Announcements
CS 103: Lecture 3 Positive and Negative
Relationships and Structural Balance
I
HW 1 posted today, due next Thursday in class
I
I
Dan Sheldon
I
I
I
Start early!
TA Office Hours
Areeba Tuesday 7–8pm, CS lounge (Clapp 222A)
Tiffany Wednesday 8–9pm, CS lounge (Clapp 222A)
Blog posts announced next week
September 17, 2015
Plan for today
Positive and Negative Relationships
So far edges in a network have been implicitly positive. But you
can have enemies too. . .
5.1. STRUCTURAL BALANCE
121
A
I
Review / finish weak ties. . .
I
Structural Balance
A
+
+
B
+
C
+
B
(a) A, B, and C are mutual friends: balanced.
C
-
+
(b) A is friends with B and C, but they don’t get
along with each other: not balanced.
Theory
of Structural
Balance
5.1.
STRUCTURAL
BALANCE
121
A
I
I
I
A
What happens if network has positive/negative edges?
+
A
What local configurations
do we expect to see?A
How does this impact the global structure of the network?
+
+
B
-
C
(c) A and B are friends with C as a mutual en-
Examples
emy:
B balanced.
Examples
C
+
B
+
+
C
-
(d) A, B, and C are mutual enemies: not balanced. B
C
-
Figure 5.1: Structural balance: Each labeled triangle must have 1 or 3 positive edges.
(a) A, B, and C are mutual friends: balanced.
5.1. STRUCTURAL BALANCE
121
A
+
B
I
I
• Similarly, there
A are sources of instability in a configuration where Aeach of A, B, and C
are mutual enemies (as in Figure 5.1(d)). In this case, there would be forces motivating
two of the three people to “team up” against the third (turning one of the three edge
+ to a +).
labels
A
+
+
C
B
+
C
+
-
(a) A, B, and C are mutual friends: balanced.
(b) A is friends with B and C, but they don’t get
Left = balanced. Mutual friendsalong with each other: not balanced.
Right = not balanced. A is stressed out
A
B
A
-
+
-
-
C
(c) A and B are friends with C as a mutual enemy: balanced.
B
Based on this reasoning, we will refer to triangles with one or three +’s as balanced, since
they are free of these sources of instability, and we will refer to triangles with zero or two
+’sBas unbalanced. The argument of
is that because unbalanced
C structural balance theorists
B
C
- of stress or psychological dissonance, people strive -to minimize them in
triangles are sources
their personal relationships, and hence they will be less abundant in real social settings than
(c) A and B are friends with C as a mutual en-
balanced.
Iemy:
Left
= balanced.
(d) A, B, and C are mutual enemies: not bal-
anced.
Two friends with
common enemy
IFigure
Right
not balanced.
Enemy
my enemy
is my
5.1: =
Structural
balance: Each
labeledoftriangle
must have
1 or 3friend
positive(two
edges.
can align against third)
friends (thus turning the B-C edge label to +); or else for A to side with one of B or
C against the other (turning one of the edge labels out of A to a ).
-
-
(b) A is friends with B and C, but they don’t get
with
each
not with
balanced.
friends (thus turning the B-C edge label toalong
+); or
else
for other:
A to side
one of B or
C against the other (turning one of the edge labels out of A to a ).
C
(d) A, B, and C are mutual enemies: not balanced.
Figure 5.1: Structural balance: Each labeled triangle must have 1 or 3 positive edges.
friends (thus turning the B-C edge label to +); or else for A to side with one of B or
C against the other (turning one of the edge labels out of A to a ).
• Similarly, there are sources of instability in a configuration where each of A, B, and C
are mutual enemies (as in Figure 5.1(d)). In this case, there would be forces motivating
two of the three people to “team up” against the third (turning one of the three edge
labels to a +).
Based on this reasoning, we will refer to triangles with one or three +’s as balanced, since
they are free of these sources of instability, and we will refer to triangles with zero or two
+’s as unbalanced. The argument of structural balance theorists is that because unbalanced
triangles are sources of stress or psychological dissonance, people strive to minimize them in
their personal relationships, and hence they will be less abundant in real social settings than
Balanced Triangles
How can we extend this to bigger graphs?
Definition:
I
I
A triangle
with 1 or 3 ‘+’ edges is balanced (left column)
5.1. STRUCTURAL BALANCE
121
A triangle with 0 or 2 ‘+’ edges is not balanced (right column)
5.1. STRUCTURAL BALANCE
A
+
A
+
+
+
B
B
+
(a) A, B, and C are mutual friends: balanced.
A
-
C
B
AH
127
-
C
B B are friends with C as a mutual
C en(c) A and
emy: balanced.
B
-
(b) A is friends with B and C, but they don’t get
A balanced.
along with each other: not
GB
Examples
emy:
Ge
balanced.
C
GB
Ge
C
balanced
-
+
+
B
+
+
D
C
-
-
B
C
D
+
-
D
not balanced
not balanced
balanced
Figure 5.2: The
labeled
four-node
complete
graph
on the left
is balanced;
one on thethe one on the
Figure
The labeled
four-node
complete
graph
on the leftthe
is balanced;
triangles
is5.2:
balanced.
right is not.
AH right is not.
balanced triangles.
balanced triangles.
Fr
Ge
Discussion
Figure 5.1: Structural balance: Each labeled triangle must have 1 or 3 positive edges.
friends (thus turning the B-C edge label to +); or else for A to side with one of B or
C against the other (turning one of the edge labels out of A to a ).
For each of these graphs, answer if it is balanced or not
Ru
D
A
- +
(d) A, B
B, and C are -mutual enemies:C not balanced.
anced.
Fr
+
A
+
-
B
+
A
Definition: a complete graph is balanced if every one of its
-
-
Figure
5.1:B Structural
balance:
triangle
1 or 3 enemies:
positivenot
edges.
(c) A and
are friends with
C as a Each
mutuallabeled
en(d) A, B,must
and Chave
are mutual
bal-
Fr
- -
+
C
(b) A isBfriends with B and C, but theyCdon’t get
along with each other: -not balanced.
-
+
B
AH
A
+
. APPLICATIONS OF STRUCTURAL
BALANCE
+
A
A
GB
CHAPTER 5. POSITIVE AND NEGATIVE RELATIONSHIPS
-
-
+
B and C are mutual friends: balanced.
C
(a) A, B,
122
+
A
+
C
122
121
A
+
Let’s focus on complete graphs (edges between every pair of nodes)
I Students in a classroom
I Diplomatic relations between countries
CHAPTER 5. POSITIVE AND NEGATIVE RELATIONSHIPS
friends (thus turning the B-C edge label to +); or else for A to side with one of B or
• Similarly, there are sources of instability in a configuration where each of A, B, and C
It
Ru one of the edge labelsIt out of A to a ).
C
against
the other (turning
Examples
on
board
are
mutual
enemies (as in Figure 5.1(d)). In this case, there would be forces motivating
Ru
of the there
three are
people
to “team
up” against
the third (turning
• two
Similarly,
sources
of instability
in a configuration
where one
eachofofthe
A, three
B, andedge
C
labels
to a +).
are mutual
enemies (as in Figure 5.1(d)). In this case, there would be forces motivating
What is the minimum number of labels to change to make this
Defining Structural
for Networks.
far we haveSo
been
DefiningBalance
Structural
Balance for So
Networks.
far talking
we haveabout
been structalking about structural balance tural
for groups
of three
nodes.of But
is easy But
to create
a definition
that
naturally that naturally
balance
for groups
threeit nodes.
it is easy
to create
a definition
generalizes this
to
complete
graphs
on
an
arbitrary
number
of
nodes,
with
edges
labeled
generalizes this to complete graphs on an arbitrary number of nodes, with by
edges labeled by
It +’s and ’s.
+’s and ’s.
Specifically, weSpecifically,
say that a we
labeled
complete
graph
is
balanced
if
every
one
of
its
triangles
say that a labeled complete graph is balanced if every one of its triangles
is balanced —isthat
is, if it—obeys
balanced
that the
is, iffollowing:
it obeys the following:
two of the three people
“team up” against the third
(turning one of the(c)
three edge
Three Emperors’
League
1872–
(b)wetoTriple
1882
Lapse 1890
Based
on this reasoning,
will refer toAlliance
triangles with one
or three +’s as balanced,German-Russian
since
graph
balanced?
GB
AH
Fr
labels to a +).
they are free of these sources of instability, and we will refer to triangles with zero or two
+’s as
unbalanced.
The argument
structural
balance
theorists
is that
because
unbalanced
Based
on this reasoning,
we willofrefer
to triangles
with
one or three
+’s
as balanced,
since
triangles
stress orofpsychological
dissonance,
people
strive to minimize
they are are
freesources
of theseofsources
instability, and
we will refer
to triangles
with zero them
or twoin
their
personal
relationships,
and hence
they willbalance
be less theorists
abundantisinthat
realbecause
social settings
than
+’s as
unbalanced.
The argument
of structural
unbalanced
GB
AH
triangles are sources of stress or psychological dissonance, people strive to minimize them in
their personal relationships, and hence they will be less abundant in real social settings than
Ge
Fr
Ge
UCTURAL BALANCE
Ru
GB
It
AH
French-Russian Alliance 1891–
GB
Fr
127
Ru
It
GB
AH
(e) Entente Cordiale 1904
Ru
Structural
BalanceaProperty:
For every
set of
nodes, if if
we every
considerone
Definition:
complete
graph
isthree
balanced
of
its the three
Structural
Balance
Property:
For
every
set of three
nodes, the
if wethree
consider
edges connecting them, either all three of these edges are labeled +, or else exactly
connecting them, either all three of these edges are labeled +, or else exactly
triangles edges
is balanced.
one of them is labeled +.
one of them is labeled +.
AH
This is an extreme definition. How can we relax it?
For example, consider the two labeled four-node networks in Figure 5.2. The one on
For example, consider the two labeled four-node networks in Figure 5.2. The one on
the left is balanced, since we can check that each set of three nodes satisfies the Structural
the left is balanced, since we can check that each set of three nodes satisfies the Structural
Not
all
pairs
arethefriends
(any
BalanceIProperty
above.
On of
the nodes
other hand,
one on the
right graph)
is not balanced, since among
Balance Property above. On the other hand, the one on the right is not balanced, since among
GeI nodes
the three
A,of
B, the
C, there
are exactly
edges labeled +, in violation of Structural
Most
triangles
aretwo
balanced
the three
nodes
A, B, C, there
are exactly two edges labeled +, in violation of Structural
Balance. (The triangle on B, C, D also violates the condition.)
Balance. (The triangle on B, C, D also violates the condition.)
Our definition of balanced networks here represents the limit of a social system that has
Our definition of balanced networks here represents the limit of a social system that has
eliminated all unbalanced triangles. As such, it is a fairly extreme definition — for example,
eliminated all unbalanced triangles. As such, it is a fairly extreme definition — for example,
It one could instead propose a definition which only required that at least some large percentage
one could instead propose a definition which only required that at least some large percentage
of all triangles were balanced, allowing a few triangles to be unbalanced. But the version
of all triangles were balanced, allowing a few triangles to be unbalanced. But the version
with all triangles balanced is a fundamental first step in thinking about this concept; and
with all triangles balanced is a fundamental first step in thinking about this concept; and
(f) British Russian Alliance 1907
Fr
Fr
Ge
ure
5.5: The evolutionGeof alliances
in Europe, 1872-1907
(the nations GB, Fr, Ru, It, Ge,
d AH are Balanced
Great Britain,
France,Intuition
Russia, Italy, Germany, and Austria-Hungary Balanced
respec- Network: Intuition
Network:
ely). Solid dark edges indicate friendship while dotted red edges indicate enmity. Note
Ru
It
Ru
It
network
“look —
like”?
say more
than
w the networkWhat
slidesdoes
intoa balanced
a balanced
labeling
andCan
intoweWorld
War
I. This figure Clearly,
and Any graph that looks like this is balanced:
5.2. CHARACTERIZING THE STRUCTURE OF BALANCED NETWORKS
triangles
are balanced”?
Discuss.[20].
mple are from“all
Antal,
Krapivsky,
and Redner
(b) Triple Alliance 1882
123
(c) German-Russian Lapse 1890
We already saw one way for a graph to be balanced.
GB
AH
GB
AH
s China’s enemy, China was India’s foe, and India had traditionally bad relations with
Fr
Fr
Ge
kistan.
Since the U.S. Ge
was at that
time improving its
relations with China, it supported
enemies of China’s enemies. Further reverberations of this strange political constellation
ame inevitable:
North
Vietnam made
friendly Itgestures toward India, Pakistan severed
Ru
It
Ru
lomatic relations with those countries of the Eastern Bloc which recognized Bangladesh,
d China
vetoed
the acceptance
Bangladesh
into the1907
U.N.”
(e) Entente
Cordiale
1904
(f) of
British
Russian Alliance
mutual friends
inside X
set X
mutual
antagonism
between
sets
mutual friends
inside Y
set Y
Figure 5.3: If a complete graph can be divided into two sets of mutual friends, with complete
sets of
mutual
with
between
them)this is the only
mutual(Two
antagonism
between
thefriends
two sets,
thenantagonim
it is balanced.
Furthermore,
Does and
this give
you any
Antal, Krapivsky,
Redner
use hints?
the shifting alliances preceding World War I asway
another
for a complete graph to be balanced.
nces in Europe, 1872-1907 (the nations GB, Fr, Ru, It, Ge,
mple of structural balance in international relations — see Figure 5.5. This also reinforces
nce, Russia, Italy, Germany, and Austria-Hungary respecthat structural
balance
not indicate
necessarily
a good
thing: since its global outcome is
efact
friendship
while dotted
red is
edges
enmity.
Note
as we will see next, it turns out to have very interesting mathematical structure that in fact
en two implacably
for balance
in a system can sometimes
alanced
labeling — opposed
and into alliances,
World Warthe
I. search
This figure
and
helps to inform the conclusions of more complicated models as well.
ky, and
[20].a hard-to-resolve opposition between two sides.
seen
as aRedner
slide into
5.2
Characterizing the Structure of Balanced Networks
At a general level, what does a balanced network (i.e. a balanced labeled complete graph)
look like? Given any specific example, we can check all triangles to make sure that they
Structural Balance Theorem (Harary 1953)
Review of Proof Idea (for notes)
5.2. CHARACTERIZING THE STRUCTURE OF BALANCED NETWORKS
125
Theorem: all balanced graphs look like the picture we just saw.
I
I
I
B
Two sets X and Y
All pairs within X, Y are friends
Everyone in X enemies with everyone in Y
+
?
?
A
+
Proof on board
D
?
-
C
Significance: a purely “local” concept (balanced triangles)
determines the global structure
E
friends of A
enemies of A
Figure 5.4: A schematic illustration of our analysis of balanced networks. (There may be
5.1. STRUCTURAL BALANCE
121
other nodes not illustrated here.)
A
(iii) Every node in X is an enemy of every node in Y .
+
Let’s argue that+each of these conditions
is in fact true+for our choice +of X and Y . This will
Generalizations
mean that
and Y do satisfy the
of the claim, and will complete the proof. The
Relax
theX Definition
of conditions
Balance
rest of the argument, establishing (i), (ii), and (iii), is illustrated schematically in Figure 5.4.
C
C
For (i), Bwe know that A is friends
with every Bother node- in X. How
about two other
+
nodes in X (let’s call them B and C) — must they be friends? We know that A is friends
Here(a) are
the
two
types
of
triangles
we
declared
imbalanced:
B, and
and CC,
are so
mutual
(b)121
A is friends
withother,
B and C,then
but they
if Bfriends:
and balanced.
C were enemies
of each
A,don’t
B, get
and C would
5.1. STRUCTURAL BALANCEwith bothA,B
along with each other: not balanced.
form a triangle with two + labels — a violation of the balance condition. Since we know
the network is balanced, this can’t happen, so it must be that B and C in fact are friends.
A names
A
A
A of any two nodes in X, we have
Since B and C were the
concluded that every two
nodes in X are friends.
+
- in Y (let’s call them
+
+
+
+
Let’s try the same kind of argument for (ii). Consider any two nodes
D and E) — must they be friends? We know that A is enemies with both D and E, so if D
and E were enemies of each other, then A, D, and E would form a triangle with no + labels
B
C
C
B
B balance condition.
C
—Ca violation
of the
Since we Bknow the network
is balanced,
this can’t
+
happen, so it must be that D and E in fact are friends. Since D and E were the names of
(c) A
and(b)B A
are
C and
as
a C,
mutual
en-that
A, B,two
and nodes
C are mutual
not baltwo
nodes
in
, we with
have
concluded
every
in Y enemies:
are friends.
(a) A, B, and C are mutual friends:any
balanced.
isYfriends
friends
with
B
but they
don’t(d)
get
emy: balanced.
anced.
along with each other: not balanced.
Finally, let’s
try condition (iii). Following the style of our arguments for (i) and (ii),
AreFigure
these
the
5.1: really
Structural
balance:
Each alabeled
mustithave
3 positive
consider
a node
in X (call
ifsame?
B) and
node triangle
in Y (call
D) 1—ormust
theyedges.
be enemies? We
A
A enemies with D, so if B and D were friends, then a, B, and
know A is friends with B and
What do you like least about this?
Generalizations:
I
I
I
Relax definition of balance (board work)
Not all edges present (main ideas)
Most, but not all triangles balanced (reading)
-
+
B
Definition: A network is weakly balanced if it has
like this:
+
B
• Similarly, there are sources of instability in a configuration where each of A, B, and C
are mutual enemies (as in Figure 5.1(d)). In this case, there would be forces motivating
two of the three people to “team up” against the third (turning
B one of the three edge
?
labels to a +).
+
C
-
+
Based on this reasoning, we will refer to triangles with one or three +’s +as balanced, since
they are free of these sources of instability, and we will refer to triangles with zero or two
?
A
+’s as unbalanced. The argument of structural balance theorists is that because unbalanced
triangles are sources of stress or psychological dissonance, people strive to minimize them in
their personal relationships, and hence they will be less abundant in real social
+ settings than
C
C are mutual friends:
balanced.
I
A is friends with B and C, but they don’t get
Two positive(b)
edges,
one negative
along with each other: not balanced.
Question: what does the global structure of a weakly balanced
network look like?
A
A
-
-
-
C
friends with C as a mutual en-
B
C
).
with one or three +’s as balanced, since
they are free of these sources of instability, and we will refer to triangles with zero or two
Figure 5.1: Structural balance: Each
triangle must
have 1 or of
3 positive
edges.
+’slabeled
as unbalanced.
The argument
structural
balance theorists is that because unbalanced
121
triangles are sources of stress or psychological dissonance, people strive to minimize them in
no triangles
Remember
our proof.
changes
this inpicture?
their personal relationships,
and What
hence they
will be less in
real social settings than
5.2.edge
CHARACTERIZING
STRUCTURE
OFabundant
BALANCED
NETWORKS
friends (thus turning the B-C
label to +); or else for ATHE
to side
with one of B or
C against the other (turning one of the edge labels out of A to a ).
A
+
C against the other (turning one of the edge labels out of A to a
• Similarly, there are sources of instability in a configuration where each of A, B, and C
are mutual enemies (as in Figure 5.1(d)). In this case, there would be forces motivating
B
C
- “team up” against the third (turning one of the three edge
two of the three people to
labels to a +).
Weakly Balanced Networks
RAL BALANCE
A
-
Intuition: three negative edges (right) is much more stable than
friends (thusone
turning
the B-C edge label to +); or else for A to side with one of B or
two positive,
- negative -(left)
(c) A and B are friends with C as a mutual en(d) A, B, and C are mutual enemies: not balBased on
this reasoning, we will refer to triangles
emy: balanced.
anced.
Weakly Balanced Networks
C
A
-
-
125
D
?
-
friends of A
E
enemies of A
Figure 5.4: A schematic illustration of our analysis of balanced networks. (There may be
other nodes not illustrated here.)
(iii) Every node in X is an enemy of every node in Y .
C
(d) A, B, and C are mutual enemies: not balanced.
Let’s argue that each of these conditions is in fact true for our choice of X and Y . This will
mean that X and Y do satisfy the conditions of the claim, and will complete the proof. The
rest of the argument, establishing (i), (ii), and (iii), is illustrated schematically in Figure 5.4.
For (i), we know that A is friends with every other node in X. How about two other
nodes in X (let’s call them B and C) — must they be friends? We know that A is friends
with both B and C, so if B and C were enemies of each other, then A, B, and C would
form a triangle with two + labels — a violation of the balance condition. Since we know
the network is balanced, this can’t happen, so it must be that B and C in fact are friends.
Weak Structural Balance
Other Networks
Theorem: a complete graph that is weakly balanced looks like this:
5.4. A WEAKER FORM OF STRUCTURAL BALANCE
129
ADVANCED MATERIAL: GENERALIZING THE DEFINITION OF STRUCTURAL BALANCE133
What if5.5.the
graph is not complete? (strong balance)
1
set V
+
+
mutual
friends
inside V
set W
set X
mutual friends
inside X
mutual friends
inside W
+
2
3
-
+
4
-
5
6
+
-
mutual
antagonism
between
all sets
-
7
+
11
-
+
12
set Y
-
10
+
+
mutual
friends
inside Y
8
-
9
14
13
mutual
friends
inside Z
-
15
set Z
Figure 5.6: A complete graph is weakly balanced precisely when it can be divided into multiple sets of
mutual friends, with complete mutual antagonism between each pair of sets.
Is it balanced?
(possible
tocomplete,
dividewe into
mutual
friends
Figure 5.8: In graphs
that are not
can stilltwo
define sets
notions of
of structural
balance
when the edges that are present have positive or negative signs indicating friend or enemy
with antagonism
between them)
relations.
Proof sketch on board
far more expert than C, and so should distrust C as well.
Ultimately, understanding how these positive and negative relationships work is important for understanding the role they play on social Web sites where users register subjective
evaluations of each other. Research is only beginning to explore these fundamental questions,
including the ways in which theories of balance — as well as related theories — can be used
to shed light on these issues in large-scale datasets [274].
Examples and Odd Cycles
5.4
A Weaker Form of Structural Balance
In studying models of positive and negative relationships on networks, researchers have also
formulated alternate notions of structural balance, by revisiting the original assumptions we
Board work: intuition, odd cycles
1. It applies only to complete graphs: we require that each person know and have an
opinion (positive or negative) on everyone else. What if only some pairs of people
know each other?
Structural Balance in General Networks
2. The Balance Theorem, showing that structural balance implies a global division of the
world into two factions [97, 204], only applies to the case in which every triangle is
balanced. Can we relax this to say that if most triangles are balanced, then the world
can be approximately divided into two factions?
5.5. ADVANCED MATERIAL: GENERALIZING THE DEFINITION OF STRUCTURAL BALANCE133
Theorem: a general network is structurally balanced if any only if
it has no odd cycles
In the two parts of this section, we discuss a pair of results that address these questions. The
1 involving the notion of breadth-first search from
first is based on a graph-theoretic analysis
Chapter 2, while the second is typical
+ of a+style of proof known as a “counting argument.”
Throughout this section, we will focus on the original definition of structural balance from
+
2
3
Sections 5.1 and 5.2, rather than the weaker version from Section 5.4.
-
Definition: an odd cycle is a cycle with an odd number of
negative edges
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+
4
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5
6
+
-
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7
8
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9
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10
+
+
+
12
11
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14
13
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15
This example
isgraphs
notthatbalanced.
Can
you
see notions
why?of structural balance
Figure 5.8: In
are not complete,
we can
still define
when the edges that are present have positive or negative signs indicating friend or enemy
relations.
1. It applies only to complete graphs: we require that each person know and have an
opinion (positive or negative) on everyone else. What if only some pairs of people
know each other?
2. The Balance Theorem, showing that structural balance implies a global division of the
world into two factions [97, 204], only applies to the case in which every triangle is
balanced. Can we relax this to say that if most triangles are balanced, then the world
can be approximately divided into two factions?
In the two parts of this section, we discuss a pair of results that address these questions. The
first is based on a graph-theoretic analysis involving the notion of breadth-first search from
Chapter 2, while the second is typical of a style of proof known as a “counting argument.”
Throughout this section, we will focus on the original definition of structural balance from
Sections 5.1 and 5.2, rather than the weaker version from Section 5.4.