Macro-Outcome measures

In order to describe the dynamics and consequences resulting from social influence in networks, moderlers have developed many different macro-level outcome measures. These measures describe the distribution of for instance opinions in a network and how this distribution changes over time. Typical outcome measures quantify (i) the degree to which consensus has been reached, (ii) the degree to which the population is fragmented into multiple distinct groups with different opinions, and (iii) the degree to which the population is polarized.

Choosing outcome measures is challenging for various reseasons. One problem is that whether a given measure is useful or not very much depends on the research question of a study. Another problem is that distributions are often hard to describe. If a certain attribute is normally distributed, for instance, only two outcome measures are needed (average and standard deviation) to perfectly describe the distribition of the attribute. Other distributions require more outcome measures and some may actually require an infinite number of measures to perfectly describe them. Another problem is that the macro-level concepts that social-influence modelers tend to study are complex and multi-dimensional. Opinion poalrization, for instance, often captures at least three aspects: (i) the existence of distinct groups, (ii) a meaningful difference between the groups, and a (iii) small number of individuals holding positions between the two groups.

A general recommendation is, therefore, to use mutliple macro-outcome measures capturing different aspects separately.

We provide here a list of measures that we have experienced as useful and point to central strengths and weaknesses for typical research problems studied by modellers of social-influence processes.

The description of an attribute's distribution depends critically on the scale on which the attribute is measured. We distinguish here between the two scales studied most commonly, nominal and continuous scales

Outcome measures for nominal attributes

Number of distinct attribute vectors

This measure counts the number of different sets of attributes represented in the population. A value of one implies consensus, a state that Axelrod (1997) called "monoculture". Higher values indicate that there is diversity. Axelrod called a population "polarized" when it consisted of subgroups who disagreed on all nominal dimensions.

Number of regions

When agents occupy positions on a lattice or a network, regions can form. A region is a set of contiguous positions with identitical arributes.

Size of the biggest cluster

The size of the biggest cluster in a network or the biggest region is very informative when it is combined with the number of distinct vectors. A value equal to the population size (N) shows that consensus has been reached. Higher values demonstrate diversity and indicate, in addtion, what form of diversity is present. For instance, when there are three clusters and the biggest has a size of N/3, then there are three equally large clusters. If the biggest cluster has a size of N-2 agents, the population consists of one big group and two isolated agents.

Outcomes measures for continuous attributes

Opinion variance

One of the most influential empirical studies on opinion polarization dynamics quantified polarization with the variance of the opinion distribution (DiMaggio et al. 1996), using a simple and easy to interpret measure. Variance is zero when the population has reached consensus. Variance is maximal when two equally large groups with maximal opinion differences have formed.

A disadvantage of this measure is that it fails to distinguish between uniform distributions and multi-modal distributions.

Literature

Esteban & Ray polarization

This is one of the most elegant and best-studied measures of polarization. It assumes that the population can be devided into distinct homogenous groups. Polarization rises when groups grow and when opinion differences between the groups grow.

This measure is very simple and is relatively easy to calculate even in huge populatons. A potential weakness for the application to social-influence dynamics of continuous attributes, however, is that the notion of groups may be problematic. Another problem is that accordingt to this measure polarization is higher with a normally distributed opinion than with a uniform opinion distribution (see the animation above), because in a normal opinion distributions, there is a dominant group in the center. Many simulation studies depart from uniform distributions and study whether opinion consensus or opinion polarization emerges. In both cases, the Esteban & Ray polarization measure grows.

Literature

  • Esteban, J. M., & Ray, D. (1994). On the measurement of polarization. Econometrica: Journal of the Econometric Society, 819-851.

  • Esteban, J., Gradín, C., & Ray, D. (2007). An extension of a measure of polarization, with an application to the income distribution of five OECD countries. The Journal of Economic Inequality, 5(1), 1-19.

Variance of distances

This measure has been developed to quantifiy opinion polarization and is defined as the variance of the Euclidian opinion distances between all pairs of agents in the population.

When the population has reached consenus, the measure adopts a value of zero. With two maximally distant and equally big subgroups, polarization adopts a value of half the range of the opinonscale.

Unlike opinion variance, this measure can distinguish between uniform and multi-modal distributions. Another advantage is that it can also describe polarization on multiple dimensions, as the pairwise distances can be based on multiple opinion dimensions. A central disadvantage of the measure is that it is computationally very demanding in big populations.

Literature

cov(fix, flex)

Sometimes, it is interesting to study whether opinions overlap with demographic differences in a population. A measure proposed for this is called cov(fix,flex). It measures the covariance between fixed (say, demographic) attributes and opinions in a population. More details can be found in Flache & Mäs (2008).

Overview papers