Assortative Social Networks and Neutral Cultural Evolution

26 Sep 2010 15:25

It is a common-place observation that there are strong relationships between cultural traits and social attributes; that different social groups accept and transmit different bits of culture. Most attempts to explain this from within the social sciences (emphatically including historical materialism and its variants) argue that this is due to some causal influence of social organization on culture. ("Social being determines consciousness" --- or, once the Hegelian gas has been released, social life shapes thought.) In these views, culture varies with social position because it's an adaptation to social position, or a reflection thereof, or an expression thereof. However, it is not clear to me that this can only be explained by a causal linkage.

A simple model to test this would be as follows. Imagine a population where each individual has a couple of social traits, which can take discrete values, and a cultural trait, which can likewise take a number of discrete values. Social traits are fixed. Now form a social network that's assortative, i.e., two individuals are more likely to be directly linked the more social traits they have in common. The cultural trait is variable over time. We start with some initial random distribution, but then, at each point in time, randomly pick one individual, who randomly copies one of their neighbors. Thus, culture is completely socially-neutral, and every cultural trait is just as well adapted as every other. My prediction is that, for reasonable-looking assortative networks, we'll see a good degree of correlation between social and cultural traits, just because people will be mostly learning from those close to them socially.

A slight refinement would be to make people uniformly more likely to adopt certain values of the cultural trait than others, independently of their social position. Then I predict that the less-popular cultural values will be concentrated in the smaller sub-networks.

(One could argue that this is still "social position" shaping thought, namely one's position in the social network. But now network structure screens-off and renders causally irrelevant the content of that social position.)

I hasten to add that such a model would be perfectly compatible with the pious hope that people have good reasons for their actions and beliefs; all that's really assumed is that there is no systematic relation between those reasons and social position. (So I'm not denying agency, rationality, etc.)

Needless to say, this would massively complicate the interpretation of opinion surveys. The typical practice of regressing responses on attributes of the responders will give you results which are weird hybrids of actual links between social status and beliefs, and the residue of diffusion.

The day after writing this, I found Hidalgo, Claro and Marquet's "Simple Dynamics on Complex Networks" (cond-mat/0411295). This looks at exactly the kind of random copying dynamic I have in mind, but divides the network into "guilds", in which all members have the same in-degree. Their surprising (to me) result is that, in equilibrium, the distribution of states (i.e., cultural traits) has to be same for all guilds. However, their guilds do not, in general, correspond to socially-defined groups, so I still have some hope my intuition is not totally and completely wrong.

Update, 21 March 2005: I should also mention (now that I've read it) V. Sood and S. Redner, "Voter Model on Heterogeneous Graphs", cond-mat/0412599 (= PRL 94 (2005): 178701). This paper's starting point is the easily-seen fact that, under the pure case of the copy-a-random-neighbor dynamics I'm considering (and which is one of several very different things called "the voter model"), everyone must come to share the same opinion. That is, the consensus states are absorbing states. Sood and Redner try to calculate the mean time to consensus as a function of properties of the social network. This is going to be useful to me, but it's not quite the same thing.

While I'm updating this, I should maybe say expand on what I hinted at above, about network structure "screening off" social status from cultural traits. There are several ways of expressing this formally, but the one I have in mind relies on our ability to decompose networks into "communities", sub-networks whose members are more closely tied to one another than to outsiders. (There are many ways of doing that, too, but I like the Newman-Girvan approach, not just because Mark is a good friend whom I can persuade to share code, but also because their algorithms make sense.) So, formally, what I'm proposing is that the dynamics I'm considering will (1) lead to strong statistical dependence between social position and cultural traits, but (2) social position and cultural traits will be (nearly) independent, conditional on community membership. (These statistical dependencies can be measured in any convenient way, e.g. through mutual information, or perhaps chi-squared to get p-values.) Of course, in the pure-copying case, this will be a transient effect, since ultimately everyone will share the same opinion. One thing I'm not sure of yet is whether it's better to just look at the transients (which Sood and Redner indicate might be very long), or to introduce some amount of perturbation (e.g., through copying errors) which will lead to a non-trivial statistical equilibrium. Maybe I should just try both and see.

22 April 2005: In conversation, Eric Smith suggests that Bill Labov's work on phonological changes in American English might have enough data to actually test such a neutral model.

Update, 16 October 2007: It works. Two social types (equiprobably), binary cultural trait (initially equiprobable). Nodes form ties with probability p if they are of the same type and probability q if they are of different types. Cultural traits change by random copying, as outlined above. I've plotted the chi-squared statistic for the association between social type and cultural trait as a function of time. The black line is a run where p=0.09, q=0.01, and the assortativity coefficient of the resulting network was r = 0.80. The grey line is a run where p=q=0.05, giving a graph with r = 0.045.