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Networks and Cycles: A Persistent Homology Approach to Complex Networks

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Proceedings of the European Conference on Complex Systems 2012

Part of the book series: Springer Proceedings in Complexity ((SPCOM))


Persistent homology is an emerging tool to identify robust topological features underlying the structure of high-dimensional data and complex dynamical systems (such as brain dynamics, molecular folding, distributed sensing).

Its central device, the filtration, embodies this by casting the analysis of the system in terms of long-lived (persistent) topological properties under the change of a scale parameter.

In the classical case of data clouds in high-dimensional metric spaces, such filtration is uniquely defined by the metric structure of the point space. On networks instead, multiple ways exists to associate a filtration. Far from being a limit, this allows to tailor the construction to the specific analysis, providing multiple perspectives on the same system.

In this work, we introduce and discuss three kinds of network filtrations, based respectively on the intrinsic network metric structure, the hierarchical structure of its cliques and—for weighted networks—the topological properties of the link weights. We show that persistent homology is robust against different choices of network metrics. Moreover, the clique complex on its own turns out to contain little information content about the underlying network. For weighted networks we propose a filtration method based on a progressive thresholding on the link weights, showing that it uncovers a richer structure than the metrical and clique complex approaches.

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  1. Newman M (2010) Networks: an introduction. Oxford University Press, New York

    MATH  Google Scholar 

  2. Albert R, Barabasi A (2002) Statistical mechanics of complex networks. Reviews of Modern Physics 74(1):47–97

    Article  MathSciNet  ADS  MATH  Google Scholar 

  3. Schaub MT, Delvenne JC, Yaliraki SN, Barahona M (2012) Markov dynamics as a zooming lens for multiscale community detection: non clique-like communities and the field-of-view limit. PloS One 7(2):e32210

    Article  ADS  Google Scholar 

  4. Barthlemy M (2011) Spatial networks. Physics Reports 499(13):1–101

    Article  MathSciNet  ADS  Google Scholar 

  5. Boguna M, Papadopoulos F, Krioukov D (2010) Sustaining the internet with hyperbolic mapping. Nature Communications 1:62

    Article  ADS  Google Scholar 

  6. Conradi C, Flockerzi D, Raisch J, Stelling J (2007) Subnetwork analysis reveals dynamic features of complex (bio)chemical networks. Proceedings of the National Academy of Sciences 104(49):19175–19180

    Article  ADS  Google Scholar 

  7. Henderson JA, Robinson PA (2011) Geometric effects on complex network structure in the cortex. Phys Rev Lett 107:018102

    Article  ADS  Google Scholar 

  8. Zomorodian A, Carlsson G (2005) Computing persistent homology. Discrete Comput Geom 33(2):249–274

    Article  MathSciNet  MATH  Google Scholar 

  9. Carlsson G (2009) Topology and data. Bulletin of the American Mathematical Society 46(2):255–308

    Article  MathSciNet  MATH  Google Scholar 

  10. Horak D, Maletic S, Rajkovic M (2009) Persistent homology of complex networks. Journal of Statistical Mechanics: Theory and Experiment 2009(03):P03034

    Article  MathSciNet  Google Scholar 

  11. Fouss F, Yen L, Pirotte A, Saerens M (2006) An experimental investigation of graph kernels on a collaborative recommendation task. In: Sixth international conference on data mining (ICDM’06), pp 863–868

    Chapter  Google Scholar 

  12. Bolch G, Greiner S, de Meer H, Trivedi KS (1998) Queueing networks and Markov chains: modeling and performance evaluation with computer science applications. Wiley-Interscience, New York

    Book  MATH  Google Scholar 

  13. Kondor R, Lafferty J (2002) Diffusion kernels on graphs and other discrete input spaces. In: Proceedings of the nineteenth international conference on machine learning (ICML’02), pp 315–322

    Google Scholar 

  14. Smola A, Kondor R (2003) Kernels and regularization on graphs. In: Learning theory and kernel machines. Lecture notes in computer science, vol 2777, pp 144–158

    Chapter  Google Scholar 

  15. Kandola J, Shawe-Taylor J, Cristianini N (2002) Learning semantic similarity. Advances in neural information processing systems 15:657–666

    Google Scholar 

  16. Yen L, Fouss F, Decaestecker C, Francq P, Saerens M (2007) Graph nodes clustering based on the commute-time kernel. In: Zhou ZH, Li H, Yang Q (eds) Advances in knowledge discovery and data mining. Lecture notes in computer science, vol 4426. Springer, Berlin, pp 1037–1045

    Chapter  Google Scholar 

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The authors acknowledge Mario Rasetti for insightful discussions and constant support.

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Petri, G., Scolamiero, M., Donato, I., Vaccarino, F. (2013). Networks and Cycles: A Persistent Homology Approach to Complex Networks. In: Gilbert, T., Kirkilionis, M., Nicolis, G. (eds) Proceedings of the European Conference on Complex Systems 2012. Springer Proceedings in Complexity. Springer, Cham.

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