Abstract
Networks are versatile representations of the interactions between entities in complex systems. Cycles on such networks represent feedback processes which play a central role in system dynamics. In this work, we introduce a measure of the importance of any individual cycle, as the fraction of the total information flow of the network passing through the cycle. This measure is computationally cheap, numerically well-conditioned, induces a centrality measure on arbitrary subgraphs and reduces to the eigenvector centrality on vertices. We demonstrate that this measure accurately reflects the impact of events on strategic ensembles of economic sectors, notably in the US economy. As a second example, we show that in the protein-interaction network of the plant Arabidopsis thaliana, a model based on cycle-centrality better accounts for pathogen activity than the state-of-art one. This translates into pathogen-targeted-proteins being concentrated in a small number of triads with high cycle-centrality. Algorithms for computing the centrality of cycles and subgraphs are available for download.
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Notes
- 1.
Since the cycle centrality and its extension to subgraphs are consistent, c(i) is also equal to the cycle centrality of a self-loop \(i\rightarrow i\) from vertex i to itself.
- 2.
In the case of \(\varSigma _{CS}(\gamma )\), we had to introduce a regularisation parameter r such that \(e^{\mathsf {A}/r}\) converges in Matlab and \(\varSigma _{CS}(\gamma )\) could be computed. We then verified that the relative variations of \(\varSigma _{CS}(\gamma )\) were qualitatively independent from r.
- 3.
How much higher exactly depends on the regularisation parameter. The smaller r, the higher the ratio of the centralities between the years 2014 and 2006. For its smallest value guaranteeing convergence \(r\sim 10^3\), the ratio is a totally improbable \(4\times 10^{23}\).
- 4.
Here the centrality of a protein is understood to be its eigenvector centrality since, by Proposition 4, this is the measure induced by the cycle-centrality on vertices.
- 5.
By contrast another protein, AT3G47620, is targeted by 29 effectors from both the bacterium and the oomycete yet has “only” degree 104 [19].
- 6.
Running the computations separately, each of these models would take the c. 30 min time to evaluate since the majority of this time is spent finding the triads.
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Acknowledgements
We thank Paul Rochet of the Laboratoire Jean-Leray, Nantes, France, for stimulating discussions. P.-L. Giscard is grateful for the financial support from the Royal Commission for the Exhibition of 1851.
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Giscard, PL., Wilson, R.C. (2018). Cycle-Centrality in Economic and Biological Networks. In: Cherifi, C., Cherifi, H., Karsai, M., Musolesi, M. (eds) Complex Networks & Their Applications VI. COMPLEX NETWORKS 2017. Studies in Computational Intelligence, vol 689. Springer, Cham. https://doi.org/10.1007/978-3-319-72150-7_2
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