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 need to study this phenomena? – Better Understanding – Promoting good behaviors/beliefs – Stopping bad behavior
Outline • Models – Complex contagions model – Power-Law and configuration model graph • Main result • Related work • A happy proof sketch
Outline • Models – Complex contagions model – Power-Law and configuration model graph • Main result • Related work • A happy proof sketch
Model of Contagions • K-Complex Contagions [GEG13; CLR 79] – Given an initial infected set 𝐽 = {𝑣, 𝑤} . a y u x v w z
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 u x v w z
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 u x v w z
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 u x v w z
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 u x v w z
Why k complex contagions? • One of most classical and simple contagions model – Threshold model [Gra 78] – Bootstrap percolation [CLR 79] • Non-submodular
Motivating Question • Do k complex contagions spread on social networks?
Question • Do k complex contagions spread on Erdos-Renyi model 𝐻 𝑜,𝑞 1 where 𝑞 = 𝑃 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?
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
Motivating Question • Do k complex contagions spread on social networks?
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?
Outline • Models – Complex contagions model – Power-Law and configuration model graph • Main result • Related work • A happy proof sketch
Power-law distribution • A power-law distribution with 𝛽 alpha=2.5 alpha=3.5 if the Pr 𝑌 = 𝑦 ~𝑦 −𝛽 0.001 0.0009 0.0008 0.0007 0.0006 0.0005 0.0004 0.0003 0.0002 0.0001 0 0 50 100 150 200
Configuration Model • Given a degree sequence Nodes with stubs deg(𝑤 1 ), deg(𝑤 2 ), … , deg(𝑤 𝑜 ) • The node 𝑤 𝑗 has deg(𝑤 𝑗 ) stubs
Configuration Model • Given a degree sequence deg(𝑤 1 ), deg(𝑤 2 ), … , deg(𝑤 𝑜 ) • The node 𝑤 𝑗 has deg(𝑤 𝑗 ) stubs • Choose a uniformly random matching on the stubs
Outline • Models – Complex contagions model – Power-Law and configuration model graph • Main result • Related work • A happy proof sketch
Theorems Main result Corollary from [Amini 10] • Configuration Model • Configuration Model – power-law degree distribution 2 < 𝛽 < 3 – power-law degree distribution 3 < 𝛽 • Initial infected node • Initial infected nodes: – the highest degree node – o(1) fraction of highest degree node • 𝑙 -complex contagions spreads to Ω(1) • 𝑙 -complex contagions spreads to o(1) fraction of nodes with high fraction of nodes with high probability. probability.
The Bottom Line Main result Corollary from [Amini 10] • Configuration Model • Configuration Model – power-law degree distribution 2 < 𝛽 < 3 – power-law degree distribution 3 < 𝛽 • 𝑙 -complex contagions • 𝑙 -complex contagions – Initial infected node: the highest degree – Initial infected node: o(1) fraction of node highest degree node • Contagions spreads to Ω(1) fraction of • Contagions spreads to o(1) fraction of nodes with high probability. nodes with high probability.
Outline • Models – Complex contagions model – Power-Law and configuration model graph • Main result • Related work • A happy proof sketch
What has been done? Random Configuration Watts-Strogatz[13] Lattice[98] regular[07] model α >3[10] Kleinberg[14] Erdos-Renyi Chung-Lu Tree[79,06] Preferential model[12] model[12] Attachment[14]
What has been done? Configuration model α >3 [10] Random Watts-Strogatz[13] Configuration Lattice[98] regular[07] Kleinberg[14] model 2< α <3[16] Erdos-Renyi Chung-Lu Tree[79,06] Preferential model[12] model[12] Attachment[14]
Start with 𝐻 𝑜,𝑞 Configuration model α >3 [10] Random Watts-Strogatz[13] Configuration Lattice[98] regular[07] Kleinberg[14] model 2< α <3[16] Erdos-Renyi Chung-Lu Tree[79,06] Preferential model[12] model[12] Attachment[14]
k- complex contagions don’t spread Physics Configuration model α >3 [10] Random Watts-Strogatz[13] Configuration Lattice[98] regular[07] Kleinberg[14] model 2< α <3[16] Erdos-Renyi Chung-Lu Tree[79,06] Preferential model[12] model[12] Attachment[14]
k-complex contagions spread Network Science Configuration model α >3 [10] Random Watts-Strogatz[13] Configuration Lattice[98] regular[07] Kleinberg[14] model 2< α <3[16] Erdos-Renyi Chung-Lu Tree[79,06] Preferential model[12] model[12] Attachment[14]
Things get complicated Physics Network Science Configuration model α >3 [10] Random Watts-Strogatz[13] Configuration Lattice[98] regular[07] Kleinberg[14] model 2< α <3[16] Erdos-Renyi Chung-Lu Tree[79,06] Preferential model[12] model[12] Attachment[14]
What have we learned? Physics Network Science Configuration model α >3 [10] Random Watts-Strogatz[13] Configuration Lattice[98] regular[07] Kleinberg[14] model 2< α <3[16] Erdos-Renyi Chung-Lu Tree[79,06] Preferential model[12] model[12] Attachment[14]
Why do we want to solve it? Configuration model α >3 Kleinberg Lattice Random regular Configuration model 2< α <3 Watts-Strogatz Tree Erdos-Renyi model Chung-Lu Preferential model Attachment
Outline • Models – Complex contagions model – Power-Law and configuration model graph • Main result • Related work • A happy proof sketch
Idea • Restrict contagions from high degree node to low degree nodes • Reveal the edges when needed
Observations • The highest degree nodes forms clique
Observations • The highest degree nodes forms clique • 𝑙 degree node has many edges to 𝑚 degree nodes where 𝑚 > 𝑙
Observations • The highest degree nodes forms clique • 𝑙 degree node has many edges to 𝑚 degree nodes where 𝑚 > 𝑙 • Inductive structure
Observations • The highest degree nodes forms clique • 𝑙 degree node has many edges to 𝑚 degree nodes where 𝑚 > 𝑙 • Inductive structure
Previous tool and challenges Physics Network Science Configuration model α >3 [10] Random Watts-Strogatz[13] Configuration Lattice[98] regular[07] Kleinberg[14] model 2< α <3[16] Erdos-Renyi Chung-Lu Tree[79,06] Preferential model[12] model[12] Attachment[14]
Inductive Structure • Partition nodes into buckets ordered by degree of nodes 𝐶 1 , 𝐶 2 , … , 𝐶 𝑚 • Induction: if infection spreads on previous buckets 𝐶 𝑗 where 𝑗 < 𝑙 , the infection also spread on bucket 𝐶 𝑙 . 𝐶 1 𝐶 3 𝐶 2
Inductive Structure • If infection spreads on previous buckets 𝐶 𝑗 where 𝑗 < 𝑙 , the infection also spread on bucket 𝐶 𝑙 . Time 0 𝐶 1 𝐶 3 𝐶 2
Inductive Structure • If infection spreads on previous buckets 𝐶 𝑗 where 𝑗 < 𝑙 , the infection also spread on bucket 𝐶 𝑙 . Time 1 𝐶 1 𝐶 3 𝐶 2
Inductive Structure • If infection spreads on previous buckets 𝐶 𝑗 where 𝑗 < 𝑙 , the infection also spread on bucket 𝐶 𝑙 . Time 2 𝐶 1 𝐶 3 𝐶 2
Inductive Structure • If infection spreads on previous buckets 𝐶 𝑗 where 𝑗 < 𝑙 , the infection also spread on bucket 𝐶 𝑙 . Time 3 𝐶 1 𝐶 3 𝐶 2
Inductive Structure • Induction: if infection spreads on previous buckets 𝐶 𝑗 where 𝑗 < 𝑙 , the infection also spread on bucket 𝐶 𝑙 . – Well connection between buckets – Infection spread in buckets 𝐶 1 𝐶 3 𝐶 2
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