In this study we present a computational approach to the generation of the major geometric structures of an idealized murine lymph node (LN). In this generation, we consider the major compartments such as the subcapsular sinus, B cell follicles, trabecular and medullar sinuses, blood vessels and the T cell zone with a primary focus on the fibroblastic reticular cell (FRC) network. Confocal microscopy data of LN macroscopic structures and structural properties of the FRC network have been generated and utilized in the present model. The methodology sets a library of modules that can be used to assemble a solid geometric LN model and subsequently generate an adaptive mesh model capable of implementing transport phenomena. Overall, based on the use of high-resolution confocal microscopy and morphological analysis of cell 3D reconstructions, we have developed a computational model of the LN geometry, suitable for further investigation in studies of fluid transport and cell migration in this immunologically essential organ.

In this study we present a benchmark of statistical distributions of the first nearest neighbors in random graphs. We consider distribution of such graphs by the number of disconnected fragments, fragments by the number of involved nodes, and nodes by their degrees. The statements about the asymptotic properties of these distributions for graphs of large dimension are proved. The problem of determining the probability of realization of a certain structure of the first nearest neighbors graph depending on the distribution function of distances between the elements of the studied set is considered. It is shown that, up to isomorphism, the graph of the first nearest neighbors does not depend on the distance distribution. This fact makes it possible to conduct numerical experiments on the construction of basic statistics based on a uniform distribution of distances and obtain tabulated data as a result of numerical modeling. A study has been conducted on the conditional extremum of the probability of realizing the distribution of graph vertices by degrees, which allows us to estimate the proportion of randomness for a particular structure resulting from clustering elements of a certain set by the nearest neighbor method.

In this paper the investigation of the structure of k-NN graph is presented. The purpose of this analysis is to match the probability of implementation particular graph under assumption of a certain distribution of distances between vertices and depending on the number of vertices. Sample distributions of the graph by the number of disconnected fragments, fragments by the number of vertices, and vertices by the degrees of incoming edges are used as the main statistics. As an example, the problem of identifying the author of a text by the n-gram method is considered.

This paper describes some statistical properties of the nearest neighbor graphs (NNGs). We study the sample distributions of graphs by the number of disconnected fragments, fragments by the number of nodes, and nodes by the degrees of incoming edges. The statements about the asymptotic properties of these distributions for graphs of a large dimension are proved and their relationship with Young’s classical diagrams and Wigner’s semicircle distribution is noted. The problem of determining the probability of realizing a certain structure of the nearest neighbors depending on the distribution of distances between the elements of the studied set is considered. It is shown that, the NNG does not depend on the distribution of distances up to an isomorphism. This fact makes it possible to construct basic statistics using a uniform distribution, and to obtain tabulated data for the statistics of the NNGs as a result of numerical modeling. A study has been conducted on the conditional extremum of the probability of realizing the distribution of graph nodes by degrees, which allows us to estimate the proportion of randomness for a particular structure, which appears from clustering elements of a certain set by the nearest neighbor method. An algorithm for collecting the sample statistics of the NNGs using the specific features of such graphs is described.

The proposed method is based on calculations of the statistics of the nearest neighbor graph (NNG) structures, which are presented as a benchmark of the probabilities of the distribution of graphs by the number of disconnected fragments. The deviation of the actually observed occurrence of connectivity from the calculated one will allow us to determine the probability that this sample can be considered a set of statistically independent variables. The statements about the independence of the NNG statistics from the distribution of distances and from the triangle inequality are proved, which allows the numerical modeling of such structures. Estimates of the accuracy of the calculated statistics for graphs and their comparison with estimates obtained by modeling random coordinates of points in d-dimensional space are carried out. It is shown that the model of the NNGs without taking into account the dimension of the space leads to fairly accurate estimates of the statistics of graph structures in spaces of dimensionality higher than five. For spaces of smaller dimensionality, the benchmark can be obtained by directly calculating the distances between points with random coordinates in a unit cube. The proposed method is applied to the problem of analyzing the level of unsteadiness of the earthquake catalog in the Kuril–Kamchatka region. The lengths of samples of time intervals between neighboring events are analyzed. It is shown that the analyzed system as a whole is interconnected with a probability of 0.91, and this dependence is fundamentally different from the lag correlation between the sample elements.