12 research outputs found
Better Bounds on the Adaptivity Gap of Influence Maximization under Full-adoption Feedback
In the influence maximization (IM) problem, we are given a social network and
a budget , and we look for a set of nodes in the network, called seeds,
that maximize the expected number of nodes that are reached by an influence
cascade generated by the seeds, according to some stochastic model for
influence diffusion. In this paper, we study the adaptive IM, where the nodes
are selected sequentially one by one, and the decision on the th seed can be
based on the observed cascade produced by the first seeds. We focus on
the full-adoption feedback in which we can observe the entire cascade of each
previously selected seed and on the independent cascade model where each edge
is associated with an independent probability of diffusing influence.
Our main result is the first sub-linear upper bound that holds for any graph.
Specifically, we show that the adaptivity gap is upper-bounded by , where is the number of nodes in the graph. Moreover, we
improve over the known upper bound for in-arborescences from
to . Finally, we
study -bounded graphs, a class of undirected graphs in which the sum of
node degrees higher than two is at most , and show that the adaptivity
gap is upper-bounded by . Moreover, we show that in
0-bounded graphs, i.e. undirected graphs in which each connected component is a
path or a cycle, the adaptivity gap is at most . To prove our bounds, we introduce new techniques to relate adaptive
policies with non-adaptive ones that might be of their own interest.Comment: 18 page
Scalable Fair Influence Maximization
Given a graph , a community structure , and a budget , the
fair influence maximization problem aims to select a seed set ()
that maximizes the influence spread while narrowing the influence gap between
different communities. While various fairness notions exist, the welfare
fairness notion, which balances fairness level and influence spread, has shown
promising effectiveness. However, the lack of efficient algorithms for
optimizing the welfare fairness objective function restricts its application to
small-scale networks with only a few hundred nodes. In this paper, we adopt the
objective function of welfare fairness to maximize the exponentially weighted
summation over the influenced fraction of all communities. We first introduce
an unbiased estimator for the fractional power of the arithmetic mean. Then, by
adapting the reverse influence sampling (RIS) approach, we convert the
optimization problem to a weighted maximum coverage problem. We also analyze
the number of reverse reachable sets needed to approximate the fair influence
at a high probability. Further, we present an efficient algorithm that
guarantees approximation
On Adaptivity Gaps of Influence Maximization Under the Independent Cascade Model with Full-Adoption Feedback
In this paper, we study the adaptivity gap of the influence maximization problem under the independent cascade model when full-adoption feedback is available. Our main results are to derive upper bounds on several families of well-studied influence graphs, including in-arborescences, out-arborescences and bipartite graphs. Especially, we prove that the adaptivity gap for the in-arborescences is between [e/(e-1), 2e/(e-1)], and for the out-arborescences the gap is between [e/(e-1), 2]. These are the first constant upper bounds in the full-adoption feedback model. Our analysis provides several novel ideas to tackle the correlated feedback appearing in adaptive stochastic optimization, which may be of independent interest
Improved Approximation Factor for Adaptive Influence Maximization via Simple Greedy Strategies
In the adaptive influence maximization problem, we are given a social network and a budget k, and we iteratively select k nodes, called seeds, in order to maximize the expected number of nodes that are reached by an influence cascade that they generate according to a stochastic model for influence diffusion. The decision on the next seed to select is based on the observed cascade of previously selected seeds. We focus on the myopic feedback model, in which we can only observe which neighbors of previously selected seeds have been influenced and on the independent cascade model, where each edge is associated with an independent probability of diffusing influence. While adaptive policies are strictly stronger than non-adaptive ones, in which all the seeds are selected beforehand, the latter are much easier to design and implement and they provide good approximation factors if the adaptivity gap, the ratio between the adaptive and the non-adaptive optima, is small. Previous works showed that the adaptivity gap is at most 4, and that simple adaptive or non-adaptive greedy algorithms guarantee an approximation of 1/4 (1-1/e) ? 0.158 for the adaptive optimum. This is the best approximation factor known so far for the adaptive influence maximization problem with myopic feedback.
In this paper, we directly analyze the approximation factor of the non-adaptive greedy algorithm, without passing through the adaptivity gap, and show an improved bound of 1/2 (1-1/e) ? 0.316. Therefore, the adaptivity gap is at most 2e/e-1 ? 3.164. To prove these bounds, we introduce a new approach to relate the greedy non-adaptive algorithm to the adaptive optimum. The new approach does not rely on multi-linear extensions or random walks on optimal decision trees, which are commonly used techniques in the field. We believe that it is of independent interest and may be used to analyze other adaptive optimization problems. Finally, we also analyze the adaptive greedy algorithm, and show that guarantees an improved approximation factor of 1-1/(?{e)}? 0.393
Adaptive Greedy versus Non-adaptive Greedy for Influence Maximization
We consider the \emph{adaptive influence maximization problem}: given a
network and a budget , iteratively select seeds in the network to
maximize the expected number of adopters. In the \emph{full-adoption feedback
model}, after selecting each seed, the seed-picker observes all the resulting
adoptions. In the \emph{myopic feedback model}, the seed-picker only observes
whether each neighbor of the chosen seed adopts. Motivated by the extreme
success of greedy-based algorithms/heuristics for influence maximization, we
propose the concept of \emph{greedy adaptivity gap}, which compares the
performance of the adaptive greedy algorithm to its non-adaptive counterpart.
Our first result shows that, for submodular influence maximization, the
adaptive greedy algorithm can perform up to a -fraction worse than the
non-adaptive greedy algorithm, and that this ratio is tight. More specifically,
on one side we provide examples where the performance of the adaptive greedy
algorithm is only a fraction of the performance of the non-adaptive
greedy algorithm in four settings: for both feedback models and both the
\emph{independent cascade model} and the \emph{linear threshold model}. On the
other side, we prove that in any submodular cascade, the adaptive greedy
algorithm always outputs a -approximation to the expected number of
adoptions in the optimal non-adaptive seed choice. Our second result shows
that, for the general submodular cascade model with full-adoption feedback, the
adaptive greedy algorithm can outperform the non-adaptive greedy algorithm by
an unbounded factor. Finally, we propose a risk-free variant of the adaptive
greedy algorithm that always performs no worse than the non-adaptive greedy
algorithm.Comment: 26 pages, 0 figure, accepted at AAAI'20: Thirty-Fourth AAAI
Conference on Artificial Intelligenc
Limitations of Greed: Influence Maximization in Undirected Networks Re-visited
We consider the influence maximization problem (selecting seeds in a
network maximizing the expected total influence) on undirected graphs under the
linear threshold model. On the one hand, we prove that the greedy algorithm
always achieves a -approximation, showing
that the greedy algorithm does slightly better on undirected graphs than the
generic bound which also applies to directed graphs. On the
other hand, we show that substantial improvement on this bound is impossible by
presenting an example where the greedy algorithm can obtain at most a approximation. This result stands in contrast to the
previous work on the independent cascade model. Like the linear threshold
model, the greedy algorithm obtains a -approximation on directed
graphs in the independent cascade model. However, Khanna and Lucier showed
that, in undirected graphs, the greedy algorithm performs substantially better:
a approximation for constant . Our results show
that, surprisingly, no such improvement occurs in the linear threshold model.
Finally, we show that, under the linear threshold model, the approximation
ratio is tight if 1) the graph is directed or 2) the
vertices are weighted. In other words, under either of these two settings, the
greedy algorithm cannot achieve a -approximation for
any positive function . The result in setting 2) is again in a sharp
contrast to Khanna and Lucier's -approximation result
for the independent cascade model, where the
approximation guarantee can be extended to the setting where vertices are
weighted. We also discuss extensions to more generalized settings including
those with edge-weighted graphs.Comment: 36 pages, 1 figure, accepted at AAMAS'20: International Conference on
Autonomous Agents and Multi-Agent System