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Networks and Stability

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Encyclopedia of Complexity and Systems Science

Definition of the Subject

Our subject is networks, and in particular, stable networks and the game theoretic underpinnings of stable networks.

Networks are pervasive. We routinely communicate over the internet, advance our careers by networking, travel to conferences over the transportationnetwork and pay for the trip using the banking network. Doing this utilizes networks in our brain. The list could go on. While network models have hada long history in sociology, the natural sciences, and engineering (e. g., in modeling social organizations, brain architecture, and electricalcircuits), the rise of the network paradigm in economics is relatively recent. Economists are now beginning to think of political and economicinteractions as network phenomena and to model everything from terrorist activities to asset market micro structures as games ofnetwork formation. This trend in economics, which began with the seminal paper by Myerson [88] on graphs and cooperation and accelerated with the...

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Abbreviations

Homogeneous networks:

A homogeneous network consists of a finite set of nodes together with a finite set of mathematical objects called links or arcs, each identifying a connection between a pair of nodes. Given finite node set N with typical element i, a homogeneous linking network G is a finite collection of sets of the form \( { \{i,i^{\prime} \} } \) called links. Link \( { \{i,i^{\prime} \}\in G } \) indicates that nodes i and \( { i^{\prime} } \) are connected in network G. A homogeneous directed network G is a finite collection of ordered pairs (\( { i,i^{\prime} } \)) called arcs. Arc \( { (i,i^{\prime})\in G } \) indicates that nodes i and i′ are connected in network G via a connection running from i to i′. In a homogeneous network (whether it be a linking network or a directed network) all connections are of the same type.

Heterogeneous networks:

A heterogeneous network consists of a finite set of nodes together with a finite set of mathematical objects called labeled links or labeled arcs, each identifying a particular type of connection between a pair of nodes. Given finite node set N with typical element i and given finite label set A with typical element a, a heterogeneous linking network G is a finite collection of ordered pairs of the form (\( { a,\{i,i^{\prime}\} } \)) called labeled links . Labeled link \( { (a,\{i,i^{\prime}\})\in G } \) indicates that nodes i and i′ are connected in network G via a type a link. A heterogeneous directed network G is a finite collection of ordered pairs of the form \( { (a,(i,i^{\prime})) } \) called labeled arcs . Labeled arc \( { (a,(i,i^{\prime}))\in G } \) indicates that nodes i and \( { i^{\prime} } \) are connected in network G via a type a arc running from i to i′. In a heterogeneous network (whether it be a linking network or a directed network) connections can differ and are distinguished by type.

Abstract game of network formation with respect to irreflexive dominance:

An abstract game of network formation with respect to irreflexive dominance consists of a feasible set of networks \( { \mathbb{G} } \) equipped with a irreflexive dominance relation >. A dominance relation on \( { \mathbb{G} } \) is a binary relation on \( { \mathbb{G} } \) such that for all G and G′ in \( { \mathbb{G} } \), \( { G^{\prime} >G } \) (read G′ dominates G) is either true or false. The dominance relation is irreflexive if \( { G > G } \) is always false.

Abstract game of network formation with respect to path dominance :

An abstract game of network formation with respect to path dominance consists of a feasible set of networks \( { \mathbb{G} } \) equipped with a path dominance relation \( { \geq _{p} } \) induced by an irreflexive dominance relation > on \( { \mathbb{G} } \). Given networks G and G′ in \( { \mathbb{G} } \), \( { G^{\prime} \geq _{p}G } \) (read G′ path dominates G) if either \( { G^{\prime} =G } \) or there is a finite sequence of networks in \( { \mathbb{G} } \) beginning with G and ending with G′ such that each network along the sequence dominates its predecessor.

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Acknowledgments

This paper was begun while Page and Wooders were visiting CERMSEM at the University of Paris 1 in Juneand October of 2007. The authors thank CERMSEM and Paris 1 for their hospitality. URLs:http://mypage.iu.edu/%7Efpage,http://www.myrnawooders.com.

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Authors and Affiliations

  1. Department of Economics, Indiana University, Bloomington, USA

    Frank H. Page Jr.

  2. Centre d'Economie de la Sorbonne, Universite Paris 1, Pantheon–Sorbonne, France

    Frank H. Page Jr.

  3. Department of Economics, Vanderbilt University, Nashville, USA

    Myrna Wooders

  4. Department of Economics, University of Warwick, Coventry, UK

    Myrna Wooders

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  1. Frank H. Page Jr.
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  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Page Jr., F.H., Wooders, M. (2009). Networks and Stability. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_355

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