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Geometric description of clustering in directed networks

Nature Physics volume 20, pages 150–156 (2024)Cite this article

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Abstract

First-principle network models are crucial to understanding the intricate topology of real complex networks. Although modelling efforts have been quite successful in undirected networks, generative models for networks with asymmetric interactions are still not well developed and unable to reproduce several basic topological properties. Progress in this direction is of particular interest, as real directed networks are the norm rather than the exception in many natural and human-made complex systems. Here we show how the network geometry paradigm can be extended to the case of directed networks. We define a maximum entropy ensemble of random geometric directed graphs with a given sequence of in-degrees and out-degrees. Beyond these local properties, the ensemble requires only two additional parameters to fix the levels of reciprocity and the frequency of the seven possible types of three-node cycles in directed networks. A systematic comparison with several representative empirical datasets shows that fixing the level of reciprocity alongside the coupling with an underlying geometry is able to reproduce the wide diversity of clustering patterns observed in real directed complex networks.

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Fig. 1: Reciprocity in real directed networks.
Fig. 2: Illustrations of the concepts behind the modelling framework.
Fig. 3: Validation of the general framework controlling reciprocity.
Fig. 4: Reproducing topological features of real directed networks with the directed-reciprocal \({{\mathbb{S}}}^{1}\) model (Dir-recip).

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Data availability

The network datasets used in the article have been made publicly available by the original authors and were downloaded from the Netzschleuder network catalogue and repository (https://networks.skewed.de).

Code availability

The scripts and the source code of the programs used to produce the figures are publicly available on Zenodo (https://doi.org/10.5281/zenodo.8264693).

References

  1. Boguñá, M. et al. Network geometry. Nat. Rev. Phys. 3, 114–135 (2021).

    Article  Google Scholar 

  2. Serrano, M. Á., Krioukov, D. & Boguñá, M. Self-similarity of complex networks and hidden metric spaces. Phys. Rev. Lett. 100, 078701 (2008).

    Article  ADS  Google Scholar 

  3. García-Pérez, G., Boguñá, M. & Serrano, M. Á. Multiscale unfolding of real networks by geometric renormalization. Nat. Phys. 14, 583–589 (2018).

    Article  Google Scholar 

  4. Zheng, M., García-Pérez, G., Boguñá, M. & Serrano, M. Á. Scaling up real networks by geometric branching growth. Proc. Natl Acad. Sci. USA 118, e2018994118 (2021).

    Article  MathSciNet  Google Scholar 

  5. Boguñá, M., Krioukov, D., Almagro, P. & Serrano, M. Á. Small worlds and clustering in spatial networks. Phys. Rev. Res. 2, 023040 (2020).

    Article  Google Scholar 

  6. Papadopoulos, F., Kitsak, M., Serrano, M. Á., Boguñá, M. & Krioukov, D. Popularity versus similarity in growing networks. Nature 489, 537–540 (2012).

    Article  ADS  Google Scholar 

  7. Allard, A., Serrano, M. Á., García-Pérez, G. & Boguñá, M. The geometric nature of weights in real complex networks. Nat. Commun. 8, 14103 (2017).

    Article  ADS  Google Scholar 

  8. Kleineberg, K.-K., Boguñá, M., Serrano, M. Á. & Papadopoulos, F. Hidden geometric correlations in real multiplex networks. Nat. Phys. 12, 1076–1081 (2016).

    Article  Google Scholar 

  9. Newman, M. E. J. Networks (Oxford Univ. Press, 2018).

  10. Allard, A., Hébert-Dufresne, L., Young, J.-G. & Dubé, L. J. General and exact approach to percolation on random graphs. Phys. Rev. E 92, 062807 (2015).

    Article  ADS  MathSciNet  Google Scholar 

  11. Gleeson, J. P. & Melnik, S. Analytical results for bond percolation and k-core sizes on clustered networks. Phys. Rev. E 80, 046121 (2009).

    Article  ADS  Google Scholar 

  12. Newman, M. E. J. Properties of highly clustered networks. Phys. Rev. E 68, 026121 (2003).

    Article  ADS  Google Scholar 

  13. Karrer, B. & Newman, M. E. J. Random graphs containing arbitrary distributions of subgraphs. Phys. Rev. E 82, 066118 (2010).

    Article  ADS  MathSciNet  Google Scholar 

  14. Miller, J. C. Percolation and epidemics in random clustered networks. Phys. Rev. E 80, 020901 (2009).

    Article  ADS  MathSciNet  Google Scholar 

  15. Battiston, F. et al. Networks beyond pairwise interactions: structure and dynamics. Phys. Rep. 874, 1–92 (2020).

    Article  ADS  MathSciNet  Google Scholar 

  16. Lee, C. & Wilkinson, D. J. A review of stochastic block models and extensions for graph clustering. Appl. Netw. Sci. 4, 122 (2019).

    Article  Google Scholar 

  17. Orsini, C. et al. Quantifying randomness in real networks. Nat. Commun. 6, 8627 (2015).

    Article  ADS  MathSciNet  Google Scholar 

  18. Serrano, M. Á. & Boguñá, M. Tuning clustering in random networks with arbitrary degree distributions. Phys. Rev. E 72, 036133 (2005).

    Article  ADS  Google Scholar 

  19. Volz, E. Random networks with tunable degree distribution and clustering. Phys. Rev. E 70, 056115 (2004).

    Article  ADS  Google Scholar 

  20. Asllani, M., Lambiotte, R. & Carletti, T. Structure and dynamical behavior of non-normal networks. Sci. Adv. 4, eaau9403 (2018).

    Article  ADS  Google Scholar 

  21. Johnson, S. Digraphs are different: why directionality matters in complex systems. J. Phys. Complex 1, 015003 (2020).

    Article  Google Scholar 

  22. Levine, S. Several measures of trophic structure applicable to complex food webs. J. Theor. Biol. 83, 195–207 (1980).

    Article  ADS  Google Scholar 

  23. Duan, C., Nishikawa, T., Eroglu, D. & Motter, A. E. Network structural origin of instabilities in large complex systems. Sci. Adv. 8, eabm8310 (2022).

    Article  Google Scholar 

  24. Johnson, S., Domínguez-García, V., Donetti, L. & Muñoz, M. A. Trophic coherence determines food-web stability. Proc. Natl Acad. Sci. USA 111, 17923–17928 (2014).

    Article  ADS  Google Scholar 

  25. Johnson, S. & Jones, N. S. Looplessness in networks is linked to trophic coherence. Proc. Natl Acad. Sci. USA 114, 5618–5623 (2017).

    Article  ADS  MathSciNet  Google Scholar 

  26. Klaise, J. & Johnson, S. From neurons to epidemics: How trophic coherence affects spreading processes. Chaos 26, 065310 (2016).

    Article  ADS  MathSciNet  Google Scholar 

  27. Nicolaou, Z. G., Nishikawa, T., Nicholson, S. B., Green, J. R. & Motter, A. E. Non-normality and non-monotonic dynamics in complex reaction networks. Phys. Rev. Res. 2, 043059 (2020).

    Article  Google Scholar 

  28. Qu, B., Li, Q., Havlin, S., Stanley, H. E. & Wang, H. Nonconsensus opinion model on directed networks. Phys. Rev. E 90, 052811 (2014).

    Article  ADS  Google Scholar 

  29. Shao, J., Havlin, S. & Stanley, H. E. Dynamic opinion model and invasion percolation. Phys. Rev. Lett. 103, 018701 (2009).

    Article  ADS  Google Scholar 

  30. Michel, J., Reddy, S., Shah, R., Silwal, S. & Movassagh, R. Directed random geometric graphs. J. Complex Netw. 7, 792–816 (2019).

    Article  MathSciNet  Google Scholar 

  31. Wolf, F., Kirsch, C. & Donner, R. V. Edge directionality properties in complex spherical networks. Phys. Rev. E 99, 012301 (2019).

    Article  ADS  Google Scholar 

  32. Wu, Z., Di, Z. & Fan, Y. An asymmetric popularity-similarity optimization method for embedding directed networks into hyperbolic space. Complexity 2020, 8372928 (2020).

    Google Scholar 

  33. Kovács, B. & Palla, G. Model-independent embedding of directed networks into Euclidean and hyperbolic spaces. Commun. Phys. 6, 28 (2023).

    Article  Google Scholar 

  34. Peralta-Martinez, K. & Méndez-Bermúdez, J. A. Directed random geometric graphs: structural and spectral properties. J. Phys. Complex. 4, 015002 (2022).

    Article  Google Scholar 

  35. Wasserman, S. & Faust, K. Social Network Analysis: Methods and Applications (Cambridge Univ. Press, 1994).

  36. Garlaschelli, D. & Loffredo, M. I. Patterns of link reciprocity in directed networks. Phys. Rev. Lett. 93, 268701 (2004).

    Article  ADS  Google Scholar 

  37. García-Pérez, G., Serrano, M. Á. & Boguñá, M. Soft communities in similarity space. J. Stat. Phys. 173, 775–782 (2018).

    Article  ADS  MathSciNet  Google Scholar 

  38. Muscoloni, A. & Cannistraci, C. V. A nonuniform popularity-similarity optimization (nPSO) model to efficiently generate realistic complex networks with communities. New J. Phys. 20, 052002 (2018).

    Article  ADS  MathSciNet  Google Scholar 

  39. Désy, B., Desrosiers, P. & Allard, A. Dimension matters when modeling network communities in hyperbolic spaces. PNAS Nexus 2, pgad136 (2023).

  40. Holland, P. W. & Leinhardt, S. An exponential family of probability distributions for directed graphs. J. Am. Stat. Assoc. 76, 33–50 (1981).

    Article  MathSciNet  Google Scholar 

  41. van der Kolk, J., Serrano, M. Á. & Boguñá, M. An anomalous topological phase transition in spatial random graphs. Commun. Phys. 5, 245 (2022).

    Article  Google Scholar 

  42. García-Pérez, G., Allard, A., Serrano, M. Á. & Boguñá, M. Mercator: uncovering faithful hyperbolic embeddings of complex networks. New J. Phys. 21, 123033 (2019).

    Article  MathSciNet  Google Scholar 

  43. Coletta, L. et al. Network structure of the mouse brain connectome with voxel resolution. Sci. Adv. 6, eabb7187 (2020).

    Article  ADS  Google Scholar 

  44. Hulse, B. K. et al. A connectome of the Drosophila central complex reveals network motifs suitable for flexible navigation and context-dependent action selection. eLife 10, e66039 (2021).

    Article  Google Scholar 

  45. Ahnert, S. E. & Fink, T. M. A. Clustering signatures classify directed networks. Phys. Rev. E 78, 036112 (2008).

    Article  ADS  Google Scholar 

  46. Jia, M., Gabrys, B. & Musial, K. Directed closure coefficient and its patterns. PLoS ONE 16, e0253822 (2021).

    Article  Google Scholar 

  47. Holland, P. W. & Leinhardt, S. Local structure in social networks. Sociol. Methodol. 7, 1–45 (1976).

    Article  Google Scholar 

  48. Bianconi, G. Entropy of network ensembles. Phys. Rev. E 79, 036114 (2009).

    Article  ADS  MathSciNet  Google Scholar 

  49. van der Hoorn, P., Lippner, G. & Krioukov, D. Sparse maximum-entropy random graphs with a given power-law degree distribution. J. Stat. Phys. 173, 806–844 (2018).

    Article  ADS  MathSciNet  Google Scholar 

  50. Chung, F. & Lu, L. Connected components in random graphs with given expected degree sequences. Ann. Comb. 6, 125–145 (2002).

    Article  MathSciNet  Google Scholar 

  51. Adamic, L. A. & Glance, N. The political blogosphere and the 2004 U.S. election: divided they blog. In Proceedings of the 3rd International Workshop on Link Discovery 36–43 (2005).

  52. Ryan, K., Lu, Z. & Meinertzhagen, I. A. The CNS connectome of a tadpole larva of Ciona intestinalis (L.) highlights sidedness in the brain of a chordate sibling. eLife 5, e16962 (2016).

    Article  Google Scholar 

  53. Martinez, N. D. Artifacts or attributes? Effects of resolution on the Little Rock Lake food web. Ecol. Monogr. 61, 367–392 (1991).

    Article  Google Scholar 

  54. De Domenico, M., Nicosia, V., Arenas, A. & Latora, V. Structural reducibility of multilayer networks. Nat. Commun. 6, 6864 (2015).

    Article  ADS  Google Scholar 

  55. Massa, P., Salvetti, M. & Tomasoni, D. Bowling alone and trust decline in social network sites. In 2009 Eighth IEEE International Conference on Dependable, Autonomic and Secure Computing 658–663 (2009).

  56. Michalski, R., Palus, S. & Kazienko, P. Matching organizational structure and social network extracted from email communication. In Business Information Systems (ed. Abramowicz, W.) 197–206 (Springer, 2011).

  57. Mastrandrea, R., Fournet, J. & Barrat, A. Contact patterns in a high school: a comparison between data collected using wearable sensors, contact diaries and friendship surveys. PLoS ONE 10, e0136497 (2015).

    Article  Google Scholar 

  58. Kosack, S. et al. Functional structures of US state governments. Proc. Natl Acad. Sci. USA 115, 11748–11753 (2018).

    Article  ADS  Google Scholar 

  59. Freeman, L. C., Webster, C. M. & Kirke, D. M. Exploring social structure using dynamic three-dimensional color images. Soc. Networks 20, 109–118 (1998).

    Article  Google Scholar 

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Acknowledgements

We are grateful to L. J. Dubé for comments and to the nursing staff at the Centre de recherche clinique et évaluative en oncologie (CRCEO) where part of this work was done. A.A. acknowledges financial support from the Sentinelle Nord initiative of the Canada First Research Excellence Fund and from the Natural Sciences and Engineering Research Council of Canada (project 2019-05183). M.A.S. and M.B. acknowledge support from Grant TED2021-129791B-I00 funded by MCIN/AEI/10.13039/501100011033 and the European Union NextGenerationEU/PRTR, Grant PID2022-137505NB-C22 funded by MCIN/AEI/10.13039/501100011033, Grant PID2019-106290GB-C22 funded by MCIN/AEI/10.13039/501100011033 and Generalitat de Catalunya grant number 2021SGR00856. M.B. acknowledges the ICREA Academia award funded by the Generalitat de Catalunya.

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Authors and Affiliations

  1. Département de physique, de génie physique et d’optique, Université Laval, Québec, Québec, Canada

    Antoine Allard

  2. Centre interdisciplinaire en modélisation mathématique, Université Laval, Québec, Québec, Canada

    Antoine Allard

  3. Departament de Física de la Matèria Condensada, Universitat de Barcelona, Barcelona, Spain

    M. Ángeles Serrano & Marián Boguñá

  4. Universitat de Barcelona Institute of Complex Systems (UBICS), Universitat de Barcelona, Barcelona, Spain

    M. Ángeles Serrano & Marián Boguñá

  5. Institució Catalana de Recerca i Estudis Avançats (ICREA), Barcelona, Spain

    M. Ángeles Serrano

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All authors designed the research. A.A. and M.B. did the analytical calculations. A.A. performed the numerical simulations. All authors discussed the results and implications and wrote the manuscript.

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Correspondence to Antoine Allard.

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Allard, A., Serrano, M.Á. & Boguñá, M. Geometric description of clustering in directed networks. Nat. Phys. 20, 150–156 (2024). https://doi.org/10.1038/s41567-023-02246-6

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