Statistical properties of telephone communication network

The directed network of telephone subscribers is considered in the article. It can be described as a dynamic network with vertices that correspond to the subscribers of the telephone network and emerging directional edges that correspond to the connections between the respective subscribers. The position of the edge and its direction is determined by the incoming and outgoing calls from the corresponding vertices. The subject of the article is the statistical properties of the connections of a certain subset of telephone network subscribers. Such connections are dynamic in nature due to their appearance and disappearance. The number of outgoing (or incoming) connections occurred during a day at a selected vertex is used as the main characteristic. The distribution density of the number of outgoing (or incoming) connections (or calls) of such a network has been analyzed using the experimental data. It has been shown that such a distribution density over the number of calls obeys the lognormal distribution density, which depends on the two parameters. The values of two parameters, namely the mean value and the variance, determining the lognormal distribution density are established. The reasons for the appearance of a lognormal distribution density over the number of incoming (or outgoing) connections have been discussed. The statistical properties of other groups of subscribers have been considered as well. In particular, the group that makes a large number of outgoing calls to various subscribers of the telephone network has been selected for a separate study. The members of this group, who create and distribute spam can be called spammers. It has been shown that these groups, spammers for example, also obeys the lognormal distribution density over the number of calls but they are characterized by the different mean value and variance.