Graphs with a Power-Law Degree Distribution Grant Schoenebeck, - - PowerPoint PPT Presentation

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Graphs with a Power-Law Degree Distribution Grant Schoenebeck, - - PowerPoint PPT Presentation

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Complex Contagions on Configuration Model Graphs with a Power-Law Degree Distribution Grant Schoenebeck, Fang-Yi Yu Contagions, diffusion, cascade Ideas, beliefs, behaviors, and technology adoption spread through network Why do we

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Complex Contagions on Configuration Model Graphs with a Power-Law Degree Distribution

Grant Schoenebeck, Fang-Yi Yu

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SLIDE 2

Contagions, diffusion, cascade…

  • Ideas, beliefs, behaviors, and

technology adoption spread through network

  • Why do we need to study this

phenomena?

– Better Understanding – Promoting good behaviors/beliefs – Stopping bad behavior

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SLIDE 3

Outline

  • Models

– Complex contagions model – Power-Law and configuration model graph

  • Main result
  • Related work
  • A happy proof sketch
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Outline

  • Models

– Complex contagions model – Power-Law and configuration model graph

  • Main result
  • Related work
  • A happy proof sketch
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Model of Contagions

  • K-Complex Contagions [GEG13; CLR 79]

– Given an initial infected set 𝐽 = {𝑣, 𝑤}.

a y z u v w x

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Model of Contagions

  • K-Complex Contagions [GEG13; CLR 79]

– Given an initial infected set 𝐽 = {𝑣, 𝑤}. – Node becomes infected if it has at least k infected neighbor

a y z u v w x

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Model of Contagions

  • K-Complex Contagions [GEG13; CLR 79]

– Given an initial infected set 𝐽 = {𝑣, 𝑤}. – Node becomes infected if it has at least k infected neighbor

a y z u v w x

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SLIDE 8

Model of Contagions

  • K-Complex Contagions [GEG13; CLR 79]

– Given an initial infected set 𝐽 = {𝑣, 𝑤}. – Node becomes infected if it has at least k infected neighbor

a y z u v w x

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Model of Contagions

  • K-Complex Contagions [GEG13; CLR 79]

– Given an initial infected set 𝐽 = {𝑣, 𝑤}. – Node becomes infected if it has at least k infected neighbor

a y z u v w x

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Why k complex contagions?

  • One of most classical and simple contagions model

– Threshold model [Gra 78] – Bootstrap percolation [CLR 79]

  • Non-submodular
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Motivating Question

  • Do k complex contagions spread on social networks?
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Question

  • Do k complex contagions spread on Erdos-Renyi model 𝐻𝑜,𝑞

where 𝑞 = 𝑃

1 𝑜 ?

– 𝑜 vertices – Each edge (𝑣, 𝑤) occurs with probability 𝑞

  • Need Ω 𝑜 (random) seeds to infect constant fraction of the

graph[JLTV89]?

  • Can we categorize all networks which spread slowly/quickly?
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What is a social network?

  • Qualitatively: special structure

– Power law degree distribution – low-diameter/small-world…

  • Quantitatively: generative model?

– Configuration model graphs – Preferential attachment model – Kleinberg’s small world model

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Motivating Question

  • Do k complex contagions spread on social networks?
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Motivating Question

  • Do k complex contagions spread on social networks?

– What properties are shared by social networks? – Do these properties alone permit complex contagion spreads?

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Outline

  • Models

– Complex contagions model – Power-Law and configuration model graph

  • Main result
  • Related work
  • A happy proof sketch
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0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007 0.0008 0.0009 0.001 50 100 150 200

alpha=2.5 alpha=3.5

Power-law distribution

  • A power-law distribution with 𝛽

if the Pr 𝑌 = 𝑦 ~𝑦−𝛽

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Configuration Model

  • Given a degree sequence

deg(𝑤1), deg(𝑤2), … , deg(𝑤𝑜)

  • The node 𝑤𝑗 has deg(𝑤𝑗) stubs

Nodes with stubs

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Configuration Model

  • Given a degree sequence

deg(𝑤1), deg(𝑤2), … , deg(𝑤𝑜)

  • The node 𝑤𝑗 has deg(𝑤𝑗) stubs
  • Choose a uniformly random

matching on the stubs

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Outline

  • Models

– Complex contagions model – Power-Law and configuration model graph

  • Main result
  • Related work
  • A happy proof sketch
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Theorems

Main result

  • Configuration Model

– power-law degree distribution 2 < 𝛽 < 3

  • Initial infected node

– the highest degree node

  • 𝑙-complex contagions spreads to Ω(1)

fraction of nodes with high probability. Corollary from [Amini 10]

  • Configuration Model

– power-law degree distribution 3 < 𝛽

  • Initial infected nodes:

– o(1) fraction of highest degree node

  • 𝑙-complex contagions spreads to
  • (1) fraction of nodes with high

probability.

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The Bottom Line

Main result

  • Configuration Model

– power-law degree distribution 2 < 𝛽 < 3

  • 𝑙-complex contagions

– Initial infected node: the highest degree node

  • Contagions spreads to Ω(1) fraction of

nodes with high probability. Corollary from [Amini 10]

  • Configuration Model

– power-law degree distribution 3 < 𝛽

  • 𝑙-complex contagions

– Initial infected node: o(1) fraction of highest degree node

  • Contagions spreads to o(1) fraction of

nodes with high probability.

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Outline

  • Models

– Complex contagions model – Power-Law and configuration model graph

  • Main result
  • Related work
  • A happy proof sketch
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What has been done?

Lattice[98] Random regular[07] Tree[79,06] Erdos-Renyi model[12] Configuration model α>3[10] Chung-Lu model[12] Watts-Strogatz[13] Kleinberg[14] Preferential Attachment[14]

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What has been done?

Lattice[98] Random regular[07] Tree[79,06] Erdos-Renyi model[12] Chung-Lu model[12] Watts-Strogatz[13] Kleinberg[14] Preferential Attachment[14] Configuration model α>3 [10] Configuration model 2<α<3[16]

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Start with 𝐻𝑜,𝑞

Lattice[98] Random regular[07] Tree[79,06] Erdos-Renyi model[12] Chung-Lu model[12] Watts-Strogatz[13] Kleinberg[14] Preferential Attachment[14] Configuration model α>3 [10] Configuration model 2<α<3[16]

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k-complex contagions don’t spread

Physics

Lattice[98] Random regular[07] Tree[79,06] Erdos-Renyi model[12] Chung-Lu model[12] Watts-Strogatz[13] Kleinberg[14] Preferential Attachment[14] Configuration model α>3 [10] Configuration model 2<α<3[16]

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k-complex contagions spread

Network Science

Lattice[98] Random regular[07] Tree[79,06] Erdos-Renyi model[12] Chung-Lu model[12] Watts-Strogatz[13] Kleinberg[14] Preferential Attachment[14] Configuration model α>3 [10] Configuration model 2<α<3[16]

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Things get complicated

Physics Network Science

Lattice[98] Random regular[07] Tree[79,06] Erdos-Renyi model[12] Chung-Lu model[12] Watts-Strogatz[13] Kleinberg[14] Preferential Attachment[14] Configuration model α>3 [10] Configuration model 2<α<3[16]

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What have we learned?

Physics Network Science

Lattice[98] Random regular[07] Tree[79,06] Erdos-Renyi model[12] Chung-Lu model[12] Watts-Strogatz[13] Kleinberg[14] Preferential Attachment[14] Configuration model α>3 [10] Configuration model 2<α<3[16]

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Why do we want to solve it?

Lattice Random regular Erdos-Renyi model Configuration model α>3 Chung-Lu model Watts-Strogatz Preferential Attachment Kleinberg Tree Configuration model 2<α<3

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Outline

  • Models

– Complex contagions model – Power-Law and configuration model graph

  • Main result
  • Related work
  • A happy proof sketch
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Idea

  • Restrict contagions from high

degree node to low degree nodes

  • Reveal the edges when needed
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Observations

  • The highest degree nodes forms

clique

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Observations

  • The highest degree nodes forms

clique

  • 𝑙 degree node has many edges

to 𝑚 degree nodes where 𝑚 > 𝑙

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SLIDE 36

Observations

  • The highest degree nodes forms

clique

  • 𝑙 degree node has many edges

to 𝑚 degree nodes where 𝑚 > 𝑙

  • Inductive structure
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Observations

  • The highest degree nodes forms

clique

  • 𝑙 degree node has many edges

to 𝑚 degree nodes where 𝑚 > 𝑙

  • Inductive structure
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Previous tool and challenges

Physics Network Science

Lattice[98] Random regular[07] Tree[79,06] Erdos-Renyi model[12] Chung-Lu model[12] Watts-Strogatz[13] Kleinberg[14] Preferential Attachment[14] Configuration model α>3 [10] Configuration model 2<α<3[16]

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Inductive Structure

  • Partition nodes into buckets ordered by degree of nodes

𝐶1, 𝐶2, … , 𝐶𝑚

  • Induction: if infection spreads on previous buckets 𝐶𝑗where

𝑗 < 𝑙 , the infection also spread on bucket 𝐶𝑙.

𝐶1 𝐶2 𝐶3

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Inductive Structure

  • If infection spreads on previous buckets 𝐶𝑗where 𝑗 < 𝑙 , the

infection also spread on bucket 𝐶𝑙.

𝐶1 𝐶2 𝐶3

Time 0

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Inductive Structure

  • If infection spreads on previous buckets 𝐶𝑗where 𝑗 < 𝑙 , the

infection also spread on bucket 𝐶𝑙.

𝐶2 𝐶3

Time 1

𝐶1

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Inductive Structure

  • If infection spreads on previous buckets 𝐶𝑗where 𝑗 < 𝑙 , the

infection also spread on bucket 𝐶𝑙.

𝐶3

Time 2

𝐶1 𝐶2

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Inductive Structure

  • If infection spreads on previous buckets 𝐶𝑗where 𝑗 < 𝑙 , the

infection also spread on bucket 𝐶𝑙.

𝐶3 𝐶1

Time 3

𝐶2

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Inductive Structure

  • Induction: if infection spreads on previous buckets 𝐶𝑗where

𝑗 < 𝑙 , the infection also spread on bucket 𝐶𝑙.

– Well connection between buckets – Infection spread in buckets 𝐶1 𝐶2 𝐶3

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Inductive Structure

  • Induction: if infection spreads on previous buckets 𝐶𝑗where

𝑗 < 𝑙 , the infection also spread on bucket 𝐶𝑙.

– Well connection between buckets: Chernoff bound – Infection spread in buckets: Chebeshev’s inequality 𝐶1 𝐶2 𝐶3

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Thanks for your listening

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How do we solve it?

  • Chebyshev’s inequality

Pr 𝑎 > 𝐹 𝑎 + 𝑢 ≤ 𝑊𝑏𝑠 𝑎 𝑢2

  • Chernoff-type bound

Pr 𝑎𝑜 > 𝑎0 + 𝑢 ≤ exp −𝑢2 𝑑