Birds of Moderately Different Feathers: Bandwagon Dynamics and the Threshold Heterogeneity of Network Neighbors

Abstract

This paper investigates bandwagon dynamics in social networks, using an extension of Granovetter's (1978) threshold model. The focus is on the pattern of social ties connecting actors with different participation thresholds. A benchmark model, in which actors' thresholds are similar to their neighbors' thresholds, is built from the principle of homophily. Computational experiments show that participation levels increase when network structure departs from pure homophily, such that actors have new neighbors with discrepant thresholds. Further increasing the heterogeneity of network neighbors causes the bandwagon effects to wane, however, suggesting that bandwagon dynamics are maximized when there is a balance of heterogeneity and homogeneity in social networks. This principle is consistent with insights from several empirical studies and consistent with the conclusions of other formal models.

Keywords:

• bandwagon
• heterogeneity
• interdependence
• network
• simulation
• threshold

Earlier drafts of this paper were presented at the 2004 meeting of the American Sociological Association and the Institutional Analysis seminar of the University of Washington. The author thanks James Kitts, Katherine Stovel, Michael Hechter, Edgar Kiser, Steven Pfaff, Nika Kabiri, Plillip Bonacich and Carter Butts for valuable feedback and encouragement.

Notes

¹Another well-known work is Schelling's (1978) tipping model.

²Granovetter's threshold model is expected to cover many kinds of collective behaviors that manifest bandwagon effects, but for simplicity he used participation in riots as an example throughout his analysis. For a similar reason, in this paper I will use participation in social events, such as the Evite example mentioned above, as a proxy for many other types of collective action of interest.

³Here “common knowledge” refers to a concept particularly used in game theory. When applied in the threshold model, it means that I know your threshold and you know my threshold. Further, I know you know my threshold and you know I know your threshold.

⁴Note that the following computational experiments allow agents to rewire ties rather than switch positions on a fixed lattice network.

⁵The simulation is implemented in MATLAB and the code is available from the author upon request.

⁶In network threshold models, threshold value (T) = threshold propensity (TP) × connectivity (k) (see Watts, 2002).

⁷The value of q is set somewhat arbitrarily here although it is important that it not be so large to obscure the effect of network threshold heterogeneity, which is possible, since a high q value means a tendency to have more low threshold agents and thus agents are more easily triggered.

⁸The bifurcation point k∗ decreases as q increases. This is because as q increases there are more low threshold agents and the effect of connectivity becomes trivial.

⁹The insight is consistent with Oliver et al. (1985) where they argued contribution to collective goods is less likely when the composition of actors is more homogeneous, especially when most of them are free riders.

¹⁰Watts' (1999) model selects new neighbors randomly.

¹¹k ≥ 4 is the requirement to construct a connected network. Cases of k ≥ 74 are also tested but not presented here as the benchmark model has achieved full participation and rewiring of ties is redundant.

¹²Note the three-dimensional plot is for the purpose of illustration and it does not mean the two coordinates β and γ are independent. Recall that γ is effective only when β is greater than 0.

¹³A note on the details of the model is worth noting here. In network threshold models threshold propensity (TP) should be distinguished from absolute threshold value (T). For instance, if one agent has five ties and a threshold two, compared to another agent who has ten ties and a threshold three, the former actually has a higher threshold propensity than the latter. In complete or regular networks where agents have an identical number of ties, the distinction between absolute threshold value and threshold propensity is unnecessary. But in networks when the number of ties varies across the population, threshold propensity is more informative (Watts, 2002)

. Given a lattice network with an equal number of ties per agent, the benchmark model is immune to the problem. All we need to do is assign a distribution of threshold propensity (TP) and threshold values will be the product of TP and connectivity (k). For simplicity the model does not change the absolute threshold values throughout the dynamics of tie rewiring. However, after ties are rewired there will be variations in the number of ties for agents and thus threshold propensity can change. One might therefore ask whether the improvements in participation levels we report are simply because the new distribution of threshold propensity is more friendly to bandwagon effects. Further scrutiny and experiments show this is not the case; quite the contrary, the overall threshold propensity is even higher after ties are rewired. These results can be obtained from the author.

¹⁴The reported value is the difference of participation level between the benchmark model and the setting after ties are rewired. For the latter, we fix the parameters (β, γ, and k) to values claimed to be optimal and run a hundred rounds of simulations. We then choose the highest participation level from the simulations.

¹⁵As Granovetter (1978) argued, people could have different distributions of thresholds for different reasons. We can consider network ties to be multi-stranded. Then, it is possible that one task requires the rewiring of ties to be homogeneously oriented, but for connections of a different task, the setting is nevertheless heterogeneous. The multiplicity of ties in networks can justify why it is possible agents are linked to heterogeneous others.

¹⁶The exact value of the Chi-Square test slightly varies across different rounds of experiments with the same parameter values. However, the level of statistical significance remains the same.