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# Correlation Robust Influence Maximization

NIPS 2020, (2020)

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Abstract

We propose a distributionally robust model for the influence maximization problem. Unlike the classic independent cascade model \citep{kempe2003maximizing}, this model's diffusion process is adversarially adapted to the choice of seed set. Hence, instead of optimizing under the assumption that all influence relationships in the network ...More

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Introduction

- Social networks are models that capture transmission of information among its members
- They find applications in testing effectiveness of policies, diffusion of medical innovations, marketing campaigns and news (Yadav et al, 2016; Tambe and Rice, 2018; Hunter and Zaman, 2018; Chen et al, 2011).

Highlights

- Social networks are models that capture transmission of information among its members
- The Independent Cascade (IC) influence maximization problem - max|S|≤k Eθic [R(c, S)] is known to be NP-hard; even evaluating f ic(S) := Eθic [R(c, S)] for a seed set S is #P-hard (Kempe et al, 2003), though several works have proposed efficient approximation methods, and a greedy algorithm provides a 1 − 1/e − approximation guarantee(Kempe et al, 2003), where > 0 accounts for sampling errors involved in the approximation of f ic(S)
- We propose to choose a k-sized seed set S that maximizes the expected number of influenced nodes with respect to the worst correlation
- (2) In Section 4, we show that finding an optimal seed set S that maximizes the worst case expected value R(c, S) is NP-hard
- While the majority of robust studies for the IC model have considered parameter uncertainty by way of the edge likelihoods, and still assume a fixed correlation structure - namely, independent edge propagation, we study the “reverse’" problem by assuming the edge likelihoods are fixed and the uncertainty lies in how they are correlated
- Using the price of correlations (POC) metric, we show instances where using a seed optimal for IC, would hurt the decision maker greatly if an adversarial diffusion process manifests

Methods

- Datasets The authors' experiments were performed on two datasets (1) wikivote: Here each node denotes a user and each edge (i, j) denotes the action of user i voting for user j to be an admin (Leskovec and Krevl, 2014).
- As in (Yan et al, 2011; Zhang et al, 2014) the authors reverse the edges so that, edge (i, j) in the original graph becomes (j, i)
- This reverse direction more aptly captures a notion of influence, as user i’s vote for j establishes that user j has influence over user i.

Conclusion

- The authors have proposed a model for influence maximization where the activation probabilities of the edges are known, but the joint distribution of these activations is unknown, adversarially chosen upon selection of a seed set.
- For measuring the utility of the model and misspecification under IC, the authors adapt the price of correlations metric for the influence maximization problem.
- Using the POC metric, the authors show instances where using a seed optimal for IC, would hurt the decision maker greatly if an adversarial diffusion process manifests.

Summary

## Introduction:

Social networks are models that capture transmission of information among its members- They find applications in testing effectiveness of policies, diffusion of medical innovations, marketing campaigns and news (Yadav et al, 2016; Tambe and Rice, 2018; Hunter and Zaman, 2018; Chen et al, 2011).
## Objectives:

The aim of this work is to address the possible pitfalls to the independence assumption in a social network, as used in the study of influence maximization## Methods:

Datasets The authors' experiments were performed on two datasets (1) wikivote: Here each node denotes a user and each edge (i, j) denotes the action of user i voting for user j to be an admin (Leskovec and Krevl, 2014).- As in (Yan et al, 2011; Zhang et al, 2014) the authors reverse the edges so that, edge (i, j) in the original graph becomes (j, i)
- This reverse direction more aptly captures a notion of influence, as user i’s vote for j establishes that user j has influence over user i.
## Conclusion:

The authors have proposed a model for influence maximization where the activation probabilities of the edges are known, but the joint distribution of these activations is unknown, adversarially chosen upon selection of a seed set.- For measuring the utility of the model and misspecification under IC, the authors adapt the price of correlations metric for the influence maximization problem.
- Using the POC metric, the authors show instances where using a seed optimal for IC, would hurt the decision maker greatly if an adversarial diffusion process manifests.

- Table1: Numerical Summaries of the Graphs from the Datasets
- Table2: Properties of Sigc and Scgorr for non-identical edge probabilities. k = 40

Related work

**Related Work and Preliminaries**

There is an extensive literature on influence maximization, including adaptive models (Peng and Chen, 2019), learning (e.g. (Narasimhan et al, 2015; He et al, 2016; Balkanski et al, 2017)), and in recent years, robustness. To the best of our knowledge, robustness in influence maximization first received attention through the parametric perturbation interval model (He and Kempe, 2014) where for each edge (i, j) ∈ E the probability pij is not known exactly, but rather lies in an interval [lij, rij] ⊆ [0, 1]. The model however still assumes all edges are independenty live. Their objective is to obtain the best seed set under the IC model, robust to the values the edge likelihoods p can take. Models of a similar spirit include (Chen et al, 2016; Kalimeris et al, 2018, 2019; Staib et al, 2019). Additionally, (Chen et al, 2016; Kalimeris et al, 2019) study robustness from the view of model misspecification; the particular objectives studied there are hard, partly due to their non-convex and non-submodular objectives. An analogous study of robustness has been performed with the linear threshold model (another diffusion process) in (Nannicini et al, 2019) where the parameters are assumed to be uncertain.

Funding

- The research of the last three authors was partly supported by the MOE Academic Research Fund Tier 2 grant MOE2019- T2-2-138, “Enhancing Robustness of Networks to Dependence via Optimization”

Study subjects and analysis

cases: 3

Properties of Seed Sets: We now demonstrate some properties of the seed sets Sigc and Scgorr for the case of non-identical probabilities. The following three cases of non-identical probabilities were studied (1) Unif(0, 1): pij drawn i.i.d. from Unif(0, 1); (2) Trivalency: pij drawn i.i.d. from Unif{0.1, 0.01, 0.001}; (3) Weighted cascade: pij = 1/deg(i), deg(i) denotes the number of edges incident to i. In Table 2 we report the mis-specification ratio under alternate diffusion processes -

Reference

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