Network Identifiability with Expander Graphs

1. Network Identifiability with Expander Graphs Hamed Firooz, Linda Bai, Sumit Roy Spring 2010

2. Outline • Identifiability definition • Identifiability using graph theory (Linda) • Identifiability using expander graph

3. Definition of Identifiability

4. 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

5. 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 router1 link2 link3 End2 End3

6. Delay Tomography We are interested in Links delay l1 l2 P1 l3 l4 l5 y=Rx

7. 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.

8. 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)

9. 1-identifiability • Delay from End1 to End2 is dl1+dl2 • It is impossible to figure out the delay of each link • In fact, there is no difference between l1 and l2 in end-to-end measurement

10. 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 dl1+dl2 or we don’t have dl1 nor dl2 l1 l2 N1 N2

11. 1-identifiable • Let’s look at routing matrix • Above statement means: if you look at columns corresponding to l1 and l2 they are both zero or one  there is two identical columns

12. 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?

13. 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

14. Expander Graphs

15. Bipartite Graph • A graph G(V,E) is called bipartite if: • Usually G(V1,V2,E) • V1 is left part, V2 is right part V1 V2

16. 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

17. 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 V2 V1

18. 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 left expands to N(S) number of nodes On right V1 V2

19. Expander graph V1 V2 V1 V2 V1 V2 V1 V2 V1 V2 V1 V2

20. 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

21. 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)

22. Example • Routing matrix P3 P1 P2 P4

23. l1 P1 l2 P2 l3 P3 l4 P4 l5 Example • This is a bipartite graph with biadjacency matrix Rt • Is this an expander?

24. l1 P1 l2 P2 l3 P3 l4 P4 l5 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

25. 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 l1 P1 l2 P2 l3 P3 l4 P4 l5

26. 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

27. l1 l2 l3 l5 l4 l6 • reverse of Theorem is not true • This network is 1-identifiable • Bipartite graph coressponding to R is not regular

28. 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

29. 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

30. 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 1 2 3 4

31. 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.

32. l1 P1 l2 P2 l3 P3 l4 P4 l5 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

33. 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

34. 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

35. Best paths P3 • 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 P2 P1 P4 P5 P6

36. 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

37. P3 • 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: l1 l1 P2 P1 P4 P5 P6

38. 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

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

40. These are Binary Integer Programming • We can solve the LP version and select the highest IPi

41. Ci is the cost of using path Pi