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]

Starter Target

Information of the Target: Name, Address, Job

An Experiment by Milgram[1967]

Starter Target

Information of the Target: Name, Address, Job

An Experiment by Milgram[1967]

Starter Target

Information of the Target: Name, Address, Job

An Experiment by Milgram[1967]

Starter Target

Information of the Target: Name, Address, Job

An Experiment by Milgram[1967]

Starter Target

Information of the Target: Name, Address, Job

Small World Model

• Six degrees of separation--- very

short paths between arbitrary pairs of nodes

NE MA

Watts/Strogatz model, Newman–Watts model

• 𝑜 people on a ring/ torus

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

Strong Ties

Weak Ties

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

Algorithmically Small World

Starter Target

Information of the Target: Name, Address, Job

Small World Model 2.0

• Six degrees of separation--- very

short paths between arbitrary pairs of nodes

• Decentralized routing---

Individuals with local information are very adept at finding these paths

NE MA

Kleinberg’s Small World Model[2000]

• 𝑜 people on a 𝑙-dimensional grid

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

Strong Ties

Weak Ties

• 𝑜 people on a 𝑙-dimensional grid
• Strong ties within distance 𝑟
• Weak ties: 𝑞𝑣𝑤~

1 𝑒 𝑣,𝑤 𝛿

𝑣

Weak Ties

• 𝑜 people on a 𝑙-dimensional grid
• Strong ties within distance 𝑟
• Weak ties: 𝑞𝑣𝑤~

1 𝑒 𝑣,𝑤 𝛿

𝑣

0.2 0.4 0.6 0.8 1 5 10 15 20

𝑞𝑣𝑤 𝑒(𝑣,𝑤)

Weak Ties with Different 𝛿

Small 𝛿 Large 𝛿

Decentralized Routing on Kleinberg’s Model

S T When 𝛿 = 2

Weak Ties with Different 𝛿

When 𝛿 < 2 When 𝛿 > 2

S T S T

Threshold Property

If 𝛿 = 2 and 𝑞, 𝑟 ≥ 1, there is a decentralized algorithm A, so that the delivery time of A is 𝑃(log2 𝑜). If 𝛿 ≠ 2, there is a constant 𝜊 > 0, so that the delivery time of any decentralized algorithm is Ω(𝑜𝜊).

S T

S T S T

0.25 0.5 0.75 1 1 2 3 PROBABILITY Γ

Histogram of γ

Threshold Property

0.25 0.5 0.75 1 1 2 3 PROBABILITY Γ

Histogram of γ

Diversity

Small World Model 2.0.1

• Six degrees of separation--- very

short paths between arbitrary pairs of nodes

• Decentralized routing---

Individuals with local information are very adept at finding these paths

NE MA

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: 𝑞𝑣𝑤~𝑒 𝑣, 𝑤 −𝛿

Nonhomogeneous Kleinberg’s 𝐼𝑓𝑢𝐿𝑞,𝑟,𝐸(𝑜)

• 𝑜 people on a 𝑙-dimensional grid
• Strong ties within distance 𝑟
• Weak ties: 𝑣 has 𝛿𝑣 from 𝐸, and

𝑞 ties sample from 𝑞𝑣𝑤~𝑒𝑣𝑤

−𝛿𝑣.

A More Natural Histogram

0.2 0.4 0.6 0.8 1 1.2 0.01 1.01 2.01 3.01 4.01 PROBABILITY Γ

Histogram of γ

0.25 0.5 0.75 1 1 2 3 4 PROBABILITY Γ

Histogram of 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

0.2 0.4 0.6 0.8 1 1.2 1 2 3 4 5

Probability γ

0.2 0.4 0.6 0.8 1 1.2 1 2 3 4 5

Probability

γ

Theorems

Upper bounds Lower bounds

2 − 𝜗 2 + 𝜗 Pr

𝛿~𝐸[ 2 − 𝛿 < 𝜗] = Ω(𝜗𝛽)

𝛽 𝛽 𝛽 𝛽 𝛽

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

T S

𝐸log 𝑜 𝐸

𝑘

𝑣

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

T S

𝐸log 𝑜 𝐸

𝑘

𝑣 𝑜𝜗/3

𝛿 = 2 − 𝜗

When 𝛿 > 2, weak ties are too short

T S

𝐸log 𝑜 𝐸

𝑘

𝑣 𝑜

1 1+𝜗

𝛿 = 2 + 𝜗

Mixture of Both

T S

𝐸log 𝑜 𝐸

𝑘

𝑣 𝑜𝜗/3 𝑜

1 1+𝜗

𝛿 = 2 + 𝜗 𝛿 = 2 − 𝜗

Mixture of Both

T S

𝐸log 𝑜 𝐸

𝑘

𝑣 𝑜

1 1+𝜗

𝑜

3+3𝜗 6+2𝜗

𝑜

1 2

𝑜𝜗/3

𝛿 = 2 + 𝜗 𝛿 = 2 − 𝜗

Mixture of Both

S

𝐸log 𝑜

𝑣

T

𝑜

3+3𝜗 6+2𝜗

𝑜

1 2

𝛿 = 2 + 𝜗 𝛿 = 2 − 𝜗

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 𝐺𝐸 2 + 𝜗 −

𝐺𝐸 2 − 𝜗 = Ω(𝜗𝛽) for any integer k > 0 and η > 0, there exists ξ = 3+α+k, such that a k-complex contagion 𝐷𝐷(𝐼𝑓𝑢𝐿𝑞,𝑟,𝐸(𝑜), 𝑙, 𝐽) starting from a k-seed cluster I and where 𝑞 > 𝑙, 𝑟2/2 ≥ k takes at most 𝑃( log𝜊𝑜) time to spread to the whole network with probability at least 1 − 𝑜 − 𝜃 over the randomness of 𝐼𝑓𝑢𝐿𝑞,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 𝛾𝑙 = 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(𝐼𝑓𝑢𝐿𝑞,𝑟,𝐸(𝑜), 𝑙, 𝐽) starting from a k- seed cluster I and 𝑞 > 𝑙, 𝑟2/2 ≥ 𝑙 is at most 𝑃(log𝜊𝑜) with probability at least 1 − 𝑜−𝜃 .

Lower Bound

• Given distribution D, constant integers 𝑙, 𝑞, 𝑟 > 0, and ε > 0

such that 𝐺𝐸 2 + 𝜗 − 𝐺𝐸 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(𝐼𝑓𝑢𝐿𝑞,𝑟,𝐸(𝑜), 𝑙, 𝐽), to infect all nodes is at least 𝑜𝜊 with probability at least 1 − 𝑃(𝑜−𝜃) over the randomness of 𝐼𝑓𝑢𝐿𝑞,𝑟,𝐸(𝑜).

Idea of Myopic Routing Upper Bound

T S

𝐸log 𝑜 𝐸

𝑘

𝑣

Idea of Complex Contagion Lower Bound

• Number of nodes within region 𝐸

𝑘

22𝑘

• Probability of node 𝑣 connecting to a node 𝑤 ∈ 𝐸

𝑘

1 𝐿2+𝜗𝑒𝑣𝑤

2+𝜗𝑣

• Probability for node 𝑣 entering region 𝐸

𝑘

Ω 𝜗 2𝑘𝜗 if 𝜗 > 0 and Ω |𝜗| 2(log 𝑜−𝑘)𝜗 if 𝜗 < 0

• Probability entering region 𝐸

𝑘

Ω න

𝜗0 𝜗

2𝑘𝜗 𝜗𝛽−1𝑒𝜗

• r

Ω න

𝜗0

𝜗 2(log 𝑜−𝑘)𝜗 𝜗𝛽−1𝑒𝜗

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

• n 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.

0.2 0.4 0.6 0.8 1 1.2 1 2 3 4 5

Probability γ