1. Eurecom, Sophia-Antipolis Thrasyvoulos Spyropoulos / spyropoul@eurecom.fr Random Graph Models: Create/Explain Complex Network Properties

2. Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis  The networks discussed are quite large!  Impossible to describe or visualize explicitly.  Consider this example:  You have a new Internet routing algorithm  You want to evaluate it, but do not have a trace of the Internet topology  You decide to create an “Internet-like” graph on which you will run your algorithm  How do you describe/create this graph??  Random graphs: local and probabilistic rules by which vertices are connected  Goal: from simple probabilistic rules to observed complexity Q: Which rules gives us (most of) the observed properties? 2

3. Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis 3

4. Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis  This is “Conway’s game of life” (many other automata)  http://www.youtube.com/watch?v=ma7dwLIEiYU&feature=relat ed (demo) http://www.youtube.com/watch?v=ma7dwLIEiYU&feature=relat ed  http://www.bitstorm.org/gameoflife/ (try your own) http://www.bitstorm.org/gameoflife/ 4 Local Rules  Each cell either white or blue  Each cell interacts with its 8 neighbors  Time is discrete (rounds) 1. Any blue cell with fewer than two live neighbors  becomes white 2. Any blue cell with two or three blue neighbors lives on to the round 3. Any blue cell with more than three blue neighbors  becomes white 4. Any white cell with exactly three blue neighbors  become blue

5. Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis  A very (very!) simple local rule: (any) two vertices are connected with probability p  Only inputs: number of vertices n and probability p  Denote this class of graphs as G(n,p) 5 Erdös-Rényi model (1960) Connect with probability p p=1/6 N=10 average degree  k  ~ 1.5

6. Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis N and p do not uniquely define the network– we can have many different realizations of it. How many? G(10,1/6) N=10 p=1/6 G(N,L): a graph with N nodes and L links The probability to form a particular graph G(N,L) is That is, each graph G(N,L) appears with probability P(G(N,L)).

7. Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis P(L): the probability to have exactly L links in a network of N nodes and probability p: The maximum number of links in a network of N nodes. Number of different ways we can choose L links among all potential links. Binomial distribution...

8. Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis P(L): the probability to have a network of exactly L links  The average number of links in a random graph  The standard deviation  Average node degree

9. Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis As the network size increases, the distribution becomes increasingly narrow—which means that we are increasingly confident that the number of links the graph has is in the vicinity of.

10. Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis  The degree distribution  average degree is = p(N-1)  variance σ 2 = p(1-p)(N-1)  Assuming z=Np is fixed, as N → ∞, B(N,k,p) is approximated by a Poisson distribution  As N → ∞  Highly concentrated around the mean  Probability of very high node degrees is exponentially small  Very different from power law! 10 Binomial

11. Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis The secret behind the small world effect – Looking at the network volume

12. Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis The secret behind the small world effect – Looking at the network volume Polynomial growth

13. Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis The secret behind the small world effect – Looking at the network volume Polynomial growth

14. Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis The secret behind the small world effect – Looking at the network volume Polynomial growthExponential growth

15. Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis Given the huge differences in scope, size, and average degree, the agreement is excellent!

16. Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis  Consider a random graph G(n,p) Q: What is the probability that two of your neighbors are also neighbors? A: It is equal to p, independent of local structure  clustering coefficient C = p  when z is fixed (sparse networks): C = z/n =O(1/n) 16

17. Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis Given the huge differences in scope, size, and average degree, there is a clear disagreement.

18. Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis  Erdos-Renyi Graphs are “small world”  path lengths are O(logn)  Erdos-Renyi Graphs are not “scale-free”  Degree distribution binomial and highly-concentrated (no power- law)  Exponentially small probability to have “hubs” (no heavy-tail)  Erdos-Renyi Graphs are not “clustered”  C  0, as N becomes larger Conclusion: ER random graphs are not a good model of real networks  BUT: still provide a great deal of insight! 18 √ X X

19. Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis Some of your neighbors neighbors are also your own Exponential growth: Clustering inhibits the small-worldness

20. Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis  Short paths must be combined with  High clustering coefficient Watts and Strogatz model [WS98]  Start with a ring, where every node is connected to the next k nodes  With probability p, rewire every edge (or, add a shortcut) to a random node 20 order randomness p = 0 p = 1 0 < p < 1

21. Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis The Watts Strogatz Model : It takes a lot of randomness to ruin the clustering, but a very small amount to overcome locality 21 log-scale in p When p = 0, C = 3(k-2)/4(k-1) ~ ¾ L = n/k For small p, C ~ ¾ L ~ logn Clustering Coefficient – Characteristic Path Length

22. Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis Nodes: online user Links: email contact, tweet, or friendship Alan Mislove, Measurement and Analysis of Online Social Networks All distributions show a fat-tail behavior: there are orders of magnitude spread in the degrees

23. Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis

24. Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis  The configuration model  input: the degree sequence [d 1,d 2,…,d n ]  process: -Create d i copies of node i; link them randomly -Take a random matching (pairing) of the copies self-loops and multiple edges are allowed 24 4 132 But: Too artificial!

25. Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis Networks continuously expand by the addition of new nodes Barabási & Albert, Science 286, 509 (1999) ER, WS models: the number of nodes, N, is fixed (static models)

26. Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis (1) Networks continuously expand by the addition of new nodes Add a new node with m links Barabási & Albert, Science 286, 509 (1999)

27. Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis Barabási & Albert, Science 286, 509 (1999) PREFERENTIAL ATTACHMENT: the probability that a node connects to a node with k links is proportional to k. A: New nodes prefer to link to highly connected nodes. Q: Where will the new node link to? ER, WS models: choose randomly.

28. Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis “The rich get richer”  First considered by [Price 65] as a model for citation networks  each new paper is generated with m citations (on average)  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 28

29. Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis  The BA model (undirected graph)  input: some initial subgraph G 0, and m the number of edges per new node  the process: -nodes arrive one at a time -each node connects to m other nodes selecting them with probability proportional to their degree -if [d 1,…,d t ] is the degree sequence at time t, the node t+1 links to node i with probability  Results in power-law with exponent α = 3  Various Problems: cannot account for every power law observed (Web), correlates age with degree, etc. 29

30. Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis 30 Path length Clust. Coeff.  Larger than ER  Still goes to 0 as N  ∞

31. Eurecom, Sophia-Antipolis Thrasyvoulos Spyropoulos / spyropoul@eurecom.fr Network Resilience or How to Break a Network

32. Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis  We saw that increasing p  denser networks  In the large N case we increase z = Np the average degree  But what really happens as p (or z) increases? 32 A random network on 50 nodes: p = 0.01  disconnected, largest component = 3

33. Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis  p = 0.03  large component appears  But almost 40% of nodes still disconnected 33

34. Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis  p = 0.05  “giant” component emerges  Only 3 nodes disconnected  Giant component  the graph “percolates” 34

35. Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis  p = 0.10  all nodes connected 35

36. Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis S: the fraction of nodes in the giant component, S=N GC /N there is a phase transition at =1:  for < 1 there is no giant component  for > 1 there is a giant component  for large the giant component contains all nodes (S=1) http://linbaba.files.wordpress.com/20 10/10/erdos-renyi.png S

37. Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis Q1: How does the degree distribution affect resilience? Q2: How does the removal strategy affect resilience? Def: network is still “functional” as long as there is a “giant component” Def: “giant component” S contains a finite percentage of all nodes n  as n  ∞ S = cn (or (Θ(n)) 37

38. Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis Def: φ = probability a vertex not having been removed  i.e. percentage of vertices present 38 φ = 1 φ = 0.7 φ = 0.3 φ = 0

39. Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis  Assume degree distribution p k  Probability of a uniformly chosen node having k neighbors  Step 1: pick random vertex i  Step 2: i not in giant cluster  none of its neighbors is in the giant cluster Def: u = average probability that a neighbor j does not connect i to giant component Result: If i has degree k  Prob(i not in S) = u k 39

40. Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis 40  φ c given at the point at which the curve is tangent to u = 1  tangent to u = 1  2 solutions for u giant component S exists u = 1: gives S = 0 u = 1: threshold

41. Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis Critical Threshold depends on mean degree ( ) and degree variability ( ) 41

42. Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis 42  Degree distribution  Average degree = c  = c(c+1)  Critical threshold  Example: for average degree c = 4  φ c = 0.25  75% vertices must be removed to “kill” the network

43. Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis 43  Degree distribution  Fact: most networks exhibit a power-law degree distribution with α in the range (2,3) Q: what is and for these networks? A: is finite but is infinite! Q: What is the critical threshold? A: it is 0!!  power-law graphs can “survive” any number of failures

44. Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis Exponential degree distribution 44 Power-law degree distribution

45. Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis  gnutella (P2P) network  20% of nodes removed 574 nodes in giant component 427 nodes in giant component

46. Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis Network attacks (good news)  Real networks (social networks, internet, P2P networks) are extremely robust to uniform removal attacks  The higher the variance of the degree distribution, the better Malware Infections and Immunization (bad news)  Epidemic occurs if a majority of nodes gets infected  Stop virus/worm/etc. from spreading  vaccinate/fix a number of nodes --- Goal: disconnect “contact” graph  Need to immunize a large majority of nodes to avoid spread 46

47. Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis  A more efficient attack: remove the highest degree nodes 47 Exponential degree distribution Power-law degree distribution

48. Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis  gnutella network,  22 most connected nodes removed (2.8% of the nodes) 301 nodes in giant component574 nodes in giant component

49. Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis Where g jk = the number of shortest paths connecting j-k, and g jk = the number that node i is on. Usually normalized by: 49 betweenness of vertex i paths between j and k that pass through i all paths between j and k

50. Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis 50 bridge

51. Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis Nodes are sized by degree, and colored by betweenness. Can you spot nodes with high betweenness but relatively low degree? What about high degree but relatively low betweenness? 51

52. Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis Connectivity a) Remove random node b) Remove high degree node c) Remove high betweeness node 52

53. Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis  The network below is a wireless network (e.g. sensor network)  Nodes run on battery  total energy E max  Each node picks a destination randomly and sends data at constant rate  every packet going through a node spends E of its energy Q: How long would it take until the first node dies out of battery? 53 S1S1 D1D1 D2D2 S2S2

54. Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis 54

55. Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis Monitoring a) Where would you place a traffic monitor in order to track the maximum number of packets (if this was your university network)? b) Where would you place traffic cameras if that was a street network? 55

56. Thrasyvoulos Spyropoulos / spyropou@eurecom.fr Eurecom, Sophia-Antipolis  Traffic Flow: Each link has capacity 1 Q: What is the maximum throughput between S-D? A: Max Flow – Min Cut theorem  max flow equal to min number of links removed to disconnect S-D  S-D throughput = 1 56 S D