Efficient Labeling Scheme for Scale-Free Networks

1
Efficient Labeling Scheme for Scale-Free Networks
Shai Carmi1, Reuven Cohen1,2 and Shlomo Havlin1 1
Minerva Center and the Department of Physics,
Bar-Ilan University, Ramat-Gan, Israel 2
Department of Computer Science and Applied
Mathematics, Weizmann Institute of Science,
Rehovot, Israel
Background and motivation
The scheme in details
Performance of the scheme First we fix the number
of hubs (to O(log(N))) and show the stretch for
various graph sizes and values of ? (? is the
exponent in the degree distribution). We present
the average stretch over all pairs, averaged over
large number of graphs Next we show
the distribution of the possible values of the
stretch (For N1000, and ? 2.5).     F
inally we fix N (8000) and show how the stretch
changes with the number of hubs, for two
different exponents.  
The internet is composed of approximately 107
routers and end-units. Those are connected mainly
through physical links. The main network protocol
is IP, which uses packet switching, that is on
each separate packet, the protocol decides what
is the best next hop. Routing Schemes Today,
routing is made using very large tables
containing a list of all the main IP addresses.
Those tables are kept at each and every router.
The main problem is that this method is poorly
scalable, that is, with the increasing size of
the internet, tables become very large since
their size is proportional to the total number of
nodes. To address this problem, new class of
algorithms have been developed. Those algorithm
do not guarantee routing through the shortest
path, but rather try to route in a smart way,
such that most of the paths will remain shortest.
The benefit is reducing the size of the routing
table considerably. The efficiency of a routing
scheme is measured in terms of its stretch factor
the ratio between the length of path computed
by the scheme and that of the shortest path. A
more sophisticated approach uses labeling as a
preprocessing part of the routing scheme.
The labeling step Fix the number of hubs, Nh.
Pick the Nh nodes with the highest degree, they
are called the hubs. For each node in the
network Start BFS with the current node being
a root, keeping for each node its predecessor.
Stop the search whenever we find one of the hubs.
The label of the current node will begin with the
hub found followed by the shortest path from that
hub to the node.
Analysis Assuming that the average path length
from each node to one of the hubs is O(log(N)),
the following is concluded The running time of
the labeling step is O(Nlog(N)). (Assuming
sparse graph). The average label size is
O(log2(N)) bits (Since the average path length is
O(log(N)), and each node on the path can be
represented with O(log(N)) bits). Next is a plot
of the average label size as a function of N (For
Nh O(log(N) and ?2.5)
Labeling Labeling is a preprocessing step of
giving new name to each node in the network. The
new name is meaningful, i.e.we can use the new
name as a source of information when making
routing decisions, thus reducing the amount of
data that must be stored in the routing tables.
For example, consider the following grid, with
meaningless name for each node. The routing table
will have to hold one entry for each node i.e.
the table will be of size O(N).
But if smarter names are used, for example the
new name is the grid coordinates, then routing
table is no longer needed at all, since each node
can route to each other node using its
coordinates only.
The routing table creation step For each hub
Perform BFS with the current hub being a root,
keeping for each node its predecessor. For each
node reachable from that hub, store its
predecessor in that nodes routing table, in the
entry that belongs to the current hub. For each
node Store all his immediate neighbors in the
table.
Current labeling strategies 4 choose a subset
of nodes, called central nodes. Each node is
assigned one of the central nodes (the closest),
and in addition a group of nodes that are in his
neighborhood (the nodes cluster). Every node
stores in his routing table the link through
which to route to all central nodes and all the
nodes in his cluster. The label of each node
consists of his central node name and the first
routing decision on the shortest path from this
central node to itself. Routing to some node v is
done by first routing to the central node of v
(using the label, everyone knows who it is, and
everyone knows how to route to him on shortest
path). Then using the label the central node
knows the next step, after that we are assured to
be in the cluster of node v, from there using the
routing tables we can route to v on shortest
path. The table size at each node is proportional
to the number of central nodes plus the cluster
size. Therefore we need to find the balance
between the sizes of those two groups.
Analysis Total run time O(NNh). (Sparse
graphs). Table size (number of entries) of each
node Nh k (Where k is the degree).
Conclusions We used the fact that the internet
forms a scale-free graph, and exploited the
unique properties of those graphs to create a new
routing scheme. Our scheme is extremely simple
and fast (both in the preprocessing and routing),
demanding labeling with very short label sizes,
and routing tables as small as we wish.
Nevertheless, the scheme performs very well,
finding almost always shortest paths, better than
current schemes 3. The scheme was also tested
on real routers network (N104) and was found
highly effective (Stretch less than 1.06 with
table size of less than 40 entries on average).
The routing procedure The routing decision made
by node v on routing request to node u If vu,
we are done. Otherwise Using the table, check
if u is one of the immediate neighbors. If it
is, send through the link (v,u) (which must
exist). Otherwise, scan us label. For each
entry in the label labeli If v is labeli,
(for some i) send through the link
(labeli,labeli1). If v is different from all
nodes in the label, extract the hub from the
label, (label1), find that hub in the routing
table, and send through the link that appears
there.
Scale Free networks In recent years it was
discovered that many natural networks, including
the internet (in both the routers and AS levels)
are scale free. Scale free networks have a power
law degree distribution, i.e. they contain some
very highly connected nodes (hubs). It was found
that in random scale-free networks 1. Most
short paths goes through one of the hubs. 2. The
average distance between nodes in the network is
ultra-small (O(log(N) or even O(log(log(N)))).
1,2 Routing schemes for scale-free graphs Due
to the above mentioned properties of scale-free
networks it is natural to propose a new labeling
scheme 1. The central nodes will be chosen to
be the hubs. 2. The label for each node will
contain its path to the closest hub.
References 1 R. Cohen and S. Havlin, "Scale
free networks are ultrasmall", Phys. Rev. Lett.
90, 058701 (2003). 2 R. Cohen, D. Dolev, S.
Havlin, T. Kalisky, O. Mokryn, and Y. Shavitt,
"On the tomography of networks and trees",
cond-mat/0305582 (2003). 3 D. Krioukov, K.
Fall, X. Yang, Compact Routing on Internet-Like
Graphs, Proc. INFOCOM 2004, Mar. 2004 4 M.
Thorup and U. Zwick, Compact routing schemes,
in Proc. Of the 13th SPAA. ACM, 2001.
Analysis Since the routing table is small, it
can be implemented as a hash-table, therefore
finding the route in constant time. The label
size in practical cases is of size O(1) also,
therefore we can scan it in constant time,
reaching total decision procedure carried out in
constant time.