Review Editor Comments to the Author:
The paper is too long. Please try to shorten the presentation. It is hard to see what are the main results.
For example it is mentioned that there is a characterization of exponential vs. polynomial growth. Please
point out the corresponding theorem(s).
The paper has been shortened from 26 pages to 18.
Reviewer: 1
See the review. Major revision is required after which recommendation to accept is likely.
Reviewer: 2
== Summary
The paper "The Growth Degree of Limits of Vertex Replacement Rules" deals
with the asymptotic growth of the size of the balls in the infinite graphs
obtained as a limit of graphs constructed by iterating a map that replaces
some vertices by a finite given graph.
More precisely, let G_0 be a graph and H be a graph whose automorphism
groups acts transitively on a subset V of vertices of size k. The authors
construct a new graph G_1, by replacing any vertex of G_0 of degree k by a
copy of H. Iterating this process and taking a convenient subsequence, we
get an infinite graph G_\infty with a marked point p\infty.
The authors compute the asymptotic growth of the balls centered at
p\infty, and give two sufficient conditions on H which ensures that this
growth is exponential or polynomial respectively. Roughly speaking, the
conditions depend on whether or not H contains a "germ" of a binary tree.
Though not very far from the preceding papers, the results seem new and
the proofs seem correct, hence I consider the paper is worth publishing.
== Comments
Here are some comments and suggestions that might make the paper easier to
read, i do not expect the author to slavishly follow them.
=== Implications for the original problem
You motivated the study of vertex replacement rules by the study of
horospheres on two dimensional branching surfaces. You should explain how
the results presented in this paper apply to this problem.
The horosphere comment has been deleted. The tie in with horospheres is somewhat
loose and so it is not possible to apply these results back to horospheres. We submit that the objects
themselves are worthy of study.
=== Exponential rate
You quantify the polynomial growth by introducing the growth degree. Why
not also introducing and computing the growth exponent in the exponential
case? This can be defined as the limit of log(f_X(y,n))/n. Such a constant
should be stable enough, and its existence should follow from usual
sub-additivity techniques (Fekete's theorem), applied to the map f(n) =
log(sup{ f_X(y,n) : y \in X }) when the degree of the considered infinite
graph is bounded (which is the case for the graphs obtained by a replacement
rule).
This is an excellent suggestion, but the resolution is not trivial, in particular the sequence is not truly
subadditive in general. We worked out the growth exponent for the replacement rule given Figure 8 where
l_1=1 and l_2=2 and found the growth degree to be the golden ratio, we needed Perron Frobenius Theory
to do this. We are currently working on the general case and hope to compute it for all replacement rules
in a future paper.
=== Formalism
In quantified formulas, some variables are not well located (so that some
quantities may move whereas they should remain fixed, and vice versa).
Here are some examples:
* Definition 4.1:
"$f_X(x,n) = |{y \in V(X) : d(x,y) \leq n, 0 \leq n}|$"
could be replaced by
"$f_X(x,n) = |{y \in V(X) : d(x,y) \leq n}|$ ($0 \leq n$)"
Changed
* Definition 4.3:
"inf_{d\in R} { d : f_X(x,n)/n^d \leq c } for some constant c"
could be replaced by something like
"inf { d\in R : \exists c \in R \forall n \geq 1 f_X(x,n)/n^d \leq c}"
Changed
* Definition of a(n), page 18
This definition looks strange since a(n) seems to depend on the choice
of u and u'. It may be worth recalling that the replacement rule is
symmetric.
Changed
=== Introduction
In the introduction, it could be a good idea to announce the main theorems
(with their precise statements), and to make a sentence that explains the
strategies of the proofs (e.g. in the exponential case, saying that you
will embed a binary tree that spans "most of" the limit graph).
We rewrote the introduction.
=== Definitions
Since you only deal with a single replacement rule, you could shorten the
definitions (8 pages for Sections 2 and 3 is quite a lot).
Done—thanks for this, it did shorten it a lot.
=== Other various comments along the text
* Abstract:
"necessary and sufficient conditions"
could be replaced by
"sufficient conditions".
Totally redone
* Proposition 3.4 (topology on the set of marked graphs)
Instead of trusting some previous paper in which a weaker version is
proved, it may be worth saying/proving that, more generally, the set of
marked graphs whose maximal degree is bounded by a given constant is
compact. Hence the existence of a limit point is just a consequence of
the fact that applying the map R many times does not increase the degree
of the graph too much.
This has been greatly simplified.
* Remark after Proposition 3.4
You write "Note that it is possible for a single R-orbit to have more
than one limit". You could give an example (for example, you can
alternate the degree of the marked point).
As per the suggestion of the other reviewer, we have restricted our attention to cases where the
distinguished point is not replaceable, which greatly simplifies issues (we get true limits instead of limit
points).
* Page 8, last sentence : "Observe that if the marked point p is not
replaceable, then the set of marked graphs (R(G), q) = {(R(G), R(p))}."
First, q is not defined and you assumed in the definition of R(G,p) that
p is replaceable (you can remove this). Second, you could make a real
sentence, e.g. "Observe that if the marked point p is not replaceable,
then the set of marked graphs R(G,p) is the singleton {(R(G), R(p))}."
This is now moot
* Definition 4.2
You could replace "\mathbb{N}" by "\mathbb{N}^*". The definitions may
cause problems when n=0, since f_X(x,0)=1.
fixed
* Page 12, first line:
"the growth function f_X is independent of the vertex x \in X"
could be replaced by
"the asymptotic behaviour of the growth function f_X is independent of
the vertex x \in X".
sentence deleted
* Lemma 4.4
The assumption "d(x,y) < \infty" could be safely removed since the
graphs are assumed to be connected.
sentence deleted
* Proof of Lemma 4.4
The constant c is not defined.
"Suppose that X has exponential growth at x"
could be replaced by
"Suppose that X has exponential growth c^n at x".
fixed
* Definition 4.5
"and has the additional condition that"
could be replaced either by
"and has the additional property that"
or by
"and satisfies the additional condition that".
deleted
* Figures 15 and 18
Instead of drawing the length l-k, you could draw the length l with a
bend arrow.
Changed
* Page 16, commutative diagram (and others)
"R" could be replaced by "R_M".
"\pi2" could be replaced by "\mathcal{R}2", with the arrow in the
opposite direction.
deleted
* Page 18, definition of b(n)
It could be replaced by something more symmetric, like
"sup { dist_{R^n(v)}(u,z) : u \in \partial R^n(v) and z in R^n(v) }"
We actually want the maximal distance of any point to the set \partial R^n(v), which
is slightly different from what is above
* Proof of Theorem 4.13
"we observe that for any integer m > 0 there exists n_m such that
a(n_m - 1) \leq m \leq a(n_m)."
could be replaced by
"Since a(n) is non-decreasing, for any integer m > 0, there exists n_m
such that a(n_m - 1) \leq m \leq a(n_m)."
deleted
This paper will eventually be a good introduction to an interesting
construction. However, it needs a fair amount of work to clean up
the exposition so that the result will be readable. Things confused me
sufficiently that by the end I pooped out on reading the proofs of the
last two, and major, results. However, I will be happy to take another
look once things are cleaned up and I expect I will then recommend
publication.
I address the authors directly.
(1) The first issue is mathematical rather than expository. For the
pointed graph construction (Do the graph theory people really call
it a marked graph? I am used to the topology lingo of a pointed
space (X, x) with x the base point) you need some discussion of the
space in which you are operating. The essential point is that for your
constructions there is a universal upper bound L on the number of edges
connected to any vertex. This means there are only a finite number of
isomorphism types of balls of radius ≤ M for each M . By a pointed
ball I’ll mean a ball with base point the center. Let XM be the finite set
of these isomorphism classes. There is an obvious retraction from XM +1
to XM , namely map the pointed ball of radius M + 1 to the concentric
ball of radius M . Then the space X which is the inverse limit of these
is the metric space you are looking at. This space is compact and so it
is misleading to compare the convergence issues here with convergence
in the Gromov-Hausdorff metric as you appear to do in Proposition 3.4
because the latter is a noncompact Polish space. Here every sequence
has limit points. The interesting issue is whether a sequence (Xn , xn ) in
X converges, i.e. has a unique limit point. It does so iff for each M the
sequence of pointed balls stabilizes, i.e. (BM (Xn , xn ), xn ) is eventually
constant in XM .
Furthermore, unless you have a reference you should include the
proof that if two locally finite pointed graphs (X, x) and (Y, y) have
distance zero, i.e. (BM (X, x), x) and (BM (Y, y), y) are isomorphic for
all M then (X, x) and (Y, y) are isomorphic. I don’t see a constructive
proof, but if you let QM be the finite set of isomorphisms of the M
balls and QM +1 → QM be the restriction map. Then the result follows
because the inverse limit of a inverse system of nonempty compact
spaces is itself nonempty.
Issues of convergence have been greatly simplified by adopting this reviewer’s suggestion (10) below and
making the limit object a lot more clear (by insisting that the distinguished point be nonreplaceable).
(2) Now for some issues of exposition. As background, it looks to me
as though when you say ”an edge e” you are thinking of the edge as
the open set which does not include the vertices that are its endpoints.
If so, say so early.
Edges are now referred to as copies of the open interval (0,1).
I guess graph theorists use the lingo ”degree” of a
vertex and so I guess you can get away with not defining it but it might
be nice to mention it for non-graph-theorists (like me).
Added
(3) Page 2: What is a length space? I don’t think you need it.
So don’t use it. Don’t introduce the Gromov-Hausdorff metric here.
First off you only use it for the construction that you are just glancing
at. Secondly, here you are talking about locally finite graphs. It is not
clear to me that the GH metric is all that nice which you leave compact
spaces and you never use it for noncompact spaces. Also, it might be
wise to point out here that isometry and isomorphism of entire spaces
are the same concept here. Furthermore, graph isomorphism is really
what you will later want when you are are looking at subgraphs. For
example, on page 5. It is true that as subsets R(K) and K need not
be isometric but the result is much stronger than homeomorphic. The
restriction of π to R(K) is a homeomorphism onto K and if K is a
subgraph then it is a graph isomorphism. Here also is the place where
you need the edge stuff mentioned above. If an edge e is open then
R(e) makes sense as an edge in R(G). You should explicitly observe
that the adjacent vertices will change if e has a replacement vertex as
an adjacent vertex.
This has been developed.. Namely the difference between isometric and graph isomorphic is more fully
developed.
(4) At the beginning of Section 3.1 you want to introduce-brieflythe Gromov metric since you don’t use it later.
Deleted GH metric except for a passing mention
(5) At the beginning of Section 3.2 you want to introduce, somewhat
more elaborately, the space X described in (1) above.
done
(6) On page 12 begin the odd announcement of conditions like d(x, y) <
∞. These are locally finite, connected graphs. The distance between
any two points is finite by definition. If you compactify you may introduce points at infinity but you never do that here.
fixed
(7) With Definition 4.5 things begin to seriously confuse me. Consider a triangle with two boundary vertices and so the third vertex is
the unique replaceable one. It seems to me that by your definition this
is exponential. Since there is a unique replaceable vertex, there continues to be just one in each Rn . Hence, R is not a standard exponential.
Furthermore, the limits have polynomial growth. What’s going on?
Definition Fixed—good catch,
(8) In the statement of Lemma 4.7 you say that wlg M looks like
Figure 15. How could it look like anything else? You didn’t say M
”contains etc etc”. You said it ”consists of”. Then the proof becomes
incoherent in the first paragraph. You say x is an infinite distance from
a replaceable vertex. After a brief try I abandoned the attempt to make
sense of this. x is a point of X a limit graph. There are no replaceable
vertices in it. All that language applies to the finite approximations.
Deleted
Furthermore, as I said, infinite distance makes no sense here. That
said, what is going on, as illustrated by Fig 16 and 17, is interesting.
(9) At Lemma 4.12 I began to lose heart. Consider the first paragraph. I assume the R(e) being an edge when e is is handled by the
open edge stuff. Even if I saw why Q(R(e)) = Q(e), which I don’t, I
would still need an argument for the new edges which have appeared
in R(p) for p replaceable. Finally when I hit the phrase on page 20: ”x
is a finite distance from a replaceable vertex” I stopped. I guess what
is going on is that while the H’s are finite. You are now allowing the
G’s to be locally finite and so speak of replaceable vertices in them. It
is, however, deeply unclear how being a replaceable vertex in the limit
graph relates to the things in the approximating sequence.
OK fixed
(10) Finally, some remarks about the set-up which don’t really apply
to the questions you are asking. To me the question of growth is
less interesting than the question of actual convergence. This in turn
causes me to feel that the amount of choice you are building in is rather
unaesthetic. If the base point in G is nonreplacement then everything is
fine because there is no choice. The further base points are determined.
One wants to include a Theorem like: If p and q are the end-point
vertices of a path containing no replaceable vertices (including them)
and (Rni (G), p) converges to (X, p) then (Rni (G), q) converges to (X, q)
where things have to be defined so as to make sense of this. I think
it should not be hard to see when (Rn (G), p) actually converges and
what the limit is. Furthermore, it would be nice to relate the limit to
the inverse limit space of the inverse system Rn (G). I suspect that the
limit space X can be embedded as a subset of the points in the inverse
limit space which have distance zero from p where I am using the limit
pseudo-metric obtained from the normalized metrics on Rn (G).
On the other hand, if the base-point is a sequence of replaceable
vertices then it is obvious that without some care the sequence might
not converge. The choice of base-points breaks the symmetry of the
boundary and so the whole construction requires some redescription in
order that a particular sequence be well-defined. Furthermore, suppose
that p1 , p2 are non-boundary replaceable vertices in H and that the
distance from each to the boundary is greater than M . If the balls
BM (p1 ), p1 ) and BM (p2 ), p2 ) in H are not isomorphic then every time
a version of p1 is chosen for Rn (G) the result has distance greater than
2−M from the Rm (G)’s when a version of p2 is chosen. This is a sort of
boring reason for nonconvergence. It would be nice to do the choices
in such a way that rules out this a priori problem.
OK fixed.. convergence is now way simpler. We plan to study replaceable base points in a later effort.