We study influence maximization in which diffusion on each step may be delayed, and the objective is to maximize influence spread within a certain deadline. Both IC and LT models are extended, and efficient algorithms are proposed and evaluated. This work appears in AAAI 2012. For the full version of the paper, please see: http://arxiv.org/abs/1204.3074

1. Time-Critical Influence Maximization in Social
Networks withTime-Delayed Diffusion Process
Wei Chen Wei Lu Ning Zhang
Microsoft ResearchAsia U. of British Columbia U. of Sci andTech of China
This work was done during the internships of Wei Lu and Ning Zhang at Microsoft Research Asia.

2. Influence in Social Networks
Thursday, July 26, 2012AAAI 2012,Toronto, Ontario.2
 We live in communities
and interact with social
acquaintances
 This forms social
networks
 In social interactions, we
influence each other

3. Kinect is great
Kinect is great
Kinect is great
Kinect is great
Kinect is great
Kinect is great
Kinect is great
Influence Diffusion & Viral Marketing
3 AAAI 2012,Toronto, Ontario.
Word-of-mouth effects

4. Social Networks as Directed Graphs
Thursday, July 26, 2012AAAI 2012,Toronto, Ontario.4
 Nodes: Individuals in the network
 Edges: Links between individuals
 Edge weight: Influence probability p(u,v) – the probability that v will be
influenced by u
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5. A Classical Influence Propagation
Model
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 Independent Cascade (IC) (Kempe, Kleinberg, andTardos 2003)
 Initially some seed nodes are activated
 At each time step (discrete), each newly-activated node u
activates its neighbor v independently with probability p(u,v)
 Influence spread: Expected number of nodes activated
 Other propagation models
 LinearThreshold (LT)
 GeneralThreshold
 etc

6. Influence Maximization
Thursday, July 26, 2012AAAI 2012,Toronto, Ontario.6
Problem
Select k individuals such that by
activating them, the expected
spread of influence is maximized.
Input
Output
A directed graph representing a
social network, with influence
probabilities on edges
Seed set of size k
NP-hard  #P-hard to compute exact influence 

7. Temporal Aspects in Influence
Diffusion
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 Influence diffusion can be time-delayed
 Heterogeneity of human activities and interactions (Iribarren and
Moro, 2009)
 Network topology and burstiness (Karsai et al., 2011)
 NOT captured in classical propagation models

8. Temporal Aspects in Influence
Diffusion
Thursday, July 26, 2012AAAI 2012,Toronto, Ontario.8
 The task of influence maximization may be time-critical.
 Xbox 360 + Kinect is on sale  inVancouver area BestBuys, for
3 days only !!!!!
 Alice has grabbed this great deal, and wants to inform Bob
 But Bob’s been away for a road trip in Banff National Park
 No stable Internet & cellphone access (Rocky mountains!) &
Uncertain return time 
 Viral marketing campaigns may have limited time horizon,
which affects the spread of word-of-mouth.
 NOT captured in current propagation models

9. Thursday, July 26, 2012AAAI 2012,Toronto, Ontario.9

10. Our Work
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 Extend the influence maximization problem to have a deadline
constraint:Time-Critical Influence Maximization
 Propose a new propagation model to reflect temporal delays
of influence diffusion: Independent Cascade with Meeting
events (IC-M)

11. Independent Cascade with Meeting
Events (The IC-M model)
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 Model parameters
 Social networks modeled as directed graphs
 Influence probability p(u,v)
 Meeting probability m(u,v), the probability that u and v
meet in each time step
 Diffusion dynamics
 Initially, a seed set is targeted and activated
 At each time step, u and v meet w.p. m(u,v).
 If u is active & is meeting v for the first time, u influences v with
prob p(u,v)

12. Time-Critical Influence Maximization
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 Original Influence Maximization
 Diffusion ends only when no more nodes can be activated
 Unconstrained time horizon
 Meeting probabilities not essential
 Time-Critical Influence Maximization
 Given an integer T << |V|, we only consider influence spread
within T time steps
 T: the deadline constraint.
 Representing limited time horizon.

13. Time-Critical Influence Maximization
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 Problem Formulation
 Input: G=(V, E), k, T
 Objective: find k seeds such that the spread of influence by the
end of time step T is maximized (under the IC-M model)
 NP-hard 

14. Greedy Approximation Algorithm
Thursday, July 26, 2012AAAI 2012,Toronto, Ontario.14
 Under the IC-M model, our objective function (spread of
influence) is monotone and submodular in the seed set.
 Monotonicity:As the seed set grows, influence is non-decreasing
 Submodularity:The law of diminishing marginal returns
 Greedy approximation algorithm
 Repeat k rounds
 In each round, select the node v that provides the largest marginal
gain in influence spread
 Approximation ratio = 1-1/e ≈ 63% (Nemhauser et al., 1978)
 #P-hard to compute influence spread exactly for IC-M 
 Can use Monte Carlo simulations to estimate, but very costly 

15. Overcome the Inefficiency of Greedy
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 MIA: Maximum InfluenceArborescence (Chen et al., 2010)
 Heuristic No.1 (MIA-M algorithm)
 Design an efficient algorithm to compute influence spread
exactly in tree structures
 Leverage it to design scalable heuristics for time-critical
influence maximization in general graphs
 Heuristic No.2 (MIA-C algorithm)
 For each pair of nodes (u,v), estimate the probability that
influence will propagate from u to v by combining p(u,v),
m(u,v), and the deadline T
 Convert the problem to one in the classical IC model, and solve
it using MIA

16. Compute Influence in Directed Trees
(In-Arborescences)
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 Activation probability ap(u,t): the probability that u
becomes active right at time step t
u

17. Calculating Activation Probability
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 Step 1: For any seed set S, Compute ap(u,t) given S via
dynamic programming
 Base cases
 The recursion: ap(u,t) =
 Step 2: By linearity of expectation, for a given seed set S,
inf(S) =

18. Computing Influence in General Graphs
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 Utilize the dynamic programming algorithm for trees
 Restrict incoming influence to a node u in a local region
 Influence from nodes far away can be ignored
 “Sparsify” the local region of a node u to an in-arborescence, by
including only the strongest influence path from other nodes
to u
 Dijkstra’s algorithm

19. Experiments: Network Datasets
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 NetHEPT:A co-authorship network from arxiv.org High Energey
PhysicsTheory section.
 WikiVote:A who-voted-whom network fromWikipedia
 Epinions:A who-trusts-whom network from the customer
reviews site Epinions.com
 DBLP:A co-authorship network from DBLP
NetHEPT WikiVote Epinions DBLP
# Nodes 15K 7.1K 75K 655K
# Edges 62K 101K 509K 2.0M
Avg. degree 4.12 26.6 13.4 6.1
Max. degree 64 1065 3079 588

20. Experimental Results
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(a). NetHEPT (b). Epinions
Graph parameters:
• Influence probability: p(u,v) = 1.0 / in-degree(v)
• Meeting probability: m(u,v) = 1.0 / out-degree(u)
Fig. Spread of influence (Y-axis) vs. Seed set size (X-axis), T = 5 and 15

21. Experimental Results
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(a). NetHEPT (b). Epinions
Spread of influence (Y-axis) vs. Seed set size (X-axis), T = 5 and 15
Graph parameters:
• Influence probability: p(u,v) = 1.0 / in-degree(v)
• Meeting probability: m(u,v) chosen uniformly at random from
{0.2, 0.3, … , 0.7, 0.8}

22. Running Time Comparisons
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 T = 5, Weighted meeting probabilities. DNP = Did Not Complete
(within 72 hours)
 In general, Greedy is too slow to use in practice
 Cannot scale to large graphs
 MIA-M, MIA-C are 2-3 orders of magnitude faster
 MIA-M is a little bit slower than MIA-C, MIA
 But has higher spread of influence
NetHEPT WikiVote Epinions DBLP
Greedy 40m 22m DNP DNP
MIA-M 1.6s 7.9s 41s 6.6m
MIA-C 0.3s 0.4s 2.7s 24s
MIA 0.3s 1.4s 12s 40s

23. Conclusions, Discussions & Future
Work
Thursday, July 26, 2012AAAI 2012,Toronto, Ontario.23
 Conclusions
 Time-Critical Influence Maximization Problem
 Independent Cascade model with meeting Events
 Approximation algorithm & heuristic solutions
 Extensions & Refinements
 LinearThreshold model with Meeting events (LT-M)
 More efficient computation of activation probabilities in tree
structures
 Details available in our full technical report:arXiv 1204.3074
 FutureWork
 Extend classical propagation models to incorporate login events
 Extend to more general propagation models

24. Thursday, July 26, 2012AAAI 2012,Toronto, Ontario.24
Thanks!!! Questions???
KDD 2012 tutorial on Information and Influence Spread in
Social Networks (Aug 12,Beijing,China)
Carlos Castillo (Qatar Computing Research Institute)
Wei Chen (Microsoft ResearchAsia)
LaksV.S. Lakshmanan (University of British Columbia)
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