[Paper Review] Harmonic Centrality in Some Graph Families

Jose Mari E. Ortega, Rolito G. Eballe|arXiv (Cornell University)|Nov 24, 2021

Complex Network Analysis Techniques6 references4 citations

TL;DR

This paper derives closed-form expressions for harmonic centrality in key graph families—such as paths, cycles, wheels, stars, and prisms—using inverse shortest-path distances normalized by graph order. It demonstrates that harmonic centrality effectively quantifies node importance even in disconnected or sparse networks, with exact values derived for symmetric graphs using harmonic numbers and distance summations.

ABSTRACT

One of the more recent measures of centrality in social network analysis is the normalized harmonic centrality. A variant of the closeness centrality, harmonic centrality sums the inverse of the geodesic distances of each node to other nodes where it is 0 if there is no path from one node to another. It is then normalized by dividing it by m-1, where m is the number of nodes of the graph. In this paper, we present notions regarding the harmonic centrality of some important classes of graphs.

Motivation & Objective

• To establish exact harmonic centrality values for fundamental graph families commonly used in network modeling.
• To address limitations of traditional centrality measures in disconnected or sparse graphs by leveraging harmonic centrality’s robustness.
• To provide analytical expressions that enable efficient centrality computation in symmetric and structured networks.
• To extend the applicability of harmonic centrality to complex graph types such as book graphs, helm graphs, and prism graphs.
• To offer a foundation for analyzing larger, real-world networks by characterizing basic building blocks via harmonic centrality.

Proposed method

• Computes the sum of reciprocals of geodesic distances from each vertex to all others, setting reciprocal to zero when no path exists.
• Normalizes the sum by dividing by (m − 1), where m is the number of vertices, to produce a normalized harmonic centrality value.
• Applies the harmonic number formula Hn = Σ(k=1 to n) 1/k to simplify distance-based summations in symmetric graphs.
• Uses structural symmetry and distance patterns in graph families (e.g., cycles, stars, Cartesian products) to derive closed-form expressions.
• Segments vertex sets based on symmetry (e.g., end vertices vs. internal vertices in paths) to compute distinct centrality values.
• Applies case analysis based on parity (odd/even m) for graphs like prism and wheel, leading to piecewise expressions.

Experimental results

Research questions

• RQ1 What is the harmonic centrality of vertices in a path graph Pm, and how does it vary between end and internal vertices?
• RQ2 How does harmonic centrality behave in cycle-based graphs such as wheels, crowns, and prisms, especially when m is odd or even?
• RQ3 What are the exact harmonic centrality values for star, book, and helm graphs, and how do they differ between central and peripheral nodes?
• RQ4 How does the normalization process affect the interpretation of centrality in graphs with varying connectivity?
• RQ5 Can closed-form expressions for harmonic centrality be derived for Cartesian product graphs like the ladder, book, and prism?

Key findings

• For the path graph Pm, harmonic centrality of end vertices (i=1 or m) is (2Hm−1 + 1/m)/(2m−1), while internal vertices i (1 < i < m) have centrality (2(Hi−1 + Hm−i) + 1/i + 1/(m−i+1) −1)/(2m−1).
• In the complete bipartite graph Km,n, harmonic centrality is (2Hn + 2Hm − 2)/(m+n−1), showing symmetric dependence on both partitions.
• For the star graph Sm, the central vertex has harmonic centrality 1, while peripheral vertices have centrality (m+1)/(2m).
• In the book graph Bm, the central vertices (u0,vj) have centrality (3m+2)/(4m+2), while peripheral vertices (ui,vj) have (5(m+2))/(6(2m+1)).
• For the helm graph Hm, the central vertex u0 has centrality 3/4, internal cycle vertices ui (1≤i≤m) have (5m+15)/(12m), and pendant vertices vj have (7m+17)/(24m).
• In the prism graph Ym, harmonic centrality depends on m’s parity: for odd m, it is (4H(m−1)/2 − (m−3)/(m+1))/(2m−1), and for even m, (4Hm/2 + 2/(m+2) − (m+2)/m)/(2m−1).