Sure Event Problem in Multicomponent Dynamical Systems with Attractive Interaction
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SURE EVENT PROBLEM IN MULTICOMPONENT DYNAMICAL SYSTEMS WITH ATTRACTIVE INTERACTION V. D. Koshmanenko1;2 and O. R. Satur3
UDC 517.9+316.4
We establish a series of sufficient conditions for the realization of a sure event as the limit (in time) state of a multicomponent dynamical system with attractive interaction. The sure event is characterized by the state of a system with finitely many positions when all coordinates of the distribution are equal to zero, except a single fixed coordinate equal to 1. The sure event can be interpreted as the state of consensus in social networks and, hence, the obtained results can be used in voter and opinion-formation models.
1. Introduction Most likely, for the first time, multicomponent dynamical systems generated by the mappings corresponding to the attractive interaction between a priori nonannihilating opponents in the n-dimensional space, n > 1; were investigated in [1]. In the cited work, the authors studied the evolution of stochastic vectors pt and rt from the space Rn ; n � 2; characterizing the state of the opponents. In particular, it was shown that these dynamical systems have equilibrium stationary states (point attractors) of the form p D .0;0; : : : ; 1; 0; : : : ; 0/; r D .0;0; : : : ; 1; 0; : : : ; 0/; where 1 may occupy any position identical for p and r: These states are stable. In numerous works devoted to applications and, in particular, in the opinion-formation and voter models (see, e.g., [2–8]), the states of this kind are interpreted as consensus. The problem of description of the basins of attraction for these attractors, i.e., the problem of description of all initial states asymptotically approaching the indicated states as t ! 1; seems to be quite urgent. Note that the pairs of vectors of the form ◆ 1 1 ; p D 0;0; : : : ; ; 0; : : : ; m m ✓
✓ ◆ 1 1 r D 0;0; : : : ; ; 0; : : : ; ; m m
1 < m n;
also correspond to fixed stationary states. However, they are unstable and connected with the problem of ambiguity in decision-making problems. On the other hand, in the theory of conflict dynamical systems developed in [9–19], the coordinates of the vectors p D .p1 ; : : : ; pn / and r D .r1 ; : : : ; rn / have the meaning of probabilities of stay of the opponents in a given position in a certain space D .!1 ; : : : ; !n /: By definition, the sure event corresponds to a vector with unit coordinate at one of the positions !i ; i D 1; n: As already indicated, for systems with attractive interaction, pairs of these vectors form stable fixed states. The problem is to determine the conditions that should be imposed on the coordinates of an arbitrary initial pair of stochastic vectors pt D0 ; rt D0 2 Rn in order to guarantee the convergence 1 Institute
of Mathematics, Ukrainian National Academy of Sciences, Tereshchenkivs’ka Str., 3, Kyiv, 01024, Ukraine; Drahomanov National Pedagogic University, Pyrohov Str., 9, Kyiv, 01030, Ukraine; “Kyyevo-Mohylyans’ka Akademiya” National University, Skovoroda Str., 2, Kyiv, 04070, Ukraine; e-mail: [email protected]. 2
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