Cost range and the stable network structures

doi:10.1016/j.socnet.2007.07.002

Social Networks

Volume 30, Issue 1, January 2008, Pages 100-101

https://doi.org/10.1016/j.socnet.2007.07.002 Get rights and content

Abstract

The work of this paper concerns with the stable structures in different cost range identified by Doreian in his paper [Doreian, P., 2006. Actor network utilities and network evolution. Social Networks 28, 137–164]. We point out some problems with his Theorem 4 and present our corrections to that theorem.

Introduction

Since Jackson and Wolinsky (1996) started with a value function that is defined on networks directly, the study concerning network formation has gained focus. A large of literature has sprung from this study based on the work of JW and here have been many significant results. Among them, the simulations of Hummon (2000) found that there were equilibrium structures that were not anticipated in the formal analyses of JW. To resolve the discrepancy, Doreian (2006) examines the cost-benefit conditions under which the different network structures emerge. By the potential evolutionary sequence with line addition and line deletion, Doreian gets three main conclusions concerning the equilibrium structures in different cost range when the number of vertexes is 3, or 4, or 5, respectively. We examine these interesting results and find out some different results for Theorem 4.

His Theorem 4 reads:

Theorem 4

For graphs with n = 5 vertices, the equilibrium structures are:

1.
For γ > δ, the null graph (G1) results;
2.
For δ > γ > δ − δ³, the star (G11), the cycle (G20), G12 and G19 result;
3.
For δ − δ³ − γ > δ−δ², the star (G11), the cycle (G20), G17 and G26 result; and
4.
If γ < δ − δ², the complete graph results.

We present three problems with this statement.

Section snippets

G17 is not the equilibrium structure for δ − δ³ > γ > δ − δ²

The cost range δ − δ³ > γ > δ − δ² implies the connecting condition δ³ < δ − γ < δ². From the utility perspective suggested by Jackson and Wolinsky (1996), it means that an indirect link with two steps is better than direct link (sometimes referred to as adjacent) and direct link is better than indirect link with three steps. Then, for vertex e and d in G17 (all relevant structures are presented in Fig. 1), the formation of line ed means that the indirect link with three steps is replaced in the calculation

The structure G19 is not the equilibrium structure for δ > γ > δ − δ³

The cost range δ > γ > δ − δ³ implies the connecting condition 0 < δ − γ < δ³ and this implies that indirect link with three steps is better than direct link. Then, for vertex b, or c, or d in G19, the severance of link bc or cd means that the direct link (bc or cd) is replaced in the calculation of the utilities. The severance of bc or cd can increase the utility δ³ − (δ − γ) for b, or c, or d so that the line bc or cd will be severed.

For example, for vertex d, the severance of link dc means that it can get

The conclusion about “1. For γ > δ, the null graph (G1) results” is incomplete

Consider the circle (G20). The utility of each vertex in G20 is 2δ − 2γ + 2δ². For γ > δ, the direct link will not provide any positive utility to vertex. So, when γ > δ, no vertex builds a new link. Then, under what conditions will a vertex in the circle sever some links when γ > δ? We analyses the problem as follows:

No matter which vertex severs a link, the structure G20 will be replaced by a line formed by vertexes a, b, c, d, and e.

Given that vertex a or c severs the link ac, it implies that the

Conclusion

By analysis, we make certain that some results of Theorem 4 in “Actor network utilities and network evolution, social networks” are inaccurate. And we think the correct conclusion is as following:

For graphs with n = 5 vertices, the equilibrium structures are:

1.
For γ > δ + δ² − δ³ − δ⁴, the null graph (G1) results;
2.
For δ + δ² − δ³ − δ⁴ > γ > δ, the null graph (G1), the cycle (G20) results;
3.
For δ > γ > δ − δ³, the star (G11), the cycle (G20), G12 result;
4.
For δ − δ³ > γ > δ − δ², the star (G11), the cycle (G20), and G26 result; and
5.
If γ <

Acknowledgements

We gratefully acknowledge the support of the National Science Foundation of China (No. 70571062 and No. 70121001).

References (3)

P. Doreian
Actor network utilities and network evolution
Social Networks
(2006)

There are more references available in the full text version of this article.

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