9 research outputs found
How to Influence People with Partial Incentives
We study the power of fractional allocations of resources to maximize
influence in a network. This work extends in a natural way the well-studied
model by Kempe, Kleinberg, and Tardos (2003), where a designer selects a
(small) seed set of nodes in a social network to influence directly, this
influence cascades when other nodes reach certain thresholds of neighbor
influence, and the goal is to maximize the final number of influenced nodes.
Despite extensive study from both practical and theoretical viewpoints, this
model limits the designer to a binary choice for each node, with no way to
apply intermediate levels of influence. This model captures some settings
precisely, e.g. exposure to an idea or pathogen, but it fails to capture very
relevant concerns in others, for example, a manufacturer promoting a new
product by distributing five "20% off" coupons instead of giving away one free
product.
While fractional versions of problems tend to be easier to solve than
integral versions, for influence maximization, we show that the two versions
have essentially the same computational complexity. On the other hand, the two
versions can have vastly different solutions: the added flexibility of
fractional allocation can lead to significantly improved influence. Our main
theoretical contribution is to show how to adapt the major positive results
from the integral case to the fractional case. Specifically, Mossel and Roch
(2006) used the submodularity of influence to obtain their integral results; we
introduce a new notion of continuous submodularity, and use this to obtain
matching fractional results. We conclude that we can achieve the same greedy
-approximation for the fractional case as the integral case.
In practice, we find that the fractional model performs substantially better
than the integral model, according to simulations on real-world social network
data
From Sets to Multisets: Provable Variational Inference for Probabilistic Integer Submodular Models
Submodular functions have been studied extensively in machine learning and
data mining. In particular, the optimization of submodular functions over the
integer lattice (integer submodular functions) has recently attracted much
interest, because this domain relates naturally to many practical problem
settings, such as multilabel graph cut, budget allocation and revenue
maximization with discrete assignments. In contrast, the use of these functions
for probabilistic modeling has received surprisingly little attention so far.
In this work, we firstly propose the Generalized Multilinear Extension, a
continuous DR-submodular extension for integer submodular functions. We study
central properties of this extension and formulate a new probabilistic model
which is defined through integer submodular functions. Then, we introduce a
block-coordinate ascent algorithm to perform approximate inference for those
class of models. Finally, we demonstrate its effectiveness and viability on
several real-world social connection graph datasets with integer submodular
objectives
Optimal Intervention in Economic Networks using Influence Maximization Methods
We consider optimal intervention in the Elliott-Golub-Jackson network model
and show that it can be transformed into an influence maximization problem,
interpreted as the reverse of a default cascade. Our analysis of the optimal
intervention problem extends well-established targeting results to the economic
network setting, which requires additional theoretical steps. We prove several
results about optimal intervention: it is NP-hard and additionally hard to
approximate to a constant factor in polynomial time. In turn, we show that
randomizing failure thresholds leads to a version of the problem which is
monotone submodular, for which existing powerful approximations in polynomial
time can be applied. In addition to optimal intervention, we also show
practical consequences of our analysis to other economic network problems: (1)
it is computationally hard to calculate expected values in the economic
network, and (2) influence maximization algorithms can enable efficient
importance sampling and stress testing of large failure scenarios. We
illustrate our results on a network of firms connected through input-output
linkages inferred from the World Input Output Database
Direct search methods for influence maximization problems
openThe influence maximization problem seeks to maximize the spread of influence in network structures. This challenging optimization task remains unsolved due to its combinatorial nature and computational complexity. In this thesis, we propose two novel direct search methods, Neighbors Search (NS) and nonmonotone NS, which leverage the network structure to enhance efficiency. We compare these methods with the state-of-the-art Custom Direct Search (CDS) method through experiments on artificial and real-world networks. Our findings show promising improvements in solving the influence maximization problem.The influence maximization problem seeks to maximize the spread of influence in network structures. This challenging optimization task remains unsolved due to its combinatorial nature and computational complexity. In this thesis, we propose two novel direct search methods, Neighbors Search (NS) and nonmonotone NS, which leverage the network structure to enhance efficiency. We compare these methods with the state-of-the-art Custom Direct Search (CDS) method through experiments on artificial and real-world networks. Our findings show promising improvements in solving the influence maximization problem
Mathematical Programming Models for Influence Maximization on Social Networks
In this dissertation, we apply mathematical programming techniques (i.e., integer programming and polyhedral combinatorics) to develop exact approaches for influence maximization on social networks. We study four combinatorial optimization problems that deal with maximizing influence at minimum cost over a social network. To our knowl- edge, all previous work to date involving influence maximization problems has focused on heuristics and approximation.
We start with the following viral marketing problem that has attracted a significant amount of interest from the computer science literature. Given a social network, find a target set of customers to seed with a product. Then, a cascade will be caused by these initial adopters and other people start to adopt this product due to the influence they re- ceive from earlier adopters. The idea is to find the minimum cost that results in the entire network adopting the product.
We first study a problem called the Weighted Target Set Selection (WTSS) Prob- lem. In the WTSS problem, the diffusion can take place over as many time periods as
needed and a free product is given out to the individuals in the target set. Restricting the number of time periods that the diffusion takes place over to be one, we obtain a problem called the Positive Influence Dominating Set (PIDS) problem. Next, incorporating partial incentives, we consider a problem called the Least Cost Influence Problem (LCIP). The fourth problem studied is the One Time Period Least Cost Influence Problem (1TPLCIP) which is identical to the LCIP except that we restrict the number of time periods that the diffusion takes place over to be one.
We apply a common research paradigm to each of these four problems. First, we work on special graphs: trees and cycles. Based on the insights we obtain from special graphs, we develop efficient methods for general graphs. On trees, first, we propose a polynomial time algorithm. More importantly, we present a tight and compact extended formulation. We also project the extended formulation onto the space of the natural vari- ables that gives the polytope on trees. Next, building upon the result for trees---we derive the polytope on cycles for the WTSS problem; as well as a polynomial time algorithm on cycles.
This leads to our contribution on general graphs. For the WTSS problem and the LCIP, using the observation that the influence propagation network must be a directed acyclic graph (DAG), the strong formulation for trees can be embedded into a formulation on general graphs. We use this to design and implement a branch-and-cut approach for the WTSS problem and the LCIP. In our computational study, we are able to obtain high quality solutions for random graph instances with up to 10,000 nodes and 20,000 edges (40,000 arcs) within a reasonable amount of time