How to Network in Online Social Network Giovanni Neglia, Xiuhui Ye - - PowerPoint PPT Presentation

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How to Network in Online Social Network Giovanni Neglia, Xiuhui Ye - - PowerPoint PPT Presentation

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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 Tards in 2003)

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How to Network in Online Social Network

Giovanni Neglia, Xiuhui Ye (Politecnico di Torino), Maksym Gabielkov, Arnaud Legout (Inria)

Maestro Team 16 January 2014

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16 January 2014

  • G. Neglia – How to Network in Online Social Networks
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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)
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Influence propagation

Recruited node Influenced node p p p p p

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t=0

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Influence propagation

Recruited node Influenced node p p p p p

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t=1

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Influence propagation

Recruited node Influenced node p p p p p

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t=2

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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

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Kempe et al 2003

  • 1. Decreasing cascade model:

q pv(u,S) = prob. that u can influence v, given that nodes in S have already tried to influence v q pv(u,S) ≥ pv(u,T) if

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v u

t=2

  • G. Neglia – How to Network in Online Social Networks

S ⊂ T

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Kempe et al 2003

  • 2. Linear Threshold Model

q Node v has a threshold θv sampled from a uniform random variable in [0,1] and link (i,j) has a weight bij q Node v is influenced if Σ biv 1(i is influenced) > θn

v s u

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t=2

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buv bsv

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Kempe et al 2003

  • 2. General Threshold Model

q Node v has a threshold θv sampled from a uniform random variable in [0,1] q Node v has a monotone activation function fv:2V->[0,1] and is influenced at t if fv(S) > θv, where S is the set of influenced nodes at t

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Kempe et al 2003

Their results:

I. Decreasing cascade model & General threshold model are equivalent

q For each {pv(u,S)}, it is possible to find {fv(S)} such that the probability distribution of φ(A) is the same

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Kempe et al 2003

Their results:

I. Decreasing cascade model & General threshold model are equivalent

q For each {pv(u,S)}, it is possible to find {wij} 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

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Monotonicity of σ(A)

q σ(A1)≤σ(A2) if

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A

1 ⊂ A2

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Submodularity of σ(A)

q σ(A1U {v}) - σ(A1) ≥ σ(A2U {v}) - σ(A2) if

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v

A

1 ⊂ A2

v

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The greedy algorithm

1: start with A={} 2: for i =1 to K 3: let vi be the node maximizing the marginal gain σ(A U {v}) - σ(A) 4: set A:=A U {vi} Question: how to calculate σ(A U {v}) - σ(A)?

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  • 2. How the problem changes in OSN

v u Tweeting node Retweeting node p p p p p

v follows u’s tweet v is a follower of u u is a following of v

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  • 2. How the problem changes in OSN

Assumption: a user can only influence people through Twitter itself

v u Tweeting node Retweeting node p p p p p

v follows u’s tweet v is a follower of u u is a following of v

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The user can only select its followers (up to K=2000)…

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  • 2. How the problem changes in OSN

v u p p p p p

v follows u’s tweet v is a follower of u u is a following of v

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The user can only select its followers (up to K=2000)… And hope that they follow back

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  • 2. How the problem changes in OSN

v u p p p p p

v follows u’s tweet v is a follower of u u is a following of v

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Our problem

Let the reciprocation probability rv be known How should the user select the set of followers A in

  • rder to maximize σ(A)=E[|φ(A)|]? (all the choices at t=0)

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v u p p p p p

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Map the new problem to the old one

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u2 u3 u1 p p p u4 p Select K followers u2 u3 u1 p p p u4 p

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Map the new problem to the old one

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u2 u3 u1 p p p u4 p Select K followers u2 u3 u1 p p p u4 p u'1 u’2 u'3 u'4 p r p r p r p r Recruit K nodes in V’ equivalent to

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Map the new problem to the old one

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Select K followers u2 u3 u1 p p p u4 p u2 u3 u1 p p p u4 p u'1 u’2 u'3 u'4 p r p r p r p r Recruit K nodes in V’ equivalent to

Greedy algorithm has the same approximation ratio

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A 2nd 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
  • 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

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An ideal policy

q Imagine to know who is going to reciprocate by T q The greedy algorithm with such knowledge would achieve an (1-1/e) approximation ratio

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u2 u3 u1 p p p u4 p u2 u3 u1 p p p u4 p u'3 u'4 p p

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A practical greedy policy

1: start with A={}, D={} i=0 2: while i ≤ K 3: let vi be the node in V-D maximizing the marginal gain σ(A U {v}) - σ(A), given that it reciprocates 5: follow vi 6: if vi reciprocates by T: 7: A:=A U {vi}, i=i+1 5: else: 6: D:=D U {vi}

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A practical greedy policy

1: start with A={}, D={} i=0 2: while i ≤ K 3: let vi be the node in V-D maximizing the marginal gain σ(A U {v}) - σ(A), given that it reciprocates 5: follow vi 6: if vi reciprocates by T: 7: A:=A U {vi}, i=i+1 5: else: 6: D:=D U {vi} practical greedy = ideal greedy

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#Readers vs #Retwitters (3rd twist)

v u Retweeting (and reading) node p p p p p Reading (non- retweeting) node

What if we consider as performance metric #readers?

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Map the new problem to the old one

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u2 u3 u1 p p p u4 p w(u''2)=1 u2 u3 u1 p p p u4 p u''1 u''2 u''3 u''4 1 1 1 1 1 1 1 1 w(u''1)=1 w(u''4)=1 w(u''3)=1 w(u1)=0 w(u4)=0 w(u3)=0 w(u2)=0

Select K nodes to maximize E[Σ w(ui) 1(ui is active)]

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An ideal policy

q Is E[Σ w(ui) 1(ui 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

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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

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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, pv(u,S), rv
  • How to calculate the marginal gain? Montecarlo

simulations…

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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)
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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

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Montecarlo simulations q Naive implementation

  • O(NKS) simulations,
  • where S is #simulations to achieve the

required confidence

  • ≈100GB to store the graph in RAM

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Trade RAM for Storage

  • Influenced node of a cascade = reachable

nodes in the pruned graph

  • Need to store S * p * 417GB
  • RAM still a problem for p≥1%

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u2 u3 u1 p p p u4 p

u2 u3 u1 u4 u2 u3 u1 u4 u2 u3 u1 u4 u2 u3 u1 u4

…

prune

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Useful preprocessing

  • Reachability can also be calculated on the

SCCs’ graph

  • For larger p we save memory, storage and

computation

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u2 u3 u1 u4 u5 S1 S2

Pruned graph SCCs’ graph u1, u2, u3 u4, u5 Calculation of the Strongly Connected Components

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How many samples? q We tried to estimate it analytically

  • Random configuration model
  • Subcritical branching process for small p
  • All-or-nothing supercritical branching process

for large p

q S≤100 for all the values of p

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Different algorithms

  • 1. Greedy
  • Know topology, probabilities
  • 2. Highest degree
  • Know nodes’ degrees
  • 3. Random
  • Know nodes’ ids

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1 20 40 60 80 100 120 140 160 180 200 0.5 1 1.5 2 2.5 3 3.5 4 4.5x 10

6

initial set size #retweets p = 0.001 greedy high−degree random 1 20 40 60 80 100 120 140 160 180 200 2 4 6 8 10 12x 10

4

initial set size #retweets p = 0.0001 greedy high−degree random

<d>p≈4*10-3 <d>p≈4*10-2

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1 20 40 60 80 100 120 140 160 180 200 0.5 1 1.5 2 2.5x 10

8

initial set size #retweets p = 0.1 greedy high−degree random 1 20 40 60 80 100 120 140 160 180 200 1 2 3 4 5 6 7x 10

7

initial set size #retweets p = 0.01 greedy high−degree random

<d>p≈0.4 <d>p≈4

High variability

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The effect of reciprocity

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1 10 20 30 40 50 60 70 80 90 100 110 1 2 3 4 5 6 7x 10

7

initial set size #retweets p = 0.01 greedy high−degree random 1 20 40 60 80 100 6.145 6.15 6.155x 10

7

r = min # followings # followers +100,1 ! " # $ % &

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#Readers vs #Retwitters

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1 5 10 15 20 25 30 35 40 0.5 1 1.5 2 2.5 3 3.5 x 10

8

initial set size #retweets or #readers p = 0.01 greedy retweets high−degree retweets random retweets greedy readers high−degree readers random readers

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Take Home Lesson q For sparse graphs, highest degree (1-hop ahead) works as well as greedy q For dense graphs, any strategy, even random, works as well as greedy q Only in the middle, greedy can outperform highest degree…

  • Remarks in Habiba and Berger-Wolf, 2011

q … but we do not observe it

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Thank you! Questions?