Serguei Osokine wrote:
Ian, Oskar, that was wonderful. I rarely see anything that has
that much aesthetic value, but the analyitical proof that the graph
rewiring caused by the random search actions actually makes it a
small-world graph with log^2(N) path length, was certainly worth the
one-year wait since last April, which is when Ian has preannounced
this paper.
It should be noted that the analytic proof in fact does not established
this. I did come up with the proof that appears in the thesis last
April, and I have since (with little success) attempted to generalize it
so it holds for the rewiring algorithm. The problem is that the proof,
like the Kleinberg proofs, assumes that edges are chosen independently
at each point, but in fact using our algorithm they will depend on one
another (in a good way, but that is hard to prove).
In fact, I haven't even managed to show that the class of distributions
for which I have a bound - those which are solutions to the system
implied by equations (2.3) and (2.4) in the text - is nonempty. Until
this is done the theorem as it stands is interesting, but from a strict
mathematical viewpoint quite meaningless. (Which is why I have held off
on publication).
<>
1. What is the difference between Freenet[1] and Sandberg/Dijjer[2]
models that makes Dijjer easier to analyze? This is mentioned a few
times in [2] and in other Freenet- and Dijjer- related places, but
I did not notice the explicit explanation of the differences
anywhere. Was it the directed links in the graph in [2]? Something
else? I honestly tried to figure this out by myself, but couldn't;
sorry... Maybe if I'd spend more time looking at both [1] and [2],
I'd figure this out, but since you guys are already here, I thought
that maybe you could answer that?
What makes Freenet (by which I mean the old Freenet routing, not the one
they are using now) much more difficult to analyze is that the data
rather than the nodes which have addresses. Data bounces around the
network dynamically, and so the markov chain implied becomes a lot more
complicated. For one thing, Freenet suffers from load balancing issues,
while the point to point routing cannot have such issues.
2. Whatever this difference is, how relevant do you think are the
Sandberg/Dijjer results[2] for the original Freenet paper[1]?
For example:
a. section 5.2 of [1] says about Fig. 3: "We can see that the
pathlength scales approximately logarithmically, with a change
of slope near 40,000 nodes."
Given what you guys know now, do you think that Fig. 3 shows
log^2(N) instead, or Freenet simply has different scaling
properties due to its different algorithms?
Paths in Freenet are affected by other things than just point to point
routing. Caching for instance will play a big role. I also don't
remember exactly how we were scaling the routing table size in those
simulations (the log^2 bound is for a constant routing table size).
I'm guessing that a lot of what was seen in those simulation were
artifacts of other factors.
b. Fig. 5 shows the link number distribution that awfully
resembles the power-law one, whereas Oskar mentioned in
another mail that Kleinberg is constant out-degree and
Poisson in-degree. Is my assumption that Sandberg/Dijjer
model is supposed to mimic the Kleinberg distibution
incorrect and in fact both Sandberg/Dijjer and Freenet
are power-law? Or is just one of them power law due to
the different algorithms used? Or maybe Fig. 5 in [1] is
actually showing a part of the Poisson distribution that
can be easily mistaken for the power law?
The model presented in the paper has fixed out-degree and (roughly)
poisson in-degree like Kleinberg's. Freenet has a fixed out-degree but
possibly a power-law in-degree, which is the cause of the aforementioned
load balancing issues (I don't have any good evidence for this, just the
feeling that their is some sort of preferential attachment going on).
c. Section 5.4 of [1] says: "In a small-world network, the
majority of nodes have only relatively few, local, connections
to other nodes, while a small number of nodes have large,
wide-ranging sets of connections. Small-world networks permit
efficient short paths between arbitrary points because of the
shortcuts provided by the well-connected nodes..."
This does not sound like anything I've been able to see in the
Oskar's thesis[2] - it has left me with a distinct impression
that the connectivity in the small world (at least in the one
analyzed there) was provided by the exactly right (Kleinberg's)
proportion of long-distance links, which Sandberg/Dijjer
rewiring model was designed to achieve, and not by the means
of any special nodes with "wide-ranging sets of connections".
Is it because Freenet and Dijjer used different algorithms, or
the original assumption about the reasons for the Freenet
connectivity was wrong?
The original paper is based on less sophisticated small-world models
than the later stuff. I think most of the small-world references in
there are hand-waving really.
If you place an arbitrary point somewhere where you say that connections
shorter than that are local and others are long distance, then
Kleinberg's model (and ours) will reflect the somewhat fuzzy statement
you quote above.
3. Why would one want to use the system with O(log^2(N)) path length
when pretty much every DHT gives a path lenght of O(log(N))? At
today's P2P nets scale (millions of nodes) the difference between
log and log^2 is non-trivial. I understand that creating the small-
world network by a simple and natural evolutionary process is very
elegant, simple, and attractive, but is this elegance worth the
increased path length? Why not simply use a DHT?
In real life, the order is much less interesting then the actual route
length. One shouldn't stair blindly at the order when the constants
could be dominating in real world situations. Also, the log^2 is with
constant routing table size, if you scale up the routing table with the
size of the network, you can roughly divide the order by the routing
table size (most log n DHTs scale the routing table with log n as well).
4. Speaking of DHTs: Oskar recently said something curious here that
I did not understand: "Actually, while the frequency of Chord links
falls exponentially with the "level" (not sure what the Chord term
is) the length of such links increases exponentially as well, so in
fact the frequency of links with certain lengths do fall
harmonically. One could see Chord as some sort of "mean field"
version of the same dynamics as Kleinberg's model."
What exactly does this mean? DHTs (including Chord) have log(N)
diameter, and Kleinberg has log^2(N), right? This looks fairly
different to me, and I totally missed the meaning of the "mean
field", and why Chord and Kleinberg would have the same dynamics;
I did not even understand what is this dynamics that you're talking
about. Oskar, could you please elaborate on this?
Again, the routing tables grow in Chord. If each node had one "chord"
link (every log n th node having a chord of the same length), then Chord
would be roughly log^2 as well.
<>
6. Both the simulation in Freenet paper[1] and Sandberg/Dijjer paper[2]
use the random requests to rewire the networks. In reality, there
will be all kinds of hotspots and uneven distributions of requests
due to the different popularity of content. Do you guys think it
would affect the results of analysis performed in [2], and if yes,
how? For example, if one particular file is extremely popular, one
could imagine that its storage node[s] will have many more links
than the average, and the search for the other (rarely requested)
data items might become ineffective, because these path lengths will
become very high.
I would imagine that the large number of files stored per node, and the
random distribution implied by using a secure hash would help even out
distribution. I'm not sure what an uneven distribution does under our
algorithm, but intuitively it should handle it just fine (it would mean
more links to the destination of popular queries, but that is a good
idea since such links are popular...)
Do you think it is a valid concern? Are there any results that would
show that this is not an issue?
Yes, and no (besides standard concentration results showing that it
should even out if there are sufficiently many documents per node).
7. While Sandberg [2] assumes that the requests are passing between
the random points, both Freenet and Dijjer seem to allow the length
of the path to be shortened when the requested content is already
cached on the intermediate node. If this is really so, what is the
effect that will have on Oskar's analysis results? Fewer links will
be rewired, so this should have some effect on the graph dynamics,
right? Any idea what it would be?
This is part of the answer to your first question about why these
situations are harder to analyze.
8. The answer to '7' above might also depend on the cache size and
replacement policies on the nodes, and on the content popularity
distribution (which I already mentioned in '6' above). All these
factors are hard to predict for the real system in advance, and I'm
wondering what is the feeling that you guys have about the overall
stability of the rewiring algorithm?
No idea.
I mean, the original Kleinberg d(x, y)^(-1) approach is extremely
sensitive to the value of this "-1". Any other value, and you either
do not have enough long distance links, or you do not have enough
short-range links once you arrive into the general vicinity of your
destination. In any case, your path length becomes polynomial instead
of polylogarithmic once you have the smallest deviation from this
"-1" number.
It isn't actually that sensitive. It is sensitive asymptotically, but
for any given network size varying the exponent a little will make
rather little difference. In fact, one can calculate numerically (I know
of no proof) that there are better exponents than -1 for any given n
(though the best exponent approaches 1 as the network grows towards
infinity).
And this makes me a bit uneasy; I do understand that the catastrophic
failure is unlikely, and the detrimental effect of all these changes
(if any) is likely to be tolerable, but still... Look at it this
way: Chapter 1 of Oskar's thesis is dedicated to proving that
"...family given by (1.1) allows for polylogarithmic routing at,
and only at, one value of the [alpha]...", then Chapter 2 proves
that this is exactly what happens with random-request rewiring, and
then Dijjer's practical implementation and deployment arrive and
throw these random requests out of the window with their caching,
different content popularity, and whatnot. So given the strict
requirements that were proven to be vital in Chapter 1, I cannot
help wondering: is Dijjer still polylogarithmic, or not?
It is certainly not proved.
I'd feel much better about all this if there would be some results
showing that the system is stable in the presence of such changes
(in a sense that reasonably small algorithm input changes should not
cause any catastrophically large changes in the algorithm results).
Do you have any results that would point in that direction? Would
you say that the data displayed on Figures 2.4-2.6 already shows
that your results are not exactly Kleinberg, but this does not
cause any catastrophic consequences, or you also have some other
sources of optimism?
I would feel better about it too if more were proven. I am optimistic
that things actually do work, but I am not very optimistic about how
much I will ever be able to show (and it isn't just me, though a better
mathematician may of course have gotten further, I have put these
problems in front of a many good researchers who seem to agree they are
difficult problems).
9. And finally, if I'm not mistaken, Dijjer description seems to imply
that the links are rewired after every search, whereas Oskar suggests
that this should be happen only with proability p, and in section 2.3
says: "...there are simple heuristic arguments for why p should
reasonably be on the order of one over the expected length of the
greedy walks". Any particular reason for that Dijjer behaviour, or
I've missed something and Dijjer does, in fact, rewire with this
probability p? (Would be interesting to hear these simple heuristic
arguments too, but that's another story...)
It won't matter so much if the routing table is large (remember the
mathematical model has one long link) though I would recommend that
Dijjer updated less often.
// oskar
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