ISSN 0032-9460, Problems of Information Transmission, 2008, Vol. 44, No. 2, pp. 156–160.
Pleiades Publishing, Inc., 2008.
Original Russian Text
A.V. Lebedev, 2008, published in Problemy Peredachi Informatsii, 2008, Vol. 44, No. 2, pp. 96–100.
COMMUNICATION NETWORK THEORY
Activity Maxima in Random Networks
in the Heavy Tail Case
A. V. Lebedev
Lomonosov Moscow State University
Received August 24, 2007
Abstract—We consider a model of information network described by an undirected random
graph, where each node has a random information activity whose distribution possesses a heavy
tail (with regular variation). We investigate the cases of networks described by classical and
power-law random graphs. We derive suﬃcient conditions under which the maximum of aggre-
gate activities (over a node and its nearest neighbors) asymptotically grows in the same way as
the maximum of individual activities and the Fr´echet limit law holds for them.
Consider an information network described by an undirected random graph [1, 2]. Let each
network node have a random information activity (rate of information production). We assume that
the activities of the nodes are independent and identically distributed and that their distribution F
has a heavy tail (with regular variation), i.e.,
F (x) ∼ x
L(x), x →∞, a>0, where L(x)is
a slowly varying function [3, Section 8.8]. This assumption is consistent with the present-day
understanding of the abundance of power laws in nature, technology, and human activity. Consider
also the aggregate activity at a node (i.e., the sum of its own activity and those of its nearest
neighbors). For example, each user of the LJ (LiveJournal) can make his own posts and read posts
of his friends, which are collected for convenience into a common “friend’s page.” We are interested
in the question of when the maximum of aggregate activities asymptotically grows in the same way
as the maximum of individual activities of the nodes. This means that large values are usually
attained not owing to the activity of a large number of nodes but due to individual nodes with high
activity. In this case it is easy to derive the Fr´echet limit law for the maxima: Φ
x>0 (see [3, Section 8.8; 4, Section 3.3.1]).
In the study of random graphs, various models are used. These are classical models, whose study
originated from , and power law (scale-free) models, whose active investigation in recent years was
initiated by . In classical random graphs, the degree of a vertex (number of its nearest neighbors)
in the limit has a Poisson distribution [2, ch. 3]. In power law graphs, the limit distribution is
power-law, i.e., p
, β>0. It was found that such models well describe many information,
engineering, and biological systems. However, it should be noted that quite diﬀerent construction
algorithms often lead to one and the same limit distribution, whereas other asymptotic properties
of graphs are model-dependent. We use a particular model of a power-law graph proposed in .
Presently, other (more complicated) models are being investigated, in particular, in Russia .
Results for the general scheme of maxima of independent random variables previously obtained
by the author  are applied in the present paper.
Supported in part by the Russian Foundation for Basic Research, project nos. 07-01-00077 and 07-01-