Graphs
Graph: Nodes and Edges
• A graph is a way of specifying relationships
among a collection of items.
• Graph G=(V, E) – ordered pair
V- set of nodes
E - set of edges
b
a
c
g
f
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e
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Graphs as Models of Networks
Graphs as Models of Networks (cont.)
Paths
• A path is a sequence of nodes with each
consecutive pair in the sequence is connected
by an edge. Length of the path is the number of
edges in the path.
V
U
X
W
Y
Z
Cycles
• A cycle is a path with at least three edges, in
which the first and last nodes are the same,
but otherwise all nodes are distinct.
Connectivity
• A graph is connected if, for every pair of nodes,
there is a path between them
Connected component
• A connected component of a graph is a subset
of the nodes such that (i) every node in the
subset has a path to every other and
(ii) the subset is not part of some larger set
Giant Components
• A giant component is a connected component
that contains a significant fraction of all nodes
Breadth-First Search
• Popular algorithm for computing distances
from a source node
S
The Small-World Phenomenon
• Social network (nodes are people, link if two
people are friends).
• Paths are surprisingly short!
• Six degrees of separation.
Six degrees of separation
• Stanley Milgram and his colleagues in the 1960s
Microsoft Instant Messenger research
Erdos number
Bacon number
• Three students at Albright College in
Pennsylvania (1994)
• Graph of movie actors and actresses: nodes
are performers, an edge connects two
performers if they’ve appeared together in a
movie
• Bacon number is the distance from the
performer to Kevin Bacon
Large Network Data Sets
• Why to study a particular network data set?
– study the actual domain that the data set comes
from;
– looking for network properties that appear to be
common across many different domains;
– proxy for a related network that is impossible to
measure.
Large Network Data Sets - Examples
• Collaboration Graphs (who works with whom
in a specific setting)
• Who-Talks-to-Whom Graphs
• Information Linkage Graphs
• Technological Networks
• Networks in the Natural World.
Exercises
A node X is pivotal for a pair of distinct nodes Y, Z
if X lies on every shortest path between
Y and Z (and X ≠ Y, Z).
Question 1
(a) Give an example of a graph in which every
node is pivotal for at least one pair of nodes.
(b) Give an example of a graph in which every
node is pivotal for at least two different pairs
of nodes.
(c) Give an example of a graph having at least
four nodes in which there is a single node X
that is pivotal for every pair of nodes (not
counting pairs that include X)
Exercises (cont.)
• A node X is a gatekeeper if for some other nodes
Y ≠ Z, every path from Y to Z passes through X.
• X is a local gatekeeper if there are two neighbors
of X, Y ≠ Z, that are not connected by an edge.
Question 2
(a) Give an example of a graph in which more
than half of all nodes are gatekeepers.
(b) Give an example of a graph in which there
are no gatekeepers, but in which every node is
a local gatekeeper.
Exercises (cont.)
• A diameter is the maximum distance between
any pair of nodes in the graph.
• An average distance is the average distance
over all pairs of nodes in the graph.
Question 3
(a) Describe an example of a graph where the
diameter is more than three times as large as the
average distance.
(b) Describe how you could extend your
construction to produce graphs in which the
diameter exceeds the average distance by as
large a factor as you’d like. (That is, for every
number c, can you produce a graph in which the
diameter is more than c times as large as the
average distance?)