How to Network in Online Social Network Giovanni Neglia, Xiuhui Ye (Politecnico di Torino), Maksym Gabielkov, Arnaud Legout (Inria) Maestro Team 16 January 2014
Outline 1. Influence maximization problem (Kempe, Kleinberg and Tardös in 2003) 2. How the problem changes for a user in an online social network 3. Simulation results on Twitter’s complete graph (2012) 16 January 2014 G. Neglia – How to Network in Online Social Networks - 2
Influence propagation p p p p p Recruited node t=0 Influenced node 16 January 2014 - 3 G. Neglia – How to Network in Online Social Networks
Influence propagation p p p p p Recruited node t=1 Influenced node 16 January 2014 - 4 G. Neglia – How to Network in Online Social Networks
Influence propagation p p p p p Recruited node t=2 Influenced node 16 January 2014 - 5 G. Neglia – How to Network in Online Social Networks
Influence maximization Recruit a set A of K nodes to maximize the expected number of influenced nodes ( σ (A)=E[| φ (A)|]) Recruited node Influenced node 16 January 2014 - 6 G. Neglia – How to Network in Online Social Networks
Kempe et al 2003 1. Decreasing cascade model: q p v (u,S) = prob. that u can influence v, given that nodes in S have already tried to influence v q p v (u,S) ≥ p v (u,T) if S ⊂ T v u t=2 16 January 2014 - 7 G. Neglia – How to Network in Online Social Networks
Kempe et al 2003 2. Linear Threshold Model q Node v has a threshold θ v sampled from a uniform random variable in [0,1] and link (i,j) has a weight b ij q Node v is influenced if Σ b iv 1 (i is influenced) > θ n b uv v u b sv s t=2 16 January 2014 - 8 G. Neglia – How to Network in Online Social Networks
Kempe et al 2003 2. General Threshold Model q Node v has a threshold θ v sampled from a uniform random variable in [0,1] q Node v has a monotone activation function f v :2 V ->[0,1] and is influenced at t if f v (S) > θ v , where S is the set of influenced nodes at t 16 January 2014 - 9 G. Neglia – How to Network in Online Social Networks
Kempe et al 2003 Their results: I. Decreasing cascade model & General threshold model are equivalent q For each {p v (u,S)}, it is possible to find {f v (S)} such that the probability distribution of φ (A) is the same 16 January 2014 - 10 G. Neglia – How to Network in Online Social Networks
Kempe et al 2003 Their results: I. Decreasing cascade model & General threshold model are equivalent q For each {p v (u,S)}, it is possible to find {w ij } such that the probability distribution of φ (A) is the same II. The greedy algorithm achieves a (1-1/e) approximation ratio q This follows from a general result proven by Nemhauser, Wolsey, Fisher in '78 for non-negative, monotone, submodular functions 16 January 2014 - 11 G. Neglia – How to Network in Online Social Networks
Monotonicity of σ (A) A 1 ⊂ A 2 q σ (A 1 ) ≤σ (A 2 ) if G. Neglia – How to Network in Online Social Networks 16 January 2014 - 12
Submodularity of σ (A) q σ (A 1 U {v}) - σ (A 1 ) ≥ σ (A 2 U {v}) - σ (A 2 ) if A 1 ⊂ A 2 v v G. Neglia – How to Network in Online Social Networks 16 January 2014 - 13
The greedy algorithm 1: start with A={} 2: for i =1 to K 3: let v i be the node maximizing the marginal gain σ (A U {v}) - σ (A) 4: set A:=A U {v i } Question : how to calculate σ (A U {v}) - σ (A)? G. Neglia – How to Network in Online Social Networks 16 January 2014 - 14
2. How the problem changes in OSN v p v follows u’s tweet u v is a follower of u p u is a following of v p p p Tweeting node Retweeting node 16 January 2014 G. Neglia – How to Network in Online Social Networks - 15
2. How the problem changes in OSN v p v follows u’s tweet u v is a follower of u p u is a following of v p p p Tweeting node Retweeting node Assumption : a user can only influence people through Twitter itself 16 January 2014 G. Neglia – How to Network in Online Social Networks - 16
2. How the problem changes in OSN v p v follows u’s tweet u v is a follower of u p u is a following of v p p p The user can only select its followers (up to K=2000)… 16 January 2014 G. Neglia – How to Network in Online Social Networks - 17
2. How the problem changes in OSN v p v follows u’s tweet u v is a follower of u p u is a following of v p p p The user can only select its followers (up to K=2000)… And hope that they follow back 16 January 2014 G. Neglia – How to Network in Online Social Networks - 18
Our problem Let the reciprocation probability r v be known How should the user select the set of followers A in order to maximize σ (A)=E[| φ (A)|]? (all the choices at t=0) v u p p p p p 16 January 2014 G. Neglia – How to Network in Online Social Networks - 19
Map the new problem to the old one p p u 2 u 2 u 1 u 1 p p p p p p u 3 u 3 u 4 u 4 Select K followers 16 January 2014 G. Neglia – How to Network in Online Social Networks - 20
Map the new problem to the old one u’ 2 u' 1 p r p r p p u 2 u 2 u 1 u 1 p p p p p p u 3 u 3 u 4 u 4 p r p r u' 3 u' 4 Recruit K nodes in V’ equivalent to Select K followers 16 January 2014 G. Neglia – How to Network in Online Social Networks - 21
Map the new problem to the old one u’ 2 u' 1 p r p r p p u 2 u 2 u 1 u 1 p p p p p p u 3 u 3 u 4 u 4 p r p r u' 3 u' 4 Recruit K nodes in V’ equivalent to Select K followers Greedy algorithm has the same approximation ratio 16 January 2014 G. Neglia – How to Network in Online Social Networks - 22
A 2 nd twist: dynamic policies q Following users is not expensive q Idea: replace non-reciprocating users q How to operate: • follow one user • if the user does not reciprocate by T o unfollow it and follow someone else q It is now possible to follow over time more than K users, but only K at a given time instant 16 January 2014 G. Neglia – How to Network in Online Social Networks - 23
An ideal policy q Imagine to know who is going to reciprocate by T p p u 2 u 2 u 1 u 1 p p p p p p u 3 u 3 u 4 u 4 p p u' 3 u' 4 q The greedy algorithm with such knowledge would achieve an (1-1/e) approximation ratio 16 January 2014 G. Neglia – How to Network in Online Social Networks - 24
A practical greedy policy 1: start with A={}, D={} i=0 2: while i ≤ K 3: let v i be the node in V-D maximizing the marginal gain σ (A U {v}) - σ (A), given that it reciprocates 5: follow v i 6: if v i reciprocates by T: 7: A:=A U {v i }, i=i+1 5: else: 6: D:=D U {v i } 16 January 2014 G. Neglia – How to Network in Online Social Networks - 25
A practical greedy policy 1: start with A={}, D={} i=0 2: while i ≤ K 3: let v i be the node in V-D maximizing the marginal gain σ (A U {v}) - σ (A), given that it reciprocates 5: follow v i 6: if v i reciprocates by T: 7: A:=A U {v i }, i=i+1 5: else: 6: D:=D U {v i } practical greedy = ideal greedy 16 January 2014 G. Neglia – How to Network in Online Social Networks - 26
#Readers vs #Retwitters (3rd twist) v p u p p p Retweeting (and reading) node p Reading (non- retweeting) node What if we consider as performance metric #readers? 16 January 2014 G. Neglia – How to Network in Online Social Networks - 27
Map the new problem to the old one u'' 2 u'' 1 1 1 w(u 1 )=0 w(u 2 )=0 w(u'' 2 )=1 p w(u'' 1 )=1 p 1 u 2 u 2 u 1 u 1 p p p p p p 1 1 u 3 u 3 u 4 u 4 w(u 3 )=0 w(u 4 )=0 1 1 1 u'' 3 u'' 4 w(u'' 4 )=1 w(u'' 3 )=1 Select K nodes to maximize E[ Σ w(u i ) 1 (u i is active)] 16 January 2014 G. Neglia – How to Network in Online Social Networks - 28
An ideal policy q Is E[ Σ w(u i ) 1 (u i is active)] submodular? - Yes it is (need to go carefully through the steps of Kempe et al) q then greedy is a (1-1/e) approximation algorithm 16 January 2014 G. Neglia – How to Network in Online Social Networks - 29
Wrap up q The point of view of a user in an OSN introduces new twists, but does not change fundamentally the problem - In particular the greedy algorithm guarantees a (1-1/e) approximation ratio 16 January 2014 G. Neglia – How to Network in Online Social Networks - 30
Wrap up q The point of view of a user in an OSN introduces new twists, but does not change fundamentally the problem - In particular the greedy algorithm guarantees a (1-1/e) approximation ratio q Limits: - need to know the whole topology, p v (u,S), r v - How to calculate the marginal gain? Montecarlo simulations… 16 January 2014 G. Neglia – How to Network in Online Social Networks - 31
Outline 1. Influence maximization problem (Kempe, Kleinberg and Tardös in 2003) 2. How the problem changes for a user in an online social network 3. Simulation results on Twitter’s complete graph (2012) 16 January 2014 G. Neglia – How to Network in Online Social Networks - 32
Know your enemy q Crawl of the whole Twitter in June 2012 q 500 million of nodes q 23 billion of arcs q 417GB as an edgelist 16 January 2014 G. Neglia – How to Network in Online Social Networks - 33
Montecarlo simulations q Naive implementation - O(NKS) simulations, • where S is #simulations to achieve the required confidence - ≈ 100GB to store the graph in RAM 16 January 2014 G. Neglia – How to Network in Online Social Networks - 34
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