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303
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Computer Science
Date
Feb 20, 2024
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Uploaded by HighnessTankLobster37
Social and Information Networks
University of Toronto CSC303
Winter/Spring 2023
Week 2: January 16-20
1/56
This week’s agenda
The Strength of Weak Ties
▶
Triadic closure
⋆
Defnition
⋆
Clustering coefcient
⋆
Driving forces
▶
Granovetter’s Thesis
⋆
Strong & Weak Ties
⋆
Bridges
⋆
Strong Triadic Closure and it’s implications
▶
Social Capital
⋆
Defnition
⋆
Bonding vs. Bridging capital
⋆
Relationship with graph structure
▶
Determining Strong Edges
⋆
Sintos & Tsaparas algorithm
⋆
Rozenshtein algorithm
2/56
Chapter 3: Strong and Weak Ties
There are two themes that run throughout this chapter.
1
Strong vs. weak ties and
“the strength of weak ties”
is the specifc
defning theme of the chapter. The chapter also starts a discussion of
how networks evolve.
2
The larger theme is in some sense
“the scientifc method”
.
▶
Formalize concepts, construct models of behaviour and relationships,
and test hypotheses.
▶
Models are not meant to be the same as reality but to
abstract the
important aspects
of a system so that it can be studied and analysed.
▶
See the discussion of the
strong triadic closure
property in section 3.2
of text (pages 53 and 56 in my online copy).
Informally
strong ties
: stronger links, corresponding to friends
weak ties
: weaker links, corresponding to acquaintances
3/56
Triadic closure (undirected graphs)
B
A
C
G
F
E
D
(a)
Before
B
-
C
edge forms.
B
A
C
G
F
E
D
(b)
After
B
-
C
edge forms.
Figure:
The formation of the edge between
B
and
C
illustrates the efects of
triadic closure, since they have a common neighbour
A
. [E&K Figure 3.1]
Triadic closure
: mutual “friends” of say
A
are more likely (than
“normally”) to become friends over time.
▶
How do we measure the extent to which triadic closure is occurring?
▶
How can we know why a new friendship tie is formed?
(Friendship ties
can range from “just knowing someone” to “a true friendship” .)
4/56
Measuring the extent of triadic closure
The
clustering coefcient
of a node
A
is a way to measure (over time)
the extent of triadic closure (perhaps without understanding why it is
occurring).
Let
E
be the set of an undirected edges of a network graph. (Forgive
the abuse of notation where in the previous and next slide
E
is a node
name.) For a node
A
, the
clustering coefcient
is the following ratio:
vextendsingle
vextendsingle
braceleftbig
(
B
,
C
)
∈
E
: (
B
,
A
)
∈
E
and (
C
,
A
)
∈
E
bracerightbigvextendsingle
vextendsingle
vextendsingle
vextendsingle
braceleftbig
{
B
,
C
}
: (
B
,
A
)
∈
E
and (
C
,
A
)
∈
E
bracerightbigvextendsingle
vextendsingle
The numerator is the number of all
edges
(
B
,
C
) in the network such
that
B
and
C
are adjacent to (i.e. mutual friends of)
A
.
The denominator is the total number of all
unordered pairs
{
B
,
C
}
such that
B
and
C
are adjacent to
A
.
5/56
Example of clustering coefcient
B
A
C
G
F
E
D
(a)
Before new edges form.
The clustering coefcient of node
A
in Fig. (a) is
1
/
6
(since there is
only
the single edge (
C
,
D
)
among the six pairs of friends:
{
B
,
C
}
,
{
B
,
D
}
,
{
B
,
E
}
,
{
C
,
D
}
,
{
C
,
E
}
, and
{
D
,
E
}
)
6/56
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