[LINK] "Lynxes and lumps"

[rfmcdonald]

At the philosophy-focused New APPS Blog, Dennis Des Chene has a post up describing how the same mathematical models can describe two very different systems.

I’ve been reading Robert Batterman’s Devil in the details, a book that packs a lot of punch in a relatively few pages. Among its themes is that of the universality of certain mathematical models. Universality is “the slightly pretentious way in which physicists denote identical behaviour in different systems” (Berry 1987:185, quoted in Batterman 13).

That requires some unpacking. Two systems exhibit “identical” behavior if that behavior can, under suitable redescription, be seen to instantiate the same mathematical system (I use the imprecise word ‘system’ rather than a more precise term because what is instantiated need not be, for example, the graph of a single equation). They are different if, as in the case of Berry’s own examples, they have different shapes, or if, as in some cases discussed by Batterman, they are made of different stuffs. We will see yet another sort of difference below.

Let me start instead with something simple: the directed graph or digraph. Family trees and citation networks instantiate that structure: draw an arrow from x to y if x is a progenitor of y or if y is cited by x. More interestingly, so-called “scale-free” networks, though arising in different real-world situations (different in the sense of being realized on quite different scales by quite different sorts of process), obey the same statistics (for example, the number of arrows entering a node—think of links to a site—obeys a power-law distribution): the probability of a node’s having n entering links is inversely proportional to some small power n k of n. Many nodes will have only a few entering links, and a very few will have many.

I would prefer to call the phenomenon “generality”. Not all networks, let alone all the things that can be modelled by digraphs, obey power-law distributions in the distribution of links; but those that do are expected to exhibit other similarities as well—for example, to have arisen by a “rich get richer” process wherein nodes that already have many entering links are more likely to receive new entering links than nodes that have just a few entering links. Were it true that scale-free networks could arise only by such processes, we might know this quite independently of knowing the physical means by which links are made, or the causes that lead, for example, one blogger to link to others. Scale-free networks or (equivalently, under the hypothesis just mentioned) “rich get richer” networks would be a genus of network, to which the mathematics of one kind of mathematical structure applied, and whose formation occurred by a process to which again the mathematics of one kind of structure applied. Not only the network structure but the process of its formation could be described independently of the stuffs and causal processes required in any instantiation of the structure. Universality or generality, so understood, offers, in Batterman’s view, a promising way to think about, among other topics, multiple realizability and emergence.

The examples Des Chene gives are populations of lynxes and snowshoe hares in northern Canada, which follow a classic boom-bust cycle with populations of the snowshoe hare prey peaking before the populations of the lynxes that eat them and with these peaks followed by a general collapse, and the distribution of particles in Saturn's F ring, which (briefly) tend to clump together into larger particles through collisions before breaking apart thanks to the kinetic energy of relatively more massive lumps.

Deep structure is interesting.