Random Graph Models of Social Networks

Paper Authors: M.E. Newman, D.J. Watts, S.H. Strogatz

Presentation presented by Jessie Riposo

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 of the generated networks by using their degree distributions

Paper Has Two Main Parts

• Modeling graphs with arbitrary degree distribution

• Modeling affiliation networks and bipartite graphs

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 degree distribution of a random graph is unlike that of real-world networks.

Known Degree Distributions

• A large random graph has a Poisson Degree distribution

• Scientific Collaboration Networks, Movie Actor Collaboration Networks, and Company Director Networks all have highly skewed degree distributions that cannot be modeled with the Poisson.

Why the Random Graph if it does not have the correct degree distribution for real-world

networks?

• The Random Graph Has Desirable Properties• Many features of its behavior can be calculated

exactly

Is it possible to create a model that matches real-world networks better than a random graph, but is still exactly solvable?

An Algorithm that Generates a Random Graph with the Desired Degree Distribution

• Given (normalized) probabilities p that a randomly chosen vertex in the network has degree k– Take N vertices– Assign to each a number k of ends (k is a random

number drawn independently of probability of k)– Chose ends randomly in pairs and connect with an

edge– If number of ends is odd throw one edge away

and generate a new one from distribution, repeating until number of ends is even.

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.

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 connected

together by edges– A giant cluster of connected vertices whose size

scales up with the size of the whole network

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)

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

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 be

normalized

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 giant component cannot exist regardless of the value of k.

• Almost all networks found in society and nature appear to be well inside these limits.

Why Affiliation Networks and Bipartite Graphs

• Affiliation networks can be used to avoid problems of:– Hard to solicit unbiased data in social network

experiments.– Data is usually limited

• Affiliation network is a network in which actors are joined together by common membership of groups

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 on– The number of directors who sit on a boards

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 pj

and 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 the random graph C = 3* Number of triangles on the graph

Number of connected triples of vertices

Network C Theory C Actual Avg. Degree Theory

Avg. Degree Actual

Company directors

.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

How Does the Theory Measure Up?

• The clustering coefficient is remarkably precise for boards of directors

• For the other networks the clustering coefficient seems to be underestimated by a factor of about two by the theory

• For the other networks the average number of collaborators is moderately accurate.

What Does This Mean?

• Remember that the graphs were created with degree distributions the same as real networks, but the connections between the nodes were generated randomly.

• Agreement between model and reality would indicate that there is no statistical difference between the real-world network and an equivalent random network.

• Differences in the models and real-world networks may be indicating some potential sociological phenomenon

The Main Contributions of This Paper Were:

• A set of Models that allow for the fact that the degree distributions of real-world social networks are often highly skewed

• The Statistical Properties of the networks are exactly solvable, once the degree distribution is specified

• A generalized theory in the case of bipartite random graphs which serve as models for affiliation networks

• Models can be applied not only to Social Networks, but to communications, transportation, distribution, and other networks