Complex Networks 1 Complex systems Made of many nonidentical elements connected by diverse interactions NETWORK Business ties in US biotechindustry ID: 560554
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Slide1
-Anurag Singh
Complex Networks
1Slide2
Complex systems
Made of many non-identical elements
connected by diverse
interactions
.
NETWORKSlide3
Business ties in US biotech-industry
Nodes: companies:
investment
pharma
research labs
public
biotech
nology
Links: financial R&D collaborations
http://ecclectic.ss.uci.edu/~drwhite/MovieSlide4
Business ties in US biotech-industry
Nodes: companies:
investment
pharma
research labs
public
biotech
nology
Links: financial R&D collaborations
http://ecclectic.ss.uci.edu/~drwhite/MovieSlide5
Red
, blue
, or
green
: departments
Yellow: consultantsGrey
: external experts
Structure of an organization
www.orgnet.comSlide6
InternetSlide7
Friendship NetworkSlide8
Network Collaboration NetworkSlide9
9-11 Terrorist (?) Network
Social Network Analysis is a mathematical methodology for
connecting the dots
-- using science to fight terrorism. Connecting multiple pairs of dots soon reveals an emergent
network
of organization. Slide10
Some interesting Problems
Consonants (Language) NetworksMarriage Networks
Collaboration Networks
Build Networks which are robust as well as efficient
Actors NetworkSlide11
Outline
Techniques to analyze networksSpecial types of networks – random networks, power law networks, small world networks
Models of network growth
Processes taking place on network – search, Slide12
Traditional vs. Complex Systems Approaches to Networks
Traditional Questions:
Social Networks:
Who is the most important person in the network?
Graph Theory:
Does there exist a cycle through the network that uses each edge exactly once?
Complex Systems Questions:
What fraction of edges have to be removed to disconnect the graph?
What kinds of structures emerge from simple growth rules?Slide13
Introduction to Complex Networks
Complex network is a network (graph) with non-trivial topological features (heavy tail in the degree distribution, a high clustering coefficient, assortavity among vertices, and community structure)Features that do not occur in simple networks
Lattices or random graphs, does not have these features..
degree
dist.
clustering
assortativity
comunity
Lattice
Random
13Slide14
Introduction to Complex Networks (contd..)
Many systems in nature can be described by models of complex networks Structures consisting of nodes or vertices connected by links or edges.
14Slide15
Introduction to Complex Networks (contd..)Examples : The Internet is a network of routers or domains.
The World Wide Web (WWW) is a network of websites The brain is a network of neurons. Social networkCitation networks
Diseases are transmitted through social networks
Man-made infrastructures, and in many physical systems such as the power grids.
15Slide16
Network structures of the Internet and the WWW.
16Slide17
Social network Citation network
17Slide18
Evolution of Complex Network researchErdös and Rényi
(ER) described a network with complex topology by a random graphMany real-life complex networks are neither completely regular nor completely random, Two significant recent discoveries are small-world effect and the scale-free nature of most complex networks.
18Slide19
Introduction to Complex Networks (Contd..)
Watts and Strogatz (WS) introduced the concept of small-world phenomenon A prominent common feature of the ER random graph and the WS small-world model is The connectivity distribution of a network peaks at an average value and decays exponentially.
Each node has about the same number of link connections.
Such networks are called “exponential networks” or “homogeneous networks,”
19Slide20
Introduction to Complex Networks (Contd..)
A significant recent discovery in the complex networks is the observation that many large-scale complex networks are scale-free, That is, their connectivity distributions are in a power-law form that is independent of the network scale . Differs from an exponential network, A scale-free network is inhomogeneous in nature
Most nodes have very few link connections and yet a few nodes have many connections.
20Slide21
Decision parameters and its definitionsAverage path length
Clustering coefficientDegree distributionDegree exponent
21Slide22
Decision parameters and its definitions (contd..)Average Path Length
In a network, the distance dij between two nodes, labeled i and j respectively, is defined as the number of edges along the shortest path connecting them.
The diameter D:
of a network, therefore, is defined to be the maximal distance among all distances between any pair of nodes in the network.
The average path length L of the network, then, is defined as the mean distance between two nodes, averaged over all pairs of nodes.
22Slide23
Decision parameters and its definitions (contd..)
L determines the effective “size” of a network, The average path length of most real complex networks
is relatively small.]
D = max l (A,B)
23Slide24
Decision parameters and its definitions (contd..)
Clustering CoefficientTwo of our friends are quite possibly friends of each other. This property refers to the clustering of the network. A clustering coefficient C as the average fraction of pairs of neighbors of a node that are also neighbors of each other.
a node
i
in the network has k
i edges they connect this node to ki other nodes (neighbors). at most
ki (ki − 1)/2 edges can exist among them
24Slide25
Decision parameters and its definitions (contd..)
The clustering coefficient :
E
i -- edges that actually exist among
ki nodesCluster coefficient C of whole network:
0 ≤ C ≤ 1In most large-scale real networks clustering coefficients are much greater than completely random
network
25Slide26
Decision parameters and its definitions (contd..)
Degree DistributionThe degree ki of a node
i
is the total number of its neighbors.
The larger the degree, the “more important” the node is in a network. The average of
ki over all i is called the average degree of the network ( < k >). The spread of node degrees over a network is characterized by a distribution function P(k)P(k) is the probability that a randomly selected node has exactly k edges.
26Slide27
A regular lattice has a simple degree sequence because all the nodes have the same number of edgesso a plot of the degree distribution contains a single sharp spike Any randomness in the network will broaden the shape of this peak. In the limiting case of a completely random network
the degree sequence obeys the familiar Poisson distribution the shape of the Poisson distribution falls off exponentially away from the peak value <k>
27Slide28
28
Small World Networks
Slide29
29
Duncan J. Watts
Six degrees - the science of a connected age, 2003, W.W. Norton.
I read somewhere that everybody on this planet is separated by only six other people. Six degrees of separation between us and everybody on this planet.
Six degrees of separation
by John GuareSlide30Slide31
31
Correct question
WHY are there short chains of acquaintances linking together arbitrary pairs of strangers???
Or
Why is this surprisingSlide32
32
Random networks
In a
random
network, if everybody has 100 friends distributed randomly in the world population, this isn’t strange
In 6 hops, you can reach 1006 people - a million million > 6,000 million (world pop.)
BUT: our social networks tend to be clustered.Slide33
33
Social networks
Not random
But
Clustered
Most of our friends come from our geographical or professional neighbourhood.
Our friends tend to have the same friends
BUT
In spite of having clustered social networks, there seem to exist short paths between any random nodes.Slide34
34
Social network research
Devise various classes of networks
Study their propertiesSlide35
35
Network parameters
Network type
Regular
Random
Natural
Size: # of nodes
Number of connexions:
average & distribution
Selection of neighboursSlide36
36
STAR
TREE
GRID
BUS
RING
REGULAR Network TopologiesSlide37
37
Connectivity in Random graphs
Nodes connected by links in a purely random fashion
How large is the largest connected component? (as a fraction of all nodes)
Depends on the number of links per node
(Erdös, Rényi 1959)Slide38
38
Connecting NodesSlide39
39
Random Network (1)
add random
pathsSlide40
40
paths
trees
Random Network (2) Slide41
41
paths
trees
networks
Random Network (3) Slide42
42
paths
trees
networks
…..
Random Network (3+) Slide43
43
paths
trees
networks
fully connected
Network Connectivity (4) Slide44
44
Connectivity of a random graph
1
1
Average number of
links per node
Fraction of all nodes
in largest component
0
Disconnected phase
Conected phaseSlide45
45
Regular or Ordered NetworkSlide46
46
Network measures
Connectivity
is not main measure.
Characteristic Path Length
(L) :
the average length of the shortest path connecting each pair of agents (nodes).
Clustering Coefficient
(C)
is a measure of local interconnection
if agent
i has
ki immediate neighbors, Ci, is the fraction of the total possible ki*(ki-1) / 2 connections that are realized between i's neighbors. C, is just the average of the Ci's. Diameter: maximum value of path lengthSlide47
47
Regular vs Random Networks
Average number of
connections/node
Diameter
Number of connections
needed to fully connect
few, clustered
Random
Regular
fewer, spread
large
moderate
many
fewer (<2/3)Slide48
Classes of Complex Networks
1.Random Graphs ModelFirst studied by Erdos and Renyi
Some properties of E-R networks:
if nodes in in graph = N
Average number of edges (= size of graph):
Let n ,vertices and connect each pair (or not) with probability p (or 1-p).
E =
p
N
(
N - 1) / 2Average degree:
〈k〉 = 2 E/N =
p (N - 1) ~
p N
48Slide49
Classes of Complex Networks (contd..)
Erdos and Renyi proposed the following model of a network :
the model called Gn,p , is the ensemble of all such graphs in which a graph having m edges appears with pm(1-p)M-m, where , ,is the maximum possible number of edges.
49Slide50
Classes of Complex Networks (contd..)
another model, called Gn,m, which is the ensemble of all graphs having n vertices and exactly m edges, each possible graph appearing with equal probability.
presence or absence of edges is independent, and hence the probability of a vertex having degree k is:
(for large n and fixed k)
Where, z =expected degree = p(n-1)
50Slide51
Classes of Complex Networks (contd..)
2.Small world Phenomenon
“Almost every pair of nodes is connected by a path with an extremely small number of steps.”
L<<N for N>>1
Small-World experiment(1960), by S. Milgram
Having people explicitly construct path through the social network defined by acquaintanceship (first-name basis)
The median length among the completed path was 6 ( six degrees of freedom)
51Slide52
Classes of Complex Networks (contd..)
For a constant k ≥ 3, if we choose uniformly at random from the set of all n-node graphs in which each node has degree exactly k, then with high probability every pair of nodes will
be joined by a path of length O(log n).
( B. Bollobas, W. F. de la Vega. The diameter of random regular graphs. Combinatorica 2 (1982) )
Path lengths is polylogarithmic in n. (bounded by a polynomial function of log n)
52Slide53
There is something missing...Small-world is not only about “short path”
A standard random graph is locally very sparse.
With reasonably high probability, none of the neighbors of a given node v are themselves neighbors of one another.
(Watts, D. J. and Strogatz, S. H., Collective dynamics of ‘small-world’ networks. Nature (1998))
Implication → The social Network appears from the local perspective of any one node to be highly clustered
.
53Slide54
Two important properties of small world networks:
Low average hop countHigh clustering coefficient
54Slide55
Classes of Complex Networks (contd..)
Small-World ModelsThe regular lattice model and the ER random model both fail to reproduce some important features of many real networks.
Most of these real-world networks are neither entirely regular nor entirely random.
People usually know their neighbors, but their circle of acquaintances may not be confined to those who live right next door, as the regular lattice model would imply.
On the other hand, cases like links among Web pages on the WWW were certainly not created at random, as the ER process would expect.
55Slide56
Classes of Complex Networks (contd..)
Watts-Strogatz ModelAdd some “random” links to a structured, high diameter network.
Most people are friends with their immediate neighbors
Everyone has one or two friends who are far away.
56Slide57
Watts and Strogatz introduced an interesting small-world network model (WS) as
.
57Slide58
WS Small-World Model Algorithm
1) Start with regular: Begin with a nearest-neighbor coupled network consisting of N nodes arranged in a ring, where each node i is adjacent to its neighbor nodes, i = 1, 2, ··· , K/2, with K being even.
2) Randomization: Randomly rewire each edge of the network with probability p; varying p in such a way that the transition between (p = 0) and randomness (p = 1) can be closely monitored.
58Slide59
Random graphs show the small-world effect, but do not show clustering.
The small-world model can be viewed as a homogeneous network.
the WS small-world network model is similar to the ER random graph model.
Analysis of WS Model
C(p) - Clustering coefficient
L(p) - average path length, considered as a function of the rewiring probability p.
A regular ring lattice (p = 0) is :highly clustered (C (0) ≈ 3/4)
L~ ln N
59Slide60
P
Fig [Courtesy of NATURE] Average
Path Length and clustering coefficient
of the WS small-world model
60Slide61
61
Example: 4096 node ring
Regular graph:
n nodes, k nearest neighbors
path length ~ n/2k
4096/16 = 256
Random graph:
path length ~ log (n)/log(k)
~ 4
Rewired graph (1% of nodes):
path length ~ random graph
clustering ~ regular graph
Small World Graph
K=4Slide62
62
Small-
world
networks
Beta network
Rewiring probability
0
1
0
1
L
CSlide63
63
More exactly …. (p =
)
Small world
behaviour
C
LSlide64
The small-world and scale-free features are common to many real-world complex networks.
Network
Size
Clustering Coefficient
Average path length
Degree exponent
Internet, domain level
32711
0.24
3.56
2.1
Internet, router level
2282980.039.512.1
WWW153127
0.11
3.1
In=2.1, out=2.45
E-mail
56969
0.03
4.95
1.81
Software
1376
0.06
6.39
2.5
Electronic Circuits
329
0.34
3.17
2.5
Movie Actors
225226
0.79
3.65
2.3
Food Web
154
0.15
3.40
1.13
Language
460902
0.437
2.67
2.7Slide65
65
Effect of short-cuts
Huge effect of just a few short-cuts.
First 5
rewirings reduces the path length by
half, regardless of size of network
Further 50% gain requires 50 more short-cutsSlide66
66
The strength of weak ties
Granovetter (1973): effective social coordination does not arise from densely interlocking strong ties, but derives from the occasional weak ties
this is because valuable information comes from these relations (it is valuable if/because it is not available to other individuals in your immediate network)Slide67
67
Two ways of constructingSlide68
68
Alpha model
Watts’ first Model (1999)
Inspired by Asimov’s
“I, Robot”
novels
R. Daneel Olivaw
Elijah Baley
Caves of Steel
(Earth)
SolariaSlide69
69
Natural networks
Between regular grids and totally random graphs
Need for parametrized models:
Regular -> natural -> random
Watts
Alpha model ( not intuitive)
Beta
rewiring
model
Slide70
Applications
Social Network of movie actors
Two actors being connected if they were cast together in the same movie.
The probability that an actor has k links (characterizing his or her popularity) has a power-law tail for large k, :
P(k) ~k-Уactor ,
where ,Уactor = 2.3 ± 0.1
C=0.79, L = 3.65so, it also follows the small world property as well as scale free network.
A more complex network with over 800 million vertices is the WWW
,
where a vertex is a document
the edges are the links pointing from one document to another.
The topology of this graph determines the Web’s connectivity Information about P(k) ,indicating that the probability that k documents point to a certain Webpage follows a power law, with Уwww = 2.1 ± 0.1.
C = 0.11, L = 3.1
70Slide71
World Wide Web
71Slide72
The electrical power grid the vertices being generators ,transformers, and substations
the edges being to the high-voltage transmission lines
For 4941 vertices :
Уpower = 4
L = 18.99
C=0.10
72Slide73
The small-world and scale-free features are common to many real-world complex networks.
Network
Size
Clustering Coefficient
Average path length
Degree exponent
Internet, domain level
32711
0.24
3.56
2.1
Internet, router level
2282980.039.512.1
WWW153127
0.11
3.1
In=2.1, out=2.45
E-mail
56969
0.03
4.95
1.81
Software
1376
0.06
6.39
2.5
Electronic Circuits
329
0.34
3.17
2.5
Movie Actors
225226
0.79
3.65
2.3
Food Web
154
0.15
3.40
1.13
Language
460902
0.437
2.67
2.7
73Slide74
74
Some more Applications Areas
Real-World Applications
• Peer-to-Peer File-sharing System
• find a file <=> find a person
• Focused Web Crawling
• how to efficiently find a webpage?• Social Network Data
• Is naturally occurring networks organized in same way?Slide75
Classes of Complex Networks (contd..)
3.Scale-Free Models (Rich get richer)
Number of large-scale complex
networks have connectivity
distributions have a power-law
form are scale free like:
the Internet,WWW, and
citation networks
The BA model suggests that two
main ingredients of a scale-free
structure are:
growth
preferential attachment
75Slide76
Classes of Complex Networks (contd..)
In most large-scale real networks the degree distribution deviates significantly from the Poisson distribution.
For a number of networks, the degree distribution can be described by a power law :
P(k) ∼ k−γ
Power-law distribution falls off more gradually than an exponential one
Allowing for a few nodes of very large degree to exist.
A network with a power-law degree distribution is called a scale-free network.
Some striking differences between an exponential network and a scale-free network can be seen by comparing a U.S. roadmap with an airline routing map, shown in Fig. in next slide
76Slide77
(Courtesy of A.L. Barabasi) US Road Map US Airline Routing map
(Exponential Network) (Power-LawDistribution)
77Slide78
Decision parameters and its definitions (contd..)
Cohen and Havlin proved that uncorrelated power law graphs having 2 < γ < 3 will also have ultrasmall
diameter
d
~ ln ln
N. So , the diameter of a growing scale-free network might be considered almost constant.
Scale free networks are still small world networks because:
(i) they have clustering coefficients much larger than random networks,
(ii) their diameter increases logarithmically with the number of vertices.
78Slide79
BA Scale-Free Model Algorithm
1) Growth: Start with a small number (m0) of nodes; at every time step, a new node is introduced and is connected to m ≤ m0 already-existing nodes.
2) Preferential Attachment: The probability Πi that a new node will be connected to node i (one of the m already-existing nodes) depends on the degree ki of node i, in such a way that Πi = ki/ Σjkj .
79Slide80
Applications
Social Network of movie actors
Two actors being connected if they were cast together in the same movie.
The probability that an actor has k links (characterizing his or her popularity) has a power-law tail for large k, :
P(k) ~k-Уactor ,
where ,Уactor = 2.3 ± 0.1
C=0.79, L = 3.65so, it also follows the small world property as well as scale free network.
A more complex network with over 800 million vertices is the WWW
,
where a vertex is a document
the edges are the links pointing from one document to another.
The topology of this graph determines the Web’s connectivity Information about P(k) ,indicating that the probability that k documents point to a certain Webpage follows a power law, with Уwww = 2.1 ± 0.1.
C = 0.11, L = 3.1
80Slide81
81
Some more Applications Areas
Real-World Applications
• Peer-to-Peer File-sharing System
• find a file <=> find a person
• Focused Web Crawling
• how to efficiently find a webpage?• Social Network Data
• Is naturally occurring networks organized in same way?Slide82
Bibliography
Reviews
Barabási, A.-L. (2002)
Linked: The New Science of Networks.
Perseus Books.
Barabási, A.-L. and Bonabeau, E. (2003) Scale-free networks. Scientific American, 288
: 60-69.Strogatz, S. H. (2001) Exploring complex networks.
Nature
,
410
(6825): 268-276.
Wang, X. F. (2002) Complex networks: topology, dynamics and synchronization. International Journal of Bifurcation and Chaos
, 12(5): 885-916.Newman M. E. J. (2003) The structure and function of complex networks. arXiv:cond-mat/0303516v1
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