MATH 17, Winter 2017
COMPLEX NETWORKS
An Introduction to Mathematics Beyond Calculus
Nishant Malik
Department of Mathematics
Dartmouth College
February 22, 2017
Effects of Homophily
Segregation along Racial and ethnic lines in cities
David Easley and Jon Kleinberg; Networks, Crowds, and Markets: Reasoning About a
Highly Connected World; Cambridge University Press (2010)
Effects of Homophily
The tendency of people to live in racially homogeneous neighborhoods produces
spatial patterns of segregation that are apparent both in everyday life and when
superimposed on a map.
David Easley and Jon Kleinberg; Networks, Crowds, and Markets: Reasoning About a
Highly Connected World; Cambridge University Press (2010)
108
CHAPTER
4. NETWORKS
IN THEIR SURROUNDING CONTEXTS
Effects
of Homophily
(a) Chicago, 1940
(b) Chicago, 1960
Effects of Homophily
NewYork in 2000
http://complexity.stanford.edu/blog/schelling-segregation-model
Schelling’s Spatial Model of Segregation
Nobel prize winner in
Economic Sciences
for "having enhanced our understanding of conflict and cooperation through game-theory analysis"’.
Thomas Crombie Schelling
(14 April 1921 – December 21 2016 )
American economist and professor of foreign policy, national
security, nuclear strategy, and arms control at the School of
Public Policy at University of Maryland, College Park
Schelling’s Spatial Model of Segregation
X
Two types individuals/agents
(rich/poor or black/white)
O
Each individual wants to have at
least t other agents of its own type as neighbors.
Schelling’s Spatial Model of Segregation
1. Unsatisfied individuals move in a sequence of rounds as
follows in each round, in a given order, each unsatisfied moves
to an unoccupied cell where it will be satisfied (details can
differ with similar qualitative behavior).
2. These new locations may cause different individuals to be
unsatisfied, leading to a new round of movement.
Schelling’s Spatial Model of Segregation
O
X
X
X
O
O
X
X
O
O
t=3
114
Schelling’s Spatial Model of Segregation
CHAPTER 4. NETWORKS IN THEIR SURROUNDING CONTEXTS
Segregation pattern
in large scale simulation of
Schelling Model
t=4
150 by 150 grid
(a) After 20 steps
(b) After 150 steps
As the rounds of movement
progress, large homogeneous
regions on the grid grow at the
expense of smaller, narrower
regions.
(c) After 350 steps
(d) After 800 steps
Schelling’s Spatial Model of Segregation
•
Spatial segregation is taking place even though no individual
agent is actively seeking it.
Schelling’s Spatial Model of Segregation
In general what Schelling’s model implies
Race/Ethnicity/Native language
Immutable
Characteristics
Decision where to live
Mutable
Characteristics
Schelling’s Spatial Model of Segregation
In general what Schelling’s model implies
Immutable
Characteristics
Mutable
Characteristics
Mutable characteristics
become highly correlated to
immutable characteristics
Segregation
Schelling’s Spatial Model of Segregation
non-spatial manifestation of the same effect, in
which beliefs and opinions become correlated across
racial or ethnic lines, and for similar underlying
reasons: as homophily draws people together along
immutable characteristics, there is a natural
tendency for mutable characteristics to change in
accordance with the network structure.
Schelling’s Spatial Model of Segregation
Rigorous mathematical analysis of the Schelling model appears
to be quite difficult, and is largely an open research question
Structural Balance:
Positive and Negative Relationships
Structural Balance: Positive and Negative Relationships
A
+
A
-
B
A and B are friends
B
A and B are enemies
Structural Balance: Positive and Negative Relationships
A
+
+
A, B and C are
mutual friends
BALANCED
C
+
B
Structural Balance: Positive and Negative Relationships
A
+
-
A and C are friends
and they have common
enemy B
BALANCED
C
-
B
Structural Balance: Positive and Negative Relationships
A is friends with B
and C but B and C
are enemies
A
+
C
+
-
B
psychological “stress”
or “instability” due to the
B—C relationship
UNBALANCED
Structural Balance: Positive and Negative Relationships
A
-
C
A,B and C are
enemies of each other
-
B
there may be forces
motivating two of the three
to “team up” against the third
UNBALANCED
Structural Balance: Positive and Negative Relationships
A
A
+
+
C
+
+
-B C-
BALANCED
+++ 3 plus
+-- 1 plus
A
-
-B
+
+
C
A
-
-
-
-B C
-
-
UNBALANCED
++- 2 plus
--- 0 plus
-B
Structural
nced
triangles.
Balance: Positive and Negative Relationships
ning Structural Balance for Networks. So far we have been talking about stru
balance for groups of three nodes. But it is easy to create a definition that natural
A complete graph is balanced if every one its triangle is
ralizes this to complete graphs on an arbitrary number of nodes, with edges labeled b
and balanced!
’s.
pecifically, we say that a labeled complete graph is balanced if every one of its triangl
lanced — that is, if it obeys the following:
Structural Balance Property: For every set of three nodes, if we consider the three
edges connecting them, either all three of these edges are labeled +, or else exactly
one of them is labeled +.
or example, consider the two labeled four-node networks in Figure 5.2. The one o
eft is balanced, since we can check that each set of three nodes satisfies the Structur
nce Property above. On the other hand, the one on the right is not balanced, since amon
hree nodes A, B, C, there are exactly two edges labeled +, in violation of Structur
nce. (The triangle on B, C, D also violates the condition.)
Our definition of balanced networks here represents the limit of a social system that h
Structural Balance: Positive and Negative Relationships
A
-
+
-
C
+ D
B
-
BALANCED
or
UNBALANCED?
Structural Balance: Positive and Negative Relationships
A
+
-
C
- D
B
+
+
BALANCED
or
UNBALANCED?
4
Structural Balance:
Positive and Negative Relationships
CHAPTER 5. POSITIVE AND NEGATIVE RELATIONSHIP
tween the groups. The surprising fact is the following: these are the only ways to ha
balanced network. We formulate this fact precisely as the following Balance Theore
oved by Frank Harary in 1953 [97, 204]:
Balance Theorem: If a labeled complete graph is balanced, then either all pairs
of nodes are friends, or else the nodes can be divided into two groups, X and Y ,
such that every pair of nodes in X like each other, every pair of nodes in Y like
each other, and everyone in X is the enemy of everyone in Y .
he Balance Theorem is not at all an obvious fact, nor should it be initially clear why
true. Essentially, we’re taking a purely local property, namely the Structural Balan
operty, which applies to only three nodes at a time, and showing that it implies a stro
bal property: either everyone gets along, or the world is divided into two battling faction
We’re now going to show why this claim in fact is true.
oving the Balance Theorem. Establishing the claim requires a proof: we’re going
ppose we have an arbitrary labeled complete graph, assume only that it is balanced, a
nclude that either everyone is friends, or that there are sets X and Y as described in t
ween the groups. The surprising fact is the following: these are the only ways to hav
alanced
network. Balance:
We formulatePositive
this fact precisely
as the following
Balance Theorem
Structural
and Negative
Relationships
ved by Frank Harary in 1953 [97, 204]:
Balance Theorem: If a labeled complete graph is balanced, then either all pairs
of nodes are friends, or else the nodes can be divided into two groups, X and Y ,
such that every pair of nodes in X like each other, every pair of nodes in Y like
each other, and everyone in X is the enemy of everyone in Y .
Balance Theorem is not at all an obvious fact, nor should it be initially clear why i
rue. Essentially, we’re taking a purely local property, namely the Structural Balanc
perty, which applies to only three nodes at a time, and showing that it implies a stron
al property: either everyone gets along, or the world is divided into two battling factions
We’re now going to show why this claim in fact is true.
Mutual
Mutual
Mutual
Antagonism
Friends
Friends
oving the Balance Theorem. Establishing
the claim requires a proof: we’re going t
Between
pose we have
an arbitrary labeled complete graph, assume only that
it is balanced, an
inside
inside
clude that either everyone is friends, or that
there are sets X and Y as described in th
Sets
m. Recall that we worked through a proof in Chapter 3 as well, when we used simpl
umptions about triadic closure in a social network to conclude all local bridges in th
work must be weak ties. Our proof here will be somewhat longer, but still very natura
Set X
Set Y
Structural Balance: Positive and Negative Relationships
Proving the balance theorem
?
B
?
C
Set X = friends of A
+
+
-
A
-
D
?
E
Set Y = enemies of A
Structural Balance: Positive and Negative Relationships
Proving the balance theorem
-
B
+
C
Set X = friends of A
+
+
-
A
-
D
+
E
Set Y = enemies of A
Applications of Structural Balance:
International Relations
USSR
?
China
?
?
?
India
-
?
USA
India and Pakistan
go to war in 1972,
whom
China, USSR and
USA will support?
Pakistan
?
Separation of
Bangladesh from
Pakistan in 1972
Applications of Structural Balance:
International Relations
-
USSR
?
-
USSR was China’s
enemy
USSR was USA’s
enemy
China and USA
were trying to be
friends
? ? +
?
India
China
-
?
USA
Pakistan
?
Separation of
Bangladesh from
Pakistan in 1972
Applications of Structural Balance:
International Relations
-
USSR
?
-
India and China
are enemies
? ? +
-
India
China
-
?
USA
Pakistan
?
Separation of
Bangladesh from
Pakistan in 1972
Applications of Structural Balance:
International Relations
-
USSR
?
-
China is
Pakistan’s friend
? + +
-
India
China
-
?
USA
Pakistan
?
Separation of
Bangladesh from
Pakistan in 1972
Applications of Structural Balance:
International Relations
-
USSR
?
-
USSR is
Pakistan’s enemy
- + +
-
India
China
-
?
USA
Pakistan
?
Separation of
Bangladesh from
Pakistan in 1972
Applications of Structural Balance:
International Relations
-
USSR
+
-
- + +
-
India
China
-
?
USA
USSR is
India’s
friend
Pakistan
?
Separation of
Bangladesh from
Pakistan in 1972
Applications of Structural Balance:
International Relations
-
USSR
+
-
- + +
-
India
China
-
-
Pakistan
+
USA
USA
supports
Pakistan
Separation of
Bangladesh from
Pakistan in 1972
Applications of Structural Balance:
International Relations
5.3. APPLICATIONS OF STRUCTURAL BALANCE
127
China
GB
AH
Fr
GB
Ge
Ru
GB
Fr
Fr
It
GB
Ru
It
Ru
It
The evolution
of alliances
in Europe,
1872-1907.
Ge
Ru
GB
Ge
It
(c) German-Russian Lapse 1890
AH
Fr
AH
Fr
It
(b) Triple Alliance 1882
AH
Ge
GB
Ge
Ru
(a) Three Emperors’ League 1872–
81
AH
AH
Fr
Ge
Ru
It
Applications of Structural Balance:
International Relations
Fr
Ge
Ru
It
Ge
Ru
It
(d) French-Russian Alliance 1891–
94
Ru
China
GB
GB
Ge
It
(e) Entente Cordiale 1904
Ge
It
(c) German-Russian Lapse 1890
AH
Fr
Ru
Fr
It
(b) Triple Alliance 1882
AH
Fr
Ge
Ru
(a) Three Emperors’ League 1872–
81
GB
Fr
AH
Fr
Ge
Ru
It
(f) British Russian Alliance 1907
Figure 5.5: The evolution of alliances in Europe, 1872-1907 (the nations GB, Fr, Ru, It, Ge,
and AH are Great Britain, France, Russia, Italy, Germany, and Austria-Hungary respectively). Solid dark edges indicate friendship while dotted red edges indicate enmity. Note
how the network slides into a balanced labeling — and into World War I. This figure and
example are from Antal, Krapivsky, and Redner [20].
The evolution of alliances in Europe, 1872-1907.
Applications of Structural Balance:
Trust and Distrust and on-Line Ratings
Epinions: Product Rating site
User can express evaluations of different products and
trust or distrust of other users
Applications of Structural Balance:
Trust and Distrust and on-Line Ratings
Epinions: Product Rating site
User
User
Trust (+)
A Distrust (-)
?
B
Applications of Structural Balance:
Trust and Distrust and on-Line Ratings
Epinions: Product Rating site
Similarities and differences with structural balance of friend
and enemy dichotomy
A
Trust (+)
Trust (+)
Prediction from
structural balance
C
B
Trust (+)
Applications of Structural Balance:
Trust and Distrust and on-Line Ratings
Epinions: Product Rating site
Similarities and differences with structural balance of friend
and enemy dichotomy
A
Distrust (-)
or
Trust (+)
?
C
Distrust (-)
B
Distrust (-)
Applications of Structural Balance:
Trust and Distrust and on-Line Ratings
Epinions: Product Rating site
Similarities and differences with structural balance of friend
and enemy dichotomy
A
Structural Balance
Trust (+)
C
Distrust (-)
B
Distrust (-)
Applications of Structural Balance:
Trust and Distrust and on-Line Ratings
Epinions: Product Rating site
Similarities and differences with structural balance of friend
and enemy dichotomy
A
Structural Balance
Trust (+)
C
Distrust (-)
B
Distrust (-)
Applications of Structural Balance:
Trust and Distrust and on-Line Ratings
Epinions: Product Rating site
Similarities and differences with structural balance of friend
and enemy dichotomy
What if A is more
knowledgeable
about a product than B
and similarly B is more
knowledgeable than C
A
Distrust (-)
Distrust (-)
C
B
Distrust (-)
Applications of Structural Balance:
Trust and Distrust and on-Line Ratings
Epinions: Product Rating site
Similarities and differences with structural balance of friend
and enemy dichotomy
If A and C has same political
view and B has alternative
views and the product is a
book
A
Distrust (-)
Trust (+)
C
B
Distrust (-)