Skip to main content
SpringerLink
Log in

An Introduction to Temporal Graphs: An Algorithmic Perspective

  • Chapter
  • First Online:
Book cover Algorithms, Probability, Networks, and Games

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9295))

Abstract

A temporal graph is, informally speaking, a graph that changes with time. When time is discrete and only the relationships between the participating entities may change and not the entities themselves, a temporal graph may be viewed as a sequence \(G_1,G_2\ldots ,G_l\) of static graphs over the same (static) set of nodes V. Though static graphs have been extensively studied, for their temporal generalization we are still far from having a concrete set of structural and algorithmic principles. Recent research shows that many graph properties and problems become radically different and usually substantially more difficult when an extra time dimension in added to them. Moreover, there is already a rich and rapidly growing set of modern systems and applications that can be naturally modeled and studied via temporal graphs. This, further motivates the need for the development of a temporal extension of graph theory. We survey here recent results on temporal graphs and temporal graph problems that have appeared in the Computer Science community.

Supported in part by the project “Foundations of Dynamic Distributed Computing Systems” (FOCUS) which is implemented under the “ARISTEIA” Action of the Operational Programme “Education and Lifelong Learning” and is co-funded by the European Union (European Social Fund) and Greek National Resources.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    In this article, we use “static” to refer to classical graphs. This is plausible as the opposite of “dynamic” that is also commonly used for temporal graphs. In any case, the terminology is still very far from being standard.

  2. 2.

    The sink is usually denoted by t in the literature. We use z instead as we reserve t to refer to time moments.

References

  1. Aaron, E., Krizanc, D., Meyerson, E.: DMVP: foremost waypoint coverage of time-varying graphs. In: Kratsch, D., Todinca, I. (eds.) WG 2014. LNCS, vol. 8747, pp. 29–41. Springer, Heidelberg (2014)

    Google Scholar 

  2. Akrida, E.C., Gasieniec, L., Mertzios, G.B., Spirakis, P.G.: Ephemeral networks with random availability of links: Diameter and connectivity. In: Proceedings of the 26th ACM symposium on Parallelism in algorithms and architectures (SPAA), pp. 267–276. ACM (2014)

    Google Scholar 

  3. Angluin, D., Aspnes, J., Diamadi, Z., Fischer, M.J., Peralta, R.: Computation in networks of passively mobile finite-state sensors. Distrib. Comput. 18, 235–253 (2006)

    Article  MATH  Google Scholar 

  4. Asadpour, A., Goemans, M.X., Madry, A., Gharan, S.O., Saberi, A.: An \(O(\log n/ \log \log n)\)-approximation algorithm for the asymmetric traveling salesman problem. In: Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 379–389. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA (2010). http://dl.acm.org/citation.cfm?id=1873601.1873633

  5. Augustine, J., Pandurangan, G., Robinson, P., Upfal, E.: Towards robust and efficient computation in dynamic peer-to-peer networks. In: Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 551–569. SIAM (2012)

    Google Scholar 

  6. Avin, C., Koucký, M., Lotker, Z.: How to explore a fast-changing world (cover time of a simple random walk on evolving graphs). In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part I. LNCS, vol. 5125, pp. 121–132. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  7. Bach, E., Shallit, J.: Algorithmic Number Theory. Efficient algorithms, vol. 1. MIT press, Cambridge (1996)

    MATH  Google Scholar 

  8. Baker, B., Shostak, R.: Gossips and telephones. Discrete Math. 2(3), 191–193 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  9. Berman, K.A.: Vulnerability of scheduled networks and a generalization of Menger’s theorem. Networks 28(3), 125–134 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bhadra, S., Ferreira, A.: Complexity of connected components in evolving graphs and the computation of multicast trees in dynamic networks. In: Pierre, S., Barbeau, M., An, H.-C. (eds.) ADHOC-NOW 2003. LNCS, vol. 2865, pp. 259–270. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  11. Bläser, M.: A 3/4-approximation algorithm for maximum ATSP with weights zero and one. In: Jansen, K., Khanna, S., Rolim, J.D.P., Ron, D. (eds.) RANDOM 2004 and APPROX 2004. LNCS, vol. 3122, pp. 61–71. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  12. Bollobás, B.: Modern Graph Theory. Graduate Texts in Mathematics. Springer, Heidelberg (1998). (Corrected edition, July 1, 1998)

    Book  MATH  Google Scholar 

  13. Bollobás, B.: Random Graphs. Cambridge Studies in Advanced Mathematics, 2nd edn. Cambridge University Press, Cambridge (2001)

    Book  MATH  Google Scholar 

  14. Broersma, H., Li, X.: Spanning trees with many or few colors in edge-colored graphs. Discuss. Math. Graph Theory 17(2), 259–269 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  15. Broersma, H., Li, X., Woeginger, G., Zhang, S.: Paths and cycles in colored graphs. Australas. J. Comb. 31, 299–311 (2005)

    MathSciNet  MATH  Google Scholar 

  16. Casteigts, A., Flocchini, P., Quattrociocchi, W., Santoro, N.: Time-varying graphs and dynamic networks. Int. J. Parallel Emerg. Distrib. Syst. 27(5), 387–408 (2012)

    Article  Google Scholar 

  17. Clementi, A.E., Macci, C., Monti, A., Pasquale, F., Silvestri, R.: Flooding time in edge-markovian dynamic graphs. In: Proceedings of the 27th ACM Symposium on Principles of Distributed Computing (PODC), pp. 213–222 (2008). http://doi.acm.org/10.1145/1400751.1400781

  18. Clementi, A.E., Pasquale, F., Monti, A., Silvestri, R.: Communication in dynamic radio networks. In: Proceedings of the Twenty-Sixth Annual ACM Symposium on Principles of Distributed Computing (PODC), pp. 205–214. ACM (2007)

    Google Scholar 

  19. Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms, 2nd edn. The MIT Press and McGraw-Hill Book Company, Cambridge (2001)

    MATH  Google Scholar 

  20. Demers, A., Greene, D., Hauser, C., Irish, W., Larson, J., Shenker, S., Sturgis, H., Swinehart, D., Terry, D.: Epidemic algorithms for replicated database maintenance. In: Proceedings of the Sixth Annual ACM Symposium on Principles of Distributed Computing (PODC), pp. 1–12. ACM (1987)

    Google Scholar 

  21. Dutta, C., Pandurangan, G., Rajaraman, R., Sun, Z., Viola, E.: On the complexity of information spreading in dynamic networks. In: Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 717–736. SIAM (2013)

    Google Scholar 

  22. Edmonds, J.: Paths, trees, and flowers. Can. J. Math. 17(3), 449–467 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  23. Erlebach, T., Hoffmann, M., Kammer, F.: On temporal graph exploration. In: Halldórsson, M.M., Iwama, K., Kobayashi, N., Speckmann, B. (eds.) ICALP 2015. LNCS, vol. 9134, pp. 444–455. Springer, Heidelberg (2015)

    Chapter  Google Scholar 

  24. Ferreira, A.: Building a reference combinatorial model for manets. IEEE Netw. 18(5), 24–29 (2004)

    Article  Google Scholar 

  25. Fleischer, L., Tardos, É.: Efficient continuous-time dynamic network flow algorithms. Oper. Res. Lett. 23(3), 71–80 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  26. Flocchini, P., Mans, B., Santoro, N.: Exploration of periodically varying graphs. In: Dong, Y., Du, D.-Z., Ibarra, O. (eds.) ISAAC 2009. LNCS, vol. 5878, pp. 534–543. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  27. Gharan, S.O., Saberi, A., Singh, M.: A randomized rounding approach to the traveling salesman problem. In: Proceedings of the IEEE 52nd Annual Symposium on Foundations of Computer Science (FOCS), pp. 550–559. IEEE Computer Society, Washington, DC (2011). http://dx.doi.org/10.1109/FOCS.2011.80

  28. Haeupler, B., Kuhn, F.: Lower bounds on information dissemination in dynamic networks. In: Aguilera, M.K. (ed.) DISC 2012. LNCS, vol. 7611, pp. 166–180. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  29. Halldórsson, M.M.: Approximating discrete collections via local improvements. In: Proceedings of the Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 160–169. Society for Industrial and Applied Mathematics (1995)

    Google Scholar 

  30. Harary, F., Gupta, G.: Dynamic graph models. Math. Comput. Model. 25(7), 79–87 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  31. Hedetniemi, S.M., Hedetniemi, S.T., Liestman, A.L.: A survey of gossiping and broadcasting in communication networks. Networks 18(4), 319–349 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  32. Holme, P., Saramäki, J.: Temporal networks. Phys. Rep. 519(3), 97–125 (2012)

    Article  Google Scholar 

  33. Karp, R., Schindelhauer, C., Shenker, S., Vocking, B.: Randomized rumor spreading. In: Proceedings of the IEEE 41st Annual Symposium on Foundations of Computer Science (FOCS), pp. 565–574. IEEE (2000)

    Google Scholar 

  34. Karpinski, M., Schmied, R.: On improved inapproximability results for the shortest superstring and related problems. In: Proceedings of 19th CATS, pp. 27–36 (2013)

    Google Scholar 

  35. Kempe, D., Kleinberg, J.: Protocols and impossibility results for gossip-based communication mechanisms. In: Proceedings of the IEEE 43rd Annual Symposium on Foundations of Computer Science (FOCS), pp. 471–480. IEEE (2002)

    Google Scholar 

  36. Kempe, D., Kleinberg, J., Kumar, A.: Connectivity and inference problems for temporal networks. In: Proceedings of the 32nd annual ACM symposium on Theory of computing (STOC), pp. 504–513 (2000). http://doi.acm.org/10.1145/335305.335364

  37. Kontogiannis, S., Michalopoulos, G., Papastavrou, G., Paraskevopoulos, A., Wagner, D., Zaroliagis, C.: Analysis and experimental evaluation of time-dependent distance oracles. In: Proceedings of the Seventeenth Workshop on Algorithm Engineering and Experiments (ALENEX), pp. 147–158 (2015)

    Google Scholar 

  38. Kontogiannis, S., Zaroliagis, C.: Distance oracles for time-dependent networks. In: Esparza, J., Fraigniaud, P., Husfeldt, T., Koutsoupias, E. (eds.) ICALP 2014. LNCS, vol. 8572, pp. 713–725. Springer, Heidelberg (2014)

    Google Scholar 

  39. Kostakos, V.: Temporal graphs. Phys. A Stat. Mech. Appl. 388(6), 1007–1023 (2009)

    Article  MathSciNet  Google Scholar 

  40. Krumke, S.O., Wirth, H.C.: On the minimum label spanning tree problem. Inf. Process. Lett. 66(2), 81–85 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  41. Kuhn, F., Lynch, N., Oshman, R.: Distributed computation in dynamic networks. In: Proceedings of the 42nd ACM symposium on Theory of Computing (STOC), pp. 513–522. ACM, New York (2010). http://doi.acm.org/10.1145/1806689.1806760

  42. Kuhn, F., Oshman, R.: Dynamic networks: models and algorithms. SIGACT News 42, 82–96 (2011). http://doi.acm.org/10.1145/1959045.1959064 (Distributed Computing Column, Editor: Idit Keidar)

  43. Leighton, F.T.: Introduction to Parallel Algorithms and Architectures, vol. 188. Morgan Kaufmann, San Francisco (1992)

    MATH  Google Scholar 

  44. Mans, B., Mathieson, L.: On the treewidth of dynamic graphs. In: Du, D.-Z., Zhang, G. (eds.) COCOON 2013. LNCS, vol. 7936, pp. 349–360. Springer, Heidelberg (2013)

    Chapter  Google Scholar 

  45. Menger, K.: Zur allgemeinen kurventheorie. Fundamenta Mathematicae 10(1), 96–115 (1927)

    MATH  Google Scholar 

  46. Mertzios, G.B., Michail, O., Chatzigiannakis, I., Spirakis, P.G.: Temporal network optimization subject to connectivity constraints. In: Fomin, F.V., Freivalds, R., Kwiatkowska, M., Peleg, D. (eds.) ICALP 2013, Part II. LNCS, vol. 7966, pp. 657–668. Springer, Heidelberg (2013)

    Chapter  Google Scholar 

  47. Mertzios, G.B., Michail, O., Spirakis, P.G.: Temporal network optimization subject to connectivity constraints. CoRR abs/1502.04382 (2015), full version of [MMCS13]

    Google Scholar 

  48. Micali, S., Vazirani, V.V.: An \({O}(\sqrt{|{V}|}\cdot |{E}|)\) algorithm for finding maximum matching in general graphs. In: Proceedings of the IEEE 21st Annual Symposium on Foundations of Computer Science (FOCS), pp. 17–27. IEEE (1980)

    Google Scholar 

  49. Michail, O.: Terminating distributed construction of shapes and patterns in a fair solution of automata. In: Proceedings of the 34th ACM Symposium on Principles of Distributed Computing (PODC) (2015) (to appear)

    Google Scholar 

  50. Michail, O., Chatzigiannakis, I., Spirakis, P.G.: Mediated population protocols. Theor. Comput. Sci. 412(22), 2434–2450 (2011). http://dx.doi.org/10.1016/j.tcs.2011.02.003

    Article  MathSciNet  MATH  Google Scholar 

  51. Michail, O., Chatzigiannakis, I., Spirakis, P.G.: New Models for Population Protocols. In: ynch, N.A. (ed.) Synthesis Lectures on Distributed Computing Theory. Morgan and Claypool (2011)

    Google Scholar 

  52. Michail, O., Chatzigiannakis, I., Spirakis, P.G.: Naming and counting in anonymous unknown dynamic networks. In: Higashino, T., Katayama, Y., Masuzawa, T., Potop-Butucaru, M., Yamashita, M. (eds.) SSS 2013. LNCS, vol. 8255, pp. 281–295. Springer, Heidelberg (2013)

    Chapter  Google Scholar 

  53. Michail, O., Chatzigiannakis, I., Spirakis, P.G.: Causality, influence, and computation in possibly disconnected synchronous dynamic networks. J. Parallel Distrib. Comput. 74(1), 2016–2026 (2014)

    Article  MATH  Google Scholar 

  54. Michail, O., Spirakis, P.G.: Unpublished work on random temporal graphs (2012)

    Google Scholar 

  55. Michail, O., Spirakis, P.G.: Simple and efficient local codes for distributed stable network construction. In: Proceedings of the 33rd ACM Symposium on Principles of Distributed Computing (PODC), pp. 76–85. ACM (2014). http://doi.acm.org/10.1145/2611462.2611466

  56. Michail, O., Spirakis, P.G.: Traveling salesman problems in temporal graphs. In: Csuhaj-Varjú, E., Dietzfelbinger, M., Ésik, Z. (eds.) MFCS 2014, Part II. LNCS, vol. 8635, pp. 553–564. Springer, Heidelberg (2014)

    Google Scholar 

  57. Molloy, M., Reed, B.: Graph Colouring and the Probabilistic Method, vol. 23. Springer, Heidelberg (2002)

    MATH  Google Scholar 

  58. Monnot, J.: The labeled perfect matching in bipartite graphs. Inf. Process. Lett. 96(3), 81–88 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  59. O’Dell, R., Wattenhofer, R.: Information dissemination in highly dynamic graphs. In: Proceedings of the 2005 Joint Workshop on Foundations of Mobile Computing (DIALM-POMC), pp. 104–110 (2005). http://doi.acm.org/10.1145/1080810.1080828

  60. Orlin, J.B.: The complexity of dynamic languages and dynamic optimization problems. In: Proceedings of the 13th Annual ACM Symposium on Theory of Computing (STOC), pp. 218–227. ACM (1981)

    Google Scholar 

  61. Papadimitriou, C.H., Yannakakis, M.: The traveling salesman problem with distances one and two. Math. Oper. Res. 18(1), 1–11 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  62. Peleg, D.: Distributed computing: a locality-sensitive approach. SIAM Monographs on Discrete Mathematics and Applications, p. 5 (2000)

    Google Scholar 

  63. Pittel, B.: On spreading a rumor. SIAM J. Appl. Math. 47(1), 213–223 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  64. Ravi, R.: Rapid rumor ramification: approximating the minimum broadcast time. In: Proceedings of the IEEE 35th Annual Symposium on Foundations of Computer Science (FOCS), pp. 202–213. IEEE (1994)

    Google Scholar 

  65. Scheideler, C.: Models and techniques for communication in dynamic networks. In: Alt, H., Ferreira, A. (eds.) STACS 2002. LNCS, vol. 2285, p. 27. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  66. Tanimoto, S.L., Itai, A., Rodeh, M.: Some matching problems for bipartite graphs. J. ACM 25(4), 517–525 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  67. Xuan, B., Ferreira, A., Jarry, A.: Computing shortest, fastest, and foremost journeys in dynamic networks. Int. J. Found. Comput. Sci. 14(02), 267–285 (2003)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Computer Technology Institute & Press “Diophantus” (CTI), N. Kazantzaki Str., Patras University Campus, Rio, P.O. Box 1382, 26504, Patras, Greece

    Othon Michail

Authors
  1. Othon Michail
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Othon Michail .

Editor information

Editors and Affiliations

  1. University of Patras, Patras, Greece

    Christos Zaroliagis

  2. Technological Educational Institute of Athens, Athens, Greece

    Grammati Pantziou

  3. University of Ioannina, Ioannina, Greece

    Spyros Kontogiannis

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Michail, O. (2015). An Introduction to Temporal Graphs: An Algorithmic Perspective. In: Zaroliagis, C., Pantziou, G., Kontogiannis, S. (eds) Algorithms, Probability, Networks, and Games. Lecture Notes in Computer Science(), vol 9295. Springer, Cham. https://doi.org/10.1007/978-3-319-24024-4_18

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-24024-4_18

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-24023-7

  • Online ISBN: 978-3-319-24024-4

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation