Discrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational Science2658-46702658-7149Peoples' Friendship University of Russia8607Research ArticleA Particular Case of a Sequential Growth of an X-GraphKruglyA LDepartment of Applied Mathematics and Computer Scienceakrugly@mail.ruScientific Research Institute for System Analysis of the Russian Academy of Science150320153617308092016Copyright © 2015,2015A particular case of discrete spacetime on a microscopic level is considered. The model is a directed acyclic dyadic graph (an x-graph). The dyadic graph means that each vertex possesses no more than two incident incoming edges and two incident outgoing edges. The sequential growth dynamics of this model is considered. This dynamics is a stochastic sequential addition of new vertices one by one. The probabilities of diﬀerent variants of addition of a new vertex depend on the structure of existed x-graph. It is proved that the algorithm to calculate probabilities of this dynamics is a unique solution that satisﬁes some principles of causality, symmetry and normalization. The algorithm of sequential growth can be represented as following tree steps. The ﬁrst step is the choice of the addition of the new vertex to the future or to the past. By deﬁnition, the probability of this choice is 1∕2 for both outcomes. The second step is the equiprobable choice of one vertex number V . Then the probability is 1∕N, where N is a cardinality of the set of vertices of the x-graph. If we choose the direction to the future, the third step is a random choice of two directed paths from the vertex number V . A new vertex is added to the ends of these paths. If we choose the direction to the past, we must randomly choose the two inversely directed paths from the vertex number V . The iterative procedure to calculate probabilities is considered.causal setrandom graphdirected graphпричинностное множествослучайный графориентированный граф[Krugly A.L. A Sequential Growth Dynamics for a Directed Acyclic Dyadic Graph // Вестник РУДН. Серия «Математика. Информатика. Физика». 2014. № 1. С. 124-138. (arXiv: 1112.1064 [gr-qc]).][Коганов А.В., Круглый А.Л. Алгоритм роста x-графа и принципы физики // Программные продукты и системы. 2012. № 3. С. 95-102.][Krugly A.L., Stepanian I.V. An Example of the Stochastic Dynamics of a Causal Set // Foundations of Probability and Physics-6 (FPP6), 12-15 June 2011, the Linnaeus University, V.axj.o, Sweden / Ed. by M. D’Ariano, S.-M. Fei, E. Haven et al. - Vol. 1424. - 2012. - Pp. 206-210. (arXiv: 1111.5474 [gr-qc]).][Pissanetzky S. The Matrix Model of Computation // Proc. 12th World Multi-Conference on Systemics, Cybernetics, and Informatics (WMSCI), June 29 -July 2, 2008, Orlando, Florida, USA. Vol. IV. 2008. Pp. 184-189.][Bolognesi T. Causal Sets from Simple Models of Computation // International Journal of Unconventional Computing. 2010. Vol. 6, No 6. Pp. 489-524. (1004.3128 [physics.comp-ph]).]