1.
Outline
Introduction
Topological Features
Philosophy
Design
Analyses
Conclusions
An Overview of Complex Networks Design and
Analyses
Sanket Patil
May 12, 2008
Sanket Patil An Overview of Complex Networks Design and Analyses
2.
Outline
Introduction
Topological Features
Philosophy
Design
Analyses
Conclusions
1 Introduction
Deﬁnition
Representation
2 Topological Features
Scale-free Nature
“Small World” properties
Other features
3 Philosophy
Emergence
Machines vs Societies
Self Interest
4 Design
Systems Design: Desiderata
Complex Systems Design
5 Analyses
6 Conclusions
Sanket Patil An Overview of Complex Networks Design and Analyses
3.
Outline
Introduction
Topological Features
Deﬁnition
Philosophy
Representation
Design
Analyses
Conclusions
Complex Networks
Not “simple networks”
Not random graphs
Have non-trivial topological features
Sanket Patil An Overview of Complex Networks Design and Analyses
4.
Outline
Introduction
Topological Features
Deﬁnition
Philosophy
Representation
Design
Analyses
Conclusions
Complex Networks
Not “simple networks”
Not random graphs
Have non-trivial topological features
Sanket Patil An Overview of Complex Networks Design and Analyses
5.
Outline
Introduction
Topological Features
Deﬁnition
Philosophy
Representation
Design
Analyses
Conclusions
Complex Networks
Not “simple networks”
Not random graphs
Have non-trivial topological features
Sanket Patil An Overview of Complex Networks Design and Analyses
6.
Outline
Introduction
Topological Features
Deﬁnition
Philosophy
Representation
Design
Analyses
Conclusions
Examples
Computer Networks, the Internet, the Web
Food-webs, protein interaction networks
Social Networks, Trade Networks
The brain
Sanket Patil An Overview of Complex Networks Design and Analyses
7.
Outline
Introduction
Topological Features
Deﬁnition
Philosophy
Representation
Design
Analyses
Conclusions
Examples
Computer Networks, the Internet, the Web
Food-webs, protein interaction networks
Social Networks, Trade Networks
The brain
Sanket Patil An Overview of Complex Networks Design and Analyses
8.
Outline
Introduction
Topological Features
Deﬁnition
Philosophy
Representation
Design
Analyses
Conclusions
Examples
Computer Networks, the Internet, the Web
Food-webs, protein interaction networks
Social Networks, Trade Networks
The brain
Sanket Patil An Overview of Complex Networks Design and Analyses
9.
Outline
Introduction
Topological Features
Deﬁnition
Philosophy
Representation
Design
Analyses
Conclusions
Examples
Computer Networks, the Internet, the Web
Food-webs, protein interaction networks
Social Networks, Trade Networks
The brain
Sanket Patil An Overview of Complex Networks Design and Analyses
10.
Outline
Introduction
Topological Features
Deﬁnition
Philosophy
Representation
Design
Analyses
Conclusions
Complex Networks
Designed (for performance)
Have evolved/emerged over time
Are teleological / have purpose of life
Sanket Patil An Overview of Complex Networks Design and Analyses
11.
Outline
Introduction
Topological Features
Deﬁnition
Philosophy
Representation
Design
Analyses
Conclusions
Complex Networks
Designed (for performance)
Have evolved/emerged over time
Are teleological / have purpose of life
Sanket Patil An Overview of Complex Networks Design and Analyses
12.
Outline
Introduction
Topological Features
Deﬁnition
Philosophy
Representation
Design
Analyses
Conclusions
Complex Networks
Designed (for performance)
Have evolved/emerged over time
Are teleological / have purpose of life
Sanket Patil An Overview of Complex Networks Design and Analyses
13.
Outline
Introduction
Topological Features
Deﬁnition
Philosophy
Representation
Design
Analyses
Conclusions
Representation
Networks are modelled as “Graphs”
A node/vertex/point represents a machine, a human, a cell
etc.
An edge/arc/line represents a relation between two nodes
Edges can be undirected or directed.
Weights are used to convey additional (extra-topological)
information
Sanket Patil An Overview of Complex Networks Design and Analyses
14.
Outline
Introduction
Topological Features
Deﬁnition
Philosophy
Representation
Design
Analyses
Conclusions
Basic Deﬁnitions
Degree: The number of edges incident on a node
Indegree (outdegree): Number of incoming (outgoing)
edges
Degree Distribution: A probability distribution of node
degrees
Sanket Patil An Overview of Complex Networks Design and Analyses
15.
Outline
Introduction
Topological Features
Deﬁnition
Philosophy
Representation
Design
Analyses
Conclusions
Basic Deﬁnitions
Degree: The number of edges incident on a node
Indegree (outdegree): Number of incoming (outgoing)
edges
Degree Distribution: A probability distribution of node
degrees
Sanket Patil An Overview of Complex Networks Design and Analyses
16.
Outline
Introduction
Topological Features
Deﬁnition
Philosophy
Representation
Design
Analyses
Conclusions
Path: A sequence of adjacent vertices
Pathlength: The number of edges in a path
Sanket Patil An Overview of Complex Networks Design and Analyses
17.
Outline
Introduction
Topological Features Scale-free Nature
Philosophy “Small World” properties
Design Other features
Analyses
Conclusions
Scale-free Nature
Network dynamics/behaviour is independent of the size
Power Law degree distributions
Sanket Patil An Overview of Complex Networks Design and Analyses
18.
Outline
Introduction
Topological Features Scale-free Nature
Philosophy “Small World” properties
Design Other features
Analyses
Conclusions
Scale-free Nature
Network dynamics/behaviour is independent of the size
Power Law degree distributions
Sanket Patil An Overview of Complex Networks Design and Analyses
19.
Outline
Introduction
Topological Features Scale-free Nature
Philosophy “Small World” properties
Design Other features
Analyses
Conclusions
Power Law Distribution
Sanket Patil An Overview of Complex Networks Design and Analyses
20.
Outline
Introduction
Topological Features Scale-free Nature
Philosophy “Small World” properties
Design Other features
Analyses
Conclusions
Properties
P[X = k] α k −p
Small number of nodes with a very high degree and a large
number of nodes with a very low degree
“heavy tail”, “long tail”, “80:20”
Income distributions, page rank, wikipedia contribution
Sanket Patil An Overview of Complex Networks Design and Analyses
21.
Outline
Introduction
Topological Features Scale-free Nature
Philosophy “Small World” properties
Design Other features
Analyses
Conclusions
Properties
P[X = k] α k −p
Small number of nodes with a very high degree and a large
number of nodes with a very low degree
“heavy tail”, “long tail”, “80:20”
Income distributions, page rank, wikipedia contribution
Sanket Patil An Overview of Complex Networks Design and Analyses
22.
Outline
Introduction
Topological Features Scale-free Nature
Philosophy “Small World” properties
Design Other features
Analyses
Conclusions
Properties
P[X = k] α k −p
Small number of nodes with a very high degree and a large
number of nodes with a very low degree
“heavy tail”, “long tail”, “80:20”
Income distributions, page rank, wikipedia contribution
Sanket Patil An Overview of Complex Networks Design and Analyses
23.
Outline
Introduction
Topological Features Scale-free Nature
Philosophy “Small World” properties
Design Other features
Analyses
Conclusions
Properties
P[X = k] α k −p
Small number of nodes with a very high degree and a large
number of nodes with a very low degree
“heavy tail”, “long tail”, “80:20”
Income distributions, page rank, wikipedia contribution
Sanket Patil An Overview of Complex Networks Design and Analyses
24.
Outline
Introduction
Topological Features Scale-free Nature
Philosophy “Small World” properties
Design Other features
Analyses
Conclusions
Preferential Attachment
Nodes prefer to attach to “popular” nodes
“Rich getting richer”
Entrenchment
Sanket Patil An Overview of Complex Networks Design and Analyses
25.
Outline
Introduction
Topological Features Scale-free Nature
Philosophy “Small World” properties
Design Other features
Analyses
Conclusions
Preferential Attachment
Nodes prefer to attach to “popular” nodes
“Rich getting richer”
Entrenchment
Sanket Patil An Overview of Complex Networks Design and Analyses
26.
Outline
Introduction
Topological Features Scale-free Nature
Philosophy “Small World” properties
Design Other features
Analyses
Conclusions
Preferential Attachment
Nodes prefer to attach to “popular” nodes
“Rich getting richer”
Entrenchment
Sanket Patil An Overview of Complex Networks Design and Analyses
27.
Outline
Introduction
Topological Features Scale-free Nature
Philosophy “Small World” properties
Design Other features
Analyses
Conclusions
“Small World” Properties
Stanley Milgram: Small world experiment
Mean Path Length 6
Critique: not a comprehensive study
“Six degrees of separation”
Watts and Strogatz
Sanket Patil An Overview of Complex Networks Design and Analyses
28.
Outline
Introduction
Topological Features Scale-free Nature
Philosophy “Small World” properties
Design Other features
Analyses
Conclusions
“Small World” Properties
Stanley Milgram: Small world experiment
Mean Path Length 6
Critique: not a comprehensive study
“Six degrees of separation”
Watts and Strogatz
Sanket Patil An Overview of Complex Networks Design and Analyses
29.
Outline
Introduction
Topological Features Scale-free Nature
Philosophy “Small World” properties
Design Other features
Analyses
Conclusions
“Small World” Properties
Stanley Milgram: Small world experiment
Mean Path Length 6
Critique: not a comprehensive study
“Six degrees of separation”
Watts and Strogatz
Sanket Patil An Overview of Complex Networks Design and Analyses
30.
Outline
Introduction
Topological Features Scale-free Nature
Philosophy “Small World” properties
Design Other features
Analyses
Conclusions
“Small World” Properties
Stanley Milgram: Small world experiment
Mean Path Length 6
Critique: not a comprehensive study
“Six degrees of separation”
Watts and Strogatz
Sanket Patil An Overview of Complex Networks Design and Analyses
31.
Outline
Introduction
Topological Features Scale-free Nature
Philosophy “Small World” properties
Design Other features
Analyses
Conclusions
“Small World” Properties
Stanley Milgram: Small world experiment
Mean Path Length 6
Critique: not a comprehensive study
“Six degrees of separation”
Watts and Strogatz
Sanket Patil An Overview of Complex Networks Design and Analyses
32.
Outline
Introduction
Topological Features Scale-free Nature
Philosophy “Small World” properties
Design Other features
Analyses
Conclusions
Saul Steinberg: Ninth Avenue
Sanket Patil An Overview of Complex Networks Design and Analyses
33.
Outline
Introduction
Topological Features Scale-free Nature
Philosophy “Small World” properties
Design Other features
Analyses
Conclusions
Social Commentary
Small world geometry
Sanket Patil An Overview of Complex Networks Design and Analyses
34.
Outline
Introduction
Topological Features Scale-free Nature
Philosophy “Small World” properties
Design Other features
Analyses
Conclusions
Social Commentary
Small world geometry
Sanket Patil An Overview of Complex Networks Design and Analyses
35.
Outline
Introduction
Topological Features Scale-free Nature
Philosophy “Small World” properties
Design Other features
Analyses
Conclusions
Kleinberg’s “small world”
Ninth Avenue as a powerful analogy
The “far” is almost as accessible as the “near”
How are your friends/acquaintances distributed?
Sanket Patil An Overview of Complex Networks Design and Analyses
36.
Outline
Introduction
Topological Features Scale-free Nature
Philosophy “Small World” properties
Design Other features
Analyses
Conclusions
Kleinberg’s “small world”
Ninth Avenue as a powerful analogy
The “far” is almost as accessible as the “near”
How are your friends/acquaintances distributed?
Sanket Patil An Overview of Complex Networks Design and Analyses
37.
Outline
Introduction
Topological Features Scale-free Nature
Philosophy “Small World” properties
Design Other features
Analyses
Conclusions
Kleinberg’s “small world”
Ninth Avenue as a powerful analogy
The “far” is almost as accessible as the “near”
How are your friends/acquaintances distributed?
Sanket Patil An Overview of Complex Networks Design and Analyses
38.
Outline
Introduction
Topological Features Scale-free Nature
Philosophy “Small World” properties
Design Other features
Analyses
Conclusions
Clustering
Sanket Patil An Overview of Complex Networks Design and Analyses
39.
Outline
Introduction
Topological Features Scale-free Nature
Philosophy “Small World” properties
Design Other features
Analyses
Conclusions
Clustering Coeﬃcient
Neighbourhood: Nodes that are adjacent to a node
no of edges in the neighbourhood
Ci = total possible edges
n
i=0 Ci
Cgraph = n
Sanket Patil An Overview of Complex Networks Design and Analyses
40.
Outline
Introduction
Topological Features Scale-free Nature
Philosophy “Small World” properties
Design Other features
Analyses
Conclusions
Clustering Coeﬃcient
Neighbourhood: Nodes that are adjacent to a node
no of edges in the neighbourhood
Ci = total possible edges
n
i=0 Ci
Cgraph = n
Sanket Patil An Overview of Complex Networks Design and Analyses
41.
Outline
Introduction
Topological Features Scale-free Nature
Philosophy “Small World” properties
Design Other features
Analyses
Conclusions
Clustering Coeﬃcient
Neighbourhood: Nodes that are adjacent to a node
no of edges in the neighbourhood
Ci = total possible edges
n
i=0 Ci
Cgraph = n
Sanket Patil An Overview of Complex Networks Design and Analyses
42.
Outline
Introduction
Topological Features Scale-free Nature
Philosophy “Small World” properties
Design Other features
Analyses
Conclusions
Small Worlds
High clustering coeﬃcient
Low average path length
Sanket Patil An Overview of Complex Networks Design and Analyses
43.
Outline
Introduction
Topological Features Scale-free Nature
Philosophy “Small World” properties
Design Other features
Analyses
Conclusions
Navigability in small worlds
Kleinberg’s metric space models (circa 2000)
Short paths do exist
But can we ﬁnd them using local information?
Sanket Patil An Overview of Complex Networks Design and Analyses
44.
Outline
Introduction
Topological Features Scale-free Nature
Philosophy “Small World” properties
Design Other features
Analyses
Conclusions
Navigability in small worlds
Kleinberg’s metric space models (circa 2000)
Short paths do exist
But can we ﬁnd them using local information?
Sanket Patil An Overview of Complex Networks Design and Analyses
45.
Outline
Introduction
Topological Features Scale-free Nature
Philosophy “Small World” properties
Design Other features
Analyses
Conclusions
Navigability in small worlds
Kleinberg’s metric space models (circa 2000)
Short paths do exist
But can we ﬁnd them using local information?
Sanket Patil An Overview of Complex Networks Design and Analyses
46.
Outline
Introduction
Topological Features Scale-free Nature
Philosophy “Small World” properties
Design Other features
Analyses
Conclusions
Short range and long range connections
Sanket Patil An Overview of Complex Networks Design and Analyses
47.
Outline
Introduction
Topological Features Scale-free Nature
Philosophy “Small World” properties
Design Other features
Analyses
Conclusions
Clustering Exponent
Connections based on distance (r) and clustering exponent α
For a node u, the probability of connecting to v is r −α
Highly clustered neighbourhood
Number of long range links decays with distance
Only when α = 2, a decentralized routing algorithm can be
found which has a log n bound
Sanket Patil An Overview of Complex Networks Design and Analyses
48.
Outline
Introduction
Topological Features Scale-free Nature
Philosophy “Small World” properties
Design Other features
Analyses
Conclusions
Clustering Exponent
Connections based on distance (r) and clustering exponent α
For a node u, the probability of connecting to v is r −α
Highly clustered neighbourhood
Number of long range links decays with distance
Only when α = 2, a decentralized routing algorithm can be
found which has a log n bound
Sanket Patil An Overview of Complex Networks Design and Analyses
49.
Outline
Introduction
Topological Features Scale-free Nature
Philosophy “Small World” properties
Design Other features
Analyses
Conclusions
Clustering Exponent
Connections based on distance (r) and clustering exponent α
For a node u, the probability of connecting to v is r −α
Highly clustered neighbourhood
Number of long range links decays with distance
Only when α = 2, a decentralized routing algorithm can be
found which has a log n bound
Sanket Patil An Overview of Complex Networks Design and Analyses
50.
Outline
Introduction
Topological Features Scale-free Nature
Philosophy “Small World” properties
Design Other features
Analyses
Conclusions
Clustering Exponent
Connections based on distance (r) and clustering exponent α
For a node u, the probability of connecting to v is r −α
Highly clustered neighbourhood
Number of long range links decays with distance
Only when α = 2, a decentralized routing algorithm can be
found which has a log n bound
Sanket Patil An Overview of Complex Networks Design and Analyses
51.
Outline
Introduction
Topological Features Scale-free Nature
Philosophy “Small World” properties
Design Other features
Analyses
Conclusions
Clustering Exponent
Connections based on distance (r) and clustering exponent α
For a node u, the probability of connecting to v is r −α
Highly clustered neighbourhood
Number of long range links decays with distance
Only when α = 2, a decentralized routing algorithm can be
found which has a log n bound
Sanket Patil An Overview of Complex Networks Design and Analyses
52.
Outline
Introduction
Topological Features Scale-free Nature
Philosophy “Small World” properties
Design Other features
Analyses
Conclusions
Finding short paths
Sanket Patil An Overview of Complex Networks Design and Analyses
53.
Outline
Introduction
Topological Features Scale-free Nature
Philosophy “Small World” properties
Design Other features
Analyses
Conclusions
Other Features
Communities
Hierarchical Structures
Sanket Patil An Overview of Complex Networks Design and Analyses
54.
Outline
Introduction
Topological Features Scale-free Nature
Philosophy “Small World” properties
Design Other features
Analyses
Conclusions
Other Features
Communities
Hierarchical Structures
Sanket Patil An Overview of Complex Networks Design and Analyses
55.
Outline
Introduction
Topological Features Emergence
Philosophy Machines vs Societies
Design Self Interest
Analyses
Conclusions
Complex Systems: Philosophy
Systems Theory
Cybernetics
Non-linear Dynamics
Multiagent Systems and AI
Network Science, Connectionism, Cognitive Psychology
Sanket Patil An Overview of Complex Networks Design and Analyses
56.
Outline
Introduction
Topological Features Emergence
Philosophy Machines vs Societies
Design Self Interest
Analyses
Conclusions
Complex Systems: Philosophy
Systems Theory
Cybernetics
Non-linear Dynamics
Multiagent Systems and AI
Network Science, Connectionism, Cognitive Psychology
Sanket Patil An Overview of Complex Networks Design and Analyses
57.
Outline
Introduction
Topological Features Emergence
Philosophy Machines vs Societies
Design Self Interest
Analyses
Conclusions
Complex Systems: Philosophy
Systems Theory
Cybernetics
Non-linear Dynamics
Multiagent Systems and AI
Network Science, Connectionism, Cognitive Psychology
Sanket Patil An Overview of Complex Networks Design and Analyses
58.
Outline
Introduction
Topological Features Emergence
Philosophy Machines vs Societies
Design Self Interest
Analyses
Conclusions
Complex Systems: Philosophy
Systems Theory
Cybernetics
Non-linear Dynamics
Multiagent Systems and AI
Network Science, Connectionism, Cognitive Psychology
Sanket Patil An Overview of Complex Networks Design and Analyses
59.
Outline
Introduction
Topological Features Emergence
Philosophy Machines vs Societies
Design Self Interest
Analyses
Conclusions
Complex Systems: Philosophy
Systems Theory
Cybernetics
Non-linear Dynamics
Multiagent Systems and AI
Network Science, Connectionism, Cognitive Psychology
Sanket Patil An Overview of Complex Networks Design and Analyses
60.
Outline
Introduction
Topological Features Emergence
Philosophy Machines vs Societies
Design Self Interest
Analyses
Conclusions
Strudy of Complex Networks
Structure or Topology
How structure governs the function
How design can be analysed teleologically
Can we use this knowledge to build better systems?
Sanket Patil An Overview of Complex Networks Design and Analyses
61.
Outline
Introduction
Topological Features Emergence
Philosophy Machines vs Societies
Design Self Interest
Analyses
Conclusions
Strudy of Complex Networks
Structure or Topology
How structure governs the function
How design can be analysed teleologically
Can we use this knowledge to build better systems?
Sanket Patil An Overview of Complex Networks Design and Analyses
62.
Outline
Introduction
Topological Features Emergence
Philosophy Machines vs Societies
Design Self Interest
Analyses
Conclusions
Strudy of Complex Networks
Structure or Topology
How structure governs the function
How design can be analysed teleologically
Can we use this knowledge to build better systems?
Sanket Patil An Overview of Complex Networks Design and Analyses
63.
Outline
Introduction
Topological Features Emergence
Philosophy Machines vs Societies
Design Self Interest
Analyses
Conclusions
Strudy of Complex Networks
Structure or Topology
How structure governs the function
How design can be analysed teleologically
Can we use this knowledge to build better systems?
Sanket Patil An Overview of Complex Networks Design and Analyses
64.
Outline
Introduction
Topological Features Emergence
Philosophy Machines vs Societies
Design Self Interest
Analyses
Conclusions
Emergence
The connections between the components is as important as
the components
Emergent behaviour of complex networks
Case Study: The Human Brain
Sanket Patil An Overview of Complex Networks Design and Analyses
65.
Outline
Introduction
Topological Features Emergence
Philosophy Machines vs Societies
Design Self Interest
Analyses
Conclusions
Emergence
The connections between the components is as important as
the components
Emergent behaviour of complex networks
Case Study: The Human Brain
Sanket Patil An Overview of Complex Networks Design and Analyses
66.
Outline
Introduction
Topological Features Emergence
Philosophy Machines vs Societies
Design Self Interest
Analyses
Conclusions
Emergence
The connections between the components is as important as
the components
Emergent behaviour of complex networks
Case Study: The Human Brain
Sanket Patil An Overview of Complex Networks Design and Analyses
67.
Outline
Introduction
Topological Features Emergence
Philosophy Machines vs Societies
Design Self Interest
Analyses
Conclusions
Machines vs Societies
Systems Thinking: parts and whole
Machines funcion based on norms
Societies are declarative
Societies are more robust
Case study: The Heart
Sanket Patil An Overview of Complex Networks Design and Analyses
68.
Outline
Introduction
Topological Features Emergence
Philosophy Machines vs Societies
Design Self Interest
Analyses
Conclusions
Machines vs Societies
Systems Thinking: parts and whole
Machines funcion based on norms
Societies are declarative
Societies are more robust
Case study: The Heart
Sanket Patil An Overview of Complex Networks Design and Analyses
69.
Outline
Introduction
Topological Features Emergence
Philosophy Machines vs Societies
Design Self Interest
Analyses
Conclusions
Machines vs Societies
Systems Thinking: parts and whole
Machines funcion based on norms
Societies are declarative
Societies are more robust
Case study: The Heart
Sanket Patil An Overview of Complex Networks Design and Analyses
70.
Outline
Introduction
Topological Features Emergence
Philosophy Machines vs Societies
Design Self Interest
Analyses
Conclusions
Machines vs Societies
Systems Thinking: parts and whole
Machines funcion based on norms
Societies are declarative
Societies are more robust
Case study: The Heart
Sanket Patil An Overview of Complex Networks Design and Analyses
71.
Outline
Introduction
Topological Features Emergence
Philosophy Machines vs Societies
Design Self Interest
Analyses
Conclusions
Machines vs Societies
Systems Thinking: parts and whole
Machines funcion based on norms
Societies are declarative
Societies are more robust
Case study: The Heart
Sanket Patil An Overview of Complex Networks Design and Analyses
72.
Outline
Introduction
Topological Features Emergence
Philosophy Machines vs Societies
Design Self Interest
Analyses
Conclusions
Self Interest
Autonomous agents
Local constraints and self interest
Global or environmental dampeners
Sanket Patil An Overview of Complex Networks Design and Analyses
73.
Outline
Introduction
Topological Features Emergence
Philosophy Machines vs Societies
Design Self Interest
Analyses
Conclusions
Self Interest
Autonomous agents
Local constraints and self interest
Global or environmental dampeners
Sanket Patil An Overview of Complex Networks Design and Analyses
74.
Outline
Introduction
Topological Features Emergence
Philosophy Machines vs Societies
Design Self Interest
Analyses
Conclusions
Self Interest
Autonomous agents
Local constraints and self interest
Global or environmental dampeners
Sanket Patil An Overview of Complex Networks Design and Analyses
75.
Outline
Introduction
Topological Features
Systems Design: Desiderata
Philosophy
Complex Systems Design
Design
Analyses
Conclusions
Systems Design: Desiderata
Safety
Resilience
Fairness
Livenss
Sanket Patil An Overview of Complex Networks Design and Analyses
76.
Outline
Introduction
Topological Features
Systems Design: Desiderata
Philosophy
Complex Systems Design
Design
Analyses
Conclusions
Systems Design: Desiderata
Safety
Resilience
Fairness
Livenss
Sanket Patil An Overview of Complex Networks Design and Analyses
77.
Outline
Introduction
Topological Features
Systems Design: Desiderata
Philosophy
Complex Systems Design
Design
Analyses
Conclusions
Systems Design: Desiderata
Safety
Resilience
Fairness
Livenss
Sanket Patil An Overview of Complex Networks Design and Analyses
78.
Outline
Introduction
Topological Features
Systems Design: Desiderata
Philosophy
Complex Systems Design
Design
Analyses
Conclusions
Systems Design: Desiderata
Safety
Resilience
Fairness
Livenss
Sanket Patil An Overview of Complex Networks Design and Analyses
79.
Outline
Introduction
Topological Features
Systems Design: Desiderata
Philosophy
Complex Systems Design
Design
Analyses
Conclusions
Complex Systems Design
Eﬃciency
Robustness
Cost
Sanket Patil An Overview of Complex Networks Design and Analyses
80.
Outline
Introduction
Topological Features
Systems Design: Desiderata
Philosophy
Complex Systems Design
Design
Analyses
Conclusions
Complex Systems Design
Eﬃciency
Robustness
Cost
Sanket Patil An Overview of Complex Networks Design and Analyses
81.
Outline
Introduction
Topological Features
Systems Design: Desiderata
Philosophy
Complex Systems Design
Design
Analyses
Conclusions
Complex Systems Design
Eﬃciency
Robustness
Cost
Sanket Patil An Overview of Complex Networks Design and Analyses
82.
Outline
Introduction
Topological Features
Systems Design: Desiderata
Philosophy
Complex Systems Design
Design
Analyses
Conclusions
An Optimization Problem
Designing complex systems is an optmization process
Search in a multidimensional space
Sanket Patil An Overview of Complex Networks Design and Analyses
83.
Outline
Introduction
Topological Features
Systems Design: Desiderata
Philosophy
Complex Systems Design
Design
Analyses
Conclusions
An Optimization Problem
Designing complex systems is an optmization process
Search in a multidimensional space
Sanket Patil An Overview of Complex Networks Design and Analyses
84.
Outline
Introduction
Topological Features
Systems Design: Desiderata
Philosophy
Complex Systems Design
Design
Analyses
Conclusions
Approaches
Mathematical Programming
Ant Colony Optimization
Swarm Intelligence
Simulated Annealing
Genetic Algorithms
Sanket Patil An Overview of Complex Networks Design and Analyses
85.
Outline
Introduction
Topological Features
Philosophy
Design
Analyses
Conclusions
Graph Theoretic Analyses
Graph thoretic properties used in design and analyses
Constraints and objectives are deﬁned in terms of graph
properties
Design is usually evolution of optimal graphs under the given
constraints
Sanket Patil An Overview of Complex Networks Design and Analyses
86.
Outline
Introduction
Topological Features
Philosophy
Design
Analyses
Conclusions
Graph Theoretic Analyses
Graph thoretic properties used in design and analyses
Constraints and objectives are deﬁned in terms of graph
properties
Design is usually evolution of optimal graphs under the given
constraints
Sanket Patil An Overview of Complex Networks Design and Analyses
87.
Outline
Introduction
Topological Features
Philosophy
Design
Analyses
Conclusions
Graph Theoretic Analyses
Graph thoretic properties used in design and analyses
Constraints and objectives are deﬁned in terms of graph
properties
Design is usually evolution of optimal graphs under the given
constraints
Sanket Patil An Overview of Complex Networks Design and Analyses
88.
Outline
Introduction
Topological Features
Philosophy
Design
Analyses
Conclusions
Eﬃciency
APL: Average of all pairs shortest paths
Diameter: The longest shortest path. An upper bound on
eﬃciency.
Sanket Patil An Overview of Complex Networks Design and Analyses
89.
Outline
Introduction
Topological Features
Philosophy
Design
Analyses
Conclusions
Eﬃciency
APL: Average of all pairs shortest paths
Diameter: The longest shortest path. An upper bound on
eﬃciency.
Sanket Patil An Overview of Complex Networks Design and Analyses
90.
Outline
Introduction
Topological Features
Philosophy
Design
Analyses
Conclusions
Local Objectives
Eccentricity: The longest shortest path for a node
The greatest separation a node suﬀers
Average eccentricity or eccentricity distribution are also good
indicators of eﬃciency
Radius: Smallest eccentricity. “Central” node
Number of “central” nodes can be another measure
Sanket Patil An Overview of Complex Networks Design and Analyses
91.
Outline
Introduction
Topological Features
Philosophy
Design
Analyses
Conclusions
Local Objectives
Eccentricity: The longest shortest path for a node
The greatest separation a node suﬀers
Average eccentricity or eccentricity distribution are also good
indicators of eﬃciency
Radius: Smallest eccentricity. “Central” node
Number of “central” nodes can be another measure
Sanket Patil An Overview of Complex Networks Design and Analyses
92.
Outline
Introduction
Topological Features
Philosophy
Design
Analyses
Conclusions
Local Objectives
Eccentricity: The longest shortest path for a node
The greatest separation a node suﬀers
Average eccentricity or eccentricity distribution are also good
indicators of eﬃciency
Radius: Smallest eccentricity. “Central” node
Number of “central” nodes can be another measure
Sanket Patil An Overview of Complex Networks Design and Analyses
93.
Outline
Introduction
Topological Features
Philosophy
Design
Analyses
Conclusions
Local Objectives
Eccentricity: The longest shortest path for a node
The greatest separation a node suﬀers
Average eccentricity or eccentricity distribution are also good
indicators of eﬃciency
Radius: Smallest eccentricity. “Central” node
Number of “central” nodes can be another measure
Sanket Patil An Overview of Complex Networks Design and Analyses
94.
Outline
Introduction
Topological Features
Philosophy
Design
Analyses
Conclusions
Local Objectives
Eccentricity: The longest shortest path for a node
The greatest separation a node suﬀers
Average eccentricity or eccentricity distribution are also good
indicators of eﬃciency
Radius: Smallest eccentricity. “Central” node
Number of “central” nodes can be another measure
Sanket Patil An Overview of Complex Networks Design and Analyses
95.
Outline
Introduction
Topological Features
Philosophy
Design
Analyses
Conclusions
Cost
Density(connectance): no of edges/no of possible edges
Edges per node
Weights, costs and other environment dependent measures
Sanket Patil An Overview of Complex Networks Design and Analyses
96.
Outline
Introduction
Topological Features
Philosophy
Design
Analyses
Conclusions
Cost
Density(connectance): no of edges/no of possible edges
Edges per node
Weights, costs and other environment dependent measures
Sanket Patil An Overview of Complex Networks Design and Analyses
97.
Outline
Introduction
Topological Features
Philosophy
Design
Analyses
Conclusions
Cost
Density(connectance): no of edges/no of possible edges
Edges per node
Weights, costs and other environment dependent measures
Sanket Patil An Overview of Complex Networks Design and Analyses
98.
Outline
Introduction
Topological Features
Philosophy
Design
Analyses
Conclusions
Robustness
Centrality Measures
Connectivity
Sanket Patil An Overview of Complex Networks Design and Analyses
99.
Outline
Introduction
Topological Features
Philosophy
Design
Analyses
Conclusions
Robustness
Centrality Measures
Connectivity
Sanket Patil An Overview of Complex Networks Design and Analyses
100.
Outline
Introduction
Topological Features
Philosophy
Design
Analyses
Conclusions
Centrality
Degree Centrality: degree distribution
Betweenness: importance of nodes based on the no. of
paths passing through them
Closeness: per node average path length
Eigenvector Centrality: importance of nodes based not just
on how many are endorsing, but also who is connected.
When a centrality distribution is uniform, the network is most
robust
Sanket Patil An Overview of Complex Networks Design and Analyses
101.
Outline
Introduction
Topological Features
Philosophy
Design
Analyses
Conclusions
Centrality
Degree Centrality: degree distribution
Betweenness: importance of nodes based on the no. of
paths passing through them
Closeness: per node average path length
Eigenvector Centrality: importance of nodes based not just
on how many are endorsing, but also who is connected.
When a centrality distribution is uniform, the network is most
robust
Sanket Patil An Overview of Complex Networks Design and Analyses
102.
Outline
Introduction
Topological Features
Philosophy
Design
Analyses
Conclusions
Centrality
Degree Centrality: degree distribution
Betweenness: importance of nodes based on the no. of
paths passing through them
Closeness: per node average path length
Eigenvector Centrality: importance of nodes based not just
on how many are endorsing, but also who is connected.
When a centrality distribution is uniform, the network is most
robust
Sanket Patil An Overview of Complex Networks Design and Analyses
103.
Outline
Introduction
Topological Features
Philosophy
Design
Analyses
Conclusions
Centrality
Degree Centrality: degree distribution
Betweenness: importance of nodes based on the no. of
paths passing through them
Closeness: per node average path length
Eigenvector Centrality: importance of nodes based not just
on how many are endorsing, but also who is connected.
When a centrality distribution is uniform, the network is most
robust
Sanket Patil An Overview of Complex Networks Design and Analyses
104.
Outline
Introduction
Topological Features
Philosophy
Design
Analyses
Conclusions
Centrality
Degree Centrality: degree distribution
Betweenness: importance of nodes based on the no. of
paths passing through them
Closeness: per node average path length
Eigenvector Centrality: importance of nodes based not just
on how many are endorsing, but also who is connected.
When a centrality distribution is uniform, the network is most
robust
Sanket Patil An Overview of Complex Networks Design and Analyses
105.
Outline
Introduction
Topological Features
Philosophy
Design
Analyses
Conclusions
Connectivity
Connected, strongly connected and weakly connected
Vertex Cut: The smallest number of vertices whose removal
renders the network disconnected
Edge Cut: The smallest number od edges whose removal
renders the network disconnected
A graph is k − connected if the size of its vertex cut is k
Higher the connectivity, more robust the network
Sanket Patil An Overview of Complex Networks Design and Analyses
106.
Outline
Introduction
Topological Features
Philosophy
Design
Analyses
Conclusions
Connectivity
Connected, strongly connected and weakly connected
Vertex Cut: The smallest number of vertices whose removal
renders the network disconnected
Edge Cut: The smallest number od edges whose removal
renders the network disconnected
A graph is k − connected if the size of its vertex cut is k
Higher the connectivity, more robust the network
Sanket Patil An Overview of Complex Networks Design and Analyses
107.
Outline
Introduction
Topological Features
Philosophy
Design
Analyses
Conclusions
Connectivity
Connected, strongly connected and weakly connected
Vertex Cut: The smallest number of vertices whose removal
renders the network disconnected
Edge Cut: The smallest number od edges whose removal
renders the network disconnected
A graph is k − connected if the size of its vertex cut is k
Higher the connectivity, more robust the network
Sanket Patil An Overview of Complex Networks Design and Analyses
108.
Outline
Introduction
Topological Features
Philosophy
Design
Analyses
Conclusions
Connectivity
Connected, strongly connected and weakly connected
Vertex Cut: The smallest number of vertices whose removal
renders the network disconnected
Edge Cut: The smallest number od edges whose removal
renders the network disconnected
A graph is k − connected if the size of its vertex cut is k
Higher the connectivity, more robust the network
Sanket Patil An Overview of Complex Networks Design and Analyses
109.
Outline
Introduction
Topological Features
Philosophy
Design
Analyses
Conclusions
Connectivity
Connected, strongly connected and weakly connected
Vertex Cut: The smallest number of vertices whose removal
renders the network disconnected
Edge Cut: The smallest number od edges whose removal
renders the network disconnected
A graph is k − connected if the size of its vertex cut is k
Higher the connectivity, more robust the network
Sanket Patil An Overview of Complex Networks Design and Analyses
110.
Outline
Introduction
Topological Features
Philosophy
Design
Analyses
Conclusions
Independent/Disjoint Paths
Two paths are vertex independent if they have no common
vertices (except the terminals)
Two paths are edge independent if they have no common
edges
Menger’s Theorem: Max-ﬂow min-cut
More independent paths implies higher robustness
Sanket Patil An Overview of Complex Networks Design and Analyses
111.
Outline
Introduction
Topological Features
Philosophy
Design
Analyses
Conclusions
Independent/Disjoint Paths
Two paths are vertex independent if they have no common
vertices (except the terminals)
Two paths are edge independent if they have no common
edges
Menger’s Theorem: Max-ﬂow min-cut
More independent paths implies higher robustness
Sanket Patil An Overview of Complex Networks Design and Analyses
112.
Outline
Introduction
Topological Features
Philosophy
Design
Analyses
Conclusions
Independent/Disjoint Paths
Two paths are vertex independent if they have no common
vertices (except the terminals)
Two paths are edge independent if they have no common
edges
Menger’s Theorem: Max-ﬂow min-cut
More independent paths implies higher robustness
Sanket Patil An Overview of Complex Networks Design and Analyses
113.
Outline
Introduction
Topological Features
Philosophy
Design
Analyses
Conclusions
Independent/Disjoint Paths
Two paths are vertex independent if they have no common
vertices (except the terminals)
Two paths are edge independent if they have no common
edges
Menger’s Theorem: Max-ﬂow min-cut
More independent paths implies higher robustness
Sanket Patil An Overview of Complex Networks Design and Analyses
114.
Outline
Introduction
Topological Features
Philosophy
Design
Analyses
Conclusions
Number of connected components
Robustness can be measured in terms of the number of
components that result due to failures
Size of the connected components also matters in many cases
Toughness: A toughness of t means, for a given number
k(> 1), at least t ∗ k nodes need to be removed to fragment
the graph into k connected components
Graph Toughness: Biggest value of t
Sanket Patil An Overview of Complex Networks Design and Analyses
115.
Outline
Introduction
Topological Features
Philosophy
Design
Analyses
Conclusions
Number of connected components
Robustness can be measured in terms of the number of
components that result due to failures
Size of the connected components also matters in many cases
Toughness: A toughness of t means, for a given number
k(> 1), at least t ∗ k nodes need to be removed to fragment
the graph into k connected components
Graph Toughness: Biggest value of t
Sanket Patil An Overview of Complex Networks Design and Analyses
116.
Outline
Introduction
Topological Features
Philosophy
Design
Analyses
Conclusions
Number of connected components
Robustness can be measured in terms of the number of
components that result due to failures
Size of the connected components also matters in many cases
Toughness: A toughness of t means, for a given number
k(> 1), at least t ∗ k nodes need to be removed to fragment
the graph into k connected components
Graph Toughness: Biggest value of t
Sanket Patil An Overview of Complex Networks Design and Analyses
117.
Outline
Introduction
Topological Features
Philosophy
Design
Analyses
Conclusions
Number of connected components
Robustness can be measured in terms of the number of
components that result due to failures
Size of the connected components also matters in many cases
Toughness: A toughness of t means, for a given number
k(> 1), at least t ∗ k nodes need to be removed to fragment
the graph into k connected components
Graph Toughness: Biggest value of t
Sanket Patil An Overview of Complex Networks Design and Analyses
118.
Outline
Introduction
Topological Features
Philosophy
Design
Analyses
Conclusions
Conclusions
Complex Networks design have non-trivial topological
properties
Structure governs function
Designed for eﬃciency under constraints of robustness and
cost
Graph thoretic measures are useful for design and analyses
Sanket Patil An Overview of Complex Networks Design and Analyses
119.
Outline
Introduction
Topological Features
Philosophy
Design
Analyses
Conclusions
Conclusions
Complex Networks design have non-trivial topological
properties
Structure governs function
Designed for eﬃciency under constraints of robustness and
cost
Graph thoretic measures are useful for design and analyses
Sanket Patil An Overview of Complex Networks Design and Analyses
120.
Outline
Introduction
Topological Features
Philosophy
Design
Analyses
Conclusions
Conclusions
Complex Networks design have non-trivial topological
properties
Structure governs function
Designed for eﬃciency under constraints of robustness and
cost
Graph thoretic measures are useful for design and analyses
Sanket Patil An Overview of Complex Networks Design and Analyses
121.
Outline
Introduction
Topological Features
Philosophy
Design
Analyses
Conclusions
Conclusions
Complex Networks design have non-trivial topological
properties
Structure governs function
Designed for eﬃciency under constraints of robustness and
cost
Graph thoretic measures are useful for design and analyses
Sanket Patil An Overview of Complex Networks Design and Analyses
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