Прикладные аспекты использования алгоритмов ранжирования для ориентированных взвешенных графов(на примере графов социальных сетей)

The article deals with the applied aspects of the preliminary vertices ranking for oriented weighted graph. In this paper, the authors observed the widespread use of this technique in developing heuristic discrete optimization algorithms. The ranking problem is directly related to the problem of social networks centrality and large real world data sets but as shown in the article ranking is explicitly or implicitly used in the development of algorithms as the initial stage of obtaining a solution for solving applied problems. Examples of such ranking application are given. The examples demonstrate the increase of efficiency for solving some optimization applied problems, which are widely used in mathematical methods of optimization, decision-making not only from the theoretical development point of view but also their applications. The article describes the structure of the first phase of the computational experiment, which is associated with the procedure of obtaining test data sets. The obtained data are presented by weighted graphs that correspond to several groups of the social network Vkontakte with the number of participants in the range from 9000 to 24 thousand. It is shown that the structural characteristics of the obtained graphs differ significantly in the number of connectivity components. Characteristics of centrality (degree's sequences), as shown, have exponential distribution. The main attention is given to the analysis of three approaches to graph vertices ranking. We propose analysis and comparison of the obtained set of ranks by the nature of their distribution. The definition of convergence for graph vertex ranking algorithms is introduced and the differences of their use in considering the data of large dimension and the need to build a solution in the presence of local changes are discussed.Рассматриваются прикладные аспекты использования предварительного ранжирования вершин ориентированного взвешенного графа. Особое внимание уделяется широкому использованию такого приема в разработке эвристических алгоритмов дискретной оптимизации. Задача ранжирования имеет непосредственное отношение к проблеме определения центральности в социальных сетях, обработке больших массивов данных реального мира, но как показано в статье, явно или косвенно используется при разработке алгоритмов решения прикладных задач в качестве начального этапа построения решения. Приводятся примеры использования предварительного ранжирования, в которых продемонстрировано повышение эффективности решения некоторых прикладных задач, имеющих широкое применение в математических методах оптимизации. Дано описание структуры первой фазы вычислительного эксперимента, которая связана с получением тестовых наборов данных. Полученные данные представлены взвешенными графами, которые соответствуют нескольким группам социальной сети ВКонтакте с числом вершин в диапазоне от 9000 до 24 тысяч участников. Показано, что структурные характеристики полученных графов по числу компонент связности существенно различаются. Продемонстрированы некоторые характеристики центральности (распределения степенных последовательностей), которые имеют экспоненциальный характер. Основное внимание уделяется анализу трех алгоритмов построения иерархии ранжирования вершин графов, предлагаются новые подходы к вычислению рангов вершин с использованием информации об активности пользователей в социальных сетях. Проводится сравнение распределений полученных совокупностей рангов. Вводится понятие сходимости алгоритмов ранжирования вершин графов, а также обсуждаются различия их использования при рассмотрении данных большой размерности и необходимости построения решения в случае учета только локальных изменений

Red Light Green Light Method for Solving Large Markov Chains

Discrete-time discrete-state finite Markov chains are versatile mathematical models for a wide range of real-life stochastic processes. One of most common tasks in studies of Markov chains is computation of the stationary distribution. We propose a new general controlled, easily distributed algorithm for this task. The algorithm includes as special cases a wide range of known, very different, and previously disconnected methods including power iterations, versions of Gauss-Southwell formerly restricted to substochastic matrices, and online distributed algorithms. We prove exponential convergence of our method, demonstrate its high efficiency, and derive straightforward control strategies that achieve convergence rates faster than state-of-the-art algorithms.</p

Supplier Ranking System and Its Effect on the Reliability of the Supply Chain

Today, due to the growing use of social media and an increase in the number of A HITS with a solution in PageRank (Massimo, 2011) sharing their opinions globally, customers can review products and services in many novel ways. However, since most reviewers lack in-depth technical knowledge, the true picture concerning product quality remains unclear. Furthermore, although product defects may come from the supplier side, making it responsible for repair cost, it is ultimately the manufacturer whose name is damaged when such defects are revealed. In this context, we need to revisit the cost vs. quality equations. Observations of customer behavior towards brand name and reputation suggest that, contrary to the currently dominant model in production where manufacturers are expected to control only Tier 1 supplier and make it responsible for all higher tiers, manufacturers should also have a better hold on the entire supply chain. Said differently, while the current system considers all parts in Tier 1 as equally important, it underestimates the importance of the impact of each piece on the final product. Another flaw of the current system is that, by commonizing the pieces in several different products, such as different care models of the same manufacturer to reduce the cost, only the supplier of the most common parts will be considered essential and thus get the most attention during quality control. To address the aforementioned concerns, in the present study, we created a parts/supplier ranking algorithm and implemented it into our supply chain system. Upon ranking all suppliers and parts, we calculated the minimum number of the elements, from Tier 1 to Tier 4, that have to be checked in our supply chain. In doing so, we prioritized keeping the cost as low as possible with most inferior possible defects

Structural Results and Applications for Perturbed Markov Chains

Each day, most of us interact with a myriad of networks: we search for information on the web, connect with friends on social media platforms, and power our homes using the electrical grid. Many of these interactions have improved our lives, but some have caused new societal issues - social media facilitating the rise of fake news, for example. The goal of this thesis is to advance our understanding of these systems, in hopes improving beneficial interactions with networks while reducing the harm of detrimental ones. Our primary contributions are threefold. First, we devise new algorithms for estimating Personalized PageRank (PPR), a measure of similarity between the nodes in a network used in applications like web search and recommendation systems. In contrast to most existing PPR estimators, our algorithms exploit local graph structure to reduce estimation complexity. We show the analysis of such algorithms is tractable for certain random graph models, and that the key insights obtained from these models hold empirically for real graphs. Our second contribution is to apply ideas from the PPR literature to two other problems. First, we show that PPR estimators can be adapted to the policy evaluation problem in reinforcement learning. More specifically, we devise policy evaluation algorithms inspired by existing PPR estimators that leverage certain side information to reduce the sample complexity of existing methods. Second, we use analytical ideas from the PPR literature to show that convergence behavior and robustness are intimately related for a certain class of Markov chains. Finally, we study social learning over networks as a model for the spread of fake news. For this model, we characterize the learning outcome in terms of a novel measure of the “density” of users spreading fake news. Using this characterization, we also devise optimal strategies for seeding fake news spreaders so as to disrupt learning. These strategies empirically outperform intuitive heuristics on real social networks (despite not being provably optimal for such graphs) and thus provide new insights regarding vulnerabilities in social learning. While the topics studied in this thesis are diverse, a unifying mathematical theme is that of perturbed Markov chains. This includes perturbations that yield useful interpretations in various applications, that provide algorithmic and analytical advantages, and that disrupt some underlying system or process. Throughout the thesis, the perturbed Markov chain theme guides our analysis and suggests more general methodologies.PHDElectrical Engineering: SystemsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/155213/1/dvial_1.pd