Abstract 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.

Abstract Networks representing social, biological, technological or other systems are often characterized by higher-order interaction involving any number of nodes. Temporal hypergraphs are given by ordered sequences of hyperedges representing sets of nodes interacting at given points in time. In this paper we discuss how a recently proposed model family for time-stamped hyperedges—relational hyperevent models (RHEM)—can be used to define tailored null distributions for temporal hypergraphs and to test and control for complex dependencies in hypergraph dynamics. RHEM can be specified with a given vector of temporal hyperedge statistics—functions that quantify the structural position of hyperedges in the history of previous hyperedges—and equate expected values of these statistics with their empirically observed values. This allows, for instance, to analyze the overrepresentation or underrepresentation of temporal hyperedge configurations in a model that reproduces the observed distributions of possibly complex sub-configurations, including but going beyond node degrees. Concrete examples include, but are not limited to, preferential attachment, repetition of subsets of any given size, triadic closure, homophily, and degree assortativity for subsets of any order.

Abstract Recruiting passive candidates, i.e. individuals not actively seeking jobs but open to compelling opportunities, remains one of the hardest challenges in digital recruitment. Motivated by a real collaboration with an industry partner, we introduce the Independent Halting Cascade (IHC) model: a simple but rich agent-based framework that couples network diffusion with the possibility of halting through job applications. Agents can either recommend vacancies to peers or apply themselves, and incentives increase the likelihood of recommendation, mobilizing otherwise passive candidates. The IHC bridges research on social network diffusion, coordinated task completion, and labor economics by modeling heterogeneous skills, job specificities, and network structures, including homophily. We derive analytical boundaries that characterize diffusion and failure regimes, and we show through simulations that the IHC reproduces the empirical chain-length distributions of Travers and Milgram and Dodds with only coarse calibration. Across synthetic (ER, BA, homophilic) and real networks (SMS, e-mail, Twitter), the IHC achieves comparable or higher success rates than direct-recommendation baselines, while requiring fewer applicants. Our findings suggest that the IHC captures core mechanisms of coordinated task completion, offering both a theoretical contribution and a practical foundation for recruitment systems designed to reach and engage passive candidates.

Abstract In this work, we present a probabilistic model for directed graphs where nodes have attributes and labels. This model serves as a generative classifier capable of predicting the labels of unseen nodes using either maximum likelihood or maximum a posteriori estimations. The predictions made by this model are highly interpretable, contrasting with some common methods for node classification, such as graph neural networks. We applied the model to two datasets, demonstrating predictive performance that is competitive with, and even superior to, state-of-the-art methods. One of the datasets considered is adapted from the Math Genealogy Project, which has not previously been utilized for this purpose. Consequently, we evaluated several classification algorithms on this dataset to compare the performance of our model and provide benchmarks for this new resource.

Abstract The robustness and resilience of complex systems are crucial for maintaining functionality amid disruptions or intentional attacks. Many such systems can be modeled as networks, where identifying structurally central nodes is essential for assessing their robustness and susceptibility to failure. Traditional centrality metrics often face challenges in identifying structurally important nodes in networks exhibiting heterogeneity at the network scale, with multilayer networks being a prime example of such networks. These metrics typically fail to balance the trade-off between capturing local layer-specific structures and integrating global multiplex connectivity. In this study, we extend DomiRank centrality, a metric that has been shown to effectively assess nodal importance across diverse monoplex topologies, to multiplex networks. Our approach combines layer-specific DomiRank calculations with a global contextualization step, incorporating multiplex-wide DomiRank scores to combine rankings. Through synthetic and real-world network studies, we demonstrate that our generalized DomiRank framework significantly improves the identification of key nodes in highly heterogeneous multiplex networks. This work advances centrality-based robustness assessments by addressing the fundamental trade-off between layer adaptability and multiplex-wide coherence.

Abstract Graph representation matrices are essential tools in graph data analysis. Recently, Hermitian adjacency matrices have been proposed to investigate directed graph structures. Previous studies have demonstrated that these matrices can extract valuable information for clustering. In this paper, we propose the complex non-backtracking matrix that integrates the properties of the Hermitian adjacency matrix and the non-backtracking matrix. The proposed matrix has similar properties with the non-backtracking matrix of undirected graphs. We reveal relationships between the complex non-backtracking matrix and the Hermitian adjacency matrix. Also, we provide intriguing insights that this matrix representation holds cluster information, particularly for sparse directed graphs.

Abstract Bipartite graphs, representing two-mode networks, arise in many research fields. These networks have two disjoint node sets representing distinct entity types, for example persons and groups, with edges representing associations between the two entity types. In bipartite graphs, the smallest possible cycle is a cycle of length four, and hence four-cycles are the smallest structure to model closure in such networks. Exponential-family random graph models (ERGMs) are a widely used model for social, and other, networks, including specifically bipartite networks. Existing ERGM terms to model four-cycles in bipartite networks, however, are relatively rarely used. In this work we demonstrate some problems with these existing terms to model four-cycles, and define new ERGM terms to help overcome these problems. The position of the new terms in the ERGM dependence hierarchy, and their interpretation, is discussed. The new terms are demonstrated in simulation experiments, and their application illustrated on a canonical example of an empirical two-mode network.

Abstract Social media platforms often foster environments where users primarily engage with content that aligns with their existing beliefs, thereby reinforcing their views and limiting exposure to opposing viewpoints. In this paper, we analyze X (formerly Twitter) discussions on climate change throughout 2019, using an unsupervised method centered on chambers –second-order information sources– to uncover ideological patterns at scale. Beyond direct connections, chambers capture shared sources of influence, revealing polarization dynamics efficiently and effectively. Analyzing retweet patterns, we identify echo chambers of climate believers and skeptics, revealing strong chamber overlap within ideological groups and minimal overlap between them, resulting in a robust bimodal structure that characterizes polarization. Our method enables us to infer the stance of high-impact users based on their audience’s chamber alignment, allowing for the classification of over half the retweeting population with minimal cross-group interaction, in what we term augmented echo chamber classification. We benchmark our approach against manual labeling and a state-of-the-art latent ideology model, finding comparable performance but with nearly four times greater coverage. Moreover, we find that echo chamber structures remain stable over time, even as their members change significantly, suggesting that these structures are a persistent and emergent property of the system. Notably, polarization decreases and climate skepticism rises during the #FridaysForFuture strikes in September 2019. This chamber-based analysis offers valuable insights into the persistence and fluidity of ideological polarization on social media.

Abstract The identification of community structure represents a central challenge in the field of complex networks. The objective is to ascertain the internal organization of agents within a network and to provide a representative network partition. Each community is presumed to comprise nodes sharing a common objective or property. The identification of these communities is typically based on the difference in connectivity density between the interior and border of a community. Indeed, nodes sharing a common purpose or property are expected to interact closely. Although this rule appears to be relevant, it nevertheless fails to address fundamental scientific problems such as disease module detection, thereby highlighting the inability to meaningfully determine communities based solely on connectivity for this situation. Consequently, another paradigm is necessary to formalize this problem accurately and detect these communities objectively. In this article, we propose a new framework to study this novel community formation property. Considering that colors represent shared properties, the problem is to maximize groups of nodes of the same color within communities. We propose a novel algorithm for detecting community structure based on a new measurement, chromatic entropy, which quantitatively assesses the community structure based on color constraint.

Abstract Detection of epileptic seizures is important for early diagnosis and treatment. It is known that the behavioral patterns of the brain in electroencephalogram (EEG) signals have huge and complex fluctuations. Diagnosing epilepsy by analyzing signals are costly process. Various methods are used to classify epileptic seizures. However, the inadequacy of these approaches in classifying signals makes it difficult to diagnose epilepsy. Complex network science produces effective solutions for analyzing interrelated structures. Using methods based on complex network analysis, it is possible to EEG signals analyze the relationship between signals and perform a classification process. In this study proposes a novel approach for classifying epileptic seizures by utilizing complex network science. In addition, unlike the studies in the literature, classification processes were carried out with lower dimensional signals by using 1-s EEG signals instead of 23.6-s full-size EEG signals. Using the topological properties of the EEG signal converted into a complex network, the classification process has been performed with the Jaccard Index method. The success of the classification process with the Jaccard Index was evaluated using Accuracy, F1 Score, Recall, and K-Fold metrics. In the results obtained, the signals of individuals with epileptic seizures were separated with an accuracy rate of 98.15%.