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Advances in Network Creation Games: Exploring Equilibrium in Graph Structures

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wells
14 diapositives · 282 Views · Aug 04, 2014

This paper delves into the dynamics of network creation games, focusing on strategies for players to minimize costs and optimize distances between nodes. We define nodes as players aiming to construct an efficient interconnection network, with various games explored, including sum and max games. Key concepts such as swap equilibrium and properties of deletion-critical and insertion-stable graphs are discussed. Our findings reveal limitations on diameters in equilibrium graphs and suggest future research directions, particularly regarding convergence in gameplay.

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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