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Property Oriented Network Models Social and Economic Networks Jafar Habibi MohammadAmin Fazli Social and Economic Networks 1 ToC Property Oriented Network Models Growing Random Network Models Models with Power law degree


  1. Property Oriented Network Models Social and Economic Networks Jafar Habibi MohammadAmin Fazli Social and Economic Networks 1

  2. ToC • Property Oriented Network Models • Growing Random Network Models • Models with Power law degree distribution • Small World models • Readings: • Chapter 5 from the Jackson book • Chapter 18 from the Kleinberg book • Chapter 20 from the Kleinberg book Social and Economic Networks 2

  3. Power Law Degree Distribution • 𝑄 𝑒 = 𝑑𝑒 −𝛿 • log 𝑄 𝑒 = log 𝑑 − 𝛿 log 𝑒 • Features: • Scale-free • Fat tail Social and Economic Networks 3

  4. Richer-Get-Richer & Preferential Attachment • In many scenarios, richers have more opportunity to get richers • More money for investment • Lower risks • More reputation to be involved in activities • … . • Preferential Attachment: richer-get-richer effect in network creation • The probability that page L experiences an increase in popularity is directly proportional to L ’ s current popularity. • In the sense that links are formed “ preferentially ” to pages that already have high popularity Social and Economic Networks 4

  5. Preferential Attachment Models • Devise models to simulate preferential attachment processes • A basic growing model: • Nodes are born over time and indexed by their date of birth i ∈ {0 , 1 , 2 . . . , t, . . . } • Upon birth each new node forms m links with pre-existing nodes • It attaches to nodes with probabilities proportional to their degrees. • the probability that an existing node i receives a new link: • The interesting fact is that these models leads to networks with power-law degree distribution Social and Economic Networks 5

  6. Growing Models • A network model dealing with adding newborn nodes instead of statically having the whole network • Consider a variation of the Poisson random random setting • Start with a complete network of m+1 nodes • Each newborn node choose m nodes from the existing ones and links to them • A natural study of degree distribution: • The expected degree of a node born at time i, at time t: 𝑛 𝑗 + 2 + ⋯ + 𝑛 𝑛 𝑗 + 1 + ⋯ + 1 1 ≈ 𝑛 1 + log 𝑢 𝑛 + 𝑗 + 1 + 𝑢 = 𝑛 1 + 𝑢 𝑗 • Degree distribution: 𝑛 1 + log 𝑢 < 𝑒 𝑗 > 𝑢𝑓 1−𝑒 𝑛 𝑗 Social and Economic Networks 6

  7. Growing Models • A natural study of degree distribution: • The nodes with expected degree less than d are those born at time 𝑢𝑓 1− 𝑒 𝑛 • This is a fraction of 1 − 𝑓 1− 𝑒 𝑛 of total t nodes • Thus 𝑢 𝑒 = 1 − e −𝑒−𝑛 𝐺 𝑛 • Another way: Mean Field Approximation Social and Economic Networks 7

  8. Mean Field Approximation • Using expected increase in the number of sth as its rate • Visiting the last example with MFA: 𝑒𝑒 𝑗 𝑢 = 𝑛 𝑢 𝑒 𝑗 𝑢 = 𝑛 + 𝑛 log 𝑢 𝑒𝑢 𝑗 𝑢 𝑒 = 𝑛 + 𝑛 log 𝑗(𝑒) 𝑗 𝑒 = 𝑓 −𝑒−𝑛 𝑛 𝑢 • With the same argumentation we have: 𝑢 𝑒 = 1 − e −𝑒−𝑛 𝐺 𝑛 Social and Economic Networks 8

  9. Basic Preferential Attachment Model • The probability that an existing node i receives a new link: 𝑒 𝑗 (𝑢) 𝑒 𝑘 (𝑢) = 𝑛 𝑒 𝑗 (𝑢) 2𝑛𝑢 = 𝑒 𝑗 𝑢 𝑛 𝑢 𝑘=1 2𝑢 • Using MFA: 𝑒𝑒 𝑗 𝑢 = 𝑒 𝑗 𝑢 𝑒𝑢 2𝑢 • With initial condition 𝑒 𝑗 𝑗 = 𝑛 we have: 1 𝑒 𝑗 𝑢 = 𝑛 𝑢 2 𝑗 Social and Economic Networks 9

  10. Basic Preferential Attachment Model • We have: 2 𝑗 𝑢 𝑒 = 𝑛 𝑢 𝑒 • Thus 𝐺 𝑢 𝑒 = 1 − 𝑛 2 𝑒 −2 𝑔 𝑢 𝑒 = 2𝑛 2 𝑒 −3 𝑒 𝑗 𝑢 • If the rate changes to 𝛿𝑢 we have: 𝑢 𝑒 = 𝛿𝑛 𝛿 𝑒 −𝛿−1 𝑔 Which is a power law disitribution Social and Economic Networks 10

  11. Hybrid Preferential Attachment Models • Mixing Random & Preferential Attachment: 𝑒𝑒 𝑗 𝑢 = 𝛽𝑛 𝑢 + 1 − 𝛽 𝑛𝑒 𝑗 𝑢 = 𝛽𝑛 𝑢 + 1 − 𝛽 𝑒 𝑗 𝑢 𝑒𝑢 2𝑛𝑢 2𝑢 • By solving the above differential equation we have: Social and Economic Networks 11

  12. Hybrid Preferential Attachment Models • By solving the above differential equation we have: • To have the degree distribution: • If 𝑒 𝑗 𝑢 = 𝜚 𝑢 𝑗 (the degree of the node with i ’ th birth) −1 𝑒 𝑢 𝑒 = 1 − 𝜚 𝑢 𝐺 𝑢 Social and Economic Networks 12

  13. Small Worlds • Six degrees of separation: although the number of edges is low, nodes are reachable from each other with small number of edges • Small diameter or Small average path length • Weak ties to close dense communities • Highly Clustered • High density of triangles • Homophily & prone to triadic closure Social and Economic Networks 13

  14. Structure + Randomness • Structure makes shortest paths • Random links make triads • It is naturally incorrect! Social and Economic Networks 14

  15. Structure + Randomness • Watts & Strogatz model • Structure makes triads • Random links make short distances: Weak ties Social and Economic Networks 15

  16. Watts-Strogatz Models for Decentralized Search • Consider a grid with additional random links each with probability 𝑒 𝑤, 𝑥 −𝑟 in which q is the clustering exponent Social and Economic Networks 16

  17. Watts-Strogatz Models for Decentralized Search • Let ’ s set the clustering coefficient q = 2 • Terms 𝑒 2 and 𝑒 −2 cancel each other and thus the probability that a random edge links into some node in this ring is approximately independent of the value of d • long-range weak ties are being formed in a way that ’ s spread roughly uniformly over all different scales of resolution Social and Economic Networks 17

  18. Watts-Strogatz Models for Decentralized Search • Rank-based friendship: • Create (weak) random links with probability 𝑠𝑏𝑜𝑙 𝑥 −𝑞 • What should p be to have a uniform spread of random links? 𝑠𝑏𝑜𝑙 approximately is 𝑒 2 , thus p should be approximately 1 Social and Economic Networks 18

  19. Watts-Strogatz Models for Decentralized Search • Some Experiments Social and Economic Networks 19

  20. Watts-Strogatz Models for Decentralized Search • Foci-based friendship: • Define the size of the smallest focal point that include both of v and w as their distance • We again draw random links with probability 𝑒𝑗𝑡 𝑤, 𝑥 𝑞 • If focal points are defined as the nearest nodes, we may again have p = 1 Social and Economic Networks 20

  21. Watts-Strogatz Models for Decentralized Search • Mathematical study of myopic decentralized search in a simple Watts- Strogatz model: • A fixed structure: a ring or a grid • Some additional random links with probability proportional to 𝑒 𝑤, 𝑥 −1 • What is the constant multiplier for link probabilities: Social and Economic Networks 21

  22. Watts-Strogatz Models for Decentralized Search • Mathematical study of myopic decentralized search in a simple Watts- Strogatz model: • Myopic search: when a node v is holding the message, it passes it to the contact that lies as close to t on the ring as possible • As the message moves from s to t, we ’ ll say that it ’ s in phase j of the search if its distance from the target is between 2 𝑘 and 2 𝑘+1 • Define 𝑌 𝑘 as the number of steps of the search spent in phase j Social and Economic Networks 22

  23. Watts-Strogatz Models for Decentralized Search • Mathematical study of myopic decentralized search in a simple Watts-Strogatz model: 𝑌 = 𝑌 1 + 𝑌 2 + ⋯ + 𝑌 log(𝑜) 𝐹 𝑌 = 𝐹 𝑌 1 + ⋯ + 𝐹[𝑌 log 𝑜 ] • Let ’ s compute an upper bound for 𝐹 𝑌 𝑘 𝐹 𝑌 𝑘 = 1 Pr 𝑌 𝑘 = 1 + 2 Pr 𝑌 𝑘 = 2 + ⋯ 𝐹 𝑌 𝑘 = Pr 𝑌 𝑘 ≥ 1 + Pr 𝑌 𝑘 ≥ 2 + ⋯ • Now Let ’ s compute an upper bound for Pr[𝑌 𝑘 ≥ 𝑗] Social and Economic Networks 23

  24. Watts-Strogatz Models for Decentralized Search • Mathematical study of myopic decentralized search in a simple Watts-Strogatz model: • The probability that v (with distance d) has a random link to some node w with distance less than d/2 is at least: Social and Economic Networks 24

  25. Watts-Strogatz Models for Decentralized Search • Mathematical study of myopic decentralized search in a simple Watts- Strogatz model: • The probability that v has some link to a node with distance less than d/2 is at least: • The probability of not having such link i times and thus being kept in phase j is at most: • Thus we have: • Thus Social and Economic Networks 25

  26. Small World Properties in Preferential Attachment Models • Low Diameter: Bollobas & Riordan- In a preferential attachment model in which each newborn node forms m ≥ 2 links, as n grows the resulting network consists of a single component with diameter log 𝑜 proportional to almost surely. log log 𝑜 • Less than Poisson random network ( log(𝑜) ) • In the hybrid preferential attachment model there is an interval log 𝑜 log log 𝑜 , log 𝑜 • But the clustering coefficient is low: 𝑢𝑛 2m • The probability of having a triad at t+1 ’ th level: = t−1 → 0 𝑢 2 • The same happens in the hybrid model but with more justifications Social and Economic Networks 26

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