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

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 $k$, and we look for a set of $k$ 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 $i$th seed can be based on the observed cascade produced by the first $i-1$ 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 $\lceil n^{1/3}\rceil$, where $n$ is the number of nodes in the graph. Moreover, we improve over the known upper bound for in-arborescences from $\frac{2e}{e-1}\approx 3.16$ to $\frac{2e^2}{e^2-1}\approx 2.31$. Finally, we study $\alpha$-bounded graphs, a class of undirected graphs in which the sum of node degrees higher than two is at most $\alpha$, and show that the adaptivity gap is upper-bounded by $\sqrt{\alpha}+O(1)$. 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 $\frac{3e^3}{e^3-1}\approx 3.16$. 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

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 $k$, iteratively select $k$ 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 $(1-1/e)$-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 $(1-1/e)$ 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 $(1-1/e)$-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

Complexity, Algorithms, and Heuristics of Influence Maximization

People often adopt improved behaviors, products, or ideas through the influence of friends. This is modeled by emph{cascades}. One way to spread such positive elements through society is to identify those most influential agents---those that cause the maximum spread, and initiate the spread by seeding them. However, this strategy has a key difficulty: finding these influential seed nodes. This is difficult even if both the network structure and the way the cascade spreads are known. In emph{the influence maximization problem}, a central planner is given a graph and a limited budget $k$, and he needs to pick $k$ seeds such that the expected total number of infected vertices in the graph at the end of the cascade is maximized. This problem plays a central role in viral marketing, outbreak detection, rumor controls, etc. This thesis focuses on computational complexity, approximability and algorithm/heuristic design aspects of the influence maximization problem, with both emph{submodular} and emph{nonsubmodular} diffusion models. The first part of the thesis studies submodular influence maximization mainly in the computational complexity and algorithm analysis aspects, which includes some breakthroughs in understanding the approximability of submodular influence maximization and the theoretical performance of the well-studied greedy algorithm. The second part of the thesis focuses on nonsubmodular influence maximization. New sociologically founded nonsubmodular diffusion models are proposed, and we show how the seeding strategy for nonsubmodular diffusion models is fundamentally different compared to submodular diffusion models.PHDComputer Science & EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/155221/1/bstao_1.pd