1. Lecture 10: Network models CS 765: Complex Networks Slides are modified from Networks: Theory and Application by Lada Adamic

2. Four structural models Regular networks Random networks Small-world networks Scale-free networks 2

3. Regular networks – fully connected 3

4. Regular networks – Lattice 4

5. Regular networks – Lattice: ring world 5

6. modeling networks: random networks Nodes connected at random Number of edges incident on each node is Poisson distributed Poisson distribution 6

7. Random networks 7

8. Random Networks 8

9. Simplest random network Erdos-Renyi random graph: each pair of nodes is equally likely to be connected, with probability p. p = 2*E/N/(N-1) Poisson degree distribution is narrowly distributed around = p*(N-1) k P(k) Poisson degree distribution

10. Random graph model The degree distribution is given by coinflips to see how many people you’ll be connected to, one coin flip per each of the (n-1) other nodes probability p, of connecting Binomial Poisson limit with large n when small p

11. Percolation threshold in Erdos-Renyi Graphs average degree size of giant component av deg = 0.99av deg = 1.18 av deg = 3.96 Percolation theshold: how many edges need to be added before the giant component appears? As the average degree increases to z = 1, a giant component suddenly appears

12. Erdos-Renyi random graphs What happens to the size of the giant component as the density of the network increases? http://ccl.northwestern.edu/netlogo/models/GiantComponent 12

13. modeling networks: small worlds Small worlds a friend of a friend is also frequently a friend but only six hops separate any two people in the world Arnold S. – thomashawk, Flickr; http://creativecommons.org/licenses/by-nc/2.0/deed.en 13

14. Small-world networks 14

15. Small world models Duncan Watts and Steven Strogatz 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. 15

16. Watts Strogatz Small World Model As you rewire more and more of the links and random, what happens to the clustering coefficient and average shortest path relative to their values for the regular lattice? http://ccl.northwestern.edu/netlogo/models/SmallWorlds 16

17. Scale-free networks 17

18. Scale-free networks Many real world networks contain hubs: highly connected nodes Usually the distribution of edges is extremely skewed many nodes with few edges fat tail: a few nodes with a very large number of edges no “typical” number of edges number of edges number of nodes with so many edges 18

19. But is it really a power-law? A power-law will appear as a straight line on a log-log plot: A deviation from a straight line could indicate a different distribution: exponential lognormal log(# edges) log(# nodes) 19

20. Scale-free networks 20 http://ccl.northwestern.edu/netlogo/models/PreferentialAttachment

21. What implications does this have? Robustness Search Spread of disease Opinion formation Spread of computer viruses Gossip

22. 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 http://www.ladamic.com/netlearn/NetLogo4/RAndPrefAttachment.html

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

24. But it depends on what you’re sharing…

25. The role of hubs in epidemics In a power-law network, a virus can persist no matter how low its infectiousness Many real world networks do exhibit power-laws: needle sharing sexual contacts email networks

26. Spread of computer viruses can be affected by the underlying network

27. Power-law networks are robust to random breakdown

28. But are especially vulnerable to targeted attack Targeting and removing hubs can quickly break up the network

29. Real world networks are often power law Sexual networks Most individuals report 1-2 partners in the past 12 months, but some… Source: The web of human sexual contacts, Liljeroset al.,Nature 411, 907-908(21 June 2001)The web of human sexual contacts

30. Preferential Attachment in Networks First considered by [Price 65] as a model for citation networks each new paper is generated with m citations (mean) new papers cite previous papers with probability proportional to their indegree (citations) what about papers without any citations? each paper is considered to have a “default” citation probability of citing a paper with degree k, proportional to k+1 Power law with exponent α = 2+1/m

31. Barabasi-Albert model Undirected model: each node connects to other nodes with probability proportional to their degree the process starts with some initial subgraph each node comes with m edges Results in power-law with exponent α = 3

32. Basic BA-model Very simple algorithm to implement start with an initial set of m 0 fully connected nodes e.g. m 0 = 3 now add new vertices one by one, each one with exactly m edges each new edge connects to an existing vertex in proportion to the number of edges that vertex already has → preferential attachment easiest if you keep track of edge endpoints in one large array and select an element from this array at random the probability of selecting any one vertex will be proportional to the number of times it appears in the array – which corresponds to its degree 12 3 1 1 2 2 2 3 3 4 5 6 6 7 8 ….

33. generating BA graphs – cont’d To start, each vertex has an equal number of edges (2) the probability of choosing any vertex is 1/3 We add a new vertex, and it will have m edges, here take m=2 draw 2 random elements from the array – suppose they are 2 and 3 Now the probabilities of selecting 1,2,3,or 4 are 1/5, 3/10, 3/10, 1/5 Add a new vertex, draw a vertex for it to connect from the array etc. 12 3 1 1 2 2 3 3 12 3 1 1 2 2 2 3 3 3 4 4 4 12 3 4 1 1 2 2 2 3 3 3 3 4 4 4 5 5 5

34. Properties of the BA graph The distribution is scale free with exponent  = 3 P(k) = 2 m 2 /k 3 The graph is connected Every vertex is born with a link (m= 1) or several links (m > 1) It connects to older vertices, which are part of the giant component The older are richer Nodes accumulate links as time goes on preferential attachment will prefer wealthier nodes, who tend to be older and had a head start

35. vertex introduced at time t=5 vertex introduced at time t=95 Time evolution of the connectivity of a vertex in the BA model Younger vertex does not stand a chance: at t=95 older vertex has ~ 20 edges, and younger vertex is starting out with 5 at t ~ 10,000 older vertex has 200 edges and younger vertex has 50 Source: Barabasi and Albert, 'Emergence of scaling in random networks’, Science 1999.

36. thoughts BA networks are not clustered. Can you think of a growth model of having preferential attachment and clustering at the same time? What would the network look like if nodes are added over time, but not attached preferentially? What other processes might give rise to power law networks?