The mean field theory of multilayer neural networks centers around a particular infinite-width scaling, in which the learning dynamics is shown to be closely tracked by the mean field limit. A random fluctuation around this infinite-width limit is expected from a large-width expansion to the next order. This fluctuation has been studied only in the case of shallow networks, where previous works employ heavily technical notions or additional formulation ideas amenable only to that case. Treatment of the multilayer case has been missing, with the chief difficulty in finding a formulation that must capture the stochastic dependency across not only time but also depth. In this work, we initiate the study of the fluctuation in the case of multilayer networks, at any network depth.

The mean field theory of multilayer neural networks centers around a particular infinite-width scaling, in which the learning dynamics is shown to be closely tracked by the mean field limit. A random fluctuation around this infinite-width limit is expected from a large-width expansion to the next order. This fluctuation has been studied only in the case of shallow networks, where previous works employ heavily technical notions or additional formulation ideas amenable only to that case. Treatment of the multilayer case has been missing, with the chief difficulty in finding a formulation that must capture the stochastic dependency across not only time but also depth.In this work, we initiate the study of the fluctuation in the case of multilayer networks, at any network depth. Leveraging on the neuronal embedding framework recently introduced by Nguyen and Pham, we systematically derive a system of dynamical equations, called the second-order mean field limit, that captures the limiting fluctuation distribution. We demonstrate through the framework the complex interaction among neurons in this second-order mean field limit, the stochasticity with cross-layer dependency and the nonlinear time evolution inherent in the limiting fluctuation. A limit theorem is proven to relate quantitatively this limit to the fluctuation realized by large-width networks.We apply the result to show a stability property of gradient descent mean field training: in the large-width regime, along the training trajectory, it progressively biases towards a solution with minimal fluctuation (in fact, vanishing fluctuation) in the learned output function, even after the network has been initialized at or has converged (sufficiently fast) to a global optimum. This extends a similar phenomenon previously shown only for shallow networks with a squared loss in the empirical risk minimization setting, to multilayer networks with a loss function that is not necessarily convex in a more general setting.

One of the most persistent challenges in network science is the development of various synthetic graph models to support subsequent analyses. Among the most notable frameworks addressing this issue is the Artificial Benchmark for Community Detection (ABCD) model, a random graph model with community structure and power-law distribution for both degrees and community sizes. The model generates graphs similar to the well-known LFR model but it is faster, more interpretable, and can be investigated analytically. In this paper, we use the underlying ingredients of ABCD and introduce its variant, mABCD, thereby addressing the gap in models capable of generating multilayer networks. The uniqueness of the proposed approach lies in its flexibility at both levels of modelling: the internal structure of individual layers and the inter-layer dependencies, which together make the network a coherent structure rather than a collection of loosely coupled graphs. In addition to the conceptual description of the framework, we provide a comprehensive analysis of its efficient Julia implementation. Finally, we illustrate the applicability of mABCD to one of the most prominent problems in the area of complex systems: spreading phenomena analysis.

Historically, bilingualism was often perceived as an additional cognitive load that could hinder linguistic and intellectual development. However, over the last three decades, this view has changed considerably. Numerous studies have aimed to model and understand the architecture of the bilingual word recognition system Dijkstra and van Heuven (2002), investigating how parallel activation operates in the brain and how one language influences another Kroll et al. (2015). Increasingly, evidence suggests that multilinguals, individuals who speak three or more languages, can perform better than monolinguals in various linguistic and cognitive tasks, such as learning an additional language Abu-Rabia and Sanitsky (2010). This research proposal focuses on the study of the mental lexicon and how it may be structured in individuals who speak multiple languages. Building on the work of Stella et al. (2018), who investigated explosive learning in humans using a multiplex model of the mental lexicon, and the Bilingual Interactive Activation (BIA+) framework proposed by Dijkstra and van Heuven (2002), the present study applies the same multilayer network principles introduced by Kivelä et al. (2014). Our experimental design extends previous research by incorporating multimodality into the multiplex model, introducing an additional layer that connects visual inputs to their corresponding lexical representations across the multilingual layers of the mental lexicon. In this research, we aim to explore how a heritage language influences the acquisition of another language. Specifically, we ask: Does the presence of visual input in a translation task influence participants' proficiency and accuracy compared to text-only conditions?

Identifying super-spreaders can be framed as a subtask of the influence maximisation problem. It seeks to pinpoint agents within a network that, if selected as single diffusion seeds, disseminate information most effectively. Multilayer networks, a specific class of heterogeneous graphs, can capture diverse types of interactions (e.g., physical-virtual or professional-social), and thus offer a more accurate representation of complex relational structures. In this work, we introduce a novel approach to identifying super-spreaders in such networks by leveraging graph neural networks. To this end, we construct a dataset by simulating information diffusion across hundreds of networks - to the best of our knowledge, the first of its kind tailored specifically to multilayer networks. We further formulate the task as a variation of the ranking prediction problem based on a four-dimensional vector that quantifies each agent's spreading potential: (i) the number of activations; (ii) the duration of the diffusion process; (iii) the peak number of activations; and (iv) the simulation step at which this peak occurs. Our model, TopSpreadersNetwork, comprises a relationship-agnostic encoder and a custom aggregation layer. This design enables generalisation to previously unseen data and adapts to varying graph sizes. In an extensive evaluation, we compare our model against classic centrality-based heuristics and competitive deep learning methods. The results, obtained across a broad spectrum of real-world and synthetic multilayer networks, demonstrate that TopSpreadersNetwork achieves superior performance in identifying high-impact nodes, while also offering improved interpretability through its structured output.

Small and Medium-sized Enterprises (SMEs) are known to play a vital role in economic growth, employment, and innovation. However, they tend to face significant challenges in accessing credit due to limited financial histories, collateral constraints, and exposure to macroeconomic shocks. These challenges make an accurate credit risk assessment by lenders crucial, particularly since SMEs frequently operate within interconnected firm networks through which default risk can propagate. This paper presents and tests a novel approach for modelling the risk of SME credit, using a unique large data set of SME loans provided by a prominent financial institution. Specifically, our approach employs Graph Neural Networks to predict SME default using multilayer network data derived from common ownership and financial transactions between firms. We show that combining this information with traditional structured data not only improves application scoring performance, but also explicitly models contagion risk between companies. Further analysis shows how the directionality and intensity of these connections influence financial risk contagion, offering a deeper understanding of the underlying processes. Our findings highlight the predictive power of network data, as well as the role of supply chain networks in exposing SMEs to correlated default risk.

Link prediction in multilayer networks is a key challenge in applications such as recommendation systems and protein-protein interaction prediction. While many techniques have been developed, most rely on assumptions about shared structures and require access to raw auxiliary data, limiting their practicality. To address these issues, we propose a novel transfer learning framework for multilayer networks using a bi-level model averaging method. A $K$-fold cross-validation criterion based on edges is used to automatically weight inter-layer and intra-layer candidate models. This enables the transfer of information from auxiliary layers while mitigating model uncertainty, even without prior knowledge of shared structures. Theoretically, we prove the optimality and weight convergence of our method under mild conditions. Computationally, our framework is efficient and privacy-preserving, as it avoids raw data sharing and supports parallel processing across multiple servers. Simulations show our method outperforms others in predictive accuracy and robustness. We further demonstrate its practical value through two real-world recommendation system applications.

Understanding the complex interactions within dynamic multilayer networks is critical for advancements in various scientific domains. Existing models often fail to capture such networks' temporal and cross-layer dynamics. This paper introduces a novel Tensor State Space Model for Dynamic Multilayer Networks (TSSDMN), utilizing a latent space model framework. TSSDMN employs a symmetric Tucker decomposition to represent latent node features, their interaction patterns, and layer transitions. Then by fixing the latent features and allowing the interaction patterns to evolve over time, TSSDMN uniquely captures both the temporal dynamics within layers and across different layers. The model identifiability conditions are discussed. By treating latent features as variables whose posterior distributions are approximated using a mean-field variational inference approach, a variational Expectation Maximization algorithm is developed for efficient model inference. Numerical simulations and case studies demonstrate the efficacy of TSSDMN for understanding dynamic multilayer networks.

Graph Neural Networks (GNNs) have been widely used for various learning tasks, ranging from node classification to link prediction. They have demonstrated excellent performance in multiple domains involving graph-structured data. However, an important category of learning tasks, namely link weight prediction, has received less emphasis due to its increased complexity compared to binary link classification. Link weight prediction becomes even more challenging when considering multilayer networks, where nodes can be interconnected across multiple layers. To address these challenges, we propose a new method named Multiplex Spatial Graph Convolution Network (MSGCN), which spatially embeds information across multiple layers to predict interlayer link weights. Extensive experiments using data with known interlayer link information show that the MSGCN model has robust, accurate, and generalizable link weight prediction performance across a wide variety of multiplex network structures.

Many researchers have considered multi-agent systems over single-layer networks as models for studying diffusion phenomena. Since real-world networks involve connections between agents with different semantics (e.g., family member, friend, colleague), the study of multi-agent systems over multilayer networks has assumed importance. Our focus is on one class of multi-agent system models over multilayer networks, namely multilayer synchronous dynamical systems (MSyDSs). We study several fundamental problems for this model. We establish properties of the phase spaces of MSyDSs and bring out interesting differences between single-layer and multilayer dynamical systems. We show that, in general, the problem of determining whether two given MSyDSs are inequivalent is NP-complete. This hardness result holds even when the only difference between the two systems is the local function at just one node in one layer. We also present efficient algorithms for the equivalence problem for restricted versions of MSyDSs (e.g., systems where each local function is a bounded-threshold function, systems where the number of layers is fixed and each local function is symmetric). In addition, we investigate the expressive power of MSyDSs based on the number of layers. In particular, we examine conditions under which a system with k >= 2 layers has an equivalent system with k-1 or fewer layers.