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Introduction to Network Science William J. Cunningham Department of - - PowerPoint PPT Presentation

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Introduction to Networks Network Dynamics Introduction to Network Science William J. Cunningham Department of Physics Network Science Institute Northeastern University November 30, 2017 Introduction to Networks Network Dynamics

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Introduction to Networks Network Dynamics

Introduction to Network Science

William J. Cunningham

Department of Physics Network Science Institute Northeastern University

November 30, 2017

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Introduction to Networks Network Dynamics

1

Introduction to Networks Basic Concepts Models of Networks Real Network Properties

2

Network Dynamics Diffusion Epidemics Navigation

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Introduction to Networks Network Dynamics Basic Concepts

What is a Network? A network is a set of nodes

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Introduction to Networks Network Dynamics Basic Concepts

What is a Network? A network is a set of nodes and links:

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Introduction to Networks Network Dynamics Basic Concepts

What is a Network? A network is a set of nodes and links: In mathematics, we call this a graph.

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Introduction to Networks Network Dynamics Basic Concepts

A Quick Problem

A chess puzzle: swap the positions

  • f black and white knights

!"#$

%&'($)

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Introduction to Networks Network Dynamics Basic Concepts

A Quick Problem

A chess puzzle: swap the positions

  • f black and white knights

!"#$

%&'($)

a b c d 1 2 3 4

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Introduction to Networks Network Dynamics Basic Concepts

A Quick Problem

A chess puzzle: swap the positions

  • f black and white knights

!"#$

%&'($)

a b c d 1 2 3 4 W W B B

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Introduction to Networks Network Dynamics Basic Concepts

What is a Network? Who cares?

Network Science

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Introduction to Networks Network Dynamics Basic Concepts

What is a Network? Who cares?

Network Science

Technological

Internet Telephone Wireless Power Grid

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Introduction to Networks Network Dynamics Basic Concepts

What is a Network? Who cares?

Network Science

Social

Facebook Twitter Instagram Citation

Technological

Internet Telephone Wireless Power Grid

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Introduction to Networks Network Dynamics Basic Concepts

What is a Network? Who cares?

Network Science

Social

Facebook Twitter Instagram Citation

Technological

Internet Telephone Wireless Power Grid

Transportation

MBTA Amtrak Airline

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Introduction to Networks Network Dynamics Basic Concepts

What is a Network? Who cares?

Network Science

Social

Facebook Twitter Instagram Citation

Technological

Internet Telephone Wireless Power Grid

Biological

Brain Proteins Food Chain

Transportation

MBTA Amtrak Airline

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Introduction to Networks Network Dynamics Basic Concepts

What is a Network? Who cares?

Network Science

Social

Facebook Twitter Instagram Citation

Technological

Internet Telephone Wireless Power Grid

Biological

Brain Proteins Food Chain

Transportation

MBTA Amtrak Airline

Theoretical

String Landscape Causal Sets Quantum Entanglement

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Introduction to Networks Network Dynamics Basic Concepts

What is a Network? Who cares?

Network Science

Social

Facebook Twitter Instagram Citation

Technological

Internet Telephone Wireless Power Grid

Biological

Brain Proteins Food Chain

Transportation

MBTA Amtrak Airline

Theoretical

String Landscape Causal Sets Quantum Entanglement

Goal: control, predict, and understand.

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Introduction to Networks Network Dynamics Models of Networks

The Erd˝

  • s-R´

enyi Random Graph

To talk about probability and statistics, we need a null model: G(N, p) or G(N, M)

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Introduction to Networks Network Dynamics Models of Networks

The Erd˝

  • s-R´

enyi Random Graph

To talk about probability and statistics, we need a null model: G(N, p) or G(N, M)

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Introduction to Networks Network Dynamics Models of Networks

The Erd˝

  • s-R´

enyi Random Graph

To talk about probability and statistics, we need a null model: G(N, p) or G(N, M)

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Introduction to Networks Network Dynamics Models of Networks

The Erd˝

  • s-R´

enyi Random Graph

To talk about probability and statistics, we need a null model: G(N, p) or G(N, M)

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Introduction to Networks Network Dynamics Models of Networks

The Erd˝

  • s-R´

enyi Random Graph

To talk about probability and statistics, we need a null model: G(N, p) or G(N, M)

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Introduction to Networks Network Dynamics Models of Networks

The Erd˝

  • s-R´

enyi Random Graph

To talk about probability and statistics, we need a null model: G(N, p) or G(N, M)

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Introduction to Networks Network Dynamics Models of Networks

The Erd˝

  • s-R´

enyi Random Graph

To talk about probability and statistics, we need a null model: G(N, p) or G(N, M)

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Introduction to Networks Network Dynamics Models of Networks

The Erd˝

  • s-R´

enyi Random Graph

To talk about probability and statistics, we need a null model: G(N, p) or G(N, M)

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Introduction to Networks Network Dynamics Models of Networks

The Erd˝

  • s-R´

enyi Random Graph

To talk about probability and statistics, we need a null model: G(N, p) or G(N, M)

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Introduction to Networks Network Dynamics Models of Networks

The Erd˝

  • s-R´

enyi Random Graph

To talk about probability and statistics, we need a null model: G(N, p) or G(N, M)

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Introduction to Networks Network Dynamics Models of Networks

The Erd˝

  • s-R´

enyi Random Graph

To talk about probability and statistics, we need a null model: G(N, p) or G(N, M)

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Introduction to Networks Network Dynamics Models of Networks

The Erd˝

  • s-R´

enyi Random Graph

To talk about probability and statistics, we need a null model: G(N, p) or G(N, M)

Nodes in real networks do not connect randomly!

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Introduction to Networks Network Dynamics Models of Networks

Basic Structural Properties How do we characterize a network? Degrees and Clustering

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Introduction to Networks Network Dynamics Models of Networks

Basic Structural Properties How do we characterize a network? Degrees and Clustering

1 2 3 2 3 2 1 2 1 3

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Introduction to Networks Network Dynamics Models of Networks

Percolation in Graphs

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Introduction to Networks Network Dynamics Models of Networks

The Barab´ asi-Albert Model

Real networks are modeled by “preferential attachment”:

Image: Bogu˜ n´ a, Papadopoulos & Krioukov, Nat. Comm. 1, 62 (2010).

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Introduction to Networks Network Dynamics Models of Networks

The Barab´ asi-Albert Model

Real networks are modeled by “preferential attachment”:

1 Start with a random graph

with m0 nodes

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Introduction to Networks Network Dynamics Models of Networks

The Barab´ asi-Albert Model

Real networks are modeled by “preferential attachment”:

1 Start with a random graph

with m0 nodes

2 Attachment Mechanism:

Π(ki) =

ki P

j kj

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Introduction to Networks Network Dynamics Models of Networks

The Barab´ asi-Albert Model

Real networks are modeled by “preferential attachment”:

1 Start with a random graph

with m0 nodes

2 Attachment Mechanism:

Π(ki) =

ki P

j kj 3 At timestep t:

N = t + m0 M = m0 + mt

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SLIDE 35

Introduction to Networks Network Dynamics Models of Networks

The Barab´ asi-Albert Model

Real networks are modeled by “preferential attachment”:

1 Start with a random graph

with m0 nodes

2 Attachment Mechanism:

Π(ki) =

ki P

j kj 3 At timestep t:

N = t + m0 M = m0 + mt

4 At late times, γ = 3

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Introduction to Networks Network Dynamics Models of Networks

The Barab´ asi-Albert Model

Real networks are modeled by “preferential attachment”:

1 Start with a random graph

with m0 nodes

2 Attachment Mechanism:

Π(ki) =

ki P

j kj 3 At timestep t:

N = t + m0 M = m0 + mt

4 At late times, γ = 3 5 The rich get richer

This isn’t the whole story...

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Introduction to Networks Network Dynamics Real Network Properties

Degree Distributions

Image: Krioukov et al., Sci. Rep. 2, 793 (2012).

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Introduction to Networks Network Dynamics Real Network Properties

Clustering

Image: Krioukov et al., Sci. Rep. 2, 793 (2012).

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Introduction to Networks Network Dynamics

1

Introduction to Networks Basic Concepts Models of Networks Real Network Properties

2

Network Dynamics Diffusion Epidemics Navigation

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Introduction to Networks Network Dynamics

Network Dynamics

What dynamic processes can we discuss?

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Introduction to Networks Network Dynamics

Network Dynamics

What dynamic processes can we discuss? Structural Changes

1 Link or Node Failure 2 Network Percolation 3 Rewiring of Links 4 Community Formation

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Introduction to Networks Network Dynamics

Network Dynamics

What dynamic processes can we discuss? Structural Changes

1 Link or Node Failure 2 Network Percolation 3 Rewiring of Links 4 Community Formation

Processes on Networks

1 Diffusion 2 Interactions 3 Navigation 4 Synchronization

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Introduction to Networks Network Dynamics

Network Dynamics

What dynamic processes can we discuss? Structural Changes

1 Link or Node Failure 2 Network Percolation 3 Rewiring of Links 4 Community Formation

Processes on Networks

1 Diffusion 2 Interactions 3 Navigation 4 Synchronization

Often we see a combination of both types.

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Introduction to Networks Network Dynamics Diffusion

Diffusion

Diffusion is modeled as a random walk.

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Introduction to Networks Network Dynamics Diffusion

Diffusion

Diffusion is modeled as a random walk.

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Introduction to Networks Network Dynamics Diffusion

Diffusion

Diffusion is modeled as a random walk.

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Introduction to Networks Network Dynamics Diffusion

Diffusion

Diffusion is modeled as a random walk.

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Introduction to Networks Network Dynamics Diffusion

Diffusion

Diffusion is modeled as a random walk.

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Introduction to Networks Network Dynamics Diffusion

Diffusion

Diffusion is modeled as a random walk.

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Introduction to Networks Network Dynamics Diffusion

Diffusion

Diffusion is modeled as a random walk.

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Introduction to Networks Network Dynamics Diffusion

Diffusion

Diffusion is modeled as a random walk.

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Introduction to Networks Network Dynamics Diffusion

Diffusion

Diffusion is modeled as a random walk.

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Introduction to Networks Network Dynamics Diffusion

Diffusion

Diffusion is modeled as a random walk.

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Introduction to Networks Network Dynamics Diffusion

Diffusion

Diffusion is modeled as a random walk.

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Introduction to Networks Network Dynamics Diffusion

Diffusion

Diffusion is modeled as a random walk. In reality, we have many objects walking, W ≫ N.

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Introduction to Networks Network Dynamics Epidemics

Epidemics: SIR Model Epidemics: diffusion and mixing over time

Susceptible

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Introduction to Networks Network Dynamics Epidemics

Epidemics: SIR Model Epidemics: diffusion and mixing over time

Susceptible Infected β

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Introduction to Networks Network Dynamics Epidemics

Epidemics: SIR Model Epidemics: diffusion and mixing over time

Susceptible Infected β Recovered µ

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Introduction to Networks Network Dynamics Epidemics

Epidemics: SIR Model Epidemics: diffusion and mixing over time

Susceptible Infected β Recovered µ

ds(t) dt

= −βki(t) [1 − r(t) − i(t)]

di(t) dt

= −µi(t) + βki(t) [1 − r(t) − i(t)]

dr(t) dt

= µi(t)

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Introduction to Networks Network Dynamics Epidemics

Epidemics: SIR Model Epidemics: diffusion and mixing over time

Susceptible Infected β Recovered µ

ds(t) dt

= −βki(t) [1 − r(t) − i(t)]

di(t) dt

= −µi(t) + βki(t) [1 − r(t) − i(t)]

dr(t) dt

= µi(t) Epidemic Timescale: τ =

k βk2−(µ+β)k

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Introduction to Networks Network Dynamics Epidemics

H1N1 Epidemic (2009)

Credit: A. Vespignani

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Introduction to Networks Network Dynamics Navigation

Navigation

Navigation uses latent geometry:

Image: D. Krioukov

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Introduction to Networks Network Dynamics Navigation

Navigation

Navigation uses latent hyperbolic geometry:

Image: Bogu˜ n´ a, Papadopoulos & Krioukov, Nat. Comm. 1, 62 (2010).

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Introduction to Networks Network Dynamics Navigation

Summary

Networks are powerful models of complex systems The same techniques are applicable to seemingly unrelated fields Computational tools are available to easily simulate/analyze networks Scalability is important for larger problems