ISSN 0032-9460, Problems of Information Transmission, 2015, Vol. 51, No. 1, pp. 66–74.
Pleiades Publishing, Inc., 2015.
Original Russian Text
A.V. Lebedev, 2015, published in Problemy Peredachi Informatsii, 2015, Vol. 51, No. 1, pp. 72–81.
COMMUNICATION NETWORK THEORY
Activity Maxima in Some Models of Information
Networks with Random Weights and Heavy Tails
A. V. Lebedev
Probability Theory Department, Faculty of Mechanics and Mathematics,
Lomonosov Moscow State University, Moscow, Russia
Received September 8, 2014
Abstract—We consider models of information networks described by random graphs and
hypergraphs where each node has a random information activity with distribution having a
heavy (regularly varying) tail. We derive suﬃcient conditions under which the maximum of the
aggregate activities (over a node and its neighbors or over communities) asymptotically grows in
the same way as the maximum of individual activities and the Fr´echet limit law holds for them.
The author’s works [1, 2] studied the behavior of maxima of aggregate activity in information
networks described by random graphs from the point of view of asymptotic equivalence to maxima
of individual activities.
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 (regularly varying) tail, i.e.,
F (x) ∼ x
L(x), x →∞, a>0,
where L(x) is a slowly varying function [3, Section 8.8]. 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 so-called Friends Page. We assume for simplicity that the activity and
the number of neighbors are independent. 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]).
Of course, this problem setting admits various modiﬁcations. For example, if we assume that
a node transmits not only his own information but also information received from its neighbors
(to other neighbors), we arrive at the aggregate activity over connected components. Then it
only makes sense to study the subcritical case; otherwise, asymptotic equivalence to maxima of
individual activities cannot occur. We do not consider this modiﬁcation here.
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 last decades
was initiated by  (and which in the Russian literature are referred to as graphs of Internet type,
Supported in part by the Russian Foundation for Basic Research, project no. 14-01-00075.