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Network Identifiability
with
Expander Graphs
Hamed Firooz, Linda Bai, Sumit Roy
Spring 2010
Outline



Identifiability definition
Identifiability using graph theory (Linda)
Identifiability using expander graph
Definition of Identifiability
Network Tomography
Network
?
Given a network,
and a limited
number of endhosts, can we infer
what’s happening
inside the network
Here our goal is to find the
links delay
Delay Tomography
Using probes that are inserted into a
data stream, end-to-end properties on
that route can be measured.
End1
Routing matrix R
link1
link 1 link 2 link 3
router1
End1  End 2
End1  End 3
End 2  End 3
link2
link3
End2
End3
1
1
0
1
0
1
0
1
1
Delay Tomography
We are interested in Links delay
l1 l2
l1
P1
l4
l2
l3
l5
1
1
R
0

0
l3 l4
l5
0
1
1
1
0
0
1
1
0
0
0
1
0 P1
0 P2
1 P3

1 P4
d P1  d l1  d l3  d l4
y=Rx
 d l1 
d 
 l2 
x   d l3 
 
 d l4 
d l 
 5
 d P1 
d 
P
y  2
 d P3 
 
d P4 
Deterministic Model
Problem: predict or estimate x from y with
y = Rx
R (N-by-M matrix) : binary routing matrix
X (M-by-1 vector) : quantity of interest, e.g, link delay
Y (N-by-1 vector) : known aggregations of X (measurements)
[3]
Identifiability: a network is identifiable if y=Rx has unique
solution [5]
• Usually, M ( # of links in network) >> N (# of measurements)
so network is generically NOT identifiable.
k-identifiability



a network is identifiable if y=Rx has unique
solution
Since this is an underdetermined system of
equations, it doesn’t have unique answer
We need side information:


k-identifiability: delay of up to k links which are
significantly higher than the others can be inferred
from end-to-end measurement y=Rx
significantly higher makes vector x k-sparse (kcompressible)
1-identifiability


Delay from End1 to End2 is
dl1+dl2
It is impossible to figure out
the delay of each link
End1
`
l1
l2
End2
`

In fact, there is no difference
between l1 and l2 in end-toend measurement
1-identifiable


A graph which has an intermediate node with
degree 2 is not 1-identifiable
In general, a graph is not 1-identifiable if and
only if:
l1, l2  E

l1  P  l2  P
In each end-to-end delay measurement we
either have the term dl1+dl2 or we don’t have
dl1 nor dl2
l1
l2
N1
N2
1-identifiable

Let’s look at routing matrix
l1, l2  E

l1  P  l2  P
Above statement means: if you look at
columns corresponding to l1 and l2 they
are both zero or one  there is two
.......l1 l2 .....
identical columns
P1 
P 2 
P3 

P4 
1 1
0 0
1 1
1 1






k-identifiable



Graph with a node (intermediate) which
has degree k+1 is not k-identifiable.
If graph is i-identifiable it is j
identifiable if j<i
Main question: given the routing
matrix of a network , is it kidentifiable?
k-identifiable


If a graph is k-identifiable then each k+1
columns of its routing matrix are independent
(necessary condition)
Is this a sufficient condition?


If every 2k columns of R are independent then
graph G is k-identifiable
if k=1 then k+1=2k=2 so identical columns
gives necessary and sufficient conditions for
1-identifiability
Expander Graphs
Bipartite Graph

A graph G(V,E) is called bipartite if:
V  V1  V2 s.t. V1  V2  
v, w V1  (v, w)  E
v, w V2  (v, w)  E
Usually G(V1,V2,E)
 V1 is left part, V2 is right
part

V1
V2
Bi-adjacency matrix


Adjacency matrix A=[aik], aik=1 iff node
i is connected to node k
Bi-adjacency matrix T=[tik], tik=1 iff
node i in V1 is connected to node k in V2
1
1
T 
1

0
0
1
0
1
0
0
0

1
0
A t
T
T

0
V1
V2
Regular Graph



A graph G(V,E) is called d-regular if
deg(v)=d for all v in V
A bipartite graph G(V1,V2,E) is called
left d-regular if for all v in V1 deg(v)=d
Number of ones in each row is d
1
1
T 
1

0
1
1
0
1
0
0
1

1
V1
V2
Expander graph



Let S  V1
Let N(S) be set of neighbors of X in V2
G(V1,V2,E) is called (s,ɛ)-expander if
S  V1, | S | s | N (S ) | (1   )d | S |
Each set of nodes on the left
expands to N(S) number of nodes
On right

V1
V2
Expander graph
V1
V2
V1
V2
V1
V2
V1
V2
V1
V1
V2
V2
Expander & Compressed
Sensing






Let G(V1,V2,E) be a (2k,ɛ)-expander
with left degree d
Let R=Tt
two vectors x and x’ have the same
projection under measurement matrix
R; i.e. Rx = Rx’
Suppose || x ||1 || x ||1
Then || x  x ||1  f ( ) || xS ||
S: set of k largest coefficients of x
c
Routing Matrix & Bipartite



Let Network N(V,E) is given with end to
end set of paths P
The routing matrix R is a |P|-by-|E|
binary matrix
It can be considered as bi-adjacency
matrix of a bipartite graph G(E,P,H)
Example

Routing matrix
l1 l2
P1 1
P2 0
R
P3 1

P4 0
l3 l4
l5
0 1 1 0
1 1 0 1
1 0 0 0

0 0 1 1
End3
End1
`
`
l1
l3
l2
l5
l4
End4
End2
`
`
P3
P1
P2
P4
Example

This is a bipartite graph with
biadjacency matrix Rt
l
Is this an expander?
1

P1
l2
P2
l3
P3
l4
P4
l5
Example

l1
P1
l2
P2
P3
l4
P4
l5
3
X  V , | X | 2 | N ( X ) | 2 | X | 1.5 | X |
4
If |X|=1, since degree each
node is 2|N(X)|=2>1.5

l3
This is (2,1/4)-expander with left
degree 2:
Example

l1
P1
l2
P2
l3
P3
l4
P4
l5
This is (2,1/4)-expander with left
degree 2:
3
X  V , | X | 2 | N ( X ) | 2 | X | 1.5 | X |
4
If |X|=1, since degree each
node is 2|N(X)|=2>1.5
 If |X|=2, it can be proved
That |N(X)|=3=1.5*2=3

1-identifiability




N(V;E) a network with paths collection P and
routing matrix R.
G(E;P;H) is a bipartite graph with biadjacency
matrix R.
x* is delay vector of N(V;E).
x is a solution to the LP optimization: min || x ||1



then
s.t.
|| x*  x ||1  f ( ) || xS c ||
Rx  Rx *
if G is a (2;d;ɛ)-expander with   1 4



reverse of Theorem is not true
This network is 1-identifiable
Bipartite graph coressponding to R
is not regular
l1
l2
l4
l3
l5
l6
1 1 0 1 0 0
R  1 0 1 0 0 1
0 0 0 1 1 1






It contains two expandersubgraphs
N(V;E) network with routing
matrix R
G(X; Y;H) bipartite graph with
bi-adjacency R
Gi(Xi;Y;Hi), i = 1; 2; …M is diregular
X   X i , H   H i , di  d j i  j
N is 1-identifiable if each Gi is an
expander
Expansion parameter




In conclusion, graph G(V,E) is k-identifiable
with routing matrix R, if R is bi-adjacency
matrix of a (2k, ɛ)-expander graph
There are lots of paper on how to construct
an expander (Used for design measurement
matrix)
Given a bipartite graph, what is its expansion
parameter?  There is no known theorem
We solve this problem for (2,ɛ)-expander; i.e.
1-identifiable


G(V,E) is a graph with adjacency matrix
H
Entry (i,j) of H2 gives number of walks
with length 2 from node i to node j
0
1
H 
1

0
1
0
1
1
1
1
0
0
0
1
0

0
2
1
H2  
1

1
1
3
1
0
1
1
2
1
1
0
1

1
1
2
3
4
2-expander


In a bipartite graph entry (i,j) of TtT
gives number of walks with length 2
from a node V1 to another node in V1
In a bipartite graph entry (i,j) of TtT
presents number of common neighbors
of nodes i and j.
0
2
H  t
T
T  0
. t

0  T
T  TT t


0  0
0 
t 
T T
Example

TtT shows that each two node have at
most 1 node in common
l
P
Each node has 2 neighbors
1



1
S  V1 , | S | 2 | N (S ) | 3
this is (2,1/4)-expander
2
1

t
T T  1

1
0
1 1 1 0
2 1 0 1
1 2 1 1

0 1 2 1
1 1 1 2
l2
P2
l3
P3
l4
P4
l5
Theorem
A bipartite graph G(V1,V2,E), with left degree d,
is (2,1/4)-expander if
1 
d
t
* J T T
 
J

2

Doesn’t have any negative entry
1 
 In conclusion, a graph G(V,E) with routing matrix
A is 1-identifiable if
d
* J  AAt
2

Doesn’t have any negative entry
1

1
Theorem
A bipartite graph G(V1,V2,E), with left degree d,
is (2, ɛ)-expander if
1 
t
2dJ  T T
J   
Doesn’t have any negative entry
1 
 In conclusion, a graph G(V,E) with routing matrix
R is 1-identifiable if

2dJ  RR t
Doesn’t have any negative entry
1

1
Best paths



There are actually 6 paths inside the
network
Obviously only 4 of them are
sufficient to figure out delay of every
link inside the network.
Question is how to select those
path?


End-to-end delay measurements using
probe transmission compels extra
burden on the network
Minimize cost of identifiability
P3
P2
P1
P5
P4
P6
Graph Covering

Suppose G(V,E) is given with set of paths
P



Question: Select a subset of P such that
every link in G belong to at least one of
the paths
Minimum number of paths that make a
link failure inside the network detectable
Is there any congested link inside the
network


1 Pi is used
Indicator function I Pi  
0 o.w.
Goal is to minimize number of
N
paths:
min  I Pi
P3
l1
P2
P1
P4
i 1


Subject to each link belong to at
least one path
link L1: Number of paths go
through it: I P1  I P3  I P5  1
P5
P6




IP=[IP1, IP2,…, IPN]
In general, ith entry of Rt .IP gives
number of paths go through link i
t
R
I P  1 componentTo cover all links
wise N
min
I
i 1
Pi
s.t. R t I P  1
I Pi  {0,1}


We know graph is 1-identifiable if R is
the bi-adjacency matrix of an 2expander graph
The condition is
3
| N ( S ) |  (deg( S ))  0
2

N
min
I
i 1
Pi
s.t. R t I P  1
I
Pj
Pj :li Pj or lk Pj
4
I
Pj
Pj :li Pj , lk Pj
 0 i  k
I Pi {0,1}


These are Binary Integer Programming
We can solve the LP version and select the
highest IPi
N

min
c I
i 1
i Pi
s.t. R t I P  1
I Pi  {0,1}
Ci is the cost of
using path Pi
N

min
c I
i 1
i Pi
s.t. R t I P  1
I
Pj
Pj :li Pj or lk Pj
I Pi {0,1}
4
I
Pj
Pj :li Pj , lk Pj
 0 i  k