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