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

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 Introduction to Networks 1 Basic Concepts Models of Networks Real Network Properties Network Dynamics 2 Diffusion Epidemics Navigation

Introduction to Networks Network Dynamics Basic Concepts What is a Network? A network is a set of nodes

Introduction to Networks Network Dynamics Basic Concepts What is a Network? A network is a set of nodes and links :

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 .

Introduction to Networks Network Dynamics Basic Concepts A Quick Problem A chess puzzle: swap the positions of black and white knights !"#$ %&'($)

Introduction to Networks Network Dynamics Basic Concepts A Quick Problem A chess puzzle: swap the positions of black and white knights a b c d 1 2 3 !"#$ %&'($) 4

Introduction to Networks Network Dynamics Basic Concepts A Quick Problem A chess puzzle: swap the positions of black and white knights a b c d 1 2 B 3 !"#$ %&'($) 4 W W B

Introduction to Networks Network Dynamics Basic Concepts What is a Network? Who cares? Network Science

Introduction to Networks Network Dynamics Basic Concepts What is a Network? Who cares? Technological Internet Telephone Wireless Power Grid Network Science

Introduction to Networks Network Dynamics Basic Concepts What is a Network? Who cares? Technological Social Internet Facebook Telephone Twitter Wireless Instagram Power Grid Citation Network Science

Introduction to Networks Network Dynamics Basic Concepts What is a Network? Who cares? Technological Social Internet Facebook Telephone Twitter Wireless Instagram Power Grid Citation Network Science Transportation MBTA Amtrak Airline

Introduction to Networks Network Dynamics Basic Concepts What is a Network? Who cares? Technological Social Internet Facebook Telephone Twitter Wireless Instagram Power Grid Citation Network Science Transportation Biological MBTA Brain Amtrak Proteins Airline Food Chain

Introduction to Networks Network Dynamics Basic Concepts What is a Network? Who cares? Technological Social Internet Facebook Telephone Twitter Wireless Instagram Power Grid Citation Network Science Transportation Biological Theoretical MBTA Brain String Landscape Amtrak Proteins Causal Sets Airline Food Chain Quantum Entanglement

Introduction to Networks Network Dynamics Basic Concepts What is a Network? Who cares? Technological Social Internet Facebook Telephone Twitter Wireless Instagram Power Grid Citation Network Science Transportation Biological Theoretical MBTA Brain String Landscape Amtrak Proteins Causal Sets Airline Food Chain Quantum Entanglement Goal: control, predict, and understand.

Introduction to Networks Network Dynamics Models of Networks The Erd˝ os-R´ enyi Random Graph To talk about probability and statistics, we need a null model: G ( N , p ) or G ( N , M )

Introduction to Networks Network Dynamics Models of Networks The Erd˝ os-R´ enyi Random Graph To talk about probability and statistics, we need a null model: G ( N , p ) or G ( N , M )

Introduction to Networks Network Dynamics Models of Networks The Erd˝ os-R´ enyi Random Graph To talk about probability and statistics, we need a null model: G ( N , p ) or G ( N , M )

Introduction to Networks Network Dynamics Models of Networks The Erd˝ os-R´ enyi Random Graph To talk about probability and statistics, we need a null model: G ( N , p ) or G ( N , M )

Introduction to Networks Network Dynamics Models of Networks The Erd˝ os-R´ enyi Random Graph To talk about probability and statistics, we need a null model: G ( N , p ) or G ( N , M )

Introduction to Networks Network Dynamics Models of Networks The Erd˝ os-R´ enyi Random Graph To talk about probability and statistics, we need a null model: G ( N , p ) or G ( N , M )

Introduction to Networks Network Dynamics Models of Networks The Erd˝ os-R´ enyi Random Graph To talk about probability and statistics, we need a null model: G ( N , p ) or G ( N , M )

Introduction to Networks Network Dynamics Models of Networks The Erd˝ os-R´ enyi Random Graph To talk about probability and statistics, we need a null model: G ( N , p ) or G ( N , M )

Introduction to Networks Network Dynamics Models of Networks The Erd˝ os-R´ enyi Random Graph To talk about probability and statistics, we need a null model: G ( N , p ) or G ( N , M )

Introduction to Networks Network Dynamics Models of Networks The Erd˝ os-R´ enyi Random Graph To talk about probability and statistics, we need a null model: G ( N , p ) or G ( N , M )

Introduction to Networks Network Dynamics Models of Networks The Erd˝ os-R´ enyi Random Graph To talk about probability and statistics, we need a null model: G ( N , p ) or G ( N , M )

Introduction to Networks Network Dynamics Models of Networks The Erd˝ os-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!

Introduction to Networks Network Dynamics Models of Networks Basic Structural Properties How do we characterize a network? Degrees and Clustering

Introduction to Networks Network Dynamics Models of Networks Basic Structural Properties How do we characterize a network? Degrees and Clustering 3 2 2 1 3 3 2 1 1 2

Introduction to Networks Network Dynamics Models of Networks Percolation in Graphs

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).

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 m 0 nodes

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 m 0 nodes 2 Attachment Mechanism: k i Π( k i ) = P j k j

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 m 0 nodes 2 Attachment Mechanism: k i Π( k i ) = P j k j 3 At timestep t : N = t + m 0 M = m 0 + mt

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 m 0 nodes 2 Attachment Mechanism: k i Π( k i ) = P j k j 3 At timestep t : N = t + m 0 M = m 0 + mt 4 At late times, γ = 3

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 m 0 nodes 2 Attachment Mechanism: k i Π( k i ) = P j k j 3 At timestep t : N = t + m 0 M = m 0 + mt 4 At late times, γ = 3 5 The rich get richer This isn’t the whole story...

Introduction to Networks Network Dynamics Real Network Properties Degree Distributions Image: Krioukov et al., Sci. Rep. 2, 793 (2012).

Introduction to Networks Network Dynamics Real Network Properties Clustering Image: Krioukov et al., Sci. Rep. 2, 793 (2012).

Introduction to Networks Network Dynamics Introduction to Networks 1 Basic Concepts Models of Networks Real Network Properties Network Dynamics 2 Diffusion Epidemics Navigation

Introduction to Networks Network Dynamics Network Dynamics What dynamic processes can we discuss?

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

Introduction to Networks Network Dynamics Network Dynamics What dynamic processes can we discuss? Structural Changes Processes on Networks 1 Link or Node Failure 1 Diffusion 2 Network Percolation 2 Interactions 3 Rewiring of Links 3 Navigation 4 Community Formation 4 Synchronization

Introduction to Networks Network Dynamics Network Dynamics What dynamic processes can we discuss? Structural Changes Processes on Networks 1 Link or Node Failure 1 Diffusion 2 Network Percolation 2 Interactions 3 Rewiring of Links 3 Navigation 4 Community Formation 4 Synchronization Often we see a combination of both types.

Introduction to Networks Network Dynamics Diffusion Diffusion Diffusion is modeled as a random walk.

Introduction to Networks Network Dynamics Diffusion Diffusion Diffusion is modeled as a random walk.

Introduction to Networks Network Dynamics Diffusion Diffusion Diffusion is modeled as a random walk.