Source detection is crucial for capturing the dynamics of real-world infectious diseases and informing effective containment strategies. Most existing approaches to source detection focus on conventional pairwise networks. However, emerging studies on the mathematical modeling and analysis of empirical contact data reveal that group-based interaction patterns, captured naturally by hypergraph representations, constitute a significant portion of infection events and are reshaping our understanding of epidemic propagation in real-world populations. In the present study, we propose a message passing algorithm, called the HDMPN, for source detection for a stochastic susceptible-infectious dynamics in which infection events within the hyperedge occur in a correlated manner. The HDMPN modifies the likelihood maximization with the use of the proportion of infectious neighbors, thus incorporating the information on hyperedges. We numerically show that, in most cases, the HDMPN outperforms benchmarks, including the likelihood maximization method without modification.

Trust and reciprocation of it form the foundation of economic, social and other interactions. While the Trust Game is widely used to study these concepts for interactions between two players, often alternating different roles (i.e., investor and trustee), its extensions to multi-player scenarios have been restricted to instances where players assume only one role. We propose a symmetric N-player Trust Game, in which players alternate between two roles, and the payoff of the player is defined as the average across their two roles and drives the evolutionary game dynamics. We find that prosocial strategies are harder to evolve with the present symmetric N-player Trust Game than with the Public Goods Game, which is well studied. In particular, trust fails to evolve regardless of payoff function nonlinearity in well-mixed populations in the case of the symmetric N-player trust game. In structured populations, nonlinear payoffs can have strong impacts on the evolution of trust. The same nonlinearity can yield substantially different outcomes, depending on the nature of the underlying network. Our results highlight the importance of considering both payoff structures and network topologies in understanding the emergence and maintenance of prosocial behaviours.

Abstract Humans are deeply social, and in early life, classroom friendships shape academic and socio-emotional development. Classroom peer ecologies describe the social structure in which individual relationships are situated and which form the context for student learning and development within these classrooms. Yet, research has only begun to explore how these global social network structures are related to classroom-level outcomes. Using a large-scale data set of 35,038 students in 1,433 classrooms, we extracted 314 social network metrics from four different network types (i.e., friendship, rejection, help, and break-time contacts) and applied machine learning to predict various classroom-level outcomes, including academic achievement, motivation, and social integration. Classroom networks predicted both academic achievement (up to 2% of variance) and well-being (up to 18% of variance), beyond school tracks, socioeconomic status, and demographic factors. However, they did not significantly improve predictions for other characteristics of the teaching and learning environment—such as instructional methods and motivational characteristics. The predictive power for different outcomes was highly dependent upon the type of network and categories of network metrics. These insights position social networks as a powerful tool for understanding aspects of educational environments and for informing network-based interventions.

Understanding the dynamics of financial transactions among people is critical for various applications such as fraud detection. One important aspect of financial transaction networks is temporality. The order and repetition of transactions can offer new insights when considered within the graph structure. Temporal motifs, defined as a set of nodes that interact with each other in a short time period, are a promising tool in this context. In this work, we study three unique temporal financial networks: transactions in Mercari, an online marketplace, payments in a synthetic network generated by J.P. Morgan Chase, and payments and friendships among Venmo users. We consider the fraud detection problem on the Mercari and J.P. Morgan Chase networks, for which the ground truth is available. We show that temporal motifs offer superior performance to several baselines, including a previous method that considers simple graph features and two node embedding techniques (LINE and node2vec), while being practical in terms of runtime performance. For the Venmo network, we investigate the interplay between financial and social relations on three tasks: friendship prediction, vendor identification, and analysis of temporal cycles. For friendship prediction, temporal motifs yield better results than general heuristics, such as Jaccard and Adamic-Adar measures. We are also able to identify vendors with high accuracy and observe interesting patterns in rare motifs, such as temporal cycles. We believe that the analysis, datasets, and lessons from this work will be beneficial for future research on financial transaction networks.

Various theoretical and empirical studies have accounted for why humans cooperate in competitive environments. Although prior work has revealed that network structure and multiplex interactions can promote cooperation, most theory assumes that individuals play similar dilemma games in all social contexts. However, real-world agents may participate in a diversity of interactions, not all of which present dilemmas. We develop an evolutionary game model on multilayer networks in which one layer supports the prisoner's dilemma game, while the other follows constant-selection dynamics, representing biased but non-dilemmatic competition, akin to opinion or fad spreading. Our theoretical analysis reveals that coupling a social dilemma layer to a non-dilemmatic constant-selection layer robustly enhances cooperation in many cases, across different multilayer networks, updating rules, and payoff schemes. These findings suggest that embedding individuals within diverse networked settings-even those unrelated to direct social dilemmas-can be a principled approach to engineering cooperation in socio-ecological and organizational systems.

Anticipating tipping points in complex systems is a fundamental challenge across domains. Traditional early warning signals (EWSs) based on critical slowing down, such as increasing sample variance, are widely used, but their ability to reliably indicate imminent bifurcations and forecast their timing remains limited. Here, we introduce TIPMOC (TIpping via Power-law fits and MOdel Comparison), a parametric framework designed to statistically detect the approach of a bifurcation and estimate its future location using only the sample variance. TIPMOC exploits the mathematical property that variance diverges with a characteristic power-law form near codimension-one bifurcations. By sequentially monitoring system variance as a control parameter changes, TIPMOC statistically adjudicates between linear and power-law divergence at each step. When evidence favors power-law divergence, TIPMOC forecasts the impending tipping point and estimates its position; otherwise, it avoids false positives. Through numerical simulations, we demonstrate TIPMOC's robustness and accuracy in both detection and timing prediction across different types of dynamics and bifurcation, whereas the accuracy of timing prediction is limited. TIPMOC shows low false positive rates and performs well even with uneven sampling and colored noise. This method thus enhances the interpretability and practical utility of classical EWSs, serving as both a transparent add-on and a stand-alone statistical tool for forecasting regime shifts in diverse complex systems.

Spatial early warning signals (EWSs) seek evidence of an approaching tipping point from a single snapshot of many interacting elements. Existing theory largely assumes spatial homogeneity, whereas networks introduce systematic differences among nodes that may obscure fluctuation-based warning signals. We develop a mathematical framework for spatial EWSs in stochastic dynamical systems on networks. We find that the expected spatial variance, a popular spatial EWS, decomposes exactly into a structural contribution from heterogeneity in the equilibrium state and a fluctuation contribution determined by the stationary covariance. Near a simple steady-state bifurcation, the potentially divergent covariance concentrates along the critical eigendirection: the left eigenvector determines how strongly noise excites the critical fluctuation, while the right eigenvector determines its spatial pattern. Consequently, the spatial variance has a divergent fluctuation contribution when the limiting critical eigendirection is noise-excited and spatially nonuniform after centering. In contrast, the spatial coefficient of variation generally saturates, while skewness, kurtosis, and Moran's I approach network-dependent limits without a universal warning direction. We also derive results for homogeneous networks, node-wise baseline subtraction as preprocessing, and Hopf bifurcations, for which the limiting distributions are qualitatively different. These results clarify when spatial EWSs provide reliable warnings and why their performance depends on network structure, noise, and preprocessing.

Real-life systems often experience regime shifts. An early warning signal (EWS) is a quantity that attempts to anticipate such a regime shift. Because complex systems of practical interest showing regime shifts are often dynamics on networks, a research interest is to design EWSs for networks, including determining sentinel nodes that are useful for constructing high-quality EWSs. Previous work has shown that the sample variance is a viable EWS including in the case of networks. We explore the use of the sample covariance of two nodes, or sentinel node pairs, for improving EWSs for networks. We perform analytical calculations in four-node networks and numerical simulations in larger networks to find that the sample covariance and its combination over node pairs is inferior to the sample variance and its combination over nodes; the latter are previously proposed EWSs based on sentinel node selection. The present results support the predominant use of diagonal entries of the covariance matrix (i.e., variance) as opposed to off-diagonal entries in EWS construction.

Many social, biological, and technological systems are recorded as sequences of time-stamped interactions. In such systems, concurrency, i.e., the tendency for an individual to participate in multiple interactions approximately at the same time, can strongly affect processes such as epidemic or information spreading. However, concurrency measures for event-based temporal network data are not established. We introduce edge-event correlation (EEC), a simple and interpretable measure that quantifies how similarly two connections are active over time. We apply EEC to empirical temporal networks and temporal hypergraphs, the latter allowing single events to involve more than two nodes. Across most datasets, pairs of edges or hyperedges that share a node show higher concurrency than pairs that do not. We further find that this elevated concurrency is mainly driven by pairs embedded in closed local structures, such as triangles in the aggregated network. EEC provides a practical tool for quantifying concurrency in event-based temporal data and may help identify network structures that facilitate rapid spreading or collective dynamics.

Temporal hypergraphs capture time-resolved group interactions among nodes. Empirical data support that time-stamped group interactions show bursty event sequences and non-trivial temporal correlations. In the present study, we introduce node-driven temporal hypergraph models in which each node stochastically alternates between low- and high-activity states, and a hyperedge produces time-stamped events with a probability that depends on the number of high-state nodes in the hyperedge. For two event-generation rules, we analytically derive interevent time distributions and autocorrelation functions of event sequences, both for hyperedges and nodes. Despite Markovian node-state dynamics, the induced event processes become mixtures of Poissonian, short-tailed components, resulting in longer-tailed interevent time distributions and slowly decaying autocorrelation. The theory further shows the dependence of these features on the size of hyperedge, which largely agrees with various empirical data. We expect our models to provide a simple, interpretable framework for connecting individual-level activity fluctuations to the timing patterns observed in real group interactions.

Segmental duplications are major drivers of evolutionary innovation, yet their dynamics across vertebrates remain poorly understood. Here, we identify segmental duplications from long-read-sequenced genomes of 117 vertebrates and one starfish, generating the largest multi-species dataset of its kind. We find that vertebrate genomes show a higher propensity for tandem duplications than for interspersed duplications. However, when focusing only on subtelomeric regions, avian and mammalian genomes show the opposite propensity toward interspersed duplications. We also observe that, across vertebrates, tandem duplications tend to be larger than interspersed duplications. Next, we construct a segmental-duplication network for each species and use network-derived metrics to quantify the duplication landscape for that species. We then compute interspecies distances for each metric and find that these distances show at most weak correlations with phylogenetic distance, indicating that segmental-duplication landscapes evolve rapidly. Functional enrichment analysis of hyper-duplicated genes reveals a strong enrichment in platypus for pheromone response, driven by the expansion of the vomeronasal pheromone receptor V1R gene family. Overall, our results uncover the general properties of vertebrate segmental duplications, demonstrate the lability of segmental-duplication landscapes, and highlight the utility of network-based approaches for studying genome evolution.

Anticipating tipping points in complex systems is difficult because many early warning signals require long time series, which are often unavailable in practice. Spatial early warning signals offer an alternative by using a single snapshot across many interacting elements, or nodes. However, their performance in heterogeneous systems is often inconsistent because raw node states reflect both dynamical changes associated with an approaching transition and static heterogeneity induced by network structure. Here, we propose a baseline-referenced framework for spatial early warning signals. The method compares each node's state with its own baseline far from the tipping point before computing a spatial statistic, thus reducing network-structure-induced variation. We evaluate baseline-referenced variants of five classical spatial early warning signals across diverse tipping scenarios and networks, and find that baseline referencing markedly improves variance-based spatial signals. The best variants increase consistently and progressively toward tipping points across different scenarios, outperform a single-node temporal variance that requires long time series, and retain high performance even when up to 80

Understanding the evolution of cooperation in structured populations remains a central challenge in multidisciplinary areas. Although previous findings suggest that structural heterogeneity in static networks hinders cooperation, real-world interactions in most natural and social systems are dynamic and best represented as temporal networks. Here, we challenge this conventional wisdom and, by developing a systematic mathematical framework, we report that structural heterogeneity in temporal networks can instead promote collective cooperation. Importantly, we reveal that such advantages depend on an often-overlooked metric-fixation time-quantifying the time required for a single cooperator to drive the entire population to cooperation. Highly heterogeneous networks accelerate this process within each subnetwork, resulting in a quantitative enhancement of cooperation in temporal networks compared to their homogeneous counterparts. By validating our results on empirical datasets through theoretical analyses and simulations, we provide a consistent framework for analysing cooperative dynamics across static and temporal networked systems.

A fundamental premise of statistical physics is that the particles in a physical system are interchangeable, and hence the state of each specific component is representative of the system as a whole. This assumption breaks down for complex networks, in which nodes may be extremely diverse, and no single component can truly represent the state of the entire system. It seems, therefore, that to observe the dynamics of social, biological or technological networks, one must extract the dynamic states of a large number of nodes-a task that is often practically prohibitive. Theoretical tools are also highly restrictive, given the analytically impenetrable combination of complex heterogeneous networks with nonlinear, often hidden, dynamics. To overcome this challenge, we use machine learning techniques to detect the network's sentinel nodes, a set of network components whose combined states can help approximate the average dynamics of the entire network. The method allows us to assess the equilibrium state of a large complex system by tracking just a small number of carefully selected nodes. We find that the sentinels are mainly determined by the network structure such that they can be extracted even with little knowledge of the system's specific interaction dynamics. Therefore, the network's sentinels offer a natural probe by which to observe the system's dynamic states. Intriguingly, sentinels tend to avoid the highly central nodes such as the hubs.

We review a class of energy landscape analysis method that uses the Ising model and takes multivariate time series data as input. The method allows one to capture dynamics of the data as trajectories of a ball from one basin to a different basin to yet another, constrained on the energy landscape specified by the estimated Ising model. While this energy landscape analysis has mostly been applied to functional magnetic resonance imaging (fMRI) data from the brain for historical reasons, there are emerging applications outside fMRI data and neuroscience. To inform such applications in various research fields, this review paper provides a detailed tutorial on each step of the analysis, terminologies, concepts underlying the method, and validation, as well as recent developments of extended and related methods.

We study synthetic temporal networks whose evolution is determined by stochastically evolving node variables-synthetic analogues of, e.g., temporal proximity networks of mobile agents. We quantify the longtimescale correlations of these evolving networks by an autocorrelative measure of network-structural memory. Several distinct patterns of autocorrelation arise, including power-law decay and exponential decay, depending on the choice of node-variable dynamics and connection probability function. Our methods are also applicable in wider contexts; our temporal network models are tractable mathematically and in simulation, and our long-term memory quantification is analytically tractable and straightforwardly computable from temporal network data.

A temporal network – a collection of snapshots recording the evolution of a network whose links appear and disappear dynamically – can be interpreted as a trajectory in graph space. In order to characterize the complex dynamics of such trajectory via the tools of time series analysis and signal processing, it is sensible to preprocess the trajectory by embedding it in a low-dimensional Euclidean space. Here we argue that, rather than the topological structure of each network snapshot, the main property of the trajectory that needs to be preserved in the embedding is the relative graph distance between snapshots. This idea naturally leads to dimensionality reduction approaches that explicitly consider relative distances, such as Multidimensional Scaling (MDS) or identifying the distance matrix as a feature matrix in which to perform Principal Component Analysis (PCA). This paper provides a comprehensible methodology that illustrates this approach. Its application to a suite of generative network trajectory models and empirical data certify that nontrivial dynamical properties of the network trajectories are preserved already in their scalar embeddings, what enables the possibility of performing time series analysis in temporal networks.

MS is a disease characterised by demyelination of the central nervous system resulting in decreased quality of life, increased anxiety, depression, fatigue, and cognitive dysfunction. Attention is frequently impaired in MS. A previous study demonstrated impairment specifically in the attentional alerting domain. However, this study did not establish whether the impairment was in intrinsic or extrinsic alertness, and did not examine the neural substrates associated with the impairment. To examine the alerting deficit in MS and establish the associated neural substrates, 40 female patients with MS and 40 age and gender match controls completed an alertness-motor paradigm designed to test both intrinsic and extrinsic alertness during functional magnetic resonance imaging. We found the MS group had a significant deficit in extrinsic alertness, which was associated with a lack of dorsal prefrontal cortex (DPFC) activation. In addition, reduced gray matter volume in the dorsal prefrontal cortex, thalamus, and basal ganglia were observed. Both fMRI and VBM correlations were observed between fatigue severity scale (FSS) scores and the DPFC. The combined results of our functional and structural MRI data analysis demonstrate that attention deficits in females with MS are a result of the complex relationship between brain tissue loss having a significant impact on functional brain regions subserving executive control of cognitive functions. Understanding the underlying mechanisms of cognitive performance and how they relate to fatigue are crucial to developing novel treatments for the symptoms of MS.

While switch-like gene expression ("on" in some individuals and "off" in others) has been linked to biological variation and disease susceptibility, a systematic analysis across tissues is lacking. Here, we analyze genomes, transcriptomes, and methylomes from 943 individuals across 27 tissues, identifying 473 switch-like genes. The identified genes are enriched for associations with cancers and immune, metabolic, and skin diseases. Only 40 (8.5%) switch-like genes show genetically controlled switch-like expression in all tissues, i.e., universally switch-like expression. The rest show switch-like expression in specific tissues. Methylation analysis suggests that genetically driven epigenetic silencing explains the universally switch-like pattern, whereas hormone-driven epigenetic modification likely underlies the tissue-specific pattern. Notably, tissue-specific switch-like genes tend to be switched on or off in unison within individuals, driven by tissue-specific master regulators. In the vagina, we identified seven concordantly switched-off genes linked to vaginal atrophy in females. Experimental analysis of vaginal tissues shows that low estrogen levels lead to decreased epithelial thickness and ALOX12 expression. We propose that switched-off driver genes in basal and parabasal epithelia suppress cell proliferation, leading to epithelial thinning and vaginal atrophy. Our findings underscore the implications of switch-like genes for diagnostic and personalized therapeutic applications.

Increasing amounts of data are available on temporal, or time-varying, networks. There have been various representations of temporal network data each of which has different advantages for downstream tasks such as mathematical analysis, visualizations, agent-based and other dynamical simulations on the temporal network, and discovery of useful structure. The tie-decay network is a representation of temporal networks whose advantages include the capability of generating continuous-time networks from discrete time-stamped contact event data with mathematical tractability and a low computational cost. However, the current framework of tie-decay networks is limited in terms of how each discrete contact event can affect the time-dependent tie strength (which we call the kernel). Here we extend the tie-decay network model in terms of the kernel. Specifically, we use a cubic spline function for modeling short-term behavior of the kernel and an exponential decay function for long-term behavior, and graft them together. This spline version of tie-decay network enables delayed and C^1-continuous interaction rates between two nodes while it only marginally increases the computational and memory burden relative to the conventional tie-decay network. We show mathematical properties of the spline tie-decay network and numerically showcase it with three tasks: network embedding, a deterministic opinion dynamics model, and a stochastic epidemic spreading model.