Models of networks(synthetic networks or generative models):Random, Small-world, Scale-free, Configuration model

and Random geometric model

By: Ralucca Gera, NPSExcellence Through Knowledge

The world around us as a network

• What do social networks look like?Watch this video

• What categories do we have for networks?Random networks (normal degree distribution)Scale free (power-law degree distribution)

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The three papers for each of the models

• “On Random Graphs I” by Paul Erdos and AlfedRenyi in Publicationes Mathematicae (1958) Times cited: 3, 517 (as of January 1, 2015)

• “Collective dynamics of ‘small-world’ networks” by Duncan Watts and Steve Strogatz in Nature, (1998) Times cited: 24, 535 (as of January 1, 2015)

• “Emergence of scaling in random networks” by László Barabási and Réka Albert in Science, (1999)Times cited: 21, 418 (as of January 1, 2015)

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Why understand the structure of networks?

• Applications such as epidemiology: Viruses propagate much faster in scale-free networks. Vaccination of random nodes in scale free does not work, but targeted vaccination is very effective

• Thus, we can create models that captures the structure which facilitates research: – Create fast networks of particular models can be quickly

and cheaply generated, instead of collecting and cleaning the data that takes time

– Promote understanding of the world around us: What effect does the degree distribution have on the behavior of the system?

Reference network: Regular Lattice

The 1-dimensional lattice is the Harary graph H(n,r) or the Circulant graph (1, 2, …, r)start with an n-cycle, and each vertex is adjacent to r/2 vertices to the left, and r/2 vertices to the right.

5Source: http://mathworld.wolfram.com/CirculantGraph.html

Reference network: Regular Lattice

6Source: http://mathworld.wolfram.com/CirculantGraph.html

a particular Circulant graph (1, 2, …, r):

Source: http://mathworld.wolfram.com/CirculantGraph.html

Reference network: Regular Lattice

• The higher dimensions are generalizations of these. An example is a hexagonal lattice is a 2-dimensional lattice:graphene, a single layer of carbon atoms with a honeycomb latticestructure.

7Source: http://phys.org/news/2013-05-intriguing-state-previously-graphene-like-materials.html

ERDŐS-RÉNYIRANDOM GRAPHS

Random graphs (Erdős-Rényi , 1959)

• RG is a model in which some specific set of parameter takes fixed values, and the network is created at random using these values.

• Two main examples:– G(n,p): fix n and probability p of the edges

between vertices. The number of edges is not fixed. This is the default construction.

– G(n, m): fix n and m

– The mean value of edges:

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G(n,m)

• To make a random network: take n nodes, m unlabeled edges, and put the edges down randomly between the n vertices

• Put the graph in a box, make another one and put it in the box, and another one…

• Pull one network at random out of the box and it will have a Normal Degree Distribution (classic degree distribution): almost everyone has the same number of friends on average

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G(n,m)

Method two and equivalent to the first:• To make a random network:

– take n nodes, – m pairs at random– place the edges between the randomly chosen nodes

• The average degree: , where is often used to denote the degree of vertex i in complex networks (enumerate the vertices, 1, 2, …)

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G(n,p)

• To make a random network: – take n nodes, – A fixed probability p– Attach edges at random to the nodes, with the

probability p

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Degree distribution is Normal

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Both for G(n,p) and G(n,m)

Erdős-Rényi random networks

• There might be some that are a bit different that don’t have this degree distribution, but there are so few of them, that you will not pull one out of this box

• The universe doesn’t produce these (they are made by us, they are mathematically constructed) rather scale-free

• We will construct them using Gephi and NetworkX. For Gephi you will need the plug-in. We will practice with NetworkX since there are more synthetic models and classes available in NetworkX

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• ER graphs are models of a network in which some specific set of parameters take fixed values, but the construction of the network is random (see below in Gephi)

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Generating Erdős-Rényi random networks.

Generating Erdős-Rényi

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Generating Erdős-Rényi random networks

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Reference for python: http://networkx.lanl.gov/reference/generated/networkx.generators.random_graphs.erdos_renyi_graph.html#networkx.generators.random_graphs.erdos_renyi_graph

WATTS-STROGATZSMALL WORLD GRAPHS(1998)

Small worlds, between perfect order and chaos

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the first graph is completely ordered (probability p =0), the graph in the middle is a "small world" graph (0 < p < 1), the graph at the right is complete random (p=1).

Source: http://www.bordalierinstitute.com/target1.html

Small world models

• Duncan Watts and Steven Strogatz small world model: a few random links in an otherwise structured graph make the network a small world: the average shortest path is short

regular lattice:my friend’s friend isalways my friend

small world:mostly structuredwith a few randomconnections

random graph:all connectionsrandom

Source: Watts, D.J., Strogatz, S.H. (1998) Collective dynamics of 'small-world' networks. Nature 393:440-442.

small worlds

Small worlds– a friend of a friend is also frequently a friend

(clustering coefficient)– but only small number of hops separate any

two people in the world (small average path)

Arnold Schwarzenegger. – thomashawk, Flickr; http://creativecommons.org/licenses/by-nc/2.0/deed.en

Generating Watts-Strogatz

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Generating Watts-Strogatz networks

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http://networkx.lanl.gov/reference/generated/networkx.generators.random_graphs.watts_strogatz_graph.html#networkx.generators.random_graphs.watts_strogatz_graph

PREFERENTIAL ATTACHMENT MODEL (WE WILL CONSIDER THE BARABASI-ALBERT EXAMPLE)

Scale-free

• Scale-free networks are a type of small world networks.

• They have – A power-law degree distribution:

– Static or evolutionary• One way to create them is through preferential

attachment, but it is not the only way. • We will particularly look at Barabasi-Albert type

(one of the most common ones)

Power law networks

• Many real world networks contain hubs: highly connected nodes

• Usually the distribution of edges is extremely skewed

many nodes with small degree

fat tail: a few nodes with a very large degree

no “typical” degree

Degree (number of edges)

num

ber o

f nod

es o

f tha

t deg

ree

But is it really a power-law?

• A power-law will appear as a straight line on a log-log plot: let be the count of vertices of degree k.

ln ln

• A deviation from a straight line could indicate a different distribution:– exponential– lognormal

log of the degree

Log

of n

umbe

r of n

odes

of t

hat d

egre

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Network growth & resulting structure

• random attachment: new node picks any existing node to attach to

• preferential attachment: new node picks from existing nodes according to their degrees (high preference for high degree)

http://projects.si.umich.edu/netlearn/NetLogo4/RAndPrefAttachment.html

Scale Free networks

• One example is the one introduced by Barabasi-Albert based on preferential attachment:– Start with a small set of nodes ( ) and no edges– Attach new nodes one at the time;

• each with the same fixed number of new edges, attaching to the existing ones in the network, with preference for high degrees (once the high degrees appear)

https://www.youtube.com/watch?v=5YdkhWB_uYQThis is not the only way to get scale–free networks!

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Generating Barabasi-Albert

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Generating Barabasi-Albert

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Generating Barabasi-Albert networks

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http://networkx.lanl.gov/reference/generated/networkx.generators.random_graphs.barabasi_albert_graph.html#networkx.generators.random_graphs.barabasi_albert_graph

Modified BA

• Many modifications of this model exists, based on:– Nodes “retiring” and losing their status– Nodes disappearing (such as website going down)– Links appearing or disappearing between the existing

nodes (called internal links)– Fitness of nodes (modeling newcomers like Google)

• Most researchers still use the standard BA model when studying new phenomena and metrics. Why? It is a simple model, and it was the first model that brought in growth (as well as preferential attachment)

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The Malloy Reed Configuration model

The configuration model

• A random graph model created based on Degree sequence of choice (can be scale free)

• Maybe more than degree sequence is needed to be controlled in order to create realistic models35

The Random Geometric model

Random Geometric Model

• Again the connections are created at random, but based on proximity rather than preferential attachment (such as ad hoc networks)

• Recall that BA was introduced based on the data obtained from the Web, where physical proximity is irrelevant.

• But if one would want to model something like the Internet, then proximity is relevant

• There is no perfect model for the world around us, not even for specific types of networks

• No model has been introduced for the Internet37

An example of a random geometric

38https://www.youtube.com/watch?v=NUisb1-INIE

A zoo of complex networks

Random, Small-World, Scale-Free

40http://noduslabs.com/radar/types-networks-random-small-world-scale-free/

Man made,“large world”:

Scale Free networks:1. High degree heterogeneity2. Various levels of modularity3. Various levels of randomness

Networks and their degree distributions

We tend to characterize networks by their degree distributions:

– Random graphs iff Poisson degree distribution– Scale free iff power-law degree distribution.But they are not! Rather:– If G is a random graphs, then G has Poisson degree

distribution– If G is scale free, then G most probably has a

power-law degree distribution.– If G was constructed using preferential attachment,

then G has a power-law degree distribution.41

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54

6367

2

94

number ofnodes found

Power-law graph

93

number ofnodes found

13

711

1519

Poisson distribution

Power-law networks are robust to random breakdown

But are especially vulnerable to targeted attack

• Targeting and removing hubs can quickly break up the network

mike

In social networks, it’s nice to be a hub

But it depends on what you’re sharing…

Python

• References to the classes that exist in python:http://networkx.lanl.gov/reference/generators.html

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