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Complexity and Networks Melanie Mitchell Portland State University and Santa Fe Institute

What are Complex Systems? Large networks of simple interacting elements, which, following simple rules, produce emergent, collective, complex behavior.

Brains

Insect Colonies

Immune Systems

Financial Markets

Central question for the sciences of complexity

How do large networks with

Central question for the sciences of complexity How do large networks with — simple components

Central question for the sciences of complexity How do large networks with — simple components — limited communication among components

Central question for the sciences of complexity How do large networks with — simple components — limited communication among components — no central control

Central question for the sciences of complexity How do large networks with — simple components — limited communication among components — no central control give rise to complex (“adaptive”, “living”, “intelligent”) behavior, involving

Central question for the sciences of complexity How do large networks with — simple components — limited communication among components — no central control give rise to complex (“adaptive”, “living”, “intelligent”) behavior, involving — information processing and computation

Central question for the sciences of complexity How do large networks with — simple components — limited communication among components — no central control give rise to complex (“adaptive”, “living”, “intelligent”) behavior, involving — information processing and computation — complex dynamics

Central question for the sciences of complexity How do large networks with — simple components — limited communication among components — no central control give rise to complex (“adaptive”, “living”, “intelligent”) behavior, involving — information processing and computation — complex dynamics — evolution and learning?

Core disciplines of the science of complexity

Dynamics: The study of continually changing structure and behavior of systems

Core disciplines of the science of complexity Dynamics: The study of continually changing structure and behavior of systems Information: The study of representation, symbols, and communication

Core disciplines of the science of complexity Dynamics: The study of continually changing structure and behavior of systems Information: The study of representation, symbols, and communication Computation: The study of how systems process information and act on the results

Core disciplines of the science of complexity Dynamics: The study of continually changing structure and behavior of systems Information: The study of representation, symbols, and communication Computation: The study of how systems process information and act on the results Evolution and learning: The study of how systems adapt to constantly changing environments

Methodologies

Goals of the Science of Complexity

Cross-disciplinary insights into complex systems

Goals of the Science of Complexity Cross-disciplinary insights into complex systems “General” theory?

Network Thinking

Neural Network (C. Elegans)

Food Web Pj45Avc/s400/food%2Bweb.bmp

Metabolic Network

Genetic Regulatory Network

Bank Network From Schweitzer et al., Science, 325, ,

Airline Routes

US Power Grid

Internet

World Wide Web (small part) From M. E. J. Newman and M. Girvin, Physical Review Letters E, 69, , 2004.

Social Network

The Science of Networks

Are there properties common to all complex networks? The Science of Networks

Are there properties common to all complex networks? If so, why? The Science of Networks

Are there properties common to all complex networks? If so, why? Can we formulate a general theory of the structure, evolution, and dynamics of networks? The Science of Networks

Observed common properties: Small world property Scale-free degree distribution Clustering and community structure Robustness to random node failure Vulnerability to targeted hub attacks Vulnerability to cascading failures

Small-World Property (Watts and Strogatz, 1998)

me

Small-World Property (Watts and Strogatz, 1998) me Barack Obama

Small-World Property (Watts and Strogatz, 1998) me Barack Obama my mother

Small-World Property (Watts and Strogatz, 1998) me Barack Obama my mother Nancy Bekavac

Small-World Property (Watts and Strogatz, 1998) me Barack Obama my mother Nancy Bekavac Hillary Clinton

Small-World Property (Watts and Strogatz, 1998) me Barack Obama my mother Nancy Bekavac Hillary Clinton

Small-World Property (Watts and Strogatz, 1998) me Barack Obama

Small-World Property (Watts and Strogatz, 1998) memy cousin Matt Dunne Barack Obama

Small-World Property (Watts and Strogatz, 1998) me Barack Obama Patrick Leahy my cousin Matt Dunne

Small-World Property (Watts and Strogatz, 1998) me Barack Obama Patrick Leahy my cousin Matt Dunne

Stanley Milgram

Nebraska farmer Boston stockbroker

Stanley Milgram Nebraska farmer Boston stockbroker

Stanley Milgram Nebraska farmer Boston stockbroker

Stanley Milgram On average: “six degrees of separation” Nebraska farmer Boston stockbroker

The Small-World Property The network has relatively few “long- distance” links but there are short paths between most pairs of nodes, usually created by “hubs”.

Most real-world complex networks seem to have the small-world property! The Small-World Property

The network has relatively few “long- distance” links but there are short paths between most pairs of nodes, usually created by “hubs”. Most real-world complex networks seem to have the small-world property! But why? The Small-World Property

And how can the shortest paths actually be found? The Small-World Property

Scale-Free Structure (Albert and Barabási, 1998)

Typical structure of World Wide Web (nodes = web pages, links = links between pages) Typical structure of a randomly connected network %20network.gif part of WWW

Concept of “Degree Distribution” A node with degree 3

Concept of “Degree Distribution” A node with degree 3

Concept of “Degree Distribution” Degree Number of Nodes A node with degree 3

part of WWW Degree Number of nodes Degree Number of nodes

part of WWW Degree Number of nodes Degree Number of nodes

The Web’s approximate Degree Distribution Number of nodes Degree

Number of nodes Degree The Web’s approximate Degree Distribution

Number of nodes The Web’s approximate Degree Distribution Number of nodes Degree

Number of nodes The Web’s approximate Degree Distribution Number of nodes Degree

The Web’s approximate Degree Distribution Number of nodes

Degree “Scale-free” distribution The Web’s approximate Degree Distribution Number of nodes

Degree “Scale-free” distribution The Web’s approximate Degree Distribution Number of nodes

Degree “Scale-free” distribution The Web’s approximate Degree Distribution Number of nodes “power law”

Degree “Scale-free” distribution The Web’s approximate Degree Distribution Number of nodes “power law” “Scale-free” distribution = “power law” distribution

Example: Human height follows a normal distribution Height Frequency

Example: Population of cities follows a power-law (“scale- free) distribution /09/350px_US_Metro_popultion_graph.png ypopulation1.png

part of WWW The scale-free structure of the Web helps to explain why Google works so well

It also explains some of the success of other scale-free networks in nature! part of WWW

Scale-Free Networks are “fractal-like”

Scale-Free Networks have high clustering part of WWW High Clustering: Low Clustering:

High-Clustering Helps in Discovering Community Structure in Networks

How are Scale-Free Networks Created?

Web pages

Preferential attachment demo (Netlogo)

Robustness of Scale-Free Networks

Vulnerable to targeted “hub” failure

Robustness of Scale-Free Networks Vulnerable to targeted “hub” failure Robust to random node failure

Robustness of Scale-Free Networks Vulnerable to targeted “hub” failure Robust to random node failure unless.... nodes can cause other nodes to fail Can result in cascading failure

August, 2003 electrical blackout in northeast US and Canada 9:29pm 1 day before 9:14pm Day of blackout images/imagerecords/3000/3719/ NE_US_OLS jpg

We see similar patterns of cascading failure in biological systems, ecological systems, computer and communication networks, wars, etc.

Normal (“bell-curve) distribution s_modeling/Random_Normal_Distribution.gif

Normal (“bell-curve) distribution s_modeling/Random_Normal_Distribution.gif “Events in ‘tail’ are highly unlikely”

Power law (“scale free”) distribution

Power law (“scale free”) distribution Notion of “heavy tail”: Events in tail are more likely than in normal distribution

Power law (“scale free”) distribution “More normal than ‘normal’ ”

Duncan Watts: “Next to the mysteries of dynamics on a network ― whether it be epidemics of disease, cascading failures in power systems, or the outbreak of revolutions ― the problems of networks that we have encountered up to now are just pebbles on the seashore.”