Engineering Agreement: The Naming Game wit ith Asymmetric and - PowerPoint PPT Presentation

Engineering Agreement: The Naming Game wit ith Asymmetric and Heterogeneous Agents Jie Gao, Bo Li, Grant Schoenebeck, Fan ang-Yi i Yu

Social Convention • Conventions are universally adopted from two or more alternatives. • Language, etiquette, or custom.

Agreement on Convention

Engineering Agreement • What can help or harm convergence? – Homogeneity or heterogeneity – Community structure • How robust are the dynamics to possible manipulations?

Naming Game [Baronchelli 06] 06] • A agent-based process on a network

Naming Game • A agent-based process on a network – Each agent has inventory of names Soft drink coke

Naming Game • A agent-based process on a network – Each agent has inventory of names coke Soda pop coke coke pop Soft drink coke Soda

Naming Game • A agent-based process on a network – Each agent has inventory of names – At each time an edge is selected at random coke pop Soda coke coke pop Soft drink coke Soda

Naming Game • A agent-based process on a network – Each agent has inventory of names – At each time an edge is selected at random, and one is speaker and the other is listener. coke pop Soda coke coke pop Soft drink coke Soda

Naming Game • A agent-based process on a network – Each agent has inventory of names – At each time an edge is selected at random, and one is speaker and the other is listener. coke pop Soda coke coke pop Soft drink coke Soda

Naming Game • A agent-based process on a network – Each agent has inventory of names – At each time an edge is selected at random, and one is speaker and the other is listener. coke • Failure pop Soda coke coke pop Soft drink coke Soda

Naming Game • A agent-based process on a network – Each agent has inventory of names – At each time an edge is selected at random, and one is speaker and the other is listener. coke • Failure: listener adds the new name pop Soda coke Soft drink coke pop Soft drink coke Soda

Naming Game • A agent-based process on a network – Each agent has inventory of names – At each time an edge is selected at random, and one is speaker and the other is listener. coke • Failure: listener adds the new name Soda pop coke Soft drink coke pop Soft drink coke Soda

Naming Game • A agent-based process on a network – Each agent has inventory of names – At each time an edge is selected at random, and one is speaker and the other is listener. coke • Failure: listener adds the new name Soda pop coke Soft drink coke pop Soft drink coke Soda

Naming Game • A agent-based process on a network – Each agent has inventory of names – At each time an edge is selected at random, and one is speaker and the other is listener. coke • Failure: listener adds the new name Soda pop • Success coke Soft drink coke pop Soft drink coke Soda

Naming Game • A agent-based process on a network – Each agent has inventory of names – At each time an edge is selected at random, and one is speaker and the other is listener. coke • Failure: listener adds the new name pop • Success: both remove all other names coke Soft drink coke Soft drink coke Soda

Naming Game • A agent-based process on a network – Each agent has inventory of names – At each time an edge is selected at random, and one is speaker and the other is listener. coke • Failure: listener adds the new name pop • Success: both remove all other names coke Soft drink • Empty coke Soft drink coke Soda

Naming Game • A agent-based process on a network – Each agent has inventory of names – At each time an edge is selected at random, and one is speaker and the other is listener. coke • Failure: listener adds the new name pop • Success: both remove all other names coke Soft drink • Empty: speaker invent a new word coke Soft drink coke Cola Soda

Naming Game • A agent-based process on a network – Each agent has inventory of names – At each time an edge is selected at random, and one is speaker and the other is listener. coke • Failure: listener adds the new name coke • Success: both remove all other names • Empty: speaker invent a new word coke – Convergence coke coke coke

Different initial states Empty in initia itial states Segregated in initi itial l states Soda coke Soda coke Soda coke

Motivating Questions • What can help or harm convergence? – Homogeneity or heterogeneity – Community structure • How robust are the dynamics to possible manipulations?

Different graphs Kleinberg star grid d-ary tree disjoint cliques complete Watts-Strogatz regular

Fast and Slo low Convergence 12000000 10000000 Consensus time 8000000 grid 6000000 complete graph regular graph 4000000 Star 2000000 0 0 5000 10000 15000 20000 25000 30000 35000 40000 # nodes

Fast and Slo low Convergence 12000000 10000000 Local Consensus time 8000000 structure grid 6000000 complete graph regular graph 4000000 Star 2000000 0 0 5000 10000 15000 20000 25000 30000 35000 40000 # nodes

Fast and Slo low Convergence 12000000 Homogeneous 10000000 Local Consensus time 8000000 structure grid 6000000 complete graph regular graph 4000000 Star 2000000 0 0 5000 10000 15000 20000 25000 30000 35000 40000 # nodes

Fast and Slo low Convergence 12000000 Homogeneous 10000000 Local Consensus time 8000000 structure grid 6000000 complete graph Heterogeneous regular graph 4000000 Star 2000000 0 0 5000 10000 15000 20000 25000 30000 35000 40000 # nodes

Heterogeneous R=1 L = 7 3 Normalized consensus time 2.5 2 1.5 10000 1 5000 R/(R+L) = 1/8 0.5 0 0 0.1 0.2 0.3 0.4 0.5 R/(R+L)

Heterogeneous R=2 L = 6 3 Normalized consensus time 2.5 2 1.5 10000 1 5000 R/(R+L) = 2/8 0.5 0 0 0.1 0.2 0.3 0.4 0.5 R/(R+L)

Heterogeneous R=3 L = 5 3 Normalized consensus time 2.5 2 1.5 10000 1 5000 R/(R+L) = 3/8 0.5 0 0 0.1 0.2 0.3 0.4 0.5 R/(R+L)

Motivating Questions • What can help or harm convergence? – Homogeneity or heterogeneity – Community structure • How robust are the dynamics to possible manipulations?

Community Structure Many edges within groups Few edges between groups

Disjoint cliques Many edges within groups Few edges between groups

Tree Structure Many edges within groups Few edges between groups

Tree Structure Many edges within groups Few edges between groups

Adding Homogeneity 1 − 𝑞 𝑞

Community Structure Empty in initia itial states Segregated in initi itial l states 1 − 𝑞 𝑞 1 − 𝑞 𝑞

Simulation on Disjo joint Cliques Empty in initia itial states Segregated in initi itial l states emp-1000: emp-5000: emp-10000: seg-1000: seg-5000: seg-10000: 1 1 Fraction of non- Fraction of non- 0.8 0.8 consensus consensus 0.6 0.6 0.4 0.4 0.2 0.2 0 0 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 p p

Simulation on Disjo joint Cliques Empty in initia itial states Segregated in initi itial l states emp-1000: emp-5000: emp-10000: seg-1000: seg-5000: seg-10000: 1 1 Fraction of non- Fraction of non- 0.8 0.8 consensus consensus 0.6 0.6 0.4 0.4 0.2 0.2 0 0 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 p p

Theoretical Analysis • Segregated start: for 𝑞 < 𝑞 0 ≈ 0.110 , consensus time = exp(Ω(𝑜)) seg-1000: seg-5000: seg-10000: 1 Fraction of non- 0.8 consensus 𝑞 0 0.6 0.4 0.2 0 0 0.1 0.2 0.3 0.4 0.5 p

Theoretical Analysis • Segregated start: for 𝑞 < 𝑞 0 ≈ 0.110 , consensus time = exp(Ω(𝑜)) 𝑞 1 − 𝑞

Theoretical Analysis • Segregated start: for 𝑞 < 𝑞 0 ≈ 0.110 , consensus time = exp(Ω(𝑜)) – Mean field approximation 𝑞 1 − 𝑞

Theoretical Analysis • Segregated start: for 𝑞 < 𝑞 0 ≈ 0.110 , consensus time = exp(Ω(𝑜)) – Mean field approximation – Stability of autonomous system • Local stability • Global stability 𝑞 1 − 𝑞

Motivating Questions • What can help or harm convergence? – Homogeneity or heterogeneity – Community structure • How robust are the dynamics to possible manipulations?

Robustness

Stubborn nodes • How and when can such nodes affect the name to which the dynamics converge?

Stubborn nodes • How and when can such nodes affect the name to which the dynamics converge? – The network topology – The time when the stubborn nodes are activated

Stubborn nodes and network Gr Graph si size = 10 1000 00 Gr Graph si size = 10 1000 000 Grid Complete graph Regular star Complete graph Regular Grid star 1.2 1.2 Fraction of stubborn opinion Fraction of stubborn 1 1 0.8 0.8 opinion 0.6 0.6 0.4 0.4 0.2 0.2 0 0 0 2 4 6 8 10 0 2 4 6 8 10 Number of stubborn nodes Number of stubborn nodes

Adding stubborn nodes aft fter consensus • After consensus: with 𝑞 < 𝑞 0 ≈ 0.108 fraction of stubborn nodes, the consensus time = exp(Ω(𝑜)) . 1 0.8 0.6 0.4 0.2 0 0 0.1 0.2 0.3 1024 10000