Vito Latora

Emanuele La Malfa, Gabriele La Malfa, Claudio Caprioli et al. · oxford

Deep Neural Networks are, from a physical perspective, graphs whose `links` and `vertices` iteratively process data and solve tasks sub-optimally. We use Complex Network Theory (CNT) to represents Deep Neural Networks (DNNs) as directed weighted graphs: within this framework, we introduce metrics to study DNNs as dynamical systems, with a granularity that spans from weights to layers, including neurons. CNT discriminates networks that differ in the number of parameters and neurons, the type of hidden layers and activations, and the objective task. We further show that our metrics discriminate low vs. high performing networks. CNT is a comprehensive method to reason about DNNs and a complementary approach to explain a model's behavior that is physically grounded to networks theory and goes beyond the well-studied input-output relation.

Randall Martyr, Benjamin Schaefer, Christian Beck et al.

As the energy transition transforms power grids across the globe, it poses several challenges regarding grid design and control. In particular, high levels of intermittent renewable generation complicate the task of continuously balancing power supply and demand, requiring sufficient control actions. Although there exist several proposals to control the grid, most of them have not demonstrated to be cost efficient in terms of optimal control theory. Here, we mathematically formulate an optimal centralized (therefore non-local) control problem for stable operation of power grids and determine the minimal amount of active power necessary to guarantee a stable service within the operational constraints, minimizing a suitable cost function at the same time. This optimal control can be used to benchmark control proposals and we demonstrate this benchmarking process by investigating the performance of three distributed controllers, two of which are fully decentralized, that have been recently studied in the physics and power systems engineering literature. Our results show that cost efficient controllers distribute the controlled response amongst all nodes in the power grid. Additionally, superior performance can be achieved by incorporating sufficient information about the disturbance causing the instability. Overall, our results can help design and benchmark secure and cost-efficient controllers.

Qi Shao, Duxin Chen, Jiawen Chen et al.

Predicting the behavior of ultra-large complex systems, from climate to biological and technological networks, is a central unsolved challenge. Existing approaches face a fundamental trade-off: equation discovery methods provide interpretability but fail to scale, while neural networks scale but operate as black boxes and often lose reliability over long times. Here, we introduce the Sparse Identification Graph Neural Network, a framework that overcome this divide by allowing to infer the governing equations of large networked systems from data. By defining symbolic discovery as edge-level information, SIGN decouples the scalability of sparse identification from network size, enabling efficient equation discovery even in large systems. SIGN allows to study networks with over 100,000 nodes while remaining robust to noise, sparse sampling, and missing data. Across diverse benchmark systems, including coupled chaotic oscillators, neural dynamics, and epidemic spreading, it recovers governing equations with high precision and sustains accurate long-term predictions. Applied to a data set of time series of temperature measurements in 71,987 sea surface positions, SIGN identifies a compact predictive network model and captures large-scale sea surface temperature conditions up to two years in advance. By enabling equation discovery at previously inaccessible scales, SIGN opens a path toward interpretable and reliable prediction of real-world complex systems.

Ziwei Zheng, Huizhi Liang, Vaclav Snasel et al.

We scrutinize the structural and operational aspects of deep learning models, particularly focusing on the nuances of learnable parameters (weight) statistics, distribution, node interaction, and visualization. By establishing correlations between variance in weight patterns and overall network performance, we investigate the varying (optimal and suboptimal) performances of various deep-learning models. Our empirical analysis extends across widely recognized datasets such as MNIST, Fashion-MNIST, and CIFAR-10, and various deep learning models such as deep neural networks (DNNs), convolutional neural networks (CNNs), and vision transformer (ViT), enabling us to pinpoint characteristics of learnable parameters that correlate with successful networks. Through extensive experiments on the diverse architectures of deep learning models, we shed light on the critical factors that influence the functionality and efficiency of DNNs. Our findings reveal that successful networks, irrespective of datasets or models, are invariably similar to other successful networks in their converged weights statistics and distribution, while poor-performing networks vary in their weights. In addition, our research shows that the learnable parameters of widely varied deep learning models such as DNN, CNN, and ViT exhibit similar learning characteristics.

Luca Gallo, Vito Latora, Alfredo Pulvirenti

Research on graph representation learning has received great attention in recent years. However, most of the studies so far have focused on the embedding of single-layer graphs. The few studies dealing with the problem of representation learning of multilayer structures rely on the strong hypothesis that the inter-layer links are known, and this limits the range of possible applications. Here we propose MultiSAGE, a generalization of the GraphSAGE algorithm that allows to embed multiplex networks. We show that MultiSAGE is capable to reconstruct both the intra-layer and the inter-layer connectivity, outperforming GraphSAGE, which has been designed for simple graphs. Next, through a comprehensive experimental analysis, we shed light also on the performance of the embedding, both in simple and in multiplex networks, showing that either the density of the graph or the randomness of the links strongly influences the quality of the embedding.

Emanuele La Malfa, Gabriele La Malfa, Giuseppe Nicosia et al. · oxford

Deep Neural Networks (DNNs) can be represented as graphs whose links and vertices iteratively process data and solve tasks sub-optimally. Complex Network Theory (CNT), merging statistical physics with graph theory, provides a method for interpreting neural networks by analysing their weights and neuron structures. However, classic works adapt CNT metrics that only permit a topological analysis as they do not account for the effect of the input data. In addition, CNT metrics have been applied to a limited range of architectures, mainly including Fully Connected neural networks. In this work, we extend the existing CNT metrics with measures that sample from the DNNs' training distribution, shifting from a purely topological analysis to one that connects with the interpretability of deep learning. For the novel metrics, in addition to the existing ones, we provide a mathematical formalisation for Fully Connected, AutoEncoder, Convolutional and Recurrent neural networks, of which we vary the activation functions and the number of hidden layers. We show that these metrics differentiate DNNs based on the architecture, the number of hidden layers, and the activation function. Our contribution provides a method rooted in physics for interpreting DNNs that offers insights beyond the traditional input-output relationship and the CNT topological analysis.

Emanuele La Malfa, Gabriele La Malfa, Giuseppe Nicosia et al.

In this paper, we interpret Deep Neural Networks with Complex Network Theory. Complex Network Theory (CNT) represents Deep Neural Networks (DNNs) as directed weighted graphs to study them as dynamical systems. We efficiently adapt CNT measures to examine the evolution of the learning process of DNNs with different initializations and architectures: we introduce metrics for nodes/neurons and layers, namely Nodes Strength and Layers Fluctuation. Our framework distills trends in the learning dynamics and separates low from high accurate networks. We characterize populations of neural networks (ensemble analysis) and single instances (individual analysis). We tackle standard problems of image recognition, for which we show that specific learning dynamics are indistinguishable when analysed through the solely Link-Weights analysis. Further, Nodes Strength and Layers Fluctuations make unprecedented behaviours emerge: accurate networks, when compared to under-trained models, show substantially divergent distributions with the greater extremity of deviations. On top of this study, we provide an efficient implementation of the CNT metrics for both Convolutional and Fully Connected Networks, to fasten the research in this direction.

A. Fiasconaro, M. Tumminello, V. Nicosia et al.

We propose here two new recommendation methods, based on the appropriate normalization of already existing similarity measures, and on the convex combination of the recommendation scores derived from similarity between users and between objects. We validate the proposed measures on three relevant data sets, and we compare their performance with several recommendation systems recently proposed in the literature. We show that the proposed similarity measures allow to attain an improvement of performances of up to 20\% with respect to existing non-parametric methods, and that the accuracy of a recommendation can vary widely from one specific bipartite network to another, which suggests that a careful choice of the most suitable method is highly relevant for an effective recommendation on a given system. Finally, we studied how an increasing presence of random links in the network affects the recommendation scores, and we found that one of the two recommendation algorithms introduced here can systematically outperform the others in noisy data sets.