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We define a semantics for conditionals in terms of stochastic graphs which gives a straightforward and simple method of evaluating the probabilities of conditionals. It seems to be a good and useful method in the cases already discussed in the literature, and it can easily be extended to cover more complex situations. In particular, it allows us to describe several possible interpretations of the conditional (the global and the local interpretation, and generalizations of them) and to formalize some intuitively valid but formally incorrect considerations concerning the probabilities of conditionals under these two interpretations. It also yields a powerful method of handling more complex issues (such as nested conditionals). The stochastic graph semantics provides a satisfactory answer to Lewis’s arguments against the PC = CP principle, and defends important intuitions which connect the notion of probability of a conditional with the (standard) notion of conditional probability. It also illustrates the general problem of finding formal explications of philosophically important notions and applying mathematical methods in analyzing philosophical issues.

Algorithms of structural (topological) integration of statuses of system depending on formulary representation of monotonous structural function of availability of network service in the form of minimal disjunctive normal form - MDNF (minimal conjunctive normal form - MCNF) in class of binary stochastic models are considered., В залежності від задач, розв'язуваних користувачами мережі передачі даних (МПД), до мережі пред'являються вимоги по структурній і (або) функціональній надійності мережевої послуги. Характеристики структурної (топологічної) надійності відображають можливість мережі надавати зв'язок абонентам у довільний проміжок часу. Характеристики функціональної надійності відображають можливості мережі надавати абонентам зв'язок у проміжку заданого інтервалу часу.

Data clustering, which is to partition the given data into different groups, has attracted much attention. Recently various effective algorithms have been developed to tackle the task. Among these methods, non-negative matrix factorization (NMF) has been demonstrated to be a powerful tool. However, there are still some problems. First, the standard NMF is sensitive to noises and outliers. Although L2,1 norm based NMF improves the robustness, it is still affected easily by large noises. Second, for most graph regularized NMF, the performance highly depends on the initial similarity graph. Third, many graph-based NMF models perform the graph construction and matrix factorization in two separated steps. Thus the learned graph structure may not be optimal. To overcome the above drawbacks, we propose a robust bi-stochastic graph regularized matrix factorization (RBSMF) framework for data clustering. Specifically, we present a general loss function, which is more robust than the commonly used L 2 and L 1 functions. Besides, instead of keeping the graph fixed, we learn an adaptive similarity graph. Furthermore, the graph updating and matrix factorization are processed simultaneously, which can make the learned graph more appropriate for clustering. Extensive experiments have shown the proposed RBSMF outperforms other state-of-the-art methods.

The subject matter of the article is the process of modeling social networks. The goal is to develop a computer model of a social network with a recommendation system. The tasks to be solved are to research the methods of generating social networks, to realize the computer model of a social network with a recommender system. The methods used are graph theory, theory of algorithms, statistics theory, probability theory, object-oriented programming. The following results: the research of existing methods for modeling social networks was conducted, in particular, such social network models as the Barabasi-Albert model, the Erdős-Renyi model and the Bollobаs-Riordan model were considered. The concept of complex networks was considered. The research of the basic properties of graphs of social networks was considered. The social network computer model with a recommender system based on the modified Barabasi-Albert model with using graph database Neo4j and programming language Python was developed. The developed model allows to model a network with users and text posts and may contain following connections "friends", "follower", "published", "viewed", "like", "similar", "recommended", and also allows testing of algorithms of recommender systems and conduct research to changes in a social network after creating and proposing recommendations. The testing of the developed computer model of virtual social network with a recommender system was conducted. Conclusions. The research of various methods of modeling social networks was conducted. The concept of complex networks was investigated. The main properties of social network graphs are considered. The computer model of a social network with a recommendation system that contains various types of nodes and connections that allow testing a recommender system algorithm has been developed. The developed model of a social network with a recommender system was tested to check its similarity with real social networks. The developed computer model of a social network has the values of network graph parameters corresponding to the values characteristic of real social networks, which allows using the developed model to research the processes that can occur in real social networks., Предметом изучения в статье является процесс моделирования социальных сетей. Целью есть создание программной модели социальной сети с рекомендательной системой. Задача: исследовать методы генерации социальных сетей, реализовать программную модель социальной сети с рекомендательной системой. Методы исследования: теория графов, теория алгоритмов, теория статистики, теория вероятностей, объектно-ориентированное программирование. Получены следующие результаты: проведено исследование существующих методов моделирования социальных сетей, в частности, рассмотрены такие модели социальных сетей как модель Барабаши-Альберт, модель Эрдеша-Раньи и модель Боллобаша-Риордана; рассмотрено понятие сложных сетей; проведено исследование основных свойств графов социальных сетей; разработана программная модель социальной сети с рекомендательной системой на основе модифицированной модели Барабаш-Альберт с применением графовой базы данных Neo4j и языка программирования Python; разработанная модель позволяет моделировать сеть с пользователями и текстовыми постами и может содержать связи типа "друзья", "подписчик", "опубликован", "просмотрен", "лайк", "похожий", "рекомендован", а также позволяет тестировать алгоритмы рекомендательных систем и исследовать изменения в социальной сети после создания и предложения рекомендаций; проведено тестирование разработанной программной модели социальной сети с рекомендательной системой. Выводы. Было проведено исследование различных методов моделирования социальных сетей. Исследовано понятие сложных сетей. Рассмотрены основные свойства графов социальных сетей. Разработана программная модель социальной сети с рекомендательной системой, содержащая различные типы узлов и связей, которые позволяют тестировать алгоритмы рекомендательных систем. Проведено тестирование разработанной модели социальной сети с рекомендательной системой для проверки ее сходства с реальными социальными сетями. Разработанная программная модель социальной сети имеет значения параметров графа сети, соответствующие значениям характерным для настоящих социальных сетей, что позволяет использовать разработанную модель для исследования процессов, которые могут происходить в реальных социальных сетях., Предметом вивчення у статті є процес моделювання соціальних мереж. Метою є створення програмної моделі соціальної мережі з рекомендаційною системою. Завдання: дослідити методи генерації соціальних мереж, реалізувати програмну модель соціальної мережі з рекомендаційною системою. Методи дослідження: теорія графів, теорія алгоритмів, теорія статистики, теорія ймовірностей, об’єктно-орієнтоване програмування. Отримані такі результати: проведено дослідження існуючих методів моделювання соціальних мереж, зокрема, розглянуті такі моделі соціальних мереж як модель Барабаши-Альберт, модель Ердеша-Ран’ї та модель Болобаша-Ріордана; розглянуто поняття складних мереж; проведено дослідження основних властивостей графів соціальних мереж; розроблено програмну модель соціальної мережі з рекомендаційною системою на основі модифікованої моделі Барабаши-Альберт з застосуванням графової бази даних Neo4j та мови програмування Python; розроблена модель дозволяє моделювати мережу з користувачами та текстовими постами та може містити зв’язки типу "друзі", "підписник", "опублікований", "переглянутий", "лайк", "схожий", "рекомендований", а також дозволяє тестувати алгоритми рекомендаційних систем та досліджувати зміни у соціальній мережі після створення та пропонування рекомендацій; проведено тестування розробленої програмної моделі соціальної мережі з рекомендаційною системою. Висновки. Було проведено дослідження різних методів моделювання соціальних мереж. Досліджено поняття складних мереж. Розглянуто основні властивості графів соціальних мереж. Розроблено програмну модель соціальної мережі з рекомендаційною системою, що містить різні типи вузлів та зв’язків, які дозволяють тестувати алгоритми рекомендаційних систем. Проведено тестування розробленої моделі соціальної мережі з рекомендаційною системою для перевірки її подібності до реальних соціальних мереж. Розроблена програмна модель соціальної мережі має значення параметрів графу мережі, що відповідають значенням характерним для справжніх соціальних мереж, що дозволяє використовувати розроблену модель для дослідження процесів, які можуть відбуватися у реальних соціальних мережах.

On electronic bulletin board systems for discussion, a topic the users argue diversifies into multiple subtopics, and the entire structure becomes complicated. It is helpful to show users summarizations of the arguments because they can help in understanding the con- tents more easily without looking over from beginning to end of the discussion forum. The purpose of this paper is to propose an automatic summarization method of a single thread considering time series, reply relationships and user information. In the proposed method, a thread is restructured in several clusters by hierarchical clustering, and important sentences compressed with linguistic relationship of predicate argument structures are selected within each cluster using LexRank, which is a stochastic graph-based method for computing the relative importance of textual units. Finally, we conducted quantitative and qualitative anal- ysis, comparing the proposed method with MMR. Both experimental results demonstrate that the proposed method can reduce redundancies more and extract fewer sentences unre- lated to the whole context of the summary than the baseline. However, the proposed method included fewer important words than the baseline.

The article describes the problem of the reflection of resulting effects in the innovative activity of hotel organisations. Despite the special attention and interest in the issue under discussion, a unified approach to the category of “resulting effects” of innovative technology application regarding hotel organisations has not been developed in the research community so far that would take into account the specific characteristics of their activities. A number of existing definitions limit innovative activity to a set of actions for the development and implementation of technologically new services without considering the effects of carrying out innovative processes, the ability to maintain a certain level of innovative activity and availability of a resource complex. The content of resulting effects is described based on the group of indicators. The authors build the stochastic graph of indicators of resulting effects in the innovative processes of hotel organisations. The proposed technique was tested in ten hotel organisations of the resort city of Sochi.

Since learning automata are suitable techniques for modelling, learning, controlling and solving dynamic and distributed problems in unknown environments, in this chapter, we present a brief description of some learning automata based algorithms for applications of social network analysis. In this regards, first we introduce complex social networks and stochastic graphs as a graph model of social networks for analysis objectives. Then, several algorithms using learning automata are introduced for solving stochastic graph problems (i.e., maximum clique problem and minimum vertex covering), computing an estimation of network centrality measures, sampling from networks and community detection. In LA-based algorithms for social networks when stochastic graphs are used as a graph model, learning automata are used by the network as a means for observing the time varying parameters of the network for the purpose of network analysis. In complex social networks, the aim of the LA-based algorithms is to collect information from the social network in order to find good estimates for the network parameters and measurements using as few numbers of samples as possible.

In this paper, we look at one of the most crucial ingredient to graph signal processing: the graph. By taking a step back on the conventional approach using Gaussian weights, we are able to obtain a better spectral representation of a stochastic graph signal. Our approach focuses on learning the weights of the graphs, thus enabling better richness in the structure by incorporating both the distance and the local structure into the weights. Our results show that the graph power spectrum we obtain is closer to what we expect, and stationarity is better preserved when going from a continuous signal to its sampled counterpart on the graph. We further validate the approach on a real weather dataset.

Given a network, the statistical ensemble of its graph-Voronoi diagrams with randomly chosen cell centers exhibits properties convertible into information on the network's large scale structures. We define a node-pair level measure called {\it Voronoi cohesion} which describes the probability for sharing the same Voronoi cell, when randomly choosing $g$ centers in the network. This measure provides information based on the global context (the network in its entirety) a type of information that is not carried by other similarity measures. We explore the mathematical background of this phenomenon and several of its potential applications. A special focus is laid on the possibilities and limitations pertaining to the exploitation of the phenomenon for community detection purposes., Comment: 14 pages,10 figures

On electronic bulletin board systems for discussion, a topic diversifies into multiple subtopics, and the entire structure becomes complicated. It is helpful to show users summarizations of the arguments because they can help in understanding the contents more easily. The purpose of this paper is to propose an automatic summarization method of a single thread considering time series, reply relationships and user information. In the proposed method, all threads are structured in several clusters by hierarchical clustering, and important sentences are selected from each cluster using LexRank, which is a stochastic graph-based method for computing the relative importance of textual units. Finally, we conducted quantitative and qualitative analysis by comparing the proposed method with MMR. The experimental results demonstrate that the proposed method can reduce redundancies more and extract fewer sentences unrelated to the whole context of the summary than MMR. However, the proposed method included fewer important words than MMR.

A stochastic graph process with a Markov property is introduced to model the flow of an infectious disease over a known contact network. The model provides a probability distribution over unobserved infectious pathways. The basic reproductive number in compartmental models is generalized to a dynamic reproductive number based on the sequence of outdegrees in the graph process. The cumulative resistance and threat associated with each individual is also measured based on the cumulative indegree and outdegree of the graph process. The model is applied to the outbreak data from the 2001 foot-and-mouth (FMD) outbreak in the United Kingdom. The Canadian Journal of Statistics 40: 55–67; 2012 © 2012 Statistical Society of Canada Nous presentons un processus stochastique graphique avec propriete markovienne afin de modeliser le flux d'une maladie infectieuse dans un reseau de contacts connu. Le modele fournit une distribution de probabilite sur les parcours inconnus de l'infection. Le taux de reproduction de base dans les modeles comportementaux est generalise a un taux de reproduction dynamique base sur une suite de degres exterieurs dans le processus graphique. La resistance cumulative et le risque associes a chaque individu sont aussi mesures a l'aide des degres interieurs et exterieurs du processus graphique. Le modele est applique aux donnees de l'epidemie en 2001 de fievre aphteuse au Royaume-Uni. La revue canadienne de statistique 40: 55–67; 2012 © 2012 Societe statistique du Canada

A stochastic-computational scheme is presented for generating as-is graph structure spanning complex local geographics. In this schematics, GPS tracks uploaded by a multitude of probe vehicles are exploited to expand a network of feasible roadway patterns. By associating the GPS tracks with a multitude of Brownian motions conditioned by landmark allocation, the graph structure is successively augmented in terms of feasible roadway segments. In reference to the augmented information, symbolic design of over-the-horizon maneuvering process can be interactively substantiated into a chain of topological minors grounded on the local geographics.

Motivation Biological network querying is a problem requiring a considerable computational effort to be solved. Given a target and a query network, it aims to find occurrences of the query in the target by considering topological and node similarities (i.e. mismatches between nodes, edges, or node labels). Querying tools that deal with similarities are crucial in biological network analysis because they provide meaningful results also in case of noisy data. In addition, as the size of available networks increases steadily, existing algorithms and tools are becoming unsuitable. This is rising new challenges for the design of more efficient and accurate solutions. Results This paper presents APPAGATO, a stochastic and parallel algorithm to find approximate occurrences of a query network in biological networks. APPAGATO handles node, edge and node label mismatches. Thanks to its randomic and parallel nature, it applies to large networks and, compared with existing tools, it provides higher performance as well as statistically significant more accurate results. Tests have been performed on protein–protein interaction networks annotated with synthetic and real gene ontology terms. Case studies have been done by querying protein complexes among different species and tissues. Availability and implementation APPAGATO has been developed on top of CUDA-C ++ Toolkit 7.0 framework. The software is available online http://profs.sci.univr.it/∼bombieri/APPAGATO. Supplementary information Supplementary data are available at Bioinformatics online.

A sharp lower bound for the smallest entries, among those corresponding to edges, of doubly stochastic matrices of trees is obtained, and the trees that attain this bound are characterized. This result is used to provide a negative answer to Merris’ question in [R. Merris, Doubly stochastic graph matrices II, Linear Multilin. Algebra 45 (1998) 275–285].

We consider a stochastic graph generated by a continuous-time birth-and-death process with exponentially distributed waiting times. The vertices are the living particles, directed edges go from mothers to daughters. The size and the structure of the connected components are investigated. Furthermore, the number of connected components is determined.

Assume a completion time T0 and a graph having edges with randomly chosen weights are specified. Each time an edge is traversed, the weight is selected at random, according to a fixed probability law indexed by that edge. Successive observations are independent, and independent of realizations at other edges. We consider the stochastic shortest path and traveling salesman problems of finding paths which maximize the probability of completion by the specified time. The methodology combines techniques of machine learning with the “k-opt” heuristic. The algorithm is supported by convergence analysis, and is also shown in computational experiments to be successful for randomized versions of TSP problems already in the literature and having from 10 to 57 vertices. To our knowledge, this success is well beyond other computational stochastic traveling salesman studies reported elsewhere. A feature of the algorithm is that the distributions of the random edge weights need not be specified, and so the technique is suited to on-line operation.

Simulation of stochastic graph transformation systems (SGTS) allows us to analyse the model's behaviour. However, complexity of models limits our capability for analysis. In this paper, we aim to simplify models by abstraction while preserving relevant trends in their global behaviour. Based on a hierarchical graph model inspired by membrane systems, structural abstraction is achieved by "zooming out" of membranes, hiding their internal state. We use Bayesian networks representing dependencies on stochastic (input) parameters, as well as causal relationships between rules, for parameter learning and inference. We demonstrate and evaluate this process via two case studies, immunological response to a viral attack and reconfiguration in P2P networks.

Given an n-vertex graph G, the matrix (G) = (In + L(G)) −1 = (!ij) is called the doubly stochastic graph matrix of G, where In is the nn identity matrix and L(G) is the Laplacian matrix of G. Let !(G) be the smallest element of (G). Zhang and Wu (X.D. Zhang and J.X. Wu. Doubly stochastic matrices of trees. Appl. Math. Lett., 18:339-343, 2005.) determined the tree T with the minimum !(T) among all the n-vertex trees. In this paper, as a continuance of the Zhang and Wu's work, we determine the first d n−1

Super nodes have been introduced to improve the performance of structured P2P networks. The resulting heterogeneity benefits efficiency without compromising the decentralised nature. However, this only works as long as there are enough super nodes and the distribution of clients among them is roughly even. With the increase in the number of users or organisations preventing the use of their clients as super nodes, the overall number of candidate super nodes is limited. Thus selection and load balancing strategies are critical, especially in voice-over-IP (VoIP) networks where poor connectivity results in immediate loss of audio quality.

In this paper we introduce the two-stage stochastic graph partitioning problem and present the stochastic mixed integer programming formulation for this problem with finite explicit scenarios. For solving this problem, we present an equivalent integer linear programming formulation where some binary variables are relaxed to continuous ones. Additionally, for some specific graphs, we present a more simplified linear programming formulation. All formulations are tested on randomly generated graphs with different densities and different numbers of scenarios.

The theory of graph games with ω-regular winning conditions is the foundation for modeling and synthesizing reactive processes. In the case of stochastic reactive processes, the corresponding stochastic graph games have three players, two of them (System and Environment) behaving adversarially, and the third (Uncertainty) behaving probabilistically. We consider two problems for stochastic graph games: the qualitative problem asks for the set of states from which a player can win with probability 1 (almost-sure winning); and the quantitative problem asks for the maximal probability of winning (optimal winning) from each state. We consider ω-regular winning conditions formalized as Muller winning conditions. We present optimal memory bounds for pure almost-sure winning and optimal winning strategies in stochastic graph games with Muller winning conditions. We also present improved memory bounds for randomized almost-sure winning and optimal strategies.

The theory of graph games with ω-regular winning conditions is the foundation for modeling and synthesizing reactive processes. In the case of stochastic reactive processes, the corresponding stochastic graph games have three players, two of them (System and Environment) behaving adversarially, and the third (Uncertainty) behaving probabilistically. We consider two problems for stochastic graph games: the qualitative problem asks for the set of states from which a player can win with probability 1 (almost-sure winning); and the quantitative problem asks for the maximal probability of winning (optimal winning) fromeach state. We consider ω-regular winning conditions formalized as Muller winning conditions. We show that both the qualitative and quantitative problem for stochastic Muller games are PSPACE-complete.We also consider two well-known sub-classes of Muller objectives, namely, upwardclosed and union-closed objectives, and show that both the qualitative and quantitative problem for these sub-classes are coNP-complete.

We consider the problem of distributing a content of finite size to a group of users connected through an overlay network that is built by a peer-to-peer application. The goal is the fastest possible diffusion of the content until it reaches all the peers. Applications like Bit-Torrent or SplitStream are examples where the problem we study is of great interest. In order to represent the content diffusion process, we model the system as a stochastic graph process and define the constraints the graph evolution is subject to. The evolution of the graph is a semi-Markov process where the sojourn times are the rewards of interest for the computation of the time needed to complete the file distribution. We discuss the general properties of the constrained stochastic graphs and we show preliminary results obtained with an ad-hoc Monte-Carlo technique.

A stochastic graph game is played by two players on a game graph with probabilistic transitions. We consider stochastic graph games with ω-regular winning conditions specified as Rabin or Streett objectives. These games are NP-complete and coNP-complete, respectively. The value of the game for a player at a state s given an objective Φ is the maximal probability with which the player can guarantee the satisfaction of Φ from s. We present a strategy-improvement algorithm to compute values in stochastic Rabin games, where an improvement step involves solving Markov decision processes (MDPs) and nonstochastic Rabin games. The algorithm also computes values for stochastic Streett games but does not directly yield an optimal strategy for Streett objectives. We then show how to obtain an optimal strategy for Streett objectives by solving certain nonstochastic Streett games.

A stochastic graph game is played by two players on a game graph with probabilistic transitions. We consider stochastic graph games with ω-regular winning conditions specified as parity objectives. These games lie in NP ∩ coNP. We present a strategy improvement algorithm for stochastic parity games; this is the first non-brute-force algorithm for solving these games. From the strategy improvement algorithm we obtain a randomized subexponential-time algorithm to solve such games.

Performance modeling plays a significant role in predicting the effects of a particular design choice or in diagnosing the cause for some observed performance behavior. Especially for complex systems such as parallel computer, typically, an intended performance cannot be achieved without recourse to some form of predictive models. In performance prediction of parallel programs we distinguish static and dynamic prediction approaches of which the choice represents a fundamental trade-off between the amount of information and its accuracy. Static techniques offer the advantage of producing analytical information on the performance effects of symbolic program/machine parameters without requiring costly execution or multiple simulation runs for each different parameter setting or input data set. However, their limitations in modeling dynamic program behavior may have a profound negative impact on prediction accuracy. Besides dynamic behavior due to scheduling and resource contention, another important source of dynamic behavior is the dependency of a program on the input data set. For such data-dependent programs, the execution time of the program can vary greatly across the space of input data sets. In this thesis we present a new approach to symbolic performance modeling of parallel programs that provides information on the distribution of execution times when considering a large space of input data sets. Our approach is based on statistical moments representation of distribution. We present a low-cost algorithm that computes the moments of the program execution time based on the moments associated with sequential, conditional and parallel composition. The novelty of our analysis technique is the combination between the general validity of the analysis with moment representation and ultra low solution complexity. The accuracy of our approach is experimentally evaluated for synthetic workloads as well as many empirical workloads measured from real parallel programs. Considering its ultra-low solution complexity, our approach provides an attractive cost-performance trade-off in analytical performance modeling of data-dependent parallel programs.

Performance modeling plays a significant role in predicting the effects of a particular design choice or in diagnosing the cause for some observed performance behavior. Especially for complex systems such as parallel computer, typically, an intended performance cannot be achieved without recourse to some form of predictive models. In performance prediction of parallel programs we distinguish static and dynamic prediction approaches of which the choice represents a fundamental trade-off between the amount of information and its accuracy. Static techniques offer the advantage of producing analytical information on the performance effects of symbolic program/machine parameters without requiring costly execution or multiple simulation runs for each different parameter setting or input data set. However, their limitations in modeling dynamic program behavior may have a profound negative impact on prediction accuracy. Besides dynamic behavior due to scheduling and resource contention, another important source of dynamic behavior is the dependency of a program on the input data set. For such data-dependent programs, the execution time of the program can vary greatly across the space of input data sets. In this thesis we present a new approach to symbolic performance modeling of parallel programs that provides information on the distribution of execution times when considering a large space of input data sets. Our approach is based on statistical moments representation of distribution. We present a low-cost algorithm that computes the moments of the program execution time based on the moments associated with sequential, conditional and parallel composition. The novelty of our analysis technique is the combination between the general validity of the analysis with moment representation and ultra low solution complexity. The accuracy of our approach is experimentally evaluated for synthetic workloads as well as many empirical workloads measured from real parallel programs. Considering its ultra-low solution complexity, our approach provides an attractive cost-performance trade-off in analytical performance modeling of data-dependent parallel programs.

There are many possibilities how to parallelize an algorithm and how to map a program onto a parallel or distributed system. Performance models help to predict which implementation and which mapping are the best for a given algorithm and for a given computer configuration. Stochastic graph modeling is an appropriate method, since the execution order of tasks, their runtime distribution, and branching probabilities are represented. In this paper the modeling capabilities and the analysis techniques implemented in our tool PEPP are presented. In order to obtain relevant modeling results, measured and not only estimated model parameters are needed. They can be obtained through monitoring existing programs. A method to carry out monitoring efficiently is model-driven monitoring: model tasks are mapped onto their corresponding program activities which allows systematic and automatic program instrumentation. Model parameters can easily be calculated since the set of events is the same in modeling and monitoring. A model without timing information can be enhanced to a performance model with realistic parameters.

For parallelizing an algorithm and for mapping a given program onto a parallel or distributed system there are generally many possibilities. Performance models can help to predict which implementation and which mapping is the best for a given algorithm and for a given computer configuration. Stochastic graph modeling is an appropriate method, since the execution order of tasks, their runtime distribution, and branching probabilities are represented.

Approaches helping to overcome the difficulties in marine system simulation problems are discussed which rest upon the aggregation of components, presentation of the model in the form of a stochastic graph and design of logical information models. The simulation techniques are exemplified with their applications to forecast problems for marine shelf ecological system behaviour.

Transitivity is a central concept for many relational structures, e.g. clusterings and partial orderings. Stochastic graph models which are used to describe uncertain relational structures can be tested for transitivity by using indices based on triad counts. The pure random variation of such counts and indices is investigated assuming simple stochastic models. Some earlier results on transitive triads in tournaments are generalized, and a modified version of a randomization model by Holland and Leinhardt which simplifies the required moment calculations is introduced.

The stochastic graph theory has been applied to define axiomatically intermolecular and intramolecular reactions, as well as the phenomenon of gelation, equations for determining the date of appearance of the phenomenon, and its domain for three topologically possible principle types of systems have been given. The experimental values for gel points in structurally different systems, being in good agreement with the theoretical (±1 to 2%), lead us to conclude that the observed slight differences near the limiting conditions of the domain are certainly due to a non-equiprobable behavior of steric origin. Assimilating crosslinking (and also vulcanization) to a process of gelation taking place in two distinct and successive phases seems to lead to results in conformity with the idea of evolution and with the values of characteristic magnitudes of these techniques in as much as their experimental conditions are not much different from those theoretical herein admitted.