1. Generating Network Topologies That Obey Power Laws
Christopher R. Palmer and
J. Gregory Steffan
School of Computer Science
Carnegie Mellon University
My name is Greg Steffan.
I’ll be talking about generating network topologies that obey power laws.
This is work done by Chris Palmer and myself at CMU.
I’d like to begin by explaining the title of the talk.

2.  the Internet obeys power laws
What is a Power Law?
y = βxα
Log
So what exactly is a power law?
Well, roughly speaking a power law is a distribution that looks like a straight
line when plotted in log-log space.
More mathematically speaking, it is some metric that follows the function
y equals beta-x to the alpha.
Work by Faloutsos showed the existence of four power laws in several internet graphs, including router topology and inter-domain topology.
And other research has shown the existence of similar power laws in different aspects of the www.
So power laws are quite prevalent in the internet.
Log
Faloutsos et al. define four power laws:
they found laws in multiple Internet graphs
others found similar laws, also for the Web
 the Internet obeys power laws

3. What is a Topology Generator?
Artificial network generation algorithm:
often used to evaluate new network schemes
Do artificial networks obey power laws?
artificial networks may not be “realistic”
conclusions could be inaccurate
Researchers often use artificial networks, randomly generated networks,
to evaluate various ideas and schemes.
What we want to know is whether the topologies from commonly used generators obey the power laws.
If they do not, then the topologies we are using may not be realistic, and could possibly cause us to draw incorrect conclusions.
Other important questions are:
how do we go about generating topologies that obey power laws
and,
does it matter? Do power law topologies impact simulated results?
 can we generate these topologies?
does it matter?

4.  Do existing generators obey power laws?
Outline
 Do existing generators obey power laws?
Can we generate graphs that obey power laws?
Do power law graphs impact results?
Related work
Conclusions
These questions basically form the outline of the rest of this talk.
First, I’m going to show you that current topology generators do not
obey all of the power laws.
Then I’ll show you our two algorithms for generating topologies that do.
We’ll see how power-law topologies impact results,
then we’ll discuss the related work and conclude.

5. Existing Topology Generators
Waxman:
place nodes randomly in 2-space
add edges with probability P(u,v)=αe-d/(βL)
There are two flavors of topology generator currently in use.
First is waxman’s generator which places nodes randomly in 2dimensional space,
then adds an edge between each pair of nodes with a probability that is inversely proportional to the distance between them.
This way, nodes that are close together will tend to be connected.
The second class of algorithms takes several
randomly generated graphs (likely by waxman), and connects them in a
hierarchy---in an attempt to model the hierarchical nature of real networks.
There are many versions of this approach of varying complexity.
So lets see how well these two techniques fare on the power laws.
N-level hierarchical:
connect random graphs in an N-level hierarchy

6.  Internet graphs obey PL #1: Out-degree vs. Rank
Power Laws 1 and 2
PL #1: Out-degree vs. Rank
compute the out-degree of all nodes
sort in descending order
PL #2: Frequency vs. Out-degree
compute the frequency of each out-degree
Starting with the first two power laws.
Power-law one is out-degree vs rank, which is computed by finding the
out-degree of all nodes, and sorting them in descending order and plotting in
log-log space.
Power-law two is frequency vs out-degree, which takes the same information but instead counts the number of nodes of each out-degree, and plots that in log-log space.
Again, Faloutsos found that internet graphs obey these two power laws.
 Internet graphs obey

7. PL #1: Out-degree vs. Rank
2-Level: ρ=0.81
Here we see out-degree vs rank for the waxman generator and a 2-level hierarchical
generator.
And apparently neither follows the power law very closely.
As a more scientific measure of how closely a curve obeys a power law, we’ve
computed the correlation roe, which is the same as measuring the quality of the
least-squares line.
A good correlation would be a value of 0.9 or higher, so neither of the generators obey this power law.
Waxman: ρ=0.80
 2-Level and Waxman do not obey

8. PL #2: Frequency vs. Out-degree
2-Level: ρ=0.23
Waxman: ρ=0.45
And on the second power law we see the both generators are way off, and in fact both plot bell-curves in log-log space rather than the straight line we would have hoped for.
And of course our correlation numbers are quire low .23 and .45.
 2-Level & Waxman REALLY do not obey!

9.  Internet graphs obey [A][vi] = λi[vi] PL #3: Hopcounts
Power Laws 3 and 4
PL #3: Hopcounts
number of pairs of nodes within i hops
PL #4: Eigenvalues
compute the largest 10 eigenvalues λi
The third power law measures the hopcounts---the number of pairs of nodes
within I hops of each other, for all the possibilities of the value I
And finally, in power law four, we compute the largest 10 eigenvalues of the adjacency matrix
as a quick reminder an eigenvalue is a constant that when multiplied with an
eigenvector v gives the same result as the adjacency matrix multplied by that
eigenvector.
Once again, Faloutsos showed that internet graphs obey these power laws by plotting near linear lines in log-log space
[A][vi] = λi[vi]
 Internet graphs obey

10.  2-Level and Waxman obey
PL #3: Hopcounts
Waxman: ρ=0.96
And it turns out that both generators do obey the third law, giving fairly high correlations,
2-Level: ρ=0.98
 2-Level and Waxman obey

11.  2-Level and Waxman obey
PL #4: Eigenvalues
2-Level: ρ=0.65
Waxman: ρ=0.98
And also obey the fourth law.
Note that the two-level generator does not have a good correlation value, but this is due to a nasty outlier, so we’ll give it credit for passing this power law anyways.
 2-Level and Waxman obey

12.  Can we generate graphs that obey power laws?
Outline
 Do existing generators obey power laws?
 Can we generate graphs that obey power laws?
Power-Law Out-Degree (PLOD)
Recursive
Do power law graphs impact results?
Related work
Conclusions
Now lets see how we can go about generating graphs that obey all four power laws.
I’m going to describe two approaches.
First, the power-law out-degree algorithm which directly manipulates the out-degree of nodes to obey that power-law,
and second a recursive algorithm based on an distribution.

13. Power-Law Out-Degree Algorithm (PLOD)
FOR i:1..N
x = uniform_random(1,N)
out_degreei = βx-α
FOR i:1..M
WHILE 1
r = uniform_random(1,N), c = uniform_random(1,N)
IF r != c AND out_degreer AND out_degreec AND !Ar,c
out_degreer--, out_degreec--
Ar,c = 1, Ac,r = 1
BREAK
Assign exponential
out-degree credits
The power-law out-degree algorithm works as follows:
First, we assign each node a number of out-degree credits, taken from an exponential distribution
Then, for as many edges as are desired in the final graph, we pick random pairs of nodes until we find a pair that has remaining out-degree credits,
and then place an edge between that pair.
So we know that this algorithm will do well on the first power law, since the graph itself is generated using this law. But will it obey the other power laws as well?
Place an edge in
the adjacency matrix

14. PLOD: Example Topology
This shows a sample topology, and again you can see the nodes of very high out-degree.
 32 nodes, 48 links

15. Recursive Topology Generator
80/20 Distribution:
80%
20% α
Our Recursive Distribution: β
Now lets look at our other approach.
we are going to generate an adjacency matrix by defining a the probability of adding a link between any pair of nodes, just like waxman.
but we start from an 80/20 distribution. it’s not a power law but has some power law characteristics. you can think of an 80/20 distribution as splitting the total space in half and saying 80% of the values in one half and 20% in the other.
You then apply this recursively.
we generalize this to an adjacency matrix by defining 4 probabilities corresponding to the 4 sub-matrices.
The full algorithm is somewhat involved, as you can read in the paper, because
care must be taken to preserve symmetry, and to manage the diagonal, but this
is the overall gist of it. γ
a+b+g+e = 1 e
 generalize to a 2D adjacency matrix

16. Recursive Topology: Generation
The picture on the left shows the probability distribution function that we just defined, where darker squares are more likely to become links.
We generate a graph by repeatedly selecting a link using this distribution.
This algorithm also generates a weighted graph:
If we select the same link more than once, we simply increase the weight of that link.
The picture on the right shows the (weighted) adjacency matrix created for 10 links. And, darker means heavier weight
Link Probabilities
10 Generated links
 darker means higher probability / weight

17. Recursive Topology: Example
We can use these weights to define different network capacities.
The experiments that we do later require both low and high-latency links, and
we can use this feature of the algorithm to define them.
Here we see an example of a random graph where the red links show the high latency connections.
With a little imagination, you can see a mini-backbone defined by these thinks.
 32 nodes, 50 low latency, 10 high latency (red) links

18. PL #1: Out-degree vs. Rank
Recursive: ρ=0.89
Now lets evaluate the performance of our techniques on the power laws.
First, we see that plod, as expected, is nearly perfectly linear, while recursive has somewhat of a non-linear tail.
PLOD: ρ=0.97
 PLOD: EXCELLENT power-law
 Recursive: good power-law tail, non-power-law start

19. PL #2: Frequency vs. Degree
Recursive: ρ=0.92
Both algorithms do very well on the second power law
PLOD: ρ=0.93
 both GOOD power-laws

20.  both EXCELLENT power-laws
PL #3: Hopcounts
Recursive: ρ=0.94
They also do quite well on the third power law
PLOD: ρ=0.98
 both EXCELLENT power-laws

21.  both EXCELLENT power-laws
PL #4: Eigenvalues
PLOD: ρ=0.98
and on the fourth power law.
Recursive: ρ=0.93
 both EXCELLENT power-laws

22. Power-Law Summary: Correlations
PL #1:
Degree
PL #2:
Deg. Freq
PL #3:
Hops
PL #4:
Eigenval
2-Level
.81
.23
.98
.65
Waxman
.80
.45
.96
.97
PLOD
.99
.93
Recursive
.89
.92
.94
So to summarize,
we see here all of the correlation values for the four different topology generators and the four power laws.
Green cells obey the power laws, and red cells do not, and patterned cells do so-so.
Overall, we conclude that our two methods generate topologies that follow power laws much more closely, and hence have more internet-like properties.
 GREEN cells obey power-laws, RED cells do not
 our generators have better Internet characteristics!

23.  Do power law graphs impact results?
Outline
 Do existing generators obey power laws?
 Can we generate graphs that obey power laws?
 Do power law graphs impact results?
Related work
Conclusions
Now, after all this work to generate topologies that obey power laws,
does it really matter:
if it impacts results.
What we have done is re-evaluate a multicast study originally done at CMU.

24. STORM Multicast Algorithm
source
client (parent)
client 
nack
repair
Called the STORM algorithm, which works as follows:
Here we see a single multicast source, and two multicast clients.
Each client dynamically chooses a nearby client to serve as its parent.
When a multicast packet is sent from the source, if a client loses the packet, it
can request a retransmission from its parent by sending the parent a nack packet,
and the parent responds with a copy of the lost packet.
This way, the storm multicast scheme is meant to scale to large numbers of clients with low protocol overheads.
 client requests repair from parent with a nack

25. Simulation Methodology
Original STORM study:
used 2-level random topology
source and clients connected to second-level
Generating comparable topologies:
equalize graph size and average out-degree
selection of high and low latency links
What impact do we expect of PL topologies?
average results will be similar
distributions will differ
The original storm study used a 2-level hierarchical topology of random graphs, and the source and client nodes are connected to nodes in the second level graphs.
We simulate STORM in action on all four of the topology generators that I’ve shown you.
For our study to be meaningful, we had to take some care to ensure that all of the graphs generated were comparable.
So we equated the graph sizes and average out-degrees.
The original storm study also made the top level graph have high-latency links,
while the lower level graphs had low latency links, and we ensured similar properties in the graphs from other topology generators.
We hypothesize, that when we evaluate the performance of STORM on all of these topologies, that the average results will remain similar. But we expect that the distributions will be different.
Let’s see.

26. STORM Average Overhead
This first graph shows ratio of overhead packets received per packet recovered
(so smaller bars are better), and each is broken down into nack or repair packets.
For each topology generator, we simulate a range from 10 to 200 clients, and average over 5 random runs each.
We see that these results stay relatively constant (near three, except for the 10-node
graphs which produce results with high variance).
This indicates that the STORM scheme scales well, across all topologies.
But these results average over all nodes in the graph. What we want to do now is look at the results at each node.
 STORM overhead averages scale for all topologies

27. STORM Overhead Distribution
2-Level
Here we’ve plotted the percent of protocol overhead for every client, sorted in descending order, and of course plotted in log-log space.
We see that the distribution of overhead for the 2-level topology is significantly different that that for the other generators.
In fact, the two distinct plateaus in the line probably indicate the two distinct levels in the topology,
perhaps indicating that the other topology generators are more natural.
 overhead distribution varies significantly by topology

28.  loss distribution also varies significantly by topology
Finally, if we look at the distribution of lost packets over all clients,
Bear in mind that since this is a log-log plot, even small variations indicate large differences in the actual results.
we see that results vary significantly by topology.
 loss distribution also varies significantly by topology

29.  BRITE uses these factors to generate graphs
Related Work
Barabási et al. (Notre Dame)
BRITE (Boston University)
What causes power laws in the Internet?
incremental growth
preferential connectivity
concurrently with our work, barabasi addressed the question:
why do power laws occur in the internet?
He observed that preferential connectivity---nodes tending to connect to higher-degree nodes, and incremental growth---as opposed to simultaneous creation,
were the properties that give the internet it’s power laws.
Both barabasi and the BRITE project from boston university suggest methods of using these properties to generate graphs,
and the BRITE people have actually conducted a study similar to ours and confirmed many of our conclusions.
 BRITE uses these factors to generate graphs

30.  moral: simulate with different generators!
Conclusions
Existing generators do not obey all power-laws
Our two topology generators do
PLOD: use power-law to generate node degrees
recursive: use 80/20 law to generate links
Do power-law topologies have any impact?
maybe: changed distributions for STORM
maybe not: averages unchanged for STORM
To conclude,
we have observed that existing topology generators do not obey all of the power laws,
and we have suggested two topology generators that do.
And we have seen that while using power-law topologies does not impact average-based results, they can have quite a large impact on distributions within the network.
So the moral of our story is to use many topology generators when conducting a simulation study, and if distributions within the network are at all important to you,
consider using topologies that obey power laws.
 moral: simulate with different generators!

31. Backup Slides

32. Generating Comparable Topologies
Equalize graph characteristics:
number of nodes
average out-degree
Ensure connectedness:
randomly connect disconnected components
Assign high/low-latency links:
Recursive algorithm provides a distinction
method for putting low-lat. links near clients