The process of growth of random graphs with the vertices lying at the points of the plane (i,j) with integral coordinates has been considered. Computer simulation of the process allowed us to establish that random graphs grow in the self-similar mode. The sectorial growth proceeds in such a way that the growth boundary in four sectors consists of two horizontal and two vertical linear segments; in the other four sectors, the boundary is a part of an ellipse; the real growth boundary lags behind the limited form to which it tends; the scatter interval along the principal growth diagonal is proportional to the cubic root of the number of the coordination sphere.
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