A.G.Chkhartishvili
1and D.N.Fedyanin
21,2Institute of Control Sciences, Moscow Russia
1[email protected] 2[email protected]
Keywords: Network, game theory, Markov model, Information control, Social networks. In the last decade interest in the study of network structures has considerably increased, in particular - to the problems of information control in active network structures [4]. First, the active network structures, as opposed to hierarchical, there is no subordination of one element by another. This significantly limits the ability of the investigated in detail the models and mechanisms in the control hierarchy (eg, organizational) structures. Second, the widespread development of information technology and, in particular, the Internet and various online networks (social networks, forums, blogs) has significantly changed and continues to change the environment of social and economic interaction. What will these changes is difficult to say right now, although researchers are trying to make predictions in the economy and policies [1]. We also note that a separate section on network structures, apparently, in the near future it will be in the textbooks on economic theory [3]. In this paper we consider a model of information control in a network structure that operates in accordance with the Markov model, under conditions of incomplete knowledge of the governing person or organization (the principal).
We describe the agents within the network structure (we also call it simply a network), set N{1, 2,..., }n Agents in the network affect each other, and the degree of influence given by the matrix A direct effect of dimension n x n where ij 0 denotes the degree of confidence in the i-th agent's j-th agent. Here and later we'll talk about how the impact and the confidence and believe that these two concepts are oppositely in
the sense that the term "trust the i-th j-th agent is » is identical in meaning to the ij expression "degree influence of the j-th agent at the i-th is ». ij
Let each agent in a certain initial time has some awareness (or, in other words,
the opinion) on certain issues. Opinion of i-th agent represents a real number 0
i
x ,
opinions of all the agents are represented by a column vector x0 of dimension n. In
accordance with a Markov model of agents in the network interacts exchanging opinions. This exchange leads to the opinion of each agent changes in accordance with the opinions of agents to which the agent trusts. Suppose a governing body - the principal - is seeking to get a maximum total value of the characteristics of the agents
x j i
. For more details please see [2]. Let’s suppose further that he is able to provide
the agents at the initial time control actions ui, changing their characteristics. Then the total change in the final characteristics of the agents is (hereu( ,..., )u1 un :
(A ) ( ) . F u j aij uj w uj j j N j N i N j N
where wi are some coefficients called influences of agents. Function F u( ) is a utility function of the principal, which he seeks to maximize. Having limited resources (for example, if only k components of vector could be different from zero and k less the dimension of vector) should be focused on agents with greater influence. This will give him greater final payoff.
We consider (in different variants of information which the principal has) the following situation: the principal can provide the control action, equal to 1, exactly on k
agents (where kn). We are interested in the following two questions: What is the optimal control strategy of the principal;
What are the networks for the principal of the most and the least profitable? Proposition 1. In the case of perfect awareness, situation in which the influence of sequence no more than k agents are different from zero is the most profitable for the principal.
Proposition 2. In the case of perfect awareness, situation in which influences of all agents are identical is the least profitable for the principal
Let’s consider the network structure, whose dynamics is given by a Markov model but in contrast to previous considerations of the preceding section, we’ll assume
that the principal does not know about the influence of specific agents, so influence to each of chosen agents has the same effect equal to 1 for the k randomly chosen agents of n (where 1 n ). As before, the principal is interested in maximizing the amount of the final characteristics of the agents.
In this case of the uninformed principal of magnitude of his influence on agents
, 1,...,
u ii n are random. Each of them takes the values from 0 to 1, while ui k
i
(Unconditional) probability of event ui 1 is k n/ , as well as the
(unconditional) the expectation:
( 1) E k.
p ui ui n
The usefulness of the principal is a random variable:
.
F w uj j j N
Proposition 3. In the case of the uninformed principal a situation in which the influences of all agents are identical is more profitable for it and situations, when the network structure has a unique element that has non-zero influence are the least favorable.
Now we consider the situation of the partial informed principal: network structure is divided into disjoint subsets, disjoint information, within each of which the principal does not distinguish between agents, but for each subset knows the number of agents and their total influence. In this case strategy of the principal is the choice of the volume impact of the information on each subset, i.e. in which he turns out to be the control action. Thus the total number of agents still to be equal to k, and each ki, of course, does not exceed the number of agents in the information subset Gi
Proposition 4. In the case of partial awareness of the principal a situation in which the total number of agents in the subset of information subsets, the average influence is that non-zero and do not exceed k is the most profitable for the principal.
Proposition 5. In the case of partial awareness of the principal a situation in which the average influence of all subsets of information islands is the same is the least profitable for the principal
In this paper we consider a model of information control in active network structure under different assumptions about knowledge of the principal. A promising
direction for further research is to analyze the various options and their impact on the effectiveness of information control in active network structures.
References
[1] Barabanov I.N, Korgin N.A., Novikov D.A., Chkhartishvili A.G. Dynamic models of an information control in social networks / / Automation and Remote Control. 2010. № 11. S. 172 - 182.
[2] Gubanov D.A., Novikov D.A, Chkhartishvili A.G. Social networks: models of influence of the information, control and confrontation - Fizmatlit, 2010. 228 p.
[3] Kuzminov Y., Bendukidze K.A, Yudkevich M.M. Course of Institutional Economics: institutions, networks, transaction costs, contracts - Moscow: Publishing house HSE, 2006. - 442.
[4] Jackson M. Social and Economic Networks - Princeton: Princeton University Press, 2008. - 520 p.