Graph theory concepts complex networks presents-rouhollah nabati

Introduction to NetworkIntroduction to Network
And SNA Theory:And SNA Theory:
Basic ConceptsBasic Concepts
R. NabatiR. Nabati
Department of Computer EngineeringDepartment of Computer Engineering
Islamic Azad University of SanandajIslamic Azad University of Sanandaj
www.rnabati.comwww.rnabati.com

What is a Network?What is a Network?
 Network = graphNetwork = graph
 Informally aInformally a graphgraph is a set of nodesis a set of nodes
joined by a set of lines or arrows.joined by a set of lines or arrows.
1 1
2 3
4 45 56 6
2 3

Graph-based representations
 Representing a problem as a graph can
provide a different point of view
 Representing a problem as a graph can
make a problem much simpler
 More accurately, it can provide the
appropriate tools for solving the problem

What is network theory?
 Network theory provides a set of techniques for
analysing graphs
 Complex systems network theory provides
techniques for analysing structure in a system of
interacting agents, represented as a network
 Applying network theory to a system means using a
graph-theoretic representation

What makes a problem graph-like?
 There are two components to a graph
 Nodes and edges
 In graph-like problems, these components
have natural correspondences to problem
elements
 Entities are nodes and interactions between
entities are edges
 Most complex systems are graph-like

Scientific collaboration network

Business ties in US biotech-
industry

Protein-Protein Interaction Networks

Graph Theory - HistoryGraph Theory - History
Leonhard Euler's paperLeonhard Euler's paper
on “on “Seven Bridges ofSeven Bridges of
Königsberg”Königsberg” ,,
published in 1736.published in 1736.

Graph Theory Ch. 1. Fundamental Concept 16
A Model
 A vertex : a region
 An edge : a path(bridge) between two
regions
e1
e2
e3
e4
e6
e5
e7
Z
Y
X
W
X
Y
Z
W

Cycles in Polyhedra
Thomas P. Kirkman William R. Hamilton
Hamiltonian cycles in Platonic graphs

Gustav Kirchhoff
Trees in Electric Circuits

Arthur Cayley James J. Sylvester George Polya
Enumeration of Chemical Isomers

Francis Guthrie Auguste DeMorgan
Four Colors of Maps

Definition: GraphDefinition: Graph
 G is an ordered triple G:=(V, E, f)G is an ordered triple G:=(V, E, f)

V is a set of nodes, points, or vertices.V is a set of nodes, points, or vertices.

E is a set, whose elements are known asE is a set, whose elements are known as
edges or lines.edges or lines.

f is a functionf is a function

maps each element of Emaps each element of E

to an unordered pair of vertices in V.to an unordered pair of vertices in V.

DefinitionsDefinitions
 VertexVertex

Basic ElementBasic Element

Drawn as aDrawn as a nodenode or aor a dotdot..

VVertex setertex set ofof GG is usually denoted byis usually denoted by VV((GG), or), or VV
 EdgeEdge

A set of two elementsA set of two elements

Drawn as a line connecting two vertices, calledDrawn as a line connecting two vertices, called
end vertices, or endpoints.end vertices, or endpoints.

The edge set of G is usually denoted by E(G), orThe edge set of G is usually denoted by E(G), or
E.E.

Example
 V:={1,2,3,4,5,6}
 E:={{1,2},{1,5},{2,3},{2,5},{3,4},{4,5},{4,6}}

Simple Graphs
Simple graphs are graphs without multiple
edges or self-loops.

Directed Graph (digraph)Directed Graph (digraph)
 Edges have directionsEdges have directions

An edge is anAn edge is an orderedordered pair ofpair of
nodesnodes
loop
node
multiple arc
arc

Weighted graphs
1 2 3
4 5 6
.5
1.2
.2
.5
1.5
.3
1
4 5 6
2 3
2
1
35
 is a graph for which each edge has an
associated weight, usually given by a
weight function w: E → R.

Structures and structural metrics
 Graph structures are used to isolate
interesting or important sections of a
graph
 Structural metrics provide a measurement
of a structural property of a graph
 Global metrics refer to a whole graph
 Local metrics refer to a single node in a graph

Graph structures
 Identify interesting sections of a graph
 Interesting because they form a significant
domain-specific structure, or because they
significantly contribute to graph properties
 A subset of the nodes and edges in a
graph that possess certain characteristics,
or relate to each other in particular ways

Connectivity
 a grapha graph is connected if
 you can get from any node to any other by
following a sequence of edges OR
 any two nodes are connected by a path.
 A directed graph is strongly connected if
there is a directed path from any node to any
other node.

ComponentComponent
 Every disconnected graph can be splitEvery disconnected graph can be split
up into a number of connectedup into a number of connected
componentscomponents..

DegreeDegree
 Number of edges incident on a nodeNumber of edges incident on a node
The degree of 5 is 3

Degree (Directed Graphs)Degree (Directed Graphs)
 In-degree: Number of edges enteringIn-degree: Number of edges entering
 Out-degree: Number of edges leavingOut-degree: Number of edges leaving
 Degree = indeg + outdegDegree = indeg + outdeg
outdeg(1)=2
indeg(1)=0
outdeg(2)=2
indeg(2)=2
outdeg(3)=1
indeg(3)=4

Degree: Simple Facts
 If G is a graph with m edges, then
Σ deg(v) = 2m = 2 |E |
 If G is a digraph then
Σ indeg(v)=Σ outdeg(v) = |E |
 Number of Odd degree Nodes is even

Walks
A walk of length k in a graph is a succession of k
(not necessarily different) edges of the form
uv,vw,wx,…,yz.
This walk is denote by uvwx…xz, and is referred to
as a walk between u and z.
A walk is closed is u=z.

Chromatic Number
 The chromatic number of a graph G,
written x(G), is the minimum number of
colors needed to label the vertices so
that adjacent vertices receive different
colors
Red
Green
Blue
Blue
x(G) = 3

PathPath
 AA pathpath is a walk in which all the edges and allis a walk in which all the edges and all
the nodes are different.the nodes are different.
Walks and Paths
1,2,5,2,3,4 1,2,5,2,3,2,1 1,2,3,4,6
walk of length 5 CW of length 6 path of length 4

Cycle
 A cycle is a closed walk in which all the
edges are different.
1,2,5,1 2,3,4,5,2
3-cycle 4-cycle

Special Types of Graphs
 Empty Graph / Edgeless graphEmpty Graph / Edgeless graph

No edgeNo edge
 Null graphNull graph

No nodesNo nodes

Obviously no edgeObviously no edge

TreesTrees
 Connected Acyclic GraphConnected Acyclic Graph
 Two nodes haveTwo nodes have exactlyexactly
one path between themone path between them

Special TreesSpecial Trees
Paths
Stars

Connected Graph
All nodes have the same
degree
Regular

Special Regular Graphs: Cycles
C3 C4 C5

BipartiteBipartite graphgraph
 VV can be partitionedcan be partitioned
into 2 setsinto 2 sets VV11 andand VV22
such that (such that (uu,,vv))∈∈EE
impliesimplies
 eithereither uu ∈∈VV11 andand vv ∈∈VV22
 OROR vv ∈∈VV11 andand uu∈∈VV2.2.

Complete GraphComplete Graph
 Every pair of vertices are adjacentEvery pair of vertices are adjacent
 Has n(n-1)/2 edgesHas n(n-1)/2 edges

Complete Bipartite GraphComplete Bipartite Graph
 Bipartite Variation of Complete GraphBipartite Variation of Complete Graph
 Every node of one set is connected toEvery node of one set is connected to
every other node on the other setevery other node on the other set
Stars

Planar GraphsPlanar Graphs
 Can be drawn on a plane such that no two edgesCan be drawn on a plane such that no two edges
intersectintersect
 KK44 is the largest complete graph that is planaris the largest complete graph that is planar

SubgraphSubgraph
 Vertex and edge sets are subsets ofVertex and edge sets are subsets of
those of Gthose of G

aa supergraphsupergraph of a graph G is a graph thatof a graph G is a graph that
contains G as a subgraph.contains G as a subgraph.

Special Subgraphs: CliquesSpecial Subgraphs: Cliques
A clique is a maximum complete
connected subgraph..
A B
D
H
FE
C
IG

Spanning subgraphSpanning subgraph
 Subgraph H has the same vertex set asSubgraph H has the same vertex set as
G.G.

Possibly not all the edgesPossibly not all the edges

““H spans G”.H spans G”.

Spanning treeSpanning tree
 Let G be a connected graph. Then aLet G be a connected graph. Then a
spanning treespanning tree in G is a subgraph of Gin G is a subgraph of G
that includes every node and is also athat includes every node and is also a
tree.tree.

IsomorphismIsomorphism
 Bijection, i.e., a one-to-one mapping:Bijection, i.e., a one-to-one mapping:
f : V(G) -> V(H)f : V(G) -> V(H)
u and v from G are adjacent if and onlyu and v from G are adjacent if and only
if f(u) and f(v) are adjacent in H.if f(u) and f(v) are adjacent in H.
 If an isomorphism can be constructedIf an isomorphism can be constructed
between two graphs, then we say thosebetween two graphs, then we say those
graphs aregraphs are isomorphicisomorphic..

Isomorphism ProblemIsomorphism Problem
 Determining whether twoDetermining whether two
graphs are isomorphicgraphs are isomorphic
 Although these graphs lookAlthough these graphs look
very different, they arevery different, they are
isomorphic; one isomorphismisomorphic; one isomorphism
between them isbetween them is
f(a)=1 f(b)=6 f(c)=8 f(d)=3f(a)=1 f(b)=6 f(c)=8 f(d)=3
f(g)=5 f(h)=2 f(i)=4 f(j)=7f(g)=5 f(h)=2 f(i)=4 f(j)=7

Components 1.2.8
 The components of a graph G are its
maximal connected subgraphs
 An isolated vertex is a vertex of
degree 0
r
q
s u v w
t p x
y z

Representation (Matrix)Representation (Matrix)
 Incidence MatrixIncidence Matrix

V x EV x E

[vertex, edges] contains the edge's data[vertex, edges] contains the edge's data
 Adjacency MatrixAdjacency Matrix

V x VV x V

Boolean values (adjacent or not)Boolean values (adjacent or not)

Or Edge WeightsOr Edge Weights

MatricesMatrices
10000006
01010105
11100004
00101003
00011012
00000111
6,45,44,35,23,25,12,1
0010006
0010115
1101004
0010103
0101012
0100101
654321

Representation (List)Representation (List)
 Edge ListEdge List

pairs (ordered if directed) of verticespairs (ordered if directed) of vertices

Optionally weight and other dataOptionally weight and other data
 Adjacency List (node list)Adjacency List (node list)

Implementation of a Graph.Implementation of a Graph.
 Adjacency-list representationAdjacency-list representation

an array of |an array of |VV | lists, one for each vertex in| lists, one for each vertex in
VV..

For eachFor each uu ∈∈ VV ,, ADJADJ [[ uu ] points to all its] points to all its
adjacent vertices.adjacent vertices.

Edge and Node ListsEdge and Node Lists
Edge List
1 2
1 2
2 3
2 5
3 3
4 3
4 5
5 3
5 4
Node List
1 2 2
2 3 5
3 3
4 3 5
5 3 4

Edge List
1 2 1.2
2 4 0.2
4 5 0.3
4 1 0.5
5 4 0.5
6 3 1.5
Edge Lists for Weighted GraphsEdge Lists for Weighted Graphs

Topological Distance
A shortest path is the minimum pathA shortest path is the minimum path
connecting two nodes.connecting two nodes.
The number of edges in the shortest pathThe number of edges in the shortest path
connectingconnecting pp andand qq is theis the topologicaltopological
distancedistance between these two nodes, dbetween these two nodes, dp,qp,q

Degree Centrality
B
ED
C
A
2
4
2
1
1
degree
00010
00010
00011
11101
00110
E
D
C
B
A
EDCBA

Betweenness centrality
 The number of shortest paths in the graph
that pass through the node divided by the
total number of shortest paths.
( ) ( )
( )
kji
ji
jki
kBC
i j
≠≠
ρ
ρ
= ∑∑ ,
,
,,

B
 Shortest paths are:
 AB, AC, ABD, ABE, BC, BD, BE,
CBD, CBE, DBE
 B has a BC of 5
A
C
D E
( ) ( )
( ) ( )
( ) ( )
( ) ( )
( ) ( ) 1,;1,,
1,;1,,
1,;1,,
1,;1,,
1,;1,,
==
==
==
==
==
EDEBD
ECEBC
DBDBC
EAEBA
DADBA
ρρ
ρρ
ρρ
ρρ
ρρ

 Nodes with a high betweenness centrality
are interesting because they
 control information flow in a network
 may be required to carry more information
 And therefore, such nodes
 may be the subject of targeted attack

Closeness centrality
( )
( )∑
−
=
j
jid
N
iCC
,
1
 The normalised inverse of the sum of
topological distances in the graph.

B
ED
C
A
02212
20212
22011
11101
22110
E
D
C
B
A
EDCBA
( )∑=
n
j
jid
1
,
6
4
6
7
7

B
ED
C
A Closeness
0.67
1.00
0.67
0.57
0.57

Node B is the most central one in spreading
information from it to the other nodes in the
network.

||VV | x || x |V |V | matrix Dmatrix D = (= ( ddijij )) such thatsuch that
ddijij is the topological distance betweenis the topological distance between ii andand jj..
0212336
2012115
1101224
2210123
3121012
3122101
654321
Distance MatrixDistance Matrix

A community is defined as a clique
in the communicability graph.
Identifying communities is reduced
to the “all cliques problem” in the
communicability graph.
Communicability Graph

Social (Friendship) Network
Communities: Example

Communities: Example
The Network
Its Communicability
Graph

Communities
Social Networks Metabolic Networks

Social Network Analysis
Introduction

22 55
4433
11
Diameter
22 55
4433
11
Average Path Length
The longest shortest path
in a graph
The average of the shortest
paths for all pairs of nodes.
76

Characterizing networks:Is everything
connected?
77

The adjacency matrix of a network with several components can be written in
a block-diagonal form, so that nonzero elements are confined to squares,
with all other elements being zero:
CONNECTIVITY OF UNDIRECTED
GRAPHS Adjacency Matrix
78

Bridges and Local Bridges0
 And edge that joins two nodes A and B in a graph is called a
bridge if deleting the edge would cause A and B to lay in two
different components
 local bridge - in real-world networks (with a giant
component) - if deleting an edge between A and B would
increase distance > 2
79

Clustering coefficient:
what fraction of your neighbors are connected?
Node i with degree ki
Ci in [0,1]
CLUSTERING COEFFICIENT
81

WWW > directed multigraph with self-interactions
Protein Interactions > undirected unweighted with self-interactions
Collaboration network > undirected multigraph or weighted.
Mobile phone calls > directed, weighted.
Facebook Friendship links > undirected, unweighted.
GRAPHOLOGY: Real networks can have multiple
characteristics
82

Undirected network
N=2,018 proteins as nodes
L=2,930 binding interactions as links.
Average degree <k>=2.90.
Not connected: 185
components
the largest (giant component)
1,647 nodes
A CASE STUDY: PROTEIN-PROTEIN INTERACTION
NETWORK
83

pk is the
probability that a
node has degree
k.
Nk = # nodes with degree
k
pk = Nk / N
NETWORK
84

dmax=14
<d>=5.61
NETWORK
85

<C>=0.12
NETWORK
86

Real network properties
 Most nodes have only a small number of neighbors
(degree), but there are some nodes with very high degree
(power-law degree distribution)
 scale-free networks
 If a node x is connected to y and z, then y and z are likely
to be connected
 high clustering coefficient
 Most nodes are just a few edges away on average.
 small world networks
 Networks from very diverse areas (from internet to
biological networks) have similar properties
 Is it possible that there is a unifying underlying generative
process?
87

The basic random graph
model
 The measurements on real networks are usually
compared against those on “random networks”
 The basic Gn,p (Erdös-Renyi) random graph model:
 n : the number of vertices
 0 ≤ p ≤ 1
 for each pair (i,j), generate the edge (i,j) independently with
probability p

Degree distributions
 Problem: find the probability distribution that best fits the
observed data
degree
frequency
k
fk
fk = fraction of nodes with degree k
p(k) = probability of a randomly
selected node to have degree k

Power-law distributions The degree distributions of most real-life networks follow a power
law
 Right-skewed/Heavy-tail distribution
 there is a non-negligible fraction of nodes that has very high degree
(hubs)
 scale-free: no characteristic scale, average is not informative
 In stark contrast with the random graph model!
 Poisson degree distribution, z=np
 highly concentrated around the mean
 the probability of very high degree nodes is exponentially small
p(k) = Ck-α
z
k
e
k!
z
z)P(k;p(k) −
==

Power-law signature
 Power-law distribution gives a line in the log-log plot
 α : power-law exponent (typically 2 ≤ α ≤ 3)
degree
frequency
log degree
log frequency α
log p(k) = -α logk + logC

Power Laws
Albert and Barabasi (1999)
Power-law distributions are straight
lines in log-log space.
-- slope being r
y=k-r
 log y = -r log k  ly= -r lk
How should random graphs be
generated to create a power-law
distribution of node degrees?
Hint:
Pareto’s* Law: Wealth
distribution follows a power law.
Power laws in real networks:
(a) WWW hyperlinks
(b) co-starring in movies
(c) co-authorship of physicists
(d) co-authorship of neuroscientists
* Same Velfredo Pareto, who defined Pareto optimality in game

Examples of degree distribution for power laws
Taken from [Newman 2003]

What is a Logarithm?
 The common or base-10 logarithm of a
number is the power to which 10 must
be raised to give the number.
 Since 100 = 102
, the logarithm of 100 is
equal to 2. This is written as:
Log(100) = 2.
 1,000,000 = 106
(one million), and
Log (1,000,000) = 6.

Years before present (YBP)
Formation of Earth 4.6 x 109
YBP
Dinosaur extinction 6.5 x 107
YBP
First hominids 2 x 106
YBP
Last great ice age 1 x 104
YBP
First irrigation of crops 6 x 103
YBP
Declaration of Independence 2 x 102
YBP
Establishment of UWB 1 x 10 YBP

Data plotted with linear scale
Events from Table I
0.E+00
1.E+09
2.E+09
3.E+09
4.E+09
5.E+09
EarthD
inosaursH
om
inids
Ice
Age
Irrigation
Independence
U
W
B
Yearsbeforepresent
All except
the first two
data points
are hidden
on the axis.

Log (YBP)
EVENT YBP Log(YBP)
Formation of Earth 4.6 x 109
9.663
Dinosaur extinction 6.5 x 107
7.813
First hominids 2 x 106
6.301
Last great ice age 1 x 104
4.000
First irrigation of crops 6 x 103
3.778
Declaration of Independence 2 x 102
2.301
Establishment of UWB 1 x 10 1.000

Plot using Logs
Events from Table I
0
2
4
6
8
10
EarthD
inosaurs
H
om
inids
Ice
Age
Irrigation
Independence
U
W
B
Log(YBP)
All data are well
represented
despite their wide
range.

Your calculator should have a button
marked LOG. Make sure you can use it
to generate this table.
N N as power of 10 Log (N)
1000 103
3.000
200 102.301
2.301
75 101.875
1.875
10 101
1.000
5 100.699
0.699

Copyright © by Houghton Mifflin
Company, Inc. All rights reserved. 101
x
y
Graph f(x) = log2
x
Since the logarithm function is the inverse of the
exponential function of the same base, its graph is the
reflection of the exponential function in the line y = x.
83
42
21
10
–1
–2
2x
x
4
1
2
1
y = log2
x
y = xy = 2x
(1, 0)
x-intercept
horizontal
asymptote y = 0
vertical asymptote
x = 0

Example
 Sketch the graph of the function y = ln x.
Solution
 We first sketch the graph of y = ex
.
11
xx
yy
11
yy ==
eexx
yy = ln= ln
xx
yy ==
xx
 The required graph isThe required graph is
thethe mirror imagemirror image of theof the
graph ofgraph of yy == eexx
withwith
respect to the linerespect to the line yy == xx::

Exponential distribution
 Observed in some technological or
collaboration networks
 Identified by a line in the log-linear plot
p(k) = λe-λk
log p(k) = - λk + log λ
degree
log frequency λ

104
An Experiment by Milgram (1967)
 Outcome revealed two fundamental
components of a social network:
 Very short paths between arbitrary pairs of nodes
 Individuals operating with purely local information
are very adept at finding these paths

105
What is the “small world” phenomenon?
 Principle that most people in a society are linked by short
chains of acquaintances
 Sometimes referred to as the “six degrees of separation”
theory

References
Aldous & Wilson, Graphs and Applications. An
Introductory Approach, Springer, 2000.
Wasserman & Faust, Social Network Analysis,
Cambridge University Press, 2008.
Estrada & Rodríguez-Velázquez, Phys. Rev. E
2005, 71, 056103.
Estrada & Hatano, Phys. Rev. E. 2008, 77,
036111.

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