## 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.

- Useful references (mostly useful for me):
- Aris Anagnostopoulos, Ravi Kumar and Mohammad Mahdian, "Influence and Correlation in Social Networks", in KDD 2008 [PDF reprint]
- Jason Kaufman, "Endogenous explanation in the sociology of
culture", Annal
Review of Sociology
**30**(2004): 335--357 [With the caveat that only the third of his three sets of approaches actually sound like "explanations" to me. PDF reprint via Prof. Kaufman.] - V. Sood and S. Redner, "Voter Model on Heterogeneous Graphs", cond-mat/0412599 (= PRL
**94**(2005): 178701 - Scott E. Page, Leonard M. Sander and Casey M. Schneider-Mizell,
"Conformity and Dissonance in Generalized Voter Models", Journal of Statistical Physics
**128**(2007): 1279--1287

- Modesty forbids me to recommend:
- CRS, "Social Media as Windows on the Social Life of the Mind", arxiv:0710.4911, for the 2008 AAAI symposium on "Social Information Processing" [The relevant parts are however mostly superseded by the next paper]
- CRS and Andrew C. Thomas, "Homophily and Contagion Are Generically Confounded in Observational Social Network Studies", arxiv:1004.4704 [Self-promoting weblog post]

- To read:
- Daron Acemoglu, Giacomo Como, Fabio Fagnani, Asuman Ozdaglar, "Opinion fluctuations and disagreement in social networks", arxiv:1009.2653
- Elena Agliari, Raffaella Burioni and Pierluigi Contucci, "A diffusive
strategic dynamics for social systems", Journal of Statistical Physics
**139**(2010): 478--491 - T. Antal, S. Redner and V. Sood, "Evolutionary Dynamics on
Degree-Heterogeneous Graphs", Physical Review Letters
**96**(2006): 188104 - G. J. Baxter, R. A. Blythe and A. J. McKane, "Fixation and
Consensus Times on a Network: A Unified
Approach", Physical Review
Letters
**101**(2008): 258701, arxiv:0801.3083 - I. J. Benczik, S. Z. Benczik, B. Schmittmann, R. K. P. Zia, "Lack of consensus in social systems", arxiv:0709.4042
- R. A. Blythe, "Ordering in voter models on networks: Exact reduction to a single-coordinate diffusion", arxiv:1006.1557 [Wow: "We study the voter model and related random-copying processes on arbitrarily complex network structures. Through a representation of the dynamics as a particle reaction process, we show that a quantity measuring the degree of order in a finite system is, under certain conditions, exactly governed by a universal diffusion equation. Whenever this reduction occurs, the details of the network structure and random-copying process affect only a single parameter in the diffusion equation. The validity of the reduction can be established with considerably less information than one might expect: it suffices to know just two characteristic timescales within the dynamics of a single pair of reacting particles. We develop methods to identify these timescales, and apply them to deterministic and random network structures. We focus in particular on how the ordering time is affected by degree correlations, since such effects are hard to access by existing theoretical approaches."]
- Claudio Castellano, "Effect of network topology on the ordering dynamics of voter models", cond-mat/0504522
- Claudio Castellano, Vittorio Loreto, Alain Barrat, Federico Cecconi
and Domenico Parisi, "Comparison of voter and Glauber oridering dynamics on
networks", Physical Review
E
**71**(2005): 066107 - Damon Centola, Juan Carlos Gonzalez-Avella, Victor M. Eguiluz and Maxi San Miguel, "Homophily, Cultural Drift and the Co-Evolution of Cultural Groups", physics/0609213
- Munmun De Choudhury, Hari Sundaram, Ajita John, Doree Duncan Seligmann, Aisling Kelliher, "``Birds of a Feather'': Does User Homophily Impact Information Diffusion in Social Media?", arxiv:1006.1702
- Andreas Flache and Michael W. Macy
- "Local Convergence and Global Diversity: From Interpersonal to Social Influence", arxiv:0808.2710
- "Local Convergence and Global Diversity: The Robustness of Cultural Homophily", arxiv:physics/0701333

- Noah E. Friedkin, A Structural Theory of Social Influence [Blurb]
- Aram Galstyan and Paul Cohen, "Cascading dynamics in modular
networks", Physical Review
E
**75**(2007): 036109 - Aram Galstyan, Vahe Musoyan and Paul Cohen, "Maximizing Influence
Propagation in Networks with Community Structure", Physical
Review E
**79**(2009): 056102, arxiv:0905.1108 - Benjamin Golub, Matthew O. Jackson, "How Homophily Affects Diffusion and Learning in Networks", arxiv:0811.4013
- Laszlo Gulyas and Elenna R. Dugundji, "Emergent Opinion Dynamics on Endogeneous Networks", physics/0610125
- Petter Holme and Andreas Gronlund, "Modelling the dynamics of youth subcultures", physics/0504181
- Petter Holme and M. E. J. Newman, "Nonequilibrium phase transition in the coevolution of networks and opinions", Physical Review E
**74**(2006): 056108 - Robert Huckfeldt, Paul E. Johnson and John Sprague, Political Disagreement: The Survival of Diverse Opinions Within Communication Networks [Sounds very cool. Blurb]
- Elihu Katz and Paul Lazarsfeld, Personal Influence: The Part Played by People in the Flow of Mass Communications
- Marcelo N. Kuperman, "Cultural propagation on social networks", nlin.AO/0509004 [The Axelrod model]
- R. Lambiotte and M. Ausloos, "Coexistence of opposite opinions in a network with communities", physics/0703266 = Journal of Statistical Mechanics (2007) P08026
- R. Lambiotte, M. Ausloos, and J. A. Hoyst, "Majority model on a
network with communities", Physical Review
E
**75**(2007): 030101 - Naoki Masuda, N. Gilbert and S. Redner, "Heterogeneous voter
models", Physical Review E
**82**(2010): 010103, arxiv:1003.0768 - Cecilia Nardini, Balazs Kozma, Alain Barrat, "Who's talking first? Consensus or lack thereof in coevolving opinion formation models", Physical Review Letters
**100**(2008): 158701, arxiv:0711.1261 - Emanuele Pugliese and Claudio Castellano, "Heterogeneous pair approximation for voter models on networks", arxiv:0903.5489
- Usha Nandini Raghavan, Reka Albert, Soundar Kumara, "Near linear time algorithm to detect community structures in large-scale networks", arxiv:0709.2938 ["every node is initialized with a unique label and at every step each node adopts the label that most of its neighbors currently have". I suspect using this definition of "community" would make it a tautology that community membership makes social position irrelevant for culture.]
- Fabio Rojas and Tom Howe, "Contact Patterns and Aggregate Opinion Levels: Results from a Simulation Study" [PDF preprint]
- Casey M. Schneider-Mizell and Leonard M. Sander, "A generalized
voter model on complex networks", Journal of Statistical Physics
**136**(2009): 59--71 - M. Angeles Serrano, Konstantin Klemm, Federico Vazquez, Victor M. Eguiluz, Maxi San Miguel, "Conservation laws for voter-like models on directed networks", arxiv:0902.1769
- Anja Sturm and Jan Swart, "Voter models with heterozygosity selection", math.PR/0701555
- Krzysztof Suchecki, Víctor M. Eguíluz, and Maxi San
Miguel, "Voter model dynamics in complex networks: Role of dimensionality,
disorder, and degree distribution", Physical Review
E
**72**(2005): 036132 - F. Vazquez, V.M. Eguiluz, M. San Miguel, "Generic absorbing transition in coevolution dynamics", arxiv:0710.4910 ["We study a coevolution voter model on a network that evolves according to the state of the nodes. In a single update, a link between opposite-state nodes is rewired with probability $p$, while with probability $1-p$ one of the nodes takes its neighbor's state." In what sense is this generic, however?]
- John R. Zaller, The Nature and Origins of Mass Opinion [Blurb]
- Alan S. Zuckerman, Josip Dasovic, Jennifer Fitzgerald, Partisan Families: The Social Logic of Bounded Partisanship in Germany and Britain [blurb]