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 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 End2End3 router1 link1 link2 link3 Routing matrix R Delay Tomography Using probes that are inserted into a data stream, end-to-end properties on that route can be measured.

y=Rx Delay Tomography P1P1 We are interested in Links delay

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. 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 d 1 +d 2 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

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 d 1 +d 2 or we don’t have d 1 nor d 2 N1N2 l1l1 l2l2

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

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(V 1,V 2,E) V 1 is left part, V 2 is right part V1V1 V2V2

Bi-adjacency matrix Adjacency matrix A=[a ik ], a ik =1 iff node i is connected to node k Bi-adjacency matrix T=[t ik ], t ik =1 iff node i in V 1 is connected to node k in V 2 V1V1 V2V2

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 V 1 deg(v)=d Number of ones in each row is d V1V1 V2V2

Expander graph Let Let N(S) be set of neighbors of X in V 2 G(V 1,V 2,E) is called (s, ɛ )-expander if Each set of nodes on the left expands to N(S) number of nodes On right V1V1 V2V2

Expander graph V1V1 V2V2 V1V1 V2V2 V1V1 V2V2 V1V1 V2V2 V1V1 V2V2 V1V1 V2V2

Expander & Compressed Sensing Let G(V 1,V 2,E) be a (2k, ɛ )-expander with left degree d Let R=T t 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

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 P1P1 P2P2 P3P3 P4P4

Example This is a bipartite graph with biadjacency matrix R t Is this an expander? l1l1 l2l2 l3l3 l4l4 l5l5 P1P1 P2P2 P3P3 P4P4

Example This is (2,1/4)-expander with left degree 2: If |X|=1, since degree each node is 2  |N(X)|=2>1.5 l1l1 l2l2 l3l3 l4l4 l5l5 P1P1 P2P2 P3P3 P4P4

Example This is (2,1/4)-expander with left degree 2: 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 l1l1 l2l2 l3l3 l4l4 l5l5 P1P1 P2P2 P3P3 P4P4

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

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

It contains two expander- subgraphs N(V;E) network with routing matrix R G(X; Y;H) bipartite graph with bi-adjacency R G i (X i ;Y;H i ), i = 1; 2; …M is d i - regular N is 1-identifiable if each G i 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 H 2 gives number of walks with length 2 from node i to node j

2-expander In a bipartite graph entry (i,j) of T t T gives number of walks with length 2 from a node V1 to another node in V1 In a bipartite graph entry (i,j) of T t T presents number of common neighbors of nodes i and j.

Example T t T shows that each two node have at most 1 node in common Each node has 2 neighbors this is (2,1/4)-expander l1l1 l2l2 l3l3 l4l4 l5l5 P1P1 P2P2 P3P3 P4P4

Theorem A bipartite graph G(V 1,V 2,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

Theorem A bipartite graph G(V 1,V 2,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

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 P1P1 P2P2 P3P3 P4P4 P5P5 P6P6

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 L 1 : Number of paths go through it: P1P1 P2P2 P3P3 P4P4 P5P5 P6P6 1 1

I P =[I P1, I P2,…, I PN ] In general, i th entry of R t. I P gives number of paths go through link i To cover all links component- wise

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

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

C i is the cost of using path P i