CMU 15-896 Social networks 2: Influence Maximization Teacher: - - PowerPoint PPT Presentation

CMU 15-896

Social networks 2: Influence Maximization

Teacher: Ariel Procaccia

15896 Spring 2016: Lecture 24

Motivation

• Firm is marketing a new product
• Collect data on the social network
• Choose set
• f early adopters and market

to them directly

• Customers in

generate a cascade of adoptions

• Question: How to choose ?

2

15896 Spring 2016: Lecture 24

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
• ver sets
• f size
• Theorem [Kempe et al. 2003]: Under the general

cascade model, influence maximization is NP- hard to approximate to a factor of for any

3

15896 Spring 2016: Lecture 24

• Proof of theorem
• SET 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 ∎

4

15896 Spring 2016: Lecture 24

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 ∗

5

15896 Spring 2016: Lecture 24

independent cascade model

• Reminder of model:
• For each

there is a weight

• When a node

becomes activated it has one chance to activate each neighbor with probability

• Theorem [Kempe et al. 2003]: Under the

independent cascade model:

• Influence maximization is NP-hard
• The influence function

is submodular

6

15896 Spring 2016: Lecture 24

Proof of NP-hardness

• Almost the same proof as before
• SET COVER: subsets
• f
• cover of size
• Bipartite graph:
• n one

side,

• n the other
• If

then there is an edge

• with weight
• Min SC of size
• Min SC of size

active

7

15896 Spring 2016: Lecture 24

Proof of submodularity

• Lemma: If

are submodular functions,

• , then
• is a

submodular function

• Proof: Let

and , then

∪ ∪

∪ ∪

• 8

15896 Spring 2016: Lecture 24

Proof of submodularity

• Key idea: for each

we flip a coin of bias in advance

• Let

denote a particular one of the || possible coin flip combinations

• activated nodes with

as seed nodes and coin flips

• iff

is reachable from via live edges

9

15896 Spring 2016: Lecture 24

Proof of submodularity

• is submodular
• ,

that is, is a nonnegative weighted sum of submodular functions

• By the lemma,

is submodular

10

15896 Spring 2016: Lecture 24

Linear threshold model

• Reminder of model:
• Nonnegative weight

for each edge

• therwise
• Assume
• Each

has threshold chosen uniformly at random in

• becomes active if
• active

11

15896 Spring 2016: Lecture 24

Linear threshold model

12

Poll 1: What is

• 1

4 1 4 1 6 1 3

15896 Spring 2016: Lecture 24

Linear threshold model

13

Poll 2: Given that is inactive, prob. it becomes active after becomes active

• 1

4 1 4 1 6 1 3

15896 Spring 2016: Lecture 24

Linear threshold model

• Theorem [Kempe et al. 2003]:

Under the linear threshold model:

• Influence maximization is NP-

hard

• The influence function

is submodular

• Difficulty: fixing the coin flips

, is not submodular

14 1/3 1/3 1/3 1/3 1/3 1/3

3 4 3 4

15896 Spring 2016: Lecture 24

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

15

15896 Spring 2016: Lecture 24

Proof of submodularity

• We sketch the equivalence of the two models
• Linear threshold:
• active nodes at end of iteration
• Pr ∈ | ∉

∑

• ∈∖

∑

• ∈
• Live edges:
• At every times step, determine whether ’s live edge

comes from current active set

• If not, the source of the live edge remains unknown,

subject to being outside the active set

• Same probability as before ∎

16

15896 Spring 2016: Lecture 24

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

17