Stream graphs and link streams for the modeling of interactions over time

  • Matthieu LatapyEmail author
  • Tiphaine Viard
  • Clémence Magnien
Original Article


Graph theory provides a language for studying the structure of relations, and it is often used to study interactions over time too. However, it poorly captures the intrinsically temporal and structural nature of interactions, which calls for a dedicated formalism. In this paper, we generalize graph concepts to cope with both aspects in a consistent way. We start with elementary concepts like density, clusters, or paths, and derive from them more advanced concepts like cliques, degrees, clustering coefficients, or connected components. We obtain a language to directly deal with interactions over time, similar to the language provided by graphs to deal with relations. This formalism is self-consistent: usual relations between different concepts are preserved. It is also consistent with graph theory: graph concepts are special cases of the ones we introduce. This makes it easy to generalize higher level objects such as quotient graphs, line graphs, k-cores, and centralities. This paper also considers discrete versus continuous time assumptions, instantaneous links, and extensions to more complex cases.


Stream graphs Link streams Temporal networks Time-varying graphs Dynamic graphs Dynamic networks Longitudinal networks Interactions Time Graphs Networks 



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) and ANR-13-CORD-0017-01 (CODDDE), by the French program “PIA - Usages, services et contenus innovants” under Grant O18062-44430 (REQUEST), and by the Ile-de-France program FUI21 under Grant 16010629 (iTRAC). We warmly thank the many colleagues and friends who read preliminary versions of this work and provided invaluable feedback.


  1. Barabási A-L, Pósfai M (2016) Network science. Cambridge University Press, CambridgeGoogle Scholar
  2. Barrat A, Cattuto C (2013) Temporal networks of face-to-face human interactions. Springer, Berlin, Heidelberg, pp 191–216Google Scholar
  3. Batagelj V, Praprotnik S (2016) An algebraic approach to temporal network analysis based on temporal quantities. Social Netw Anal Mining 6(1):28:1–28:22zbMATHGoogle Scholar
  4. Berge C (1962) The theory of graphs and its applications. Wiley, New YorkzbMATHGoogle Scholar
  5. Bhadra S, Ferreira A (2012) Computing multicast trees in dynamic networks and the complexity of connected components in evolving graphs. J Internet Serv Appl 3(3):269–275Google Scholar
  6. Blondel VD, Decuyper A, Krings G (2015) A survey of results on mobile phone datasets analysis. EPJ Data Sci 4(1):10Google Scholar
  7. Blonder B, Wey TW, Dornhaus A, James R, Sih A (2012) Temporal dynamics and network analysis. Methods Ecol Evol 3(6):958–972Google Scholar
  8. Bondy JA (1976) Graph Theory Appl. Elsevier Science Ltd., OxfordGoogle Scholar
  9. Bui-Xuan B-M, Ferreira A, Jarry A (2003) Computing shortest, fastest, and foremost journeys in dynamic networks. Int J Found Comput Sci 14(2):267–285MathSciNetzbMATHGoogle Scholar
  10. Butts CT (2008) A relational event framework for social action. Sociol Methodol 38(1):155–200Google Scholar
  11. Caceres RS, Berger-Wolf T (2013) Temporal scale of dynamic networks. Springer, Berlin, Heidelberg, pp 65–94Google Scholar
  12. Casteigts A, Flocchini P, Mans B, Santoro N (2015) Shortest, fastest, and foremost broadcast in dynamic networks. Int J Found Comput Sci 26(4):499–522MathSciNetzbMATHGoogle Scholar
  13. Casteigts A, Flocchini P, Quattrociocchi W, Santoro N (2012) Time-varying graphs and dynamic networks. IJPEDS 27(5):387–408Google Scholar
  14. Chinelate CE, Borges VA, Klaus W, Artur Z, da Silva APC (2015) Time centrality in dynamic complex networks. Adv Complex Syst 18(7–8):1550023MathSciNetGoogle Scholar
  15. Conan V, Leguay J, Friedman T (2007) Characterizing pairwise inter-contact patterns in delay tolerant networks. In: Proceedings of the 1st international conference on autonomic computing and communication systems, autonomics ’07, pages 19:1–19:9, ICST, Brussels, Belgium, Belgium. ICST (Institute for Computer Sciences, Social-Informatics and Telecommunications Engineering)Google Scholar
  16. Corneli M, Latouche P, Rossi F (2016) Block modelling in dynamic networks with non-homogeneous poisson processes and exact icl. Social Netw Anal Mining 6(1):55Google Scholar
  17. Corneli M, Latouche P, Rossi F (2015) Modelling time evolving interactions in networks through a non stationary extension of stochastic block models. In: ASONAM '15 Proceedings of the 2015 IEEE/ACM international conference on advances in social networks analysis and mining 2015, pp 1590–1591Google Scholar
  18. David E, Jon K (2010) Networks, crowds, and markets: reasoning about a highly connected world. Cambridge University Press, New YorkzbMATHGoogle Scholar
  19. Diestel R (2012) Graph theory, 4th edn, volume 173 of Graduate texts in mathematics. Springer, New YorkGoogle Scholar
  20. Doreian P, Stokman F (1997) Evolution of social networks. The journal of mathematical sociology, vol 1. Gordon and Breach Publishers, AmsterdamzbMATHGoogle Scholar
  21. Dutta BL, Ezanno P, Vergu E (2014) Characteristics of the spatio-temporal network of cattle movements in France over a 5-year period. Prev Vet Med 117(1):79–94Google Scholar
  22. Flores J, Romance M (2017) On eigenvector-like centralities for temporal networks: discrete vs. continuous time scales. J Comput Appl Math 330:1041–1051MathSciNetzbMATHGoogle Scholar
  23. Gaumont N, Magnien C, Latapy M (2016a) Finding remarkably dense sequences of contacts in link streams. Social Netw Anal Mining 6(1):87:1–87:14Google Scholar
  24. Gaumont N, Viard T, Fournier-S’niehotta R, Wang Q, Latapy M (2016b) Analysis of the temporal and structural features of threads in a mailing-list. Springer International Publishing, Cham, pp 107–118Google Scholar
  25. Gauvin L, Panisson A, Cattuto C (2014) Detecting the community structure and activity patterns of temporal networks: a non-negative tensor factorization approach. PLOS One 9(1):1–13Google Scholar
  26. George B, Kim S (2013) Spatio-temporal Networks: modeling and algorithms. SpringerBriefs in Computer Science. Springer, New YorkzbMATHGoogle Scholar
  27. Gomes LH, Almeida VAF, Almeida JM, Castro FDO, Bettencourt LA (2009) Quantifying social and opportunistic behavior in email networks. Adv Complex Syst 12(01):99–112Google Scholar
  28. Gulyás L, Kampis G, Legendi RO (2013) Elementary models of dynamic networks. Eur Phys J Spec Top 222(6):1311–1333Google Scholar
  29. Hamon R, Borgnat P, Flandrin P, Robardet C (2015) Duality between temporal networks and signals: Extraction of the temporal network structures. CoRR.
  30. Harshaw CR, Bridges RA, Iannacone MD, Reed JW, Goodall JR (2016) Graphprints: towards a graph analytic method for network anomaly detection. In: Proceedings of the 11th annual cyber and information security research conference, CISRC ’16, ACM, New York, NY, USA, pp 15:1–15:4Google Scholar
  31. Hernández-Orallo E, Cano JC, Calafate CT, Manzoni P (2016) New approaches for characterizing inter-contact times in opportunistic networks. Ad Hoc Netw 52:160–172 (Modeling and Performance Evaluation of Wireless Ad Hoc Networks) Google Scholar
  32. Holme P (2015) Modern temporal network theory: a colloquium. Eur Phys J B 88(9):234Google Scholar
  33. Holme P, Saramäki J (2012) Temporal networks. Phys Rep 519(3):97–125 (Temporal Networks) Google Scholar
  34. Huanhuan W, Cheng J, Huang S, Ke Y, Yi L, Yanyan X (2014) Path problems in temporal graphs. Proc VLDB Endow 7(9):721–732Google Scholar
  35. Hulovatyy Y, Chen H, Milenkovic T (2016) Exploring the structure and function of temporal networks with dynamic graphlets. Bioinformatics 32(15):2402Google Scholar
  36. Karsai M, Kivelä M, Pan RK, Kaski K, Kertész J, Barabási A-L, Saramäki J (2011) Small but slow world: How network topology and burstiness slow down spreading. Phys Rev E 83:025102Google Scholar
  37. Kivelä M, Arenas A, Barthelemy M, Gleeson JP, Moreno Y, Porter MA (2014) Multilayer networks. J Complex Netw 2(3):203–271Google Scholar
  38. Kivelä M, Cambe J, Saramäki J, Karsai M (2018) Mapping temporal-network percolation to weighted, static event graphs. Sci Rep 8:12357. Google Scholar
  39. Kostakos V (2009) Temporal graphs. Physica A Stat Mech Appl 388(6):1007–1023MathSciNetGoogle Scholar
  40. Kovanen L, Karsai M, Kaski K, Kertész J, Saramäki J (2011) Temporal motifs in time-dependent networks. J Stat Mech Theory Exp 2011(11):P11005Google Scholar
  41. Kovanen L, Kaski K, Kertész J, Saramäki J (2013) Temporal motifs reveal homophily, gender-specific patterns, and group talk in call sequences. PNAS 110(45):18070–18075. Google Scholar
  42. Krings G, Karsai M, Bernhardsson S, Blondel VD, Saramäki J (2012) Effects of time window size and placement on the structure of an aggregated communication network. EPJ Data Sci 1(1):4Google Scholar
  43. Latapy M, Magnien C, Del Vecchio N (2008) Basic notions for the analysis of large two-mode networks. Soc Netw 30(1):31–48Google Scholar
  44. Laurent G, Saramäki J, Karsai M (2015) From calls to communities: a model for time-varying social networks. Eur Phys J B 88(11):301MathSciNetGoogle Scholar
  45. Leskovec J, Kleinberg JM, Faloutsos C (2007) Graph evolution: densification and shrinking diameters. TKDD 1(1):2Google Scholar
  46. Leskovec J, Backstrom L, Kumar R, Tomkins A (2008) Microscopic evolution of social networks. In: Ying L, Bing L, Sunita S (eds) Proceedings of the 14th ACM SIGKDD international conference on knowledge discovery and data mining, ACM, Las Vegas, Nevada, USA, Aug 24–27, pp 462–470Google Scholar
  47. Léo Y, Crespelle C, Fleury E (2015) Non-altering time scales for aggregation of dynamic networks into series of graphs. In Felipe Huici and Giuseppe Bianchi, editors, Proceedings of the 11th ACM conference on emerging networking experiments and technologies, CoNEXT 2015, ACM, Heidelberg, Germany, Dec 1–4, 2015, pp 29:1–29:7Google Scholar
  48. Magnien C, Tarissan F (2015) Time evolution of the importance of nodes in dynamic networks. In: ASONAM '15 Proceedings of the 2015 IEEE/ACM international conference on advances in social networks analysis and mining 2015. ACM, New York, pp 1200–1207Google Scholar
  49. Martinet L, Crespelle C, Fleury E (2014) Dynamic contact network analysis in hospital wards. Springer International Publishing, Cham, pp 241–249Google Scholar
  50. Masuda N, Lambiotte R (2016) A guide to temporal networks. Series on complexity science, vol 4. World Scientific, UKzbMATHGoogle Scholar
  51. Matias C, Miele V (2017) Statistical clustering of temporal networks through a dynamic stochastic block model. J R Stat Soc Ser B 79:1119–1141MathSciNetzbMATHGoogle Scholar
  52. Michail O (2015) An introduction to temporal graphs: an algorithmic perspective. Springer International Publishing, Cham, pp 308–343zbMATHGoogle Scholar
  53. Newman MEJ (2001) Clustering and preferential attachment in growing networks. Phys Rev E 64:025102Google Scholar
  54. Newman M (2010) Networks: an introduction. Oxford University Press Inc, New YorkzbMATHGoogle Scholar
  55. Nicosia V, Tang J, Musolesi M, Russo G, Mascolo C, Latora V (2012) Components in time-varying graphs. Chaos: an Interdisciplinary. J Nonlinear Sci 22(2):023101zbMATHGoogle Scholar
  56. Nicosia V, Tang J, Mascolo C, Musolesi M, Russo G, Latora V (2013) Graph metrics for temporal networks. Springer Berlin Heidelberg, Berlin, Heidelberg, pp. 15–40Google Scholar
  57. Paranjape A, Benson AR, Leskovec J (2017) Motifs in temporal networks. In: de Rijke M, Shokouhi M, Tomkins A, Zhang M (eds) Proceedings of the Tenth ACM international conference on web search and data mining, WSDM 2017, Cambridge, United Kingdom, February 6–10, pp 601–610Google Scholar
  58. Payen A, Tabourier L, Latapy M (2017) Impact of temporal features of cattle exchanges on the size and speed of epidemic outbreaks. In: Osvaldo G, Beniamino M, Sanjay M, Giuseppe B, Carmelo Maria T, Ana Maria ACR, David T, Bernady OA, Elena NS, Alfredo C (eds) Computational science and its applications—ICCSA 2017—17th International Conference, Trieste, Italy, July 3-6, 2017, Proceedings, Part II, volume 10405 of Lecture Notes in Computer Science. Springer, New York, pp 84–97Google Scholar
  59. Pei J, Silvestri F, Tang J (eds) (2015) Proceedings of the 2015 IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining, ASONAM 2015, ACM. Paris, France, August 25–28, 2015Google Scholar
  60. Perry PO, Wolfe PJ (2013) Point process modelling for directed interaction networks. J R Stat Soc Ser B (Statistical Methodology) 75(5):821–849MathSciNetGoogle Scholar
  61. Praprotnik S, Batagelj V (2016) Semirings for temporal network analysis. CoRR
  62. Ribeiro B, Perra N, Baronchelli A (2013) Quantifying the effect of temporal resolution on time-varying networks. Sci Rep 3:3006Google Scholar
  63. Rocha LEC, Masuda N (2014) Random walk centrality for temporal networks. N J Phys 16(6):063023MathSciNetGoogle Scholar
  64. Rozenshtein P, Tatti N, Gionis A (2017) Finding dynamic dense subgraphs. ACM Trans Knowl Discov Data 11(3):271–2730Google Scholar
  65. Santoro N, Quattrociocchi W, Flocchini P, Casteigts A, Amblard F (2011) Time-varying graphs and social network analysis: Temporal indicators and metrics. In: 3rd AISB social networks and multiagent systems symposium (SNAMAS), pp 32–38Google Scholar
  66. Saramäki J, Holme P (2015) Exploring temporal networks with greedy walks. Eur Phys J B 88(12):334Google Scholar
  67. Scholtes I, Wider N, Garas A (2016) Higher-order aggregate networks in the analysis of temporal networks: path structures and centralities. Eur Phys J B 89(3):61Google Scholar
  68. Scott J (2017) Social network analysis. SAGE Publications, Thousand OaksGoogle Scholar
  69. Sikdar S, Ganguly N, Mukherjee A (2015) Time series analysis of temporal networks. CoRR
  70. Sizemore AE, Bassett DS (2018) Dynamic graph metrics: tutorial, toolbox, and tale. NeuroImage 180:417–427Google Scholar
  71. Snijders TAB (2001) The statistical evaluation of social network dynamics. Sociol Methodol 31(1):361–395Google Scholar
  72. Snijders TAB, van de Bunt GG, Steglich. CEG (2010) Introduction to stochastic actor-based models for network dynamics. Soc Netw 32(1):44–60 (Dynamics of Social Networks) Google Scholar
  73. Stadtfeld C, Block P (2017) Interactions, actors, and time: dynamic network actor models for relational events. Sociol Sci 4:318–352Google Scholar
  74. Stadtfeld C, Hollway J, Block P (2017) Dynamic network actor models: investigating coordination ties through time. Sociol Methodol 47(1):1–40Google Scholar
  75. Starnini M, Baronchelli A, Barrat A, Pastor-Satorras R (2012) Random walks on temporal networks. Phys Rev E 85:056115Google Scholar
  76. Sun J, Tao D, Faloutsos C (2006) Beyond streams and graphs: dynamic tensor analysis. In: Tina ER, Lyle HU, Mark C, Dimitrios G (eds) Proceedings of the Twelfth ACM SIGKDD international conference on knowledge discovery and data mining, ACM, Philadelphia, PA, USA, Aug 20–23, 2006, pp 374–383Google Scholar
  77. Takaguchi T, Yano Y, Yoshida Y (2016) Coverage centralities for temporal networks. Eur Phys J B 89(2):35Google Scholar
  78. Tang J, Musolesi M, Mascolo C, Latora V (2010) Characterising temporal distance and reachability in mobile and online social networks. SIGCOMM Comput Commun Rev 40(1):118–124Google Scholar
  79. Tang J, Scellato S, Musolesi M, Mascolo C, Latora V (2010) Small-world behavior in time-varying graphs. Phys Rev E 81:055101Google Scholar
  80. Tang J, Musolesi M, Mascolo C, Latora V, Nicosia V (2010) Analysing information flows and key mediators through temporal centrality metrics. In: Proceedings of the 3rd workshop on social network systems, SNS ’10, ACM. New York, NY, USA, pp 3:1–3:6Google Scholar
  81. Taylor D, Myers SA, Clauset A, Porter MA, Mucha PJ (2017) Eigenvector-based centrality measures for temporal networks. Multiscale Model Simul 15(1):537–574MathSciNetzbMATHGoogle Scholar
  82. Thompson WH, Brantefors P, Fransson P (2017) From static to temporal network theory: applications to functional brain connectivity. Netw Neurosci 1(2):69–99Google Scholar
  83. Tiphaine V, Raphaël FS (2018) Movie rating prediction using content-based and link stream features. CoRR, abs/1805.02893Google Scholar
  84. Uddin S, Hossain L, Wigand RT (2014) New direction in degree centrality measure: towards a time-variant approach. Int J Inform Technol Decis Making 13(4):865Google Scholar
  85. Uddin S, Khan A, Piraveenan M (2016) A set of measures to quantify the dynamicity of longitudinal social networks. Complexity 21(6):309–320MathSciNetGoogle Scholar
  86. Uddin MS, Mahendra P, Chung KSK, Hossain L (2013) Topological analysis of longitudinal networks. In: 46th Hawaii International Conference on System Sciences, HICSS 2013, Wailea, HI, USA, 7–10 January, 2013. pp 3931–3940Google Scholar
  87. Viard T, Latapy M, Magnien C (2016) Computing maximal cliques in link streams. Theor Comput Sci 609:245–252MathSciNetzbMATHGoogle Scholar
  88. Viard T, Fournier-S’niehotta R, Magnien C, Latapy M (2018) Discovering patterns of interest in IP traffic using cliques in bipartite link streams. In: Cornelius S, Coronges K, Gonçalves B, Sinatra R, Vespignani A (eds) Complex networks IX. CompleNet 2018. Springer proceedings in complexity. Springer, ChamGoogle Scholar
  89. Viard T, Latapy M (2014) Identifying roles in an IP network with temporal and structural density. In: 2014 Proceedings IEEE INFOCOM Workshops, IEEE, Toronto, ON, Canada, April 27–May 2, 2014, pp 801–806Google Scholar
  90. Viard T, Latapy M, Magnien Cl (2015) Revealing contact patterns among high-school students using maximal cliques in link streams. In: ASONAM '15 Proceedings of the 2015 IEEE/ACM international conference on advances in social networks analysis and mining 2015. ACM, New York, pp 1517–1522Google Scholar
  91. Wasserman S, Faust K (1994) Social network analysis: Methods and applications, vol 8. Cambridge University Press, CambridgezbMATHGoogle Scholar
  92. Wehmuth K, Fleury É, Ziviani A (2016) Multiaspect graphs: algebraic representation and algorithms. Algorithms 10(4):1MathSciNetzbMATHGoogle Scholar
  93. Wehmuth K, Fleury É, Ziviani A (2016) On multiaspect graphs. Theor Comput Sci 651:50–61MathSciNetzbMATHGoogle Scholar
  94. Wehmuth K, Ziviani A, Fleury E (2015) A unifying model for representing time-varying graphs. In: 2015 IEEE international conference on data science and advanced analytics, DSAA 2015, Campus des Cordeliers, IEEE, Paris, France, October 19–21, 2015, pp 1–10Google Scholar
  95. West DB (2000) Introduction to graph theory, 2nd edn. Prentice Hall, PrenticeGoogle Scholar
  96. Whitbeck J, de Amorim MD, Conan V, Guillaume JL (2012) Temporal reachability graphs. In: Özgür BA, Eylem E, Lili Q, Alex CS (eds) The 18th annual international conference on mobile computing and networking, Mobicom’12, ACM, Istanbul, Turkey, Aug 22–26, 2012, pp 377–388Google Scholar
  97. Wilmet A, Viard T, Latapy M, Lamarche-PR (2018) Degree-based outliers detection within ip traffic modelled as a link stream. In: Proceedings of the 2nd network traffic measurement and analysis conference (TMA)Google Scholar
  98. Xu KS, Hero AO (2013) Dynamic stochastic blockmodels: statistical models for time-evolving networks. Springer, Berlin, Heidelberg, pp 201–210Google Scholar
  99. Zhao K, Karsai M, Bianconi G (2013) Models, entropy and information of temporal social networks. Springer, Berlin, Heidelberg, pp 95–117Google Scholar
  100. Zignani M, Gaito S, Rossi GP, Zhao X, Zheng H, Zhao BY (2014) Link and triadic closure delay: temporal metrics for social network dynamics. In Eytan A, Paul R, Munmun De C, Bernie H, Alice HH (eds) Proceedings of the eighth international conference on weblogs and social media, ICWSM 2014, The AAAI Press, Ann Arbor, Michigan, USA, June 1–4, 2014Google Scholar

Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Sorbonne Universités, UPMC Univ Paris 06, CNRS UMR 7606, LIP6ParisFrance

Personalised recommendations