Models of networks synthetic networks or generative models

Models of networks (synthetic networks or generative models) Prof. Ralucca Gera, Applied Mathematics Dept. Naval Postgraduate School Monterey, California rgera@nps. edu 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 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 care? • Epidemiology: – 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 • 5 Source: http: //mathworld. wolfram. com/Circulant. Graph. html

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

Erdős-Rényi Random Graphs (1959) 8

Random graphs (Erdős-Rényi , 1959) • 9

Creating G(n, m) • 10

Creating G(n, m) – method 2 • 11

Creating G(n, p) • 12

Results about E-R graphs: • 13

Generating Erdős-Rényi ER(n, p) • 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) 14

Generating Erdős-Rényi ER(n, m) 15

Generating Erdős-Rényi random networks Reference for python: http: //networkx. lanl. gov/reference/generated/networkx. generators. random_graphs. erdos_renyi_graph. html#networkx. generators. random_graphs. erdos_renyi_graph 16

The Random Geometric model 17

Random Geometric Model • 18

An example of a random geometric 19 https: //www. youtube. com/watch? v=NUisb 1 -INIE

Creating it in Python https: //networkx. github. io/documentation/networkx 1. 10/reference/generated/networkx. generators. geometric. random_geometric_graph. html#networkx. generators. geometric. random_geometric_graph 20

The Malloy Reed Configuration model (1995) 21

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 22

The MR configuration model • A random graph model created based on a degree sequence of choice: 4, 3, 2, 2, 2, 1, 1, 1 Step 1: Step 2: Or this step 2: 23

Mathematical properties • 24

Mathematical properties (parallel edges) • 25 http: //tuvalu. santafe. edu/~aaronc/courses/5352/csci 5352_2017_L 4. pdf

Mathematical properties (loops) • 26 http: //tuvalu. santafe. edu/~aaronc/courses/5352/csci 5352_2017_L 4. pdf

Generating it in Python https: //networkx. github. io/documentation/networkx-1. 10/reference/generated/networkx. generators. degree_seq. configuration_model. html 27

Part 2 28

Coding it in Co. Calc • Go to www. Co. Calc. com and create an account using your NPS email • Create your new folder to copy the code • Open “MA 4404 -2019” folder to copy its contents to your new folder. 29

Copy contents to NEW folder 30

Make a copy • Choose “Create. Synthetic. Networks. ipynb” • Notice projects, folders & files 31

Create ER networks 32

Watts-Strogatz Small World Graphs (1998) 33

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 type of structure): my friend’s friend is always my friend small world: mostly structured with a few random connections Source: Watts, D. J. , Strogatz, S. H. (1998) Collective dynamics of 'small-world' networks. Nature 393: 440 -442. random graph: all connections happen at random

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

Variations of avg path and clustering as a function of the rewiring probability p 36 https: //pdfs. semanticscholar. org/8 c 4 c/455 de 44 fa 99 e 73 e 79 d 6 fddf 008 ca 6 ae 0 f 9 aa. pdf

Generating Watts-Strogatz WS (n, k, alpha) Alpha is the rewiring probability 37

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 38

Barabási-Albert Scale free model (1999) 39

Network growth & resulting structure • Random attachment: new node picks any existing node to attach to • Preferential/fitness attachment: new node picks from existing nodes according to their degrees/fitness (high preference for high degree/fitness) http: //projects. si. umich. edu/netlearn/Net. Logo 4/RAnd. Pref. Attachment. html

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 No “typical” degree node fat tail: a few nodes with a very large degree Degree (number of edges)

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

Fitting distributions Node (frame) and edge (inset) counts of European Airline Transportation Network's layers with distribution fitting. 44 http: //faculty. nps. edu/rgera/ANGEL. html

Fitting distributions European Airline Transportation Network's multilayer network: Degree histogram of the multiplexes with the log scale in the inset. Upper right: average shortest path, lower right: centrality coefficient, per node http: //faculty. nps. edu/rgera/ANGEL. html 45

Scale Free networks • 46

Generating Barabasi-Albert 47

Generating Barabasi-Albert networks http: //networkx. lanl. gov/reference/generated/networkx. generators. random_graphs. barabasi_albert_graph. html#networkx. generators. random_graphs. barabasi_albert_graph 48

Modified BA • Many modifications of this model exists, based on: – Nodes “retiring” and losing their status/outdated – 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. – It is a simple model (allows consistent research) that has growth and preferential attachment – One can add more conditions to this basic model, in 49 order to mimic reality

A zoo of complex networks 50

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

Main References • Newman “The structure and function of complex networks” (2003) • Estrada “The structure of complex Networks” (2012) • Barabasi “Network Science” (online: http: //barabasi. com/networksciencebook/) • References to the classes that exist in python: http: //networkx. lanl. gov/reference/generators. html 52

Back to coding in Co. Calc 53