1. Diffusion in Networks
Dr. Henry Hexmoor Department of Computer Science Southern Illinois University Carbondale
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2. Diffusion
The way in which an individual’s choices depend on what other people do.
Modeling the processes by which new ideas and innovations are adopted by a population.
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3. 𝐹 𝑡 =𝐹 𝑡−1 +𝑝 1−𝐹 𝑡−1 +𝑞 1−𝐹 𝑡−1 ×𝐹 𝑡−1
The Bass Model
F(t) = Fraction of population who have adopted a new idea or product.
p = rate of innovation (lead, generate new ideas)
q = rate of imitation (follow, imitate)
𝐹 𝑡 =𝐹 𝑡−1 +𝑝 1−𝐹 𝑡−1 +𝑞 1−𝐹 𝑡−1 ×𝐹 𝑡−1
Innovation
Imitation
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4. 𝐹 𝑡 =𝐹 𝑡−1 +𝑝 1−𝐹 𝑡−1 +𝑞 1−𝐹 𝑡−1 1−𝐹 𝑡−1 ×𝐹 𝑡−1
The Bass Model
Innovation
𝐹 𝑡 =𝐹 𝑡−1 +𝑝 1−𝐹 𝑡−1 +𝑞 1−𝐹 𝑡−1 1−𝐹 𝑡−1 ×𝐹 𝑡−1
𝑑𝐹(𝑡) 𝑑𝑡 = 𝑝+𝑞 ×𝐹 𝑡 1−𝐹 𝑡
𝐼𝑓 𝐹 0 = 0 𝑎𝑛𝑑 𝑝 > 0
𝑡ℎ𝑒𝑛,
𝐹 𝑡 = 1 − 𝑒 − 𝑝+𝑞 𝑡 1+ 𝑞 𝑝 𝑒 − 𝑝+𝑞 𝑡
Imitation
Mostly adopted, e.g., Soda
Rarely adopted
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5. S: Susceptible, I: Infected , R: Removed/Recovered.
Epidemic Models
They are similar to each other and all use the infection life cycle, e.g, SIR, SIS, SI and SIRE.
S: Susceptible, I: Infected , R: Removed/Recovered.
SIR
SIS
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6. SIR Model A disease that is caught once in a node’s lifetime.
Divide population into S and I.
Each I node will remain I for 𝑡 1 time period.
During 𝑡 1 time steps, each node has probability, p, of passing disease to on S neighbor.
After 𝑡 1 expires, all I nodes become R.
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7. SIR Model Contagiousness is constant at each step of I phase.
Contagion probability is uniformly fixed to p.
Certain links may provide a more rapid contagion due to nature of contacts.
SIR can be interpreted as a percolation model if and only if links are pipes.
SIR saturates after time - no more infections.
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8. SIS Model A disease that is caught repeatedly by a node.
Divide population into S and I.
Each I node will remain I for 𝑡 1 time period.
During 𝑡 1 time steps, each node has probability, p, of passing disease to on S neighbor.
After 𝑡 1 expires, each I node becomes S, as before.
In SIS, if all nodes are simultaneously free of disease (S), then the epidemic has ended.
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9. The Cascade Model Can be captured by using a coordination game.
Consider v and w are neighbors playing the given coordination game.
Let p be the fraction of v’s neighbors playing A and (1−𝑝) fraction play B. Let v have d neighbors.
v’s payoff for
𝐴=𝑝 ×𝑑×𝑎
𝐵= 1−𝑝 ×𝑑×𝑏
Behavior A is preferred Iff 𝑝𝑑𝑎≥ 1−𝑝 𝑑𝑏
which is equivalent to 𝑝≥ 𝑏 𝑎+𝑏 =𝑞
v’s preference
threshold
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10. The Cascade Model
𝑝≥ 𝑏 𝑎+𝑏 =𝑞
If at least q fraction of its neighbors adopt A, v adopts A as well.
In other words, if q is small, v adopts A. If q is large, then v adopts B.
Behavior adoption spreads in the network by successive behavior adoption. This is the basic for viral spreading.
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11. A Cluster of Density p
A set of nodes such that each node in the set has at least a fraction p of its neighbors in the set is a cluster of density p.
A cluster of density 1 is all the nodes in the network.
For each i and j clusters of density p, i ∪ j is also a cluster of density p.
A cascade halts when it encounters a dense cluster. Clusters are the only obstacles to cascades.
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12. Cascade Model Revised
Individuals (i.e., v and w) value behaviors A and B differently.
If v has d neighbors of when p fraction have behavior A, (1−𝑝) have behavior B.
If 𝑛𝑒𝑖𝑔ℎ𝑏𝑜𝑟𝑠 ℎ𝑎𝑣𝑒 𝐴 𝑑 =𝑝≥ 𝑏 𝑣 𝑎 𝑣 + 𝑏 𝑣 = 𝑞 𝑣 (v’s threshold for adopting A),
then A is the better choice.
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13. An Example 1 3 since 0.1 < 1/3. Then, 3 2, 3 5, 2 4, 5 6
Cascade stops at node 4 and 6.
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14. References
D. Easly, J. Kleinberg, Networks, Crowds, and Markets, Cambridge University press.
2. M. Jackson, Social and Economic Networks, Princeton University press.
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