Graphs, Social Networks and Foodwebs

1. Graphs, Social Networks and Foodwebs Professor Martin Everett 21 June 2001

2. 6 5 3 7 8 1 2 4 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}

3. 3 4 5 2 1 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}

4. Social Networks use graphs derived from the social sciences Can be graphs or digraphs

7. Social Role • Captures idea of role concepts • Manager of • Brother/Sister • Teacher, Nurse

8. 8 3 6 1 7 5 2 4 8 3 6 1 7 5 2 4 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))

9. 3 4 5 1 2 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))

10. 2 1 3 4 {1,2,3,4} {1,4}, {2,3} {1,2,3} {4} {1} {2,3,4} {1,2}, {3,4} {1,3}, {2,4} {1}, {2,3}, {4} {1,4}, {2}, {3} {1}, {2}, {3}, {4}

11. 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.

22. 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.

27. 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.

28. Food Webs

30. Malaysian Pitcher Plant Food Web Beaver (1985)

31. Multi-dimensional Scaling of the REGE coefficients, with color classes determined by clustering Image Graph of the food web

32. 3-D MDS of REGE coefficients Green– producers Yellow– suspension feeders Gray– detritvores Blue – fishes and benthos consumers Red– top predators

33. Conclusions • Mathematics has a role in the Social Sciences • Social Sciences can provide some interesting mathematical problems • Mathematics is universal