Musing: connecting dynamics to statistical inference

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The emphasis in this seminar, Anthropological Methods and Models 2008, is on inferential statistics. which is concerned with probability model testing of inferences about the processes generating your data.

One type of classical ethnography was static snapshot description, though another set the ethnography in a historical context. Archeology and history provide the time dimension. We have seen ways to introduce the longitudinal dimension even in something as simple as a community-level genealogy. The genealogical method applies not only to kinship but to other domains. Some students were surprised to hear that Victor Turner's ethnography, Schism and Continuity, combined the genealogical approaches to kinship and to community level rituals as the data for his study of how crises in communities and how a particular crisis was resolved.

A problem of inferential statistics is to understand something about dynamics -- historical, ethnographic, archaeological, sociological, political -- from outcomes (i.e., data) not only through qualitative models and narrative but through quantitative data. We can also use simulations to study possible or better, plausible dynamics. Laurent Tambayong's research on alliance formation is an example: two parameters, one function guiding change, and a whole host of longitudinal and equilibrium outcomes that match a large variety of emergent patterns that we see in the case studies on business alliances.

This week are looking for the first time (long deferred) at structural cohesion, one of the key group-level variables composed by aggregating the data in networks to show units that are likely (testably) to have causal consequences, as demonstrated in a number of studies.

Why was I so interested in this strange distribution, the q-exponential, in relation to networks? The idea is simple and very general. Structurally cohesive groups -- which generate a host of downstream consequences -- are composed of intermeshed cycles that provide the redundancies of operational, social, distributed-cognitive, cultural integration. Modeling the dynamics of cycle formation ought to be a critical step in social theory. The longitudinal outcomes of these dynamics will show a large range of variability, but are thee common processes that can be parameterized in a few fundamental ways to generate this range of variability?

[edit] Simplicity in complexity

Cosma Shalizi's [http://intersci.ss.uci.edu/wiki/htm/talks.htm talk last Friday emphasized that complex systems involve many variables interacting in complex ways. Yes, that is what he and many complexity scientists study. But if you have the right aggregated variables things are much simpler. The example this week is structural cohesion, matched with q-exponential scaling and dynamics.

[edit] Generative models

Two generative models have been constructed for this process, based on two different ideas of how cycles are formed:

1) a links to b to ... and eventually back to a. When this process is only a to b back to a we have different types of connected dyads, those in which links are reciprocated or unreciprocated. We might call these strong and weak. The intentionality of the connection is evident in the case of strong ties, but not in the weak. Georg Simmel (1909) argued that social life begins with triples or triads: when a dyadic tie (strong or weak) is embedded in a cycle of length 3 (strong if directed, a -> b -> c -> a, weak if not, complete if all ties are reciprocated). David Krackhardt and Mark S. Handcock] (2007) have demonstrated the "Simmelian stabilization effect" using inferential statistics on longitudinal data that show (in this case) dyads embedded in triads to be more stable over time than unembedded dyads. Harary, White, Moody, Powell and others (in articles with diverging coauthorships) have shown a large number of "structural k-cohesion effects" (including stability) for groups with k multiple independent cycles connecting every pair of members. In all these studies and the "Social circles" simulations of White, Tsallis et al. (2006), cycles are formed in the ordinary way.

2) A later simulation by Thurner and Tsallis (2007) forms cycles by merger of nodes that are already connected, like Tambayong's "alliance formation" merging connecting firms into cliques (completely connected).

The interesting thing about the Simmelian and structural k-cohesion effects is that they are not self-amplifying is size. Structurally k-cohesive groups, however, are scalable in the sense that they can increase in size independently of the level of k-cohesion. This is explored historically in my article for the Encyclopedia of Complexity and Systems Science. External conflict and aggression against a meta-ethnically distinct structurally k-cohesive group, for example, can provoke the growth in extent or size of the group by recruitment with the intensive variable k (level of cohesion) either remaining constant or growing slightly but without much cost in an increased number of cohesive (within-group) ties per person.

The generality of such processes as stability, growth, or dissolution of structurally k-cohesive groups leads to some interesting modeling problems.

For example, let c = the number of independent cycles in a network, and for a network of n nodes and e edges count this number as c = e - n + 1

then observe empirically the change in c relative to n:

δc / δn = cq

This is the dynamic of the 'q'-exponential.

EMPIRICALLY, with c from different types of relations, you can study where interesting dynamics are coming from, according to which independent variables, and study which form q-exponentials.

SIMULATING, you can build interaction models among these variables.

This is what we are doing with our kinship and community studies, slated for presentation December 2008.

[edit] References

1. Krackhardt, David, and Mark S. Handcock 2007 Heider vs Simmel: Emergent Features in Dynamic Structures
2. Moody and White (2003)

$Powell, White... (2005)

1. Thurner and Tsallis (2007)
2. White and Harary (2001)
3. White, Powell, Moody (2005)