On Tue, 9 Mar 2004, Einar Stefferud wrote:
It might be interesting to view the Internet through the contextual lens of
spherical geometry concepts which I think fit as well as anything, contrary
to some of our historical internautical terminology. For example, in
spherical
Geometry, a manifold has no edges, and has no center, while IETF folk insist
that the Internet has an edge somewhere (just one) but I have not heard any
claims that it has a surface, or that it has a center.
Not to be picky, but the geometry isn't spherical. In fact, the
geometry of the Internet is a network -- a network IS a geometry
consisting of nodes (locations) connected by links. The mathematics of
a network is called graph theory. The network geometry of the Internet
isn't horribly well ordered or simple and is highly dynamic. It
certainly isn't (hyper)spherical in any dimensionality -- spherical
geometries have certain properties that the network lacks, although of
course there exists a projection of the physical network onto the
physical sphere (the globe) that provides some useful information.
Less than one might think, of course. The network isn't necessarily
simply connected, for example, as I could go upstairs and unplug my
router and create a network fragment disconnected (transiently) from the
rest of the Internet. The metrics are not obviously connected to real
space geometry on any but a very local scale. For example, I am LESS
than two physical miles away from my office at Duke as I type this.
However, I'm 17 network hops away from my desktop there, and traceroute
reveals that the packets go through Atlanta and Raleigh (it can be worse
depending on congestion and dynamic routing -- I've seen as many as 30
hops).
The network geometry is multidimensional and nodal. One can define a
surface (of a simply connected nodal set) -- the union of all nodes with
a single entry/exit route (link). Similarly, it has an interior (all
nodes with multiple links). It has a norm that permits a discrete
measure of distance to be constructed -- the "hop" from one node to
another (the information revealed by traceroute measures a normed
distance between nodes, albeit quite possibly a transient one and one
where physical distance is nearly irrelevant). It even has a center --
one could usefully define it to be the union of all interior nodes that
are a weighted MINIMUM distance, on average, from the entire surface --
the so called "backbone" -- although this isn't a sharp concept and may
not even be all of that useful because of details of the network.
For example, one can generate a variety of renormalized views of the
Internet where nodes are THEMSELVES networks (or the routers/gateways
that isolate them) -- "rgb.private.net" (my home LAN might be one) --
and the relevant network links are ones that connect routers, ignoring
the edge nodes served by the routers. Then there are aggregations of
LANs (such as duke.edu) which may have multiple links as well as LAN
aggregations that have just a single link. Nowadays although one can
still talk about a network "backbone" people also speak of "clouds" and
use other metaphors to more accurately describe the core connectivity.
A lot of this topology is built into both the internet addressing scheme
and the underlying routing schema. "Usually" a surface node has a
single IP number and is part of a IP LAN that is at least reasonably
spatially contiguous. "Usually" interior nodes have multiple IP
numbers. "Usually" routing attempts to dynamically solve a problem in
the topology such as "how to I get a packet from this node to that node
with a minimal number of hops, strictly less than the TTL value, no
loops, no dropped packets". Even here one has to be somewhat fuzzy as
there are multiple protocols in use in layers -- what does one call an
ethernet bridge, for example, and how do you describe entities such as
compute cluster nodes that might have a proprietary non-ethernet non-IP
interface, or various devices that link to nodes. There are even cost
functions that have to be applied, as some of the intermediary links may
charge a de facto "toll" for transit.
Naturally, all of this has been studied extensively by mathematicians
since Euler and the Seven Bridges of Konigsburg (which more or less
invented the subject), and work continues today. Equally naturally, all
of this has been studied by computer scientists and network engineers
from the pre-Internet beginning, and was very intelligently incoded into
the network as we know it today. Their dynamic solution for routing and
addressability may not be theoretically optimal -- I'm not an expert in
graph theory but I'd be surprised if it was -- but it has proven
evolutionarily to be amazingly robust and more than "good enough" at the
scales it has worked with so far. Note that there are plenty of
networks that do NOT scale -- decnet, appletalk, raw ethernet -- and
that TCP/IP is actually one of the greatest human accomplishments of all
time -- a true wonder of the world -- if one looks at it a certain way.
I think that one of the major questions associated with IPv6 is going to
be whether or not that robustness and scalability persists in the
new/extended model. It is not obvious to me that it will, only because
(as a colleague of mine who works in complex systems is wont to say)
"more is different" -- new structures emerge, often nonlinearly, when
you make something bigger and potentially more complex. I'm optimistic
though, and humans are pretty good at fixing things that don't work so
even where problems emerge I expect that we'll fix them. I'm also
optimistic that a lot of the new structures that emerge will be GOOD
ones -- the additional intrinsic complexity will permit us to make
amazing extensions to the network, IF they scale in application.
Surely, some of you will be quite upset about my observations, but I ask you
to
stay cool and just ponder it all for a while to see of things don't start to
look different from this point of view, hopefully yielding some useful new
insights.
Enjoy;-)...\Stef
Why would anybody be upset? They are "a" way of viewing the network,
possibly a somewhat projective and naive view, but as you say, it can
still yield certain insights. However, from an engineering perspective
they aren't horribly useful. Check out network/graph theory -- there
are plenty of sites you can google, and some good books on the subject.
Then you'll have a better grasp of the actual underlying mathematics
(which is really quite lovely and can be extended all the way down to
the network of nerves that is generating the HIGHLY nonlinearly
organized impulses that are typing this reply and the network of traces
through which flowing electrons are encoding and processing my typing so
that it can be sent out over a much simpler network (the one we are
discussing) to you.
rgb
--
Robert G. Brown http://www.phy.duke.edu/~rgb/
Duke University Dept. of Physics, Box 90305
Durham, N.C. 27708-0305
Phone: 1-919-660-2567 Fax: 919-660-2525
email:rgb(_at_)phy(_dot_)duke(_dot_)edu