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A problem of finding of an invariant measure of irreducible discrete-time Markov chain with a finite state space is considered. There is a unique invariant measure for such Markov chain that can be multiplied by an arbitrary constant. A representation of a Markov chain by a directed graph is considered. Each state is represented by a vertex, and each conditional transition probability is represented by a directed edge. It is proved that an invariant measure for each state is a sum of $n^{n−2}$ non-negative summands, where $n$ is a cardinality of state space. Each summand is a product of $n − 1$ conditional transition probabilities and is represented by an opposite directed tree that includes all vertices. The root represents the considered state. The edges are directed to the root. This result leads to methods of analyses and calculation of an invariant measure that is based on a graph theory.

This article describes a method to model the course of forest ecosystem succession to the climax state by means of a Markov chain. In contrast to traditional methods of forest succession modelling based on changes of vegetation types, several variants of the vertical structure of communities formed by late-successional tree species are taken as the transition states of the model. Durations of succession courses from any stage are not set in absolute time units, but calculated as the average number of steps before reaching the climax in a unified time scale. The regularities of succession courses are revealed in the proper time of forest ecosystems shaping. The evidences are obtained that internal features of the spatial and population structure do stochastically determine the course and the pace of forest succession. The property of developing vegetation of forest communities is defined as an attribute of stochastic determinism in the course of autogenous succession.

We consider two versions of the maximum cross section method for the solutions of the stationary equation of radiative transfer in dimensional inhomogeneous medium. Both are based on the application Monte-Carlo method to the summation of the Neumann series for the solution transport equation. First modification is traditional and second is based on the use of branching Markov chains. We carried out numerical comparison of these algorithms.

The article deals with the problem of a distributed system operation reliability. The system core is an open integration platform that provides interaction of varied software for modeling gas transportation. Some of them provide an access through thin clients on the cloud technology “software as a service”. Mathematical models of operation, transmission and computing are to ensure the operation of an automated dispatching system for oil and gas transportation. The paper presents a system solution based on the theory of Markov random processes and considers the stable operation stage. The stationary operation mode of the Markov chain with continuous time and discrete states is described by a system of Chapman–Kolmogorov equations with respect to the average numbers (mathematical expectations) of the objects in certain states. The objects of research are both system elements that are present in a large number – thin clients and computing modules, and individual ones – a server, a network manager (message broker). Together, they are interacting Markov random processes. The interaction is determined by the fact that the transition probabilities in one group of elements depend on the average numbers of other elements groups.

The authors propose a multi-criteria dispersion model of risk assessment for such systems (both in the broad and narrow sense, in accordance with the IEC standard). The risk is the standard deviation of estimated object parameter from its average value. The dispersion risk model makes possible to define optimality criteria and whole system functioning risks. In particular, for a thin client, the following is calculated: the loss profit risk, the total risk of losses due to non-productive element states, and the total risk of all system states losses.

Finally the paper proposes compromise schemes for solving the multi-criteria problem of choosing the optimal operation strategy based on the selected set of compromise criteria.

Finding PageRank vector is of great scientific and practical interest due to its applicability to modern search engines. Despite the fact that this problem is reduced to finding the eigenvector of the stochastic matrix $P$, the need for new algorithms is justified by a large size of the input data. To achieve no more than linear execution time, various randomized methods have been proposed, returning the expected result only with some probability close enough to one. We will consider two of them by reducing the problem of calculating the PageRank vector to the problem of finding equilibrium in an antagonistic matrix game, which is then solved using the Grigoriadis – Khachiyan algorithm. This implementation works effectively under the assumption of sparsity of the input matrix. As far as we know, there are no successful implementations of neither the Grigoriadis – Khachiyan algorithm nor its application to the task of calculating the PageRank vector. The purpose of this paper is to fill this gap. The article describes an algorithm giving pseudocode and some details of the implementation. In addition, it discusses another randomized method of calculating the PageRank vector, namely, Markov chain Monte Carlo (MCMC), in order to compare the results of these algorithms on matrices with different values of the spectral gap. The latter is of particular interest, since the magnitude of the spectral gap strongly affects the convergence rate of MCMC and does not affect the other two approaches at all. The comparison was carried out on two types of generated graphs: chains and $d$-dimensional cubes. The experiments, as predicted by the theory, demonstrated the effectiveness of the Grigoriadis – Khachiyan algorithm in comparison with MCMC for sparse graphs with a small spectral gap value. The written code is publicly available, so everyone can reproduce the results themselves or use this implementation for their own needs. The work has a purely practical orientation, no theoretical results were obtained.

In this paper construct the mathematical model for consensus in technical committees for standardization (TC), based on the consensus model proposed DeGroot. The basic problems of achieving consensus in the development of consensus standards in terms of the proposed model are discussed. The results of statistical modeling characterizing the dependence of time to reach consensus on the number of members of the TC and their authoritarianism are presented. It has been shown that increasing the number of TC experts and authoritarianism negative impact on the time to reach a consensus and increase fragmentation of the TC.

A spatially distributed mathematical model is proposed for the evolution of five myocardial tissue states — healthy myocardium, inflammatory infiltrate, necrosis, replacement fibrosis, and interstitial fibrosis — in the form of a multicomponent parabolic reaction – diffusion system. At each spatial point the tissue state is described by a vector of Dirichlet concentration parameters, which allows one to encode both the expected tissue composition and the uncertainty of classification associated with transition zones of damage. For a truncated system that coincides with the original one inside an invariant region, local Lipschitz continuity and quasi-positivity of the extended reaction operator are established. Global existence of a weak solution, non-negativity, an explicit a priori upper bound, and a quantitative exponential lower bound are proved. These results imply invariance of the admissible region and hence existence of a global weak solution to the original model. Uniqueness is established in a strengthened class of solutions characterized by additional gradient regularity. For an auxiliary reversible transition matrix, a formal entropy balance is derived for smooth positive solutions; for the clinically motivated irreversible transition matrix, a modal spectral stability condition is formulated for the linearized problem. On the numerical side, a positivity condition is derived for the reaction substep of a splitting scheme relative to an adaptive upper trajectory bound, and numerical experiments are conducted for a reproducible calibrated computational variant. In the one-dimensional scenario, by day $60$ the replacement-fibrosis fraction at the lesion center increases from $0.091$ to $0.855$, while the total concentration parameter $\alpha_0$ rises from $11.0$ to $15.9$; in the two-dimensional scenario, on day $14$ the mean value of $\overline p_4^{}$ is $0.431$ in the lesion core versus $0.097$ in the healthy region. These results confirm the mathematical well-posedness of the proposed model and its applicability for the quantitative description of post-infarction scar formation and heterogeneous transition zones.

In many social groups, for example, in technical committees for standardization, at the international, regional and national levels, in European communities, managers of ecovillages, social movements (occupy), international organizations, decision-making is based on the consensus of the group members. Instead of voting, where the majority wins over the minority, consensus allows for a solution that each member of the group supports, or at least considers acceptable. This approach ensures that all group members’ opinions, ideas and needs are taken into account. At the same time, it is noted that reaching consensus takes a long time, since it is necessary to ensure agreement within the group, regardless of its size. It was shown that in some situations the number of iterations (agreements, negotiations) is very significant. Moreover, in the decision-making process, there is always a risk of blocking the decision by the minority in the group, which not only delays the decisionmaking time, but makes it impossible. Typically, such a minority is one or two odious people in the group. At the same time, such a member of the group tries to dominate in the discussion, always remaining in his opinion, ignoring the position of other colleagues. This leads to a delay in the decision-making process, on the one hand, and a deterioration in the quality of consensus, on the other, since only the opinion of the dominant member of the group has to be taken into account. To overcome the crisis in this situation, it was proposed to make a decision on the principle of «consensus minus one» or «consensus minus two», that is, do not take into account the opinion of one or two odious members of the group.

The article, based on modeling consensus using the model of regular Markov chains, examines the question of how much the decision-making time according to the «consensus minus one» rule is reduced, when the position of the dominant member of the group is not taken into account.

The general conclusion that follows from the simulation results is that the rule of thumb for making decisions on the principle of «consensus minus one» has a corresponding mathematical justification. The simulation results showed that the application of the «consensus minus one» rule can reduce the time to reach consensus in the group by 76–95%, which is important for practice.

The average number of agreements hyperbolically depends on the average authoritarianism of the group members (excluding the authoritarian one), which means the possibility of delaying the agreement process at high values of the authoritarianism of the group members.

The theoretical study of consensus makes it possible to analyze the various situations that social groups that make decisions in this way have to face in real life, abstracting from the specific characteristics of the groups. It is relevant for practice to study the dynamics of a social group consisting of loyal experts who, in the process of seeking consensus, yield to each other. In this case, psychological “traps” such as false consensus or groupthink are possible, which can sometimes lead to managerial decisions with dire consequences.

The article builds a mathematical consensus model for a group of loyal experts based on modeling using regular Markov chains. Analysis of the model showed that with an increase in the loyalty (decrease in authoritarianism) of group members, the time to reach consensus increases exponentially (the number of agreements increases), which is apparently due to the lack of desire among experts to take part of the responsibility for the decision being made. An increase in the size of such a group leads (ceteris paribus):

– to reduce the number of approvals to consensus in the conditions of striving for absolute loyalty of members, i. e. each additional loyal member adds less and less “strength” to the group;

– to a logarithmic increase in the number of approvals in the context of an increase in the average authoritarianism of members. It is shown that in a small group (two people), the time for reaching consensus can increase by more than 10 times compared to a group of 5 or more members), in the group there is a transfer of responsibility for making decisions.

It is proved that in the case of a group of two absolutely loyal members, consensus is unattainable.

A reasonable conclusion is made that consensus in a group of loyal experts is a special (special) case of consensus, since the dependence of the time until consensus is reached on the authoritarianism of experts and their number in the group is described by different curves than in the case of a regular group of experts.