# The degree measure as utility function over positions in networks

*René Van Den Brink* () and
*Agnieszka Rusinowska*

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René Van Den Brink: Department of Econometrics and Tinbergen Institute - VU University Amsterdam

Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) from HAL

**Abstract:**
In this paper, we connect the social network theory on centrality measures to the economic theory of preferences and utility. Using the fact that networks form a special class of cooperative TU-games, we provide a foundation for the degree measure as a von Neumann-Morgenstern expected utility function reflecting preferences over being in different positions in different networks. The famous degree measure assigns to every position in a weighted network the sum of the weights of all links with its neighbours. A crucial property of a preference relation over network positions is neutrality to ordinary risk. If an expected utility function over network positions satisfies this property and some regularity properties, then it must be represented by a utility function that is a multiple of the degree centrality measure. We show this in three steps. First, we characterize the degree measure as a centrality measure for weighted networks using four natural axioms. Second, we relate these network centrality axioms to properties of preference relations over positions in networks. Third, we show that the expected utility function is equal to a multiple of the degree measure if and only if it represents a regular preference relation that is neutral to ordinary risk. Similarly, we characterize a class of affine combinations of the outdegree and indegree measure in weighted directed networks and deliver its interpretation as a von Neumann-Morgenstern expected utility function.

**Keywords:** Weighted network; network centrality; utility function; degree centrality; von Neumann-Morgenstern expected utility function; cooperative TU-game; weighted directed network; Réseau pondéré; centralité; fonction d'utilité; centralité de degré; fonction d'utilité attendue de von Neumann-Morgenstern; jeu coopératif; réseau pondéré orienté (search for similar items in EconPapers)

**Date:** 2017-07

**New Economics Papers:** this item is included in nep-gth, nep-mic and nep-upt

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**Published** in 2017

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**Related works:**

Working Paper: The degree measure as utility function over positions in networks (2017)

Working Paper: The degree measure as utility function over positions in networks (2017)

Working Paper: The Degree Measure as Utility Function over Positions in Networks (2017)

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**Persistent link:** https://EconPapers.repec.org/RePEc:hal:cesptp:halshs-01592181

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