Rank-Refining Seed Selection Methods for Budget Constrained Influence Maximisation in Multilayer Networks under Linear Threshold Model

TL;DR

We address budget-constrained influence maximisation on multilayer networks using the Multilayer Linear Threshold Model (MLTM). The authors adapt and evaluate sixteen rank-refining seed heuristics, including VoteRank variants, across twelve networks, and introduce Gain $G$ and Diffusion Length $DL$ as principal diffusion metrics, highlighting that no single method dominates and that VoteRank-based seeds (notably $v$-rnk-$m$) generally perform best under challenging conditions. They prove MLTM is not submodular, propose efficiency-curve analysis to relate model parameters to diffusion outcomes, and provide a rigorous, large-scale experimental framework with code and data for reproducibility. The study offers practical guidance on seed selection under budget constraints in multilayer networks and establishes a robust baseline for future work in centrality-based diffusion on complex networks.

Abstract

The problem of selecting an optimal seed set to maximise influence in networks has been a subject of intense research in recent years. However, despite numerous works addressing this area, it remains a topic that requires further elaboration. Most often, it is considered within the scope of classically defined graphs with a spreading model in the form of Independent Cascades. In this work, we focus on the problem of budget-constrained influence maximisation in multilayer networks using a Linear Threshold Model. Both the graph model and the spreading process we employ are less prevalent in the literature, even though their application allows for a more precise representation of the opinion dynamics in social networks. This paper aims to answer which of the sixteen evaluated seed selection methods is the most effective and how similar they are. Additionally, we focus our analysis on the impact of spreading model parameters, network characteristics, a budget, and the seed selection methods on the diffusion effectiveness in multilayer networks. Our contribution also includes extending several centrality measures and heuristics to the case of such graphs. The results indicate that all the factors mentioned above collectively contribute to the effectiveness of influence maximisation. Moreover, there is no seed selection method which always provides the best results. However, the seeds chosen with VoteRank-based methods (especially with the $v-rnk-m$ variant we propose) usually provide the most extensive diffusion.

Figures (15)

• Figure 1: Schematic depiction of the influence maximisation's components. Light blue ellipses --- input data, dark blue ellipses --- output data, green rectangles --- functions, arrows --- order of execution.
• Figure 2: Example of a multilayer network.
• Figure 3: Example of spreading of MLTM through toy network with protocol $AND$.
• Figure 4: Example of spreading of MLTM through toy network with protocol $OR$.
• Figure 5: Mean pairwise Jaccard similarity with standard deviation between seed sets constructed with the ranking lists from Tab. \ref{['tab:rankings_toy_net']} in a function of the budget size.

Theorems & Definitions (10)

• Definition 2.1: Network
• Definition 2.2: Multilayer Network
• Definition 2.3: Multiplex Network
• Definition 2.4: Binary Discrete System
• Definition 2.5: Seeds in the Binary Discrete System
• Definition 2.6: Linear Threshold Model
• Definition 2.7: Multilayer Linear Threshold Model
• Definition 2.8: Protocol Function
• Definition 2.9: Submodular Function
• Definition 2.10: Budget Constrained Influence Maximisation under MLTM