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Combinatorial algebra and random graphs
V. E. Stepanov Moscow
Abstract:
Let
$A$ be a finite set of vertices and
$\lambda_a>0$ be the intensity of the vertex
$a\in A$. A random time-dependent graph
$\mathscr G_L(A\mid t)$ is defined as follows: at time
$t=0$ all the vertices are isolated; the probability that at time
$t>0$ vertices
$a$ and
$b$ are connected equals
$1-e^{-\lambda}a^\lambda b^t$, and the connections appear independently for different pairs, let
$\mathbf P_L(A\mid t)$ be the probability that the random graph
$\mathscr G_L(A\mid t)$ is connected.
In the paper, an explicit expression for
$\mathbf P_L(A\mid t)$ is found, a number of combinatorial relations including the probabilities
$\mathbf P_L(A\mid t)$ is obtained, and it is proved that if the set of vertices
$A$, intensities of vertices
$\lambda_a$, and time
$t$ are changed in a certain way, then, under some conditions,
$\mathbf P_L(A\mid t)e^{\mu(A\mid t)}\to1$, where
$$
\mu(A\mid t)=\sum_{a\in A}\exp\{-t\lambda_aL(A)\}\quad\text{and}\quad L(A)=\sum_{a\in A}\lambda_a.
$$
Received: 05.11.1967