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Basic Network Creation Games

Basic Network Creation Games. Noga Alon, Erik D. Demaine, MohammadTaghi Hajiaghayi, Tom Leighton. Presented by: Miki Shifman. Definition. Nodes = Players The players attempt to builds an efficient network that interconnects everyone Each player has 2 (selfish) goals:

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Basic Network Creation Games

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  1. Basic Network Creation Games Noga Alon, Erik D. Demaine, MohammadTaghi Hajiaghayi, Tom Leighton Presented by: Miki Shifman

  2. Definition • Nodes = Players • The players attempt to builds an efficient network that interconnects everyone • Each player has 2 (selfish) goals: • Minimize the cost of building links • Minimize the distance (average or maximal) to all other nodes

  3. Definition • Most models define the cost of building the links as • Various results depend heavily on specific ranges of and for some ranges of it, the behavior is poorly understood.

  4. Related Works • Fabrikant et. al, 03 • Demaine et. al, 07

  5. Our model • Doesn’t use • Cost of any edge is equal • Each agent can perform edge swap – that means swapping an incident edge with another incident edge, whenever it improves his state in the game

  6. Swap Equilibrium • No edge replacement by the rules of replacement we’ve defined, decreases agent’s usage cost. • usage cost is defined for a vertex v as:

  7. Sum games • A graph is in Sum Equilibrium if for every edge vw and every node w’ swapping edge vw with edge vw’ does not decrease the total sum of distances from v to all other

  8. Max Games • Local diameter – for v it’s the max distance from v to any other node • Deletion-critical graph – a deletion of every edge increases the local diameter in both its endpoints • Insertion-stable graph – insertion of every edge doesn’t decrease the local diameter of both its endpoints. • A graph which is both Deletion-critical & Insertion stable is in Max equilibrium

  9. Results on Equilibrium trees • Sum games: • The diameter is 2 • Max games: • The diameter is 3

  10. Results on Equilibrium trees • Theorem: If a sum equilibrium graph in the basic network-creation game is a tree, then it has diameter of length 2.

  11. Results on Equilibrium trees Proof: • Suppose for contradiction that an equilibrium tree has a diameter of 3 • Perform swaps in the way illustrated:

  12. Results on Equilibrium trees • In order that the swaps wouldn’t be a net win there should exist: • Combined: and so the contradiction ( are both defined as

  13. Result on general networks • Max games: • A lower bound • Sum games: • lower bound for diameter is 3. • Upper bound for diameter is

  14. Future Work • Upper bound for max games on general graphs • Computing the POA in our model and proving bounds on it • How many steps it takes for convergence in our model, if only best responses played

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