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CMU 15-896 Social networks 2: Influence Maximization Teacher: Ariel Procaccia Motivation Firm is marketing a new product Collect data on the social network Choose set of early adopters and market to them directly Customers in

  1. CMU 15-896 Social networks 2: Influence Maximization Teacher: Ariel Procaccia

  2. Motivation • Firm is marketing a new product • Collect data on the social network • Choose set of early adopters and market to them directly • Customers in generate a cascade of adoptions • Question: How to choose ? 15896 Spring 2016: Lecture 24 2

  3. Influence functions • Assume: finite graph, progressive process • Fixing a cascade model, define influence function expected #active nodes at the end of the • process starting with • Maximize over sets of size • Theorem [Kempe et al. 2003]: Under the general cascade model, influence maximization is NP- hard to approximate to a factor of ��� for any 15896 Spring 2016: Lecture 24 3

  4. Proof of theorem � � � � � � � � � � • S ET COVER : subsets � � , … , � � of � � � � � � � � , … , � � ; cover of size �? � � � � � � • Bipartite graph: � � , … , � � on one side, � � � � � � � � , … , � � and � � , … , � � for T � � � on the other • � � becomes active if � � ∋ � � is active � � � � � � � � • � � becomes active if � � , … , � � are active � � � � � � � � • Min set cover of size � ⇒ � � � � � � � � � active � � � � � � � � • Min set cover of size � � ⇒ � � � � active ∎ � � � � � � � � � � � � � � � � 15896 Spring 2016: Lecture 24 4

  5. Submodularity for approximation • Try to identify broad subclasses where good approx is possible • � is submodular if for � ⊆ �, � ∉ �, � � ∪ � � � � � � � ∪ � � ���� • � is monotone if for � ⊆ �, � � � ���� • Reduction gives � that is not submodular • Theorem [Nemhauser et al. 1978]: � monotone and submodular, � ∗ optimal � -element subset, � obtained by greedily adding � elements that maximize marginal increase; then 1 � 1 � ��� ∗ � � � � 15896 Spring 2016: Lecture 24 5

  6. independent cascade model • Reminder of model: For each there is a weight �� o When a node becomes activated it has one o chance to activate each neighbor with probability �� • Theorem [Kempe et al. 2003]: Under the independent cascade model: Influence maximization is NP-hard o The influence function is submodular o 15896 Spring 2016: Lecture 24 6

  7. Proof of NP-hardness � � � � � � � � � � • Almost the same proof as before � � � � • S ET COVER : subsets � � of � � � � � � � � � � cover of size � � � � • Bipartite graph: � � on one side, � � on the other � � � � � � � � • If � � then there is an edge � � � � with weight � � � � � � � � � � • Min SC of size � � � � • Min SC of size � � � � � � � � active 15896 Spring 2016: Lecture 24 7

  8. Proof of submodularity • Lemma: If � � are submodular functions, � , then is a � � � � ��� submodular function • Proof: Let and , then � � ∪ � � � � � �� � ∪ � � � � � � � � � � � � � ∪ � � � � � � �� � � ∪ � � � � � � � 0 ��� 15896 Spring 2016: Lecture 24 8

  9. Proof of submodularity • Key idea: for each we flip a coin of bias �� in advance denote a particular one of the |�| • Let possible coin flip combinations activated nodes with as seed • � nodes and coin flips iff is reachable from via live • � edges 15896 Spring 2016: Lecture 24 9

  10. Proof of submodularity � is submodular • , • � � that is, is a nonnegative weighted sum of submodular functions • By the lemma, is submodular 15896 Spring 2016: Lecture 24 10

  11. Linear threshold model • Reminder of model: Nonnegative weight �� for each edge o otherwise �� Assume � �� o Each has threshold � chosen o uniformly at random in becomes active if o �� � active � 15896 Spring 2016: Lecture 24 11

  12. Linear threshold model Poll 1: What is � 1 � 3 � � � 1 1 1 � 4 4 6 �� � 15896 Spring 2016: Lecture 24 12

  13. Linear threshold model Poll 2: Given that is inactive, prob. it becomes active after 1 3 becomes active � 1 1 1 4 4 6 � 15896 Spring 2016: Lecture 24 13

  14. Linear threshold model • Theorem [Kempe et al. 2003]: 1/3 � � 3 Under the linear threshold 4 1/3 model: 1/3 Influence maximization is NP- o hard The influence function is o 1/3 submodular � � 3 4 1/3 • Difficulty: fixing the coin flips 1/3 , � is not submodular 15896 Spring 2016: Lecture 24 14

  15. Proof of submodularity • Each chooses at most one of its incoming edges at random; selected with prob. �� , and none with prob. � �� • If we can show that these choices of live edges induce the same influence function as the linear threshold model, then the theorem follows from the same arguments as before 15896 Spring 2016: Lecture 24 15

  16. Proof of submodularity • We sketch the equivalence of the two models • Linear threshold: � � � active nodes at end of iteration � o ∑ � �� �∈��∖���� Pr � ∈ � ��� | � ∉ � � � o ��∑ � �� �∈���� • Live edges: At every times step, determine whether � ’s live edge o comes from current active set If not, the source of the live edge remains unknown, o subject to being outside the active set Same probability as before ∎ o 15896 Spring 2016: Lecture 24 16

  17. Progressive vs. nonprogressive � � • Nonprogressive threshold model is identical except that � � � � � at each round � chooses � � u.a.r. in �0,1� � � � � • Suppose process runs for � steps � � � � � � � � � � � � • At each step � � � , can target � � � � � � � � � � � � � for activation; � interventions overall � � � � � � � � � � � � • Goal: ∑ # rounds � was active � � � � � � � � � � � � � • Reduces to progressive case � � � � � � � � � � � � 15896 Spring 2016: Lecture 24 17

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