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 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
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
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
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
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
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
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
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
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
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
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
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
Structure + Randomness • Structure makes shortest paths • Random links make triads • It is naturally incorrect! Social and Economic Networks 14
Structure + Randomness • Watts & Strogatz model • Structure makes triads • Random links make short distances: Weak ties Social and Economic Networks 15
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
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
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
Watts-Strogatz Models for Decentralized Search • Some Experiments Social and Economic Networks 19
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
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
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
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
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
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
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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