How To Explore a Fast Changing World

(On the Cover Time of Dynamic Graphs)

(On the Cover Time of Dynamic Graphs)

Chen AvinBen Gurion University

Joint work with Michal Koucky & Zvi Lotker (ICALP-08)

Israeli Networking Seminar 29-May-2008 2

MotivationMotivationToday’s communication networks are dynamic: mobility, communication fluctuations, duty cycles, clients joining and leaving, etc.

Structure-base schemes (e.g., spanning tress, routing tables) are thus problematic.

Turning attention to structure-free solutions.

Random-walk-based protocols are simple, local, distributed and robust to topology changes.

Robust to topology changes ??!!Robust to topology changes ??!!

Israeli Networking Seminar 29-May-2008 3

RW on Static GraphsRW on Static GraphsThe Simple Random Walk on Graph.

Cover Time, hitting time arebounded by n3.

Random walk can be efficient for some applications/networks, i.e., the time to visit a subset of N nodes, can be linear in N.

Partial cover time.

Tempting to use on dynamic networks

Israeli Networking Seminar 29-May-2008 4

Main ResultsMain ResultsQuestionQuestion: What: What will be the expected number of steps for a random walk on dynamic network to visit every node in the network (i.e., Cover Time).

Answers in short:

Bad, very bad (compare to static network).

Can be fixed by the “Lazy Random Walk”.

Israeli Networking Seminar 29-May-2008 5

Dynamic ModelDynamic ModelEvolving Graphs:

Random walk on dynamic graph

Worst case analysis: a game between the walker and an oblivious adversary that controls the network dynamics.

G1 G2 G3 G4 G5 ...

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The adversary has a simple (deterministic) strategy to increase h(1,n):

The Cover Time of this dynamic graph is exponential!

“Sisyphus Wheel” “Sisyphus Wheel”

1

2

3

n-3

n-2

n

...

n-1

Israeli Networking Seminar 29-May-2008 7

The Lazy Random WalkThe Lazy Random WalkLazy random walk: At each step of the walk pick a vertex v from V(G) uniformly at random and if there is an edge from the current vertex to the vertex v then move to v otherwise stay at the current vertex.

Theorem: For any connected evolving graphG the cover time of the lazy random walk onG is O(n5ln2n).

?? Slower is faster ?? :-)

Israeli Networking Seminar 29-May-2008 8

Summary Summary Demonstrate that the cover time of the simple random walk on dynamic graphs is significantly different from the case of static graphs: exponential vs. polynomial.

The cover time is bounded to be polynomial by the use of lazy random walk.

Gives some theoretical justification for the use of random-walks-techniques in dynamic networks, but careful attention is required.

Thank You!Thank You!