Assessing the Robustness and Reducibility of Multiplex Networks with Embedding-Aided Interlayer Similarities

TL;DR

The paper tackles the challenge of measuring interlayer similarity in multiplex networks by introducing Embedding Aided inTerlayer Similarity (EATSim), which combines intralayer structural cues from Pairwise Euclidean Distance (PED) loss and cross-layer alignment cues from Aligned Euclidean Distance (AED) loss using node2vec embeddings. The framework defines $D = \omega \mathcal{L}_{\mathrm{PED}} + (1-\omega) \mathcal{L}_{\mathrm{AED}}$ and $EATSim = 1 - D$ with $\omega = 0.5$, and evaluates it on synthetic BA rewiring and Geometric Multiplex Model data as well as nine real multiplex networks. Empirical results show EATSim correlates strongly with network robustness quantified by $\Omega$ (synthetic $\approx 0.882$, real $\approx 0.856$) and achieves state-of-the-art performance in network reducibility via the distinguishability metric $q$, outperforming several baselines. The work provides a unified, embedding-based metric that captures both local and global interlayer similarities, enabling better design and management of robust, compact multiplex systems in domains such as transportation, biology, and social networks.

Abstract

The study of interlayer similarity of multiplex networks helps to understand the intrinsic structure of complex systems, revealing how changes in one layer can propagate and affect others, thus enabling broad implications for transportation, social, and biological systems. Existing algorithms that measure similarity between network layers typically encode only partial information, which limits their effectiveness in capturing the full complexity inherent in multiplex networks. To address this limitation, we propose a novel interlayer similarity measuring approach named Embedding Aided inTerlayer Similarity (EATSim). EATSim concurrently incorporates intralayer structural similarity and cross-layer anchor node alignment consistency, providing a more comprehensive framework for analyzing interconnected systems. Extensive experiments on both synthetic and real-world networks demonstrate that EATSim effectively captures the underlying geometric similarities between interconnected networks, significantly improving the accuracy of interlayer similarity measurement. Moreover, EATSim achieves state-of-the-art performance in two downstream applications: predicting network robustness and network reducibility, showing its great potential in enhancing the understanding and management of complex systems.

Figures (5)

• Figure 1: A toy example illustrating the effectiveness of PED and AED.(a) Diagram of two networks with the same number of nodes but different topologies. (b) Visualization of network embeddings after dimensionality reduction to 2 using the Uniform Manifold Approximation and Projection (UMAP) method. (c) Visualization of network embeddings after aligning anchor nodes using an orthogonal transformation.
• Figure 2: Quantifying interlayer similarity in synthetic networks with EATSim. (a) EATSim values between the original network and 19 additional networks with varying rewiring probabilities. (b) Heatmap illustrating EATSim similarity across 20 BA networks, each labeled with its respective rewiring percentage. (c) EATSim shows a strong correlation with angular similarity in synthetic interconnected networks generated with GMM, with higher angular similarity corresponding to larger EATSim values and more similar topologies. (d) Similar to (c), but for varying radial correlation values with fixed angular correlation.
• Figure 3: Robustness $\Omega$ of multiplexes as a function of their EATSim values.(a) The correlation between robustness and EATSim for synthetic multiplex networks generated by the GMM, with differing sizes ranging from 2,000 to 5,000 for each interlayer angular correlation $g$. (b) The correlation between robustness and EATSim for 17 selected real-world multiplex networks.
• Figure 4: Network reducibility of four genetic networks evaluated using the distinguishability metric $q$.(a) In the Candida network, which consists of seven layers, reducing the layer number $m$ from seven to four enables EATSim to achieve the highest $q$ value of 0.711, outperforming the other four methods. (b) In the Gallus network, which comprises six layers, EATSim is one of the methods that increases the $q$ value, reaching a maximum of 0.577 after a single aggregation step. (c) and (d) For the Bos and Human-Herpes4 networks, each with four layers, EATSim attains the maximum $q$ values of 0.493 and 0.384, respectively, both achieved after the first aggregation step.
• Figure 5: Network reduction process on synthetic multilayer network.(a) EATSim can capture the similarity between layers with relatively large differences in rewiring rates, , rather than always aggregating layers with adjacent rewiring rates. (b) The hierarchical clustering procedure of JSD-based network reduction merges layers with adjacent rewiring rates preferentially. (c) The $q$ function is decreasing throughout the entire reduction process of both JSD and our EATSim, with the network reaching its optimal state when no layers are aggregated.