Local Structural Properties and Attribute Characteristics in 2-mode Networks: p∗ Models to Map Choices of Theater Events¹

This research was partly supported by the Research Foundation Flanders (FWO Vlaanderen) and the “Bijzonder Onderzoeksfonds” from Ghent University. We would like to thank two anonymous reviewers for their helpful comments.

Abstract

Choices of plays made by theatergoers can be considered as a 2-mode or affiliation network. In this article we illustrate how p∗ models (an exponential family of distributions for random graphs) can be used to uncover patterns of choices. Based on audience research in three theater institutions in Ghent (Belgium), we analyze the loyalty for an institution, the diversity in audience composition as well as more complex local structural patterns. The results seem to indicate that theater audiences are generally loyal to an institution. However, when controlling for more complex structural patterns, such as the co-attendance of identical plays, the tendency toward mobility across institutions becomes more pronounced. By integrating both attribute-specific patterns and more complex structural patterns in the same model social network analysis offers the possibility to investigate differences in co-attendance, loyalty and audience diversity.

Keywords:

• affiliation network
• arts participation
• 2-mode network
• network analysis
• p∗ models

Notes

²Like we mentioned before, this chance is higher for people with a preference for experimental drama: they attend plays more frequently than members of the audience with more conventional aesthetic expectations (Roose and Waege, 2003).

³This is a pseudo-likelihood method and there is some uncertainty about the properties of the estimation (cf. Snijders, 2002; Snijders et al., 2006). Hence, the results from such pseudo-likelihood methods need to be treated with considerable caution. Nevertheless, using the results of pseudo-likelihood estimation is quite common (e.g. Moody, 2001; Faust and Skvoretz, 2002; Lubbers, 2003; Baerman et al., 2004; Contractor et al., 2006). More recently, simulation techniques have been proposed, which open the way for a more reliable parameter estimation (e.g., Snijders, 2002; Snijders et al., 2006). A number of packages are available to estimate parameters for 1-mode networks, such as Stocnet (Boer et al., 2003), Statnet (Handcock et al., 2003), and PNet (Wang et al., 2006). However, up until this date no such package is generally available for 2-mode networks that allows us to test the full model needed here, although considerable improvement has been made in recent years.

⁴As one reviewer aptly remarks, this sampling strategy will result in an overrepresentation of the frequent visitors in our sample (cf. Roose et al., 2003). However, since the sample was proportional to the audience attending the institution we will make the assumption that frequent visitors are equally overrepresented in all three institutions, which would result in a bias for the overall variance only. Of course, it would have been more convenient if we could have selected all spectators of all the plays within a certain period. Yet, such an approach is very expensive and will come at the cost of generalizability.

∗∗∗p < 0.001; ∗∗p < 0.01; ∗p < 0.05.

^a significance compared to model 1.1.

∗∗∗p < 0.001; ∗∗p < 0.01; ∗p < 0.05 (N = 4165).

Note: Parameters for grey part of table are obtained by using different reference group.

For actors (subscript before comma): 1 = traditional/2 = experimental.

For events (subscript after comma): 1 = Vooruit/2 = Nieuwpoort/3 = NTGent.

∗∗∗p < 0.001; ∗∗p < 0.01; ∗p < 0.05.

∗∗∗p < 0.001; ∗∗p < 0.01; ∗p < 0.05.

∗∗∗p < 0.001; ∗∗p < 0.01; ∗p < 0.05.

⁵This parameter now becomes the inclination if the value for the actor 2-star effect is 0 (i.e., no other events are chosen). This explains the low density value when 2-star effects are included.

∗∗∗p < 0.001; ∗∗p < 0.01; ∗p < 0.05.

∗∗∗p < 0.001; ∗∗p < 0.01; ∗p < 0.05.

∗∗∗p < 0.001; ∗∗p < 0.01; ∗p < 0.05.

∗∗∗p < 0.001; ∗∗p < 0.01; ∗p < 0.05.

∗∗∗p < 0.001; ∗∗p < 0.01; ∗p < 0.05.

∗∗∗p < 0.001; ∗∗p < 0.01; ∗p < 0.05.

⁶Due to systematic sampling of spectators at the entrance it is very unlikely that both partners of a couple are included in the sample. Hence, this can be ruled out as one of the reasons for identical co-nomination.

⁷Because we include the overall effects, (U + W − 1) of the actor-event-specific patterns (C_u,w) and one pattern of the actor-specific (C _u,X ) and event-specific (C _X,w ) patterns are linear dependent on the other patterns of the same type, with U the number of categories for the actor-attribute, and W the number of categories for the event-attribute. Hence the parameters where the actor takes the first value and/or the event belongs to category 1 are excluded from the logistic regression analysis.

⁸Although S_X,wX and S_X,Xy can be differentiated at a dyadic level for the calculation of the change score, they represent one common parameter in the analysis without a difference between w and y, since they correspond to the same clique in the moral graph (Robins and Pattison, 2005).

This research was partly supported by the Research Foundation Flanders (FWO Vlaanderen) and the “Bijzonder Onderzoeksfonds” from Ghent University. We would like to thank two anonymous reviewers for their helpful comments.