🚀 go-pugleaf

On Sat, 8 Apr 2000, "Star Dancing" wrote:

> A)    In S. J. Gould's book Full House, chap. 14, he displays two
> curves relating "frequency of occurrence" to "complexity" of species.
> Though he does not elaborate on the definition of frequency, it
> could be reasonably argued as total number of species, individuals,
                      ^^^^^^
                     construed?

> biomass or even energy embodied by species.  

It might be helpful to know three "complexity measures", two of which have
been widely used in ecology, and all three of which have been frequently
discussed in bionet.info-theory.

The first was introduced by George Gaylord Simpson (and independently, in
cryptography, by William F. Friedman).  Suppose we have a set of
individuals X partitioned into a finite set of classes A_j.  
Equivalently, suppose we have an equivalent relation ~ on X.  Then the
quotient X/~ is the set of classes X/~ = {A_1, .. A_s}.  If X is finite,
there is an obvious way to assign each A_j a frequency

 f_j = |A_j|/|X|

Then Simpson's "entropy" is

 S(X/~) = sum_j f_j^2

where f_j is the frequency of the j-th class, where of course we require
that

 f_j > 0   sum_j f_j = 1.

Here, S(X/~) is of course nothing other than the probability that two
individuals picked at random belong to the same class.

A second complexity measure arises by counting the number of partitions
yielding classes of the same sizes as in a given partition; this is given
by the multinomial coefficient

         n!
 ------------------
 n_1! n_2! ... n_s!

where |X| = n and |A_j| = n_j.  Then Boltzmann's entropy is

               [      n!       ]
 H(X/~) =  log [ ------------- ]
               [ n_1! ... n_s! ]

Notice that this has some natural "formal properties" we would expect in
any measure of the difficulty of picking out a single class in X/~ from
the others.   In particular, if = is a refinement of ~, i.e. if each class
in X/= is a subset of some class in X/~, then

 H(X/=) = H(X/~) + sum_j H(A_j/=)

where X/~ = {A_1, A_2, .. A_s}.  That is, the complexity of the whole is
the sum of the complexity of each part, plus the complexity of the
partition into the parts.  For this reason alone, Boltzmann's entropy is
far more meaningful than Simpson's entropy.

Even if X is not finite, we might be able to assign a probability to each
class. For example, if mu is a probability measure on X then we can set
p_j = mu(A_j) where now

 p_j >= 0   sum_j p_j = 1.

In this situation, a third "complexity measure", or rather, a third
measure of "variety" or "ambiguity" or the "difficulty of description" is
Shannon's discrete entropy

 H_p(X/~) = -sum_j p_j log p_j

Shannon's entropy and Boltzmann's entropy satisfy essentially the same
formal properties; in particular, if = refines ~ then

 H(X/=) = H(X/~) + sum_j p_j H(A_j/=)

That is, the variety of X/= is the variety of the partition into "clumps"
formed from the classes of =, plus the -average- variety of each clump.  
To see the formal analogy, notice that the "normalized" Boltzmann entropy

 N(X/=) = H(X/=)/|X|

satisfies

 N(X/=) = N(X/~) + sum_j |A_j|/|X| N(A_j/=)

where clearly p_j is analogous to |A_j|/|X|.

In ecology, in the fifties, X was taken to be the set of individual
organisms in an ecosystem and the equivalence relation was x~y if x,y
belong to the same biological species (a tricky concept, as Simpson was
well aware, but never mind).  However, one could just as easily take X to
be the set of biomolecules in an individual organism, and x~y iff x,y are
chemically the same (e.g. a particular sugar, or a particular enzyme).

There are of course more sophisticated notions of "biocomplexity".  For
example, suppose we wanted for formulate a theory of the biocomplexity of
cells which would allow us to compare (say) "the biocomplexity of a
paramecium" with "the biocomplexity of a yeast cell".

As a rule, the best way of proceeding is to formulate a graded theory of
complexity. At the top, we define a high grade complexity, a more or less
complete mathematical model of a bacterium (say).

For example, we could model each biomolecule as an ideal in a polynomial
ring with a very large number of variables.  The ideal mathematically
defines a variety which represents the "configuration space" of the
molecule, including the vibrational and rotational modes in each molecular
bond, global self avoidance, and so forth.  We could then pass by functors
(structure preserving transformations) to categories of simpler
mathematical structures, which yield "lower grade invariants" which are
less faithful mathematical representations (involve idealizations and
simplifications) but which are easier to compute.  For example, we could
represent the molecule as a graph G with edge and vertex colorings.

We could pass from the graph to a still lower grade invariant, the Tutte
polynomial or dichromatic polynomial of the graph Z_G(x,y). The Tutte
polynomial has the nice property that the number Z_G(n,-1) gives the
number of vertex colorings by n colors, but this is only the beginning.  
It can be defined recursively by three laws:

1. Z_(*)(x,y) = 1, where * = the one vertex trivial graph (the final
object in the category of graphs)

2. Z_(* v G)(x,y) = x Z_G(x,y), where * v G is the graph obtained by
adding a new vertex but no new edges to G,

3. Z_G)(x,y) = Z_(G-e)(x,y) + y Z_(G/e)(x,y), where 

(a) G is a graph containing an edge, 

(b) G-e is the graph obtained by removing the edge e ("cut"),

(c) G/e is the graph obtained by collapsing e to a vertex ("fuse").

In a word, the variable y keeps track of the number of cut/fuse
decompositions as we break up the graph into progressively simpler pieces,
and x keeps track of the number of isolated vertices which are created in
this process.  This is an elegant way of definiting the Tutte polynomial,
but it can be computed in other ways.  The Tutte polynomial turns out to
contain a great deal of combinatorial information about the graph (see
Bollobas, Modern Graph Theory, Springer, 1999).

Next, we can consider the bacterium as the configuration space obtained by
amalgamating the configuration space for each biomolecule (not each
species of biomolecule, but each individual biomolecule).  Then the
corresponding collection of labeled graphs (possibly topologically linked)
is a high grade complexity measure, and the Tutte polynomial of the
bacterium is a slightly lower grade complexity measure.  Any polynomial in
many variables has a "Newton polytope", and the volume of this Newton
polytope can be taken as a low grade complexity

 K(f) = vol(NP(f))

Now, in my eprint "A Formal Theory of Information", I described a number
of formal properties shared by Boltzmann's normalized entropy of finite
partitions, Shannon's discrete entropy of finite probability spaces, the
dimension of affine subspaces, the Hausdorff dimension of Borel sets,
metric entropy of a measure-preserving transformation, topological entropy
of a continuous tranformation, etc.  In all of these situations, there is
a binary operation called join, written v, defined on the "atoms" A, B,...
of a "globule". -Entropies- or measures of the difficulty of
distinguishing "elements" of globules, satisfy -subadditivity-,

 H(AvB) <= H(A) + H(B)

IOW, "the ambiguity of the whole is at most the sum of the ambiguity of
its parts" (usually less, because of "correlations" between the parts;
equality is achieved iff the parts are "independent").  -Complexities-, on
the other hand, should however be -superadditive-:

 K(AvB) >= K(A) + K(B)

IOW, "the complexity of the whole is no less than the sum of the
complexity of its parts" (usually more, because of the "interactions" of
the parts, which -decrease- the ambiguity of the whole systems, but
-increase- its complexity).

In particular, K(f) is superadditive if we define the "join" of two
polynomials to be the product fg; then

 NP(fg) = NP(f) + NP(g)

(Minkowksi sum of two convex polytopes, a new convex polytope) and since
vol(A+B) >= vol(A) + vol(B) by Minkowski's inequality,

 KP(fg) >= K(f) + K(g)

Thus, while the Boltzmann entropy and Shannon entropy are actually
(numerical) -entropies-, K(f) is a (numerical) -complexity- and thus
suitable as a measure of biocomplexity.

In information theory, the -deficit-

 MH(A,B) = H(A) + H(B) - H(AvB) = H(A) + H(B/A) = H(B) + H(A/B)

or mutual information plays a crucial role.  In algebraic geometry, the
-excess-

 MK(A,B) = K(AvB) - K(A) - K(B) = K(A) + K(B/A) = K(B) + K(A/B)

or "mixed volume" plays an important role, in the case where K(f) is
defined as above.  On the other hand, the affine dimension of the variety
defined by an ideal behaves like an entropy (as any dimension does;
entropies arise as a natural generalization of the notion of Hausdorff
dimension, as was shown by Yakov Pesin, and Hausdorff dimension is in turn
a natural generalization of topological dimension, which is generalization
of the dimension of a Riemannian manifold, which is a generalization of
the dimension of an affine subspace). 

BTW, entropies satisfy many other nice formal properties such as the "data
processing principle"

 H(A'/B) <= H(A/B) <= H(A/B')

where A' is any "coarsening" of A and B' is any "coarsening" of B.  That
is, if we "process" the information in B, the processed version B' gives
us less information about A.  OTH, if we "process" the information in A,
the processed version A' requires less information to describe.

> Complexity is not given a rigorous definition but can be construed as
> a measure of elaboration of cellular organelles, cells, tissues,
> organs etc.

"Elaboration"?  Does Gould use that word?
 
> The first curve represents that for all life well before the Cambrian,
> essentially bacteria and unicells and resembles a single smooth hill
> starting at the left wall of "minimum possible complexity", peaking at
> the dominant bacteria and sloping quickly to the right (no 'higher or-
> ganisms').

You mean "decreasing rapidly to the right"?
 
> The second curve, however, resembles the first on its left but is
> higher, bigger and slopes very slowly to the far right toward zero.  
> This shows the present day distribution, with all the variously modern
> species added on, vertebrates in the high tail.  This curve is
> striking to me because it bears a strong resemblance to the well known
> Maxwell-Boltzman distribution of velocities for molecules in a gas at
> equilibrium at a given temperature.  And it begs a similar
> interpretation.

Groan.

1.  Consider Zipf's law.  Consider in particular the slightly modified
Zipf law derived by Mandelbrot.  Conclude that if the ranked frequencies
of the classes roughly conform to Zipf's law (logarithmic decay), there is
no cause for surprise, much less for the assumption of some "underlying
grammar" as Zipf erroneously thought.

2. Consider the wave equation in R^3.  Consider in particular the
applications to wiggling jump ropes, EM waves, and QFT (decomposition into
plane waves).  Observe that jump ropes and EM waves do not involve the
same physical phenomenon, they are just governed (in part) by the same
equation.

3. Consider second order ODE's in physics as discussed in Feynman's
lectures.  Same lesson: the same mathematical theory might apply to
physical phenomena which involve completely different fundamental physics.

4. Consider the Potts model in statistical mechanics, in which a state s
is defined by coloring the vertices of a graph G, using n colors.  The
energy of s is defined, for a given temperature T, by

 E(s) = sum_v sum_{neighbors w of v) d(v,w)

where d(v,w) = 1 if the colors of w,v are different.  Then the partition
function, which completely defines the thermodynamics of the model, is
define to be

 P_G(n,T) = sum_s exp[-E(s)/(kT)]

But this can be rewritten as

 P_G(n,T) = Z_G(n, exp[-1/(kT)]-1)

(see Bollobas).

So I'd caution you against reading too much into the same of a curve which
looks superficially like the Maxwell-Boltzmann distribution, because your
intuition can be made precise by a number of rather general mathematical
concepts.  Almost anything can give rise to a unimodal distribution!

> Does anyone have familiarity with these bio-curves and can you refer
> me to sources?

See my entropy pages for a few books on similar topics.
 
> B)  I have a hunch that the resemblance to the M-B curve is not a
> coincidence.

You see a curve which looks like the Maxwell-Boltzmann distribution.  
What if I see a curve which looks like the Planck black-body distribution?  
Or just a generic unimodal distribution?

> It seems possible that at any point in the history of life there is a
> very slow, long term dynamic balance, 

If you are trying to draw a comparison with "dynamic equilibrium" in
statistical mechanics, be aware that the whole point is that this a near
maximal entropy state is achieved -quickly-, not very slowly.

I think you're mixing baseballs and budworms :-/

> on average, between the stability of all species at any point on the
> curve and species to either side of that point, especially on the
> left.  Call this a meta-ecology.
> 
> Just as very high energy (velocity) molecules interact most often with
> molecules in the relatively low mean-energy region, more complex life
> forms may interact most often with those of much lower complexity, in
> terms of individual to individual interactions and with more
> consequences for their survival.  We interact far more often with
> bacteria, fungi, protozoans and tiny plants and metazoans than we do
> with larger critters and such.  Call these left-critical interactions
> because they are generally more important for survival than those with
> 'peers' or even more complex life, though these are the more obvious
> and visible interactions.
> 
> I obviously haven't worked this out much, but if true these curves
> represent not acausal random dribblings of species evolution away from
> the left wall as Gould argues, but an ongoing very long term
> equilibrium/stabilization process which actually supports the
> development of exponentially decreasing populations further and
> further to the right, analogous to populations of high energy
> molecules existing in ever decreasing numbers as you look at the high
> tail of the M-B curve.

Ecologists certainly noticed early in this century that "more complex"
species are less numerous, and explained this in terms of the
thermodynamics of the food chain.  See e.g. the book Why Big Fierce
Animals Are Rare by Colinvaux.

But this isn't the Maxwell-Boltzmann distribution!
 
> To recapitulate, within certain limits, complexity of organisms seems
> to correlate positively with their success in the environment. 

Bacteria aren't "successful"?  Fungi aren't "successful"?  

> It is becoming clearer that evolution of larger, more complex forms
> occurs to some degree through symbiotic co-evolution and even merging
> (see L. Margulies et al).
>
> It may follow that as organism complexity increases (along with its
> mass and energy correlates) in the main body of the curve (bacteria,
> fungi etc), the capacity for symbiosis , both negative and positive,
> increases viv-a-vis life higher up in the complexity tail. 

It is frequently assumed that "the biocomplexity of the biosphere"
increases monotonically due to the process of evolution by natural
selection, or that "the biocomplexity of a human being" is larger than
"the complexity of a redwood tree".  However, these assumptions do not
rest upon any computable or even estimable definition of "biocomplexity"
and I'd guess that both assumptions are -wrong- if you define
"biocomplexity" in a reasonable way, such as definition of the
biocomplexity of an unicellular organism offered above.

> There would also be an action going leftward:  organisms on the right,
> which even by random mutation become slightly more complex,

This is another hypothesis which is almost certainly wrong.  Populations
-adapt-, or die out, when ecological conditions change, but the first
alternative does not necessarily imply that they become "more complex".  
The second probably does imply that their constituent molecules become
"more disordered" :-/ (not the same thing as "less complex").

> will provide that much more niche-space, as it were, for all kinds of
> symbiosis for life on the left.  Again, in the long run, success in
> life in one region of the curve would facilitate success in life in
> other regions.
> 
> I'm sure this could have been stated more sharply and it may even
> seem self evident to old hands but I would appreciate any feedback
> I can get.

I don't think it's self evident: I think it's wrong.

FWIW,

Chris Hillman

Home Page: http://www.math.washington.edu/~hillman/personal.html