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How Fast Can We Reach a Target Vertex in Stochastic Temporal Graphs?

Authors Eleni C. Akrida , George B. Mertzios , Sotiris Nikoletseas, Christoforos Raptopoulos , Paul G. Spirakis , Viktor Zamaraev

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Author Details

Eleni C. Akrida
  • Department of Computer Science, University of Liverpool, UK
George B. Mertzios
  • Department of Computer Science, Durham University, UK
Sotiris Nikoletseas
  • Computer Engineering & Informatics Department, University of Patras, and CTI, Greece
Christoforos Raptopoulos
  • Computer Engineering & Informatics Department, University of Patras, and CTI, Greece
Paul G. Spirakis
  • Department of Computer Science, University of Liverpool, UK
  • Computer Engineering & Informatics Department, University of Patras, Greece
Viktor Zamaraev
  • Department of Computer Science, Durham University, UK

Funding

This work was supported by the NeST initiative of the EEE/CS School of the University of Liverpool and by the EPSRC grants EP/P020372/1 and EP/P02002X/1.

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Eleni C. Akrida, George B. Mertzios, Sotiris Nikoletseas, Christoforos Raptopoulos, Paul G. Spirakis, and Viktor Zamaraev. How Fast Can We Reach a Target Vertex in Stochastic Temporal Graphs?. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 131:1-131:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.ICALP.2019.131

Abstract

Temporal graphs are used to abstractly model real-life networks that are inherently dynamic in nature, in the sense that the network structure undergoes discrete changes over time. Given a static underlying graph G=(V,E), a temporal graph on G is a sequence of snapshots {G_t=(V,E_t) subseteq G: t in N}, one for each time step t >= 1. In this paper we study stochastic temporal graphs, i.e. stochastic processes G={G_t subseteq G: t in N} whose random variables are the snapshots of a temporal graph on G. A natural feature of stochastic temporal graphs which can be observed in various real-life scenarios is a memory effect in the appearance probabilities of particular edges; that is, the probability an edge e in E appears at time step t depends on its appearance (or absence) at the previous k steps. In this paper we study the hierarchy of models memory-k, k >= 0, which address this memory effect in an edge-centric network evolution: every edge of G has its own probability distribution for its appearance over time, independently of all other edges. Clearly, for every k >= 1, memory-(k-1) is a special case of memory-k. However, in this paper we make a clear distinction between the values k=0 ("no memory") and k >= 1 ("some memory"), as in some cases these models exhibit a fundamentally different computational behavior for these values of k, as our results indicate. For every k >= 0 we investigate the computational complexity of two naturally related, but fundamentally different, temporal path (or journey) problems: {Minimum Arrival} and {Best Policy}. In the first problem we are looking for the expected arrival time of a foremost journey between two designated vertices {s},{y}. In the second one we are looking for the expected arrival time of the best policy for actually choosing a particular {s}-{y} journey. We present a detailed investigation of the computational landscape of both problems for the different values of memory k. Among other results we prove that, surprisingly, {Minimum Arrival} is strictly harder than {Best Policy}; in fact, for k=0, {Minimum Arrival} is #P-hard while {Best Policy} is solvable in O(n^2) time.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph theory
  • Mathematics of computing → Graph algorithms
  • Mathematics of computing → Paths and connectivity problems
Keywords
  • Temporal network
  • stochastic temporal graph
  • temporal path
  • #P-hard problem
  • polynomial-time approximation scheme

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References

  1. E. Aaron, D. Krizanc, and E. Meyerson. DMVP: foremost waypoint coverage of time-varying graphs. In International Workshop on Graph-Theoretic Concepts in Computer Science (WG), pages 29-41, 2014. Google Scholar
  2. E. Akrida, G.B. Mertzios, S. Nikoletseas, C. Raptopoulos, P.G. Spirakis, and V. Zamaraev. How fast can we reach a target vertex in stochastic temporal graphs?, 2019. Technical Report available at URL: https://arxiv.org/abs/1903.03636.
  3. E. C. Akrida, L. Gasieniec, G. B. Mertzios, and P. G. Spirakis. Ephemeral networks with random availability of links: The case of fast networks. Journal of Parallel and Distributed Computing, 87:109-120, 2016. Google Scholar
  4. E. C. Akrida, L. Gasieniec, G. B. Mertzios, and P. G. Spirakis. The complexity of optimal design of temporally connected graphs. Theory of Computing Systems, 61(3):907-944, 2017. Google Scholar
  5. E.C. Akrida, G.B. Mertzios, P.G. Spirakis, and V. Zamaraev. Temporal vertex cover with a sliding time window. In Proceedings of the 45th International Colloquium on Automata, Languages, and Programming (ICALP), pages 148:1-148:14, 2018. Google Scholar
  6. A. Anagnostopoulos, J. Lacki, S. Lattanzi, S. Leonardi, and M. Mahdian. Community Detection on Evolving Graphs. In Advances in Neural Information Processing Systems 29: Annual Conference on Neural Information Processing Systems, pages 3522-3530, 2016. Google Scholar
  7. C. Avin, M. Koucký, and Z. Lotker. How to Explore a Fast-Changing World (Cover Time of a Simple Random Walk on Evolving Graphs). In International Colloquium on Automata, Languages and Programming, ICALP, pages 121-132, 2008. Google Scholar
  8. P. Basu, A. Bar-Noy, R. Ramanathan, and M. P. Johnson. Modeling and Analysis of Time-Varying Graphs. CoRR, abs/1012.0260, 2010. Google Scholar
  9. P. Basu, S. Guha, A. Swami, and D. Towsley. Percolation phenomena in networks under random dynamics. In International Conference on Communication Systems and Networks, COMSNETS, pages 1-10, 2012. Google Scholar
  10. P. Basu, F. Yu, A. Bar-Noy, and D. Rawitz. To Sample or To Smash? Estimating reachability in large time-varying graphs. In SIAM International Conference on Data Mining, pages 983-991, 2014. Google Scholar
  11. P. Basu, F. Yu, M. P. Johnson, and A. Bar-Noy. Low expected latency routing in dynamic networks. In IEEE International Conference on Mobile Ad Hoc and Sensor Systems, MASS, pages 267-271, 2014. Google Scholar
  12. A. Casteigts and P. Flocchini. Deterministic Algorithms in Dynamic Networks: Formal Models and Metrics. Technical report, Defence R&D Canada, April 2013. URL: https://hal.archives-ouvertes.fr/hal-00865762.
  13. A. Casteigts and P. Flocchini. Deterministic Algorithms in Dynamic Networks: Problems, Analysis, and Algorithmic Tools. Technical report, Defence R&D Canada, April 2013. URL: https://hal.archives-ouvertes.fr/hal-00865764.
  14. A. Casteigts, P. Flocchini, E. Godard, N. Santoro, and M. Yamashita. On the expressivity of time-varying graphs. Theoretical Computer Science, 590:27-37, 2015. Google Scholar
  15. A. Clementi, A. Monti, F. Pasquale, and R. Silvestri. Information Spreading in Stationary Markovian Evolving Graphs. IEEE Transactions on Parallel and Distributed Systems, 22(9):1425-1432, 2011. Google Scholar
  16. A.E.F. Clementi, C. Macci, A. Monti, F. Pasquale, and R. Silvestri. Flooding Time of Edge-Markovian Evolving Graphs. SIAM Journal on Discrete Mathematics (SIDMA), 24(4):1694-1712, 2010. Google Scholar
  17. R. Durrett. Probability: Theory and examples, 2011. Google Scholar
  18. J. Enright, K. Meeks, G.B. Mertzios, and V. Zamaraev. Deleting edges to restrict the size of an epidemic in temporal networks, 2018. Technical Report available at URL: https://arxiv.org/abs/1805.06836.
  19. T. Erlebach, M. Hoffmann, and F. Kammer. On Temporal Graph Exploration. In Proceedings of the 42nd International Colloquium on Automata, Languages, and Programming (ICALP), pages 444-455, 2015. Google Scholar
  20. A. Ferreira. Building a Reference Combinatorial Model for MANETs. IEEE Network, 18(5):24-29, 2004. Google Scholar
  21. P. Flocchini, B. Mans, and N. Santoro. On the exploration of time-varying networks. Theoretical Computer Science, 469:53-68, 2013. Google Scholar
  22. M. L. Fredman and R. E. Tarjan. Fibonacci heaps and their uses in improved network optimization algorithms. Journal of the ACM (JACM), 34(3):596-615, 1987. Google Scholar
  23. S. Henri, S. Shneer, and P. Thiran. On the Delays in Time-Varying Networks: Does Larger Service-Rate Variance Imply Larger Delays? In ACM International Symposium on Mobile Ad Hoc Networking and Computing, MobiHoc, pages 201-210, 2018. Google Scholar
  24. A.-S. Himmel, H. Molter, R. Niedermeier, and M. Sorge. Adapting the Bron-Kerbosch algorithm for enumerating maximal cliques in temporal graphs. Social Network Analysis and Mining, 7(1):35:1-35:16, 2017. Google Scholar
  25. P. Holme and J. Saramäki, editors. Temporal Networks. Springer, 2013. Google Scholar
  26. S. Janson. Tail bounds for sums of geometric and exponential variables. Statistics &Probability Letters, 135:1-6, 2018. Google Scholar
  27. D. Kempe, J. M. Kleinberg, and A. Kumar. Connectivity and inference problems for temporal networks. In ACM Symp. on Theory of Comp. (STOC), pages 504-513, 2000. Google Scholar
  28. I. Lamprou, R. Martin, and P. G. Spirakis. Cover Time in Edge-Uniform Stochastically-Evolving Graphs. Algorithms, 11(10):149, 2018. Google Scholar
  29. G.B. Mertzios, O. Michail, I. Chatzigiannakis, and P.G. Spirakis. Temporal network optimization subject to connectivity constraints. In International Colloquium on Automata, Languages and Programming (ICALP), Part II, pages 657-668, 2013. Google Scholar
  30. G.B. Mertzios, H. Molter, and V. Zamaraev. Sliding window temporal graph coloring. In Proceedings of the 33st AAAI Conference on Artificial Intelligence (AAAI), 2019. To appear. Google Scholar
  31. O. Michail and P.G. Spirakis. Elements of the Theory of Dynamic Networks. Communications of the ACM, 61(2):72-72, January 2018. Google Scholar
  32. P. Nain, D. Towsley, M. P. Johnson, P. Basu, A. Bar-Noy, and F. Yu. Computing Traversal Times on Dynamic Markovian Paths, 2013. Technical Report available at URL: http://arxiv.org/abs/1303.3660.
  33. A. Orda and R. Rom. Distributed Shortest-Path Protocols for Time-Dependent Networks. Distributed Computing, 10(1):49-62, 1996. Google Scholar
  34. B. Pittel. On Spreading a Rumor. SIAM Journal on Applied Mathematics, 47(1):213-223, 1987. Google Scholar
  35. J.S. Provan and M.O. Ball. The complexity of counting cuts and of computing the probability that a graph is connected. SIAM Journal on Computing, 12(4):777-788, 1983. Google Scholar
  36. N. Santoro. Computing in Time-Varying Networks. In International Symposium on Stabilization, Safety, and Security of Distributed Systems SSS, page 4, 2011. Google Scholar
  37. C. Scheideler. Models and Techniques for Communication in Dynamic Networks. In Annual Symposium on Theoretical Aspects of Computer Science (STACS), pages 27-49, 2002. Google Scholar
  38. T. Viard, M. Latapy, and C. Magnien. Computing maximal cliques in link streams. Theoretical Computer Science, 609:245-252, 2016. Google Scholar
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