A Particular Case of a Sequential Growth of an X-Graph

A 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 set, random graph, directed graph