A guided tour of Chernoff bounds
Information Processing Letters
Authoritative sources in a hyperlinked environment
Proceedings of the ninth annual ACM-SIAM symposium on Discrete algorithms
On power-law relationships of the Internet topology
Proceedings of the conference on Applications, technologies, architectures, and protocols for computer communication
Efficient schemes for nearest neighbor load balancing
Parallel Computing - Special issue on parallelization techniques for numerical modelling
The degree sequence of a scale-free random graph process
Random Structures & Algorithms
Fast computation of low rank matrix approximations
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
Spectral Partitioning with Indefinite Kernels Using the Nyström Extension
ECCV '02 Proceedings of the 7th European Conference on Computer Vision-Part III
RANDOM '02 Proceedings of the 6th International Workshop on Randomization and Approximation Techniques
Random Evolution in Massive Graphs
FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
The web as a graph: measurements, models, and methods
COCOON'99 Proceedings of the 5th annual international conference on Computing and combinatorics
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Many graphs arising in various real world networks exhibit the so called “power law” behavior, i.e., the number of vertices of degree i is proportional to i−β, where β 2 is a constant (for most real world networks β≤ 3). Recently, Faloutsos et al. [18] conjectured a power law distribution for the eigenvalues of power law graphs. In this paper, we show that the eigenvalues of the Laplacian of certain random power law graphs are close to a power law distribution. First we consider the generalized random graph model G(d) =(V,E), where d=(d1, ..., dn) is a given sequence of expected degrees, and two nodes vi, vj ∈V share an edge in G(d) with probability pi, j=didj /$\sum^{n}_{k=1}$dk, independently [9]. We show that if the degree sequence d follows a power law distribution, then some largest Θ(n1/β) eigenvalues of L(d) are distributed according to the same power law, where L(d) represents the Laplacian of G(d). Furthermore, we determine for the case β ∈(2,3) the number of Laplacian eigenvalues being larger than i, for any i = ω(1), and compute how many of them are in some range (i,(1+ε) i), where i=ω(1) and ε0 is a constant. Please note that the previously described results are guaranteed with probability 1–o(1/n). We also analyze the eigenvalues of the Laplacian of certain dynamically constructed power law graphs defined in [2,3], and discuss the applicability of our methods in these graphs.