Another extension to networks with disconnected components has been proposed by Opsahl (2010) and later studied by Boldi and Vigna (2014) in general directed graphs:
The harmonic mean H of the positive real numbers x1, x2, ..., xn > 0 is defined to be:
See Information Centrality
Putman, K.L., Boekhout, H.D. and Takes, F.W., 2019, August. Fast incremental computation of harmonic closeness centrality in directed weighted networks. In 2019 IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining (ASONAM) (pp. 1018-1025). IEEE.
From BOLDI 2014
Yannick Rochat, in a talk at Application of Social Network Analysys (ASNA 2009) observed that in an undirected graph with several disconnected components, the inverse of the harmonic mean of distances offers a better notion of centrality than closeness, because it weights less those elements that belong to smaller components. Tore Opsahl made the same observation in a March 2010 blog posting. Raj Kumar Pan and Jari Saram¨aki deviated from the classical definition of closeness in [Pan and Saram¨aki 11], using, in practice, harmonic centrality, with the motivation of better handling disconnected nodes (albeit with no reference to the harmonic mean). Edith Cohen and Haim Kaplan defined in [Cohen and Kaplan 07] the notion of a spatially decaying aggregate, of which harmonic centrality is a particular instance.
- BOLDI, P. & VIGNA, S. 2014. Axioms for centrality. Internet Mathematics, 00-00.
- MARCHIORI, M. & LATORA, V. 2000. Harmony in the small-world. Physica A: Statistical Mechanics and its Applications, 285, 539-546.
- OPSAHL, T., AGNEESSENS, F. & SKVORETZ, J. 2010. Node centrality in weighted networks: Generalizing degree and shortest paths. Social Networks, 32, 245-251.
- OPSAHL, T. 2010. Closeness centrality in networks with disconnected components (http://toreopsahl.com/2010/03/20/closeness-centrality-in-networks-with-disconnected-components/)
See Holme & Ghoshal, Phys. Rev. Lett. 96, 098701 (2006), Eq. (1) for an earlier reference.
(But I don't think it is such a great measure. It is an arbitrary combination of two aspects of networks—the component size distribution and the distances within the components—that rather should be kept separate.)
|Add Replay||written March 24, 2018, 12:14 pm by Petter Holme|