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Popularity Ratio Maximization: Surpassing Competitors through Influence Propagation

Hao Liao, Sheng Bi, Jiao Wu, Wei Zhang, Mingyang Zhou, Rui Mao, Wei Chen

TL;DR

An algorithmic study on how to surpass competitors in popularity by strategic promotions in social networks and devise a surrogate objective function that empirically and theoretically is very close to the original objective function while theoretically, it is monotone but not submodular.

Abstract

In this paper, we present an algorithmic study on how to surpass competitors in popularity by strategic promotions in social networks. We first propose a novel model, in which we integrate the Preferential Attachment (PA) model for popularity growth with the Independent Cascade (IC) model for influence propagation in social networks called PA-IC model. In PA-IC, a popular item and a novice item grab shares of popularity from the natural popularity growth via the PA model, while the novice item tries to gain extra popularity via influence cascade in a social network. The popularity ratio is defined as the ratio of the popularity measure between the novice item and the popular item. We formulate Popularity Ratio Maximization (PRM) as the problem of selecting seeds in multiple rounds to maximize the popularity ratio in the end. We analyze the popularity ratio and show that it is monotone but not submodular. To provide an effective solution, we devise a surrogate objective function and show that empirically it is very close to the original objective function while theoretically, it is monotone and submodular. We design two efficient algorithms, one for the overlapping influence and non-overlapping seeds (across rounds) setting and the other for the non-overlapping influence and overlapping seed setting, and further discuss how to deal with other models and problem variants. Our empirical evaluation further demonstrates that the proposed PRM-IMM method consistently achieves the best popularity promotion compared to other methods. Our theoretical and empirical analyses shed light on the interplay between influence maximization and preferential attachment in social networks.

Popularity Ratio Maximization: Surpassing Competitors through Influence Propagation

TL;DR

An algorithmic study on how to surpass competitors in popularity by strategic promotions in social networks and devise a surrogate objective function that empirically and theoretically is very close to the original objective function while theoretically, it is monotone but not submodular.

Abstract

In this paper, we present an algorithmic study on how to surpass competitors in popularity by strategic promotions in social networks. We first propose a novel model, in which we integrate the Preferential Attachment (PA) model for popularity growth with the Independent Cascade (IC) model for influence propagation in social networks called PA-IC model. In PA-IC, a popular item and a novice item grab shares of popularity from the natural popularity growth via the PA model, while the novice item tries to gain extra popularity via influence cascade in a social network. The popularity ratio is defined as the ratio of the popularity measure between the novice item and the popular item. We formulate Popularity Ratio Maximization (PRM) as the problem of selecting seeds in multiple rounds to maximize the popularity ratio in the end. We analyze the popularity ratio and show that it is monotone but not submodular. To provide an effective solution, we devise a surrogate objective function and show that empirically it is very close to the original objective function while theoretically, it is monotone and submodular. We design two efficient algorithms, one for the overlapping influence and non-overlapping seeds (across rounds) setting and the other for the non-overlapping influence and overlapping seed setting, and further discuss how to deal with other models and problem variants. Our empirical evaluation further demonstrates that the proposed PRM-IMM method consistently achieves the best popularity promotion compared to other methods. Our theoretical and empirical analyses shed light on the interplay between influence maximization and preferential attachment in social networks.
Paper Structure (37 sections, 11 theorems, 40 equations, 8 figures, 7 tables, 4 algorithms)

This paper contains 37 sections, 11 theorems, 40 equations, 8 figures, 7 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Related Works
  3. Preferential Attachment
  4. Model and Problem Definition
  5. PA-IC Model
  6. Popularity Ratio Maximization
  7. Results on the OINS Setting
  8. Objective Function under the OINS Setting
  9. Round-Weighted Influence Maximization
  10. Pair-wise Reverse Reachable Sets
  11. Popularity Ratio Maximization IMM
  12. Results on the NIOS Setting
  13. Objective Function under the NIOS Setting
  14. Efficient Algorithm for NIOS setting
  15. Other Variants of PRM Problem
  16. ...and 22 more sections

Key Result

lemma 1

For the OINS setting, the popularity ratio function at the end of round $T$ is: where $r_0 = d_0^n/d_0^p$ is the initial popularity ratio, $\mathcal{S} = \bigcup_{t=1}^T S_t \times \{t\}$.

Figures (8)

  • Figure 1: A numerical example of the PA-IC model with the overlapping influence setting. In this example, we show the changes in popularity measure of two items, within two rounds. In this directed graph, every edge's activation probability is equal to 1. Black nodes is the "active" nodes in each round.
  • Figure 2: The popularity ratio vs. budgets for different algorithms in OINS setting.
  • Figure 3: Model justification. (a) Comparing between the random growth (red) and average growth (blue); (b) Comparing between the original objective function and the surrogate function.
  • Figure 4: Parameter analysis for PRM under FX01.
  • Figure 5: Different parameter's impact on seeds allocation, in the FX01 dataset.
  • ...and 3 more figures

Theorems & Definitions (13)

  • Definition 1: Popularity Ratio Maximization
  • lemma 1
  • lemma 2
  • Definition 2: Round-weighted Influence Maximization in the OINS Setting
  • lemma 3
  • lemma 4
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • ...and 3 more