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Wei Chen
Microsoft Research Asia
In collaboration with
Yifei Yuan University of Pennsylvania
Li Zhang Microsoft Research Silicon Valley
ICDM'10, Dec. 15, 2010 1
Scalable Influence Maximization in Social Networks under the Linear Threshold Model
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Background and problem definition
Influence computation in the linear threshold model
Local directed acyclic graph (LDAG) heuristic
Experimental evaluations
Future directions
Outline
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Explosion of Online Social Media
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Social Influence Social influence is when the actions or thoughts of
individual(s) are changed by other individual(s).
Social influence is everywhere
How can we extract social influence pattern from rich online social media?
How can we utilize social influence in online social media? --- focus of this paper
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A Hypothetical Example of Viral Marketing
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Xbox Kinect is great
Xbox Kinect is great
Kinect is great
Xbox Kinect is great
Kinect is great
Kinect is great
Kinect is great
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Given a social network
Given a dynamic influence cascade model
From an initial seed set, a stochastic process propagates node activation (influence) to part of the network
independent cascade (IC) model
linear threshold (LT) model
Influence maximization:
finding a seed set with size at most k,
such that the expected number of activated nodes (called influence spread) is the largest
The Problem of Influence Maximization
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Social influence graph vertices are individuals
links are social relationships
link (𝑢, 𝑣) has weight 𝑤 𝑢, 𝑣 : 𝑤 𝑢, 𝑣 ≤ 1𝑢
Linear threshold model each node 𝑣 selects a threshold 𝜆𝑣 ∈ [0,1] uniformly at random
initially some seed nodes are activated
At each step, node 𝑣 checks if the weighted sum of its active neighbors is greater than its threshold 𝜆𝑣, if so 𝑣 is activated
Linear Threshold Model
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0.3
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0.5 0.6
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Influence maximization as a discrete optimization problem proposed by Kempe, Kleinberg, and Tardos, in KDD’2003
Approximation algorithms Greedy approximation algorithm in [KKT’03], 63% approximation of the optimal solution
Several improvements on running time [Leskovec, et al. 2007, Chen et al. 2009]
very slow, not scalable: > 3 hrs on a 30k node graph for 50 seeds
Heuristic algorithms SPIM [Kimura and Saito, 2006], SPIN [Narayanam and Narahari, 2008], not scalable
PMIA [Chen et al. 2010] scalable and good performance, but only for IC model
Lack of scalable solution for the LT model
Research Background
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Influence spread computation in the LT model
Computing exact influence spread in a general graph given a seed set is #P-hard (counting hardness)
Reduced from counting the number of simple paths in a graph
resolve an open problem in [KKT’03]
indicate the intrinsic difficulty of computing influence spread
Computing exact influence spread in a DAG (directed acyclic graph) can be done in linear time
Influence maximization heuristic for the LT model
LDAG (local directed acyclic graph) heuristic
specifically designed for the LT model
103 speedup --- from hours to seconds (or days to minutes)
influence spread close to that of the greedy algorithm of [KKT’03]
Our Work
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Setup
DAG 𝐷 = (𝑉, 𝐸, 𝑤)
Seed set 𝑆
activation probability of node 𝑣: 𝑎𝑝(𝑣)
Influence spread = 𝑎𝑝(𝑣)𝑣∈𝑉
Computing activation probability in 𝐷:
Follow the DAG partial order to compute all 𝑎𝑝(𝑣)’s
Computing Influence Spread in a DAG
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𝑎𝑝 𝑣 = 𝑎𝑝 𝑢 ⋅ 𝑤(𝑢, 𝑣)
𝑢∈𝑉\{𝑣}
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Compute local DAGs (LDAGs) surrounding every node
idea 1: restrict influence computation at local region
Compute incremental influence of every node based on LDAGs
idea 2: influence computation in DAGs is fast
Select k seeds one by one with largest incremental influence
select new seed 𝑠 with the largest incremental influence
update incremental influence of all nodes 𝑢 sharing LDAGs with 𝑠
idea 3: batch updates reducing running time from 𝑂(|𝐷|2) to 𝑂(𝐷)
Main Structure of the LDAG Heuristic
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Want:
a DAG D surrounding 𝑣, with 𝑣 as a sink in 𝐷 --- LDAG rooted at 𝑣
the LDAG is local the influence of every node 𝑢 in 𝐷 to 𝑣 is above some threshold 𝜃
the LDAG covers a significant portion of influence the sum of influence of all nodes 𝑢 in 𝐷 to 𝑣 is as large as possible
Exact maximization problem is NP-hard
Greedy approach (similar to Dijkstra’s shortest-path algorithm)
select a node x with the largest influence to v 𝑥 = argmax 𝑢 𝐼𝑛𝑓(𝑢, 𝑣)
after x is selected, update the influence of the in-neighbors u of x, based on the linear relationship
𝐼𝑛𝑓 𝑢, 𝑣 += 𝑤 𝑢, 𝑥 ⋅ 𝐼𝑛𝑓(𝑥, 𝑣)
Repeat above two steps until no node has influence greater than 𝜃
Finding an LDAG Influencing Node 𝑣
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For an LDAG 𝐷 rooted at 𝑣, if a node 𝑢 in 𝐷 is selected as an additional seed, the incremental influence of 𝑢 to 𝑣 in 𝐷 is
1 − 𝑎𝑝 𝑢 ⋅ 𝛼𝑣(𝑢)
𝛼𝑣 𝑢 ’s for all 𝑢 in 𝐷 can be computed in linear time
time reduced from 𝑂(|𝐷|2) to 𝑂(𝐷)
After selecting seed 𝑠, update 𝛼𝑣 𝑤 for all 𝑤’s that are in the same LDAGs as 𝑠
Efficient Batch Updates on Activation Probabilities
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Networks
Real-world datasets: collaboration networks: arXiv (31K), DBLP (2M)
trust network: Epinions (509K)
product co-purchasing network: Amazon (1.2M)
Synthetic datasets: generated from power-law random graphs
Influence weights
uniform: for node 𝑣 with degree 𝑑𝑣, every incoming edge has weight 1/𝑑𝑣
random
Algorithms tested: LDAG, Greedy, SPIN, PageRank, DegreeDiscount
Experimental Evaluation
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Experiment Results on Influence Spread
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LDAG always among the best
matches with Greedy
PageRank / DegreeDiscount not stable
arXiv Epinions
Amazon DBLP random weights
PageRank is not stable
DegreeDiscount is not stable
DegreeDiscount is not stable
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Running Time on Real-World Networks
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Running time is for selecting 50 seeds
>103 times speed up
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Scalability of LDAG on Synthetic Graphs
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LDAG is scalable
𝑦 ∝ 𝑥1.02, almost linear
Scalable Influence Maximization for Prevalent Viral Marketing in Large-Scale Social Networks
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Compare with Greedy with Different Number of Simulated Cascade Runs
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Greedy cannot maintain high influence spread when reducing the number of simulations
running time comparable with LDAG, but influence significantly worse
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Compare with Random LDAG Construction
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random LDAG construction is significantly worse
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Theoretical problem: efficient approximation algorithms:
How to efficiently approximate influence spread given a seed set?
Practical problem:
Influence analysis from online social media: How to mine the influence graph?
Influence maximization in other settings
Future Directions
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Thanks! and questions?
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