Skip to main content

Networks and Cycles: A Persistent Homology Approach to Complex Networks

  • Conference paper

Part of the Springer Proceedings in Complexity book series (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.


  • Complex networks
  • Persistent homology
  • Metrics
  • Computational topology

This is a preview of subscription content, access via your institution.

Buying options

USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-319-00395-5_15
  • Chapter length: 7 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
USD   219.00
Price excludes VAT (USA)
  • ISBN: 978-3-319-00395-5
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   279.99
Price excludes VAT (USA)
Hardcover Book
USD   329.99
Price excludes VAT (USA)
Fig. 15.1
Fig. 15.2


  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

    CrossRef  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

    CrossRef  ADS  Google Scholar 

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

    CrossRef  MathSciNet  ADS  Google Scholar 

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

    CrossRef  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

    CrossRef  ADS  Google Scholar 

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

    CrossRef  ADS  Google Scholar 

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

    CrossRef  MathSciNet  MATH  Google Scholar 

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

    CrossRef  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

    CrossRef  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

    CrossRef  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

    CrossRef  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

    CrossRef  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

    CrossRef  Google Scholar 

Download references


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

Author information

Authors and Affiliations


Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2013 Springer International Publishing Switzerland

About this paper

Cite this paper

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.

Download citation