In this KDD'13 paper we study competitive viral marketing from the host perspective. Competitive viral marketing refers to the case in which two or more players compete with comparable products over the same market. By host we mean the social networking platform owner, which offers viral marketing as a service to its clients. From the host’s perspective, it is important not only to choose the seed nodes to target in the campaigns in such a way to maximize the collective expected number of adoptions across all companies, but also to allocate seeds to companies in a way that guarantees the ''bang for the buck'' for all companies is nearly the same. Intuitively, the bang for the buck for a company is the cost benefit ratio between the expected number of adopters of its product over its number of seeds.

1. 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
welu@cs.ubc.ca
Yahoo! Research
bonchi@yahoo-inc.com
Twitter
goyal@cs.ubc.ca
Univ. of British Columbia
laks@cs.ubc.ca
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, … , 𝑆 𝐾, such that:
* 𝑆𝑖 = 𝑏𝑖 (budget)
* 𝑆𝑖 ∩ 𝑆𝑗 = ∅, ∀𝑖 ≠ 𝑗 and 𝑆𝑖 = 𝑆𝐾
𝑖=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