Abstract
In the layered-graph query model of network discovery, a query at a node v of an undirected graph G discovers all edges and non-edges whose endpoints have different distance from v. We study the number of queries at randomly selected nodes that are needed for approximate network discovery in Erdős-Rényi random graphs G n,p. We show that a constant number of queries is sufficient if p is a constant, while Ω(n α) queries are needed if p = n ε/n, for arbitrarily small choices of ε = 3 / (6 ·i + 5) with i ∈ ℕ. Note that α> 0 is a constant depending only on ε. Our proof of the latter result yields also a somewhat surprising result on pairwise distances in random graphs which may be of independent interest: We show that for a random graph G n,p with p = n ε/n, for arbitrarily small choices of ε> 0 as above, in any constant cardinality subset of the nodes the pairwise distances are all identical with high probability.
Work partially supported by European Commission - Fet Open project DELIS IST-001907 Dynamically Evolving Large Scale Information Systems, for which funding in Switzerland is provided by SBF grant 03.0378-1.
Access provided by Autonomous University of Puebla. Download to read the full chapter text
Chapter PDF
Similar content being viewed by others
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
References
DIMES: Mapping the Internet with the help of a volunteer community (2004), http://www.netdimes.org
Oregon RouteViews: University of Oregon RouteViews project (1997), http://www.routeviews.org
Beerliová, Z., Eberhard, F., Erlebach, T., Hall, A., Hoffmann, M., Mihal’ák, M., Ram, L.S.: Network discovery and verification. In: Kratsch, D. (ed.) WG 2005. LNCS, vol. 3787, pp. 127–138. Springer, Heidelberg (2005)
Barrat, A., Hall, A., Mihal’ák, M.: Network discovery on snapshots of the Internet graph. Technical Report DELIS-TR-465, DELIS – Dynamically Evolving, Large-Scale Information Systems (2006)
Beerliová, Z., Eberhard, F., Erlebach, T., Hall, A., Hoffmann, M., Mihal’ák, M., Ram, L.S.: Network discovery and verification. IEEE Journal on Selected Areas in Communications 24(12), 2168–2181 (2006)
Barabási, A.L., Albert, R.: Emergence of scaling in random networks. Science 286, 509–512 (1999)
Cheswick, B., Burch, H.: Internet mapping project (1998), http://www.cs.bell-labs.com/who/ches/map/
Govindan, R., Reddy, A.: An analysis of Internet inter-domain topology and route stability. In: Proc. IEEE INFOCOM, April 1997, pp. 850–857. IEEE Computer Society Press, Los Alamitos (1997)
Govindan, R., Tangmunarunkit, H.: Heuristics for Internet map discovery. In: Proc. IEEE INFOCOM, March 2000, pp. 1371–1380. IEEE Computer Society Press, Los Alamitos (2000)
Gao, L.: On inferring autonomous system relationships in the Internet. IEEE/ACM Trans. Networking 9(6), 733–745 (2001)
Barford, P., Bestavros, A., Byers, J., Crovella, M.: On the marginal utility of deploying measurement infrastructure. In: Proc. ACM SIGCOMM Internet Measurement Workshop, November 2001, ACM Press, New York (2001)
Subramanian, L., Agarwal, S., Rexford, J., Katz, R.: Characterizing the Internet hierarchy from multiple vantage points. In: Proc. IEEE INFOCOM, IEEE Computer Society Press, Los Alamitos (2002)
Di Battista, G., Erlebach, T., Hall, A., Patrignani, M., Pizzonia, M., Schank, T.: Computing the types of the relationships between autonomous systems. IEEE/ACM Transactions on Networking 15(2), 267–280 (2007)
Achlioptas, D., Clauset, A., Kempe, D., Moore, C.: On the bias of traceroute sampling; or, power-law degree distributions in regular graphs. In: Proceedings of the 37th Annual ACM Symposium on Theory of Computing (STOC 2005), pp. 694–703. ACM Press, New York (2005)
Dall’Asta, L., Alvarez-Hamelin, I., Barrat, A., Vázquez, A., Vespignani, A.: Statistical theory of Internet exploration. Phys. Rev. E 71 (2005)
Dall’Asta, L., Alvarez-Hamelin, I., Barrat, A., Vázquez, A., Vespignani, A.: Exploring networks with traceroute-like probes: theory and simulations. Theoret. Comput. Sci. 355(1), 6–24 (2006)
Khuller, S., Raghavachari, B., Rosenfeld, A.: Landmarks in graphs. Discrete Appl. Math. 70, 217–229 (1996)
Erlebach, T., Hall, A., Hoffmann, M., Mihal’ák, M.: Network discovery and verification with distance queries. In: Calamoneri, T., Finocchi, I., Italiano, G.F. (eds.) CIAC 2006. LNCS, vol. 3998, pp. 69–80. Springer, Heidelberg (2006)
Harary, F., Melter, R.A.: The metric dimension of a graph. Ars Combin., 191–195 (1976)
Chartrand, G., Zhang, P.: The theory and applications of resolvability in graphs: A survey. Congr. Numer. 160, 47–68 (2003)
Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, San Francisco (1979)
Sebő, A., Tannier, E.: On metric generators of graphs. Math. Oper. Res. 29(2), 383–393 (2004)
Lindström, B.: On a combinatory detection problem I. Magyar Tud. Akad. Mat. Kutató Int. Közl. 9, 195–207 (1964)
Cáceres, J., Hernando, C., Mora, M., Pelayo, I.M., Puertas, M.L., Seara, C., Wood, D.R.: On the metric dimension of cartesian products of graphs. SIAM J. Discrete Math. 21(2), 423–441 (2007)
Chung, F., Lu, L.: The diameter of random sparse graphs. Adv. in Appl. Math. 26, 257–279 (2001)
Bollobás, B.: Random graphs. Academic Press, New York (1985)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Erlebach, T., Hall, A., Mihal’ák, M. (2007). Approximate Discovery of Random Graphs. In: Hromkovič, J., Královič, R., Nunkesser, M., Widmayer, P. (eds) Stochastic Algorithms: Foundations and Applications. SAGA 2007. Lecture Notes in Computer Science, vol 4665. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74871-7_8
Download citation
DOI: https://doi.org/10.1007/978-3-540-74871-7_8
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-74870-0
Online ISBN: 978-3-540-74871-7
eBook Packages: Computer ScienceComputer Science (R0)