Models of networks synthetic networks or generative models

Models of networks (synthetic networks or generative models): Random, Small-world, Scale-free, Configuration model and Random geometric model By: Ralucca Gera, NPS Excellence Through Knowledge

The world around us as a network • What do social networks look like? ØWatch this video • What categories do we have for networks? ØRandom networks (normal degree distribution) ØScale free (power-law degree distribution) 2

The three papers for each of the models • “On Random Graphs I” by Paul Erdos and Alfed Renyi 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) 3

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 • Source: http: //mathworld. wolfram. com/Circulant. Graph. html 5

Reference network: Regular Lattice Source: http: //mathworld. wolfram. com/Circulant. Graph. html 6

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

ERDŐS-RÉNYI RANDOM GRAPHS

Random graphs (Erdős-Rényi , 1959) • 9

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 10

G(n, m) • 11

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 12

Degree distribution is Normal Both for G(n, p) and G(n, m) 13

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 Network. X. For Gephi you will need the plug-in. We will practice with Network. X since there are more synthetic models and classes available in Network. X 14

Generating Erdős-Rényi random networks. • 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) 15

Generating Erdős-Rényi 16

Generating Erdős-Rényi random networks Reference for python: http: //networkx. lanl. gov/reference/generated/networkx. generators. random_graphs. erdos_r enyi_graph. html#networkx. generators. random_graphs. erdos_renyi_graph 17

WATTS-STROGATZ SMALL WORLD GRAPHS (1998)

Small worlds, between perfect order and chaos 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/target 1. html 19

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 is always my friend small world: mostly structured with a few random connections random graph: all connections random 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 22

Generating Watts-Strogatz networks http: //networkx. lanl. gov/reference/generated/networkx. generators. random_graphs. watts_strogatz_graph. html#networkx. generators. random_graphs. watts_strogatz_ graph 23

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

Scale-free •

Power law networks number of nodes of that degree • 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 Degree (number of edges) no “typical” degree

But is it really a power-law? Log of number of nodes of that degree • log of the degree

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/Net. Logo 4/RAnd. Pref. Attachment. html

Scale Free networks • 29

Generating Barabasi-Albert 30

Generating Barabasi-Albert 31

Generating Barabasi-Albert networks http: //networkx. lanl. gov/reference/generated/networkx. generators. random_gra phs. barabasi_albert_graph. html#networkx. generators. random_graphs. baraba si_albert_graph 32

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) 33

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 models 35

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 Internet 37

An example of a random geometric https: //www. youtube. com/watch? v=NUisb 1 -INIE 38

A zoo of complex networks

Random, Small-World, Scale-Free Scale Free networks: 1. High degree heterogeneity 2. Various levels of modularity 3. Various levels of randomness Man made, “large world”: 40 http: //noduslabs. com/radar/types-networks-random-small-world-scale-free/

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

Power-law graph number of nodes found 94 67 63 54 2 6 1

Poisson distribution number of nodes found 93 19 11 3 15 7 1

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

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

But it depends on what you’re sharing…

Python • References to the classes that exist in python: http: //networkx. lanl. gov/reference/generators. html 48