proof of Lemmas

• The first-order degree mass is the degree sum of a node and its neighbors, and its distribution is built from neighbor contributions.The proof uses the generating function's powers property to combine contributions from k independent neighbor realizations.Source: C Proof of Lemmas · pp. 18–19
• For ER networks, the node degree has a binomial generating function based on N and p.The proof also introduces the generating function for the degree of a node reached by following a randomly chosen link.Source: C Proof of Lemmas · pp. 18–19
• The proof invokes the generating function's moments property to derive moments of the first-order degree mass.The stated variance expression for the second-order degree mass is given in equation (4).Source: C Proof of Lemmas · pp. 18–19
• The second-order degree-mass distribution is generated by nesting the degree generating function through neighboring nodes.The proof gives its generating function and then obtains expressions for its average and variance.Source: C Proof of Lemmas · pp. 18–19

C.2 Proof of Lemma2

• The proof relates the principal eigenvector to degree-mass vectors and compares their Pearson correlations through a ratio.For large ER graphs, it uses asymptotic expressions for degree moments and variance.Source: §C.2 · p. 19
• In sparse random graphs with N →∞ and Np = ς, ρ(D(1), X1) ≥ ρ(D, X1).Under this condition, the spectral radius λ1 →ς.Source: §C.2 · p. 20
• For large sparse random networks, ρ(D(2), X1) ≥ ρ(D(1), X1).This establishes the second-order degree-mass comparison with the principal eigenvector.Source: §C.2 · p. 20