Competition Graphs of Semiorders

1. Competition Graphs of Semiorders Fred Roberts, Rutgers University Joint work with Suh-Ryung Kim, Seoul National University

2. Happy Birthday Joel!

3. RAND Corporation Santa Monica, CA 1968-1971

5. Table of Contents: • Preference** • II. Scrambling** • k-suitable sets • III. Transitive Subtournaments • IV. Matrices and Line Shifts

6. Searching for More Information about Joel The results of my Google search

11. Semiorders The notion of semiorder arose from problems in utility/preference theory and psychophysics involving thresholds. V = finite set, R = binary relation on V (V,R) is a semiorder if there is a real-valued function f on V and a real number  > 0 so that for all x, y  V, (x,y)  R  f(x) > f(y) + 

12. Semiorders • Of course, semiorders are special types of partial orders. • Theorem (Scott and Suppes 1954): A digraph (with no loops) is a semiorder iff the following conditions hold: • aRb & cRd  aRd or cRb • (2) aRbRc  aRd or dRc

13. aRb & cRd  aRd or cRb c a d b

14. aRb & cRd  aRd or cRb a c b d

15. aRb & cRd  aRd or cRb a c d b

16. a aRbRc  aRd or dRc d b c

17. a b d aRbRc  aRd or dRc c

18. a b d aRbRc  aRd or dRc c

19. Competition Graphs The notion of competition graph arose from a problem of ecology. Key idea: Two species compete if they have a common prey.

20. Competition Graphs of Food Webs bird fox insect grass deer Food Webs Let the vertices of a digraph be species in an ecosystem. Include an arc from x to y if x preys on y.

21. Competition Graphs of Food Webs Consider a corresponding undirected graph. Vertices = the species in the ecosystem Edge between a and b if they have a common prey, i.e., if there is some x so that there are arcs from a to x and b to x.

22. bird insect bird deer grass fox fox insect grass deer

23. Competition Graphs More generally: Given a digraph D = (V,A). The competition graph C(D) has vertex set V and an edge between a and b if there is an x with (a,x)  A and (b,x)  A.

24. Competition Graphs: Other Applications • Other Applications: • Coding • Channel assignment in communications • Modeling of complex systems arising from study of energy and economic systems • Spread of opinions/influence in decisionmaking situations • Information transmission in computer and communication networks

25. Competition Graphs: Communication Application • Digraph D: • Vertices are transmitters and • receivers. • Arc x to y if message sent at x • can be received at y. • Competition graph C(D): • a and b “compete” if there is a receiver x so that messages from a and b can both be received at x. • In this case, the transmitters a and b interfere.

26. Competition Graphs: Influence Application • Digraph D: • Vertices are people • Arc x to y if opinion of x • influences opinion of y. • Competition graph C(D): • a and b “compete” if there is a person x so that opinions from a and b can both influence x.

27. Structure of Competition Graphs In studying competition graphs in ecology, Joel Cohen (at the RAND Corporation) observed in 1968 that the competition graphs of real food webs that he had studied were always interval graphs. Interval graph: Undirected graph. We can assign a real interval to each vertex so that x and y are neighbors in the graph iff their intervals overlap.

28. Interval Graphs c a b e a b d e d c

29. Structure of Competition Graphs Cohen asked if competition graphs of food webs are always interval graphs. It is simple to show that purely graph-theoretically, you can get essentially every graph as a competition graph if a food web can be some arbitrary directed graph. It turned out that there are real food webs whose competition graphs are not interval graphs, but typically not for “homogeneous” ecosystems.

30. Aside: Boxicity and k-Suitable Sets of Arrangements More generally, Cohen studied ways to represent competition graphs as the intersection graphs of boxes in Euclidean space. The boxicity of G is the smallest p so that we can assign to each vertex of G a box in Euclidean p-space so that two vertices are neighbors iff their boxes overlap. Well-defined but hard to compute.

31. Aside: Boxicity and k-Suitable Sets of Arrangements A set L of linear orders on a set A of n elements is called k-suitable if among every k elements a1, a2, …, ak in A, for every i, there is a linear order in L in which ai follows all other aj. N(n,k) = size of smallest k-suitable set L on A. Notion due to Dushnik who applied it to calculate dimension of certain partial orders. Main results about N(n,k) due to Spencer (in his thesis).

32. Aside: Boxicity and k-Suitable Sets of Arrangements Let G be a graph and A be a set of q vertices. A is q-suitable if for every subset B of A with q-2 vertices, if a in A-B, there is a vertex x in G adjacent to all vertices of B and not to a. Theorem (Cozzens and Roberts 1984): If G has a 2p-suitable set of vertices, then boxicity of G is at least p. Proof uses N(2p,2p-1).

33. Aside: Boxicity and k-Suitable Sets of Arrangements Let G be a graph and A be a set of r vertices. A is (r,s)-suitable if for every subset B of A with s vertices, if a in A-B, there is a vertex x in G adjacent to all vertices of B and not to a. Theorem (Cozzens and Roberts 1984): If G has an (r,s)-suitable set of vertices, then boxicity of G is at least ceiling[N(r,s+1)/2].

34. Structure of Competition Graphs The remarkable empirical observation of Cohen’s that real-world competition graphs are usually interval graphs has led to a great deal of research on the structure of competition graphs and on the relation between the structure of digraphs and their corresponding competition graphs, with some very useful insights obtained. Competition graphs of many kinds of digraphs have been studied. In many of the applications of interest, the digraphs studied are acyclic.

35. Structure of Competition Graphs • We are interested in finding out what graphs are the competition graphs arising from semiorders.

36. Competition Graphs of Semiorders • Let (V,R) be a semiorder. • In the communication application: Transmitters and receivers in a linear corridor and messages can only be transmitted from right to left. • Because of local interference (“jamming”) a message sent at x can only be received at y if y is sufficiently far to the left of x.

37. Competition Graphs of Semiorders • In the computer/communication network application: Think of a hierarchical architecture for the network. • A computer can only communicate with a computer that is sufficiently far below it in the hierarchy.

38. Competition Graphs of Semiorders • The influence application involves a similar model -- the linear corridor is a bit far-fetched, but the hierarchy model is not. • We will consider more general situations soon. • Note that semiorders are acyclic. • So: What graphs are competition graphs of semiorders?

39. Graph-Theoretical Notation I7 Iq is the graph with q vertices and no edges:

40. Competition Graphs of Semiorders K5 U I7 Theorem: A graph G is the competition graph of a semiorder iff G = Iq for q > 0 or G = Kr Iq for r >1, q > 0. Proof: straightforward.

41. Competition Graphs of Semiorders • So: Is this interesting?

42. Boring!

43. Really boring!

44. Competition Graphs of Interval Orders A similar theorem holds for interval orders. D = (V,A) is an interval order if there is an assignment of a (closed) real interval J(x) to each vertex x in V so that for all x, y  V, (x,y)  A  J(x) is strictly to the right of J(y). Semiorders are a special case of interval orders where every interval has the same length.

45. Competition Graphs of Interval Orders Interval orders are digraphs without loops satisfying the first semiorder axiom: aRb & cRd  aRd or cRb

46. Competition Graphs of Interval Orders Theorem: A graph G is the competition graph of an interval order iff G = Iq for q > 0 or G = Kr Iq for r >1, q > 0. Corollary: A graph is the competition graph of an interval order iff it is the competition graph of a semiorder. Note that the competition graphs obtained from semiorders and interval orders are always interval graphs. We are led to generalizations.

47. The Weak Order Associated with a Semiorder Given a binary relation (V,R), define a new binary relation (V,) as follows: ab  (u)[bRu  aRu & uRa  uRb] It is well known that if (V,R) is a semiorder, then (V,) is a weak order. This “associated weak order” plays an important role in the analysis of semiorders.

48. The Condition C(p) We will be interested in a related relation (V,W): aWb  (u)[bRu  aRu] Condition C(p), p  2 A digraph D = (V,A) satisfies condition C(p) if whenever S is a subset of V of p vertices, there is a vertex x in S so that yWx for all y  S – {x}. Such an x is called a foot of set S.

49. The Condition C(p) Condition C(p) does seem to be an interesting restriction in its own right when it comes to influence. It is a strong requirement: Given any set S of p individuals in a group, there is an individual x in S so that whenever x has influence over individual u, then so do all individuals in S.

50. a b c e f d The Condition C(p) Note that aWc. If S = {a,b,c}, foot of S is c: we have aWc, bWc