Leverage Centrality

• Leverage centrality is defined as a local measure that assesses a node’s importance by comparing its degree with those of its immediate neighbors.
• It computes an average of normalized degree differences, highlighting nodes with local dominance even when absolute degree is moderate.
• Empirical studies reveal that leverage centrality excels in identifying dispersed influential node sets, particularly under SIR dynamical models.

Searching arXiv for recent and foundational papers on leverage centrality and its comparative analysis. Leverage centrality is a vertex centrality measure that evaluates a node’s importance relative to the connectivity of its immediate neighborhood rather than by absolute degree alone. Introduced by Joyce, Laurienti, Burdette, and Hayasaka for studying functional brain networks and later analyzed both empirically and graph-theoretically, it compares the degree of a node to the degrees of its neighbors, thereby quantifying a local advantage or disadvantage in connectivity (Jr. et al., 2017). In recent comparative work on 80 real-world networks, leverage centrality has been identified as one of the most behaviorally distinctive centrality measures and as one of the strongest methods for identifying influential node sets under SIR dynamics, even though it is not among the best methods for identifying the most influential single node (Bi et al., 13 Aug 2025).

1. Formal definition and local interpretation

Leverage centrality is defined as a measure of the relationship between the degree of a node and the degree of each of its neighbors, averaged over all neighbors. In graph-theoretic notation, for a vertex $v$ with neighborhood $N(v)$, the definition is

$$\ell(v)=\frac{1}{\deg(v)}\sum_{u\in N(v)}\frac{\deg(v)-\deg(u)}{\deg(v)+\deg(u)}.$$

This formula appears in equivalent forms in both the graph-theoretic treatment and the comparative centrality literature (Jr. et al., 2017, Li et al., 2014).

The measure is inherently local. Computation depends only on the degree of the focal node and the degrees of its nearest neighbors. This distinguishes leverage centrality from global metrics such as betweenness, closeness, and eigenvector centrality, which require shortest-path or spectral information (Li et al., 2014). In the comparative study of 16 representative centrality measures, its computational complexity is listed as $O(M)$, where $M$ is the number of edges, placing it among the cheapest measures studied (Bi et al., 13 Aug 2025).

Its interpretation is relative rather than absolute. If a node has higher degree than most of its neighbors, then the normalized degree-difference terms are positive and leverage centrality is positive; if it has lower degree than its neighbors, the value is negative; if its degree is locally balanced against the neighborhood, the value is near zero (Bi et al., 13 Aug 2025, Jr. et al., 2017). This means that a node can have high degree but low leverage if its neighbors are also highly connected, and conversely a moderate-degree node can attain high leverage if it dominates a weaker local neighborhood (Li et al., 2014, Jr. et al., 2017).

The papers consistently characterize leverage centrality as a measure of local relative dominance. One formulation states that it “measures the extent of the connectivity of a node relative to the connectivity of its nearest neighbors,” while another states that it “measures the relative advantage of a node by comparing its degree to the degrees of its neighbors” (Li et al., 2014, Bi et al., 13 Aug 2025). This makes leverage centrality conceptually distinct from degree centrality, which depends only on $\deg(v)$, and from neighbor-prestige measures such as eigenvector or Katz centrality, which reward attachment to important neighbors rather than local degree advantage (Bi et al., 13 Aug 2025).

2. Mathematical properties and graph-theoretic structure

Several basic properties of leverage centrality have been established in the graph-theoretic literature. For a graph with $n$ vertices, the bound

$|l(v)|\le 1-\frac{2}{n}$

holds for every vertex, and these bounds are tight in the cases of stars and complete graphs (Jr. et al., 2017). A related formulation gives the range in a connected network as

$[-1+2/N,\;1-2/N],$

with equality occurring in star graphs and complete graphs $K_N$ (Li et al., 2014).

Regular graphs provide the canonical zero case. If all vertices have the same degree, then every summand in the leverage expression is zero, so

$N(v)$0

The graph-theoretic analysis states that $N(v)$1 for every vertex if and only if the graph is regular (Jr. et al., 2017). Cycles and complete graphs are immediate examples.

A notable global identity concerns the signed sum of leverage centralities. For any graph,

$N(v)$2

with equality for regular graphs and strict negativity for non-regular graphs (Jr. et al., 2017). The proof is obtained by regrouping contributions edge-by-edge: every edge joining vertices of unequal degree contributes negatively to the total. This establishes that leverage centrality does not, in general, average to zero over a graph.

The sign structure is constrained by extremal degree classes. A vertex of minimum degree cannot have positive leverage centrality, and a vertex of maximum degree cannot have negative leverage centrality (Jr. et al., 2017). Moreover, the maximum number of vertices with positive leverage centrality is $N(v)$3, while the star $N(v)$4 shows that almost all vertices can instead be negative (Jr. et al., 2017).

The graph-theoretic treatment also emphasizes that degree alone does not determine leverage centrality. It constructs an infinite family in which a vertex $N(v)$5 has much larger degree than a vertex $N(v)$6, yet $N(v)$7. In the stated example, $N(v)$8 and

$N(v)$9

while a degree-$$\ell(v)=\frac{1}{\deg(v)}\sum_{u\in N(v)}\frac{\deg(v)-\deg(u)}{\deg(v)+\deg(u)}.$$0 vertex satisfies

$$\ell(v)=\frac{1}{\deg(v)}\sum_{u\in N(v)}\frac{\deg(v)-\deg(u)}{\deg(v)+\deg(u)}.$$1

so for all $$\ell(v)=\frac{1}{\deg(v)}\sum_{u\in N(v)}\frac{\deg(v)-\deg(u)}{\deg(v)+\deg(u)}.$$2, $$\ell(v)=\frac{1}{\deg(v)}\sum_{u\in N(v)}\frac{\deg(v)-\deg(u)}{\deg(v)+\deg(u)}.$$3 but $$\ell(v)=\frac{1}{\deg(v)}\sum_{u\in N(v)}\frac{\deg(v)-\deg(u)}{\deg(v)+\deg(u)}.$$4 (Jr. et al., 2017). This is a precise demonstration that leverage centrality is not reducible to high degree.

3. Exact values on standard graph families

The behavior of leverage centrality on standard graph families is explicit and instructive. On the star $$\ell(v)=\frac{1}{\deg(v)}\sum_{u\in N(v)}\frac{\deg(v)-\deg(u)}{\deg(v)+\deg(u)}.$$5, the center attains

$$\ell(v)=\frac{1}{\deg(v)}\sum_{u\in N(v)}\frac{\deg(v)-\deg(u)}{\deg(v)+\deg(u)}.$$6

while every leaf attains

$$\ell(v)=\frac{1}{\deg(v)}\sum_{u\in N(v)}\frac{\deg(v)-\deg(u)}{\deg(v)+\deg(u)}.$$7

These realize the global upper and lower bounds (Jr. et al., 2017). In complete graphs $$\ell(v)=\frac{1}{\deg(v)}\sum_{u\in N(v)}\frac{\deg(v)-\deg(u)}{\deg(v)+\deg(u)}.$$8, every vertex has leverage centrality $$\ell(v)=\frac{1}{\deg(v)}\sum_{u\in N(v)}\frac{\deg(v)-\deg(u)}{\deg(v)+\deg(u)}.$$9, as in all regular graphs (Jr. et al., 2017).

Paths exhibit a three-level structure for $O(M)$0. The endpoints satisfy

$O(M)$1

the next-to-endpoints satisfy

$O(M)$2

and all deeper interior vertices satisfy

$O(M)$3

This shows that the path’s boundary vertices are locally subordinate, the near-boundary vertices are locally advantaged, and the deeper interior is locally balanced (Jr. et al., 2017).

Complete multipartite graphs admit a closed formula. For

$O(M)$4

a vertex $O(M)$5 in part $O(M)$6 has degree $O(M)$7, and its leverage centrality is

$O(M)$8

Here the sign of each contribution is governed by comparisons among part sizes, since the degree differences reduce to $O(M)$9 (Jr. et al., 2017).

A further product formula holds when one factor is regular. If $M$0 is an $M$1-regular graph and $M$2 denotes the Cartesian product, then for $M$3,

$M$4

where $M$5 and $M$6 are degrees in $M$7 (Jr. et al., 2017). This shows that adjoining a regular factor shifts the degrees while preserving the underlying degree-difference structure of leverage.

The longest graph-theoretic analysis concerns the lattice

$M$8

In this family, corner vertices minimize leverage and satisfy

$M$9

while inner corner vertices maximize leverage and satisfy

$\deg(v)$0

Accordingly, every vertex satisfies

$\deg(v)$1

and as $\deg(v)$2, the leverage centralities of all vertices converge to $\deg(v)$3 (Jr. et al., 2017). The paper also identifies a combinatorial pattern: for $\deg(v)$4, the number of distinct leverage centralities in $\deg(v)$5 is at most

$\deg(v)$6

with equality for $\deg(v)$7 and computational verification through $\deg(v)$8, yielding a connection to triangle numbers (Jr. et al., 2017).

4. Position among centrality measures and correlation structure

Leverage centrality occupies different positions in two distinct comparative frameworks. In the broad 2025 study of 16 representative centrality measures across 80 real-world networks, pairwise similarity was measured using average Kendall’s $\deg(v)$9, and the measures were partitioned into three groups using hierarchical clustering (Bi et al., 13 Aug 2025). Leverage centrality belongs to the third group, together with Collective Influence, Cycle Ratio, Betweenness, and Eccentricity. This group is described as the “most intriguing” because its pairwise correlations are relatively low and its average correlation strengths with other measures are significantly lower than the overall average (Bi et al., 13 Aug 2025).

This distinctiveness is numerically pronounced. Across all $n$0 distinct pairs of measures, the average correlation is about $n$1, whereas the average correlations of leverage centrality and eccentricity with other measures are both below $n$2 (Bi et al., 13 Aug 2025). The minimum pairwise correlation reported in the entire study is

$n$3

The study interprets this weak alignment as indicating higher information value, suggesting that these measures capture node properties not reflected by the more strongly correlated families (Bi et al., 13 Aug 2025).

The same paper identifies two relatively coherent communities from which leverage centrality is separated: one consisting of $n$4, and another consisting of $n$5 (Bi et al., 13 Aug 2025). In this framework, leverage centrality stands apart from both the degree/coreness-like family and the eigenvector/subgraph-like family.

An earlier correlation study presents a different picture because it focuses on Pearson correlation and ranking overlap among a smaller collection of centrality metrics, and because leverage centrality there is considered relative to degree, betweenness, closeness, eigenvector centrality, $n$6-shell index, and degree masses (Li et al., 2014). That paper does not identify leverage as a universal Pearson surrogate for any one global metric. Instead, it reports heterogeneous Pearson correlations across 34 real-world networks. For example, $n$7 is $n$8 in Karate and $n$9 in American football, but $|l(v)|\le 1-\frac{2}{n}$0 in WordAdjEnglish and $|l(v)|\le 1-\frac{2}{n}$1 in Electric s420 (Li et al., 2014). Correlations with betweenness, closeness, and eigenvector centrality are similarly mixed.

What that earlier study emphasizes is ranking similarity rather than uniform linear correlation. Its key claim is that, in most real-world networks, either $|l(v)|\le 1-\frac{2}{n}$2 or $|l(v)|\le 1-\frac{2}{n}$3 is the largest among the centrality similarities between leverage and all other metrics (Li et al., 2014). Thus, among top-ranked nodes, leverage centrality often behaves most similarly to degree or betweenness. This suggests that leverage can be degree-like in many networks, but with a local relative correction determined by neighbor degrees.

A plausible implication is that leverage centrality is structurally hybrid across comparative settings: it can overlap strongly with degree or betweenness at the top of the ranking, yet still remain behaviorally distinctive at the full-ranking level when evaluated across a broader and more heterogeneous set of centralities and networks.

5. Dynamical validation: single influential nodes and influential node sets

The most detailed dynamical evaluation of leverage centrality in the provided literature uses the Susceptible-Infected-Recovered (SIR) spreading model on 80 real-world networks (Bi et al., 13 Aug 2025). Two tasks are distinguished: identifying the most influential single node, and identifying the most influential node set. The paper stresses that these seemingly similar tasks produce very different performance rankings for centrality measures.

For a single seed node $|l(v)|\le 1-\frac{2}{n}$4, spreading influence $|l(v)|\le 1-\frac{2}{n}$5 is defined as the average number of ultimately recovered nodes over repeated SIR simulations starting from node $|l(v)|\le 1-\frac{2}{n}$6 alone (Bi et al., 13 Aug 2025). Performance is assessed using precision of overlap between top-$|l(v)|\le 1-\frac{2}{n}$7 nodes by centrality and top-$|l(v)|\le 1-\frac{2}{n}$8 nodes by empirical influence, and also by Kendall’s $|l(v)|\le 1-\frac{2}{n}$9 between centrality scores and empirical influences (Bi et al., 13 Aug 2025). In this task, the best methods are reported to be LocalRank, Subgraph Centrality, and Katz Centrality (Bi et al., 13 Aug 2025). Leverage centrality is not among the top methods for identifying the most influential single spreader.

For node sets, the performance criterion changes. If $[-1+2/N,\;1-2/N],$0 nodes are selected as seeds, the final infection rate is

$[-1+2/N,\;1-2/N],$1

where $[-1+2/N,\;1-2/N],$2 is the number of recovered nodes at the end of the SIR process and $[-1+2/N,\;1-2/N],$3 is the total number of nodes (Bi et al., 13 Aug 2025). The operational choice is the top-$[-1+2/N,\;1-2/N],$4 nodes under a centrality ranking, with iterative recalculation and removal used for Collective Influence (Bi et al., 13 Aug 2025). In this task, leverage centrality performs extremely well: the abstract states that “Leverage Centrality, Collective Influence, and Cycle Ratio excel in identifying the most influential node sets,” and the detailed results specify that “the two best-performing measures are LC and CI” (Bi et al., 13 Aug 2025).

The evaluation protocol is fixed across the 80 networks. The infection probability is set to the epidemic threshold

$[-1+2/N,\;1-2/N],$5

the recovery probability is

$[-1+2/N,\;1-2/N],$6

and averages are taken over 10 independent runs per network (Bi et al., 13 Aug 2025). Because the conclusion is aggregated over a heterogeneous real-world corpus, the reported strength of leverage centrality for node-set identification is not tied to a single network class.

One of the study’s most striking findings is that the rankings of the 16 centrality measures for the two SIR tasks are negatively correlated: $[-1+2/N,\;1-2/N],$7 Leverage centrality is one of the clearest examples of this cross-task inversion: it is not top-tier for identifying the most influential single node, but it is top-tier for identifying influential node sets (Bi et al., 13 Aug 2025).

6. Dispersion, selection behavior, and applications to opinion competition

The 2025 comparative study attributes the node-set strength of leverage centrality in large part to the spatial distribution of the nodes it selects (Bi et al., 13 Aug 2025). Different centrality measures produce top-ranked nodes that are either clustered together or topologically dispersed. To quantify this effect, the paper defines the average pairwise shortest-path distance among the top-$[-1+2/N,\;1-2/N],$8 selected nodes: $[-1+2/N,\;1-2/N],$9 Averaging this quantity over the 80 networks and 10 runs yields $K_N$0 (Bi et al., 13 Aug 2025).

In this analysis, leverage centrality is the most dispersed of all 16 measures. The paper states that the five measures with the most dispersed top-ranked node distributions are LC, CR, Betweenness, Eccentricity, and CI, “with LC showing a significantly higher $K_N$1 than the other four” (Bi et al., 13 Aug 2025). This is directly connected to the observed node-set performance: dispersed seeds reduce overlap among their influential regions, so their collective coverage under SIR can be larger. The paper therefore concludes that measures that identify influential nodes with larger topological distances between them tend to perform better in detecting influential node sets (Bi et al., 13 Aug 2025).

At the same time, the paper explicitly cautions that high $K_N$2 is not sufficient by itself. Eccentricity also yields highly dispersed selections but poor node-set performance, so dispersion is only an approximate explanatory factor rather than a complete account (Bi et al., 13 Aug 2025). A plausible implication is that leverage centrality benefits from combining dispersion with a meaningful local notion of dominance, not from dispersion alone.

A second application area appears in the inflexible contrarian opinion model, a two-opinion competition process studied in a separate paper (Li et al., 2014). In that model, leverage centrality is used to choose which opinion-$K_N$3 nodes are converted into permanent opinion-$K_N$4 contrarians. The outcomes tracked are the sizes $K_N$5 and $K_N$6 of the largest and second-largest opinion-$K_N$7 clusters, the critical threshold $K_N$8, and the destruction threshold $K_N$9 (Li et al., 2014).

Across ER networks, SF networks, and three real-world social networks—CondMat 95–99, CondMat 95–03, and Astro_Ph—the leverage-based strategy is reported to be among the most effective intervention rules (Li et al., 2014). In the model comparisons for ER and SF networks, the paper states that the efficiency ranking is, in decreasing order, Leverage, Degree, Betweenness, $N(v)$00st-order Degree mass, $N(v)$01nd-order Degree mass, $N(v)$02-shell index, Principal Eigenvector, and Random (Li et al., 2014). In the conclusion, it states that “The leverage $N(v)$03 turns out to be the most efficient strategy in both network models and real-world networks” (Li et al., 2014).

This applied result is interpreted through the correlation structure reported in the same paper. Because leverage has high centrality similarity with degree and often betweenness, the three strategies tend to select many of the same critical nodes, and strongly correlated centrality metrics tend to perform similarly in the opinion-control task (Li et al., 2014). In this setting, leverage centrality functions as a low-complexity intervention rule with performance comparable to or better than more computationally demanding alternatives.

7. Scope, limitations, and technical significance

The collected evidence presents leverage centrality as a local relative-degree centrality with a technically unusual profile. It is simple to compute, but it is not merely a surrogate for degree. The graph-theoretic analysis proves that high degree does not imply high leverage, that leverage values need not sum to zero, and that exact formulas and extremal structures can be obtained for broad graph families including stars, complete multipartite graphs, and Cartesian products of paths (Jr. et al., 2017). These results make leverage centrality mathematically tractable while preserving nontrivial dependence on local degree heterogeneity.

In comparative empirical analysis, leverage centrality is distinctive rather than redundant. On the 80-network benchmark of 16 representative centrality measures, its average correlation with other measures is below $N(v)$04, far below the overall pairwise average of about $N(v)$05, and it belongs to the weak-correlation group alongside Collective Influence, Cycle Ratio, Betweenness, and Eccentricity (Bi et al., 13 Aug 2025). This suggests that leverage centrality captures node properties not represented by more tightly correlated degree-like or eigenvector-like families.

Its practical strengths are task-dependent. For identifying the most influential single spreader under SIR, the evidence favors LocalRank, Subgraph Centrality, and Katz Centrality rather than leverage centrality (Bi et al., 13 Aug 2025). For identifying influential node sets, leverage centrality is one of the strongest methods studied and is closely associated with unusually large topological dispersion among selected seeds (Bi et al., 13 Aug 2025). In opinion competition, leverage-based selection of inflexible contrarians is reported as the most efficient strategy among the tested centralities (Li et al., 2014).

Several limitations are explicit in the literature. There is no universally best centrality measure across tasks, and even within one dynamical process, the rankings of methods can reverse when the objective changes from single-node to node-set influence (Bi et al., 13 Aug 2025). High dispersion alone does not guarantee good performance for node sets (Bi et al., 13 Aug 2025). In the graph-theoretic direction, necessary and sufficient conditions for $N(v)$06, especially when neighbors have distinct degrees, remain an open problem (Jr. et al., 2017). The comparative SIR study also suggests that future methods should explicitly penalize redundancy among selected nodes, implying that leverage centrality succeeds partly because it incidentally yields dispersed top-$N(v)$07 selections rather than because it was designed for explicit redundancy control (Bi et al., 13 Aug 2025).

Taken together, these results place leverage centrality at the intersection of local structural analysis, comparative centrality theory, and dynamical intervention. It is best understood not as a generic replacement for degree or betweenness, but as a measure of local dominance whose empirical value depends strongly on the operational definition of “influence” and on whether the target is a single spreader, a seed set, or a strategic intervention set (Bi et al., 13 Aug 2025, Li et al., 2014, Jr. et al., 2017).