E.g. the Internet at the level
of Autonomous Systems
supports the critical
BGP routing protocol.
“Erdos and the Internet”
The Internet is a remarkable phenomenon
Milena
Mihail
that involves graph
theory
in a natural way
Tech. and models.
and gives rise toGeorgia
new questions
1
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.
2
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
3
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
Is it
or
?
or
?
How fast can you crawl the WWW?
Can you search a P2P network with low overhead?
Searching
Graph on
Are there strategies to improve crawling
and searching?
Is it
How can you maintain a well connected topology?
Total flow
nodes.
Route 1 unit of flow
between each pair of nodes.
How about distributed and dynamic networks?
Design
.
Congestion
link
Is it= flow on most loaded
or
under optimal routing.
4
?
Relevant metric:
Conductance
“bottlenecks”
Alon 85
Jerrum & Sinclair 88
Leighton & Rao 95
5
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
6
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).
7
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.
8
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
9
CONFIGURATIONAL MODEL
Given
edge multiplicity
O(log n) , a.s.
Choose random perfect matching over
connected, a.s.
minivertices
10
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
11
Structural Model, Proof Idea:
Difficulty: Non homogeneity in state-space
Worst case is when all vertices have degree 3.
12
Growth with Preferential Connectivity Model, Proof Idea:
Difficulty:
Arrival Time Dependencies
Shifting Argument
13
first
last
last
first
14
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.
15
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
?
16
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).
17
Conductance, Mixing and Cover Time
For
Cover Time
Rapid Mixing of Random Walk
“mixing” in
Alon 85
Jerrum & Sinclair 88
18
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.
19
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
20
Cover Time with Look-Ahead Two
Theorem [MM,Saberi,Tetali 05]:
In the configurational model
with
can discover
in
vertices
steps.
Proof
21
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
?
22
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
23
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.
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.
24
P2P Network Topology Maintenance by Constant Randomization
Theorem [Cooper, Frieze & Greenhill 04]:
The Markov chain corresponding to a 2-link switch on d-regular graphs is rapidly mixing.
Theorem
[Feder,
Saberi
06]: “pick” a random 2-link switch?
Question:
HowGuetz,
doesM,the
network
The
Markovthe
chain
d-regularingraphs
is rapidly
mixing,
In reality,
linksoninvolved
a switch
are within
constant distance.
even under local 2-link switches or flips.
25
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
Define a mapping
from
to
such that
(a)
(b) each edge in
maps to a path of constant length in
26
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
27
Open Problems
The Internet topology has constant second eigenvalue,
but larger than the second eigenvalue of random graphs.
Can we develop random graph models
(with powerlaw degree distributions)
and with varying values of the second eigenvalue ?
Preliminary work by Flaxman, Frieze & Vera
Routing on the Internet is done according to shortest paths.
Can we characterize congestion under shortest path routing?
How can we maintain a P2P topology with good connectivity
under dynamic settings or arriving and departing nodes?
Can we develop efficient distributed algorithms
that discover critical links in the network?
Preliminary work by Boyd, Diaconis & Xiao.
28