12 influence maximization in networks

Announcements: 1) Project milestones due in one week 2) Honor code and code submission CS224W: Analysis of Networks Jure Leskovec, Stanford University http://cs224w.stanford.edu ¡ We are more influenced by our friends than strangers ă 68% of consumers consult friends and family before purchasing home electronics ă50% research online before purchasing electronics 11/1/18 Jure Leskovec, Stanford CS224W: Analysis of Networks, http://cs224w.stanford.edu Identify influential customers Convince them to adopt the product – Offer discount or free samples 11/1/18 These customers endorse the product among their friends Jure Leskovec, Stanford CS224W: Analysis of Networks, http://cs224w.stanford.edu Kate Middleton effect The trend effect that Kate, Duchess of Cambridge has on others, from cosmetic surgery for brides, to sales of coral-colored jeans.” 11/1/18 Jure Leskovec, Stanford CS224W: Analysis of Networks, http://cs224w.stanford.edu n According to Newsweek, "The Kate Effect may be worth £1 billion to the UK fashion industry." n Tony DiMasso, L K Bennett’s US president, stated in 2012, " when she does wear something, it always seems to go on a waiting list." 11/1/18 Jure Leskovec, Stanford CS224W: Analysis of Networks, http://cs224w.stanford.edu 11/1/18 ¡ Influential persons often have many friends ¡ Kate is one of the persons that have many friends in this social network ¡ For more Kates, it’s not as easy as you might think! Jure Leskovec, Stanford CS224W: Analysis of Networks, http://cs224w.stanford.edu 11/1/18 ¡ Given a directed graph and k>0, ¡ Find k seeds (Kates) to maximize the number of influenced people (possibly in many steps) Jure Leskovec, Stanford CS224W: Analysis of Networks, http://cs224w.stanford.edu ¡ Linear Threshold Model ¡ Independent Cascade Model 11/1/18 Jure Leskovec, Stanford CS224W: Analysis of Networks, http://cs224w.stanford.edu ¡ A node v has random threshold !" ~ U[0,1] ¡ A node v is influenced by each neighbor w according to a weight #",% such that å bv ,w £ w neighbor of v ¡ A node v becomes active when at least (weighted) !" fraction of its neighbors are active å bv ,w ³ q v w active neighbor of v 11/1/18 Jure Leskovec, Stanford CS224W: Analysis of Networks, http://cs224w.stanford.edu Inactive Node 0.6 0.3 Active Node 0.2 X Threshold 0.2 Active neighbors 0.1 0.4 U 0.5 w 11/1/18 0.3 0.5 Stop! 0.2 v Jure Leskovec, Stanford CS224W: Analysis of Networks, http://cs224w.stanford.edu 10 ¡ Thus, the spread of a node can be computed by enumerating simple paths starting from the node 0.4 x z 0.2 0.3 0.1 0.5 y Total influence of node x is 1.96 Influence Spread of node x is = + (0.3 + 0.4 * 0.5) + (0.4 + 0.3 * 0.2) = 1.96 11/1/18 Jure Leskovec, Stanford CS224W: Analysis of Networks, http://cs224w.stanford.edu 69 0.4 x z 0.2 0.3 0.1 0.5 y Influence of node x in a subgraph that does not contain y Let the seed set S = {x,y}, then influence spread of S is s ( S ) = s V - y ( x) + s V - x ( y ) = + 0.4 + + 0.2 = 2.6 11/1/18 Jure Leskovec, Stanford CS224W: Analysis of Networks, http://cs224w.stanford.edu 71 0.4 x z 0.2 0.3 0.1 0.5 y Influence of node y in a subgraph that does not contain x Let the seed set S = {x,y}, then influence spread of S is s ( S ) = s V - y ( x) + s V - x ( y ) = + 0.4 + + 0.2 = 2.6 11/1/18 Jure Leskovec, Stanford CS224W: Analysis of Networks, http://cs224w.stanford.edu 72 0.4 x z 0.2 0.3 0.1 Total influence of the seed set {x, y} is 2.6 0.5 y Let the seed set S = {x,y}, then influence spread of S is s ( S ) = s V - y ( x) + s V - x ( y ) = + 0.4 + + 0.2 = 2.6 11/1/18 Jure Leskovec, Stanford CS224W: Analysis of Networks, http://cs224w.stanford.edu 73 Thus, influence can be estimated by enumerating all simple paths starting from the seed set On slightly different subgraphs The majority of influence flows in a small neighborhood 11/1/18 Jure Leskovec, Stanford CS224W: Analysis of Networks, http://cs224w.stanford.edu 74 ¡ Through a parameter η, we can control the size of the neighborhood: § That is, stop enumerating paths when the influence weight drops below η § Direct trade-off between accuracy of spread estimation and running time ¡ 11/1/18 Backtracking algorithm to enumerate simple paths smartly Jure Leskovec, Stanford CS224W: Analysis of Networks, http://cs224w.stanford.edu 75 In each iteration, add to the seed set, the node providing the maximum marginal gain in spread Simpath-Spread Compute marginal gain by enumerating simple paths 11/1/18 Jure Leskovec, Stanford CS224W: Analysis of Networks, http://cs224w.stanford.edu 76 ¡ As the seed set grows, the time spent in estimating spread increases § More paths to enumerate A lot of paths are repeated though The optimization avoids this repetition intelligently ¡ A look ahead parameter ‘l’ ¡ ¡ 11/1/18 Jure Leskovec, Stanford CS224W: Analysis of Networks, http://cs224w.stanford.edu 77 78 l = here Seed Set Si after iteration i Let y and x be prospective seeds from CELF queue y x Compute Compute spread spread achieved by by S+y S+x A lot1.2.of paths areachieved enumerated repeatedly 11/1/18 Jure Leskovec, Stanford CS224W: Analysis of Networks, http://cs224w.stanford.edu ¡ A collaboration network: co-authorships in papers of the arXiv high-energy physics theory: § 10,748 nodes, 53,000 edges § Example cascade process: Spread of new scientific terminology/method or new research area ¡ Independent Cascade Model: § Each user’s threshold is uniform random on [0,1] § Case 1: Uniform probability p on each edge § Case 2: Edge from v to w has probability 1/deg(w) of activating w 11/1/18 Jure Leskovec, Stanford CS224W: Analysis of Networks, http://cs224w.stanford.edu 80 ¡ Simulate the process 10,000 times for each targeted set § Every time re-choosing edge outcomes randomly ¡ Compare with other common heuristics § Degree centrality: Pick nodes with highest degree § Closeness centrality: Pick nodes in the “center” of the network § Random nodes: Pick a random set of nodes 11/1/18 Jure Leskovec, Stanford CS224W: Analysis of Networks, http://cs224w.stanford.edu 81 f(Sk) f(Sk) k puv = 0.01 k puv = 0.10 Uniform edge firing probability puv 11/1/18 Jure Leskovec, Stanford CS224W: Analysis of Networks, http://cs224w.stanford.edu 82 f(Sk) k puv=1/deg(v) 11/1/18 Non-uniform edge firing probability puv Jure Leskovec, Stanford CS224W: Analysis of Networks, http://cs224w.stanford.edu 83 ¡ More realistic viral marketing: § Different marketing actions increase likelihood of initial activation, for several nodes at once ¡ Study more general influence models: § Find trade-offs between generality and feasibility ¡ Deal with negative influences: § Model competing ideas ¡ 11/1/18 Obtain more data (better models) about how activations occur in real social networks Jure Leskovec, Stanford CS224W: Analysis of Networks, http://cs224w.stanford.edu 84