Signed Graphs - PowerPoint Presentation

Slide1

Signed Graphs

Christopher Muir

CS 494Slide2

Table of Contents

Motivation

*

Definitions

*

History

*

Theory

*

Open Problems

*

Applications

*

Homework

*

ReferencesSlide3

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?Slide4

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 graphThe sign of a cycle or path is defined as the product of its edgesA graph can have a marking on its nodes, assigned by the function μ: V (+,-)

 Slide5

HistorySlide6

Fritz Heider

Austrian Psychologist

Creator of Balance Theory

“The enemy of my enemy is my friend

”

P-O-X Model

The Psychology of Interpersonal RelationsSlide7

Frank Harary

American Mathematician

Father of Signed Graph Theory

Extended the work of

Heider

to

a theory of Balanced GraphsFundamental Theorem of Signed GraphsWrote one of the earliest textbooks on graph theorySlide8

Good Will Hunting

Find all

homeomorphically

irreducible trees on 10 verticesSlide9

TheorySlide10

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 triadSlide11

P-O-X modelSlide12

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?

Slide13

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. Slide14

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?Slide15

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.

 Slide16

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.  Slide17

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. Slide18

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.

Slide19

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, E1

E2 = E.

For any two vertices

u,w

E1 we have one of two casesCase 1: u=v or w=v, the edge uw is positive by definitionCase 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 positiveFor 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

 Slide20

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 balancedSlide21

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.Slide22

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.Slide23

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. Slide24

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 setA 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

)

 Slide25

Switching Example

Show that the two graphs are switching equivalentSlide26

Switching Example

τ

(1,2,3,4) = (+,-,+,-)Slide27

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) = +-- = +Slide28

IS a graph Balanced?

(

Harary

and

Kabell

1979)

Proposed a polynomial time algorithm to determine whether a graph is balanced.

Correspondence TheoremFor 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 Slide29

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 balancedSlide30

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)Slide31

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

 Slide32

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 Slide33

ILP Approach

is the weight of the corresponding

edge

and

/

are binary variables

 Slide34

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 ∑

 Slide35

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  Slide36

What is the maximum frustration of any Cycle?Slide37

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

 Slide38

Open ProblemsSlide39

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

 Slide40

ApplicationsSlide41

International Relations

Political scientists use the original ideas of

Heider

to help explain how relations between countries evolve overtime.Slide42

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 factionsSlide43

Portfolio Balancing

Signed graphs are used to analyze the level of hedging in a portfolio.

V

ertices 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. Slide44

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 clustersSlide45

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 indexSlide46

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 perturbationsSlide47

Homework and ReferencesSlide48

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

 Slide49

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_Hararyhttps://en.wikipedia.org/wiki/Spin_glasshttps://en.wikipedia.org/wiki/Ising_modelHarary, Frank. On the notion of balance of a signed graph. Michigan Math. J. 2 (1953)Slide50

References

Structural balance: a generalization of

Heider's

theory. Cartwright,

Dorwin

;

Harary

, Frank Psychological Review, Vol 63(5), Sep 1956http://math.sfsu.edu/beck/papers/signedgraphs.slides.pdfR. Crowston, G. Gutin, M. Jones and G. Muciaccia, Maximum Balanced Subgraph Problem Parameterized Above Lower BoundF. H¨uffner, N. Betzler, and R. Niedermeier. Optimal edge deletions for signed graph balancingB. DasGupta, G. A. Enciso, E. D. Sontag, and Y. Zhang. Algorithmic and complexity results for decompositions of biological networks into monotone subsystemsSlide51

References

Garry

Bowlin

, Maximum Frustration in Bipartite Signed Graphs

Networks

, Crowds, and Markets: Reasoning about a Highly Connected World. By David

Easley

and Jon Kleinberg Chapter 5T. Antal,P. L. Krapivsky, and S. Redner, Dynamics of Social Balance on NetworksOsamu Katai and Sousuke Iwai Studies on the Balancing, the Minimal Balancing, and the Minimum Balancing Processes for Social Groups with Planar and Nonplanar Graph StructuresFrank Harary, Ming-Hiot Lim, and Donald C. Wunsch, Signed graphs for portfolio analysis in risk management, IMA J Management Math (2002)Slide52

References

Hila Becker A Survey of Correlation Clustering

Falk

H¨uffner

,

Nadja

Betzler, and Rolf Niedermeier, Separator-Based Data Reduction for Signed GraphThomas Zaslavsky, Balanced Decompositions of a Signed Graph http://www.openproblemgarden.org/op/signing_a_graph_to_have_small_magnitude_eigenvaluesYezhou Wu, Dong Ye, Wenan Yang, and Cun-Quan Zhang Nowhere-zero 3 flows in signed graphs