On Balanced Signed Graphs and Consistent Marked Graphs

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On Balanced Signed Graphs and Consistent Marked
Graphs

• Fred S. Roberts
• DIMACS, Rutgers University
• Piscataway, NJ, USA

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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.

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Balanced
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Every cycle uses two vertices from the set of
vertices. Therefore, consistent.
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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 .
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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)
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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.
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Motivation Heiders Experiments
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unbalanced
balanced
balanced
unbalanced
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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.
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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
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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.
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It is easy to find the two sets if they exist.
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This can be made into a linear time algorithm to
check for balance (Maybee and Maybee 1983, Hansen
1978).
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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.
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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.
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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.
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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. So, balance theory is a type of
coloring theory.
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Testing the Balance Model
How does one test the model? Morisettes
experiments.
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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.
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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.
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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.
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Main Agenda of this Talk

• 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?

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Connection Among Balance, Consistency, and other
Graph-theoretical 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.
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So problem of checking for balance can be
reduced to problem of checking for consistency.
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Balance and Bipartiteness Graph G. Put a
sign on each edge, obtaining signed graph
H. Then G is bipartite iff H is balanced.
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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.
So, testing balancedness of signed graphs is
equivalent to testing bipartiteness of graphs.
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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.
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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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Thus, checking for consistency of marked graphs
is equivalent to checking for double
balancedness of bipartite graphs.
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Characterization of Consistency I GV
subgraph induced by all vertices GV
subgraph induced by all vertices. Theorem
(Acharya 1984, Rao 1984). If marked graph G is
consistent, then GV is bipartite. Moreover,
there is at most one edge between each component
of GV and each set in the bipartition of
each component of GV.
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GV is bipartite. There are 2 edges between
bipartite class b1,b2 and component
c1,c2 Therefore, not consistent.
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Characterization of Consistency I Theorem (R
and Xu 2003). Let G be a 2-connected marked
graph satisfying the necessary conditions of the
Theorem of Acharya and Rao. Shrink each component
of GV 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.
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Fundamental Cycles
G any graph. T a spanning tree.
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.
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h

a
g
-
-
T
i
f
b
-



e
c
d
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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.
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• 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.

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Characterization of Consistency II Hoedes
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.
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Characterization of Consistency II Variant of
Hoedes 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.
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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 congruent to 0
mod 4.
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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.
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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.
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C1, C2, C3 is a cycle basis
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Characterization of Consistency III The set of
fundamental cycles relative to a given spanning
tree forms a cycle basis. Here is a
generalization of Hoedes 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 consistency.
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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.
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3-Connected Markable Graphs Theorem (R 1995).
If graph G is 3-connected, then G is markable
iff it is bipartite. Proof Straightforward
using Mengers Theorem. Thus, we may
concentrate on graphs that are not 3-connected.
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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.
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Markable Blocks
Observation A graph is markable if every
block is markable. (Trivial by induction on
number of blocks.) The converse is false.
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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.
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G
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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 .
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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
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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.
This is markable
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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.

a
e
b
d
c
p vertices
q vertices

This is markable
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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.
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Open Questions
1. Give a structural characterization of
markable blocks with longer cycles.
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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.)
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Open Questions
3. Introduce similar degrees of markability.
E.g. What is smallest of vertices whose
removal results in a markable graph?
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