Harmonic Centrality in Networks

• Harmonic centrality for directed graph G is a measure that assigns vertices a importance measure equal to the reciprocal of their shortest-path distance from other vertices, rewarding reachability and short paths.
• Harmonic centrality is closely related to closeness centrality but differs by treating unreachable vertices and in the sum of distances taken.
• This measure is significant in network-optimization problems, provides applications in social network analysis such as collaboration networks, and can assess the robustness and efficiency of directed and weighted networks.

Harmonic centrality is a distance-based network centrality that assigns each vertex a contribution equal to the reciprocal of its shortest-path distance from another vertex, with unreachable vertices contributing zero. For a directed graph $G=(N,A)$, the incoming form is

$$C_{\mathrm{harm}}(x)=\sum_{y\neq x}\frac{1}{d(y,x)},\qquad \frac{1}{\infty}=0,$$

where $d(y,x)$ is the length of a shortest directed path from $y$ to $x$. An outgoing form instead uses $d(x,y)$. The measure rewards both broad reachability and short paths: distance $1$ contributes $1$, distance $2$ contributes $1/2$, distance $$C_{\mathrm{harm}}(x)=\sum_{y\neq x}\frac{1}{d(y,x)},\qquad \frac{1}{\infty}=0,$$0 contributes $$C_{\mathrm{harm}}(x)=\sum_{y\neq x}\frac{1}{d(y,x)},\qquad \frac{1}{\infty}=0,$$1, and an unreachable vertex contributes $$C_{\mathrm{harm}}(x)=\sum_{y\neq x}\frac{1}{d(y,x)},\qquad \frac{1}{\infty}=0,$$2. Some literature uses the unnormalized sum, while other work defines normalized harmonic centrality by dividing by $$C_{\mathrm{harm}}(x)=\sum_{y\neq x}\frac{1}{d(y,x)},\qquad \frac{1}{\infty}=0,$$3, where $$C_{\mathrm{harm}}(x)=\sum_{y\neq x}\frac{1}{d(y,x)},\qquad \frac{1}{\infty}=0,$$4 is the number of vertices. Harmonic centrality is closely related to closeness centrality but remains meaningful on disconnected and directed graphs.

1. Definition, orientation, and relation to closeness

Let $$C_{\mathrm{harm}}(x)=\sum_{y\neq x}\frac{1}{d(y,x)},\qquad \frac{1}{\infty}=0,$$5 be a directed graph with $$C_{\mathrm{harm}}(x)=\sum_{y\neq x}\frac{1}{d(y,x)},\qquad \frac{1}{\infty}=0,$$6. The directed distance $$C_{\mathrm{harm}}(x)=\sum_{y\neq x}\frac{1}{d(y,x)},\qquad \frac{1}{\infty}=0,$$7 is the length of a shortest directed path from $$C_{\mathrm{harm}}(x)=\sum_{y\neq x}\frac{1}{d(y,x)},\qquad \frac{1}{\infty}=0,$$8 to $$C_{\mathrm{harm}}(x)=\sum_{y\neq x}\frac{1}{d(y,x)},\qquad \frac{1}{\infty}=0,$$9, and $d(y,x)$0 when no such path exists. A vertex $d(y,x)$1 is reachable from $d(y,x)$2 if $d(y,x)$3.

The original axiomatic formulation uses the incoming, or “negative,” version

$d(y,x)$4

It measures how efficiently other vertices can reach $d(y,x)$5. The outgoing version is

$d(y,x)$6

and is equivalent to applying incoming harmonic centrality to the transposed graph. In undirected graphs, the two orientations coincide.

The unnormalized definition contains no division by the number of vertices, the number of reachable vertices, or a harmonic-mean denominator. A normalized version frequently used for finite undirected graphs is

$d(y,x)$7

It satisfies $d(y,x)$8, with $d(y,x)$9 exactly when $y$0 is adjacent to every other vertex (Ortega et al., 2021, Ortega et al., 2022).

Bavelas’s closeness centrality, in the incoming orientation, is

$y$1

On a strongly connected graph this is well-defined. If unreachable vertices are assigned infinite distance, the denominator becomes infinite and the score degenerates to zero for every vertex that is not reachable from all other vertices. A common repair omits infinite distances: $y$2 This patched form can favor vertices in small components because it omits all vertices outside the reachable set. Harmonic centrality instead retains every vertex conceptually and assigns zero only to unreachable pairs. This prevents a vertex from receiving an artificially high score merely because it belongs to a small component (Boldi et al., 2013).

The two measures also aggregate distances differently. Closeness takes the reciprocal of an aggregate distance, whereas harmonic centrality aggregates reciprocal distances: $y$3 Consequently, harmonic centrality is more sensitive to local density and to the number of reachable vertices, while avoiding the domination of the score by many distant vertices.

2. Combinatorial interpretation and axiomatic properties

A node’s harmonic centrality is the sum of contributions from distance layers. A neighbor contributes $y$4, a vertex at distance $y$5 contributes $y$6, and so forth. The measure therefore combines reachability and proximity while ensuring that distant vertices have diminishing influence.

Its relationship with harmonic means follows from considering all ordered pairs of distinct vertices: $y$7 The total harmonic centrality is the reciprocal of a denormalized harmonic mean of distances. Infinite distances contribute zero reciprocal distance rather than making the aggregate infinite.

This distinction is evident in a perfect matching consisting of $y$8 disconnected pairs. Only $y$9 ordered pairs have finite distance, each equal to $x$0. Averaging only finite distances yields $x$1, although almost all ordered pairs are unreachable. The harmonic mean over all ordered pairs is

$x$2

reflecting the disconnected structure.

For a node in a $x$3-clique,

$x$4

For a node in a directed $x$5-cycle, the incoming distances are $x$6, giving

$x$7

Since $x$8, a cycle must be exponentially larger than a clique to overcome the clique’s linear score.

The axiomatic framework of “Axioms for Centrality” assumes invariance under graph isomorphism and proposes three requirements (Boldi et al., 2013):

1. Size: a sufficiently large community must eventually outrank a fixed community, while a sufficiently large dense community must eventually outrank a fixed sparse one.
2. Density: increasing local density while holding size and connectivity structure fixed must increase the score of the affected vertex.
3. Score monotonicity: if $x$9, adding $d(x,y)$0 must strictly increase the centrality of $d(x,y)$1.

Harmonic centrality is the only one of the eleven examined measures satisfying all three axioms. For the size construction consisting of a $d(x,y)$2-clique and a directed $d(x,y)$3-cycle, its scores are $d(x,y)$4 and $d(x,y)$5, respectively. Thus the cycle eventually overtakes any fixed clique, while a sufficiently large clique eventually overtakes any fixed cycle.

For score monotonicity, adding $d(x,y)$6 cannot increase any directed distance into $d(x,y)$7. It may shorten existing paths and may make previously unreachable vertices reachable. Every reciprocal-distance contribution therefore weakly increases, and the contribution from $d(x,y)$8 changes from zero or a smaller value to $d(x,y)$9.

The result contrasts with several alternatives. Degree centrality sees only distance $1$0 and fails the size axiom. Closeness, Lin’s index, and betweenness fail one or more axioms. Spectral methods may concentrate mass in one component or fail to increase the score of a newly connected vertex. Katz satisfies score monotonicity within its convergence range but fails the size axiom. PageRank satisfies score monotonicity under stated conditions but fails the size axiom because of normalization and component effects.

3. Graph families, centers, and centralization

For finite simple undirected graphs, normalized harmonic centrality is

$1$1

The distance distribution directly determines the value. Paths, cycles, fans, wheels, complete bipartite graphs, ladders, crowns, prisms, stars, books, and helms admit explicit formulas (Ortega et al., 2021).

For a path $1$2,

$1$3

The maximum occurs at the central vertex or vertices. For a cycle, all vertices have equal harmonic centrality because of vertex transitivity. For a star $1$4 with $1$5 leaves, the center has value $1$6, while each leaf has

$1$7

For a wheel $1$8, the universal hub has value $1$9, and each rim vertex has

$1$0

In a graph of diameter $1$1,

$1$2

Thus harmonic centrality is monotone in degree in this class. More generally, the harmonic center is the induced subgraph on vertices attaining maximum harmonic centrality, and the harmonic periphery is the induced subgraph on vertices attaining minimum harmonic centrality (Madaras et al., 25 Sep 2026).

The center can behave differently from the ordinary closeness center because reciprocal distances disproportionately weight nearby vertices. In trees, every pendant vertex has lower harmonic centrality than its neighbor, so the harmonic center lies in the leaf-pruned trunk. However, pendant vertices need not form the entire harmonic periphery: in a graph consisting of a nine-vertex path with many pendant vertices attached at both ends, a middle path vertex can have lower harmonic centrality than the pendant vertices.

Harmonic centralization is a graph-level measure of the inequality of vertex centralities. Using Freeman’s normalization,

$1$3

where $1$4. The denominator is realized by the star $1$5, so

$1$6

Complete graphs, cycles, crowns, and prisms have $1$7 because all vertices have equal harmonic centrality. A star has $1$8. Thus large vertex-level scores do not imply high centralization: centralization measures concentration or inequality, not absolute importance (Ortega et al., 2022).

Harmonic centers can have complex block structure. For graphs of diameter at most $1$9, the harmonic center is contained in a single block. Sufficiently dense graphs are therefore also subject to block localization. Conversely, graphs with large diameter can have harmonic centers arbitrarily far apart, and graphs with approximately $2$0 edges can still have center vertices in different blocks (Madaras et al., 25 Sep 2026).

The same work establishes realizability results: every connected graph can occur as the harmonic center of a suitable larger connected graph, with a construction of order at most $2$1; every connected graph can likewise occur as a harmonic periphery in a graph of order at most $2$2. Harmonic-uniform graphs can also have cut vertices, although small-diameter harmonic-uniform graphs are strongly constrained.

4. Monotonicity under network modification

Adding an edge has especially transparent effects on harmonic centrality because shortest-path distances cannot increase. If $2$3 is obtained from $2$4 by adding an edge $2$5, then

$2$6

for all vertices, and therefore

$2$7

The endpoints strictly gain in the connected undirected setting because their mutual distance changes from $2$8 to $2$9.

Score monotonicity does not automatically imply rank monotonicity: another vertex may gain more and overtake the endpoint. A weaker property, semi-monotonicity, requires that at least one endpoint of a newly added edge improve its score or preserve its rank. Harmonic centrality satisfies score semi-monotonicity and, more strongly, strict rank semi-monotonicity on connected undirected graphs (Boldi et al., 2023, Boldi et al., 21 Sep 2026).

The key proof mechanism is basin dominance. For a newly added edge $1/2$0, vertices are assigned to basins according to whether they are at least as close to $1/2$1 as to $1/2$2, or conversely. The endpoint associated with a basin receives a score increase at least as large as that of every other vertex in the basin. For harmonic centrality, the reciprocal-distance improvement is strict for at least one relevant destination, yielding strict rank semi-monotonicity.

At least one of $1/2$3 or $1/2$4 therefore satisfies

$1/2$5

for every $1/2$6, or the analogous relation holds for $1/2$7. This does not assert that both endpoints become strictly superior to every other vertex; it asserts that one endpoint’s former ties and lower-ranked competitors become strictly lower after the edge is added.

The distinction from other centralities is substantive. Closeness is rank semi-monotone but not strictly rank semi-monotone. Betweenness is rank semi-monotone but not score semi-monotone. Spectral measures have different behavior because normalization and global mass redistribution can cause an endpoint’s score to decrease even when a new edge is added. Harmonic centrality avoids this numerical-score problem because it is an unnormalized sum of nondecreasing distance contributions.

5. Influence-based and dynamic generalizations

Harmonic centrality has several extensions that retain its harmonic interpretation while changing the underlying notion of distance.

Harmonic influence centrality

Harmonic Influence Centrality (HIC) is not the ordinary reciprocal-distance sum. It is defined in a social-influence model with stubborn agents. A fixed set $1/2$8 holds opinion $1/2$9, and a candidate node $$C_{\mathrm{harm}}(x)=\sum_{y\neq x}\frac{1}{d(y,x)},\qquad \frac{1}{\infty}=0,$$00 is converted into a stubborn agent holding opinion $$C_{\mathrm{harm}}(x)=\sum_{y\neq x}\frac{1}{d(y,x)},\qquad \frac{1}{\infty}=0,$$01. Regular agents update according to a stochastic consensus matrix $$C_{\mathrm{harm}}(x)=\sum_{y\neq x}\frac{1}{d(y,x)},\qquad \frac{1}{\infty}=0,$$02: $$C_{\mathrm{harm}}(x)=\sum_{y\neq x}\frac{1}{d(y,x)},\qquad \frac{1}{\infty}=0,$$03 Under the stated stability conditions,

$$C_{\mathrm{harm}}(x)=\sum_{y\neq x}\frac{1}{d(y,x)},\qquad \frac{1}{\infty}=0,$$04

The HIC of $$C_{\mathrm{harm}}(x)=\sum_{y\neq x}\frac{1}{d(y,x)},\qquad \frac{1}{\infty}=0,$$05 is the total limiting opinion

$$C_{\mathrm{harm}}(x)=\sum_{y\neq x}\frac{1}{d(y,x)},\qquad \frac{1}{\infty}=0,$$06

Under reciprocity, $$C_{\mathrm{harm}}(x)=\sum_{y\neq x}\frac{1}{d(y,x)},\qquad \frac{1}{\infty}=0,$$07 can be represented using symmetric conductances $$C_{\mathrm{harm}}(x)=\sum_{y\neq x}\frac{1}{d(y,x)},\qquad \frac{1}{\infty}=0,$$08. The limiting opinion becomes an electrical voltage satisfying a discrete Dirichlet problem: $$C_{\mathrm{harm}}(x)=\sum_{y\neq x}\frac{1}{d(y,x)},\qquad \frac{1}{\infty}=0,$$09 at regular nodes. Then

$$C_{\mathrm{harm}}(x)=\sum_{y\neq x}\frac{1}{d(y,x)},\qquad \frac{1}{\infty}=0,$$10

HIC is therefore a boundary-conditioned harmonic potential, not a reciprocal-distance centrality. It depends on stubborn-agent placement, edge weights, random-walk dynamics, and effective resistance (Vassio et al., 2013).

On trees, exact series and parallel resistance reductions yield a decentralized message-passing algorithm. The algorithm uses transmission messages and accumulated influence messages, converges in at most the tree diameter, and has total message count $$C_{\mathrm{harm}}(x)=\sum_{y\neq x}\frac{1}{d(y,x)},\qquad \frac{1}{\infty}=0,$$11. For bounded-degree trees, its total computation is essentially linear in the number of nodes. On cyclic graphs, the same recursions operate on computation trees and may overcount repeated paths. Convergence is proved for regular graphs with unit resistances, but correctness with respect to the original cyclic graph is not guaranteed.

Community structure can produce upward bias and overestimate local leaders. In a Facebook ego-network experiment, the approximate values exceeded exact Dirichlet values, while rankings remained broadly useful. The algorithm also adapted smoothly after topology changes when messages on retained edges were preserved and messages on new edges were initialized locally (Rossi et al., 2018).

Influence-based harmonic centrality under cascades

A different generalization replaces deterministic shortest-path distance with random activation time. An influence instance specifies a probability distribution over cascading sequences

$$C_{\mathrm{harm}}(x)=\sum_{y\neq x}\frac{1}{d(y,x)},\qquad \frac{1}{\infty}=0,$$12

where $$C_{\mathrm{harm}}(x)=\sum_{y\neq x}\frac{1}{d(y,x)},\qquad \frac{1}{\infty}=0,$$13 is the set active by time $$C_{\mathrm{harm}}(x)=\sum_{y\neq x}\frac{1}{d(y,x)},\qquad \frac{1}{\infty}=0,$$14. The cascading distance $$C_{\mathrm{harm}}(x)=\sum_{y\neq x}\frac{1}{d(y,x)},\qquad \frac{1}{\infty}=0,$$15 is the random time at which $$C_{\mathrm{harm}}(x)=\sum_{y\neq x}\frac{1}{d(y,x)},\qquad \frac{1}{\infty}=0,$$16 becomes active from singleton seed $$C_{\mathrm{harm}}(x)=\sum_{y\neq x}\frac{1}{d(y,x)},\qquad \frac{1}{\infty}=0,$$17, with $$C_{\mathrm{harm}}(x)=\sum_{y\neq x}\frac{1}{d(y,x)},\qquad \frac{1}{\infty}=0,$$18 if activation never occurs.

Influence-based harmonic centrality is

$$C_{\mathrm{harm}}(x)=\sum_{y\neq x}\frac{1}{d(y,x)},\qquad \frac{1}{\infty}=0,$$19

A node activated at time $$C_{\mathrm{harm}}(x)=\sum_{y\neq x}\frac{1}{d(y,x)},\qquad \frac{1}{\infty}=0,$$20 contributes $$C_{\mathrm{harm}}(x)=\sum_{y\neq x}\frac{1}{d(y,x)},\qquad \frac{1}{\infty}=0,$$21, unlike influence spread, which gives every activated node weight $$C_{\mathrm{harm}}(x)=\sum_{y\neq x}\frac{1}{d(y,x)},\qquad \frac{1}{\infty}=0,$$22. Deterministic breadth-first propagation reduces exactly to ordinary harmonic centrality (Chen et al., 2018).

The measure is characterized as the unique anonymous and Bayesian influence centrality that conforms to ordinary harmonic centrality on deterministic BFS instances. The same construction extends to groups of seed nodes. Group harmonic centrality measures the reciprocal activation times from a coalition, and Shapley harmonic centrality assigns each node its expected marginal contribution to group harmonic influence.

For triggering models, reverse-reachable sets yield an unbiased estimator: $$C_{\mathrm{harm}}(x)=\sum_{y\neq x}\frac{1}{d(y,x)},\qquad \frac{1}{\infty}=0,$$23 The framework provides relative-error guarantees for sufficiently high-centrality nodes and supports individual, group, and Shapley variants.

6. Computation and approximation

Exact harmonic centrality for every vertex requires extensive shortest-path computation. In an unweighted sparse graph, breadth-first search from every source has typical cost

$$C_{\mathrm{harm}}(x)=\sum_{y\neq x}\frac{1}{d(y,x)},\qquad \frac{1}{\infty}=0,$$24

and dense-matrix methods such as Floyd–Warshall require $$C_{\mathrm{harm}}(x)=\sum_{y\neq x}\frac{1}{d(y,x)},\qquad \frac{1}{\infty}=0,$$25. The all-pairs requirement is especially burdensome in large directed networks.

Several approximation strategies have been developed.

Heuristic estimation from local degree

QuickCent estimates harmonic centrality from in-degree using nested binary clues. For thresholds $$C_{\mathrm{harm}}(x)=\sum_{y\neq x}\frac{1}{d(y,x)},\qquad \frac{1}{\infty}=0,$$26,

$$C_{\mathrm{harm}}(x)=\sum_{y\neq x}\frac{1}{d(y,x)},\qquad \frac{1}{\infty}=0,$$27

Nodes are assigned to degree intervals and receive representative harmonic-centrality values, often estimated from a fitted power-law distribution. The method stops at the first absent clue and returns the representative value for the corresponding interval (Plana et al., 2023).

QuickCent is designed for preferential-attachment-like networks in which in-degree correlates with both local density and the accessible size of the surrounding network. In synthetic preferential-attachment graphs with $$C_{\mathrm{harm}}(x)=\sum_{y\neq x}\frac{1}{d(y,x)},\qquad \frac{1}{\infty}=0,$$28 nodes, it was generally competitive with linear regression, regression trees, and neural networks for attachment exponents $$C_{\mathrm{harm}}(x)=\sum_{y\neq x}\frac{1}{d(y,x)},\qquad \frac{1}{\infty}=0,$$29 and $$C_{\mathrm{harm}}(x)=\sum_{y\neq x}\frac{1}{d(y,x)},\qquad \frac{1}{\infty}=0,$$30. It often had lower dispersion and remained relatively stable when only $$C_{\mathrm{harm}}(x)=\sum_{y\neq x}\frac{1}{d(y,x)},\qquad \frac{1}{\infty}=0,$$31 of nodes were used for training.

The method is not universal. Its assumptions can fail under community bottlenecks, degree-preserving randomization, near-percolation Erdős–Rényi structure, superlinear attachment, non-heavy-tailed centrality distributions, or weak empirical degree–centrality correlations. It is a low-cost structural proxy rather than a shortest-path approximation with formal error guarantees.

Uncertain graphs

In an uncertain graph, every candidate edge exists independently with a specified probability. A possible world is one deterministic graph sampled from the uncertain graph. Exact expected harmonic closeness requires summing over exponentially many possible worlds.

A possible-shortest-path heuristic explores shortest paths that occur in at least one possible world. It estimates distance distributions using path existence probabilities and then forms an expected-reliable distance

$$C_{\mathrm{harm}}(x)=\sum_{y\neq x}\frac{1}{d(y,x)},\qquad \frac{1}{\infty}=0,$$32

The resulting PSP-harmonic score is

$$C_{\mathrm{harm}}(x)=\sum_{y\neq x}\frac{1}{d(y,x)},\qquad \frac{1}{\infty}=0,$$33

This differs from the expected harmonic centrality

$$C_{\mathrm{harm}}(x)=\sum_{y\neq x}\frac{1}{d(y,x)},\qquad \frac{1}{\infty}=0,$$34

because in general $$C_{\mathrm{harm}}(x)=\sum_{y\neq x}\frac{1}{d(y,x)},\qquad \frac{1}{\infty}=0,$$35 (Ketels, 2024).

The approach can outperform large Monte Carlo runs on some synthetic and real-world graphs, with strong rank correlations on Barabási–Albert and Erdős–Rényi instances. However, shared edges make path events dependent, approximate probability masses can exceed one, and the number of shortest paths can be exponential. Large-scale tests exhausted approximately $$C_{\mathrm{harm}}(x)=\sum_{y\neq x}\frac{1}{d(y,x)},\qquad \frac{1}{\infty}=0,$$36 GB of memory, so the method is a heuristic whose applicability depends strongly on graph structure and the exploration threshold.

Harmonic centrality in applications

Harmonic centrality has been used to characterize directed weighted collaboration networks. In student homework-assistance networks, edge weights count the number of assignments on which one student helped another, and distance is the inverse edge weight. Harmonic centrality was one of four measures—along with in-strength, out-strength, and closeness—that correlated significantly with homework scores across three upper-division physics courses. Its interpretation was that students with broad, short-path access to collaborators tended to achieve higher homework scores. The study was correlational, and exam-score relationships were weaker and more course-dependent (Vargas et al., 2018).

In Web-domain networks, harmonic centrality has been applied to directed hyperlink graphs containing hundreds of thousands of Australian domains. Domain-level score distributions were multimodal and were partitioned into six ordered clusters. Higher-centrality clusters were associated descriptively with larger organizations, greater revenue, technology spending, employee counts, social links, domain age, and organizational prominence. Across OECD national domain spaces, median harmonic centrality showed a reported correlation of approximately $$C_{\mathrm{harm}}(x)=\sum_{y\neq x}\frac{1}{d(y,x)},\qquad \frac{1}{\infty}=0,$$37 with the World Justice Project Rule of Law Index. These results are exploratory and do not establish causal relationships; the analysis is affected by domain-registration changes, missing values, temporal mismatch between Web snapshots and organizational data, and the absence of multivariate controls (Gong et al., 2024).

In scientific collaboration networks, a proposed variant called Harmonic Closeness with Temporal and Collaboration Count Decay (HCTCD) modifies reciprocal-distance contributions using collaboration history and temporal terms. In a study of $$C_{\mathrm{harm}}(x)=\sum_{y\neq x}\frac{1}{d(y,x)},\qquad \frac{1}{\infty}=0,$$38 papers from NeurIPS, ICML, and ICLR, long-term closeness-like measures and HCTCD were positively associated with citation percentiles after content and metadata controls. HCTCD and closeness were highly correlated, and the exact HCTCD formula contains unresolved sign and normalization ambiguities in the supplied manuscript. The results therefore support an association between scholarly network embeddedness and citation performance, not a causal effect (Jie et al., 26 Dec 2025).

7. Extensions, variants, and interpretation

“Harmonic centrality” denotes several related but nonidentical constructs. The standard graph-theoretic measure is the reciprocal-shortest-path sum. Its key properties are:

• unreachable vertices contribute zero;
• nearby vertices receive disproportionately large contributions;
• the measure is naturally compatible with directed and disconnected graphs;
• incoming and outgoing versions differ in directed networks;
• normalization by $$C_{\mathrm{harm}}(x)=\sum_{y\neq x}\frac{1}{d(y,x)},\qquad \frac{1}{\infty}=0,$$39 is optional and changes scale but not rankings.

Influence-based harmonic centrality replaces shortest-path distance with random activation time. Harmonic Influence Centrality replaces distance with a Dirichlet voltage generated by a stubborn-agent boundary condition. Harmonic centralization measures inequality among vertex scores. HCTCD adds temporal and collaboration-count terms. PSP-harmonic closeness replaces deterministic distances with heuristic expected-reliable distances in uncertain graphs.

These variants should not be conflated. Standard harmonic centrality is purely topological and depends only on shortest distances. HIC depends on boundary conditions and electrical or consensus dynamics. Influence-based harmonic centrality depends on a stochastic cascade model. Harmonic centralization is graph-level rather than vertex-level. Domain- and author-network applications often use additional normalization, weighting, or temporal conventions.

The principal conceptual contribution of harmonic centrality is the treatment of unreachable vertices. By replacing the reciprocal of an aggregate distance with an aggregate of reciprocal distances,

$$C_{\mathrm{harm}}(x)=\sum_{y\neq x}\frac{1}{d(y,x)},\qquad \frac{1}{\infty}=0,$$40

the measure remains finite on arbitrary directed graphs and disconnected networks. It rewards both the size of the reachable region and its internal geometric proximity, avoids the small-component bias of patched closeness, and has strong monotonicity properties under edge addition.

At the same time, harmonic centrality is not a universal measure of importance. It does not capture brokerage in the manner of betweenness, recursive prestige in the manner of PageRank or eigenvector centrality, or influence dynamics in the manner of HIC. Its interpretation is specifically geometric: a vertex is central when many vertices can reach it—or, in the outgoing orientation, when it can reach many vertices—through short paths.