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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 distribution
• Small World models
• Readings:
• Chapter 5 from the Jackson book
• Chapter 18 from the Kleinberg book
• Chapter 20 from the Kleinberg book

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Power Law Degree Distribution

• 𝑄 𝑒 = 𝑑𝑒−𝛿
• log 𝑄 𝑒

= log 𝑑 − 𝛿 log 𝑒

• Features:
• Scale-free
• Fat tail

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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

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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

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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 + 𝑛 𝑗 + 1 + 𝑛 𝑗 + 2 + ⋯ + 𝑛 𝑢 = 𝑛 1 + 1 𝑗 + 1 + ⋯ + 1 𝑢 ≈ 𝑛 1 + log 𝑢 𝑗

• Degree distribution:

𝑛 1 + log 𝑢 𝑗 < 𝑒 𝑗 > 𝑢𝑓1−𝑒

𝑛

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Growing Models

• A natural study of degree distribution:
• The nodes with expected degree less than d are those born at time 𝑢𝑓1− 𝑒

𝑛

• This is a fraction of 1 − 𝑓1− 𝑒

𝑛 of total t nodes

• Thus

𝐺

𝑢 𝑒 = 1 − e−𝑒−𝑛 𝑛

• Another way: Mean Field Approximation

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Mean Field Approximation

• Using expected increase in the number of sth as its rate
• Visiting the last example with MFA:

𝑒𝑒𝑗 𝑢 𝑒𝑢 = 𝑛 𝑢 𝑒𝑗 𝑢 = 𝑛 + 𝑛 log 𝑢 𝑗 𝑒 = 𝑛 + 𝑛 log 𝑢 𝑗(𝑒) 𝑗 𝑒 𝑢 = 𝑓−𝑒−𝑛

𝑛

• With the same argumentation we have:

𝐺

𝑢 𝑒 = 1 − e−𝑒−𝑛 𝑛

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Basic Preferential Attachment Model

• The probability that an existing node i receives a new link:

𝑛 𝑒𝑗(𝑢) 𝑘=1

𝑢

𝑒𝑘(𝑢) = 𝑛 𝑒𝑗(𝑢) 2𝑛𝑢 = 𝑒𝑗 𝑢 2𝑢

• Using MFA:

𝑒𝑒𝑗 𝑢 𝑒𝑢 = 𝑒𝑗 𝑢 2𝑢

• With initial condition 𝑒𝑗 𝑗 = 𝑛 we have:

𝑒𝑗 𝑢 = 𝑛 𝑢 𝑗

1 2

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Basic Preferential Attachment Model

• We have:

𝑗𝑢 𝑒 𝑢 = 𝑛 𝑒

2

• Thus

𝐺𝑢 𝑒 = 1 − 𝑛2𝑒−2 𝑔

𝑢 𝑒 = 2𝑛2𝑒−3

• If the rate changes to

𝑒𝑗 𝑢 𝛿𝑢 we have:

𝑔

𝑢 𝑒 = 𝛿𝑛𝛿𝑒−𝛿−1

Which is a power law disitribution

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Hybrid Preferential Attachment Models

• Mixing Random & Preferential Attachment:

𝑒𝑒𝑗 𝑢 𝑒𝑢 = 𝛽𝑛 𝑢 + 1 − 𝛽 𝑛𝑒𝑗 𝑢 2𝑛𝑢 = 𝛽𝑛 𝑢 + 1 − 𝛽 𝑒𝑗 𝑢 2𝑢

• By solving the above differential equation we have:

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Hybrid Preferential Attachment Models

• By solving the above differential equation we have:
• To have the degree distribution:
• If 𝑒𝑗 𝑢 = 𝜚𝑢 𝑗 (the degree of the node with i’th birth)

𝐺

𝑢 𝑒 = 1 − 𝜚𝑢 −1 𝑒

𝑢

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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

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Structure + Randomness

• Structure makes shortest paths
• Random links make triads
• It is naturally incorrect!

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Structure + Randomness

• Watts & Strogatz model
• Structure makes triads
• Random links make short distances: Weak ties

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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

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Watts-Strogatz Models for Decentralized Search

• Let’s set the clustering coefficient q = 2
• Terms 𝑒2 and 𝑒−2 cancel each other

and thus the probability that a random edge links into some node in this ring is approximately independent

• f the value of d
• long-range weak ties are being formed

in a way that’s spread roughly uniformly

• ver all different scales of resolution

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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 𝑒2, thus p should be approximately 1

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Watts-Strogatz Models for Decentralized Search

• Some Experiments

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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

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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:

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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 2𝑘 and 2𝑘+1

• Define 𝑌

𝑘 as the number of steps of the

search spent in phase j

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Watts-Strogatz Models for Decentralized Search

• Mathematical study of myopic decentralized search in a simple

Watts-Strogatz model: 𝑌 = 𝑌1 + 𝑌2 + ⋯ + 𝑌log(𝑜) 𝐹 𝑌 = 𝐹 𝑌1 + ⋯ + 𝐹[𝑌log 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]

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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:

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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

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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 proportional to

log 𝑜 log log 𝑜

almost surely.

• 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:
• The probability of having a triad at t+1’th level:

𝑢𝑛

𝑢 2

=

2m t−1 → 0

• The same happens in the hybrid model but with more justifications

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Small World Properties in Preferential Attachment Models

• Core Periphery structure:
• Homophily suggests that high-status people will mainly know other high-

status people, and low-status people will mainly know other low-status people, but this does not imply that the two groups occupy symmetric or interchangeable positions in the social network

• The high-status people are linked in a densely-connected core, while the low-

status people are atomized around the periphery of the network

• One source of small worlds: High status people have the resources to travel

widely

• Preferential attachment model has core periphery structure.

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Small World Properties in Preferential Attachment Models

• Core Periphery structure:
• Jackson & Rogers Theorem: Consider a growing hybrid random network-

formation process. Under the mean-field estimate, a node i’s degree is larger than a node j’s degree at time t after both are born, if and only if i is older than j. In that case, if α > 0, then the estimated distribution of i’s neighbors’ degrees strictly first-order stochastically dominates that of j’s at each time t > j relative to younger nodes; that is, 𝐺

𝑗 𝑢 𝑒 < 𝐺 𝑘 𝑢(𝑒) for all 𝑒 < 𝑒𝑘(𝑢) (𝐺 𝑗 𝑢 𝑒

denote the fraction of node i’s neighbors at time t who have degree d or less)

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Meeting-based Network Growing Model

• Each new node meets mr > 0 nodes uniformly at random and forms directed links to
• them. Then the new node randomly chooses mn of the out-links from the first group of

nodes and follows those links to meet new nodes and form additional links.

• Because a node i is chosen in the second iteration with probability:

𝑒𝑗

𝑗𝑜 𝑢 × 𝑛𝑠

𝑢 × 𝑛𝑜 𝑛𝑠𝑛

• Thus we have:

with 𝑠 = 𝑛𝑜

𝑛𝑠

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Clustering in the Meeting-based Growing Model

• Jackson & Rogers Theorem: Under the mean field approximation the

fraction of transitive triples 𝐷𝑚𝑈𝑈 tends to 1 𝑛 𝑠 + 1 , 𝑠 ≥ 1 𝑛 − 1 𝑠 𝑛 𝑛 − 1 1 + 𝑠 𝑠 − 𝑛 𝑠 − 1 , 𝑠 < 1

• Proof: see the blackboard

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