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12 pages, 484 KiB

Open AccessArticle

A Generative Model for Correlated Graph Signals

by Pavel Loskot

Mathematics 2021, 9(23), 3078; https://doi.org/10.3390/math9233078 - 29 Nov 2021

Viewed by 1588

A graph signal is a random vector with a partially known statistical description. The observations are usually sufficient to determine marginal distributions of graph node variables and their pairwise correlations representing the graph edges. However, the curse of dimensionality often prevents estimating a [...] Read more.

A graph signal is a random vector with a partially known statistical description. The observations are usually sufficient to determine marginal distributions of graph node variables and their pairwise correlations representing the graph edges. However, the curse of dimensionality often prevents estimating a full joint distribution of all variables from the available observations. This paper introduces a computationally effective generative model to sample from arbitrary but known marginal distributions with defined pairwise correlations. Numerical experiments show that the proposed generative model is generally accurate for correlation coefficients with magnitudes up to about 0.3, whilst larger correlations can be obtained at the cost of distribution approximation accuracy. The generative models of graph signals can also be used to sample multivariate distributions for which closed-form mathematical expressions are not known or are too complex. Full article

(This article belongs to the Special Issue Random Processes on Graphs)

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14 pages, 2497 KiB

Open AccessArticle

Construction and Analysis of Queuing and Reliability Models Using Random Graphs

by Gurami Tsitsiashvili

Mathematics 2021, 9(19), 2511; https://doi.org/10.3390/math9192511 - 07 Oct 2021

Cited by 1 | Viewed by 1513

Abstract

In this paper, the use of the construction of random processes on graphs allows us to expand the models of the theory of queuing and reliability by constructing. These problems are important because the emphasis on the legal component largely determines functioning of [...] Read more.

In this paper, the use of the construction of random processes on graphs allows us to expand the models of the theory of queuing and reliability by constructing. These problems are important because the emphasis on the legal component largely determines functioning of these models. The considered models are reliability and queuing. Reliability models arranged according to the modular principle and reliability networks in the form of planar graphs. The queuing models considered here are queuing networks with multi server nodes and failures, changing the parameters of the queuing system in a random environment with absorbing states, and the process of growth of a random network. This is determined by the possibility of using, as traditional probability methods, mathematical logic theorems, geometric images of a queuing network, dual graphs to planar graphs, and a solution to the Dirichlet problem. Full article

(This article belongs to the Special Issue Random Processes on Graphs)

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24 pages, 607 KiB

Open AccessArticle

Polynomial Representations of High-Dimensional Observations of Random Processes

by Pavel Loskot

Mathematics 2021, 9(2), 123; https://doi.org/10.3390/math9020123 - 07 Jan 2021

Cited by 3 | Viewed by 2494

Abstract

The paper investigates the problem of performing a correlation analysis when the number of observations is large. In such a case, it is often necessary to combine random observations to achieve dimensionality reduction of the problem. A novel class of statistical measures is [...] Read more.

The paper investigates the problem of performing a correlation analysis when the number of observations is large. In such a case, it is often necessary to combine random observations to achieve dimensionality reduction of the problem. A novel class of statistical measures is obtained by approximating the Taylor expansion of a general multivariate scalar symmetric function by a univariate polynomial in the variable given as a simple sum of the original random variables. The mean value of the polynomial is then a weighted sum of statistical central sum-moments with the weights being application dependent. Computing the sum-moments is computationally efficient and amenable to mathematical analysis, provided that the distribution of the sum of random variables can be obtained. Among several auxiliary results also obtained, the first order sum-moments corresponding to sample means are used to reduce the numerical complexity of linear regression by partitioning the data into disjoint subsets. Illustrative examples provided assume the first and the second order Markov processes. Full article

(This article belongs to the Special Issue Random Processes on Graphs)

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18 pages, 2258 KiB

Open AccessArticle

Graph Theory for Modeling and Analysis of the Human Lymphatic System

by Rostislav Savinkov, Dmitry Grebennikov, Darya Puchkova, Valery Chereshnev, Igor Sazonov and Gennady Bocharov

Mathematics 2020, 8(12), 2236; https://doi.org/10.3390/math8122236 - 17 Dec 2020

Cited by 11 | Viewed by 3193

Abstract

The human lymphatic system (HLS) is a complex network of lymphatic organs linked through the lymphatic vessels. We present a graph theory-based approach to model and analyze the human lymphatic network. Two different methods of building a graph are considered: the method using [...] Read more.

The human lymphatic system (HLS) is a complex network of lymphatic organs linked through the lymphatic vessels. We present a graph theory-based approach to model and analyze the human lymphatic network. Two different methods of building a graph are considered: the method using anatomical data directly and the method based on a system of rules derived from structural analysis of HLS. A simple anatomical data-based graph is converted to an oriented graph by quantifying the steady-state fluid balance in the lymphatic network with the use of the Poiseuille equation in vessels and the mass conservation at vessel junctions. A computational algorithm for the generation of the rule-based random graph is developed and implemented. Some fundamental characteristics of the two types of HLS graph models are analyzed using different metrics such as graph energy, clustering, robustness, etc. Full article

(This article belongs to the Special Issue Random Processes on Graphs)

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11 pages, 686 KiB

Open AccessArticle

Update-Based Machine Learning Classification of Hierarchical Symbols in a Slowly Varying Two-Way Relay Channel

by Jakub Kolář, Jan Sýkora and Petr Hron

Mathematics 2020, 8(11), 2007; https://doi.org/10.3390/math8112007 - 11 Nov 2020

Viewed by 1189

Abstract

This paper presents a stochastic inference problem suited to a classification approach in a time-varying observation model with continuous-valued unknown parameterization. The utilization of an artificial neural network (ANN)-based classifier is considered, and the concept of a training process via the backpropagation algorithm [...] Read more.

This paper presents a stochastic inference problem suited to a classification approach in a time-varying observation model with continuous-valued unknown parameterization. The utilization of an artificial neural network (ANN)-based classifier is considered, and the concept of a training process via the backpropagation algorithm is used. The main objective is the minimization of resources required for the training of the classifier in the parametric observation model. To reach this, it is proposed that the weights of the ANN classifier vary continuously with the change of the observation model parameters. This behavior is then used in an update-based backpropagation algorithm. This proposed idea is demonstrated on several procedures, which re-use previously trained weights as prior information when updating the classifier after a channel phase change. This approach successfully saves resources needed for re-training the ANN. The new approach is verified via a simulation on an example communication system with the two-way relay slowly fading channel. Full article

(This article belongs to the Special Issue Random Processes on Graphs)

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21 pages, 459 KiB

Open AccessArticle

The Combined Estimator for Stochastic Equations on Graphs with Fractional Noise

by Pavel Kříž and Leszek Szała

Mathematics 2020, 8(10), 1766; https://doi.org/10.3390/math8101766 - 13 Oct 2020

Cited by 2 | Viewed by 1282

Abstract

In the present paper, we study the problem of estimating a drift parameter in stochastic evolution equations on graphs. We focus on equations driven by fractional Brownian motions, which are particularly useful e.g., in biology or neuroscience. We derive a novel estimator (the [...] Read more.

In the present paper, we study the problem of estimating a drift parameter in stochastic evolution equations on graphs. We focus on equations driven by fractional Brownian motions, which are particularly useful e.g., in biology or neuroscience. We derive a novel estimator (the combined estimator) and prove its strong consistency in the long-span asymptotic regime with a discrete-time sampling scheme. The promising performance of the combined estimator for finite samples is examined under various scenarios by Monte Carlo simulations. Full article

(This article belongs to the Special Issue Random Processes on Graphs)

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21 pages, 5039 KiB

Open AccessArticle

Viral Infection Dynamics Model Based on a Markov Process with Time Delay between Cell Infection and Progeny Production

by Igor Sazonov, Dmitry Grebennikov, Mark Kelbert, Andreas Meyerhans and Gennady Bocharov

Mathematics 2020, 8(8), 1207; https://doi.org/10.3390/math8081207 - 22 Jul 2020

Cited by 7 | Viewed by 2196

Abstract

Many human virus infections including those with the human immunodeficiency virus type 1 (HIV) are initiated by low numbers of founder viruses. Therefore, random effects have a strong influence on the initial infection dynamics, e.g., extinction versus spread. In this study, we considered [...] Read more.

Many human virus infections including those with the human immunodeficiency virus type 1 (HIV) are initiated by low numbers of founder viruses. Therefore, random effects have a strong influence on the initial infection dynamics, e.g., extinction versus spread. In this study, we considered the simplest (so-called, ‘consensus’) virus dynamics model and incorporated a delay between infection of a cell and virus progeny release from the infected cell. We then developed an equivalent stochastic virus dynamics model that accounts for this delay in the description of the random interactions between the model components. The new model is used to study the statistical characteristics of virus and target cell populations. It predicts the probability of infection spread as a function of the number of transmitted viruses. A hybrid algorithm is suggested to compute efficiently the system dynamics in state space domain characterized by the mix of small and large species densities. Full article

(This article belongs to the Special Issue Random Processes on Graphs)

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16 pages, 636 KiB

Open AccessFeature PaperArticle

d-Path Laplacians and Quantum Transport on Graphs

by Ernesto Estrada

Mathematics 2020, 8(4), 527; https://doi.org/10.3390/math8040527 - 03 Apr 2020

Cited by 3 | Viewed by 2080

Abstract

We generalize the Schrödinger equation on graphs to include long-range interactions (LRI) by means of the Mellin-transformed d-path Laplacian operators. We find analytical expressions for the transition and return probabilities of a quantum particle at the nodes of a ring graph. We [...] Read more.

We generalize the Schrödinger equation on graphs to include long-range interactions (LRI) by means of the Mellin-transformed d-path Laplacian operators. We find analytical expressions for the transition and return probabilities of a quantum particle at the nodes of a ring graph. We show that the average return probability in ring graphs decays as a power law with time when LRI is present. In contrast, we prove analytically that the transition and return probabilities on a complete and start graphs oscillate around a constant value. This allowed us to infer that in a barbell graph—a graph consisting of two cliques separated by a path—the quantum particle get trapped and oscillates across the nodes of the path without visiting the nodes of the cliques. We then compare the use of the Mellin-transformed d-path Laplacian operators versus the use of fractional powers of the combinatorial Laplacian to account for LRI. Apart from some important differences observed at the limit of the strongest LRI, the d-path Laplacian operators produces the emergence of new phenomena related to the location of the wave packet in graphs with barriers, which are not observed neither for the Schrödinger equation without LRI nor for the one using fractional powers of the Laplacian. Full article

(This article belongs to the Special Issue Random Processes on Graphs)

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