Quantitative Models of Emergence

in Light of Mathematical Theory of Human Systems

Part 3:  Probabilistic Mathematical Models of New Quality Emergence



By Pavel Barseghyan, PhD

Texas and Armenia




The first two parts of the article were devoted to linear and nonlinear mathematical models of emergense, which describe the generation of human activity results and new qualities at the level of averages of their parameters.

Within the framework of such deterministic approach, many phenomena related to human activity can be described, interpreted and predicted, but these approaches encounter serious difficulties when it is necessary to take into account the various kinds of random events that accompany human activity.

The difficulties associated with such random factors are much greater when performing activities at the upper limits of human abilities.

On the other hand, it is at the upper limits of human abilities and human systems capabilities that the most important events and emergencies related to human progress and safety take place.

An adequate quantitative description of such phenomena is possible only within the framework of probabilistic approaches, which will be discussed in the third part of the article, which includes mathematical models of the emergence at the level of probability distributions of the characteristic parameters of human activity.


Although non-linear models of emergence have wide opportunities to adequately reflect different aspects of this important phenomenon, there is one area where they as deterministic models are not capable of describing randomness as an integral part of human activity.

The thing is that the emergence itself is a highly random phenomenon, especially in the areas where it is really about creating, giving birth, and emerging new phenomena and processes.

Emergence deterministic models describe the phenomena and processes in question at the mean level of the parameters, which does not allow them to form an understanding of the likelihood or risk of successful completion of human activities.

The need for probabilistic approaches in this area is due to the fact that some parameters in the equations of state for describing the emergence phenomenon are of a pronounced random character.

To illustrate the essence of the question more precisely, consider the following simplest equation of state of person’s activity [1, 2]


This equation describes the realization by a person of an activity of magnitude  with motivation  over a period of time  .

The value  in equation (1) is the number of successfully performed actions by a person per unit time, and the value  is the degree of difficulty of the actions performed.

Given that one of the main causes of failures in human activity is a delay in the performance of their actions, which gradually accumulates in order to exceed the time allotted for activities, let’s consider the use of equation (1) to analyze the duration of an individual’s activity:

In principle, all four parameters contained in this expression can be random values, so that the duration  of activity will also be random.

Depending on many circumstances, these four random variables may have different distribution functions: symmetric, asymmetric, with a heavy tail, etc.

This means that the distribution of activity duration  will be directly determined by the distributions of these four quantities, which in simple linear cases can be symmetric and have a small dispersion, and in complex nonlinear cases it can be asymmetric and have a large dispersion, and in more complex cases – heavy tail.

Given these circumstances, a methodology based on hypothetical asymmetric triangular or other distributions of  values, which are often used for simulation in the field of project management, needs serious adjustments.



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How to cite this paper: Barseghyan, P. (2020). Quantitative Models of Emergence in Light of Mathematical Theory of Human Systems – Part 3:  Probabilistic Mathematical Models of New Quality Emergence; PM World Journal, Vol. IX, Issue V, May. Available online at https://pmworldlibrary.net/wp-content/uploads/2020/04/pmwj93-May2020-Barseghyan-Emergence-Part3-probabalistic-mathematical-models.pdf



About the Author


Pavel Barseghyan, PhD

Yerevan, Armenia
Plano, Texas, USA


Dr. Pavel Barseghyan is a consultant in the field of quantitative project management, project data mining and organizational science. Has over 45 years’ experience in academia, the electronics industry, the EDA industry and Project Management Research and tools development. During the period of 1999-2010 he was the Vice President of Research for Numetrics Management Systems. Prior to joining Numetrics, Dr. Barseghyan worked as an R&D manager at Infinite Technology Corp. in Texas. He was also a founder and the president of an EDA start-up company, DAN Technologies, Ltd. that focused on high-level chip design planning and RTL structural floor planning technologies. Before joining ITC, Dr. Barseghyan was head of the Electronic Design and CAD department at the State Engineering University of Armenia, focusing on development of the Theory of Massively Interconnected Systems and its applications to electronic design. During the period of 1975-1990, he was also a member of the University Educational Policy Commission for Electronic Design and CAD Direction in the Higher Education Ministry of the former USSR. Earlier in his career he was a senior researcher in Yerevan Research and Development Institute of Mathematical Machines (Armenia). He is an author of nine monographs and textbooks and more than 100 scientific articles in the area of quantitative project management, mathematical theory of human work, electronic design and EDA methodologies, and tools development. More than 10 Ph.D. degrees have been awarded under his supervision. Dr. Barseghyan holds an MS in Electrical Engineering (1967) and Ph.D. (1972) and Doctor of Technical Sciences (1990) in Computer Engineering from Yerevan Polytechnic Institute (Armenia).

Pavel’s publications can be found here: http://www.scribd.com/pbarseghyan and here: http://pavelbarseghyan.wordpress.com/.  Pavel can be contacted at terbpl@gmail.com