On Balanced Signed Graphs
and Consistent Marked Graphs
Fred S. Roberts
DIMACS, Rutgers University
Piscataway, NJ, USA
Signed Graphs and Marked Graphs
Data in the social sciences can often be modeled
using a signed graph: A graph where every edge
has a sign + or –.
Less widely used in the social sciences is a
marked graph, where every vertex has a sign +
or –.
A signed graph is balanced if every cycle has an
even number of – signs.
A marked graph is consistent if every cycle has
an even number of – signs.
-
a
c
+
-
b
d
Balanced
3
+
+
Every cycle uses two vertices from the set of –
vertices.
Therefore, consistent.
4
We will speak of the sign of a path or cycle as
being + if it has an even number of – signs, and –
otherwise .
So: a signed graph is balanced iff every cycle is +.
A marked graph is consistent iff every cycle is +.
5
Balance: Sociological Motivation
Small group is “balanced” if
it works well together, lacks
tension.
Signed graphs used to
“explicate” this concept.
Vertices = people
Edges = strong relationship
Sign = positive or negative
(likes/dislikes, lies/tells truth to,
associates with/avoids)
6
Balance: Sociological Motivation
Balanced signed graphs introduced as model for
balanced small groups by Cartwright and Harary
in early 1950s.
Evidence that small group is balanced iff its
corresponding signed graph is balanced.
7
Motivation: Heider’s Experiments
I:
+
+
II:
-
+
III:
-
+
+
+
IV:
-
-
-
8
unbalanced
balanced
I:
+
+
II:
-
+
III:
-
-
+
+
IV:
-
-
+
-
balanced
unbalanced
9
Balance: Other Applications
Political science: international
relations
Vertices = countries, signs =
allies/enemies
Analysis of literature: At point of
“tension,” tension is resolved by
changing to balance.
10
Balance: Other Applications
Sociology: social justice,
analysis of inequities.
Economics: Analysis of
structure of mathematical
models for large complex
systems such as those used to
analyze energy and economic
systems
11
Characterization of Balanced
Signed Graphs
Theorem (Harary 1954): A signed graph G is
balanced iff the set of vertices of G can be
partitioned into two disjoint sets such that each +
edge joins vertices in the same set and each – edge
joins vertices in different sets.
12
a
-
+
b
+
c
+
+
-
d
g
e
-
-
f
+
+
h
13
a
-
+
b
+
c
+
+
-
d
g
e
-
-
f
+
+
h
14
It is easy to find the two sets if they exist.
a
-
+
b
+
c
+
+
-
d
g
e
-
-
f
+
+
h
15
This can be made into a linear time algorithm to
check for balance (Maybee and Maybee 1983,
Hansen 1978).
16
Idealized Political Party Structure
Idealized party structure:
Whenever members of the
same party have a dialogue,
they agree; whenever members
of different parties have a
dialogue, they disagree.
17
Idealized Political Party Structure
Idealized party structure:
Whenever members of the
same party have a dialogue,
they agree; whenever members
of different parties have a
dialogue, they disagree.
Theorem: A political system is balanced iff it has
an idealized two party structure.
18
Idealized Political Party Structure
Idealized party structure:
Whenever members of the
same party have a dialogue,
they agree; whenever members
of different parties have a
dialogue, they disagree.
Theorem: A political system is balanced iff it has
an idealized two party structure.
One party could be empty.
19
Balance and Graph Coloring
Balanced signed graphs are a generalization of
bipartite graphs or 2-colorable graphs. (Given
undirected graph, let all signs be –.)
There is (as in political party example) also
interest in finding ways to partition the vertices of
a signed graph into more than two sets (more than
two colors) so that all + edges join vertices of the
same set and all – edges join vertices of different
sets.
20
So, balance theory is a type of coloring theory.
Testing the Balance Model
How does one test the model?
Morisette’s experiments.
21
Consistent Marked Graphs:
Communication Networks Motivation
Binary messages sent through
a network.
Messages reversed at – vertices.
In a consistent marked graph: If a
message is sent from x to y through two
different vertex-disjoint paths and x and y have
the same sign, then y will receive the same
message no matter which path is followed.
22
-
+
x
+
+
y
-
Assuming that x and y also either reverse the
message or keep it intact, the hypothesis that x
and y have the same sign can be removed.
23
Consistent Marked Graphs
Social Networks Interpretation: Vertices
represent people who always lie or always tell the
truth.
Characterization Problem: There is no simple
structural characterization of consistent marked
graphs analogous to the 2-class structural theorem
for balance.
24
Main Agenda of this Talk
1. Results about graphs that have
consistent markings
2. Efficient Algorithms for
determining if a marked graph has
a consistent marking
3. The “markability problem”: When can we mark
a graph with signs on vertices (at least one –) to
obtain consistency?
25
Connection Among Balance,
Consistency, and other Graphtheoretical Notions:
Balance and Consistency
Signed graph G.
Put a + sign on each vertex and insert a vertex
with – sign in each negative edge.
Get a marked graph H.
G is balanced iff H is consistent.
26
+
G
+
-
+
-
+
+
+
-
H
+
-
+
+
+
So: problem of checking for balance can be
reduced to problem of checking for
consistency.
27
Balance and Bipartiteness
Graph G.
Put a – sign on each edge, obtaining signed graph H.
Then G is bipartite iff H is balanced.
G
H
- 28
Balance and Bipartiteness
Signed graph G.
Replace each + edge by two consecutive – edges to
get signed graph H.
Then G is balanced iff H is balanced iff H is
bipartite.
+
G
+
-
+
-
+
H
So, testing balancedness of
signed graphs is equivalent to
testing bipartiteness of graphs.
- - - - - 29
Double Balancedness
Bipartite graph G. If each cycle has length 0 mod
4, say G is double balanced.
Bipartite graph G. Assign + signs to one bipartite
class and – signs to the other, getting marked graph
H.
Then, G is double balanced iff H is consistent.
30
Double Balancedness
Marked graph G.
Insert a + vertex in each edge incident to 2 –
vertices and insert 3 vertices with signs –, +, – in
each edge incident to 2 + vertices, getting marked
graph H.
Then H is bipartite with one part + and the other
part –.
Moreover, G is consistent iff H is double
balanced.
Proof is by induction on number of edges of G
that join vertices of the same sign.
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G
-
-
+
+
-
-
+
H
+
-
+
+
- +
-
Thus, checking for consistency of marked graphs is
equivalent to checking for double balancedness of
bipartite graphs.
32
Characterization of Consistency I
G[V+] = subgraph induced by all + vertices
G[V–] = subgraph induced by all – vertices.
Theorem (Acharya 1984, Rao 1984). If marked
graph G is consistent, then G[V–] is bipartite.
Moreover, there is at most one edge between each
component of G[V+] and each set in the
bipartition of each component of G[V–].
33
a-
b1
1
-
+c
1
G
a2
-
-
b2
c2
+
G[V–] is bipartite.
There are 2 edges between bipartite class {b1,b2}
and component {c1,c2}
Therefore, not consistent.
34
Characterization of Consistency I
Theorem (R and Xu 2003). Let G be a 2connected marked graph satisfying the necessary
conditions of the Theorem of Acharya and Rao.
Shrink each component of G[V–] that is not a
single vertex into a single edge, to get marked
graph H. Then G is consistent iff H is.
This is sometimes helpful in reducing size of
graphs in checking for consistency.
35
Fundamental Cycles
G any graph. T a spanning tree.
h+
+a
G
-
f
d+
e+
i
b
+
c
-
+g
-
Adding any edge of G joining 2 vertices of T
gives rise to a unique cycle of G called a
fundamental cycle relative to T.
36
h+
+a
T
-
f
d+
e+
i
b
+
c
-
+g
-
37
Characterization of Consistency II
Theorem (Hoede 1992). Let G be a marked graph
and T be a spanning tree of G. Then G is
consistent iff
1). Every fundamental cycle relative to T is +;
and
2). Each common path of a pair of fundamental
cycles relative to T has end vertices with the same
sign.
38
Characterization of Consistency II
In the example:
•The 2 fundamental cycles relative to T are +
•There is only one common path of the pair of
fundamental cycles, namely, d, i, h.
•This path has both end vertices with the same
sign.
Thus, G is consistent.
39
Characterization of Consistency II
Hoede’s Theorem provides an O(m2n) algorithm
to check if a marked graph is consistent.
(m = number of edges, n = number of vertices)
R and Xu (2003) give an O(mn) algorithm.
40
Characterization of Consistency II
Variant of Hoede’s Theorem:
Theorem (R and Xu 2003). Let G be a marked
graph and T be a spanning tree of G. Then G is
consistent iff
1). Every fundamental cycle relative to T is +;
and
2). Each 3-connected vertex pair in G has the
same sign.
In example: h and d are the only 3-connected
pair.
41
Characterization of Consistency II
Because checking for consistency of a marked graph
is equivalent to checking for double balancedness of a
bipartite graph, the following can be thought of as a
bipartite analogue of R-Xu Theorem:
Theorem (Conforti and Rao 1987). Let G be a
bipartite graph and T be a spanning tree of G. Then
G is double balanced iff
1). Every fundamental cycle relative to T has length
congruent to 0 mod 4 ;
and
2). Any cycle that is a symmetric difference of 2
fundamental cycles relative to T has length
42
congruent to 0 mod 4.
Characterization of Consistency II
Conforti-Rao Theorem leads to an O(m2n)
algorithm to determine if a bipartite graph is
double balanced.
R-Xu (2003) provide an O(mn) algorithm.
43
Cycle Bases
Recall that a set K of cycles in a graph is a cycle
basis if every cycle of G can be expressed as a
symmetric difference of cycles in K and K is
minimal.
44
a
c
b
c
e
d
a
b
b
bd
d
C1
e
d
G
a
c
c
==

=CC
C121
CC
C233
c
a
b
c
b
c
d
e
d
e
C2
C3
C1, C2, C3 is a cycle basis
45
Characterization of Consistency III
The set of fundamental cycles relative to a given
spanning tree forms a cycle basis. Here is a
generalization of Hoede’s Theorem:
Theorem (R and Xu, 2003). Let G be a marked graph
and B be any cycle basis of G. Then G is consistent
iff
1). Every cycle in B is +;
and
2). Each 3-connected vertex pair in G has the same
sign.
This theorem leads to the O(mn) algorithm to test for
46
consistency.
The Markability Problem
Given G unmarked.
Can always mark it consistently: Use all + signs.
What if at least one – sign is required?
Then even K4 cannot be consistently marked.
G is markable if it can be consistently marked
using at least one – sign.
Problem: When is a graph markable?
Problem: Find a “structure theorem” that
characterizes markable graphs.
47
3-Connected Markable Graphs
Theorem (R 1995). If graph G is 3-connected,
then G is markable iff it is bipartite.
Proof: Straightforward using Menger’s Theorem.
Thus, we may concentrate on graphs that are not 3connected.
48
Markable Blocks
Recall that a block is a connected graph with more
than one vertex and no cutpoints.
A block in a graph is a maximal subgraph that is a
block.
A graph is 2-connected iff it is a block consisting
of more than one edge.
49
Markable Blocks
Observation: A graph is markable if every block is
markable. (Trivial by induction on number of
blocks.)
The converse is false.
50
+
+
G
+
+
-
+
G is markable
There
is a non-markable block K4
51
Markable Blocks
Observation (Trotter): A structure theorem for
markable graphs that are not blocks is impossible.
Given any graph G, G is an induced subgraph of
a markable graph.
52
-
-
+
+
+
+
+ + +
+ +
G
53
Markable Blocks
Some examples of Markable Blocks.
K(p,q)
Complete bipartite graph with p vertices in one
class and q in the other.
K(2,q) is markable.
Make the class of 2 vertices + and the other class –
.
54
K(2,q) + e2
Start with K(2,q). Add an edge between the vertices in the
class of 2
-
-
+
+
…
+++
q vertices
+
This is markable
55
J(p,q): Start with 4-cycle a,b,c,d,a. Add q vertices
adjacent to a and d and p vertices adjacent to a and c.
+
+
…
q vertices
-
d
c
a
b
+
-
- -
-
This is markable
56
L(p,q): Start with 5-cycle a,b,c,d,e,a. Add p
vertices adj. to a and c and q vertices adj. to c and
e.
-
…
p vertices
+
--
a
b
c
+
This is markable
+
e
d
-
-
57
Markable Blocks
Theorem (R): Suppose that G is a block with no
cycle of length greater than 5. Then G is
markable iff G is
K2;
K3 ;
K(2,q) for q  2;
K(2,q) + e2 for q  2;
J(p,q) for p  0, q  1;
or
L(p,q) for p,q  0.
58
Open Questions
1. Give a structural characterization of markable
blocks with longer cycles.
59
Open Questions
2. Lots of work has been done on degrees of
balance. Introduce similar notions of degree of
consistency.
(E.g.: line index for balance = smallest # edges
whose removal gives balance. Vertex index of
consistency = smallest # vertices whose
removal gives consistency.)
60
Open Questions
3. Introduce similar degrees of markability. E.g.:
What is smallest # of vertices whose removal
results in a markable graph?
61