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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.

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how to explore a fast changing world
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
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
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
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

G1 G2 G3 G4 G5 ...

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
“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
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
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.
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