Graphs, Social Networks and Foodwebs

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Graphs, Social Networks and Foodwebs

• Professor Martin Everett
• 21 June 2001

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A graph consists of a set of vertices and a set
of edges.
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
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Digraphs Directed Graphs. The edges are
replaced by arcs.
We have in and out neighborhoods denoted by
Ni and No respectively. No(1) 2,5 Ni(4)
2,3,4,5
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Social Networks use graphs derived from the
social sciences
Nodes Edges
People Friends of
Employees Is the manager of
Organisations Does business with
Countries Trades with
Monkeys Grooms
Can be graphs or digraphs
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• Social Role
• Captures idea of role concepts
• Manager of
• Brother/Sister
• Teacher, Nurse

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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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A coloration which uses k colours is called a
k-coloration
In 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.
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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.
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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.
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Food Webs
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Malaysian Pitcher Plant Food Web Beaver (1985)
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Multi-dimensional Scaling of the REGE
coefficients, with color classes determined by
clustering
Image Graph of the food web
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3-D MDS of REGE coefficients Green producers
Yellow suspension feeders Gray
detritvores Blue fishes and benthos
consumers Red top predators
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Conclusions

• Mathematics has a role in the Social Sciences

• Social Sciences can provide some interesting
  mathematical problems

• Mathematics is universal