Local Update Algorithms for Random Graphs Romaric Duvignau ees AL - PowerPoint PPT Presentation

Local Update Algorithms for Random Graphs Romaric Duvignau ees AL´ Journ´ EA 2014 March 17, 2014 Joint Work with Philippe Duchon

Overview Introduction Update Algorithms Deletion Algorithms Insertion Algorithms Conclusion Romaric Duvignau Local Update Algorithms for Random Graphs 0 / 11

General Setting Context and Motivations Peer-to-Peer Network (structure maintained locally) Insertion and Deletion : dependence on the update sequence Malicious update sequence may perturb the network Difficulty in designing/analysing update algorithms Our suggestion : maintain exactly a given distribution for the network Romaric Duvignau Local Update Algorithms for Random Graphs 1 / 11

General Setting Context and Motivations Peer-to-Peer Network (structure maintained locally) Insertion and Deletion : dependence on the update sequence Malicious update sequence may perturb the network Difficulty in designing/analysing update algorithms Our suggestion : maintain exactly a given distribution for the network Distribution-preserving update algorithms The network is modelled as a random graph G For each possible vertex set V , G should follow a given target distribution µ V , which is preserved through updates : Insertion : If G ∼ µ V and u �∈ V then I ( G , u ) ∼ µ V ∪{ u } Deletion : If G ∼ µ V and u ∈ V then D ( G , u ) ∼ µ V \{ u } No probabilistic model for update sequences Romaric Duvignau Local Update Algorithms for Random Graphs 1 / 11

Our graph model k -out graphs Simple directed graphs with vertices of outdegree exactly k Good properties (low distances, etc) under the uniform distribution ν V : All N + G ( v ) are independent and each N + G ( v ) is a uniform k -subset of V − v Romaric Duvignau Local Update Algorithms for Random Graphs 2 / 11

Our graph model k -out graphs Simple directed graphs with vertices of outdegree exactly k Good properties (low distances, etc) under the uniform distribution ν V : All N + G ( v ) are independent and each N + G ( v ) is a uniform k -subset of V − v Some properties of uniform n -vertices k -out graphs k Indegrees follow the Binomial( n − 1 , n − 1 ) distribution Connected with asymptotic probability 1 Romaric Duvignau Local Update Algorithms for Random Graphs 2 / 11

Our graph model k -out graphs Simple directed graphs with vertices of outdegree exactly k Good properties (low distances, etc) under the uniform distribution ν V : All N + G ( v ) are independent and each N + G ( v ) is a uniform k -subset of V − v Some properties of uniform n -vertices k -out graphs k Indegrees follow the Binomial( n − 1 , n − 1 ) distribution Connected with asymptotic probability 1 Our goal: Local update algorithms Given a uniform k -out graph G over V : Deletion of u ∈ V : build a uniform k -out graph over V \ { u } Insertion of u �∈ V : build a uniform k -out graph over V ∪ { u } Romaric Duvignau Local Update Algorithms for Random Graphs 2 / 11

Our decentralized and cost model Decentralized model Only use local knowledge and the size of the graph Access to a global primitive RandomVertex() that returns a uniform node of the vertex set Romaric Duvignau Local Update Algorithms for Random Graphs 3 / 11

Our decentralized and cost model Decentralized model Only use local knowledge and the size of the graph Access to a global primitive RandomVertex() that returns a uniform node of the vertex set RandomVertex() : RV for short Costly primitive Cost of an update algorithm = expected number of calls to RV Romaric Duvignau Local Update Algorithms for Random Graphs 3 / 11

Our decentralized and cost model Decentralized model Only use local knowledge and the size of the graph Access to a global primitive RandomVertex() that returns a uniform node of the vertex set RandomVertex() : RV for short Costly primitive Cost of an update algorithm = expected number of calls to RV Our algorithms Minimize the symmetric difference between the input G and the output G ′ Constant expected time Romaric Duvignau Local Update Algorithms for Random Graphs 3 / 11

Our results (best algorithms) Optimal local update algorithms: Deletion algorithm o (1) calls to RV Insertion algorithm k calls to RV Romaric Duvignau Local Update Algorithms for Random Graphs 4 / 11

Overview Introduction Update Algorithms Deletion Algorithms Insertion Algorithms Conclusion Romaric Duvignau Local Update Algorithms for Random Graphs 4 / 11

Deletion Algorithm: Simple Algorithm Deletion of vertex 2 2 1 0 5 4 3 Romaric Duvignau Local Update Algorithms for Random Graphs 5 / 11

Deletion Algorithm: Simple Algorithm Deletion of vertex 2 2 1 0 5 4 3 Romaric Duvignau Local Update Algorithms for Random Graphs 5 / 11

Deletion Algorithm: Simple Algorithm Deletion of vertex 2 2 1 0 5 4 3 Romaric Duvignau Local Update Algorithms for Random Graphs 5 / 11

Deletion Algorithm: Simple Algorithm Deletion of vertex 2 2 1 0 5 4 3 Romaric Duvignau Local Update Algorithms for Random Graphs 5 / 11

Deletion Algorithm: Simple Algorithm Deletion of vertex 2 RV 2 1 RV 0 5 RV 4 3 Romaric Duvignau Local Update Algorithms for Random Graphs 5 / 11

Deletion Algorithm: Simple Algorithm Deletion of vertex 2 RV 2 1 RV 0 5 RV 4 3 The simple solution Randomly redirect loose edges, RVA RVA u avoiding incompatible choices Asymptotic cost k Romaric Duvignau Local Update Algorithms for Random Graphs 5 / 11

Deletion Algorithm: A better algorithm ? Deletion of vertex 2 2 1 0 5 4 3 Romaric Duvignau Local Update Algorithms for Random Graphs 6 / 11

Deletion Algorithm: A better algorithm ? Deletion of vertex 2 2 1 0 5 4 3 Romaric Duvignau Local Update Algorithms for Random Graphs 6 / 11

Deletion Algorithm: A better algorithm ? Deletion of vertex 2 Try 0 Try 1 2 1 0 5 4 3 Romaric Duvignau Local Update Algorithms for Random Graphs 6 / 11

Deletion Algorithm: A better algorithm ? Deletion of vertex 2 2 1 0 5 RV 4 3 Romaric Duvignau Local Update Algorithms for Random Graphs 6 / 11

Deletion Algorithm: A better algorithm ? Deletion of vertex 2 2 1 0 5 RV 4 3 Suggestions must be independent, and can be made so Romaric Duvignau Local Update Algorithms for Random Graphs 6 / 11

Deletion Algorithm: A better algorithm ? Deletion of vertex 2 2 1 0 5 RV 4 3 Suggestions must be independent, and can be made so Second algorithm Use N + ( u ) to save calls to RV , while preserving independence between suggestions � � e − k · k k � k Asymptotic Cost: k · ≃ k ! 2 π Romaric Duvignau Local Update Algorithms for Random Graphs 6 / 11

Deletion Algorithm: what about re-using N − ( u ) ? Best Algorithm: the typical case P 4 P 3 S 3 u S 2 P 2 S 1 P 1 Romaric Duvignau Local Update Algorithms for Random Graphs 7 / 11

Deletion Algorithm: how to redirect loose edges ? L = | N − G ( u ) | and 1 ≤ i ≤ L − 1 P 2 P i − 1 u P i P i − 1 P i − 2 . . . P 2 P 1 Romaric Duvignau Local Update Algorithms for Random Graphs 8 / 11

Deletion Algorithm: how to redirect loose edges ? L = | N − G ( u ) | and 1 ≤ i ≤ L − 1 P 2 P i − 1 u P i P i − 1 P i − 2 . . . P 2 P 1 Romaric Duvignau Local Update Algorithms for Random Graphs 8 / 11

Deletion Algorithm: how to redirect loose edges ? L = | N − G ( u ) | and 1 ≤ i ≤ L − 1 P 2 P i − 1 u P i P i − 1 P i − 2 . . . P 2 P 1 Romaric Duvignau Local Update Algorithms for Random Graphs 8 / 11

Deletion Algorithm: how to redirect loose edges ? L = | N − G ( u ) | and 1 ≤ i ≤ L − 1 P 2 P i − 1 u P i P i − 1 1 n − 1 − k P i − 2 . . . P 2 P 1 Romaric Duvignau Local Update Algorithms for Random Graphs 8 / 11

Deletion Algorithm: how to redirect loose edges ? L = | N − G ( u ) | and 1 ≤ i ≤ L − 1 P 2 P i − 1 u P i P i +1 P i − 1 1 n − 1 − k P i − 2 u P i . . P i +1 �∈ N + . G ( P i ) P 2 P 1 Romaric Duvignau Local Update Algorithms for Random Graphs 8 / 11

Deletion Algorithm: how to redirect loose edges ? L = | N − G ( u ) | and 1 ≤ i ≤ L − 1 P 2 P i − 1 u P i P i +1 P i − 1 1 n − 1 − k P i − 2 u P i . . P i +1 �∈ N + . G ( P i ) P 2 P 1 Romaric Duvignau Local Update Algorithms for Random Graphs 8 / 11

Deletion Algorithm: how to redirect loose edges ? L = | N − G ( u ) | and 1 ≤ i ≤ L − 1 P 2 P i − 1 u P i P i +1 P i +1 P i − 1 1 n − 1 − k P i − 2 u u P i P i . . P i +1 �∈ N + P i +1 �∈ N + . G ( P i ) G ( P i ) P 2 P 1 Romaric Duvignau Local Update Algorithms for Random Graphs 8 / 11

Deletion Algorithm: how to redirect loose edges ? L = | N − G ( u ) | and 1 ≤ i ≤ L − 1 P 2 P i − 1 u P i P i +1 P i +1 P i − 1 RVA 1 n − 1 − k P i − 2 u u P i P i . . P i +1 �∈ N + P i +1 ∈ N + . G ( P i ) G ( P i ) P 2 P 1 Romaric Duvignau Local Update Algorithms for Random Graphs 8 / 11

Deletion Algorithm: how to redirect loose edges ? L = | N − G ( u ) | and 1 ≤ i ≤ L − 1 P 2 P i − 1 P i +1 P i +1 u P i RVA P i − 1 1 u u P i P i n − 1 − k P i − 2 P i +1 �∈ N + P i +1 ∈ N + G ( P i ) G ( P i ) . . . S 3 P 2 P L u S 2 S 1 P 1 Romaric Duvignau Local Update Algorithms for Random Graphs 8 / 11