Small worlds: the dynamics of networks between order and randomness:

*“One obvious candidate for the most highly clustered graph possible is the complete graph, in which every vertex is adjacent to every other vertex. However, this construction automatically violates the sparseness condition, as it necessarily has k = n — 1. What is required, then, is a graph that is globally sparse but locally dense. that is, with k << n and y ~ 1.*

*A better solution is what might be termed the caveman graph, which consists of a number of fully connected clusters (or "caves") in which every member is adjacent with every other (Fig. 4.1). Every vertex has degree ky and so each cluster must consist of n_local = k + 1 vertices, and there must be n_global = n/(k + 1) such clusters. Note also that all edges are part of multiple triads, so no edge is a shortcut. Hence the caveman graph satisfies the sparseness condition and still has y = 1, but it fails to satisfy another essential condition: that it be connected. The best solution, then, appears to be a close approximation to the caveman graph that is both periodic and connected.“*

The graph on the right: http://i.imgur.com/6XNkb.png

From Justin’s article: http://www.ncbi.nlm.nih.gov/pubmed/21697174

Looks like a continuous (hybrid?) approximation of this: http://mathworld.wolfram.com/CavemanGraph.html

Which is interesting in light of this: http://www.ncbi.nlm.nih.gov/pubmed/22197359

…

Also-also: http://www.scribd.com/doc/74944514/

]]>“Standing variation harbored in natural population provides important raw material for selection to act upon, in particular after a sudden change in environments or hybridization events (20)”

]]>http://www.ncbi.nlm.nih.gov/pubmed/21697174

“”The attained ﬁtness is maximal at K = 3 to 5, from which we infer

that an intermediate amount of epistasis and pleiotropy is most

conducive to adaptation”

using the NK model:

“NK model of genetic interactions [31–33] consists of circular, binary

sequences encoding the alleles at N loci, where each locus contributes

to the ﬁtness of the haplotype via an interaction with K other loci.””

http://www.ncbi.nlm.nih.gov/pubmed/19366665

Neher RA, Shraiman BI., Kavli Institute for Theoretical Physics, University of California, Santa Barbara

“We demonstrate that a large number of polymorphic interacting loci can, despite frequent recombination, exhibit cooperative behavior that locks alleles into favorable genotypes leading to a population consisting of a set of competing clones. …Our results demonstrate that the collective effect of many weak epistatic interactions can have dramatic effects on the population structure.”

Which is precisely what you’d expect if the cows evolved to be spherical. *Some* of our DNA has clearly been modularized (i.e. selected specifically for a lack of polygenicity). It’s like an approximation of a local function (say, a bell curve) that consists of a discrete (but infinite) sum of frequencies (sinuosoids of some sort). A finite approximation will repeat eventually (periodicitypolygenicity), but an infinite (and non-repeating) one runs into all these silly “philosophical issues” when you try to apply it outside theoretical mathematics. Thankfully thermodynamics accounts for the fuzzyness and, lo, functional modularity exists (except perhaps in the eyes of a few naysayers).

Evidently this is not the entire story.

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