Algorithmic Performance
in Complex Networks
Milena Mihail
Georgia Tech.
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E.g. the Internet at the level
of Autonomous Systems
supports the critical
BGP routing protocol.
The Internet is a remarkable phenomenon
that involves graph theory in a natural way
and gives rise to new questions and models.
2
Search and routing networks,
like the WWW, the internet, P2P networks,
ad-hoc (mobile, wireless, sensor) networks
are pervasive and scale at an unprecedented rate.
Performance analysis/evaluation in networking:
measure parameters hopefully predictive of performance.
Important in network simulation and design.
3
Want
Sparse
metrics
small-world
predictive
graphs
orwith
explanatory
large degree-variance.
of network function.
, but
frequency
no sharp concentration
Erdos-Renyi
2
4
10
100
degree
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Networking questions
How does delay scale in routing?
Does packet drop (blocking) scale?
Are network resources used efficiently?
Is there load balancing?
Does the network evolve towards monopolies?
Routing Congestion
How fast can you crawl the WWW?
Can you search a P2P network with low overhead?
Graph on
Are there strategies to improve crawling
and searching?
nodes.
Route 1 unit of flow
between each pair of nodes.
How can you maintain a well connected topology?
How about distributed and dynamic networks?
Searching
Total flow
Design
.
Congestion = flow on most loaded link
under optimal routing.
5
Relevant metric:
Conductance
“bottlenecks”
Alon 85
Jerrum & Sinclair 88
Leighton & Rao 95
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computationally soft
Second eigenvalue
Matlab does 1-2M node sparse graphs
of the lazy random walk associated with the adjacency matrix
closely approximates conductance:
+
+
+
+
This also says that congestion
under link capacities, search time
and sampling time scale smoothly
Internet
-
-
is also another point of view
- This
of the small-world phenomenon
Plots at 700 nodes,
3000 nodes,
and 15000 nodes.
Eigenvectors associated with large eigenvalues
are “shadows” of sets with bad conductance.
Random Graph
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100 largest eigenvalues
Beyond today, we need network models
to predict future behavior.
What are suitable network models?
The Internet grows anarchically,
so random graphs are good canditates.
Current network models are random graphs
which produce power law degree sequences
(thus also matching this important observed data).
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EVOLUTIONARY:Growth & Preferential Attachment
One vertex at a time
New vertex attaches to
existing vertices
Simon 55, Barabasi & Albert 99, Kumar et al 00,
Bollobas & Riordan 01,
Bollobas, Riordan, Spencer & Tusnady 01.
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CONFIGURATIONAL aka structural MODEL
“Random” graph with given “power law” degree sequence.
Given
choose random perfect matching over
minivertices
Bollobas 80s, Molloy & Reed 90s, Aiello, Chung & Lu 00s, Sigcomm/Infocom 00s
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CONFIGURATIONAL MODEL
Given
edge multiplicity
O(log n) , a.s.
Choose random perfect matching over
connected, a.s.
minivertices
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Bounds on Conductance
Technique: Probabilistic Counting Arguments & Combinatorics.
Difficulty: Non homogeneity in state-space, Dependencies.
Theorem [M, Papadimitriou, Saberi 03]: For a random graph grown with
preferential attachment with
,
, a.s.
Previously:
Cooper & Frieze 02
Theorem [Gkantsidis, M, Saberi 03]: For a random graph in the configurational
model arising from degree sequence
,
, a.s.
Independent:
for a different structural random graph model and
Chung,Lu&Vu 03
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Structural Model, Proof Idea:
Difficulty: Non homogeneity in state-space
But all vertices do not have the same degree.
Worst case is when all vertices have degree 3.
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Growth with Preferential Connectivity Model, Proof Idea:
1st
2nd
3rd
4th
5th
6th
7th
Difficulty: whether there is edge depends upon arrival order of all vertices!
Key Observation: To bound conductance of S, suffices to study combinatorics
of how these two sequences interleave.
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Growth with Preferential Connectivity Model, Proof Idea:
1st
2nd
3rd
4th
5th
6th
7th
Shifting Argument
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Theorem [MM, Papadimitriou, Saberi 03]: For a random graph grown with
preferential attachment with
there is a poly time computable flow that routes
demand
between all vertices
and
with max link
congestion
, a.s.
Theorem [Gkantsidis,MM, Saberi 03]: For a random graph in the structural
model arising from degree sequence
there is a poly
time computable flow that routes demand
between all vertices and with
max link congestion
a.s.
Note: Why is demand
?
Each vertex with degree
in the network core
serves
customers from the network periphery.
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Networking questions
How does delay scale in routing?
Does packet drop (blocking) scale?
Are network resources used efficiently?
Is there load balancing?
Does the network evolve towards monopolies?
Routing Congestion
It is
How fast can you crawl the WWW?
Can you search a P2P network with low overhead?
Are there strategies to improve crawling
and searching?
How can you maintain a well connected topology?
How about distributed and dynamic networks?
Searching
Is it
or
?
Design
Is it
or
?
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Searching, Cover Time and Mixing Time
Graph on
nodes.
Search the graph by random walk.
Cover time = expected time
to visit all nodes.
Mixing time = time to reach
stationary distribution
(arbitrarily close).
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Conductance, Mixing and Cover Time
For
Cover Time
Rapid Mixing of Random Walk
“mixing” in
Alon 85
Jerrum & Sinclair 88
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Extensions of Cover Time
In practice, when crawling the WWW or searching a P2P network,
when a node is visited,
all nodes incident to the node are also visited.
This can be implemented by one-step local replication of information.
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Cover Time with Look-Ahead One
Theorem [MM,Saberi,Tetali 05]:
In the configurational model
with
Proof
can discover
vertices
in
steps.
Adamic et al 02
Chawathe et al 03
Gkanstidis, MM, Saberi 05
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Cover Time with Look-Ahead Two
Theorem [MM,Saberi,Tetali 05]:
In the configurational model
with
can discover
in
vertices
steps.
Proof
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Networking questions
How does delay scale in routing?
Does packet drop (blocking) scale?
Are network resources used efficiently?
Is there load balancing?
Does the network evolve towards monopolies?
Routing Congestion
It is
How fast can you crawl the WWW?
Can you search a P2P network with low overhead?
Searching Cover time
Are there strategies to improve crawling
and searching?
It is
and
local replication offers
substantial improvement
How can you maintain a well connected topology?
How about distributed and dynamic networks?
Design
Is it
or
?
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The case of Peer-to-Peer Networks
Must maintain well connected topology,
e.g. a graph with good concuctance, a random graph
Distributed, decentralized
n nodes, d-regular graph
Each node has resources O(polylogn)
and knows a very small size
neighborhood around itself
?
Search for content,
e.g. by flooding or random walk
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P2P networks are constantly randomizing their links
Gnutella: constantly drops existing connections
and replaces them with new connections
There are between 5 and 30 requests for new connections per second
per client.
About 1% of these requests are satisfied and existing links are dropped.
The network is working “in panic” trying to randomize
thus avoiding network configurations with bottlenecks
and trying to maintain high conductance.
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P2P Network Topology Maintenance by Constant Randomization
Theorem [Cooper, Frieze & Greenhill 04]:
The Markov chain corresponding to a general 2-link switch on d-regular graphs is rapidly mixing.
LOCALITY:
In reality, network can only switch links that are within constant distance.
Theorem [Feder, Guetz, M, Saberi 06]: Rapid mixing even under local 2-link switches or flips.
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The proof is a Markov chain comparison argument
Space of connected d-regular graphs
local Flip Markov chain
Space of d-regular graphs
general 2-link switch Markov chain
Map the transitions of S to the transitions of SC, with small load.
Load = max # transitions of S mapped to single transition of SC.
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The proof is a Markov chain comparison argument
Space of connected d-regular graphs
local Flip Markov chain
Space of d-regular graphs
general 2-link switch Markov chain
Natural mapping from S to SC: map switch between u, v to path of local flip switches.
Problem: length of path unbounded.
Key Construction: mapping such that each edge in S maps to a constant number
of edges in SC
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P2P Dynamic Network Construction Problem:
?
?
?
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P2P Dynamic Network Topology Construction by Random Walk
?
?
?
Theorem [Law & Siu ‘03]: Construct a constant expander on n vertices
with overhead O( log n) per node addition.
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P2P Dynamic Network Topology Construction by Random Walk
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P2P Dynamic Network Topology Construction by Random Walk
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P2P Dynamic Network Topology Construction by Random Walk
Gkantsidis, MM, Saberi ’04
Heuristic reminiscent of saving random bits in simulation of BPP [AKS87,ZI89,G95]
Overhead O(1) per new node addition.
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Networking questions
Congestion
It is
How fast can you crawl the WWW?
Can you search a P2P network with low overhead?
Are there strategies to improve crawling
and searching?
How can you maintain a well connected topology?
How about distributed and dynamic networks?
Cover time
It is
Conductance
How does delay scale in routing?
Does packet drop (blocking) scale?
Are network resources used efficiently?
Is there load balancing?
Does the network evolve towards monopolies?
Mixing time
It is
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Networking is growing at an unprecedented rate
and it is rich with algorithmic questions.
In particular,
it raises novel new questions related to expander graphs.
How can we maintain an expander in a distributed way
under dynamic settings or arriving and departing nodes?
Can we develop efficient distributed algorithms
that discover critical links in the network?
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