Signed Graphs

1. Signed Graphs Christopher Muir CS 494

2. Table of Contents Motivation * Definitions * History * Theory * Open Problems * Applications * Homework * References

3. Motivation Graphs show the relationships between different objects Different types of graphs exist to show different types of relations, directed graphs for example show directed relations What about when the objects demonstrate opposite types of relations between members?

4. What are signed graphs? • A signed graph is a graph in which a sign is mapped to every edge • Denoted normally with a +/- sign on edges, with solid and dotted lines, or / signs on the edges • Typically denoted by ∑ = G(V, E, σ) • σ is a function such that σ: E (+,-) referred to as the signature of the graph • The sign of a cycle or path is defined as the product of its edges • A graph can have a marking on its nodes, assigned by the function μ: V (+,-)

5. History

6. Fritz Heider • Austrian Psychologist • Creator of Balance Theory • “The enemy of my enemy is my friend” • P-O-X Model • The Psychology of Interpersonal Relations

7. Frank Harary • American Mathematician • Father of Signed Graph Theory • Extended the work of Heider to • a theory of Balanced Graphs • Fundamental Theorem of • Signed Graphs • Wrote one of the earliest textbooks on • graph theory

8. Good Will Hunting Find all homeomorphically irreducible trees on 10 vertices

9. Theory

10. P-O-x Model • Originally used to describe the way psychological consistency • is obtained • P, X, and O represent some combination of individuals and objects • The three agents have different relations to each other, either positive or negative • Typically referred to as a triad

11. P-O-X model

12. P-o-x model Example Imagine that you are person P and that O is someone, whom you think highly of, now imagine X is a presidential candidate you dislike, but X vehemently endorsees O. What do you suspect would happen?

13. Cont. Heider describes this situation as imbalanced, and he suggests that a system such as this will change to achieve balance In this case, he suggests that you will either accept your friends endorsement of candidate X or you will come to dislike O because of his endorsement, which ever is the easiest way to obtain balance.

14. What is Balance? • A cycle is said to be balanced if it has a positive sign • A cycle has a positive if it has an even number of - edges • A graph is said to be balanced if all of its cycles are positive • When is an all negative graph balanced?

15. Harary’s Theorem (1). A signed graph is balanced if and only if, for every u,v V, all paths connecting u and v have the same sign. (2). A signed graph is balanced if and only if, V can be partitioned into two subgraphs, such that vertices within a subgraph are connected by a positive edge and vertices in separate subgraphs are connected by a negative edge.

16. Proof of 1 Necessity Let u,v V and p,q are paths connecting these two points. The removal of any common edges in these paths results in a collection of edge disjoint cycles. Divide these cycles into two paths, p1 and p2. Since the graph is balanced the two paths must have the same sign. Now adding these subpaths with the shared edges in p and q, the resulting paths will have the same sign.

17. Proof of 1 Sufficiency Given every u,v V, all paths p,q connecting u and v have the same sign. All cycles containing u and v will be positive. Meaning that all cycles will be balanced.

18. Proof of 2 To prove this you first prove the following A complete signed graph is balanced if and only if, V can be partitioned into two subgraphs, such that vertices within a subgraph are connected by a positive edge and vertices in separate subgraphs are connected by a negative edge.

19. Proof of 2 Necessity Take a vertex v, define E1 as the set of all vertices positively connected to v and E2 as the set of all vertices negatively connected to v, E1E2 = E. For any two vertices u,w E1 we have one of two cases Case 1: u=v or w=v, the edge uw is positive by definition Case 2: v≠w and v≠u, by definition edges uv and wv are positive, so for the 3-cycle to be balanced uw must also be positive For any two vertices u,w E2 edges uv and wv are negative. It follows that edge uw must be positive for the 3-cycle to be balanced

20. Proof of 2 Sufficiency If the graph meets the conditions of the theorem, it is clear that for every cycle in ∑, there will be an even number of E1-E2 edges. Lemma The subgraph of a balanced graph is balanced

21. Proof of 2 Now we are properly equipped to prove the theorem Necessity Imagine a graph partitioned into two sets of vertices. For size 0 and 1, we can partition this according to the theorem. Now take a graph with some number of edges connecting vertices as stated in the theorem that and is also balanced. Adding an edge to two non adjacent vertices as the theorem prescribes will not result in an unbalanced graph as all cycles will still have an even number of edges connecting the two sets.

22. Proof of 2 Sufficiency Take a graph, partition it as the theorem prescribes. Now you can add edges of the appropriate sign to the graph until it a complete graph. From the previous proof this graph is balanced and from the lemma, the original graph must also be balanced.

23. Other Balance Theorems (Sampathkumar 1984) A signed graph is balanced if and only if there exists a marking μ such that for all uv E, σ(uv) = μ(u)μ(v) (Zaslavsky 1984) A signed graph can be switched to an all positive signed graph if and only if it is balanced.

24. Switching • A switching function τ : V (+,-), is a marking on G, such that (uv) = τ(v)(uv)τ(u) • Another view is taking a subset of the vertices U, and forming a cut • [, and switching the sign of all edges in the cut set • A graph switched by τ is denoted as • Two graphs and are switching equivalent, ~ if they have the same underlying graph and there exists a τ such that = • A switching class for a ∑ := (: ~ ∑ for some τ)

25. Switching Example Show that the two graphs are switching equivalent

26. Switching Example τ(1,2,3,4) = (+,-,+,-)

27. Switching Example τ(1)σ(12)τ(2) = ++- = - τ(2)σ(24)τ(4) = --+ = + τ(3)σ(34)τ(4) = -++ = - τ(1) σ(13)τ(3) = +-- = +

28. IS a graph Balanced? (Harary and Kabell 1979) Proposed a polynomial time algorithm to determine whether a graph is balanced. Correspondence Theorem For each marked graph M, their exists a single balanced signed graph S. For each connected S, their exists two marking M and M`, which are signed reversals of each other

29. Harary-Kabell Algorithm Input: Signed graph S Step 1: Select spanning tree T Step 2: Root T at an arbitrary point v Step 3: Mark v positive Step 4: Select an unsigned point adjacent in T to a signed point Step 5: Mark this point the sign of the product of the sign of the previously signed point to which it is adjacent in T and the sign of the edge connecting them Step 6: Are their remaining unsigned vertices in T? Yes- Go to step 4 No- Go to step 7 Step 7: Is there an untested edge of S – E(T) Yes- Go to step 8 No- Go to step 11 Step 8: Select an untested edge of S – E(T) Step 9: Is the sign of the edge equal to the product of the signs of its vertices Yes- Go to step 7 No- Go to step 10 Step 10: Stop, S is unbalanced Step 11: Stop, S is balanced

30. Frustration Index • The frustration index is the minimum number of edges whose deletion from ∑ results in a balanced graph • Denoted I(∑) = n, where n represents the number of edges that need removal • At least as hard as the maximum cut problem, if the graph is all negative the problems are equivalent • Solvable in polynomial time if the graph is planar or embeddable on the torus • (Barahona 1982) and (Katai and Iwai 1978)

31. Maximum Balanced Subgraph Problem • Complement of the frustration index problem, the removal of the minimum number of frustrated edges results in a maximum balanced subgraph • NP-Hard • Every ∑ with n vertices and m edges has a balanced subgraph with at least edges

32. Maximum Balanced Subgraph Problem (DasGupta 2007) Determined a polynomial time approximation algorithm that solves approximately within 87.9% optimality , where L is the number of - edges (Hüffner 2007) Developed a data reduction scheme and utilized a method based on a parameterized algorithm for the edge bipartization problem to find exact solutions to instances approximated by DasGuspta , k is the maximum amount of edge deletions

33. ILP Approach is the weight of the corresponding edge and / are binary variables

34. ILP Approach The program can be further refined in the following manner This adds further cutting planes by marking all of the odd cycles of length n in ∑

35. Most frustrated graphs Find the maximum I(∑) over all possible σ (Petersdorf 1966) has a uniquely maximum frustration index of , achieved when has an all negative signing

36. What is the maximum frustration of any Cycle?

37. Most frustrated Graphs (Bowlin 2012) Upper bound for complete bipartite graphs equality if r is a positive integer multiple of Also found exact solutions for

38. Open Problems

39. Open Problems • For a k-regular graph, is there a signing, replacing some 1’s in the adjacency matrix with -1’s, such that the eigenvalues have an upper bound of • Every oriented signed graph that allows for a nowhere-zero integer flow allows for a nowhere-zero 6 flow • What other genus allow for polynomial time answers to the frustration index problem

40. Applications

41. International Relations Political scientists use the original ideas of Heider to help explain how relations between countries evolve overtime.

42. Antal, Krapivsky, and Redner Model (Antal, Krapivsky, and Redner 2005) Local Triad Dynamics: using some probability p that represents whether or not its easier to gain negative or positive relations, uses time steps to show how triads attempt to attain balance Constrained Triad Dynamics: randomly selects edges in a graph, either switching the sign if it makes it more balanced, switching it if it is neutral with probability p = ½, and nothing if changing the sign would result in a less balanced graph In both models, over a long time for large N, graphs enter a state of “paradise” or form two opposed factions

43. Portfolio Balancing Signed graphs are used to analyze the level of hedging in a portfolio. Vertices represent securities and edges represent the positive or negative correlations between the securities. To protect from sudden swings in value, it is desirable to have a balanced graph with at least one negative edge, the specific ratio of + and – edges depends on the investor.

44. Data Clustering Signed graphs appear in data clustering under the idea of correlation clustering. Correlation clustering is a form of data clustering in which the data is partitioned into clusters that maximizes the number of positive edges within the partitions and the number of negative edges between clusters This is different from other methods in that it doesn’t require a predetermined number of clusters

45. Spin Glasses An Ising model is a lattice where each vertex represents an atom and each edge represents the interaction between that atom and its neighbors in the lattice A spin glass is a special case where a combination of + and – signs are on the edges The lowest energy configuration is one that has the minimum frustration index

46. Gene Regulatory Networks Some claim that regulatory networks, where inhibiting connections between genes are negative edges and activating connections are positive, form balanced graphs By breaking down a regulatory network into a monotone subsystem, a maximum balanced subgraph, it is possible to study well behaved reactions to perturbations

47. Homework and References

48. Homework • Find the six switching classes of the Petersen graph • Find the most frustrated signing of • Prove or disprove, the frustration index of a graph is equal to the sum of the frustration index of its blocks

49. References https://www.math.binghamton.edu/zaslav/Bsg/sgbgprobs.html https://en.wikipedia.org/wiki/Signed_graph https://en.wikipedia.org/wiki/Fritz_Heider https://en.wikipedia.org/wiki/Frank_Harary https://en.wikipedia.org/wiki/Spin_glass https://en.wikipedia.org/wiki/Ising_model Harary, Frank. On the notion of balance of a signed graph. Michigan Math. J. 2 (1953)

50. References Structural balance: a generalization of Heider's theory. Cartwright, Dorwin; Harary, Frank Psychological Review, Vol 63(5), Sep 1956 http://math.sfsu.edu/beck/papers/signedgraphs.slides.pdf R. Crowston, G. Gutin, M. Jones and G. Muciaccia, Maximum Balanced Subgraph Problem Parameterized Above Lower Bound F. H¨uffner, N. Betzler, and R. Niedermeier. Optimal edge deletions for signed graph balancing B. DasGupta, G. A. Enciso, E. D. Sontag, and Y. Zhang. Algorithmic and complexity results for decompositions of biological networks into monotone subsystems