02 newman randomgraphs social scalefree slides

7/27/2019 02 Newman Randomgraphs Social Scalefree Slides

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Random Graph Models of Social

Networks

Paper Authors: M.E. Newman, D.J. Watts,

S.H. Strogatz

Presentation presented by Jessie Riposo


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This Paper Focuses on New Techniques for

Generating Social networks

This paper focuses on how to generate random

graphs that will give degree distributions of real

world networks and how to calculate properties ofthe generated networks by using their degree

distributions


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Paper Has Two Main Parts

Modeling graphs with arbitrary degree

distribution

Modeling affiliation networks and bipartite

graphs


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Modeling Graphs with Arbitrary Degree

Distributions Using Random Graphs to model real world

networks has some serious short-comings

Specifically the fact that the natural degreedistribution of a random graph is unlike that of

real-world networks.


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Known Degree Distributions

A large random graph has a Poisson Degree

distribution

Scientific Collaboration Networks, Movie ActorCollaboration Networks, and Company Director

Networks all have highly skewed degree

distributions that cannot be modeled with the

Poisson.


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Why the Random Graph if it does not have the

correct degree distribution for real-worldnetworks?

The Random Graph Has Desirable Properties

Many features of its behavior can be calculated

exactly


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Is it possible to create a model that

matches real-world networks better

than a random graph, but is still

exactly solvable?


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An Algorithm that Generates a Random Graph

with the Desired Degree Distribution

Given (normalized) probabilities p that a randomlychosen vertex in the network has degree k

Take N verticesAssign to each a number k of ends (k is a

random number drawn independently ofprobability of k)

Chose ends randomly in pairs and connect withan edge

If number of ends is odd throw one edge awayand generate a new one from distribution,

repeating until number of ends is even.


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Properties of the Network Model are

Exactly Solvable in the limit of large N

The trick is to use the generating function instead

of working directly with the degree distribution

Generating Function = SUM (p*x^k) (k=0 to 100)For example:

Avg. Degree of a vertex = Derivative of the

GF evaluated at 1.


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From Experimentation in Social

Networks There are Two Regimes Depending upon the exact probability distribution

of the degrees there are two different regimes:

Many small clusters of vertices connectedtogether by edges

A giant cluster of connected vertices whose size

scales up with the size of the whole network


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If Degree Distribution is Known,

Moment Functions are Used to Calculate

Size of Giant Cluster

Generating function is used to calculate the sizes

of the giant component and average components.

The fraction of the networks which is filled by

the giant component, is given by S=1-G(u)

Where u is the smallest non-neg. real solution

of G(1)u=G(u)


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The Existence (or not) of a Giant Component is

Important in Social Networks If there is no giant component then

communication can only take place within small

groups of people If there is a giant component then a large fraction

of network can all communicate with one another


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A Sample Problem was Derived to Test

the Models The distribution used was a power-law distribution

characterized by

P= CK^(-t)e^(-K/k)Exponent t

Cutoff length k

C is a constant fixed by the requirement to benormalized


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The Results Show that Giant Components

Exist Only at Specific t and k When k is below.9102 a giant component can

never exist regardless of the value of t.

For values of t larger than 3.4788 a giantcomponent cannot exist regardless of the value of

k.

Almost all networks found in society and nature

appear to be well inside these limits.


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Why Affiliation Networks and Bipartite

Graphs Affiliation networks can be used to avoid

problems of:

Hard to solicit unbiased data in social networkexperiments.

Data is usually limited

Affiliation network is a network in which actors

are joined together by common membership of

groups


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For an Affiliation Network There are

Two Different Degree Distributions For example if looking at directors and boards the

distributions would be:

The number of boards that directors sit onThe number of directors who sit on a boards


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Mathematically the Networks are

Generated as Random Graphs, But There are now two moment functions

One for each distribution

Let probability that a director sits on j boards equal pjand probability that a board has k members equal qk.

f(x)=Sum (pj(x^j)), g(x)=sum(qk(x^k))

j k

Clustering coefficient is different from that of therandom graph

C = 3* Number of triangles on the graph

Number of connected triples of vertices


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Network C Theory C Actual Avg. Degree

Theory

Avg. Degree

Actual

Companydirectors .59 .588 14.53 14.44

Movie

actors

.084 .199 125.6 113.4

Physics .192 .452 16.74 9.27

Biomedicine .042 .088 18.02 16.93

Results of Experimentation


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How Does the Theory Measure Up?

The clustering coefficient is remarkably precise

for boards of directors

For the other networks the clustering coefficientseems to be underestimated by a factor of about

two by the theory

For the other networks the average number of

collaborators is moderately accurate.


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What Does This Mean?

Remember that the graphs were created withdegree distributions the same as real networks, butthe connections between the nodes were generatedrandomly.

Agreement between model and reality wouldindicate that there is no statistical difference

between the real-world network and an equivalentrandom network.

Differences in the models and real-world networksmay be indicating some potential sociological

phenomenon


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The Main Contributions of This Paper

Were:

A set of Models that allow for the fact that thedegree distributions of real-world social networksare often highly skewed

The Statistical Properties of the networks areexactly solvable, once the degree distribution isspecified

A generalized theory in the case of bipartite

random graphs which serve as models foraffiliation networks

Models can be applied not only to SocialNetworks, but to communications, transportation,

distribution, and other networks

02 newman randomgraphs social scalefree slides

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