Discrete Mathematics & Theoretical Computer Science |

- 1 Karelian Research Center of the Russian Academy of Science

Consider random graph with $N+ 1$ vertices as follows. The degrees of vertices $1,2,\ldots, N$ are the independent identically distributed random variables $\xi_1, \xi_2, \ldots , \xi_N$ with distribution $\mathbf{P}\{\xi_1 \geq k\}=k^{− \tau},$ $k= 1,2,\ldots,$ $\tau \in (1,2)$,(1) and the vertex $N+1$ has degree $0$, if the sum $\zeta_N=\xi_1+ \ldots +\xi_N$ is even, else degree is $1$. From (1) we get that $p_k=\mathbf{P}\{\xi_1=k\}=k^{−\tau}−(k+ 1)^{−\tau}$, $k= 1,2,\ldots$ Let $G(k_1, \ldots , k_N)$ be a set of graphs with $\xi_1=k_1,\ldots, \xi_N=k_N$. If $g$ is a realization of random graph then $\mathbf{P}\{g \in G(k_1, \ldots , k_N)\}=p_{k_1} \cdot \ldots \cdot p_{k_N}$. The probability distribution on the set of graph is defined such that for a vector $(k_1, \ldots, k_N)$ all graphs, lying in $G(k_1, \ldots , k_N)$, are equiprobable. Studies of the past few years show that such graphs are good random graph models for Internet and other networks topology description (see, for example, H. Reittu and I. Norros (2004)).To build the graph, we have $N$ numbered vertices and incident to vertex $i \xi_i$ stubs, $i= 1, \ldots , N$.All stubs need to be connected to another stub to construct the graph. The stubs are numbered in an arbitrary order from $1$ to $\zeta_N$. Let $\eta_{(N)}$ be the maximum degree of the vertices.

Source: HAL:hal-01184692v1

Volume: DMTCS Proceedings vol. AG, Fourth Colloquium on Mathematics and Computer Science Algorithms, Trees, Combinatorics and Probabilities

Section: Proceedings

Published on: January 1, 2006

Imported on: May 10, 2017

Keywords: random graph,web graph,giant component,[INFO.INFO-DS] Computer Science [cs]/Data Structures and Algorithms [cs.DS],[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM],[MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO],[INFO.INFO-HC] Computer Science [cs]/Human-Computer Interaction [cs.HC]

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