Maintenance of random logical networks Romaric Duvignau DCS seminar, Chalmers October 4, 2017
Plan 1 A Quick Example of P2P Networks 2 A New Model of Evolution 3 A Concrete Example: Uniform k -out Random Graphs 4 A More General Question Romaric Duvignau Maintenance of random logical networks 0 / 23
Motivation: modelling of P2P networks 1/2 Romaric Duvignau Maintenance of random logical networks 1 / 23
Motivation: modelling of P2P networks 1/2 Romaric Duvignau Maintenance of random logical networks 1 / 23
Motivation: modelling of P2P networks 1/2 Romaric Duvignau Maintenance of random logical networks 1 / 23
Motivation: modelling of P2P networks 1/2 < 200 ko/s (or kr) Romaric Duvignau Maintenance of random logical networks 1 / 23
Motivation: modelling of P2P networks 1/2 < 200 ko/s (or kr) Romaric Duvignau Maintenance of random logical networks 1 / 23
Motivation: modelling of P2P networks 1/2 < 200 ko/s (or kr) Romaric Duvignau Maintenance of random logical networks 1 / 23
Motivation: modelling of P2P networks 2/2 Romaric Duvignau Maintenance of random logical networks 2 / 23
Motivation: modelling of P2P networks 2/2 Romaric Duvignau Maintenance of random logical networks 2 / 23
Motivation: modelling of P2P networks 2/2 Romaric Duvignau Maintenance of random logical networks 2 / 23
Motivation: modelling of P2P networks 2/2 Romaric Duvignau Maintenance of random logical networks 2 / 23
Motivation: modelling of P2P networks 2/2 Romaric Duvignau Maintenance of random logical networks 2 / 23
Motivation: modelling of P2P networks 2/2 Romaric Duvignau Maintenance of random logical networks 2 / 23
Motivation: modelling of P2P networks 2/2 Romaric Duvignau Maintenance of random logical networks 2 / 23
Plan 1 A Quick Example of P2P Networks 2 A New Model of Evolution 3 A Concrete Example: Uniform k -out Random Graphs 4 A More General Question Romaric Duvignau Maintenance of random logical networks 2 / 23
Introduction: Logical, Decentralized, Dynamic networks C B A U E G Romaric Duvignau Maintenance of random logical networks 3 / 23
Introduction: Logical, Decentralized, Dynamic networks C B A U E G Romaric Duvignau Maintenance of random logical networks 3 / 23
Introduction: Logical, Decentralized, Dynamic networks C B A U E G Why should we look for good models ? analysis of the evolution of some concrete networks analysis of distributed algorithms running over such networks simulations of distributed algorithms operating over such networks (including adaptive algorithms working over dynamic networks) Romaric Duvignau Maintenance of random logical networks 3 / 23
Our model of evolution Constrain the evolution to always stick to the target model Romaric Duvignau Maintenance of random logical networks 4 / 23
Our model of evolution Constrain the evolution to always stick to the target model delete(c) insert(z) c c b b b a a a z i i i g g g e e e µ V − c µ V µ V + z Romaric Duvignau Maintenance of random logical networks 4 / 23
Introduction: Why such an evolutionary paradigm ? Context : evolution of P2P networks In the literature , some good properties of the network are maintained over time, but implies : difficulties in update alogrithms’ conception leading to complex procedures dynamicity modelled by a probabilistic process (Poisson) : non realistic [Pouwelse et al , 2005] difficult analysis without those hypothesis (analysis under simplistic update models : insertion only, fifo) Typical good properties small degree, small diameter, high connectivity, etc Romaric Duvignau Maintenance of random logical networks 5 / 23
Introduction: Why such an evolutionary paradigm ? Context : evolution of P2P networks In the literature , some good properties of the network are maintained over time, but implies : difficulties in update alogrithms’ conception dynamicity modelled by a probabilistic process (Poisson) : non realistic difficult analysis without those hypothesis Typical good properties small degree, small diameter, high connectivity, etc Romaric Duvignau Maintenance of random logical networks 5 / 23
Introduction: Why such an evolutionary paradigm ? Context : evolution of P2P networks In the literature , some Our solution : randomness good properties of the preservation network are maintained over time, but implies : difficulties in update alogrithms’ conception dynamicity modelled by a probabilistic process (Poisson) : non realistic difficult analysis without those hypothesis Typical good properties small degree, small diameter, high connectivity, etc Romaric Duvignau Maintenance of random logical networks 5 / 23
Introduction: Why such an evolutionary paradigm ? Context : evolution of P2P networks In the literature , some Our solution : randomness good properties of the preservation , answers those network are maintained problems: over time, but implies : properties are always difficulties in update maintained alogrithms’ conception analysis is simplified dynamicity modelled by a it is not influenced by an probabilistic process adversarial sequence of (Poisson) : non realistic updates difficult analysis without no drift phenomena those hypothesis Typical good properties small degree, small diameter, high connectivity, etc Romaric Duvignau Maintenance of random logical networks 5 / 23
Introduction: An Optimistic First Contact Model Local update algorithms LOCAL model (synchronous, error-free, message passing) Two submodels: exact size of the network known or unknown to the participating nodes Romaric Duvignau Maintenance of random logical networks 6 / 23
Introduction: An Optimistic First Contact Model Local update algorithms LOCAL model (synchronous, error-free, message passing) Two submodels: exact size of the network known or unknown to the participating nodes “First Contact”: Every node has access to a global primitive which samples uniformly a node over the entire network RandomVertex () Romaric Duvignau Maintenance of random logical networks 6 / 23
Introduction: An Optimistic First Contact Model Local update algorithms LOCAL model (synchronous, error-free, message passing) Two submodels: exact size of the network known or unknown to the participating nodes “First Contact”: Every node has access to a global primitive which samples uniformly a node over the entire network RandomVertex () An optimistic first contact but... Optimistic: uniformity, reusability, availability Romaric Duvignau Maintenance of random logical networks 6 / 23
Introduction: An Optimistic First Contact Model Local update algorithms LOCAL model (synchronous, error-free, message passing) Two submodels: exact size of the network known or unknown to the participating nodes “First Contact”: Every node has access to a global primitive which samples uniformly a node over the entire network RandomVertex () An optimistic first contact but... Optimistic: uniformity, reusability, availability May be approximated in practice: uniform hash ( Chord ), centralized cache, random walks, dissemination of tokens, etc Romaric Duvignau Maintenance of random logical networks 6 / 23
Introduction: An Optimistic First Contact Model Local update algorithms LOCAL model (synchronous, error-free, message passing) Two submodels: exact size of the network known or unknown to the participating nodes “First Contact”: Every node has access to a global primitive which samples uniformly a node over the entire network RandomVertex () An optimistic first contact but... Optimistic: uniformity, reusability, availability May be approximated in practice: uniform hash ( Chord ), centralized cache, random walks, dissemination of tokens, etc What about the cost model ? Romaric Duvignau Maintenance of random logical networks 6 / 23
Plan 1 A Quick Example of P2P Networks 2 A New Model of Evolution 3 A Concrete Example: Uniform k -out Random Graphs 4 A More General Question Romaric Duvignau Maintenance of random logical networks 6 / 23
Uniform k -out graphs Example of a 2-out graph C B A I E G Romaric Duvignau Maintenance of random logical networks 7 / 23
Uniform k -out graphs Example of a 2-out graph C B A I E G Directed graphs with no loops and where each vertex has exactly k out-neighbours Romaric Duvignau Maintenance of random logical networks 7 / 23
Uniform k -out graphs Example of a 2-out graph C B A I E G Directed graphs with no loops and where each vertex has exactly k out-neighbours The uniform distribution over vertex set V is equivalent to: For each v ∈ V , the outgoing neighbourhood of v is a uniform k -subset of V − v All outgoing neighbourhood are independent Romaric Duvignau Maintenance of random logical networks 7 / 23
Why uniform k -out graphs ? Figure : Some statistics of 2-out random graphs. 0.3 0.35 0.3 0.25 0.25 0.2 0.2 degrés 0.15 2+Poi(2) 0.15 0.1 0.1 0.05 0.05 0 0 0 2 4 6 8 10 12 14 16 0 2 4 6 8 10 12 (a) Degrees distribution for G 2 (b) Distances distribution for G 2 10 7 10 4 Romaric Duvignau Maintenance of random logical networks 8 / 23
Recommend
More recommend
