Models of networks (synthetic networks or generative models)

Prof. Ralucca Gera, Applied Dept. Naval Postgraduate School Monterey, California [email protected]

Excellence Through Knowledge Learning Outcomes

• Identify network models and explain their structures; • Contrast networks and synthetic models; • Understand how to design new network models (based on the existing ones and on the collected data) • Distinguish methodologies used in analyzing networks. The three papers for each of the models

Synthetic models are used as reference/null models to compare against and build new complex networks

•“On Random Graphs I” by Paul Erdős 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 , (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 care?

• : – A virus propagates much faster in scale-free networks. – Vaccination of random nodes in scale free does not work, but targeted vaccination is very effective • Create synthetic networks to be used as null models: – What effect does the degree distribution alone have on the behavior of the system? (answered by comparing to the configuration model) • Create networks of different sizes – Networks of particular sizes and structures can be quickly and cheaply generated, instead of collecting and cleaning the data that takes time 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.

5 Source: http://mathworld.wolfram.com/CirculantGraph.html Reference network: Regular Lattice

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

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

7 Source: http://phys.org/news/2013-05-intriguing-state-previously-graphene-like-materials.html Watts-Strogatz Small World Graphs (1998)

8 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 (one small world: random graph: type of structure): mostly structured all connections my friend’s friend is with a few random happen at always my friend connections random

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

High clustering: .75 Low clustering: p (probability) High average path: Low average path: Small worlds

the graph on the left has order (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 Avg path and avg clustering

Variations of avg path and clustering as a function of the rewiring probability p

11 https://pdfs.semanticscholar.org/8c4c/455de44fa99e73e79d6fddf008ca6ae0f9aa.pdf Generating Watts-Strogatz networks

.15 is the rewiring probability

http://networkx.lanl.gov/reference/generated/networkx.generators.random_graphs.watts_strogatz_graph.html#networkx.generators.random_graphs.watts_strogatz_graph 12 Let’s practice in CoCalc

13 Coding in CoCalc

14 Main References

• Newman “The structure and function of complex networks” (2003) • Estrada “The structure of complex Networks” (2012) • Barabasi “” (online: http://barabasi.com/networksciencebook/) • References to the classes that exist in python: http://networkx.lanl.gov/reference/generators.html

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