Comment and Response on "Choosing the 'β' Parameter when Using the Bonacich Power Measure"

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  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  parameter affects the power indices the Bonacich measure generates. I consider three network properties which might affect the way the choice of  influences the Bonacich power indices. I find that in high density networks with few internal ‘chains’ and few pendants, the choice of  is largely immaterial. Conversely, in sparse networks, those with a high proportion of pendant nodes, or those with many chains, the value of  has a substantial effect on the power indices the measure generates. Next I consider whether power indices produced by interior values of  might be represented as a linear combination of “pure” vectors, those generated with values of  at either end of the parameter range and  = 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 , making the choice of  largely moot. Finally, in the unlikely case that this disaggregation is inappropriate, I discuss the question of determining an appropriate value of  empirically.