Stein (1985), Paladin et al. (1985), Qiang (1986)]. Finally, a third related process is the random walk on the backbone of a tree, which has been used to.
SLAC-PUB 4077 March 1987 0’)
COMPLEXITY
AND ULTRADIFFUSION*
_. -.
Constantin Stanford -
P. Bachas
t
Linear Accelerator Stanford University Stanford,
Center
CA 94305
-B. A. Huberman Xerox Palo Alto Research Center Palo Alto, CA 94304
ABSTRACT
.
We present
the exact solution
hierarchical
space. We derive rigorous
exponent
describing
-- -
function.
and randomly
a class of highly
unbalanced
We conclude that the speed of relaxation
complexity,
or lack of self-similarity
complexity
may be revealed
exponent,
and in particular
power-lay
decay.
Submitted
in an arbitrary
upper and lower bounds for the dynamic
by both uniformly
trees, and identify
the lower bound.
of diffusion
the decay of the autocorrelation
the upper bound is saturated hierarchical
to the problem
of the underlyng
by the temperature by the nature
to Journal
multifurcating
trees that saturate
---
is a measure of the
tree.
We point out that
dependence
of the transition
We show that
of the dynamic
from exponential
to
of Physics A --
*Work
supported
f
Present
of Energy,
contract
DE-ACOS-
and by ONR contract NOOOl-14-82-0699.
76SF00515 _.-
in part by the Department
Address:
Palaiseau,
France
Centre
de Physique
Theorique,
Ecole Polytechnique,
91128
I . 2
1. INTRODUCTION A large variety
;
hierarchial
of natural
structure
from the reporting are built -
constituents
More
ts.
organization
et al. (1986)l.
schemes in social organizations,
spins can be collected
ferromagne
systems have an exact or approximate
[Simon (1962), Rammal
out of atomic
- individual
and artificial
recently
out of cells, to the way
blocks at the critical
it has been
of states appears spontaneously
range
to the way macromolecules
or organisms into larger
Examples
realized
that
point
of
a hierarchical
in the low-temperature
phase of
the mean-field
theory spin-glass [Sherrington and Kirkpatrick (1975), Mezard et al. (1984)], and it has been conjectured that the same holds true for glasses, hard combinatorial
optimization
Bachas (1985), Bouchaud Anderson
(1986)],
problems
[Kirkpatrick
and Le Doussal (1986), Sorkin
the conformational
substates
different perturbed
-
(1984)].
time scales, they appear [Kohlrausch
(1847),
processes
on hierarchical
Huberman
and Kerszberg
hoppings
Paladin
process is the random Ott (1985), Grossman
- .-
The
Qiang
variant
analyzed
by [Teitel
diffusive by
and Domany
diffusing
over a
for long-rang
ultrametric
space, which
(1985), Ogielski
Finally,
of a tree, which
in the chaotic transport
et al.
proposed
Allowing
(1986)].
walk on the backbone
a third
and
related
has been used to
of particles
[Meiss and
et al. (1985)l.
of all these treatments hierarchical
structures,
hierarchy-level
similarity
of such trees allows so that
earliest
[Schreckenberg
when
Palmer
have considered
in a truly
many
exponentially (1970),
in one dimension.
of authors
et al. (1985),
any given techniques,
authors
of diffusion
model the observed stickiness
case of uniform
than
and Stella (1986)], consisted of a particle
leads to the problem
A_major limitation
slower
(1985) and further
array of energy barriers
(1985),
et al.
systems.
and Watts
several
structures.
has been analyzed by a number Stein
to relax
Williams
To model this behavior
hierarchical
[Ansari
of all these systems is that because of the existence’of
(1985), and Maritan -
(1985),
et al. (1986), Fu and
of proteins
(1985), Stein (1985)], and other complex frustrated A common feature
and Toulouse
are totally
ultradiffusion
is that they only apply to the simple described
by trees whose branches
indistinguishable.
for the application is in this
The intrinsic
of renormalization
case only
a variant
at
selfgroup of the
---
3 4
-.
-‘.
extensively Orbach
studied
metastable different
states of a spin-glass weights
_ precisely
[Mezard
or the appearance
of the hierarchy,
that
of different
accounts
organisms
interactions
for the complexity
of
and social organizations.
It
to determine how hierarchical dynamics may depend on _ This is the subject of this paper, a summary of tree structure.
which has already been published We will restrict assumptions
concerning
in particular;
elsewhere
[Bachas and Huberman
ourselves here to the problem of diffusion
In section 2 we will
solve this problem
the structure
a closed expression
whose asymptotic A central
by the indistinguishable
interesting
the underlying
complexity the absence information
decay has the qualitative by Huberman
of self-similarity,
content of the underlying
the
We will obtain, function,
v that characterizes
of a measure
and Hogg (1986): than
any
of the system.
exponent
features
rather
tree.
making
autocorrelation
decay describes the rate of relaxation
introduced
without
of the underlying
for the average
(1986)].
in a truly ultrametric
exactly,
thesis of this paper, is that the dynamic
this asymptotic
-. -
and
free energy valleys are known to carry et al. (1985)l. A n d in a more general context, it is
systems such as biological
is therefore
-
Alexander
tree, when different
new level
hierarchical
-
(1976),
could be represented
the lack of self-similarity,
at every
space.
[deGennes
and Toulouse (1983)]. Hierarchical structures need For instance, it is hard to imagine how the be self-similar.
leaves of a uniform -
on fractals
(1982), Rammal
not, however,
--
diffusion
of physical
namely it is sensitive
randomness,
to
or detailed
tree. To be precise, we derive a rigorous
upper bound for v, and show in sections 3 and 4, that it is saturated
by both
uniformly
optimal
and randomly
multifurcating
in that they lead to fastest relaxation.
trees. Such trees are therefore
In section 5 we also derive a lower bound
for v, and identify
a class of non-self-similar
All
have
these
results
percolation
threshold
complexity
of games.
recently
unbalanced
been shown
to hold
trees that saturate
it.
also for the critical
by [Bachas and Wolff (1987)1, which can be related to the
-
-A corollary temperature - .-
of our results dependence
with the underlying
is that
in thermally
of the dynamic
tree structure:
exponent
in particular
activated
processes
is not universal, the transition
the
but varies
to instability
is
I
l
4
continuous is explained --
-
:: ;:i
-
-. -
4
-.
-‘. for self-similar
trees, and discontinuous
for unbalanced
in section 6 which also contains some concluding
ones. This
remarks.
I .
5 .+
-.
.
2. THE GENERAL Ultradiffusion
SOLUTION
is described by the dynamical
equation
-. dP,
1V
dt=,+
1
(2.1) E.lJ pJ
-
. . . ,N labels the states of the system, Pi(t) is the probability
where i=l, finding
the system at state i at time t, and the symmetric
satisfies the ultrametric
transition
of
matrix
c
property
(2.2)
- for any three distinct
states i, j and k. If E represents
sites i and j, E-q. (2.2) implies that the two smallest are equal.
The diagonal
transition
probability
be conserved, i.e.;
elements
hopping
rates between
hopping rates out of a triplet
are fixed by the requirement
that
“ii= - L
(2.3)
LJ
Jti
-. -
The ultrametric
property
system can be represented rate between
(2.2) is equivalent
to saying
as the leaves of a generic
that
the states of the
tree, where the hopping
any two of them, say i and j, is only a function
of the nearest
common ancestor A of i and j on the tree
Eij = Eji = EACij,
with
cA decreasing
root. By appropriately
‘.
of generality - .-
monotonically stretching
that cx = exp[-h,],
as one climbs
along any path towards
the
the tree, we may always assume without
loss
where h, is the height of the branching
from the bottom of the tree, as shown in Figure
point A
1. We shall refer to such a tree
,,
I. *
6
-.
-‘. as a “metric -branches
tree”, to stress the fact that both its topology matter.
There
ultradiffusion
problem
An important
remark
is one hierarchical
and the heights
transition
matrix
of its
and one
for every metric tree.
--
transition
probabilities
_ we can effectively appropriately
increase
father,
XI(B) =
1 if i 1s a descendant 0 otherwzse
assume that branchings adjustable
height
h =m*Ah
of final
B (B,=B
descendants
by convention,
or tree-leaves
and S, for the number the characteristic
or root).
nth
B, is the N, will
generated
by B
of sons, or immediate
function
ofB
Ah, and will
as the mth generation.
define the silhouette height
we shall denote by B, the unique
may only occur at integral
interval
-
it multifurcate
(2.1), let us first introduce
. . ., N runs over the leaves of the tree.
where i=l,
height
of equation
and so on, up the the patriarch
a leaf),
We also introduce
this is not the case, since
of any state by letting
or tree-leaf
B; the grandfather,
offsprings.
of symmetric
as shown in figure 1.
of any branch-point
(Ns= 1 if B is itself
the assumption
restrictive
and te&&nology:
stand for the total number
-. -
the weight
to an exact solution
some useful notation
although
may seem overly
at low altitude
Before proceeding ancestor
is in order here:
slopes, which
For convenience
multiples
we will
of some minimum
occasionally refer to all branches at If n(h) is their total number, we
measure
the rate of population
growth
at
h by:
s(h)= -
A log n(h) 1 =ah Ah
(2.4a) bgin(:hih,I
We - shall refer to its average asymptotic
value
---
I . 7
h’
S --
h
lim S’h%++m
(2.4b)
(h’-h)
as simply
the tree’s
correspond
to asymptotically
Large
silhouette.
and
small
values
of silhouettes
fat and thin trees, respectively.
We are now ready to obtain the complete set of eigenvectors and eigenvalues of any hierarchical transition matrix C. To this end consider first the action of c ~_.
on the characteristic
$
ei;XJ
function
of some branch-point
or tree-leaf
B:
5 J=l cti [ I-X;iBl]=
(B)=-
j=l
-h NB.
e
(2.5) (A( z,B,
if XI(B) =O
= -
root 1 (NB
-NB II
II=1
).
if
eehBn
Xi(B)=1
n-l
---
where A(i,B) -- -
is the nearest common ancestor of i and B, and by slight
abuse of
notation
root 1
stands for a summation including the root.
n=l
over
all of B’s ancestors,
up to and
We next define the vector
-
- .-
V$B,$=
2-x X,(B) - No
j) B
I (B7
(2.6)
B
for any two brothers
B and .B. Since B n = Bn for all n >O, and A(i,B) = A(i,B)
any i that is neither
a descendant
for
of B nor of B, we easily deduce from Eq. (2.5)
I
I . 8 1
-.
..
that V(B,B)
is an eigenvector
of c, with eigenvalue
A that only depends on the
common father of B and fi, i.e.: --hB
1 h(B,)=-
-
e
= -N,
’
(2.7)
n
1
n=2
zB1
-
degenerate eigenvectors of type (2.6), to all pairs of sons of Bl. A convenient basis for the subspace
Consequently,
there
corresponding
are +SB,(SB,-~)
they span consists of the following
-
vectors, one for each son of Bl:
X;(B)-
+-
XI (B,)
(2.8)
B1
- These satisfy the linear relationship
..
-1
N, V(B) = 0
-brothers B
-
so that only (SB,-1) of them are linearly
independent.
Since ---
1.’
-- -
(S,-1)=&l
branch poznts B
there
is a single
corresponds
missing
eigenvector
to the steady-state
of the NxN
of equal
probability
transition
matrix
e. This
for all sites, which
we
denote by
(2.9)
Going back to equation - .-
given tree leaf L. Writing
(2.1), consider a particle the initial
condition
that starts out at-time zero at a as
9 -.
root x vc by
Pc(t=0)=6Li= --
n=O
we immediately
deduce that at later times
-t/CL
root
P,(t)= z
1
“+;
Vi (L,- 1) e
(2.10)
n=l
From- this
we can exactly
concentrate particle
on the autocorrelation
returns
n=l
L
..
conditions
PW= ;
any
be mainly
n-l
1
quantity
function,
to its point of departure;
- root1 (,(t)=Jj + 1 cN-We will
calculate
of interest.
We will
i.e., the probability
that
this reads
-t/XL e ) n
NL n
interested
the
in this
(2.11)
quantity
averaged
over
all
initial ---
L, which can be written:
(2.12)
rB
+ branch pocnts B
where, for later ease of reference,
1 -=N, -
-hB
e -hB
e
autocorrelation
(2.13)
’
n-l
xB
From Eqs. (2.12) and (2.13).it - .-
we recall that
function
easily follows that for finite trees the decay of the
to its equilibrium
value
is always
exponential,
and
-
10
dominated
by the largest characteristic
1
--
e
=-. lJ root N
For infinite -
+h
time,
root
trees, on the other
hand,
either
- scenarios may take place: a) Some characteristic that relaxation
is unstable
to slower
asymptotic
~_.
than
behavior
by the asymptotic -
-
CL branch points B
N
as -c-+00. In the following ..
which relaxation
times may vanish,
times may accumulate
relaxation
of the autocorrelation
s,-1
p(t)=
exponential
behavior
of the following indicating
-
in part or all of the tree, which can thus be collapsed
to a single state, and b) some characteristic leading
one or both
at long
function
times.
to infinity, The leading
is in this case determined
of the spectral density
(2.14)
F(MBB)
sections we will
is everywhere
analyze
this behavior
for trees on
stable. \
-
-- -
-. -
11 4
-.
.‘.
3. UNIFORM We begin ;
with
multifurcating offsprings. already _ Paladin
-
the simplest
TREES
example
tree, for which
of an infinite,
each member
This is shown in Figure
uniformly
of every generation
produces
b
2. Some of the results of this section have
been derived
in [Schreckenberg
et al. (1985),
Qiang
successive generations,
regular,
(1986)].
the silhouette,
(1985),
Ogielski
If Ah is the height
and Stein interval
(1985), _
between
(eqs. (2.4 a,b)) is given by
(3.1)
Using inverse
characteristic
generation,
..
we can easily
the fact that NB~= b”-N,,
time corresponding
calculate
from Eq. (2.13) the
to any branch
point
B of the mth
with the result
-
provided
e ah> b, that
relaxation function, --
is unstable.
is s< 1.
For sz 1 all characteristic
Assuming
s< 1, we finally
times
obtain
vanish
and
the autocorrelation
Eq. (2.12), in the form
-
6
--v t
unrform
(3.2a)
where
V
-
-and _
--
untform
=-
S
l-s
(3.2b)
-
---
12
(3.2~)
with the corrections at large t [Erdelyi Note that,
to the asymptotic
in contrast
Ah+a-Ah.
off exponentially
to the prefactor
D, the dynamical
of the silhouette
For instance,
critical
and is therefore
a tree that tetrafurcates
interval
(see Figure
- members
2). One may also relax the condition
intervals
of a given
bounding
be generation-independent
generation
can be defined
still
by expression
given
acquire oscillatory By allowing
(3.2b).
non-constant
height
can also study the limiting the height
a l/f-frequency
refer to such trees with that ultradiffusion tree is a broom. leading
_ -h,=
function spectrum vanishing
to finite characteristic
mlogb
intervals
+alogm,
D however,
exponent
is
can in this case
among successive generations, or unit silhouette.
grows faster than linearly decays
slower
than
(up to logarithmic silhouette
may lead to a l/f-like (b) At the other
to show that if the
time-dependence.
cases of vanishing
hm of the mth generation
s=O> the autocorrelation implying
By appropriately
0 and 1, the dynamic
The prefactor
and/or logarithmic
every half unit
(but still insist that all
be indistinguishable).
and lies between
height
that the branching
p(t) from above and below, it is straightforward
silhouettes
under
at every unit
has the same power law decay as one that bifurcates
and height
exponent
invariant
interval ratio
-. -
(3.2a) falling
(195611.
vuniform is only a function b+ba,
behavior
extreme,
any power
of time,
we will
We shall later
show
only when the underlying
the slowest
rate of growth
times is given by
witha>l
It is clear that in this case s= 1, and the autocorrelation
(a) if
with m (so that
corrections).
as “brooms”;
spectrum
Indeed:
one
function
reads:
of hm
I 13
(3.3)
--
where “2 1; this is as will become clear in a moment,
If this were also true
would obey the homogeneity
small
relation
for the w-integrations,
the
22
S- n+l(t)
=
-1sQte-4h)
from which we (z=t.n.e-n*4h)
could that
easily
deduce
by
a change
of
variables
m F(t)=
2
-” s;(t)
- t
(4.8)
random
n=t
with
We will now show that a more careful leads to at most logarithmic
modifications
To this end we first prove the following
-
Lemma 2:
-. -
Proof:
p(w) is either
bounded,
as w+O.
’
Assume
p(w)-c-w-a
normalizable).
treatment
of the above power-law
or else diverges
with
and hence * t.
.
P(l)C
(
w.
this into the stationarity
-a
decay.
at most logarithmically,
l>a>O
then obtains
c f w-‘l=
cutoff,
lemma:
as w-+0,
Plugging
of the w-integration
) i
1 +o(W’-a)
I
(since
p must
condition
---
be
(4.4) one
23
but
1;
=P(l)+2P(2)+3P(3)+
. .s
P(1)+2(1
-p(l))
so that + a-1 52. This cannot be satisfied in turn implies -
that we have derived a contradiction.
- faster than logarithmically
Thus p(w) cannot diverse
as w-*0.
Note that one can likewise ti+O, which justifies
by any > 1 which
show that I?(*) diverges
our throwing
at most logarithmically
away the ti-integration
as
cutoffs.
Converting the sum over generations to an integral, and changing variables z = t. 2 X > n.e-n.Ah, we can write the average autocorrelation function
to at
large in the form
F(t) -t
-” pzrform
j
m 0
..
dz
-2 2
‘uniform
dw - p(w)Az,w)
‘w I
v (f)
untform
w
I
-
with f(z,w> a bounded function is integrable
that decays exponentially
as w+m, and has at most a logarithmic
easily conclude
that
time-dependence
the term in square brackets
at large z. Since p(w) singularity
as w+O, we
has at most a logarithmic
at large t.
-. This completes our demonstration dynamic this
exponent
assertion
function Similar
as completely
directly
that uniformly
random
ordered uniform
trees.
by calculating
the exact
in the special case of the exponential results have been obtained
by Kumar
trees have the same One could also verify
average
distributions
autocorrelation given
and Shenoy (1986).
by (4.5).
--
I 24
5: COMPLEX Both the uniform
-.
HIERARCHICAL
and the totally
random
systems, whose parts (or subtrees)
-
whole.
Huberman
equally
simple
and Hogg
structures,
of self-similarity, contrasted defined
trees are self-similar
which
have
should
or to diversity
argued
that
minimize
Complexity
to the information
they
theoretic
any physically
measure
algorithm
maximum
value
describing
- fastest relaxation.
decay. Thus complexity
the rate of relaxation,
f.
results Wolff
have also been obtained (1987);
operational -
Consider
exponent
with
the word
meaning. a particular
structures sensible
in the context
“complexity”
diversity
is reflected
measure
of it.
in that
in
Similar
by Bachas and context
a precise --
unbalanced
tree, constructed
trifurcate,
by allowing
while the right
by renormalization-group
doubles at every step, its silhouette
the left half
half members
to a single son each, as shown in Fig. 3a. This is clearly population
do indeed lead to a
.
members of every generation and cannot be analyzed
to a
lower bound for v,
of percolation
given
leads
v, and hence guarantees
or structural
and -v is a physically
an
Indeed, we have
self-similarity
show that non-self-similar
entropy
how to construct
In this section we will obtain a saturated
and will in particular slower power-law
critical
This is to be
by random trees.
tree-silhouette,
for the dynamic
to lack
given by Shannon’s
Our results place the above ideas in a precise physical context. for a given
relevant
is in this sense tantamount
exact replica of a given tree, and hence maximized
that
to the
are therefore
at all levels of the hierarchy.
by the size of the smallest
demonstrated
hierarchical
are at least on the average identical
(1986)
_ measure of a tree’s complexity.
STRUCTURES
give rise
a non-self-similar
techniques.
tree
Since the total
is given by
-
As we will relaxation __ .-
now show, its dynamic
critical
is slower than for the corresponding
we previously
found
exponent
is v =s, meaning
self-similar
structure
that
for which
25
v
=-
uniform
s
.
1 --s
Intuitively,
this is because different
different
of an unbalanced
rates, and it is the slowest processes that dominate
To calculate
the average
autocorrelation
assume first that the total number the end).
We will
generation
by the coordinates triangle,
the silhouette
times
the intergeneration
trifurcate
either until
right-half
of their generation,
following
at long times.
P(t) for our tree of Fig. 3a, is M (M will be taken to ~0in
as shown in Fig. 3b, with
now the descendants
respectively,
of generations
at
(x = M-i, y = log j); the tree is then represented
Consider
dead branches
function
tree relax
label the jth from the left node of the ith from the bottom
an orthogonal
T(x,y)
parts
until
of a given
a hypotenuse
height
interval:
node (x,y)
slope equal to log 2 = sehh.
with
yz0;
they reach the bottom of the tree, or until
these
in Fig. 3b. The number
of fertile
will
they enter the
from which point on they will continue
the end; these two cases correspond
by
to regions
(i.e., trifurcating)
as single 1 and 2
generations
(x,y) is
M-x rnregion (X- l)log2-y
=
1
(5.1)
.
--
f 1 rn regzon 2
log 32
:i
-. -
where the f 1 ambiguity descendants
is due to the fact that
of (x,y) may be partly
sequel neglect this ambiguity, exponent.
Clearly,
the number
fertile
(at most) one generation
and partly
infertile.
of
We will in the
since it does not affect the result for the dynamic of final descendants
or tree-leaves
generated
by
node (x,y) is
-
(5.2a)
N(x,y)=3T’xJ)
For y = 0 and in region 2, the number
of final descendants
but we may still express its rate of change with x as:
is actually
modified
,
I 26
z
We are finally corresponding -
- ancestors find:
(5.2b)
3T(x’o)
(x,0) = 2 .
in a position
to calculate
the inverse
characteristic
to the node (x,y), as given by eq. (2.13). Converting
to a line integral,
time
the sum over
as shown in Fig. 3b, and using eqs. (5.1, 5.2) we
-
(5.3)
in region 1
-
xlog2-y -r;-‘(x,y)=3
z log2-y+cx-z~log3
x
log3/2
e-Ah(.v-xJ+
y
&
,-AhcM-d3’
)
log 3/2
I x- log 3
-.
X--
-
Y log3
+
(5.4) ---
riog 2 &
e-AhtM-r)
3’Og3a
in region 2
I 0
where
once again
we have dropped
which cannot modify spectral density,
the leading
eq. (2.14).
.-
-
2M
ify=O.
,-Ah(M
-x)
power-law
finite
multiplicative
in the asymptotic
constants,
behavior
of the
Note also that the first term on the left-hand-side
eq. (5.4) should be changed to
_
various
of
27
These expressions simplify considerably if one takes the limit M+m, and notes that for almost all nodes (but a set of measure zero) both x and y > > 1. If
_ one then finds easily
-
that relaxation
that all inverse
is unstable
everywhere
Up to finite multiplicative terms, we then find: -A.h,M-x
and exponentially
e
-A&M-x)
autocorrelation
eq. (2.12), which can be written
function,
dx
= Regions1 and 2
=
‘+D,t
D,t
suppressed
dy 2.
the asymptotic
behaviour
of the. average
as:
e-th(xy’
(5.54
’
2
where the two terms in the last expression regions 1 and 2 respectively.
correspond
to the integration
log2
;
over
One finds:
s vl=
additive
-”
-”
-
consider
in regzon 2
to analyze
II
implying
in region 1
It is not straightforward
P(t)
-.
times diverge,
on the tree. Let us therefore
_constants
t70g2-y 3 logm
characteristic
(5.6)
v2=s
--s log3
-
-The second of these two exponents, thus describes the asymptotic - .-
We have underlined
being smaller,
dominates
at long times and
decay of the average autocorrelation the word “average”, because for any given initial
function. condition
28
decay can be much faster; if for example our particle the left-half
of its generation,
dies out with
1;
exponent
tree that
trifurcates
actually
mislead
conditions
it can be shown that the autocorrelation
VI, which uniformly
the reader
should dominate
starts out on a tree-leaf
incidentally
is the exponent
every
height
to think
that
interval
Ah.
function
obtained Figure
such “fastly-decaying”
in
for a
3b may initial
the average, but this is not the case*: the prefactor
_ D2 does not vanish.
-
The dynamic
exponent
for the unbalanced
satisfies
tree of this
section
obviously
_ S
vfs
Thus
’
= l-s
rearranging
silhouette,
...
Vunrform
a limit
the branches
of a uniform
i.e. adding new branches)
to this slowing-down
-
following
result answers this question:
Theorem
2: The dynamic
by its silhouette: -. -
VIS.
Proof: From equation
-1>, ‘B -
(without
changing
can indeed slow down relaxation.
its
Is there
effect or can we go all the way to a logarithmic
time decay (l/f noise) by appropriately . ..
tree
critical
complexifying
exponent
the tree’s structure?
The
of any tree is bounded from below -- -
. (2.13) it follows easily, since NBZ 1, NB, >NB,-, , that:
-hB
for all tree-nodes
B. Thus we can bound the average autocorrelation
function
as follows:
-*?&ote that in Fig. 3b, y is a logarithmic scale, so that practically the entire population of tree-leaves is concentrated in the lower left corner of the triangle.
29
-hB J-(sB-l)e-t.
e
=
tree nodes B
=
-
i
(e~’
AhmIIe-n.
S.
Ah,
,-teen’
Oh
n=l
z
t
-S
Q.E.D
-.. -.
This simple theorem saturatesthe allowed unique: - b-furcates
..
unstable,
-
that the unbalanced
lower bound for the dynamic
relaxation. (with
exponent,
tree of this section
and leads to the slowest
We should point out that this slowest-relaxing
for instance
dead branches prefactor
thus demonstrates
any tree for which
b some integer),
while
to the end, would
D, as well
as the critical
the bth-fraction the remaining
of each generation members
give the same dynamic silhouette
above
tree is not
which
continue
exponent. relaxation
as The is
do however depend on b.
---- -
I 30
6. DISCUSSION Figure
1;
4 summarizes
both uniform
and random
percolation maximized therefore entropy
of the underlying
quantitative .
relations
than
tree.
among
by uniform is sensitive
to the structural
It would
and
We may complexity
noise or Shannon
be interesting
these various
trees
(1987)].
to the physical
for
of winning
and random
and Wolff
and
to see whether
complexity
measures
can be
obtained. Besides slowing has-another,
-
rather
of non-
threshold
as the complexity
5 [Bachas
say that the rate of relaxation
or lack of self-similarity,
the number
it has been shown that the critical
by the trees of-section
trees of
of the tree as shown by -Huberman
in a game, is also minimized
by
if instead of v one plots a
by counting
on a tree, which can be interpretated
strategies
v is maximized
by the unbalanced
is obtained
defined
pieces at every generation
Hogg (1986) . More recently
-_
behavior
of the tree’s complexity
_ isomorphic
exponent
trees, and minimized
section 5. The same qualitative measure
The dynamic
our results.
down relaxation,
more spectacular
processes. Assuming
the complexity
or absence of self-similarity
effect, when one considers
thermally
in this case that the hopping rates are given as
-Avij. eij = e .’
activated
(6.1)
T
-. -
with
AV,j an energy barrier,
the silhouette
is proportional
dependence of the dynamic
-. T V
we easily conclude from the definition to the temperature.
Therefore
(2.4) that
the temperature
exponent is given by:
,
TT ’ c
1
(uniform
or random)
trees, and
---
31
i’ c.
V
;
most
complex
(n
=
T
TT’
(6.2b)
c
for the most complex trees of section 5. This is illustrated particular
-
that the transition
discontinuous Before
is continuous
in the former case and
in the latter.
proceeding
we should
model of the relaxational actually
to instability
in Fig. 5a,b. Note in
considered
point out that ultradiffusion
dynamics
is not just a naive
of systems with many time scales. It can be
as a universal
description
of such dynamics,
in two
different-ways:
a)
In real space one may describe the spreading together
in larger
..
by lumping
blocks the degrees of freedom that relax at times ‘ci < t2 < . . . , as has been suggested by Palmer et al.
characteristic (1984).
of an excitation
and larger
Note that for systems with disorder
is in general
the ensuing
hierarchical
tree
non-uniform.
b)
In configuration
space one may describe relaxation
of an ensemble
of particles
[Dotsenko between
-. -
(1985)].
in a valley
At sufficiently
landscape
low temperature,
as a stochastic
motion
of metastable
states
the hopping
rate
two such states is:
- mlnmm
(LJ)
T ‘V
= e
where minmax
(ij) is the minimum
energy
encountered
barrier
along
over all paths from i to j, of the maximum This satisfies the ultrametric the path.
property
minmax
(iJ) 5 max
1
minmax
(i,k); minman
(j, k)
I
---
32 +
-.
.‘. since one can always structure
go from i to j via k, and hence defines
in the space of metastable
Mezard (private
states.
a hierarchical
We owe this argument
to Marc
communication).
It is this second description
of relaxation
as ultradiffusion
in the space of
metastable
states that is relevant for studying the temperature dependence of The simple assumption (6.1), however, is a priori the dynamic exponent. justified
only in the limit
general invalidated and hopping
of vanishing
temperature.
for several reasons:
that ultrametricity
of the transition
matrix
hope that the ultrametricity
of equilibrium
field spin-glass
and Kirkpatrick
conjectured an exact
[Sherrington
in a variety
metastable
by a maximum
hierarchy
comes into play
energy barrier,
may be destroyed.
so
One may simply
states, demonstrated
in the mean-
(19751, Mezard et al. (198411, and
of other systems [Palmer
or approximate
As T is raised, it is in
to begin with, entropy
rates are not simply determined
*
(preprint
of hopping
rates
1986)], implies among
also
long-lived
states.
Even if this is so, free-energy the silhouette
is not simply
barriers
would in general be T-dependent,
proportional
so that
to T. Eqs. (6.2a,b) should therefore
be
replaced by:
(6.3a)
-c
V
c
and
V
most complex
(n
s(T)
;
TT’
c
(6.3b)
C
where s(T,) = 1 in the first case, and s(T,‘) = log 2/lag 3 for the trees of section 5. -The precise form of v(T) now depends however, the
the silhouette
transition
discontinuous
on the unknown
changes continuously
is robust: for complex
with temperature,
it is continuous trees.
Note
function
for
incidentally
If,
the nature
self-similar that
S(T). trees
the
of and
transition
,,
*
described i-n eq. (6.2) is unphysical, critical
temperature.
hoppings,
-.
33
-.
.‘.
This is, however,
and can be easily
independent
became unstable
only due to the long-range by introducing
rectified:
cutoff that respects the ultrametric
ensure that the silhouette transition
since relaxation
nature
an expontial
structure
of T-
of the space, we can
is bounded from above at all temperatures,
is to a region of exponential
above the
and the
relaxation.
-
It is tempting known-for
to compare
the mean-field
[Sompolinsky
these predictions spin glass:
and Zippelius
exponential
(1981)
of ultradiffusion
with
as shown by Sompolinsky and (1982)]
above the glassy transition,
relaxation
what
is
and Zippelius in the latter
is
and changes over to a power law with
exponent:
2 v(T)=
;
C
Does this
below. ..
Kirkpatrick -
imply
that
model [Sherrington
the hierarchical and Kirkpatrick
tree
in the
Sherrington-
(1975), Mezard et al. (1984)]
is complex? More work properties
is necessary
to answer
such questions,
of systems that defy the simplicity
and to unravel
the rich
of scaling laws.
-- -
ACKNOWLEDGEMENT This was work supported 76SF00515
-
in part by the Department
and by ONR contract N00014-82-0699.
of Energy,
contract
DE-AC03-
--c
I 34 -+
_.
.‘.
REFERENCES Alexander, Ansari,
S. and Orbach, R. (1982) J. Physique A., Benendzen,
Shyamsunder,
J., Botine,
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S., Frauenfelder,
H., Iben,
E., and Young, R. (1985) Proc. Nat. Acad. SC. (U.S.A.)
I., Sauke,
T.,
82,500O.
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.._ Bachas, C. (1985) Phys. Rev. Lett. 54,53. Bachas, C. and Huberman,
B. A. (1986) Phys. Rev. Lett. 57,1965.
Bachas, C. and Wolff, W. F. (1987) J. Phys. A20, L39. Bouchaud,
J. P. and Le Doussal, P. (1986) Europhysics
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de‘Gennes, P. G., (1976) Recherche 7,919. Dotsenko, f
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Erdelyi,
V. (1985) J. Phys. C 18,6023. A. (1956) Asymptotic
Fu, Y. and Anderson, - Grossmann,
Harris,
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P. W. (1986) J. Phys. A19,1605.
SI, Wegner,
F. and Hoffmann,
T. E. (1963) The Theory of Branching
- Huberman,
---
K. (1985) J. Physique Processes, Springer,
B. A. and Hogg, T. (1986) Physica 22D, 376.
Huberman,
B. A. and Kerzberg,
Kirkpatrick, -
S. and Toulouse,
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R. (1847) Ann. Phys. (Leipzig)
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A. and Stella, A. L. (1986) Phys. Rev. Lett 56, 1754; (1986) J. Phys. A19,
L269.
1;
Meiss, J. and Ott, E. (1985) Phys. Rev. Lett. 55,274l; _
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M . (1985) J. Physique Lett. 46, L217.
G., Sourlas, N., Toulouse,
G. and Virasoro,
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M . (1984) Phys. Rev.
(1984) J. Physique 45,843.
Ogielski,
A. T. and Stein, D. L. (1985) Phys. Rev. Lett. 55,1634.
Paladin,
G., Mezard, M . and de Dominicis,
Palmer,
R. G. (preprint
Dynamics.
f
(1986) Physica 20D, 387.
C. (1985) J. Physique Lett. 46, L985.
1986) to appear
in Heidelberg
Colloquium
on Glassy
_
Palmer,.T.,
Stein, D., Abrahams,
E. and Anderson,
P. W . (1984) Phys. Rev. Lett. 53,
. 958. Qiang, 2. (1986) Acad. Sinica preprint. - Rammal,
Rammal,
---
R. and Toulouse, G. (1983) ibid 44, L-13 , and reference therein. R., Toulouse, G., Virasoro,
-Schreckenberg, Sherrington,
M . A. (1986) Rev. Mod. Phys. 58,765.
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Sot. 66,80.
37
FIGURES Fig. 1: A generic
tree illustrating
our notation.
The node A has three sons
1;
(SA =3) and six final descendants grandfather,
(NA=~).
A1 is his father
and A2 the
that also happens to be the root of the tree. The height of a
node as measured from the bottom leaves, is minus the logarithm -
corresponding
transition
a state can be effectively it multifurcate Fig. 2: Regular
rate. The node B illustrates
uniformly
(a) bifurcating
rate of population
yield the same dynamic Fig. 3: (a) An example lower-bound
and (b) tetrafurcating growth
critical
for the dynamic
when parametrized
critical
is the same, these
tree that
exponent.
triangle
trifurcating
of their generation
members
the nodes of this tree,
The root lies at the origin.
subtree,
trifurcate
while
region
for awhile,
and continue
2 contains
but eventually as dead branches
The broken line is the line of ancestors of a typical
which
we integrate
Aylhx = log 3, until point on it continues
to calculate
it hits the left-most straight
the
give rise to a single
containing
in the text.
saturates
The left-half
while the right-half
those nodes whose descendants enter the right-half
height
unbalanced
as explained
1 is a regularly
with
trees; since the
exponent.
trifurcate,
son each. (b) The orthogonal
along
T;~; its slope
branch
node A,
is initially
of the tree, from which
up to the root.
Fig. 4: A schematic plot of the dynamic exponent v versus the Shannon Entropy, or detailed information content of the underlying tree. Here s is the silhouette -
- .-
are rigorous
-
and hence, as discussed in the text,
of a maximally
of every generation
thereafter.
of
increased (by a factor 8 in the figure) by letting
two trees have the same silhouette,
Region
how the weight
at low altitude.
(asymptotic)
f
of the
of the tree, which is held constant,
and the broken lines
bounds.
Fig. 5.: The temperature dependence of the dynamic exponent, assuming hopping rates are given by eq. (6.1), for a) self-similar and-b) maximally
--
I 38 -.
complex trees. Note that the transition and discontinuous
is continuous
in the former case
in the latter.
-.
,.
---
--
A1_ A2
A
II.
.x- h
B --
--I
---
f
Fig. 1 ---
-- -
-.
f
Fig. 2 -- -
---
--
03)
M log 2
0
w
y
I I ---
-- -
Fig. 3
I t
S
UNIFORM
COMPLEX
RANDOM
ShannonEntropy Fig. 4
I _
.~ 4
Y
(a 1 Self-similar
Y
(b) Complex ---
-- -
Fig. 5
