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Tadeusz Sozański: Exchange Networks |
STRUKTURA WYMIANA WŁADZA (in English) (in Polish) / THE MATHE- MATICS OF EXCHANGE NETWORKS |
I owe my interest in exchange networks to Professor Jacek Szmatka (1950-2001), the late head of the Chair of Research on Group Processes he founded in 1989 (as Microsociological Laboratory) at the Jagiellonian University, Institute of Sociology (the chair ceased to exist in 2006). When I joined his research team in 1990, Jacek was doing his first network exchange experiments he had designed with his American colleague David Willer from the University of South Carolina. The aim of those experiments was to test some hypotheses on power distribution in simple exchange networks first studied by Willer (Theory and Experimental Investigation of Social Structures,1987). Willer’s Elementary Theory of social relations deals, in particular, with exchange systems endowed with network structure. In an exchange network, transactions (mutually agreed-on bilateral flows of valued resources or divisions of a resource pool) are subject to certain structural constraints absent in a free market, namely: (1) a transaction can be concluded solely by two actors whose positions are connected with each other in a fixed communication network; (2) not all configurations of transactions are permitted in a single "negotiation round" but only those consistent with a definite rule, such as the k-exchange rule which requires of every actor to make in every round no more one deal with each of at most k different partners. / This section of my website is to document my involvement in mathematical and experimental research on exchange networks. You will find here information on the book Struktura, wymiana, władza (Structure, Exchange, and Power; in Polish). It is a collection of papers I edited with Jacek Szmatka and Marian Kempny as 2nd and 3rd co-editors. The volume contains my first two papers on exchange networks. The first of them (A Tentative Formalization of Network Exchange Theory) deals with conceptual foundations of network exchange paradigm, the other one (Hierarchical Exchange Systems. A Replication of an Experiment by David Willer) reports on the first computer-aided exchange experiment I did in 1990 in our Lab (we had then just one 8-bit computer, but soon, owing to a research grant won by Professor Szmatka, the Lab was equipped with a computer network similar to that used in South Carolina). Both papers are included in the list of my most important publications. Unfortunately, I was too busy with other work to prepare their English versions. I decided to write all future works in mathematical sociology in English. / When Structure, Exchange, and Power appeared in print in 1993, I started work on the second book, The Mathematics of Exchange Networks. My aim was to present in it my own mathematical results in each of three main formal approaches to network exchange (Willer-Markovsky-Skvoretz's approach based on the notion of exclusionary power; equal dependence approach started by Emerson and developed by Cook and Yamagishi; applying the multi-person games to exchange networks, as proposed by Bienenstock and Bonacich). Part II, which consists of Chapters 3, 4, 5, was ready in August 2004. The preprint (some 200 pages with bibliography) had since then been available here until September 2011. The three chapters will reappear on my website as soon as I reedit them so as to adjust their content and layout to preceding Chapters 1 and 2. If you want to see chapters 3,4, and 5 in their current form, send me a request by email. Most results given in Chapter 3, in particular, theorems on "strong" and "weak" variety of exclusionary power, remain unpublished. They will be shown to the world for the first time in my book. Chapter 4 also contains many unpublished results along with those coming from the paper Toward a Formal Theory of Equilibrium in Network Exchange Systems - my contribution to the volume Status, Network, and Structure. Theory Development in Group Processes (Stanford University Press, 1997) edited by Jacek Szmatka, John Skvoretz, and Joseph Berger. Most theorems given in Chapter 5 have already been presented in my papers: A Graph-Theoretic Characterization of the Core in a Homogeneous Generalized Assignment game (Banach Center Publications, vol. 71, 2006) and On the Core of Characteristic Function Games Associated with Exchange Networks (Social Networks 28, 2006/4). /
While Chapter 2 is still being written, Chapter 1, which I completed in August 2011, is now made available on my website. Please contact me by email if you have any questions or comments or would like to make a reference to this text. The initial chapter of my book is very long (nearly 90 pages). It was being written incredibly long as I was unable to resist the temptation to present in there my meta-theoretical reflections and conclusions concerning the nature of scientific knowledge, mathematical and sociological structuralism, and "foundational problems of the social science." / Below you can see the table of contents of my book. The fragments which have been completed are printed in blue, those still in process in red. THE MATHEMATICS OF EXCHANGE NETWORKS Table of contents
Preface Part I: STRUCTURAL MODELS Chapter 1: Structural Mathematical Sociology 1.1. Basics of the methodology of the basic sciences 1.2. Exact sciences 1.3. Foundations of social science. Social mathematics and mathematical sociology 1.4. "Structure" and "structuralism" in the social sciences 1.5. "Structure" and "structuralism" in mathematics 1.6. Mathematical modeling of empirical systems
Chapter 2: Exchange Relations and Network Exchange Systems 2.1. The mathematics of social action 2.2. The Elementary Theory of social relations 2.3. Exchange as a resource flow. The Edgeworth Box 2.4. The Nash bargaining model 2.5. The concept of an exchange network (Emerson) 2.6. An introduction to graph theory 2.7. A formal model of a network exchange system
Part II: MATHEMATICAL THEORIES OF POWER IN EXCHANGE NETWORKS
Chapter 3: Exclusion and Power 3. 1. First attempts to define power in exchange networks 3. 2. Graph-theoretic Power Index 3. 3. The definition of exclusion and elementary power relation 3. 4. Equipower relation. Extending elementary relations 3. 5. Power parameters 3. 6. Probability power indices 3. 7. Assessing the size of power in a network line 3. 8. Troubles with defining strong and weak power 3. 9. Structural types of positions. Equipower networks 3.10. Exchange-seek relation generated by the elementary relations 3.11. Strong and weak power: how the definition was sought and found 3.12. The typology of power components 3.13. Structural types of power networks. Subnetworks in power networks 3.14. Optimal transaction sets 3.15. Characterizations of strong power networks 3.16. Adding and deleting links in strong power networks 3.17. The size of strong and weak power 3.18. Theories predicting the size of payoff advantage 3.19. Expected value models of network exchange (Friedkin) 3.20. Iterative resistance theory Chapter 4: The Principle of Equal Dependence
4.1. Out of Emerson's definition of dyadic dependence to a concept of networkwide equilibrium 4.2. Profit demand matrices and their properties: equal dependence and balance 4.3. Exchange-seek relation and power 4.4. Two dynamical systems associated with a profit pool network 4.5. More theorems on power distribution 4.6. The problem of power order indeterminacy. Power in the context of equidependence theory: conclusions Chapter 5: Game Theory and Exchange Networks 5.1. Multiperson games in characteristic function form 5.2. The multiperson game associated with an exchange network 5.3. Elementary properties of the core in generalized assignment games 5.4. The point covering number of a graph and the core of the game associated with a homogenous one-exchange network 5.5. A linear-algebraic criterion of the existence of the core for the Bienenstock-Bonacich game. 5.6. Structural types of game-indecomposable networks 5.7. Decomposition of a one-exchange network into game-components 5.8. Stability in network games 5.9. The Shapley value 5.10. The kernel 5.11. Concluding remarks on the mathematics of exchange networks Part III: EXPERIMENTAL RESEARCH ON EXCHANGE NETWORKS
Chapter 6: Experimental Studies of Simplest One-Exchange Networks
Appendices: Definitions of NET mathematical terms; List of figures; List of tables Subject and name index Bibliography Tadeusz Sozański October 2004, September 2008, September 20, 2011, October 11, 2013 |
