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 end-hosts, can we infer what’s happening inside the network

Here our goal is to find the links delay

End1

End2 End3

router1

link1

link2

link3 110321013101121

321

EndEndEndEndEndEnd

linklinklink

Routing matrix R

Delay TomographyUsing probes that are inserted into a data stream, end-to-end properties on that route can be measured.

4

3

2

1

54321

11000101100001101101

PPPP

R

lllll

y=Rx

Delay Tomography

5

4

3

2

1

l

l

l

l

l

ddddd

x

4

3

2

1

P

P

P

P

dddd

y4311 lllP dddd

1 2

3

4 5

P1

We are interested in Links delay

Problem: predict or estimate x from y with y = Rx

R (N-by-M matrix) : binary routing matrixX (M-by-1 vector) : quantity of interest, e.g, link delayY (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.

Deterministic Model

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 (k-compressible)

1-identifiability Delay from End1 to End2 is

d1+d2

It is impossible to figure out the delay of each link

In fact, there is no difference between 1 and 2 in end-to-end measurement

`

`

l1

l2

End1

End2

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:

In each end-to-end delay measurement we either have the term d1+d2 or we don’t have d1 nor d2

PlPlEll 2121,

N1 N2l1 l2

1-identifiable Let’s look at routing matrix

Above statement means: if you look at columns corresponding to 1 and 2 they are both zero or one there is two identical columns

11110011

4321

............ 21

PPPP

ll

PlPlEll 2121,

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 k-identifiable?

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:

Usually G(V1,V2,E) V1 is left part, V2 is rightpart

EwvVwvEwvVwvVVtsVVV

),(,),(,

..

2

1

2121

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

V1 V2

110001011001

T

00tTT

A

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

V1V2

110101011011

T

Expander graph Let Let N(S) be set of neighbors of X in V2 G(V1,V2,E) is called (s,ɛ)-expander if

Each set of nodes on the leftexpands to N(S) number of nodesOn right

1VS

||)1(|)(|||,1 SdSNsSVS

V1 V2

Expander graph

V1 V2 V1 V2 V1 V2

V1 V2 V1 V2 V1 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 Then S: set of k largest coefficients of x

||||)(|||| 1 cSxfxx

11 |||||||| xx

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

End1

End2

`

`

End3

End4

l3

l4l5

11000000111011001101

4

3

2

1

54321

PPPP

R

lllll

P1 P2

P3

P4

Example This is a bipartite graph with

biadjacency matrix Rt

Is this an expander?l1

l2

l3

l4

l5

P1

P2

P3

P4

Example This is (2,1/4)-expander with

left degree 2:

If |X|=1, since degree eachnode is 2|N(X)|=2>1.5

l1

l2

l3

l4

l5

P1

P2

P3

P4

||5.1||243|)(|2||, XXXNXVX

Example This is (2,1/4)-expander with

left degree 2:

If |X|=1, since degree eachnode is 2|N(X)|=2>1.5 If |X|=2, it can be provedThat |N(X)|=3=1.5*2=3

l1

l2

l3

l4

l5

P1

P2

P3

P4

||5.1||243|)(|2||, XXXNXVX

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:

then if G is a (2;d;ɛ)-expander with

*

1

s.t.||||min

RxRx

x

||||)(|||| 1

*cS

xfxx 41

reverse of Theorem is not true This network is 1-identifiable Bipartite graph coressponding

to R is not regular 1

2 3

45

6

111000100101001011

R

It contains two expander-subgraphs

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 di-regular N is 1-identifiable if each Gi is an expander

jiddHHXX jiii ,,

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

0010001111010110

H1

2

3 4

1101121101311112

2H

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.

TTTT

TT

TT

Ht

t

tt 00

00

.0

02

Example TtT shows that each two node have

at most 1 node in common Each node has 2 neighbors this is (2,1/4)-expander

l1

l2

l3

l4

l5

P1

P2

P3

P4

2111012101112111012101112

TT t

3|)(|2||,1 SNSVS

Theorem A bipartite graph G(V1,V2,E), with left

degree d, is (2,1/4)-expander if

Doesn’t have any negative entry In conclusion, a graph G(V,E) with routing

matrix A is 1-identifiable if

Doesn’t have any negative entry

TTJd t*2

11

11

J

tAAJd*

2

Theorem A bipartite graph G(V1,V2,E), with left

degree d, is (2, ɛ)-expander if

Doesn’t have any negative entry In conclusion, a graph G(V,E) with routing

matrix R is 1-identifiable if

Doesn’t have any negative entry

TTJd t2

11

11

J

tRRJd 2

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

P1P2

P3

P4

P5P6

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

Indicator function

Goal is to minimize number of paths:

Subject to each link belong to at least one path

link L1: Number of paths go through it:

N

iPiI

1

min

P1P2

P3

P4

P5P6

1531 PPP III

o.w.0used is 1 i

P

PI

i 11

IP=[IP1, IP2,…, IPN] In general, ith entry of Rt .IP gives

number of paths go through link i To cover all links

component-wise

1Pt IR

}1,0{1..

min1

i

i

P

Pt

N

iP

IIRts

I

We know graph is 1-identifiable if R is the bi-adjacency matrix of an 2-expander graph

The condition is 0))(deg(

23|)(| SSN

These are Binary Integer Programming We can solve the LP version and select

the highest IPi

}1,0{

04

1..

min

,::

1

i

jkjij

j

jkjij

j

i

P

PlPlPP

PlorPlPP

Pt

N

iP

I

kiII

IRts

I

}1,0{

1..

min1

i

i

P

Pt

N

iPi

IIRts

Ic

}1,0{

04

1..

min

,::

1

i

jkjij

j

jkjij

j

i

P

PlPlPP

PlorPlPP

Pt

N

iPi

I

kiII

IRts

Ic

Ci is the cost of using path Pi