Synthetic Networks Luke Osterritter losterritter@cmu.edu Center - - PDF document

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Synthetic Networks Luke Osterritter losterritter@cmu.edu Center - - PDF document

<Your Name> Synthetic Networks Luke Osterritter losterritter@cmu.edu Center for Computational Analysis of Social and Organizational Systems http://www.casos.cs.cmu.edu/ Overview Network researchers have identified some


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Center for Computational Analysis of Social and Organizational Systems http://www.casos.cs.cmu.edu/

Synthetic Networks

Luke Osterritter

losterritter@cmu.edu

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Overview

  • Network researchers have identified some

“stylized” network structures

  • Stylized networks in ORA

– Ring Lattice – Small world – Erdos-Renyi – Core-Periphery – Scale Free – Cellular

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Why Synthetic Networks?

  • Synthetic networks generated using

random network generators

– Easy to obtain – Can vary parameters when running experiments

  • Real-life (empirical) networks often

compared to stylized networks

  • Have characteristics of real social

networks

– Clustering – Degree distribution

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Structural metrics: Average path length

Slide by Kraemer & Barabasi, Bonabeau (SciAm’03)

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Structural Metrics: Clustering coefficient

Slide by Kraemer & Barabasi, Bonabeau (SciAm’03)

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ORA – Getting Started

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  • Purely random
  • Most common form studied
  • Properties

– Simplest network – Short distances – No local structure (clustering) (till a threshold) – Very different than real-world networks – Rich theory, explains small diameter and giant component

Erdos-Renyi (Random)

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  • Purely random
  • Most common form

studied

  • Properties

– Short distances – No local structure – Very different than real-world networks

Erdos-Renyi

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  • Input: approx.

density of resulting network

Erdos-Renyi

Approximate density of resulting network

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  • Input: approximate density of

resulting network

Erdos-Renyi

Approximate density of resulting network

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Erdos-Renyi

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Erdos-Renyi – bi-modal

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Erdos-Renyi – bi-modal

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  • Most common form studied
  • Statistical tests to decide if your

network is random

  • Easy to generate
  • Good mathematical properties
  • Very different than real world

networks

Erdos-Renyi Notes

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Ring Lattice

  • Nodes laid out in a circle and

connected to their K-closest neighbors

  • Properties

– High clustering – High average path length

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Ring Lattice

  • Number of agents:

network size

  • Number of

neighbors: number

  • f neighbors each

node is connected to

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Ring Lattice

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Small World

  • Three Steps

– Begin with a lattice, e.g. 2D grid with k-nearest neighbors connected – Randomly remove connections – Randomly add long-distance connections

  • Properties

– High local structure (clustering) – Short maximum distances

Examples: Telephone call graphs, electric power grids

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Small World

  • Number of neighbors:

Dimensionality of embedding space/lattice. Alternatively, average degree.

  • Probability of removing a

neighbor: Remove any local structure?

  • Probability of adding a neighbor:

Add long-range connections?

  • Power law exponent: How much

should long-range connections ignore local structure? How far should they be?

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Small World

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Core-Periphery

  • Two kinds of nodes

– Core of interconnected nodes – Periphery of pendants with single connection to Core

  • Properties

– Short distances – Some local structure (core vs non-core)

Examples: Observed trade flows, diplomatic ties among countries

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Core-Periphery

  • Proportion of core

nodes

  • Density of core

nodes: How dense should the within- core network be?

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Core-Periphery

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Cellular

  • Small number of

tight clusters with few links between clusters

  • Properties

– Large distances – High local structure (clustering)

Examples: Terrorist networks

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Cellular

  • Number of cells: How

many cells?

  • Inner Density: how dense

should the network within each cell be?

  • Outer Density: how dense

should connections between cells be?

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Cellular

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Scale-Free Networks – Degree Distribution

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Scale-Free

  • Process

– Some initial network – New nodes more likely to connect to existing nodes with high degree

  • Properties

– Short distances – No local structure

Examples: Social networks, Computer networks (WWW)

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Scale-Free

  • How likely is it for new

nodes to connect to the core?

  • Initial node count: How big

should the initial network be?

  • Initial density: How dense

should the initial network be?

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Scale-Free

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Summary

  • Erdos-Reny networks

– Random network, IID links

  • Ring Lattice

– Circle layout, k-closest neighbors

  • Scale free

– The degree distribution obeys a power law

  • Small-world

– Ring with a few extra hubs

  • Cellular

– Highly connected cells connected by a link to few other cells

  • Core periphery

– Single strong component with high level of peripherals

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Summary