Graphs, Social Networks and Foodwebs

Professor Martin Everett21 June 2001

A graph consists of a set of vertices and a set of edges.

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The neighborhood N(v) of a vertex v is the set of vertices it is connected to.

N(3) = {2,4,5}

N(8) = {7}

Digraphs = Directed Graphs. The edges are replaced by arcs.

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We have ‘in’ and ‘out’ neighborhoods denoted by Ni and No respectively.

No(1) = {2,5}

Ni(4) = {2,3,4,5}

Social Networks use graphs derived from the social sciences

Can be graphs or digraphs

Nodes Edges

People Friends of

Employees Is the manager of

Organisations Does business with

Countries Trades with

Monkeys Grooms

Social Role

Captures idea of role concepts

• Manager of

• Brother/Sister

• Teacher, Nurse

A coloration C of a graph is an assignment of colours to the vertices.

A coloration of a graph is regular if whenever

C(x) = C(y) then

C(N(x)) = C(N(y))

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For a directed graph we consider in and out neighborhoods seperately if C(x) = C(y) then

C(Ni (x)) = C(Ni(y)) and

C(No (x)) = C(No(y))

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{1,2,3,4}

{1,2,3} {4} {1} {2,3,4}

{1}, {2,3}, {4}

{1,4}, {2,3}

{1,4}, {2}, {3}

{1,3}, {2,4} {1,2}, {3,4}

{1}, {2}, {3}, {4}

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A coloration which uses k colours is called a k-colorationIn a graph with n vertices (and no isolated vertices) a 1-coloration and an n-coloration are both regular.

A source is a vertex whose in-neighborhood is the empty set. A sink has an empty out-neighborhood.

In a digraph with no sources or sinks and n vertices a 1-coloration and an n-coloration are both regular.

These colorations are called trivial colorations.

A graph or digraph with no non-trivial colorations is called role primitive.

The regular coloration which uses the fewest colours is called the maximal regular coloration.

A concept of importance for digraphs with sources or sinks.

There is a simple algorithm which finds the maximal regular coloration.

Image graph of a coloration has colours as vertices and an edge between two colours if there is is an edge connecting these colours in the original graph.

Note a coloration is regular iff

C(N(x)) = N(C(x))

Image graph captures the role structure.

Food Webs

Malaysian Pitcher Plant Food Web

Beaver (1985)

Multi-dimensional Scaling of the REGE

coefficients, with color classes determined by

clustering

Image Graph of the food web

3-D MDS of REGE coefficients

Green – producers

Yellow – suspension feeders

Gray – detritvores

Blue – fishes and benthos consumers

Red – top predators

Conclusions

Mathematics has a role in the Social Sciences

Social Sciences can provide some interesting mathematical problems

Mathematics is universal