02 measuring networks, and random graph model

CS224W: Analysis of Networks Jure Leskovec, Stanford University http://cs224w.stanford.edu Degree distribution: P(k) Path length: h Clustering coefficient: C Connected components: s 10/3/18 Jure Leskovec, Stanford CS224W: Analysis of Networks, http://cs224w.stanford.edu Degree distribution P(k): Probability that a randomly chosen node has degree k Nk = # nodes with degree k P(k) ¡ Normalized histogram: P(k) = Nk / N ➔ plot ¡ 0.6 0.5 0.4 0.3 0.2 0.1 k Nk k 10/3/18 Jure Leskovec, Stanford CS224W: Analysis of Networks, http://cs224w.stanford.edu ¡ A path is a sequence of nodes in which each node is linked to the next one Pn = {i0,i1,i2, ,in } ¡ Pn = {(i0 ,i1),(i1 ,i2 ),(i2 ,i3 ), ,(in-1,in )} Path can intersect itself and pass through the same edge multiple times § E.g.: ACBDCDEG § In a directed graph a path can only follow the direction of the “arrow” 10/3/18 B F A D E G C X Jure Leskovec, Stanford CS224W: Analysis of Networks, http://cs224w.stanford.edu H D ¡ between a pair of nodes is defined as A X the number of edges along the C shortest path connecting the nodes B § *If the two nodes are not connected, the hB,D = hA,X = ∞ distance is usually defined as infinite D ¡ In directed graphs paths need to follow the direction of the arrows A C B hB,C = 1, hC,B = 10/3/18 Distance (shortest path, geodesic) § Consequence: Distance is not symmetric: hB,C ≠ hC, B Jure Leskovec, Stanford CS224W: Analysis of Networks, http://cs224w.stanford.edu ¡ Diameter: The maximum (shortest path) distance between any pair of nodes in a graph ¡ Average path length for a connected graph (component) or a strongly connected (component of a) directed graph where h is the distance from node i to node j h= hij E is max number of edges (total number of å Emax i , j ¹i node pairs) = n(n-1)/2 ij max § Many times we compute the average only over the connected pairs of nodes (that is, we ignore “infinite” length paths) 10/3/18 Jure Leskovec, Stanford CS224W: Analysis of Networks, http://cs224w.stanford.edu ¡ Clustering coefficient: § What portion of i’s neighbors are connected? § Node i with degree ki § Ci Î [0,1] § where ei is the number of edges between the neighbors of node i ¡ Average clustering coefficient: C = N 10/3/18 Jure Leskovec, Stanford CS224W: Analysis of Networks, http://cs224w.stanford.edu N åC i i ¡ Clustering coefficient: § What portion of i’s neighbors are connected? § Node i with degree ki § where ei is the number of edges between the neighbors of node i B F A D E G C H 10/3/18 kB=2, eB=1, CB=2/2 = kD=4, eD=2, CD=4/12 = 1/3 Avg clustering: C=0.33 Jure Leskovec, Stanford CS224W: Analysis of Networks, http://cs224w.stanford.edu ¡ Size of the largest connected component § Largest set where any two vertices can be joined by a path ¡ Largest component = Giant component B A D F C H How to find connected components: • Start from random node and perform Breadth First Search (BFS) • Label the nodes BFS visited • If all nodes are visited, the network is connected • Otherwise find an unvisited node and repeat BFS G 10/3/18 Jure Leskovec, Stanford CS224W: Analysis of Networks, http://cs224w.stanford.edu 10 ¡ Assume each human is connected to 100 other people Then: § § § § § ¡ Step 1: reach 100 people Step 2: reach 100*100 = 10,000 people Step 3: reach 100*100*100 = 1,000,000 people Step 4: reach 100*100*100*100 = 100M people In steps we can reach 10 billion people s What’s wrong here? We ignore clustering! § Not all edges point to new people § 92% of FB friendships happen through a friend-of-a-friend 10/3/18 Jure Leskovec, Stanford CS224W: Analysis of Networks, http://cs224w.stanford.edu 49 MSN network has orders of magnitude larger clustering than the corresponding Gnp! ¡ Other examples: ¡ Actor Collaborations (IMDB): N = 225,226 nodes, avg degree k = 61 Electrical power grid: N = 4,941 nodes, k = 2.67 Network of neurons: N = 282 nodes, k = 14 Network hactual hrandom Crandom Film actors 3.65 2.99 Power Grid 18.70 12.40 0.005 C elegans 2.65 2.25 0.05 0.00027 h Average shortest path length C Average clustering coefficient “actual” … real network “random” … random graph with same avg degree 10/3/18 Jure Leskovec, Stanford CS224W: Analysis of Networks, http://cs224w.stanford.edu 50 ¡ Consequence of expansion: § Short paths: O(log n) § This is the smallest diameter we can get if we have a constant degree § But clustering is low! ¡ But networks have “local” structure: § Triadic closure: Friend of a friend is my friend § High clustering but diameter is also high ¡ 10/3/18 How can we have both? Low diameter Low clustering coefficient High clustering coefficient High diameter Jure Leskovec, Stanford CS224W: Analysis of Networks, http://cs224w.stanford.edu 51 ¡ Could a network with high clustering also be a small world (log $ dimeter)? § How can we at the same time have high clustering and small diameter? High clustering High diameter Low clustering Low diameter § Clustering implies edge “locality” § Randomness enables “shortcuts” 10/3/18 Jure Leskovec, Stanford CS224W: Analysis of Networks, http://cs224w.stanford.edu 52 [Watts-Strogatz, ‘98] Small-World Model [Watts-Strogatz ‘98] Two components to the model: ¡ (1) Start with a low-dimensional regular lattice § (In our case we are using a ring as a lattice) § Has high clustering coefficient ¡ Now introduce randomness (“shortcuts”) ¡ (2) 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 node 10/3/18 Jure Leskovec, Stanford CS224W: Analysis of Networks, http://cs224w.stanford.edu 53 [Watts-Strogatz, ‘98] High clustering High diameter h= N 2k C= 31 42 High clustering Low diameter Low clustering Low diameter h= log N log a C= k N Rewiring allows us to “interpolate” between a regular lattice and a random graph 10/3/18 Jure Leskovec, Stanford CS224W: Analysis of Networks, http://cs224w.stanford.edu 54 Parameter region of high clustering and low path length (scaled) Average Path Length # $ Clustering Coefficient, ! = ∑ !& Intuition: It takes a lot of randomness to ruin the clustering, but a very small amount to create shortcuts Prob of rewiring, p 10/3/18 Jure Leskovec, Stanford CS224W: Analysis of Networks, http://cs224w.stanford.edu 55 ¡ Could a network with high clustering be at the same time a small world? § Yes! You don’t need more than a few random links ¡ The Watts Strogatz Model: § Provides insight on the interplay between clustering and the small-world § Captures the structure of many realistic networks § Accounts for the high clustering of real networks § Does not lead to the correct degree distribution 10/3/18 Jure Leskovec, Stanford CS224W: Analysis of Networks, http://cs224w.stanford.edu 58 ¡ 10/3/18 What mechanisms people use to navigate networks and find the target? Jure Leskovec, Stanford CS224W: Analysis of Networks, http://cs224w.stanford.edu 59 The setting: ¡ s only knows locations of its friends and location of the target t ¡ s does not know links of anyone else but itself ¡ Geographic Navigation: s “navigates” to a node geographically closest to t ¡ Search time T: Number of steps to reach t s t 10/3/18 Jure Leskovec, Stanford CS224W: Analysis of Networks, http://cs224w.stanford.edu 60 Searchable Search time T: Not searchable Search time T: b O((log n) ) Kleinberg’s model O((log n) ) a O(n ) Watts-Strogatz O(n ) Erdős–Rényi Note: We know these graphs have diameter O(log n) So in Kleinberg’s model search time is polynomial in log n, while in Watts-Strogatz it is exponential (in log n) 10/3/18 O(n) Jure Leskovec, Stanford CS224W: Analysis of Networks, http://cs224w.stanford.edu 61 Watts-Strogatz graphs are not searchable ¡ How we make a searchable small-world graph? ¡ Intuition: ¡ § Our long range links are not random § They follow geography! Saul Steinberg, “View of the World from 9th Avenue” 10/3/18 Jure Leskovec, Stanford CS224W: Analysis of Networks, http://cs224w.stanford.edu 62 Model [Kleinberg, Nature ‘01] ¡ § Nodes still on a grid § Node has one long range link § Prob of long link to node v: -α P(u → v) ~ d(u,v) 10/3/18 d α=1 P(u®v) α=0 P(u®v) P(u®v) § d(u,v) … grid distance between u and v § α … parameter ≥ d Jure Leskovec, Stanford CS224W: Analysis of Networks, http://cs224w.stanford.edu d(u, v)-a P(u ® v) = å d(u, w)-a w¹ u α >> d 63 ¡ We know: § α = (i.e., Watts-Strogatz): We need O( n ) steps CHAPTER 20 THE SMALL-WORLD PHENOMENON § α = 1: We need O(log(n)2) steps 20 7.0 Search time ln T 6.0 5.0 0.0 10/3/18 1.0 2.0 exponent Exponent αq Jure Leskovec, Stanford CS224W: Analysis of Networks, http://cs224w.stanford.edu 64 igure 20.6: Simulation of decentralized search in the grid-based model with clustering xponent q Each point is the average of 1000 runs on (a slight variant of) a grid with 400 Small α: too many long links 10/3/18 Big α: too many short links Jure Leskovec, Stanford CS224W: Analysis of Networks, http://cs224w.stanford.edu 65