VLDB'16 Research Paper. Influence maximization is a well-studied problem that asks for a small set of influential users from a social network, such that by targeting them as early adopters, the expected total adoption through influence cascades over the network is maximized. However, almost all prior work focuses on cascades of a single propagating entity or purely-competitive entities. In this work, we propose the Comparative Independent Cascade (Com-IC) model that covers the full spectrum of entity interactions from competition to complementarity. In Com-IC, users’ adoption decisions depend not only on edge-level information propagation, but also on a node-level automaton whose behavior is governed by a set of model parameters, enabling our model to capture not only competition, but also complementarity, to any possible degree. We study two natural optimization problems, Self Influence Maximization and Complementary Influence Maximization, in a novel setting with complementary entities. Both problems are NP-hard, and we devise efficient and effective approximation algorithms via non-trivial techniques based on reverse-reachable sets and a novel “sandwich approximation” strategy. The applicability of both techniques extends beyond our model and problems. Our experiments show that the proposed algorithms consistently outperform intuitive baselines on four real-world social networks, often by a significant margin. In addition, we learn model parameters from real user action logs.

1. Comparative Influence Maximization:
From Competition to Complementarity
Wei Lu (LinkedIn)
Wei Chen (Microsoft Research)
Laks V.S. Lakshmanan (UBC)
NDA’16 Workshop, SIGMOD
To appear in VLDB’16, New Delhi, India

2. Social influence
• Ubiquitous in life
• Fueled by the widespread popularity of online
social networks and social media
• Computational Social Influence (CSI)
– Viral Marketing
– Influence Maximization
– The applications and extensions to the above

3. Computational Social Influence
• Social networks with edge weights (influence
probabilities or weights)
• Stochastic influence/information propagation models
– Single-item vs. Multiple-item models
• Diffusion dynamics depend heavily on the
relationship of the propagating entities
• Pure Competition: Each user adopts at most one
item
– Competitive Independent Cascade Model (CIC)
– K-LT Model
– WPCLT Model …

4. Limitations of Pure Competition
Models: Example

5. Item Relationships
• Propagating items can be of any relationship:
– Compete (iPhone vs Nexus)
– Complement (iPhone vs Apple Watch, iPhone vs
iPhone cases)
• Natural and well-studied in economics
– Substitute goods and complementary goods
• Item relationship may be asymmetric
• Item relationship may be to an arbitrary degree
(not “pure”)

6. Motivations and Challenges
“One model that works for all kinds of item
relationships”: Not existent until this work
Challenges:
• Unified model with great expressive power
• Compact and manageable representation
• Allows room to develop tractable solutions for
natural influence optimization problems
• Model validation, data

7. Main Contributions
• Comparative Independent Cascade (ComIC):
Capturing both competition and
complementarity, to any arbitrary degree
• Problem: Self Influence Maximization
• Problem: Complementary Influence
Maximization
• Algorithm: Generalized Reverse Reachable Sets
• Algorithm: Sandwich Approximation

8. Model Overview
• Focusing on two items
– Challenges abundant already
– Future work: extended to an arbitrary number of items
• Edge-level influence/information propagation
– Similar to the classic IC model
• Node-level Decision-making controlled by
Node-Level Automata (NLA)
– Global Adoption Probabilities (GAP)

9. Global Adoption Probabilities
• Key parameters measuring the degree to which two
items compete with or complement each other
• q(A|0): probability of adopting A when the user has
not yet adopted any other items
• q(A|B): probability of adopting A when the user has
already adopted B
• q(A|0) >= q(A|B): B competes with A
• q(A|0) <= q(A|B): B complements A

10. Transition diagram
For each item, each node may be of the following status:
• Idle (inactive)
• Informed (influenced)
• Suspended / Adopted / Rejected

11. Diffusion dynamics
• Initially,every node is inactive/idle wrt both items
• When any node adopts the first item, its
outgoing edges are tested for information
propagation to neighbors (“info channel”)
– Each edge (u,v) becomes open w.p.p(u,v)
• If u is A-adopted, and info channel on edge (u,v)
is open, then v decides to adopt A based on:
– w.p. q(A|0) if v has not adopted B
– w.p. q(A|B) if v has adopted B

12. Node tie-breaking
• What if there are multiple in-neighbors active in
the last time step t-1?
• Generate a random permutation of those in-
neighbors, and follow that order to test activation
• If one such neighbor adopted both items at t-1,
following the same order for informing
• If a seed is targeted with both items, decide the
order randomly (0.5 and 0.5 prob.)

13. Node Reconsideration
• Suppose B complements A: q(A|0) <= q(A|B)
• User v was informed of A, but did not adopt with
probability 1 – q(A|0)
• Once v adopts B, since B complements A, user
may want to revisit the decision with a
reconsideration probability:

14. General Properties of ComIC
model
• Neither submodularity nor monotonicity holds in
an arbitrary instance of the model
• Influence maximization may be intractable
• Overall strategy:
– Identify a parameter subspace such that
submodularity is satisfied
– Develop efficient approximation algorithm
(Generalized RR-set) for submodular cases
– “Sandwich Approximation” for non-submodular
cases

15. Submodularity: Complementary Case

16. Possible World Definition
• An equivalent representation of the model and
the propagation dynamics
– Propagation in a possible world is deterministic, easy
to reason about
• Equivalent Possible World model for ComIC
– For each edge (u,v), remove w.p. 1-p(u,v)
– For each node v, randomly generate α(v,A) and α(v,B)
for testing with adoption probabilities.
– Adoption happens when α <= adoption prob.

17. Influence Maximization Problems
• Self Influence Maximization (SIM): Fix B-seed
set, find the best A-seed set of size k to
maximize A’s expected influence spread
• Complementary Influence Maximization
(CIM): Fix A-seed set, find the best B-seed set of
size k to maximize the boost B gives to A’s
expected influence spread
• Both NP-hard under ComIC model

18. Algorithm Design for SIM and CIM
• Generalized Reverse-Reachable Set (RR-set):
RR-set based algorithms are the state-of-the-art
for classical influence maximization with single-
item propagation models (IC and LT)
• Sandwich Approximation to achieve
approximation guarantees in non-submodular
cases
• Both techniques are generic and applicable to
any non-submodular maximization problems

19. Recap: Reverse-Reachable Set
• If u can reach v (in a deterministic directed
graph), then u is in a RR-set rooted at v [Borgs et
al., SODA’14]
• Random RR-set: root v is randomly chosen
• Two-phase Inf. Max. (TIM) [Tang et al 2014]
– Estimate the minimum number of random RR-sets
required, for probabilistic approx. guarantees
• 1-1/e-ε: smaller ε requires more RR-sets to be generated
– Generate random RR-sets using backward BFS
– Seed selection (deterministic max-cover problem)

20. Recap: TIM Algorithm
• (1-1/e-ε)-approximation with high probability
– Same as greedy, modulo probabilistic part
• Orders of magnitude faster than Greedy + Monte
Carlo simulations
• Scalable to billion-edge graphs
• Applies to a large family of stochastic
propagation models

21. Generalized RR-set and TIM
Algorithms
• Works for any stochastic propagation models
satisfying monotonicity and submodularity
– Has (1-1/e-ε)-approximation with high probability
• General RR-set (in a deterministic possible
world): u belongs to the RR-set rooted at v if the
singleton seed set {u} can activate v
– Note difference from “reaching”
– Random RR-set: root v is sampled uniformly at
random from the graph

22. RR-set generation for SIM (RR-
SIM)
• Problem definition and submodular setting
– Fix B-seed set, find A-seed set (size k)
– A is complemented by B: q(A|0) <= q(A|B)
– B is indifferent to A: q(B|0) = q(B|A)
• Phase 1: Forward Labeling: Start from B-seed
set, label node status w.r.t. B
• Phase 2: Backward BFS (details next)

23. Phase 2: Backward BFS
• Randomly choose root v from the graph
• Enqueue v into a FIFO queue Q
• Until empty, repeatedly dequeue from Q
• Let’s say we get a node u from Q
• Enqueue u’s in-neighbours (with edge test) if
either is true
– u is B-adopted and α(A,u) <= q(A|B)
– u is not B-adopted and α(A,u) <= q(A|0)

24. RR-Set generation for CIM (RR-
CIM)
• Given A-seed set, find best complementing B set
• Cross-submodularity holds q(B|A) = 1
• Forward Labeling: Start from A-seed set, identify
nodes can be A-adopted potentially
• Backward BFS: Two passes required

25. Sandwich Approximation
• Given any non-submodular set functions, how to
leverage submodular maximization (e.g., greedy,
local search) to achieve provable approximation
guarantees?
• Answer:
– Derive upper/lower bound submodular functions
(“sandwiched”)
– Use the best of the three solutions, which gives a
data-dependent approximation ratio

26. Sandwich Approximation
non-submodular, function we
want to maximize
lower bound, submodular
upper bound, submodular

27. Remarks
• Applicable to any non-submodular function
maximization
• If monotone, run Greedy on the upper bound,
lower bound, and the actual function
• If non-monotone, run Local Search
• Upper/lower bound should be reasonably tight to
be meaningful

28. Experiments: Datasets
Also have synthetic dataset up to 1 million nodes

29. Learning Global Adoption Probabilities
Dataset: Flixster
• Signals for adoption: rated a movie
• Signals for informed: “Want to See”, “Not Interested”

30. Effects of εin General TIM algorithm: Tradeoff
between seed set quality and running time

31. SIM experiments: spread

32. CIM experiments: spread

33. Running time

34. Sandwich Approximation Bounds

35. Thank you!
See you in VLDB’16!