Outline Random Networks Basics Basics Basics Definitions - PowerPoint PPT Presentation

Random Networks Random Networks Outline Random Networks Basics Basics Basics Definitions Definitions How to build Definitions How to build Complex Networks, Course 295A, Spring, 2008 Some visual examples Some visual examples How to build Structure Structure Clustering Clustering Some visual examples Degree distributions Degree distributions Prof. Peter Dodds Configuration model Configuration model Largest component Largest component Structure Generating Generating Department of Mathematics & Statistics Functions Clustering Functions Definitions Definitions University of Vermont Degree distributions Properties Properties References References Configuration model Largest component Generating Functions Definitions Properties References Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License . Frame 1/83 Frame 2/83 Random Networks Random Networks Random networks Random networks Basics Basics Some features: Definitions Definitions How to build How to build Pure, abstract random networks: Some visual examples Some visual examples ◮ Number of possible edges: Structure Structure Clustering Clustering ◮ Consider set of all networks with N labelled nodes � N � Degree distributions Degree distributions = N ( N − 1 ) Configuration model Configuration model and m edges. 0 ≤ m ≤ Largest component Largest component 2 2 ◮ Standard random network = randomly chosen Generating Generating Functions Functions network from this set. Definitions � ( N 2 ) � Definitions ◮ Given m edges, there are different possible Properties Properties m ◮ To be clear: each network is equally probable. References References networks. � N � ◮ Sometimes equiprobability is a good assumption, but ◮ Crazy factorial explosion for 1 ≪ m ≪ . 2 it is always an assumption. ◮ Limit of m = 0: empty graph. ◮ Known as Erdös-Rényi random networks or ER � N � ◮ Limit of m = : complete or fully-connected graph. 2 graphs. ◮ Real world: links are usually costly so real networks are almost always sparse. Frame 4/83 Frame 5/83

Random Networks Random Networks Random networks Random networks A few more things: Basics Basics How to build standard random networks: Definitions Definitions ◮ For method 1, # links is probablistic: How to build How to build Some visual examples Some visual examples ◮ Given N and m . Structure � N � Structure = p 1 Clustering Clustering ◮ Two probablistic methods (we’ll see a third later on) � m � = p 2 N ( N − 1 ) Degree distributions Degree distributions 2 Configuration model Configuration model Largest component Largest component � N � 1. Connect each of the pairs with appropriate Generating Generating 2 Functions Functions probability p . ◮ So the expected or average degree is Definitions Definitions Properties Properties ◮ Useful for theoretical work. References References � k � = 2 � m � 2. Take N nodes and add exactly m links by selecting N edges without replacement. ◮ Algorithm: Randomly choose a pair of nodes i and j , 2 N ( N − 1 ) = ✓ = 2 N p 1 N p 1 2 � 2 � i � = j , and connect if unconnected; repeat until all m N ( N − 1 ) = p ( N − 1 ) . � � ✓ edges are allocated. ◮ Best for adding small numbers of links (most cases). ◮ 1 and 2 are effectively equivalent for large N . ◮ Which is what it should be... ◮ If we keep � k � constant then p ∝ 1 / N → 0 as Frame 7/83 Frame 8/83 N → ∞ . Random Networks Random Networks Random networks: examples for N =500 Random networks: largest components Basics Basics Definitions Definitions How to build How to build Some visual examples Some visual examples Structure Structure Clustering Clustering Degree distributions Degree distributions Configuration model Configuration model Largest component Largest component Generating Generating Functions Functions m = 100 m = 200 m = 230 m = 240 m = 250 m = 100 m = 200 m = 230 m = 240 m = 250 � k � = 0.4 � k � = 0.8 � k � = 0.92 � k � = 0.96 � k � = 1 Definitions � k � = 0.4 � k � = 0.8 � k � = 0.92 � k � = 0.96 � k � = 1 Definitions Properties Properties References References m = 260 m = 280 m = 300 m = 500 m = 1000 m = 260 m = 280 m = 300 m = 500 m = 1000 � k � = 1.04 � k � = 1.12 � k � = 1.2 � k � = 2 � k � = 4 � k � = 1.04 � k � = 1.12 � k � = 1.2 � k � = 2 � k � = 4 Frame 21/83 Frame 22/83

Random Networks Random Networks Random networks: examples for N =500 Random networks: largest components Basics Basics Definitions Definitions How to build How to build Some visual examples Some visual examples Structure Structure Clustering Clustering Degree distributions Degree distributions Configuration model Configuration model Largest component Largest component Generating Generating Functions Functions m = 250 m = 250 m = 250 m = 250 m = 250 m = 250 m = 250 m = 250 m = 250 m = 250 � k � = 1 � k � = 1 � k � = 1 � k � = 1 � k � = 1 Definitions � k � = 1 � k � = 1 � k � = 1 � k � = 1 � k � = 1 Definitions Properties Properties References References m = 250 m = 250 m = 250 m = 250 m = 250 m = 250 m = 250 m = 250 m = 250 m = 250 � k � = 1 � k � = 1 � k � = 1 � k � = 1 � k � = 1 � k � = 1 � k � = 1 � k � = 1 � k � = 1 � k � = 1 Frame 23/83 Frame 24/83 Random Networks Random Networks Random networks Random networks Clustering: Basics Basics Definitions Definitions How to build How to build ◮ For method 1, what is the clustering coefficient for a Some visual examples Some visual examples Structure Structure finite network? Clustering Clustering Degree distributions Degree distributions ◮ Consider triangle/triple clustering coefficient Clustering: Configuration model Configuration model (Newman [1] ): Largest component Largest component Generating ◮ So for large random networks ( N → ∞ ), clustering Generating Functions Functions C 2 = 3 × # triangles Definitions drops to zero. Definitions Properties Properties # triples ◮ Key structural feature of random networks is that References References they locally look like branching networks (no loops). ◮ Recall: C 2 = probability that two nodes are connected given they have a friend in common. ◮ For standard random networks, we have simply that C 2 = p . Frame 26/83 Frame 27/83

Random Networks Random Networks Random networks Random networks Degree distribution: Basics Basics Definitions Definitions ◮ Recall p k = probability that a randomly selected node How to build How to build Limiting form of P ( k ; p , N ) : Some visual examples Some visual examples has degree k . Structure Structure ◮ Consider method 1 for constructing random Clustering ◮ Our degree distribution: Clustering Degree distributions Degree distributions � N − 1 � p k ( 1 − p ) N − 1 − k . Configuration model Configuration model networks: each possible link is realized with P ( k ; p , N ) = Largest component k Largest component probability p . ◮ What happens as N → ∞ ? Generating Generating Functions Functions ◮ Now consider one node: there are ‘ N choose k ’ ways Definitions ◮ We must end up with the normal distribution right? Definitions Properties Properties the node can be connected to k of the other N − 1 ◮ If p is fixed, then we would end up with a Gaussian References References nodes. with average degree � k � ≃ pN → ∞ . ◮ Each connection occurs with probability p , each ◮ But we want to keep � k � fixed... non-connection with probability ( 1 − p ) . ◮ So examine limit of P ( k ; p , N ) when p → 0 and ◮ Therefore have a binomial distribution: N → ∞ with � k � = p ( N − 1 ) = constant. � N − 1 � p k ( 1 − p ) N − 1 − k . P ( k ; p , N ) = k Frame 29/83 Frame 30/83 Random Networks Random Networks Limiting form of P ( k ; p , N ) : Limiting form of P ( k ; p , N ) : Basics Basics ◮ We are now here: Definitions Definitions � k � ◮ Substitute p = N − 1 into P ( k ; p , N ) and hold k fixed: How to build How to build Some visual examples � � N − 1 − k Some visual examples P ( k ; p , N ) ≃ � k � k � k � � � � k � Structure 1 − Structure � N − 1 � k � � N − 1 − k � k � k ! N − 1 Clustering Clustering P ( k ; p , N ) = 1 − Degree distributions Degree distributions k N − 1 N − 1 Configuration model Configuration model Largest component ◮ Now use the excellent result: Largest component Generating Generating � � N − 1 − k � k � k Functions � � n Functions 1 + x ( N − 1 )! � k � = e x . lim Definitions Definitions = 1 − n ( N − 1 ) k Properties Properties k !( N − 1 − k )! N − 1 n →∞ References References (Use l’Hôpital’s rule to prove.) � � N − 1 − k � k � k ◮ Identifying n = N − 1 and x = −� k � : = ( N − 1 )( N − 2 ) · · · ( N − k ) � k � 1 − ( N − 1 ) k k ! N − 1 � � − k P ( k ; � k � ) ≃ � k � k → � k � k � k � k ! e −� k � k ! e −� k � 1 − N − 1 ◮ This is a Poisson distribution ( ⊞ ) with mean � k � . Frame 31/83 Frame 32/83