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LOGO
Identity and Searchin
Social Networks
D.J.Watts, P.S. Dodds, M.E.J. Newman
Maryam Fazel-Zarandi
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LOGO Outlines
Introduction
The Hierarchical Model
Discussion
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LOGO
Introduction
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LOGO Milgram’s Experiment
Short chains of acquaintances exist. People are able to find these chains using
only local information.
Source
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LOGO Results in Literature
Connected random networks have short average path lengths:
xij log(N)
N = population size, xij = distance between nodes i and j.
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LOGO Results in Literature
Kleinberg (2000) demonstrated that emergence of the second phenomenon requires special topological structure.
For each node i: local edges d(i,j) ≤ p long-range directed edges
to q random nodes
Pr(ij) ~ d(i,j)-a
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LOGO Results in Literature
If networks have a certain fraction of hubs can also search well.
Basic idea: get to hubs first
Hubs in social networks are limited.
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LOGO
The Hierarchical Model
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LOGO Hierarchical Model – Why? How?
Basic idea: impose some high-level structure, and fill in details at random.
Incorporate identity.
Need some measure of distance between individuals.
Some possible knowledge: Target's identity, friends' identities, friends' popularity,
where the message has been.
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LOGO Hierarchical Network Construction
xij = the height of the lowest common ancestor level between i and j
z connections for each node with probability:p(x) = ce-αx
Hierarchical template for the network Network constructed from template
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LOGO Hierarchical Network Construction
Individuals hierarchically partition the social world in more than one way. h = 1, …, H hierarchies
Identity vector is position of node i in hierarchy h.
Social distance:
iv
hiv
hij
hij x y min
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LOGO Directing Messages
At each step the holder i of the message passes it to one of its friends who is closest to the target t in terms of social distance.
Individuals know the identity vectors of: themselves, their friends, the target.
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LOGO Expected Number of Steps
What is the expected number of steps to forward a message from a random source to a random target?
Define q as probability of an arbitrary message chain reaching a target.
Searchable network: Any network for which
q ≥ r
for a desired r.
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LOGO Number of Steps - Results
If message chains fail at each node with probability p, require
where L = length of message chain.
Approximation:
L ln r / ln (1 - p)
q = (1 - p)L ≥ r
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LOGO Searchable Network Regions
In H-α space
p = 0.25, r = 0.05 b = 2 g = 100, z = 99
N=102400 N=204800 N=409600
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LOGO Probability of Message Completion
α = 0 (squares) versus α = 2 (circles) N = 102400
q ≥ r
q < rr = 0.05
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LOGO Milgram's Data
N = 108
b = 10 g = 100 z = 300 Lmodel 6.7
Ldata 6.5
α = 1, H = 2
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LOGO
Discussion
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LOGO Is this an acceptable model?
Simple greedy algorithm.
Represents properties present in real social networks: Considers local clustering. Reflects the notion of locality.
High-level structure + random links.
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LOGO Can the Model be Extended?
Should we consider other parameters such as friend’s popularity information in addition to homophily? Allow variation in node degrees?
Assume correlation between hierarchies?
Are all hierarchies equally important?
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LOGO Applications
Can solutions to sociology problems inform other areas of research?
Suggested applications: Construction of peer-to-peer networks. Search in databases.
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LOGO
Thank You!
Any Questions???