Christopher Muir CS 494 Table of Contents Motivation Definitions History Theory Open Problems Applications Homework References Motivation Graphs show the relationships between different objects ID: 559777
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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