Weighted, Bipartite, or Directed Stream Graphs for the Modeling of Temporal Networks

  • Matthieu LatapyEmail author
  • Clémence Magnien
  • Tiphaine Viard
Part of the Computational Social Sciences book series (CSS)


We recently introduced a formalism for the modeling of temporal networks, that we call stream graphs. It emphasizes the streaming nature of data and allows rigorous definitions of many important concepts generalizing classical graphs. This includes in particular size, density, clique, neighborhood, degree, clustering coefficient, and transitivity. In this contribution, we show that, like graphs, stream graphs may be extended to cope with bipartite structures, with node and link weights, or with link directions. We review the main bipartite, weighted or directed graph concepts proposed in the literature, we generalize them to the cases of bipartite, weighted, or directed stream graphs, and we show that obtained concepts are consistent with graph and stream graph ones. This provides a formal ground for an accurate modeling of the many temporal networks that have one or several of these features.


Temporal networks Link streams Stream graphs Weighted links Directed links Bipartite graphs 



This work is funded in part by the European Commission H2020 FETPROACT 2016-2017 program under grant 732942 (ODYCCEUS), by the ANR (French National Agency of Research) under grants ANR-15-CE38-0001 (AlgoDiv), by the Ile-de-France Region and its program FUI21 under grant 16010629 (iTRAC).


  1. 1.
    Ahnert, S.E., Garlaschelli, D., Fink, T.M.A., Caldarelli, G.: Ensemble approach to the analysis of weighted networks. Phys. Rev. E 76, 016101 (2007)ADSCrossRefGoogle Scholar
  2. 2.
    Alstott, J., Panzarasa, P., Rubinov, M., Bullmore, E.T., Vértes, P.E.: A unifying framework for measuring weighted rich clubs. Sci. Rep. 4, 7258 (2014)ADSCrossRefGoogle Scholar
  3. 3.
    Amano, S.I., Ogawa, K.I., Miyake, Y.: Node property of weighted networks considering connectability to nodes within two degrees of separation. Sci. Rep. 8, 8464 (2018)ADSCrossRefGoogle Scholar
  4. 4.
    Antoniou, I.E., Tsompa, E.T.: Statistical analysis of weighted networks. Discret. Dyn. Nat. Soc. 2008, 375452 (2008)Google Scholar
  5. 5.
    Barrat, A., Barthélemy, M., Pastor-Satorras, R., Vespignani, A.: The architecture of complex weighted networks. Proc. Natl. Acad. Sci. 101(11), 3747–3752 (2004)ADSCrossRefGoogle Scholar
  6. 6.
    Battiston, S., Catanzaro, M.: Statistical properties of corporate board and director networks. Eur. Phys. J. B 38, 345–352 (2004)ADSCrossRefGoogle Scholar
  7. 7.
    Bernardes, D., Diaby, M., Fournier, R., Françoise, F., Viennet, E.: A social formalism and survey for recommender systems. SIGKDD Explorations 16(2), 20–37 (2014)CrossRefGoogle Scholar
  8. 8.
    Bonacich, P: Technique for analyzing overlapping memberships. Sociol. Methodol. 4, 176–185 (1972)CrossRefGoogle Scholar
  9. 9.
    Borgatti, S.P., Everett, M.G.: Network analysis of 2-mode data. Soc. Netw. 19(3), 243–269 (1997)CrossRefGoogle Scholar
  10. 10.
    Breiger, R.L.: The duality of persons and groups. Soc. Forces 53(2), 181–190 (1974)CrossRefGoogle Scholar
  11. 11.
    Butts, C.T.: A relational event framework for social action. Sociol. Methodol. 38(1), 155–200 (2008)CrossRefGoogle Scholar
  12. 12.
    Candeloro, L., Savini, L.: A new weighted degree centrality measure: the application in an animal disease epidemic. PLoS One 11, e0165781 (2016)CrossRefGoogle Scholar
  13. 13.
    Casteigts, A., Flocchini, P., Quattrociocchi, W., Santoro, N.: Time-varying graphs and dynamic networks. IJPEDS 27(5), 387–408 (2012)Google Scholar
  14. 14.
    Clemente, G.P., Grassi, R.: Directed clustering in weighted networks: a new perspective. Chaos, Solitons Fractals 107, 26–38 (2018)ADSMathSciNetCrossRefGoogle Scholar
  15. 15.
    Curzel, J.L., Lüders, R., Fonseca, K.V.O. and Rosa, M.O.: Temporal performance analysis of bus transportation using link streams. Math. Probl. Eng. 2019, 6139379 (2019)Google Scholar
  16. 16.
    Doreian, P.: A note on the detection of cliques in valued graphs. Sociometry 32, 237–242 (1969)CrossRefGoogle Scholar
  17. 17.
    Esfahlani, F.Z., Sayama, H.: A percolation-based thresholding method with applications in functional connectivity analysis. In: Cornelius, S., Coronges, K., Gonçalves, B., Sinatra, R., Vespignani, A. (eds.) Complex Networks IX, pp. 221–231. Springer, Cham (2018)CrossRefGoogle Scholar
  18. 18.
    Fagiolo, G.: Clustering in complex directed networks. Phys. Rev. E 76, 026107 (2007)ADSCrossRefGoogle Scholar
  19. 19.
    Faust, K.: Centrality in affiliation networks. Soc. Netw. 19, 157–191 (1997)CrossRefGoogle Scholar
  20. 20.
    Grindrod, P.: Range-dependent random graphs and their application to modeling large small-world proteome datasets. Phys. Rev. E 66, 066702 (2002)ADSCrossRefGoogle Scholar
  21. 21.
    Guillaume, J.L., Le Blond, S., Latapy, M.: Statistical analysis of a P2P query graph based on degrees and their time-evolution. In: Proceedings of the 6th International Workshop on Distributed Computing (IWDC). Lecture Notes in Computer Sciences (LNCS). Springer, Berlin (2004)CrossRefGoogle Scholar
  22. 22.
    Guillaume, J.L., Le Blond, S., Latapy, M.: Clustering in P2P exchanges and consequences on performances. In: Proceedings of the 4th International Workshop on Peer-to-Peer Systems (IPTPS). Lecture Notes in Computer Sciences (LNCS). Springer, Berlin (2005)Google Scholar
  23. 23.
    Guillaume, J.-L., Latapy, M.: Bipartite structure of all complex networks. Inf. Process. Lett. 90(5), 215–221 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Hakimi, S.L.: On the degrees of the vertices of a directed graph. J. Frankl. Inst. 279(4), 290–308 (1965)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Kalna, G., Higham, D.J.: A clustering coefficient for weighted networks, with application to gene expression data. AI Commun. 20, 263–271 (2007)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Latapy, M., Magnien, C., Del Vecchio, N.: Basic notions for the analysis of large two-mode networks. Soc. Netw. 30(1), 31–48 (2008)CrossRefGoogle Scholar
  27. 27.
    Latapy, M., Viard, T., Magnien, C.: Stream graphs and link streams for the modeling of interactions over time. Soc. Netw. Anal. Mining 8(1), 1–61 (2018)zbMATHCrossRefGoogle Scholar
  28. 28.
    Lind, P.G., González, M.C., Herrmann, H.J.: Cycles and clustering in bipartite networks. Phys. Rev. E 72, 056127 (2005)ADSCrossRefGoogle Scholar
  29. 29.
    Lioma, C., Tarissan, F., Simonsen, J.G., Petersen, C., Larsen, B.: Exploiting the bipartite structure of entity grids for document coherence and retrieval. In: Proceedings of the 2016 ACM International Conference on the Theory of Information Retrieval (ICTIR ’16), pp. 11–20. ACM, New York (2016)Google Scholar
  30. 30.
    Mazel, J., Casas, P., Fontugne, R., Fukuda, K., Owezarski, P.: Hunting attacks in the dark: clustering and correlation analysis for unsupervised anomaly detection. Int. J. Netw. Manag. 25(5), 283–305 (2015)CrossRefGoogle Scholar
  31. 31.
    Meusel, R., Vigna, S., Lehmberg, O., Bizer, C.: The graph structure in the web—analyzed on different aggregation levels. J. Web Sci. 1, 33–47 (2015)CrossRefGoogle Scholar
  32. 32.
    Mislove, A., Marcon, M., Gummadi, K.P., Druschel, P. and Bhattacharjee, B.: Measurement and analysis of online social networks. In: Proceedings of the 7th ACM SIGCOMM Conference on Internet Measurement, pp. 29–42. ACM, New York (2007)Google Scholar
  33. 33.
    Newman, M.E., Strogatz, S.H., Watts, D.J.: Random graphs with arbitrary degree distributions and their applications. Phys. Rev. E 64, 026118 (2001)ADSCrossRefGoogle Scholar
  34. 34.
    Newman, M.E.J.: Scientific collaboration networks: I. Network construction and fundamental results. Phys. Rev. E 64, 016131 (2001)Google Scholar
  35. 35.
    Newman, M.E.J.: Scientific collaboration networks: II. Shortest paths, weighted networks, and centrality. Phys. Rev. E 64, 016132 (2001)Google Scholar
  36. 36.
    Newman, M.E.J.: Analysis of weighted networks. Phys. Rev. E 70, 056131 (2004)ADSCrossRefGoogle Scholar
  37. 37.
    Onnela, J.-P., Saramäki, J., Kertész, J., Kaski, K.: Intensity and coherence of motifs in weighted complex networks. Phys. Rev. E 71, 065103 (2005)ADSCrossRefGoogle Scholar
  38. 38.
    Opsahl, T.: Triadic closure in two-mode networks: Redefining the global and local clustering coefficients. Soc. Netw. 35(2), 159–167 (2013)CrossRefGoogle Scholar
  39. 39.
    Opsahl, T., Agneessens, F., Skvoretz, J.: Node centrality in weighted networks: generalizing degree and shortest paths. Soc. Netw. 32(3), 245–251 (2010)CrossRefGoogle Scholar
  40. 40.
    Opsahl, T., Colizza, V., Panzarasa, P., Ramasco, J.J.: Prominence and control: the weighted rich-club effect. Phys. Rev. Lett. 101, 168702 (2008)ADSCrossRefGoogle Scholar
  41. 41.
    Opsahl, T, Panzarasa, P.: Clustering in weighted networks. Soc. Netw. 31(2), 155–163 (2009)CrossRefGoogle Scholar
  42. 42.
    Panzarasa, P., Opsahl, T., Carley, K.M.: Patterns and dynamics of users’ behavior and interaction: network analysis of an online community. J. Am. Soc. Inf. Sci. Technol. 60(5), 911–932 (2009)CrossRefGoogle Scholar
  43. 43.
    Robins, G., Alexander, M.: Small worlds among interlocking directors: network structure and distance in bipartite graphs. Comput. Math. Organ. Theory 10(1), 69–94 (2004)zbMATHCrossRefGoogle Scholar
  44. 44.
    Saramäki, J., Kivelä, M., Onnela, J.P., Kaski, K., Kertesz, J.: Generalizations of the clustering coefficient to weighted complex networks. Phys. Rev. E 75(2), 027105 (2007)ADSCrossRefGoogle Scholar
  45. 45.
    Serrano, M.Á., Boguná, M., Vespignani, A.: Extracting the multiscale backbone of complex weighted networks. Proc. Natl. Acad. Sci. 106(16), 6483–6488 (2009)ADSCrossRefGoogle Scholar
  46. 46.
    Smith, K., Azami, H., Parra, M.A., Starr, J.M., Escudero, J.: Cluster-span threshold: an unbiased threshold for binarising weighted complete networks in functional connectivity analysis. In: 2015 37th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (EMBC), pp. 2840–2843. IEEE, Piscataway (2015)Google Scholar
  47. 47.
    Stadtfeld, C., Block, P.: Interactions, actors, and time: Dynamic network actor models for relational events. Sociol. Sci. 4, 318–352 (2017)CrossRefGoogle Scholar
  48. 48.
    Viard, T., Fournier-S’niehotta, R., Magnien, C., Latapy, M.: Discovering patterns of interest in IP traffic using cliques in bipartite link streams. In: Proceedings of Complex Networks IX, pp. 233–241. Springer, Cham (2018)CrossRefGoogle Scholar
  49. 49.
    Wang, Y., Ghumare, E., Vandenberghe, R., Dupont, P.: Comparison of different generalizations of clustering coefficient and local efficiency for weighted undirected graphs. Neural Comput. 29, 313–331 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  50. 50.
    Wasserman, S., Faust, K.: Social Network Analysis: Methods and Applications. Cambridge University Press, Cambridge (1994)zbMATHCrossRefGoogle Scholar
  51. 51.
    Watts, D., Strogatz, S.: Collective dynamics of small-world networks. Nature 393, 440–442 (1998)ADSzbMATHCrossRefGoogle Scholar
  52. 52.
    Wehmuth, K., Fleury, E., Ziviani, A.: On multiaspect graphs. Theor. Comput. Sci. 651, 50–61 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  53. 53.
    Wehmuth, K., Ziviani, A., Fleury, E.: A unifying model for representing time-varying graphs. In:2015 IEEE International Conference on Data Science and Advanced Analytics (DSAA 2015), Campus des Cordeliers, Paris, pp. 1–10 (2015)Google Scholar
  54. 54.
    Zhang, B., Horvath, S.: A general framework for weighted gene co-expression network analysis. Stat. Appl. Genet. Mol. Biol. 4, 1544–6115 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  55. 55.
    Zhang, P., Wang, J., Li, X., Li, M., Di, Z., Fan, Y.: Clustering coefficient and community structure of bipartite networks. Phys. A Stat. Mech. Appl. 387(27), 6869–6875 (2008)CrossRefGoogle Scholar
  56. 56.
    Zlatic, V., Bianconi, G., Díaz-Guilera, A., Garlaschelli, D., Rao, F., Caldarelli, G.: On the rich-club effect in dense and weighted networks. Eur. Phys. J. B 67(3), 271–275 (2009)ADSCrossRefGoogle Scholar
  57. 57.
    Zou, Z.: Polynomial-time algorithm for finding densest subgraphs in uncertain graphs. In: Procedings of MLG Workshop (2013)Google Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Matthieu Latapy
    • 1
    Email author
  • Clémence Magnien
  • Tiphaine Viard
  1. 1.CNRSLaboratoire d’Informatique de Paris 6ParisFrance

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