Small World Model: Cascades and Myopic Routing Jie Gao, Grant - PowerPoint PPT Presentation

Nonhomogeneous Kleinberg’s Small World Model: Cascades and Myopic Routing Jie Gao, Grant Schoenebeck, Fang-Yi Yu

What is a social network? • Social network models interactions between individuals – Individuals behave freely. – Society shows special properties.

Outline • Background – Milgram’s Experiment – Kleinberg's Small World Model • Nonhomogeneous Kleinberg’s Small World Model • Myopic Routing – Theorem – Proof Outline • 𝑙 -Complex Contagions Model

An Experiment by Milgram[1967] Target Starter Information of the Target: Name, Address, Job

An Experiment by Milgram[1967] Target Starter Information of the Target: Name, Address, Job

An Experiment by Milgram[1967] Target Starter Information of the Target: Name, Address, Job

An Experiment by Milgram[1967] Target Starter Information of the Target: Name, Address, Job

An Experiment by Milgram[1967] Target Starter Information of the Target: Name, Address, Job

Small World Model • Six degrees of separation--- very short paths between arbitrary MA pairs of nodes NE

Watts/Strogatz model, Newman – Watts model • 𝑜 people on a ring/ torus

Strong Ties • 𝑜 people on a ring/ torus • Strong ties within distance 𝑟

Weak Ties • 𝑜 people on a ring/ torus • Strong ties within distance 𝑟 • Weak ties: 𝑞 𝑣𝑤 = 𝑞

Algorithmically Small World Target Starter Information of the Target: Name, Address, Job

Small World Model 2.0 • Six degrees of separation--- very short paths between arbitrary MA pairs of nodes • Decentralized routing--- NE Individuals with local information are very adept at finding these paths

Kleinberg’s Small World Model[2000] • 𝑜 people on a 𝑙 -dimensional grid

Strong Ties • 𝑜 people on a 𝑙 -dimensional grid • Strong ties within distance 𝑟

Weak Ties • 𝑜 people on a 𝑙 -dimensional grid • Strong ties within distance 𝑟 1 • Weak ties: 𝑞 𝑣𝑤 ~ 𝑒 𝑣,𝑤 𝛿 𝑣

Weak Ties • 𝑜 people on a 𝑙 -dimensional grid • Strong ties within distance 𝑟 1 • Weak ties: 𝑞 𝑣𝑤 ~ 𝑒 𝑣,𝑤 𝛿 1 𝑣 0.8 0.6 𝑞 𝑣𝑤 0.4 0.2 0 0 5 10 15 20 𝑒 ( 𝑣 , 𝑤 )

Weak Ties with Different 𝛿 Small 𝛿 Large 𝛿

Decentralized Routing on Kleinberg’s Model T When 𝛿 = 2 S

Weak Ties with Different 𝛿 When 𝛿 < 2 When 𝛿 > 2 T T S S

Threshold Property If 𝛿 = 2 and 𝑞, 𝑟 ≥ 1 , there is a If 𝛿 ≠ 2 , there is a constant 𝜊 > 0 , decentralized algorithm A, so that so that the delivery time of any the delivery time of A is 𝑃(log 2 𝑜) . decentralized algorithm is Ω(𝑜 𝜊 ) . T T T S S S

Threshold Property Histogram of γ 1 0.75 PROBABILITY 0.5 0.25 0 0 1 2 3 Γ

Diversity Histogram of γ 1 0.75 PROBABILITY 0.5 0.25 0 0 1 2 3 Γ

Small World Model 2.0.1 • Six degrees of separation--- very short paths between arbitrary MA pairs of nodes • Decentralized routing--- NE Individuals with local information are very adept at finding these paths

Outline • Background – Milgram’s Experiment – Kleinberg's Small World Model • Nonhomogeneous Kleinberg’s Small World Model • Myopic Routing – Theorem – Proof Outline • 𝑙 -Complex Contagions Model

Recall: Kleinberg’s Small World Model • 𝑜 people on a 𝑙 -dimensional grid • Strong ties within distance 𝑟 • Weak ties: 𝑞 𝑣𝑤 ~𝑒 𝑣, 𝑤 −𝛿

Nonhomogeneous Kleinberg’s 𝐼𝑓𝑢𝐿 𝑞,𝑟,𝐸 (𝑜) • 𝑜 people on a 𝑙 -dimensional grid • Strong ties within distance 𝑟 • Weak ties: 𝑣 has 𝛿 𝑣 from 𝐸 , and −𝛿 𝑣 . 𝑞 ties sample from 𝑞 𝑣𝑤 ~𝑒 𝑣𝑤

A More Natural Histogram Histogram of γ Histogram of γ 1 1.2 1 0.75 0.8 PROBABILITY PROBABILITY 0.5 0.6 0.4 0.25 0.2 0 0 0 1 2 3 4 0.01 1.01 2.01 3.01 4.01 Γ Γ

Outline • Background – Milgram’s Experiment – Kleinberg's Small World Model • Nonhomogeneous Kleinberg’s Small World Model • Myopic Routing – Theorems – Proof Outline • 𝑙 -Complex Contagions Model

Theorems Upper bounds Lower bounds 1.2 𝛽 𝛿~𝐸 [ 2 − 𝛿 < 𝜗] = Ω(𝜗 𝛽 ) Pr 1.2 𝛽 2 − 𝜗 2 + 𝜗 𝛽 1 𝛽 1 𝛽 0.8 0.8 Probability Probability 0.6 0.6 0.4 0.4 0.2 0.2 0 0 0 1 2 3 4 5 0 1 2 3 4 5 γ γ

Outline • Background – Milgram’s Experiment – Kleinberg's Small World Model • Nonhomogeneous Kleinberg’s Small World Model • Myopic Routing – Theorem – Proof Outline (upper bound) • 𝑙 -Complex Contagions Model

When 𝛿 = 2 𝐸 log 𝑜 T 𝐸 𝑘 𝑣 S

Outline • Background – Milgram’s Experiment – Kleinberg's Small World Model • Nonhomogeneous Kleinberg’s Small World Model • Myopic Routing – Theorem – Proof Outline (lower bound) • 𝑙 -Complex Contagions Model

When 𝛿 < 2 , weak ties are too random 𝐸 log 𝑜 𝑜 𝜗/3 T 𝐸 𝛿 = 2 − 𝜗 𝑘 𝑣 S

When 𝛿 > 2 , weak ties are too short 𝐸 log 𝑜 T 𝐸 𝛿 = 2 + 𝜗 𝑘 𝑣 1 𝑜 1+𝜗 S

Mixture of Both 𝐸 log 𝑜 𝑜 𝜗/3 T 𝐸 𝛿 = 2 − 𝜗 𝑘 𝑣 𝛿 = 2 + 𝜗 1 𝑜 1+𝜗 S

Mixture of Both 𝐸 log 𝑜 3+3𝜗 𝑜 6+2𝜗 𝑜 𝜗/3 T 𝐸 𝛿 = 2 − 𝜗 𝑘 𝑣 1 𝛿 = 2 + 𝜗 𝑜 2 1 𝑜 1+𝜗 S

Mixture of Both 3+3𝜗 𝐸 log 𝑜 𝑜 6+2𝜗 T 𝛿 = 2 − 𝜗 𝑣 1 𝛿 = 2 + 𝜗 𝑜 2 S

Outline • Background – Milgram’s Experiment – Kleinberg's Small World Model • Nonhomogeneous Kleinberg’s Small World Model • Myopic Routing – Theorem – Proof Outline • 𝑙 -Complex Contagions Model

Thanks for your listening

Upper Bound — Non-negligible Mass Near 2 • Fixed a distribution D with constant α ≥ 0 where 𝐺 𝐸 2 + 𝜗 − 𝐺 𝐸 2 − 𝜗 = Ω(𝜗 𝛽 ) for any integer k > 0 and η > 0, there exists ξ = 3+α+k, such that a k -complex contagion 𝐷𝐷(𝐼𝑓𝑢𝐿 𝑞,𝑟,𝐸(𝑜) , 𝑙, 𝐽) starting from a k-seed cluster I and where 𝑞 > 𝑙, 𝑟 2 /2 ≥ k takes at most 𝑃( log 𝜊 𝑜) time to spread to the whole network with probability at least 1 − 𝑜 − 𝜃 over the randomness of 𝐼𝑓𝑢𝐿 𝑞,𝑟,𝐸(𝑜) .

Upper Bound — Fixed k • Given a distribution D and an integer k > 0, such that Pr 𝛿←𝐸 [𝛿 ∈ [2, 𝛾𝑙)] > 0 where 𝛾 𝑙 = 2(𝑙 + 1) , for all 𝜃 > 0 there exists ξ > 0 depending on D and k such that, the speed of a k- complex contagion 𝐷𝐷(𝐼𝑓𝑢𝐿 𝑞,𝑟,𝐸(𝑜) , 𝑙, 𝐽) starting from a k- seed cluster I and 𝑞 > 𝑙, 𝑟2/2 ≥ 𝑙 is at most 𝑃(log 𝜊 𝑜) with probability at least 1 − 𝑜 −𝜃 .

Lower Bound • Given distribution D, constant integers 𝑙, 𝑞, 𝑟 > 0 , and ε > 0 such that 𝐺 𝐸 2 + 𝜗 − 𝐺 𝐸 2 − 𝜗 = 0 , then there exist constants 𝜊, 𝜃 > 0 depending on D and k, such that the time it takes a k-contagion starting at seed-cluster I, 𝐷𝐷(𝐼𝑓𝑢𝐿 𝑞,𝑟,𝐸(𝑜) , 𝑙, 𝐽) , to infect all nodes is at least 𝑜 𝜊 with probability at least 1 − 𝑃(𝑜 −𝜃 ) over the randomness of 𝐼𝑓𝑢𝐿 𝑞,𝑟,𝐸(𝑜) .

Idea of Myopic Routing Upper Bound 𝐸 log 𝑜 T 𝐸 𝑘 𝑣 S

Idea of Complex Contagion Lower Bound

• Number of nodes within region 𝐸 𝑘 2 2𝑘 • Probability of node 𝑣 connecting to a node 𝑤 ∈ 𝐸 𝑘 1 2+𝜗 𝑣 𝐿 2+𝜗 𝑒 𝑣𝑤 • Probability for node 𝑣 entering region 𝐸 𝑘 𝜗 |𝜗| Ω 2 𝑘𝜗 if 𝜗 > 0 and Ω if 𝜗 < 0 2 (log 𝑜−𝑘)𝜗

• Probability entering region 𝐸 𝑘 𝜗 0 𝜗 2 𝑘𝜗 𝜗 𝛽−1 𝑒𝜗 Ω න 0 or 𝜗 0 𝜗 2 (log 𝑜−𝑘)𝜗 𝜗 𝛽−1 𝑒𝜗 Ω න 0

Proof Sketch for lower bound • 𝛿 > 2 the weak ties will be too short (concentrated edges) • 𝛿 < 2 the weak ties will be too random (diffuse edges)

A Very Brief Summary — History • Kleinberg’s small world model models social networks with both strong and weak ties, and the distribution of weak-ties, parameterized by γ . – He showed how value of γ influences the efficacy of myopic routing on the network. – Recent work on social influence by k-complex contagion models discovered that the value of γ also impacts the spreading rate

A Very Brief Summary — Our Work • A natural generalization of Kleinberg’s small world model to allow node heterogeneity is proposed, and – We show this model enables myopic routing and k-complex contagions on a large range of the parameter space. – Moreover, we show that our generalization is supported by real- world data.

1.2 1 0.8 Probability 0.6 0.4 0.2 0 0 1 2 3 4 5 γ