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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???