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The Bang for the Buck: Fair Competitive Viral Marketing from the Host Perspective
Wei Lu Francesco Bonchi Amit Goyal Laks V.S. Lakshmanan
Univ. of British Columbia
Yahoo! Research
Univ. of British Columbia
Background
Viral marketing Use word-of-mouth effects to improve product awareness and adoptions through social networks
Influence maximization problem Identify 𝑘 most influential users in a social network such that by targeting them as early adopters, the spread of influence is maximized
Previous research
Ignore competitions: one advertiser, one product Or, focus on the best strategy of one competing company Assume companies have free access to network!
However, in reality…
Competitions are everywhere!!! Network graphs are owned by service provider (host)
without whose permission no viral marketing campaigns would be possible!
K-LT (Linear Threshold) Propagation Model
Model specifications Each node in graph has a random activation threshold;
each edge has an influence weight 𝐾 competing companies, each targeting a seed set Activation phase 1: a node becomes active if influence
weights from active neighbors exceeds threshold Activation phase 2: it chooses a company out of those
chosen by its neighbors in the previous time step
Model properties
Monotoncity and submodularity hold for both total spread function and individual spread functions (unlike previous models)
Intuitive and natural
Problem Statements
Overall Influence Maximization
Given a graph and budgets of all companies, maximize the collective influence spread
Shown equivalent to influence maximization under classical LT model (no competition)
NP-hard, but can be approximated within (1 −1
𝑒− 𝜖)
Algorithm: treat companies as a giant one with budget = sum of all budgets, and apply the greedy algorithm:
Starts with an empty set, and in each iteration, adds the element providing the largest marginal gain in total influence spread
Next question: How to allocate seeds?
Individual budget constraints must be satisfied Allocation needs to be fair: ensure bang for the buck for
companies as balanced as possible Maintain good reputations of host’s business
How do we define fairness?
Min-max fairness: the happiest one should not be happier than others by a lot
Fair Seed Allocation Problem
Bang for the buck: influence spread per seed
influence spread of company 𝑖
budget of company 𝑖
Optimization problem: Given the global seed set 𝑆, partition it into 𝐾 subsets 𝑆1, 𝑆2, … , 𝑆𝐾, such that:
* 𝑆𝑖 = 𝑏𝑖 (budget) * 𝑆𝑖 ∩ 𝑆𝑗 = ∅, ∀𝑖 ≠ 𝑗 and 𝑆𝑖 = 𝑆
𝐾𝑖=1
* max. bang for the buck is minimized (min-max) Hardness results: * Strongly NP-hard in general (reduction from the NP-complete 3-Partition problem) * Weakly NP-hard when 𝐾 = 2 (from Partition) No free-ride: low budget company will not benefit from
higher budget competitors Other fairness objectives also possible: max-min, etc.
Fair Allocation Algorithms
Adjusted Marginal Gain of Seeds
Definition: the spread of a seed (by itself) on the subgraph induced by nodes excluding other seeds
Theorem: In K-LT model: spread for a company = sum of adjusted marginal gains of seeds allocated to it
Needy-Greedy Algorithm
Sort seeds in decreasing order of adjusted marginal gains Assign seeds in the sorted order: in each iteration, assign
to the company with the smallest bang for the buck amongst all of which the budget is not yet exhausted.
Dynamic Programming: Solve the problem optimally for 2-company instances in pseudo-polynomial time.
Experimental Results
Network Datasets: arXiv, Epinions, Flixster
Baselines Algorithms: random and round-robin
Evaluation Metrics: compare max. bang for the buck with theoretical lower bound: total spread/total budget
Conclusions & Future Work
Viral marketing in a more realistic setting: competitions and host selling viral marketing as a service
Fair seed allocation: a new challenge for hosts, solved Future work: other business models for hosts, game-
theoretical models, etc.
Our Contributions
Study competitive viral marketing from the host perspective
Propose a competition-aware propagation model Propose the Fair Seed Allocation problem Design efficient and effective fair allocation algorithms
