ISSN 0005-1179, Automation and Remote Control, 2017, Vol. 78, No. 8, pp. 1489–1499.
Pleiades Publishing, Ltd., 2017.
Original Russian Text
V.M. Bure, E.M. Parilina, A.A. Sedakov, 2016, published in Problemy Upravleniya, 2016, No. 1, pp. 21–28.
Consensus in a Social Network with Two Principals
V. M. Bure,
E. M. Parilina,
St. Petersburg State University, St. Petersburg, Russia
Received November 12, 2015
Abstract—This paper considers a model of opinion dynamics in a social network with two
principals, in which the members may aﬀect the opinions of each other and their opinions evolve
according to a time-homogeneous Markov chain. We study the existence of a consensus in this
network for two types of inﬂuence models, namely, when the principals may or may not aﬀect
the opinions of each other directly. In addition, we ﬁnd the values of social network parameters
under which a consensus is reached. For the cases without a consensus in its standard deﬁnition,
we introduce the notion of a consensus of the majority and ﬁnd the parameter values under
which it is reached. Two numerical examples illustrate the obtained theoretical results.
The mathematical model of opinion dynamics in a social network that uses the theory of Markov
processes was pioneered by De Groot . This model proceeds from the assumption that the opinion
of a given member can be aﬀected by other members with ﬁxed constant weights. The member’s
opinion about some unknown event at the next step represents a linear combination of all opinions
of members at the current step. De Groot introduced the notion of consensus and deﬁned suﬃcient
conditions for its existence. In the paper , De Groot’s model was extended by the notion of wise
society and the issue of reaching the “truth” was explored in the case when the number of members
grows inﬁnitely. As an example, the authors considered a social network where a certain member
is a principal whereas all other members are symmetrical. In this case, the inﬂuence matrix of
the members that describes opinion dynamics has a special form. It was proved that there exists
the limiting inﬂuence matrix, and its elements were obtained in explicit form. Buechel et al. 
proposed an alternative model of opinions formation, with the main hypothesis that the network
members observe only the declared opinions (not necessarily their actual opinions). In this model,
the members know for sure only their own opinions at each step. Next, the paper  presented a
model of the opinion formation in a social group, e.g., employees’ opinions about an event suggested
by their company manager. The model characterizes the mutual inﬂuence of the employees, as well
as the manager’s inﬂuence on the employees and vice versa. The role of a principal was assigned
to the company manager.
The paper  and the book  considered a controlled modiﬁcation of De Groot’s model, i.e.,
a game-theoretic model of informational confrontation with the opinion dynamics described by a
homogeneous Markov chain. Particularly, it was assumed that a principal may aﬀect the opinions
of the network members as his control is incorporated in the equation of the opinion dynamics.
A game-theoretic approach for modeling the opinion dynamics was also adopted in . In order
to study opinions convergence, the authors involved graphs reﬂecting the structure of interactions
among the network members, and assumed that their opinions evolve by aggregating the opinions
of all network members. And then convergence analysis was performed. Note that the hierarchical
structures of interaction among network members were also examined in , with a single principal