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

2. A graph consists of a set of vertices and a set of edges. 3 5 6 7 8 421 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. Digraphs = Directed Graphs. The edges are replaced by arcs. 4 5 1 2 3 We have ‘in’ and ‘out’ neighborhoods denoted by N i and N o respectively. N o (1) = {2,5} N i (4) = {2,3,4,5}

4. Social Networks use graphs derived from the social sciences Can be graphs or digraphs NodesEdges PeopleFriends of EmployeesIs the manager of OrganisationsDoes business with CountriesTrades with MonkeysGrooms

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

8. 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)) 1 2 3 45 6 7 8 1 2 3 45 6 7 8

9. For a directed graph we consider in and out neighborhoods seperately if C(x) = C(y) then C(N i (x)) = C(N i (y)) and C(N o (x)) = C(N o (y)) 3 2 1 5 4

10. {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} 1 2 34

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