Optimal Multiphase Investment Strategies for Influencing Opinions in a Social Network

We study the problem of optimally investing in nodes of a social network in a competitive setting, where two camps aim to maximize adoption of their opinions by the population. In particular, we consider the possibility of campaigning in multiple phases, where the final opinion of a node in a phase acts as its initial biased opinion for the following phase. Using an extension of the popular DeGroot-Friedkin model, we formulate the utility functions of the camps, and show that they involve what can be interpreted as multiphase Katz centrality. Focusing on two phases, we analytically derive Nash equilibrium investment strategies, and the extent of loss that a camp would incur if it acted myopically. Our simulation study affirms that nodes attributing higher weightage to initial biases necessitate higher investment in the first phase, so as to influence these biases for the terminal phase. We then study the setting in which a camp's influence on a node depends on its initial bias. For single camp, we present a polynomial time algorithm for determining an optimal way to split the budget between the two phases. For competing camps, we show the existence of Nash equilibria under reasonable assumptions, and that they can be computed in polynomial time

Optimal multiphase investment strategies for influencing opinions in a social network

International audienceWe study the problem of two competing camps aiming to maximize the adoption of their respective opinions, by optimally investing in nodes of a social network in multiple phases. The final opinion of a node in a phase acts as its biased opinion in the following phase. Using an extension of Friedkin-Johnsen model, we formulate the camps' utility functions, which we show to involve what can be interpreted as multiphase Katz centrality. We hence present optimal investment strategies of the camps, and the loss incurred if myopic strategy is employed. Simulations affirm that nodes attributing higher weightage to bias necessitate higher investment in initial phase. The extended version of this paper analyzes a setting where a camp's influence on a node depends on the node's bias; we show existence and polynomial time computability of Nash equilibrium

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

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

Effectiveness of Diffusing Information through a Social Network in Multiple Phases

We study the effectiveness of using multiple phases for maximizing the extent of information diffusion through a social network, and present insights while considering various aspects. In particular, we focus on the independent cascade model with the possibility of adaptively selecting seed nodes in multiple phases based on the observed diffusion in preceding phases, and conduct a detailed simulation study on real-world network datasets and various values of seeding budgets. We first present a negative result that more phases do not guarantee a better spread, however the adaptability advantage of more phases generally leads to a better spread in practice, as observed on real-world datasets. We study how diffusing in multiple phases affects the mean and standard deviation of the distribution representing the extent of diffusion. We then study how the number of phases impacts the effectiveness of multiphase diffusion, how the diffusion progresses phase-by-phase, and what is an optimal way to split the total seeding budget across phases. Our experiments suggest a significant gain when we move from single phase to two phases, and an appreciable gain when we further move to three phases, but the marginal gain thereafter is usually not very significant. Our main conclusion is that, given the number of phases, an optimal way to split the budget across phases is such that the number of nodes influenced in each phase is almost the same.Comment: This paper is under revie

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

A Two Phase Investment Game for Competitive Opinion Dynamics in Social Networks

We propose a setting for two-phase opinion dynamics in social networks, where a node's final opinion in the first phase acts as its initial biased opinion in the second phase. In this setting, we study the problem of two camps aiming to maximize adoption of their respective opinions, by strategically investing on nodes in the two phases. A node's initial opinion in the second phase naturally plays a key role in determining the final opinion of that node, and hence also of other nodes in the network due to its influence on them. More importantly, this bias also determines the effectiveness of a camp's investment on that node in the second phase. To formalize this two-phase investment setting, we propose an extension of Friedkin-Johnsen model, and hence formulate the utility functions of the camps. There is a tradeoff while splitting the budget between the two phases. A lower investment in the first phase results in worse initial biases for the second phase, while a higher investment spares a lower available budget for the second phase. We first analyze the non-competitive case where only one camp invests, for which we present a polynomial time algorithm for determining an optimal way to split the camp's budget between the two phases. We then analyze the case of competing camps, where we show the existence of Nash equilibrium and that it can be computed in polynomial time under reasonable assumptions. We conclude our study with simulations on real-world network datasets, in order to quantify the effects of the initial biases and the weightage attributed by nodes to their initial biases, as well as that of a camp deviating from its equilibrium strategy. Our main conclusion is that, if nodes attribute high weightage to their initial biases, it is advantageous to have a high investment in the first phase, so as to effectively influence the biases to be harnessed in the second phase