Choosing the ‘β’ Parameter When Using the Bonacich Power Measure

Abstract

Abstract Bonacich (1987) suggested a family of centrality measures that provide a useful way of modeling questions of power and network constraint. However, the literature offers little guidance regarding the choice of β, the parameter which alters the way the measure accounts for the effect of having powerful contacts in ones network. In this paper I explore the way the choice of the β parameter affects the power indices the Bonacich measure generates. I consider three network properties which might affect the way the choice of β influences the Bonacich power indices. I find that in high density networks with few internal ‘chains’ and few pendants, the choice of β is largely immaterial. Conversely, in sparse networks, those with a high proportion of pendant nodes, or those with many chains, the value of β has a substantial effect on the power indices the measure generates. Next I consider whether power indices produced by interior values of β might be represented as a linear combination of “pure” vectors, those generated with values of β at either end of the parameter range and β = 0. I find that in the vast majority of cases a linear combination of “pure” vectors power is equivalent to using indices produced by interior values of β, making the choice of β largely moot. Finally, in the unlikely case that this disaggregation is inappropriate, I discuss the question of determining an appropriate value of β empirically.