1. How To Explore a Fast Changing World (On the Cover Time of Dynamic Graphs) Chen Avin Ben Gurion University Joint work with Michal Koucky & Zvi Lotker (ICALP-08)

2. Israeli Networking Seminar 29-May-2008 2 Motivation Today’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 ??!!

3. Israeli Networking Seminar 29-May-2008 3 RW on Static Graphs The Simple Random Walk on Graph. Cover Time, hitting time are bounded by n 3. 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

4. Israeli Networking Seminar 29-May-2008 4 Main Results Question: What Question: 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”.

5. Israeli Networking Seminar 29-May-2008 5 Dynamic Model Evolving Graphs: Random walk on dynamic graph Worst case analysis: a game between the walker and an oblivious adversary that controls the network dynamics. G 1 G 2 G 3 G 4 G 5...

6. Israeli Networking Seminar 29-May-2008 6 The adversary has a simple (deterministic) strategy to increase h(1,n): The Cover Time of this dynamic graph is exponential! “Sisyphus Wheel” 1 2 3 n-3 n-2 n... n-1

7. Israeli Networking Seminar 29-May-2008 7 The Lazy Random Walk Lazy 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 graph G the cover time of the lazy random walk on G is O(n 5 ln 2 n). ?? Slower is faster ?? :-)

8. Israeli Networking Seminar 29-May-2008 8 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.

9. Thank You!