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Unformatted text preview: tworks As Graphs
We will typically (mathematically) represent networks with graphs,
which formalize the patterns of links between diﬀerent units, or nodes.
Graphs can be directed or undirected, depending on what kinds of
relationship they represent. For example, blog links are directed.
They can also be weighted or unweighted, depending on whether links
diﬀer in terms of their importance, capacity, likelihood of
materializing, etc.. For example, weak vs. strong ties in referrals.
At the simplest level, a directed (unweighted) graph is
G = (N , E )
N = the set of nodes in the graph (e.g., in the blogs example, the nodes are the weblogs) E = the set of edges, linking nodes in the graph. 25 Networks: Lecture 1 A Little Bit of Analysis Networks As Graphs (continued)
We write j ∈ N if j is a node in this network, and (i , j ) ∈ E if there is a link from i to j . If this is a directed graph, then this does not necessarily imply that (j , i ) ∈ E . For undirected graphs, sometimes we use the notation {i , j } ∈ E to denote an edge between i and j , but we will not do so in this lecture. We can also use the notation gij = 1 if (i , j ) ∈ E and gij = 0
otherwise (and use g as the matrix of gij ’s to do matrix algebra to
derive properties of networks).
For a weighted graph, we could also use the notation gij > 0 if
(i , j ) ∈ E and gij = 0 otherwise.
In this case, the magnitude of gij would correspond to the strength of
the link. 26 Networks: Lecture 1 A Little Bit of Analysis The Strong Triadic Closure
Recall job referral patterns.
Let us represent a weighted (undirected) graph in an economical
fashion as “augmented” undirected graph, G = (N , E , E � ), where
E � ⊂ E represents “strong ties”. Thus, (i , j ) ∈ E means that i and j
are acquaintances, while (i , j ) ∈ E � means that i and j are close
friends.
The strong triadic closure property is the following:
if (i , j ) ∈ E � and (i , k ) ∈ E � , then (j , k ) ∈ E . (i , j ) ∈ E �
j i (i , k ) ∈ E � (j , k ) ∈ E
k Figure: Triadic Closure
27 Networks: Lecture 1 A Little Bit of Analysis The Strong Triadic Closure (continued)
Naturally, this property in this strong form is often violated, so one
may wish to have a “probabilistic” version of this, where we would
say that the conditional probability that (j , k ) ∈ E given (i , j ) ∈ E �
and (i , k ) ∈ E � is greater than the unconditional probability that
(j , k ) ∈ E , i.e.,
�
�
P (j , k ) ∈ E  (i , j ) ∈ E � and (i , k ) ∈ E � > P ((j , k ) ∈ E ) .
An alternative probabilistic version would be
P((j , k ) ∈ E and (i , j ) ∈ E �  (i , k ) ∈ E � ) > P((j , k ) ∈ E and (i , j ) ∈ E �  (i , k ) ∈ E \ E � ) 28 Networks: Lecture 1 A Little Bit of Analysis The Strong Triadic Closure (continued)
Now suppose that i can get a job with manager k with a referral
through j if j is close friends with k but i and k do not know each
other.
Assume the probabilistic version of the strong triadic closure. Then close friends are less useful for ﬁnding jobs than acquaintances. More formally, let {(i , j ) ∈ R } be the event that i obtained the job through a r...
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This document was uploaded on 03/18/2014 for the course EECS 6.207J at MIT.
 Fall '09
 Acemoglu

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