How Works Harmonic Centrality?

Meaning

The network measurement harmonic centrality measures the 'average' distance of a node to the other nodes in the network. This can be used to find nodes in a network that can quickly communicate with other nodes. Harmonic and closeness centrality use the length of the shortest path between a pair of nodes, but define it differently.

Closeness centrality

The closeness of a node is the inverse of the sum of lengths. Closeness centrality has a constraint, there must be a path between each pair of nodes. If not, these unreachable and undefined nodes give some complications, which leads to wrong interpretations.

Harmonic centrality

Harmonic centrality inverts the sum and reciprocal operations in the definition of closeness centrality: the sum of the inverse lengths. Beauchamp introduced this idea in 1965 based on the property that 1/infinity is (almost) 0 [2]. This allows this centrality to deal with infinite values.

Based on this idea harmonic centrality was proposed in 2000 (!) by Marchiori and Latora [3] and later other authors. Using the harmonic centrality avoids cases where an infinite distance outweighs the others and treats this limitation of closeness centrally. Thus, harmonic centrality can be applied to disconnected networks.

The harmonic centrality can be calculated for nodes in undirected, directed and weighted networks.

Closeness vs. harmonic: similarities

In most networks, both centralities are strong correlated. Similarities are:

• measure length shortest path;
• related to entire network;
• find broadcasters;
• same computing time, O(n|E|).

Closeness vs. harmonic: differences

But there also differences. Both centralities not always correlated; the results can even be at odds with each other. Figure 2 shows an example where the Spearman's correlation ?=-0.2; both centralities have not the same order and be at odds with each other.

Other differences:

• definition:
  • closeness: 1/sum(length);
  • harmonic: sum(1/length);
• Normalisation, multiply by:
  • closeness: (n-1);
  • harmonic: 1/(n-1);
• applicability networks: closeness has a constraint all nodes must be reachable;
• closeness is more sensitive to changes in a network;
• closeness is more popular.

To calculate closeness centrality, we need the total network, this makes this centrality sensitive to changes in this network. The way of defining makes harmonic centrality less sensitive to this.

Alternatives

Harmonic centrality is just one of many measures of centrality. There are also other alternatives that can that deal with the constraint of closeness centrality and can be applied to disconnected networks: information centrality, effective distance-based closeness centrality, random walk closeness centrality, hierarchical closeness centrality, forest distance closeness centrality, influence range closeness centrality and more.

Some of these contains also other improvements. Most alternatives have similar use cases.

Conclusion

Harmonic centrality is very similar to closeness centrality. Closeness centrality is perhaps a popular centrality, but the harmonic centrality is in most cases a better measurement, but in some cases, they measure different centralities.

References

[1] Vignery, K., & Laurier, W. (2020). A methodology and theoretical taxonomy for centrality measures: What are the best centrality indicators for student networks?. PLoS One, 15(12), e0244377.

[2] Beauchamp, M. A. (1965). An improved index of centrality. Behavioural science, 10(2), 161-163.

[3] Marchiori, M., & Latora, V. (2000). Harmony in the small-world. Physica A: Statistical Mechanics and its Applications, 285(3-4), 539-546.