How To Explore a Fast Changing World

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

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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)

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 ??!!
RW on Static Graphs
• The 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
Main Results
• 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”.
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.
“Sisyphus Wheel”
• The adversary has a simple (deterministic) strategy to increase h(1,n):
• The Cover Time of this dynamic graph is exponential!

...

3

n-1

2

n-3

1

n-2

n

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 graphG the cover time of the lazy random walk onG is O(n5ln2n).
• ?? Slower is faster ?? :-)
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.