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Estimating network parameters using random walks

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Abstract

Sampling from large graphs is an area of great interest, especially since the emergence of huge structures such as Online Social Networks and the World Wide Web (WWW). The large scale properties of a network can be summarized in terms of parameters of the underlying graph, such as the total number of vertices, edges and triangles. However, the large si ze of these networks makes it computationally expensive to obtain such structural properties of the underlying graph by exhaustive search. If we can estimate these properties by taking small but representative samples from the network, then size is no longer such a problem. In this paper we present a general framework to estimate network properties using random walks. These methods work under the assumption we are able to obtain local characteristics of a vertex during each step of the random walk, for example the number of its neighbours, and their labels. As examples of this approach, we present practical methods to estimate the total number of edges/links m, number of vertices/nodes n and number of connected triads of vertices (triangles) t in graphs. We also give a general method to count any type of small connected subgraph, of which vertices, edges and triangles are specific examples. Additionally we present experimental estimates for n, m, t we obtained using our methods on real or synthetic networks. The synthetic networks were random graphs with power-law degree distributions and designed to have a large number of triangles. We used these graphs as they tend to correspond to the structure of large online networks. The real networks were samples of the WWW and social networks obtained from the SNAP database. In order to test that the methods are indeed practical, the total number of steps made by the walk was limited to at most the size n of the network. In fact the estimates appear to converge to the correct value at a lower number of steps, indicating that our proposed methods are feasible in practice.

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References

  • Aldous D, Fill JA (1995) Reversible markov chains and random walks on graphs. http://stat-www.berkeley.edu/pub/users/aldous/RWG/book.html

  • Barabási A, Albert R (1999) Emergence of scaling in random networks. Science 286(5439):509–512

    Google Scholar 

  • Bawa M, Garcia-Molina H, Gionis A, Motwani R (2003) Estimating aggregates on a peer-to-peer network. Technical Report 2003-24, Stanford InfoLab. http://ilpubs.stanford.edu:8090/586/

  • Bollobás B, Riordan O (2002) Mathematical results on scale-free graphs. In: Bornholdt S, Schuster H (eds) Handbook of graphs and networks. Wiley-VCH, Weinheim, pp 1–32

  • Bollobás B, Riordan O (2004) The diameter of a scale-free random graph. Combinatorica 24:5–34. http://dl.acm.org/citation.cfm?id=1005121.1005123

    Google Scholar 

  • Bollobás B, Riordan O, Spencer J, Tusnády G (2001) The degree sequence of a scale-free random graph process. Random Struct Algorithms 18:279–290

    Article  MATH  Google Scholar 

  • Broder AZ, Kumar R, Maghoul F, Raghavan P, Rajagopalan S, Stata R, Tomkins A, Wiener JL (2000) Graph structure in the web. Comput Netw 33(1–6):309–320

    Article  Google Scholar 

  • Cheeger J (1971) A lower bound for the smallest eigenvalue of the laplacian. Probl Anal 195–199 (papers dedicated to Salomon Bochner, 1969)

  • Cooper C (2002) Classifying special interest groups in web graphs. In: RANDOM 2002: Randomization and Approximation Techniques in Computer Science, pp 263–275

  • Cooper C (2006) The age specific degree distribution of web-graphs. Comb Probab Comput 15:637–661

    Article  MATH  Google Scholar 

  • Cooper C, Frieze A (2003) A general model web graphs. Random Struct Algorithms 22(3):311–335

    Google Scholar 

  • Cooper C, Frieze A (2007) The cover time of the preferential attachment graphs. J Comb Theory B 97:269–290

    Article  MathSciNet  MATH  Google Scholar 

  • Cooper C, Frieze AM (2008) Random walks on random graphs. In: Cheng MX (ed) NanoNet. Lecture notes of the institute for computer sciences, social informatics and telecommunications engineering, vol 3. Springer, Berlin, pp 95–106

  • Cooper C, Radzik T, Siantos Y (2012) Estimating network parameters using random walks. In: CASoN. IEEE, pp 33–40

  • Cooper C, Radzik T, Siantos Y (2012) A fast algorithm to find all high degree vertices in graphs with a power law degree sequence. In: WAW 2012 Proceedings. LNCS, vol 7323. Springer, Berlin, pp 165–178

  • Drinea E, Enachescu M, Mitzenmacher M (2001) Technical report: variations on random graph models for the web. Tech. rep., Harvard University, Department of Computer Science

  • Ganesh A, Kermarrec A, Le Merrer E, Massouli L (2007) Peer counting and sampling in overlay networks based on random walks. Distrib Comput 20:267–278. doi:10.1007/s00446-007-0027-z

    Google Scholar 

  • Gjoka M, Kurant M, Butts CT, Markopoulou A (2009) A walk in facebook: uniform sampling of users in online social networks. CoRR. abs/0906.0060

  • Hastings WK (1970) Monte carlo sampling methods using markov chains and their applications. Biometrika 57(1):97–109

    Article  MATH  Google Scholar 

  • Katzir L, Liberty E, Somekh O (2011) Estimating sizes of social networks via biased sampling. In: Srinivasan S, Ramamritham K, Kumar A, Ravindra MP, Bertino V, Kumar R (eds.) WWW. ACM, pp 597–606

  • Leskovec J (2009) Stanford network analysis package. http://snap.stanford.edu/

  • Leskovec J, Backstrom L, Kumar R, Tomkins A (2008) Microscopic evolution of social networks. In: Proceeding of the 14th ACM SIGKDD international conference on knowledge discovery and data mining, KDD’08, ACM, New York, NY, USA, pp 462–470. doi:10.1145/1401890.1401948

  • Leskovec J, Lang KJ, Dasgupta A, Mahoney MW (2008) Community structure in large networks: Natural cluster sizes and the absence of large well-defined clusters. CoRR. abs/0810.1355

  • Lovász L (1996) Random walks on graphs: a survey. Bolyai Soc Math Stud 2:353–397

    Google Scholar 

  • Massoulié L, Merrer EL, Kermarrec AM, Ganesh AJ (2006) Peer counting and sampling in overlay networks: random walk methods. In: Ruppert E, Malkhi D (eds.) PODC. ACM, pp 123–132

  • Mckinney EH (1966) Generalized birthday problem. Am Math Mon 73(4):385–387. http://www.jstor.org/stable/2315408

    Google Scholar 

  • Metropolis N, Rosenbluth AW, Rosenbluth MN, Teller AH, Teller E (1953) Equation of state calculations by fast computing machines. J Chem Phys 21(6):1087–1092

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Informatics, King’s College, London, UK

    Colin Cooper, Tomasz Radzik & Yiannis Siantos

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  1. Colin Cooper
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  2. Tomasz Radzik
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  3. Yiannis Siantos
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Correspondence to Yiannis Siantos.

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Cooper, C., Radzik, T. & Siantos, Y. Estimating network parameters using random walks. Soc. Netw. Anal. Min. 4, 168 (2014). https://doi.org/10.1007/s13278-014-0168-6

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  • DOI: https://doi.org/10.1007/s13278-014-0168-6

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