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1 CS224W: Social and Information Network Analysis Jure Leskovec, Stanford University, y

2 Due in 1 week: Oct 4 in class! The idea of the reaction papers is: To familiarize yourselves more in depth with the material covered in class Do reading beyond what htwas covered. You should be thinking beyond what you just read, and not just take other people's work for granted. Can be done in groups of 2 3 students Read at least 3 papers: Anything from course website, last year s website Anything from Easley Kleinberg How to submit: File: PDF or DOC with SUNetIds of team members: Eg E.g., if 2 members then: <SUNetId> <SUNnetId> <SUNnetId>.pdf Upload to Dropbox folder at 9/28/2010 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 2

3 On 3 5 pages answer the following questions: 1 page: Summary What is main technical content of the papers? How do they fit in the field, and what you have learned in class so far? What is the connection between the papers you are discussing? 1 page: Critique Why is it interesting in relation to the corresponding section of the course? What were the authors missing? Was anything particularly unrealistic? 1 page: pg Brainstorming What are promising further research questions in the direction of the papers? How could they be pursued? An idea of a better model for something? A better algorithm? A test of a model or algorithm on a dataset or simulated data? 9/28/2010 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 3

4 Erdos Renyi Random Graph model [Erdos Renyi Renyi, 60] aka.: Poisson/Bernoulli random graphs Not perfect model but interesting calculations Two variants: G np n,p: undirected graph on n nodes and each edge (u,v) appears i.i.d. with probability p So a graph with m edges appears with prob.: (M choose m)p m (1-p) M-m, where M=n(n-1)/2 is the max number of edges G n,m : undirected graph with n nodes, m uniformly at random picked edges What kinds of networks does such model produce? 9/28/2010 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 4

5 What is expected degree of a node? Let LtX v be a random var. measuring the degree of the node v (# of incident edges): E[X v ]= j jp(x v =j) Linearity of expectation: For any random variables Y 1,Y 2,,Y k If Y=Y 1 +Y 2 + Y k, then E[Y]= i E[Y i ] Easier way: decompose X v in X v = X v1 +X v2 + +X vn where X vu is a {0,1} random variable which tells if edge (v,u) exists or not. So: E[X v ]= u E[X vu ] = p(n-1) How to think about it: Prob. of node u linking to node v is p u can link (flips a coin) for all of (n-1) remaining nodes Thus, the expected degree of node u is: p(n-1) 9/28/2010 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 5

6 We want E[Xv ] be independent of n So let: p=const/(n-1) Observation: If we build random graph G(n,p) with ih p=c/(n-1) we have many isolated nodes Why? P[v has degree 0]= 9/28/2010 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 6

7 How big do we have to make p before we re likely to have no isolated nodes? We know: P[v has degree 0] < e -c Event we are asking about is: I=some node is isolated I = v I v where I v is the event that v is isolated We have: P(I)= P( v I v ) v P(I v ) = ne -c 9/28/2010 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 7

8 So, P(I) = ne c Let s try: c=ln(n) then: ne -c =ne -ln n =n1/n n 1/n = 1 c=2ln(n) then: ne -2ln n = n 1/n 2 = 1/n So if: p=ln(n) then P(I)=1 p=2ln(n) then P(I)=1/n 0 as n 9/28/2010 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 8

9 Graph structure as p changes: 1/(n 1) const/(n 1) log(n)/(n 1) 2*log(n)/(n 1) Giant component Avg. deg const. Fewer isolated No isolated nodes. 0 1 appears Lots of isolated nodes. Complete nodes. 0 edges graph Emergence of a giant component: avg. degree k=2m/n or p=k/(n-1) k=1-ε: all components are of size Ω(log n) k=1+ε: 1 component of size Ω(n), others have size Ω(log n) Demo! p 9/28/2010 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 9

10 Degree distribution is Binomial. Let p k denote a fraction of nodes with degree k: p Mean=np Var=np(1-p) k n p k (1 p k ) n k 9/28/2010 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 10

11 Assume each node has d spokes (half edges): d=1: set of pairs d=2: d2: set of cycles d=3: arbitrarily complicated graphs Randomly pair them up 9/28/2010 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 11

12 Configuration model: Nodes with spokes Nodes with mini n0desn0des Assume a degree sequence d 1, d 2, d n Useful for social networks because we have control over the degree sequence 9/28/2010 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 12

13 G(V, E) has expansion α: if S V: #edges leaving S α min( S, V-S ) Or equivalently: α is the minimum ratio: # edges leaving S min( S, V S ) over all sets S ie i.e, every set of nodes has a high surface to volume ratio 9/28/2010 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 13

14 Expansion gives us a measure of robustness if we want to disconnect l nodes, we need to cut α l edges Low expansion: α = High expansion: α = Social networks: Communities 9/28/2010 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 14

15 d regular graph (every node has deg. d): Expansions it at best d. (when S is 1 node) d) Is there a graph on n nodes (n ), max deg. d (const), so that expansion α remains const? Examples: Grid: d=4: α =2n/(n 2 /4) 0 (n/2 by n/2 square in the center) Complete binary tree: α 0 for S =(n/2)-1 Fact: for a random 3 regular graph on n nodes, there is some const α (α>0, indep. of n) such that w.h.p. the expansion of the graph is α 9/28/2010 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 15

16 Fact: In a graph on n nodes with expansion α for all pairs of nodes s and t there is a path connecting them of O((log n) / α) edges. Proof: Let S j be a set of all nodes found within j steps of BFS from t. Then: S j+1 S j + α S j /d = S j (1+ α/d) In how many steps of BFS we reach >n/2 nodes? Need j so that: (1+ α/d) j >n/2 n/2, set j=d log(n)/α So, in O(log n) steps S j grows to Θ(n). And dthe diameter of G is O(log(n)/ ( α) ) 9/28/2010 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 16

17 Consequence of expansion: Short paths: O(log n) between each pair Working definition of a short path : O(log n) This is the best we can do if the graphhashas constant degree and n nodes But social networks have local structure: Triadic closure: Friend of a friend is my friend Maybe grid is a better model? Pure exponential growth Triadic closure reduces growth rate 9/28/2010 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 17

18 Just before the edge (u,v) (uv) is placed how many hops is between u and v? 9/28/2010 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 18

19 Just before the edge (u,v) (uv) is placed how many hops is between u and v? (D) (L) (F) G np (A) Fraction of triad closing edges Network u % Δ F 66% D 28% A 23% L 50% 9/28/2010 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 19 v w

20 [Watts Strogatz Nature 98] How to have local edges (lots of triangles) and small diameter? Small world model [Watts Strogatz 98]: Start with a low dimensional regular lattice Rewire: Add/remove edges to create shortcuts to join remote parts of the lattice For each edge with prob. p move the other end to a random vertex 9/28/2010 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 20

21 [Watts Strogatz Nature 98] High clustering High clustering Low clustering High diameter Low diameter Low diameter Rewiring allows to interpolate between regular lattice and a random graph 9/28/2010 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 21

22 Measure of clustering (local structure, i.e., ie triangles in a graph) Clustering coefficient C i of node i is: k i degree of node i C i =0 C i =1/3 C i =1 Clustering coefficient: C =1/n C i 9/28/2010 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 22

23 1/n C i ng coefficient, C = 1 Clusterin Prob. of rewiring, p 9/28/2010 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 23

24 Collaborations between actors (IMDB): 225,226 nodes, avg. degree k=61 Electrical power grid: 4,941 nodes, k=2.67 Network of neurons 282 nodes, k=14 L... Average shortest path length C... Average clustering coefficient 9/28/2010 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 24

25 When we add one random connection out of each node we get short paths. Why? Suppose we build random edges by giving every node half edge and randomly pair them Consider a graph where we contract t 2x2 subgraphs into supernodes Now we have 4 edges sticking out of each supernode From Thm. we have short paths between super nodes, we can turn this into a path in a real graph by adding at most 2 steps per hop: O(2log n) 9/28/2010 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 25

26 Ok, so paths are short And people are able to find them! (without theglobal knowledge of the network) 9/28/2010 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 26

27 s only knows locations of its friends and location of the target t s does not know links of anyone but itself Geographic navigation: s forwards the message to the node closest to t 9/28/2010 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 27

28 Model: Grid where each node has one random edge This is a small world. Fact: A decentralized algorithm in Watts Strogatz model needs n 2/3 steps to reach t in expectation (even though paths of length log(n) exist). Proof: Let s do this in 1 dim. n nodes on a ring plus one random directed d edge per node. Lower bound on search time is now n 1/2 Lower bound for d dim.: n d/d+1 9/28/2010 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 28