The bibliography of scientific papers lists items with variable degree of relevance for the contents of the paper itself. If we could identify the sources, i.e., the works that actually inspired the paper, their citations can help us uncover the genesis of scientific projects and would be more representative of the actual importance of papers and authors than the standard citation counts, when all references are considered. Here we present an analysis of the backbone of science, i.e., the network of citations between papers and their sources. The latter are extracted from the full body of papers via Large Language Models (LLMs), which are currently very capable of correctly identifying the context in which a paper is cited. Using two different but related prompts, we find that the LLMs select only a small set of references, not taken at random, and that the resulting backbone networks are quite similar to each other with respect to their in-degree distributions, modularity, transitivity, and degree correlations. Backbone networks have higher heterogeneity in their in-degree distributions, compared to the full network, but the most cited papers are usually the same, with some important exceptions. Citation rankings among authors are also remarkably stable. We conclude that the full citation network, despite its redundancy with respect to the backbones, presents a reliable picture of the relative citation impact of papers and authors.

When searching for communities in networks, domain experts may have some prior expectations about the size of communities. Yet, community detection methods normally do not optimize communities under cluster size constraints. Multi-resolution techniques allow users to indirectly control the average community size through changing a resolution parameter, but this practice does not control the size of individual communities. We here study the problem of size-constrained community detection, where the size of all communities is limited to a user-specified range of values, in the context of modularity optimization. We propose a heuristic for modularity optimization under community size constraints. To demonstrate the reliability of our proposed heuristic, we also formulate an exact integer optimization model and use its results as a baseline. Our analysis based on synthetic benchmarks and real networks demonstrate the issues with the currently common practice of changing resolution parameters and reveal the advantages of the proposed methods as a principled way of obtaining size-constrained communities. The proposed method is publicly available in the Python Leiden algorithm package.

Single cell RNA-seq (scRNA-seq) technologies provide unprecedented resolution representing transcriptomics at the level of single cell. One of the biggest challenges in scRNA-seq data analysis is the cell type annotation, which is usually inferred by cell separation approaches. In-silico algorithms that accurately identify individual cell types in ongoing single-cell sequencing studies are crucial for unlocking cellular heterogeneity and understanding the biological basis of diseases. In this study, we focus on robustly identifying cell types in single-cell RNA sequencing data; we conduct a comparative analysis using methods established in biology, like Seurat, Leiden, and WGCNA, as well as network-based methods Infomap, statistical inference via Stochastic Block Models (SBM), and single-cell Graph Neural Networks (scGNN). We also analyze preprocessing pipelines to identify and optimize key components in the process, explicitly considering their role in mitigating inherent data noise and potential batch effects for robust cell type identification. Leveraging three independent datasets, PBMC, ROSMAP, and MOp, we employ clustering algorithms on cell-cell networks derived from gene expression data. Our findings reveal that clusters identified by multiresolution Infomap and Leiden show a closer alignment, with Infomap standing out as a particularly effective approach. Infomap notably offers valuable insights for the precise characterization of cellular landscapes related to neurodegeneration and immunology in scRNA-seq.

Multilayer network science has emerged as a central framework for analysing interconnected and interdependent complex systems. Its relevance has grown substantially with the increasing availability of rich, heterogeneous data, which makes it possible to uncover and exploit the inherently multilayered organisation of many real-world networks. In this review, we summarise recent developments in the field. On the theoretical and methodological front, we outline core concepts and survey advances in community detection, dynamical processes, temporal networks, higher-order interactions, and machine-learning-based approaches. On the application side, we discuss progress across diverse domains, including interdependent infrastructures, spreading dynamics, computational social science, economic and financial systems, ecological and climate networks, science-of-science studies, network medicine, and network neuroscience. We conclude with a forward-looking perspective, emphasizing the need for standardised datasets and software, deeper integration of temporal and higher-order structures, and a transition toward genuinely predictive models of complex systems.

Neural networks excel across a wide range of tasks, yet remain black boxes. In particular, how their internal representations are shaped by the complexity of the input data and the problems they solve remains obscure. In this work, we introduce a suite of five data-agnostic probes-pruning, binarization, noise injection, sign flipping, and bipartite network randomization-to quantify how task difficulty influences the topology and robustness of representations in multilayer perceptrons (MLPs). MLPs are represented as signed, weighted bipartite graphs from a network science perspective. We contrast easy and hard classification tasks on the MNIST and Fashion-MNIST datasets. We show that binarizing weights in hard-task models collapses accuracy to chance, whereas easy-task models remain robust. We also find that pruning low-magnitude edges in binarized hard-task models reveals a sharp phase-transition in performance. Moreover, moderate noise injection can enhance accuracy, resembling a stochastic-resonance effect linked to optimal sign flips of small-magnitude weights. Finally, preserving only the sign structure-instead of precise weight magnitudes-through bipartite network randomizations suffices to maintain high accuracy. These phenomena define a model- and modality-agnostic measure of task complexity: the performance gap between full-precision and binarized or shuffled neural network performance. Our findings highlight the crucial role of signed bipartite topology in learned representations and suggest practical strategies for model compression and interpretability that align with task complexity.

We investigate the robustness of sparse artificial neural networks trained with adaptive topology. We focus on a simple yet effective architecture consisting of three sparse layers with 99

Most networks encountered in nature, society, and technology have weighted edges, representing the strength of the interaction or association between their vertices. Randomizing the structure of a network is a classic procedure used to estimate the statistical significance of properties of the network such as transitivity, centrality, and community structure. Randomization of weighted networks has traditionally been done via the weighted configuration model (WCM), a simple extension of the configuration model, where weights are interpreted as bundles of edges. It has previously been shown that the ensemble of randomizations provided by the WCM is affected by the specific scale used to compute the weights, but the consequences for statistical significance were unclear. Here we find that statistical significance based on the WCM is scale dependent, whereas in most cases results should be independent of the choice of the scale. A two-step approach, originally introduced for network reconstruction, in which one first randomizes the structure and then the weights, with a suitable distribution, restores scale invariance and allows us to conduct unbiased assessments of significance on weighted networks.

In the attention economy, understanding how individuals manage limited attention is critical. We introduce a simple model describing the decay of a user’s engagement when facing multiple inputs. Assuming that active commenters receive more attention, we analytically show that individual attention decay is determined by the overall duration of interactions, not their number or user activity. Our model is validated using data from Reddit’s Change My View subreddit, where the user’s attention dynamics is explicitly traceable. Despite its simplicity, our model offers a crucial microscopic perspective complementing macroscopic studies. In contemporary digital environments, it is important to understand how an individual’s attention decays with time when confronted with numerous online social interactions. Here, the authors show that this decay is fully driven by the total duration of the interactions, independent of thread size and user activity.

We study the necessary condition to detect, by means of spectral modularity optimization, the ground-truth partition in networks generated according to the weighted planted-partition model with two equally sized communities. We analytically derive a general expression for the maximum level of mixing tolerated by the algorithm to retrieve community structure, showing that the value of this detectability threshold depends on the first two moments of the distributions of node degree and edge weight. We focus on the standard case of Poisson-distributed node degrees and compare the detectability thresholds of five edge-weight distributions: Dirac, Poisson, exponential, geometric, and signed Bernoulli. We show that Dirac distributed weights yield the smallest detectability threshold, while exponentially distributed weights increase the threshold by a factor of sqrt[2], with other distributions exhibiting distinct behaviors that depend, either or both, on the average values of the degree and weight distributions. Our results indicate that larger variability in edge weights can make communities less detectable. In cases where edge weights carry no information about community structure, incorporating edge weights in community detection is detrimental.

Human mobility plays a crucial role in the spread of human diseases, but is rarely quantified in plant disease epidemics. To address this gap, we integrate a unique, high-resolution network of human movements in New Zealand with a metapopulation model to mechanistically simulate pathogen transmission. We calibrate the model on the nationwide 2010 kiwifruit vine disease (Psa-V) outbreak, and show that it accurately reproduces the observed spatiotemporal spread, confirming that the human mobility network is a strong foundation for modeling transmission dynamics. By analyzing spatial infection trends, we find that most dispersal occurs locally, as often illustrated in the plant-outbreak literature. However, sporadic long-range connections are necessary to model a nationwide outbreak. Using the model as an in-silico laboratory, we demonstrate that enhanced surveillance accelerates detection and that outbreak severity is highly sensitive to the timing and location of initial disease importation. We observe a potential causal link between seasonal labor patterns and epidemic risk in high-traffic seasons. This study provides a robust, data-driven framework for modeling and predicting the spatiotemporal spread of agricultural pathogens. It underscores the importance of leveraging human mobility networks to design timely interventions and surveillance systems, protecting global food security.

A key challenge in network science is the detection of communities, which are sets of nodes in a network that are densely connected internally but sparsely connected to the rest of the network. A fundamental result in community detection is the existence of a nontrivial threshold for community detectability on sparse graphs that are generated by the planted partition model (PPM). Below this so-called “detectability limit”, no community-detection method can perform better than random chance. Spectral methods for community detection fail before this detectability limit because the eigenvalues corresponding to the eigenvectors that are relevant for community detection can be absorbed by the bulk of the spectrum. One can bypass the detectability problem by using special matrices, like the non-backtracking matrix, but this requires one to consider higher-dimensional matrices. In this paper, we show that the difference in graph energy between a PPM and an Erdős–Rényi (ER) network has a distinct transition at the detectability threshold even for the adjacency matrices of the underlying networks. The graph energy is based on the full spectrum of an adjacency matrix, so our result suggests that standard graph matrices still allow one to separate the parameter regions with detectable and undetectable communities.

We develop an algorithm that finds the consensus among many different clustering solutions of a graph. We formulate the problem as a median set partitioning problem and propose a greedy optimization technique. Unlike other approaches that find median set partitions, our algorithm takes graph structure into account and finds a comparable quality solution much faster than the other approaches. For graphs with known communities, our consensus partition captures the actual community structure more accurately than alternative approaches. To make it applicable to large graphs, we remove sequential dependencies from our algorithm and design a parallel algorithm. Our parallel algorithm achieves 35x speedup when utilizing 64 processing cores for large real-world graphs representing mass cytometry data from single-cell experiments.

Recent advances in machine learning offer new ways to represent and study scholarly works and the space of knowledge. Graph and text embeddings provide a convenient vector representation of scholarly works based on citations and text. Yet, it is unclear whether their representations are consistent or provide different views of the structure of science. Here, we compare graph and text embedding by testing their ability to capture the hierarchical structure of the Physics and Astronomy Classification Scheme (PACS) of papers published by the American Physical Society (APS). We also provide a qualitative comparison of the overall structure of the graph and text embeddings for reference. We find that neural network-based methods outperform traditional methods and graph embedding methods such as node2vec are better than other methods at capturing the PACS structure. Our results call for further investigations into how different contexts of scientific papers are captured by different methods, and how we can combine and leverage such information in an interpretable manner.

The COVID-19 pandemic shifted academic collaboration from in-person to remote interactions. This study explores, for the first time, the effects on scientific collaborations and impact of such a shift, comparing research output before, during, and after the pandemic. Using large-scale bibliometric data, we track the evolution of collaboration networks and the resulting impact of research over time. Our findings are twofold: first, the geographic distribution of collaborations significantly shifted, with a notable increase in cross-border partnerships after 2020, indicating a reduction in the constraints of geographic proximity. Second, despite the expansion of collaboration networks, there was a concerning decline in citation impact, suggesting that the absence of spontaneous in-person interactions-which traditionally foster deep discussions and idea exchange-negatively affected research quality. As hybrid work models in academia gain traction, this study highlights the need for universities and research organizations to carefully consider the balance between remote and in-person engagement.

Understanding how misinformation affects the spread of disease is crucial for public health, especially given recent research indicating that misinformation can increase vaccine hesitancy and discourage vaccine uptake. However, it is difficult to investigate the interaction between misinformation and epidemic outcomes due to the dearth of data-informed holistic epidemic models. Here, we employ an epidemic model that incorporates a large, mobility-informed physical contact network as well as the distribution of misinformed individuals across counties derived from social media data. The model allows us to simulate various scenarios to understand how epidemic spreading can be affected by misinformation spreading through one particular social media platform. Using this model, we compare a worst-case scenario, in which individuals become misinformed after a single exposure to low-credibility content, to a best-case scenario where the population is highly resilient to misinformation. We estimate the additional portion of the U.S. population that would become infected over the course of the COVID-19 epidemic in the worst-case scenario. This work can provide policymakers with insights about the potential harms of exposure to online vaccine misinformation.

We study the structure of personal relationships among 1068 high school students using a dataset that contains the network of self-reported friendly and conflictive relationships, with information on their directionality and intensity. We analyse the resulting weighted, directed, and signed network using a Bayesian stochastic block model framework, which enables the inference of group structure without imposing prior assumptions on the role of negative or asymmetric ties. While a full model incorporating all edge attributes yields statistically coherent clusters, these do not align with socially meaningful communities. To address this, we focus first on the network backbone of mutual affinities, and we characterize its group organization. Many communities display an assortative structure, often embedded within larger cohesive configurations, but we also observe more diverse patterns such as core-periphery structure and isolated nodes. We then examine how relationship intensity, directionality, and conflict shape group structure. Asymmetric ties, though often occurring between communities, are frequently present within them, revealing the stabilizing effect of group membership on non-mutual relationships. Furthermore, the presence of asymmetric ties does not inherently imply a hierarchical structure, given that all groups both receive and report significant levels of non-reciprocal ties. More intense ties play a disproportionate role in shaping community structure. Finally, negative ties tend to bridge communities, but we find that groups feature a significant level of internal conflict. Our research offers a new perspective on the study of group organization when rich information about the directionality, the intensity and the sign of ties is considered, with implications for identifying social vulnerability and designing targeted interventions.

The network density matrix (NDM) framework, enabling an information-theoretic and multiscale treatment of network flow, has been gaining momentum over the last decade. Benefiting from the counterparts of physical functions such as free energy and entropy, NDM's applications range from estimating how nodes influence network flows across scales the centrality of nodes at the local level to explaining the emergence of structural and functional order. Here, we introduce a generalized notion of the network internal energy E_τ, where τ denotes a temporal hyperparameter allowing for multi-resolution analysis, showing how it measures the leakage of dynamical correlations from arbitrary partitions, where the minimally leaky subsystems have minimal E_τ. Moreover, we analytically demonstrate that E_τ reduces to the well-known modularity function at the smallest temporal scale τ= 0. We investigate this peculiar resemblance by comparing the communities minimizing E_τ, with those detected by widely used methods like multiscale modularity and Markov stability. Our work provides a detailed analytical and computational picture of network generalized internal energy, and explores its effectiveness in detecting communities in synthetic and empirical networks within a unifying framework.

Deep neural networks and brains both learn and share superficial similarities: processing nodes are likened to neurons and adjustable weights are likened to modifiable synapses. But can a unified theoretical framework be found to underlie them both? Here we show that the equations used to describe neuronal avalanches in living brains can also be applied to cascades of activity in deep neural networks. These equations are derived from non-equilibrium statistical physics and show that deep neural networks learn best when poised between absorbing and active phases. Because these networks are strongly driven by inputs, however, they do not operate at a true critical point but within a quasi-critical regime – one that still approximately satisfies crackling noise scaling relations. By training networks with different initializations, we show that maximal susceptibility is a more reliable predictor of learning than proximity to the critical point itself. This provides a blueprint for engineering improved network performance. Finally, using finite-size scaling we identify distinct universality classes, including Barkhausen noise and directed percolation. This theoretical framework demonstrates that universal features are shared by both biological and artificial neural networks.

Since the COVID-19 pandemic, online grocery shopping has rapidly reshaped consumer behavior worldwide, fueled by ever-faster delivery promises aimed at maximizing convenience. Yet, this growth has also substantially increased urban traffic congestion, emissions, and pollution. Despite extensive research on urban delivery optimization, little is known about the trade-off between individual convenience and these societal costs. In this study, we investigate the value of marginal extensions in delivery times, termed customer patience, in mitigating the traffic burden caused by grocery deliveries. We first conceptualize the problem and present a mathematical model that highlights a convex relationship between patience and traffic congestion. The theoretical predictions are confirmed by an extensive, network-science based analysis leveraging two large-scale datasets encompassing over 8 million grocery orders in Dubai. Our findings reveal that allowing just five additional minutes in delivery time reduces daily delivery mileage by approximately 30 percent and life-cycle CO2 emissions by 20 percent. Beyond ten minutes of added patience, however, marginal benefits diminish significantly. These results highlight that modest increases in consumer patience can deliver substantial gains in traffic reduction and sustainability, offering a scalable strategy to balance individual convenience with societal welfare in urban delivery systems.

Collaboration is a key driver of science and innovation. Mainly motivated by the need to leverage different capacities and expertise to solve a scientific problem, collaboration is also an excellent source of information about the future behavior of scholars. In particular, it allows us to infer the likelihood that scientists choose future research directions via the intertwined mechanisms of selection and social influence. Here we thoroughly investigate the interplay between collaboration and topic switches. We find that the probability for a scholar to start working on a new topic increases with the number of previous collaborators, with a pattern showing that the effects of individual collaborators are not independent. The higher the productivity and the impact of authors, the more likely their coworkers will start working on new topics. The average number of coauthors per paper is also inversely related to the topic switch probability, suggesting a dilution of this effect as the number of collaborators increases.