Minimal toy topology

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Minimal toy topology
Regional POPs
Core network
Customers
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• Simplifying assumptions
• Physical and IP connectivity are identical (No
  level 2 or MPLS)
• Minimal geometry ( topology plus link speeds
  and locations)
• Aim to capture essence of topology
• Add complexity back in later

Line thickness roughly represents link bandwidth.
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Minimal toy topology
Regional POPs
Core network
Customers
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Router bandwidth is constrained.
Technically infeasible
Technically feasible
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Customers with a variety of local connectivity
speeds.
high connectivity
Gateway routers
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high connectivity
high speed
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high speed
Technically infeasible
high connectivity
high speed
high connectivity
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2
9
3
6
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9
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5
10
1
10
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Total of nodes 56
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2
56
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3
6
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9
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5
10
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Connectivity at edges?
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2
56
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3
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9
9
?
?
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5
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Note Assume some unspecified local connectivity.
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We are interested in distributions of core and
gateways so will largely ignore local
connectivity for now
Local
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Core
?
?
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9
6
5
4
log(rank)
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2
Gateways
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1
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Scaling (Mandelbrot) or Power law
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Do real networks have power law
connectivity? (There are people here better
qualified to answer than me, but very roughly
yes. And power laws per se are not important,
but heavy tails are.)
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1
10
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How do we define a simple, computable, yet
reasonable performance measure for such networks?
Proposal we define desired flows between
customers, which are then to be implemented by
flows in the network. We then can measure the
efficiency and perhaps robustness of the network
in producing these flows.
Well come back to this after a qualitative
discussion.
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Varied customer demand
Conjecture
? Power law connectivity
56
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Bandwidth constraints
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Varied customer demand
What is the null hypothesis?
Interpret statistics as a probabilistic model.
? Power law connectivity
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Eliminate design constraints.
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Bandwidth constraints
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Total of nodes 56
Total of links 72
Compute the degree distribution for random graph
with 56 nodes and 72 links. This has some
probability distribution (needs to be looked up).
Probability that there exists a node with degree
20 is vanishingly small.
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Total of nodes 56
Therefore, vanishingly unlikely to have this
distribution in a purely random graph.
Total of links 72
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10
0
Degree distribution for random graph with 56
nodes and 72 links????
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-1
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-2
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Probability exists node with degree 20 is less
than ?1e-12????
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Can reject simplest null hypothesis.
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0
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A more sophisticated null hypothesis.
? Power law connectivity

• Assume this scaling degree distribution but
  otherwise random
• Find typical or generic cases
• Standard statistical physics approach to complex
  systems
• Yields scale-free networks

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10
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What is the null hypothesis?
Interpret statistics as a probabilistic model.
? Power law connectivity

• Compare designed network with most likely
  (maximum likelihood) random graph with same
  connectivity degree

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10
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Null hypothesis
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Total of links 72
Total of nodes 56
Total of links 72
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Null hypothesis
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2
9
Total of links 72
Most likely, highly connected nodes are connected
to each other.
10
1
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• This is one of many possible definitions of
  scale-free
• Simply fill in links using the most probable
  ones, in order, this will be the maximum
  likelihood graph
• Rearrange at random until it reaches statistical
  steady state
• others???
• All should give nearly the same thing, but the
  max likelihood could be used as the canonical
  scale-free network, and the likelihood (or some
  related function) could serve as a measure of
  nearness to scale-free.

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• Note that there are two distinct notions here
• Max likelihood given a degree distribution (or
  presumably any attributes, but were focusing on
  degree distribution), identify the most likely
  graph with that distribution among otherwise
  random graphs
• Scaling graphs with power law degree
  distributions
• Scale-free graphs are the max likelihood
  scaling graphs.
• Lets see what this looks like.

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Redraw slightly and eliminate distinctions
between lines.
Reconnect edges to increase probability
This will quickly morph into a scale-free
graph. Completely random reconnections will get
there much more slowly.
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Ignore the connections at the periphery for now.
Reconnect edges to increase probability
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Reconnect edges to increase probability
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Reconnect edges to increase probability
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Reconnect edges to increase probability
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Reconnect edges to increase probability
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This is the scale-free network.
One problem is that it doesnt even connect to
all the nodes.
What to do at this point?
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Recall that we had not specified additional local
connectivity. Could try to add that in, making
it try to get a fully connected graph.
Alternatively, we could just try to keep a power
law degree distribution and add links until it
becomes fully connected. Have to sort that out,
but will consider the qualitative features next.
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Redrawn with core routers back in the middle.
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Reconnect edges with same probability but
increase connectivity
And assume some local connections complete a
fully connected graph.
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56
Note local connections not shown
These are extremely different, but have the same
connectivity distribution.
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• Scale-rich
• Highly structured
• Self-dissimilar
• Efficient
• Robust
• Designed

• Scale-free
• Unstructured
• Self-similar
• Wasteful
• Fragile
• Emergent

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• Scale-rich
• Each level has very different characteristics
• Pieces need not resemble the whole

• Scale-free
• Each level is similar
• Pieces resemble the whole

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• Delete the highly connected node
• Only local loss

• Delete the highly connected node
• Global loss
• Remaining parts are still scale-free

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• Delete the worst case node
• Relatively local loss

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• The Internet does have huge fragilities
• Denial of service, DNS, BGP, etc
• The one fragility it doesnt have is deletion of
  routers

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• These are completely opposite extremes
• Scale-rich, Highly structured, Self-dissimilar,
  Efficient, Robust, Designed
• Vs. Scale-free, Unstructured, Self-similar,
  Wasteful, Fragile, Emergent

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• This is very different from the real Internet
• More importantly, it is very different from what
  any Internet could possibly be

• Of course, no one would seriously propose that
  the Internet really does look like this
• Surely this is just a purely hypothetical null
  hypothesis against which to compare more
  sensible explanations.
• Right?

• Scale-free
• Unstructured
• Self-similar
• Wasteful
• Fragile
• Emergent

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• Typical of physics-based approaches to complex
  networks
• This approach dominates mainstream science
  literature
• Assume statistics describe an otherwise generic
  configuration
• Similar to edge-of-chaos, order for free,
  criticality, order-disorder transitions,
  emergence,
• Essentially the same arguments and results hold
  for biological networks

• Scale-free
• Unstructured
• Self-similar
• Wasteful
• Fragile
• Emergent

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Scaling
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10
Scale-free
1
10
1
Scale-rich

• Power laws (scaling) are consistent with either
  scale-rich or scale-free
• Scale-free is a natural and sophisticated null
  hypothesis and is clearly refuted

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Varied customer demand
Designed, Scale-rich, Highly structured,
Self-dissimilar, Efficient, Robust
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? Power law connectivity
Bandwidth constraints
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high speed
high connectivity
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10
1
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More realism

• Redundancy can easily give additional robustness
• Level 2 (ATM, Ethernet) and MPLS increase IP
  versus physical connectivity
• Real networks are obviously much more
  complicated
• Does this capture the essence of the degree
  distribution?
• What is a simple way to quantify efficiency and
  robustness?
• Could customer demand plus bandwidth constraints
  give useful way to generate realistic
  geometries ( topology plus locations plus
  bandwidths)?

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ASx
ASx
AS graph
ASy
ASy
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• Every level is much more complicated.

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Message Theory should help explain what is
observed and suggest what is necessary vs what is
accident.
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10
Power laws
1
Scale-rich
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1
high speed
Constraints
Scale-free
high connectivity
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How do we define a simple, computable, yet
reasonable performance measure for such networks?
Proposal we define desired flows between
customers, which are then to be implemented by
flows in the network. We then can measure the
efficiency and perhaps robustness of the network
in producing these flows.
Well come back to this after a qualitative
discussion.
52
How do we define a simple, computable, yet
reasonable performance measure for such
networks? Below is a network with r5 routers and
n8 hosts in c3 clusters. There would be
n(n-1)/287/228 possible flows between all
hosts, which might get too big for larger
networks. So we can assume perhaps that there
will be 1337 local flows and then all
combinations of aggregated flows from the
clusters of local groups, which in this case
would be c(c-1)/23 flows.
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For example.
3 aggregate flows
3 local flows
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R has rows which correspond to nodes and columns
which correspond to links.
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local flows
aggregate flows
matrix of flows on links
hosts
routers
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representative set of flows
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least squares (pseudo-inverse)
some norm?
absolute values
weighting matrix
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• Could put link capacities into R perhaps, but
  maybe want to use J to figure out what capacities
  are needed and what the cost would be.
• J as given here is a matrix with each column
  giving the least squares link flows to realize
  the desired traffic in the corresponding column
  of X.
• It might be better to solve linear programs with
  constraints, but that will be more expensive, but
  we should consider that. But something like this
  should be ok.

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• This has essentially the same features as
  stoichiometry in biochemistry but is much
  simpler, because
• there is only one conserved quantity (flows or
  packets)
• the reactions are links, and thus all binary,
  and thus representable as a graph

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Stoichiometry

• We can define something similar for a biochemical
  network, where R is the stoichiometry
• The bigger problem is that we dont have an
  obvious notion of scale-free here, since
  reducing stoichiometry to a graph loses all
  biological meaning
• What we need to do is correctly characterize what
  we mean here by
• scaling
• max likelihood

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• For both metabolism and the Internet graphs we
  want to have a picture that looks something like
  this.
• Here bad performance means wasteful and fragile.

Empty
Scale-free
High
Likelihood
Possible, but to be avoided
optimal
Low
Good
Bad
Performance
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• But also we want to make the point that this is a
  very general picture, and applies to
  edge-of-chaos, criticality, and self-organized
  criticality as well.

Empty
EOC/SOC
High
Likelihood
optimal
Low
Good
Bad
Performance