Abstract
The preferential attachment graph is a random graph formed by adding a new vertex at each time step, with a single edge which points to a vertex selected at random with probability proportional to its degree. Every m steps the most recently added m vertices are contracted into a single vertex, so at time t there are roughly t/m vertices and exactly t edges. This process yields a graph which has been proposed as a simple model of the world wide web [BA99]. For any constant k, let Δ1 ≥ Δ2 ≥ ⋯ ≥ Δ k be the degrees of the k highest degree vertices. We show that at time t, for any function f with f(t)→ ∞ as t→ ∞, \(\frac{t^{1/2}}{f(t)} \leq \Delta_1 \leq t^{1/2}f(t),\) and for i = 2,..., k, \(\frac{t^{1/2}}{f(t)} \leq \Delta_i \leq \Delta_{i-1} -- \frac{t^{1/2}}{f(t)},\) with high probability (whp). We use this to show that at time t the largest k eigenvalues of the adjacency matrix of this graph have λ k = (1± o(1))Δ k 1/2 whp.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Albert, R., Barabási, A., Jeong, H.: Diameter of the world wide web. Nature 401, 103–131 (1999)
Aiello, W., Chung, F.R.K., Lu, L.: A random graph model for massive graphs. In: Proc. of the 32nd Annual ACM Symposium on the Theory of Computing, pp. 171–180 (2000)
Barabási, A., Albert, R.: Emergence of scaling in random networks. Science 286, 509–512 (1999)
Broder, A., Kumar, R., Maghoul, F., Raghavan, P., Rajagopalan, S., Stata, R., Tomkins, A., Wiener, J.: Graph structure in the web. In: Proc. of the 9th Intl. World Wide Web Conference, pp. 309–320 (2000)
Buckley, G., Osthus, D.: Popularity based random graph models leading to a scale-free degree distribution (2001)
Bollobás, B., Riordan, O.: The diameter of a scale-free random graph (to appear)
Bollobás, B., Riordan, O.: Mathematical results on scale-free random graphs. In: Handbook of Graphs and Networks, Wiley-VCH, Berlin (2002)
Bollobás, B., Riordan, O., Spencer, J., Tusanády, G.: The degree sequence of a scale-free random graph process. Random Structures and Algorithms 18, 279–290 (2001)
Cooper, C., Frieze, A.M.: A general model of undirected Web graphs. In: Proc. of ESA, pp. 500–511 (2001)
Chung, F.R.K., Lu, L., Vu, V.: Eigenvalues of random power law graphs (to appear)
Drinea, E., Enachescu, M., Mitzenmacher, M.: Variations on random graph models for the web. Technical report, Harvard University (2001)
Erdös, P., Rényi, A.: On random graphs I. Publicationes Mathematicae Debrecen 6, 290–297 (1959)
Faloutsos, M., Faloutsos, P., Faloutsos, C.: On power-law relationships of the internet topology. In: SIGCOMM, pp. 251–262 (1999)
Hayes, B.: Graph theory in practice: Part II. American Scientist 88, 104–109 (2000)
Kleinberg, J.M., Kumar, R., Raghavan, P., Rajagopalan, S., Tomkins, A.S.: The Web as a graph: Measurements, models and methods. In: Asano, T., Imai, H., Lee, D.T., Nakano, S.-i., Tokuyama, T. (eds.) COCOON 1999. LNCS, vol. 1627, p. 1. Springer, Heidelberg (1999)
Kumar, R., Raghavan, P., Rajagopalan, S., Sivakumar, D., Tomkins, A., Upfal, E.: Stochastic models for the web graph. In: FOCS: IEEE Symposium on Foundations of Computer Science, FOCS (2000)
Kumar, R., Raghavan, P., Rajagopalan, S., Sivakumar, D., Tomkins, A., Upfal, E.: The Web as a graph. In: Proc. 19th ACM SIGACT SIGMOD-AIGART Symp. Principles of Database Systems, PODS, pp. 1–10. ACM Press, New York (2000)
Kumar, R., Raghavan, P., Rajagopalan, S., Tomkins, A.: Trawling the Web for emerging cyber-communities. Computer Networks (Amsterdam, Netherlands: 1999) 31(11-16), 1481–1493 (1999)
Mitzenmacher, M.: A brief history of generative models for power law and lognormal distributions. In: Proc. of the 39th Annual Allerton Conf. on Communication, Control, and Computing, pp. 182–191 (2001)
Mihail, M., Papadimitriou, C.H.: On the eigenvalue power law. In: Proc. of 6th Intl. Workshop on Randomization and Approximation Techniques, pp. 254–262 (2002)
Simon, H.A.: On a class of skew distribution functions. Biometrika 42(3/4), 425–440 (1955)
Strang, G.: Linear algebra and its applications. Hardcourt Brace Jovanovich Publishing, New York (1988)
Watts, D.J.: Small Worlds: The Dynamics of Networks between Order and Randomness. Princeton University Press, Princeton (1988)
Yule, G.: A mathematical theory of evolution based on the conclusions of Dr. J.C. Willis. Philosophical Transactions of the Royal Society of London (Series B) 213, 21–87 (1925)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Flaxman, A., Frieze, A., Fenner, T. (2003). High Degree Vertices and Eigenvalues in the Preferential Attachment Graph. In: Arora, S., Jansen, K., Rolim, J.D.P., Sahai, A. (eds) Approximation, Randomization, and Combinatorial Optimization.. Algorithms and Techniques. RANDOM APPROX 2003 2003. Lecture Notes in Computer Science, vol 2764. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-45198-3_23
Download citation
DOI: https://doi.org/10.1007/978-3-540-45198-3_23
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-40770-6
Online ISBN: 978-3-540-45198-3
eBook Packages: Springer Book Archive