Viral Marketing and Outbreak Detection. Fang Jin Yao Zhang
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1 Viral Marketing and Outbreak Detection Fang Jin Yao Zhang
2 Paper 1: Maximizing the Spread of Influence through a Social Network Authors: David Kempe, Jon Kleinberg, Éva Tardos KDD 2003
3 Outline Problem Statement Two models Properties - Independent Cascade - Linear Threshold Algorithms Applications (Experiments)
4 Problem Statement Problem: - If we try to convince a subset of individuals to adopt a new product, whom to target? Goal: - Trigger a large cascade of further adoptions Given - a limited budget B Assumption - A directed graph - An active node may trigger activation of neighboring nodes - Active nodes never deactivate
5 Independent Cascade Model Each active node only has a single chance to activate its inactive neighbors with probability p vw. Y 0.6 H Inactive Node Active Node Stop! 0.4 w X v U 0.2 Newly active node Successful attempt Unsuccessful attempt
6 Linear Threshold Model All the nodes are given a random threshold θ v ~ U[0,1] Each active node has a random weight b vw ~ U[0,1] to activate its neighbors, such that A node will become active when w active neighbor of v, w The weight from its neighbors will be accumulated. v b w neighbor of v v b vw, 1
7 Example Inactive Node Y 0.6 H Active Node Threshold 0.4 X 0.1 U Active neighbors Stop! w 0.5 v
8 Submodular f(s) properties Monotone: f ( S v) f ( S) Submodular: Let N be a finite set A set function f :2 N is submodular. S T N, v N \ T, f ( S v) f ( S) f ( T v) f ( T )
9 Influence Maximization Problem Influence function σ(a): expected number of active nodes at the end Influence Maximization Problem: Given a parameter k (budget), find a k-node set S to maximize f(s) Set cover problem: given a universe U and a family S of subsets of U, select k of the subsets whose union is equals to U. Can be reduced from set cover problem Suppose we have select k nodes from subsets, if there is a set A of k nodes in this graph, with σ(a) >= n + k The influence maximization problem can be reduced from set cover problem
10 Influence Maximization Problem Reduction from Set Cover. Since set cover is NP complete problem, so influence maximization is NP-hard problem. Sets, S i p uv =1 Elements in set S i
11 Greedy Algorithm Let S be a set obtained by selecting elements one at a time For k iterations, seeking the largest marginal increase: Add node v to S that maximizes f(s +v) - f(s). Performance f(s) >= (1 1/e). f(s*) The greedy algorithm is a (1 1/e) approximation. The resulting set S activates at least (1-1/e) > 63% of the number of nodes that best k-sized set could activate.
12 Experiment A collaboration graph from co-authorships in papers Resulting graph: nodes, distinct edges Simulate the process times for each targeted set Compare with other 3 common heuristics degree centrality: choose the node with highest degree distribution distance centrality: choose the nodeswith shortest paths to other nodes random nodes.
13 Results: linear threshold model Greedy Algorithm
14 Independent Cascade Model Greedy Algorithm Greedy Algorithm P = 1% P = 10%
15 Conclusion Propose linear threshold and independent cascade model Formulize influence maximization problem Prove it is NP hard Propose a greedy algorithm based on submodular property
16 Paper 2: Cost-effective Outbreak Detection in Networks Jure Leskovec, Andreas Krause, Carlos Guestrin, Christos Faloutsos, Jeanne VanBriesen, Natalie Glance KDD 2007
17 Contributions Take the cost into the consideration of the submodular based greedy algorithm Each node has the cost (e.g. Reading big blogs is more time consuming) Speed up greedy algorithm for submodular function Proposed a fast algorithm called CELF (Cost Effective Lazy Forward)
18 Application 1: Water Network Given a water distribution network, and data on how contaminants spread over the network Problem: how to select nodes to place sensors to efficiently detect the all possible contaminations?
19 Applications 2: Cascades in blogs Posts Blogs Time ordered hyperlinks Information cascade Slides from Jure Leskovec et al KDD 2006
20 General Problem Given cascades over the network, selecting a set of nodes to detect the outbreak of cascades Measurements of outbreak: 1) Minimize time to detection 2) Maximize number of detected propagations 3) Minimize number of infected people
21 Problem Formulation Given: a graph G(V,E) a budget B for sensors Cascades Select a subset of nodes A that maximize the expected reward R: s.t. cost(a)<b
22 Reward Function R(A) Submodular! For unit cost Same greedy algorithm as the Influence maximization problem Each time select a node with the maximum marginal gain (1-1/e)-approximation algorithm
23 Variable Cost Still use greedy algorithm Two objective functions A_GCB: benefit-cost greedy algorithm A_GUC: unit-cost greedy algorithm
24 Online Bound How close between R(A_k) and R(A_k-1) In practice, it is tighter than offline bound (1-1/e)
25 CELF: Speed up Greedy Algorithm CELF (Cost Effective Lazy Forward) Idea: marginal gain decrease as the solution size increases: Each time sort the marginal gain If, we can make sure is the maximum marginal gain at t Lazy evaluations! Evaluating only top values of marginal gain
26 Summary of CELF Algorithm
27 Experiments Evaluation of CELF algorithm Online bound is much tighter! Blog network Water network
28 Experiments Comparison of algorithms: CELF wins! Blog network Water network
29 Experiments Running time Blog network Water network
30 Conclusion Formulate the problem of outbreak detection Minimize time to detection Maximize number of detected propagations Minimize number of infected people Propose a fast algorithm called CELF Variable-cost algorithm with near optimal guarantee Online bound Speed up submodular based greedy algorithm
31 Paper 3: How to Win Friends and Influence People, Truthfully: Influence Maximization Mechanisms for Social Networks Yaron Singer WSDM 2012
32 Outline Problem Statement Properties Algorithms Analysis Applications (Experiments)
33 Problem Statement Given a limited budget, whom to target for the max influence? Design framework which advocate for allocation Core problems: Elicit individuals costs Optimize the influence given the budget How the propagation proceed from the selected nodes
34 Properties: Monotonicity A mechanism M=( A, p), A is an algorithm, p is a payment rule. (Payment vector describes the payments to each agent in the subset) is the subset selected by A inf (infimum), is defined to be the biggest real number that is smaller than or equal to every number in set
35 Properties Incentive compatible mechanisms: A mechanism is incentive compatible if and only if it is monotone and use threshold payments We want to find such mechanisms
36 Well-studied models Coverage model Independent cascade model Linear threshold model Coverage model is a special case of independent cascade model
37 Look back: Linear Threshold Model A node v has random threshold θ v ~ U[0,1] A node v is influenced by each neighbor w according to a weight b vw such that w neighbor of v A node v becomes active when at least (weighted) θv fraction of its neighbors are active b w active neighbor of v vw, Coverage model can transfer into independent cascade model b 1 v, w v
38 Look back: Independent Cascade Model When node v becomes active, it has a single chance of activating each currently inactive neighbor w. The activation attempt succeeds with probability p vw. coverage model can transfer into independent cascade model for each interaction, can construct a reachability graph with node v and the subset of nodes it influences.
39 Coverage Model Coverage Model: each agent a_i in the network is associated with a subset of nodes R_i The coverage function is submodular
40 Coverage Model Model Subset of nodes R, Coverage Function influence is f(s),
41 Coverage Model Good mechanism: select agents that yield high coverage while the payments <= budget Idea: greedy algorithm: greedily selecting agents based on their marginal gains and enforcing an appropriate stopping condition
42 Weighted Marginal contribution sorting o Selection rules: every time maximize: o Stop constrain: when selecting a_i, make sure: using B/2, to guaranteed the payments do not exceed budget
43 Integer programming for coverage optimization problem Z_j: variables representing the agents we wish to cover X_i: variables representing the agents we can select,
44 Relaxation We denote this relaxation as L. L(x) to denote a solution that respects the constraints
45 Influence Maximization in coverage The incentive compatible payments are bounded by a constant factor (<=2) of the proportional contributions, when using B/2 in the stopping condition, the payments do not violate the budget constraints.
46 Performance Guarantee OPT, the optimal solution the output of the algorithm
47 Experiment Advertise for a travel agency Ad method: posting a message with commercial content in their Facebook page Need to specify $$ and $ of friends on FB Amazon s Mechanical Turk platform for 3 months Reward: Each worker who participated in the competition was paid o The worker who won the competition received a bonus reword at least as high as their bid
48 Correlation = 0.04
49 Facebook graph propagate time steps Limited to 5 (10% Independent cascade) Limited to 10 (1% IC) Limited to 25 (linear threshold)
50 Influence models The coverage model (p=1): Use each agent s friends as its reachable set Independent cascade model: P=1% P=10% Linear threshold model: Weight = 1/d, d means degree
51
52
53 Reference Adapted from: e_through_a_social_network.pdf uencepeople.pdf slides of Cost-effective Outbreak Detection in Networks
54
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