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Graph theory concepts complex networks presents-rouhollah nabati
1. 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
2. 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
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
4. 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
5. 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
14. 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.
15.
16. 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
17. Graph Theory - HistoryGraph Theory - History
Cycles in Polyhedra
Thomas P. Kirkman William R. Hamilton
Hamiltonian cycles in Platonic graphs
18. Graph Theory - HistoryGraph Theory - History
Gustav Kirchhoff
Trees in Electric Circuits
19. Graph Theory - HistoryGraph Theory - History
Arthur Cayley James J. Sylvester George Polya
Enumeration of Chemical Isomers
20. Graph Theory - HistoryGraph Theory - History
Francis Guthrie Auguste DeMorgan
Four Colors of Maps
21. 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.
22. 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.
25. 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
26. 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.
27. 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
28. 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
29. 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.
30. ComponentComponent
Every disconnected graph can be splitEvery disconnected graph can be split
up into a number of connectedup into a number of connected
componentscomponents..
31. DegreeDegree
Number of edges incident on a nodeNumber of edges incident on a node
The degree of 5 is 3
32. 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
33. 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
34. 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.
35. Graph Theory Ch. 1. Fundamental Concept 35
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
36. 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
37. 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
38. 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
39. TreesTrees
Connected Acyclic GraphConnected Acyclic Graph
Two nodes haveTwo nodes have exactlyexactly
one path between themone path between them
43. 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.
44. 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
45. 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
46. 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
48. 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.
50. 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”.
51. 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.
52. 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..
53. 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
54. Graph Theory Ch. 1. Fundamental Concept 54
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
55. 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
57. 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)
58. 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.
60. 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
61. 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
63. 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
= ∑∑ ,
,
,,
64. Betweenness centrality
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
65. Betweenness centrality
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
66. Closeness centrality
( )
( )∑
−
=
j
jid
N
iCC
,
1
The normalised inverse of the sum of
topological distances in the graph.
69. Node B is the most central one in spreading
information from it to the other nodes in the
network.
Closeness centrality
70. ||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
71. 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
78. 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
79. 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
82. 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
83. 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
84. pk is the
probability that a
node has degree
k.
Nk = # nodes with degree
k
pk = Nk / N
A CASE STUDY: PROTEIN-PROTEIN INTERACTION
NETWORK
84
87. 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
88. 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
89. 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
90. 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) −
==
91. 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
92. 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
95. 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.
96. 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
97. 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.
98. 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
99. 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.
100. 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
102. 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::
103. 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. 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. 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
106. 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.
Here is another pair of useful definitions. Look at the lest picture.
The person labeled A has four friends in this picture, but one of her friendships is qualitatively different from the others: A&apos;s links to C, D, and E connect her to a tightly-knit group of friends who all know each other, while the link to B seems to reach into a different part of the network.
To make precise the sense in which the A-B link is unusual, we introduce the following definition. We say that an edge joining two nodes A and B in a graph is a bridge if deleting the edge would cause A and B to lie in two different components. In other words, this edge is literally the only route between its endpoints, the nodes A and B.
Now, lets remember out discussion about giant components and small-world properties. Bridges are extremely rare in real social networks. You may have a friend from a very different background, and it may seem that your friendship is the only thing that bridges your world and his, but one expects in reality that there will be other, hard-to-discover, multi-step paths that also span these worlds. In other words, if we were to look at the left picture as it is embedded in a larger, real social network, we would likely see a picture that looks like the one on the right. Here, the A-B edge isn&apos;t the only path that connects its two endpoints; though they may not realize it, A and B are also connected by a longer path through F, G, and H. We say that an edge joining two nodes A and B in a graph is a local bridge if its endpoints A and B have no friends in common. in other words, if deleting the edge would increase the distance between A and B to a value strictly more than two.
Local bridges, especially those with reasonably large span, still play roughly the same role that bridges do, though in a less extreme way | they provide their endpoints with access to parts of the network, and hence sources of information, that they would otherwise be far away from.
Let us discuss some of characteristics of this network, relying on the quantities we introduced so far. The undirected network has $N=2,018$ proteins as nodes and $L=2,930$ binding interactions as links.
Hence the average degree is $2.90$, suggesting that a typical protein interacts with approximately three other proteins. Yet, this number is somewhat misleading.
The breath first search algorithm will also convince us that the protein interaction network is not connected, but consists of 185 components, shown as isolated clusters in Fig. \ref{F-G-PPI-Overall}a. The largest, called the giant component, contains 1,647 of the 2,018 nodes; all other components are tiny compared to it. As we will see in the coming chapters, such fragmentation is common in model networks.
Finally, a visual inspection reveals an interesting pattern: hubs have a tendency to connect to small nodes, giving the network a hub and spoke character. This is a consequence of \textit{degree correlations}, reflecting the {\it dissasortative } nature of the protein interaction network. Degree correlations influence a number of network characteristics, from the spread of ideas and viruses in social networks to the number of driver nodes needed to control a network.
Indeed, the degree distribution $p_k$ shown in Fig. \ref{F-G-PPI-Overall}b indicates that the vast majority of nodes have only a few links. To be precise, in this network 69% of nodes have fewer than three links, i.e. for these $k &lt; \langle k \rangle$. They coexist with a few highly connected nodes, or hubs, the largest having as many as 91 links. Such wide differences in node degrees is a consequence of the network&apos;s {\it scale-free property}, a property encountered in many real networks. We will see that the shape of the degree distribution determines a wide range of network properties, from a network&apos;s robustness to node failures to the spread of viruses.
If we use the breath-first-search algorithm, we can determine the network&apos;s diameter, finding $d_{max}=14$. We might be tempted to expect wide variations in $d$, as some nodes are close to each other, others, however, maybe quite far. The distance distribution (Fig. \ref{F-G-PPI-Overall}c), indicates otherwise: $p_d$ has a prominent peak around $\langle d \rangle=5.61$, indicating that most distances are rather short, being in the vicinity of $\langle d \rangle$. Also, $p_d$ decays fast for large $\langle d \rangle$, suggesting that large distances are essentially absent. These are manifestations of the {\it small world property}, another common feature of real networks, indicating that most nodes are close to each other.
The breath first search algorithm will also convince us that the protein interaction network is not connected, but consists of 185 components, shown as isolated clusters in Fig. \ref{F-G-PPI-Overall}a. The largest, called the giant component, contains 1,647 of the 2,018 nodes; all other components are tiny compared to it. As we will see in the coming chapters, such fragmentation is common in model networks.
The average clustering coefficient of the network is $\langle C \rangle=0.12$, which, as we will come to appreciate, is rather large, indicating a significant degree of local clustering in the network. A further caveat is provided by the dependence of the clustering coefficient on the node&apos;s degree, or the $C(k)$ function (Fig. \ref{F-G-PPI-Overall}d), which indicates that the clustering coefficient of the small nodes is significantly higher than the clustering coefficient of the hubs. This suggests that the small degree nodes tend to be part of dense local neighborhoods, while the neighborhood of the hubs is much more sparse. This is a consequence of {\it network hierarchy}, another widely shared network property.
Finally, a visual inspection reveals an interesting pattern: hubs have a tendency to connect to small nodes, giving the network a hub and spoke character. This is a consequence of \textit{degree correlations}, reflecting the {\it dissasortative } nature of the protein interaction network. Degree correlations influence a number of network characteristics, from the spread of ideas and viruses in social networks to the number of driver nodes needed to control a network.
Taken together, Fig. \ref{F-G-PPI-Overall} illustrates that the quantities we introduced in this chapter can help us diagnose several key properties of real networks. The purpose of the coming chapters is to study systematically these network characteristics, understanding what they tell us about the behavior of complex systems.
If we use the breath-first-search algorithm, we can determine the network&apos;s diameter, finding $d_{max}=14$. We might be tempted to expect wide variations in $d$, as some nodes are close to each other, others, however, maybe quite far. The distance distribution (Fig. \ref{F-G-PPI-Overall}c), indicates otherwise: $p_d$ has a prominent peak around $\langle d \rangle=5.61$, indicating that most distances are rather short, being in the vicinity of $\langle d \rangle$. Also, $p_d$ decays fast for large $\langle d \rangle$, suggesting that large distances are essentially absent. These are manifestations of the {\it small world property}, another common feature of real networks, indicating that most nodes are close to each other.
The breath first search algorithm will also convince us that the protein interaction network is not connected, but consists of 185 components, shown as isolated clusters in Fig. \ref{F-G-PPI-Overall}a. The largest, called the giant component, contains 1,647 of the 2,018 nodes; all other components are tiny compared to it. As we will see in the coming chapters, such fragmentation is common in model networks.
The average clustering coefficient of the network is $\langle C \rangle=0.12$, which, as we will come to appreciate, is rather large, indicating a significant degree of local clustering in the network. A further caveat is provided by the dependence of the clustering coefficient on the node&apos;s degree, or the $C(k)$ function (Fig. \ref{F-G-PPI-Overall}d), which indicates that the clustering coefficient of the small nodes is significantly higher than the clustering coefficient of the hubs. This suggests that the small degree nodes tend to be part of dense local neighborhoods, while the neighborhood of the hubs is much more sparse. This is a consequence of {\it network hierarchy}, another widely shared network property.
Finally, a visual inspection reveals an interesting pattern: hubs have a tendency to connect to small nodes, giving the network a hub and spoke character. This is a consequence of \textit{degree correlations}, reflecting the {\it dissasortative } nature of the protein interaction network. Degree correlations influence a number of network characteristics, from the spread of ideas and viruses in social networks to the number of driver nodes needed to control a network.
Taken together, Fig. \ref{F-G-PPI-Overall} illustrates that the quantities we introduced in this chapter can help us diagnose several key properties of real networks. The purpose of the coming chapters is to study systematically these network characteristics, understanding what they tell us about the behavior of complex systems.
The average clustering coefficient of the network is $\langle C \rangle=0.12$, which, as we will come to appreciate, is rather large, indicating a significant degree of local clustering in the network. A further caveat is provided by the dependence of the clustering coefficient on the node&apos;s degree, or the $C(k)$ function (Fig. \ref{F-G-PPI-Overall}d), which indicates that the clustering coefficient of the small nodes is significantly higher than the clustering coefficient of the hubs. This suggests that the small degree nodes tend to be part of dense local neighborhoods, while the neighborhood of the hubs is much more sparse. This is a consequence of {\it network hierarchy}, another widely shared network property.
Taken together, Fig. \ref{F-G-PPI-Overall} illustrates that the quantities we introduced in this chapter can help us diagnose several key properties of real networks. The purpose of the coming chapters is to study systematically these network characteristics, understanding what they tell us about the behavior of complex systems.
Source: Albert and Barabasi, “Statistical mechanics of complex networks.” Review of Modern Physics. 74:48-94. (2002)