A mechanistic framework for complex microbe-host symbioses.

Virtually all multicellular organisms on Earth live in symbiotic associations with complex microbial communities: the microbiome. This ancient relationship is of fundamental importance for both the host and the microbiome. Recently, the analyses of numerous microbiomes have revealed an incredible diversity and complexity of symbionts, with different mechanisms identified as potential drivers of this diversity. However, the interplay of ecological and evolutionary forces generating these complex associations is still poorly understood. Here we explore and summarise the suite of ecological and evolutionary mechanisms identified as relevant to different aspects of microbiome complexity and diversity. We argue that microbiome assembly is a dynamic product of ecology and evolution at various spatio-temporal scales. We propose a theoretical framework to classify mechanisms and build mechanistic host-microbiome models to link them to empirical patterns. We develop a cohesive foundation for the theoretical understanding of the combined effects of ecology and evolution on the assembly of complex symbioses.

Controllability in attention deficit hyperactivity disorder brains.

The role of network metrics in exploring brain networks of mental illness is crucial. This study focuses on quantifying a node controllability index (CA-scores) and developing a novel framework for studying the dysfunction of attention deficit hyperactivity disorder (ADHD) brains. By analyzing fMRI data from 143 healthy controls and 102 ADHD patients, the controllability metric reveals distinct differences in nodes (brain regions) and subsystems (functional modules). There are significantly atypical CA-scores in the Rolandic operculum, superior medial orbitofrontal cortex, insula, posterior cingulate gyrus, supramarginal gyrus, angular gyrus, precuneus, heschl gyrus, and superior temporal gyrus of ADHD patients. A comparison with measures of connection strength, eigenvector centrality, and topology entropy suggests that the controllability index may be more effective in identifying abnormal regions in ADHD brains. Furthermore, our controllability index could be extended to investigate functional networks associated with other psychiatric disorders. Supplementary Information: The online version contains supplementary material available at 10.1007/s11571-023-10063-z.

Reinforcement learning-based pinning control for synchronization suppression in complex networks.

Synchronization in complex networks is a ubiquitous and important phenomenon with implications in various fields. Excessive synchronization may lead to undesired consequences, making desynchronization techniques essential. Exploiting the Proximal Policy Optimization algorithm, this work studies reinforcement learning-based pinning control strategies for synchronization suppression in global coupling networks and two types of irregular coupling networks: the Watts-Strogatz small-world networks and the Barabási-Albert scale-free networks. We investigate the impact of the ratio of controlled nodes and the role of key nodes selected by the LeaderRank algorithm on the performance of synchronization suppression. Numerical results demonstrate the effectiveness of the reinforcement learning-based pinning control strategy in different coupling schemes of the complex networks, revealing a critical ratio of the pinned nodes and the superior performance of a newly proposed hybrid pinning strategy. The results provide valuable insights for suppressing and optimizing network synchronization behavior efficiently.

DoS attacks resilience of heterogeneous complex networks via dynamic event-triggered impulsive scheme for secure quasi-synchronization.

This paper addresses the secure quasi-synchronization issue of heterogeneous complex networks (HCNs) under aperiodic denial-of-service (DoS) attacks with dynamic event-triggered impulsive scheme (ETIS). The heterogeneity of networks and the aperiodic DoS attacks, which hinder communication channels and synchronization goals, present challenges to the analysis of secure quasi-synchronization. The ETIS leverages impulsive control and dynamic event-triggered scheme (ETS) to handle the network heterogeneity and the DoS attacks. We give specific bounds on the attack duration and frequency that the network can endure, and obtain synchronization criteria that relate to event parameters, attack duration, attack frequency, and impulsive gain by the variation of parameter formula and recursive methods. Moreover, we prove that the dynamic ETS significantly reduces the controller updates, saves energy without sacrificing the system decay rate, and prevents the Zeno phenomenon. Finally, we validate our control scheme with a numerical example.

The Ihara zeta function as a partition function for network structure characterisation.

Statistical characterizations of complex network structures can be obtained from both the Ihara Zeta function (in terms of prime cycle frequencies) and the partition function from statistical mechanics. However, these two representations are usually regarded as separate tools for network analysis, without exploiting the potential synergies between them. In this paper, we establish a link between the Ihara Zeta function from algebraic graph theory and the partition function from statistical mechanics, and exploit this relationship to obtain a deeper structural characterisation of network structure. Specifically, the relationship allows us to explore the connection between the microscopic structure and the macroscopic characterisation of a network. We derive thermodynamic quantities describing the network, such as entropy, and show how these are related to the frequencies of prime cycles of various lengths. In particular, the n-th order partial derivative of the Ihara Zeta function can be used to compute the number of prime cycles in a network, which in turn is related to the partition function of Bose-Einstein statistics. The corresponding derived entropy allows us to explore a phase transition in the network structure with critical points at high and low-temperature limits. Numerical experiments and empirical data are presented to evaluate the qualitative and quantitative performance of the resulting structural network characterisations.

An Attacker-Defender Game Model with Constrained Strategies.

Recently, research interest in the field of infrastructure attack and defense scenarios has increased. Numerous methods have been proposed for studying strategy interactions that combine complex network theory and game theory. However, the unavoidable effect of constrained strategies in complex situations has not been considered in previous studies. This study introduces a novel approach to analyzing these interactions by including the effects of constrained strategies, a factor often neglected in traditional analyses. First, we introduce the rule of constraints on strategies, which depends on the average distance between selected nodes. As the average distance increases, the probability of choosing the corresponding strategy decreases. Second, we establish an attacker-defender game model with constrained strategies based on the above rule and using information theory to evaluate the uncertainty of these strategies. Finally, we present a method for solving this problem and conduct experiments based on a target network. The results highlight the unique characteristics of the Nash equilibrium when setting constraints, as these constraints influence decision makers' Nash equilibria. When considering the constrained strategies, both the attacker and the defender tend to select strategies with lower average distances. The effect of the constraints on their strategies becomes less apparent as the number of attackable or defendable nodes increases. This research advances the field by introducing a novel framework for examining strategic interactions in infrastructure defense and attack scenarios. By incorporating strategy constraints, our work offers a new perspective on the critical area of infrastructure security.

Optimizing Contact Network Topological Parameters of Urban Populations Using the Genetic Algorithm.

This paper explores the application of complex network models and genetic algorithms in epidemiological modeling. By considering the small-world and Barabási-Albert network models, we aim to replicate the dynamics of disease spread in urban environments. This study emphasizes the importance of accurately mapping individual contacts and social networks to forecast disease progression. Using a genetic algorithm, we estimate the input parameters for network construction, thereby simulating disease transmission within these networks. Our results demonstrate the networks' resemblance to real social interactions, highlighting their potential in predicting disease spread. This study underscores the significance of complex network models and genetic algorithms in understanding and managing public health crises.

An Evaluation Model for Node Influence Based on Heuristic Spatiotemporal Features.

The accurate assessment of node influence is of vital significance for enhancing system stability. Given the structural redundancy problem triggered by the network topology deviation when an empirical network is copied, as well as the dynamic characteristics of the empirical network itself, it is difficult for traditional static assessment methods to effectively capture the dynamic evolution of node influence. Therefore, we propose a heuristic-based spatiotemporal feature node influence assessment model (HEIST). First, the zero-model method is applied to optimize the network-copying process and reduce the noise interference caused by network structure redundancy. Second, the copied network is divided into subnets, and feature modeling is performed to enhance the node influence differentiation. Third, node influence is quantified based on the spatiotemporal depth-perception module, which has a built-in local and global two-layer structure. At the local level, a graph convolutional neural network (GCN) is used to improve the spatial perception of node influence; it fuses the feature changes of the nodes in the subnetwork variation, combining this method with a long- and short-term memory network (LSTM) to enhance its ability to capture the depth evolution of node influence and improve the robustness of the assessment. Finally, a heuristic assessment algorithm is used to jointly optimize the influence strength of the nodes at different stages and quantify the node influence via a nonlinear optimization function. The experiments show that the Kendall coefficients exceed 90% in multiple datasets, proving that the model has good generalization performance in empirical networks.

F-Deepwalk: A Community Detection Model for Transport Networks.

The design of transportation networks is generally performed on the basis of the division of a metropolitan region into communities. With the combination of the scale, population density, and travel characteristics of each community, the transportation routes and stations can be more precisely determined to meet the travel demand of residents within each of the communities as well as the transportation links among communities. To accurately divide urban communities, the original word vector sampling method is improved on the classic Deepwalk model, proposing a Random Walk (RW) algorithm in which the sampling is modified with the generalized travel cost and improved logit model. Urban spatial community detection is realized with the K-means algorithm, building the F-Deepwalk model. Using the basic road network as an example, the experimental results show that the Deepwalk model, which considers the generalized travel cost of residents, has a higher profile coefficient, and the performance of the model improves with the reduction of random walk length. At the same time, taking the Shijiazhuang urban rail transit network as an example, the accuracy of the model is further verified.

Aperiodic intermittent dynamic event-triggered synchronization control for stochastic delayed multi-links complex networks.

In this work, the exponential synchronization issue of stochastic complex networks with time delays and time-varying multi-links (SCNTM) is discussed via a novel aperiodic intermittent dynamic event-triggered control (AIDE-TC). The AIDE-TC is designed by combining intermittent control with an exponential function and dynamic event-triggered control, aiming to minimize the number of the required triggers. Then, based on the proposed control strategy, the sufficient conditions for exponential synchronization in mean square of SCNTM are obtained by adopting graph theoretic approach and Lyapunov function method. In the meanwhile, it is proven that the Zeno behavior can be excluded under the AIDE-TC, which ensures the feasibility of the control mechanism to realize the synchronization of SCNTM. Finally, we provide a numerical simulation on islanded microgrid systems to validate the effectiveness of main results and the simulation comparison results show that the AIDE-TC can reduce the number of event triggers.

Dynamics and bifurcations in genetic circuits with fibration symmetries.

Circuit building blocks of gene regulatory networks (GRN) have been identified through the fibration symmetries of the underlying biological graph. Here, we analyse analytically six of these circuits that occur as functional and synchronous building blocks in these networks. Of these, the lock-on, toggle switch, Smolen oscillator, feed-forward fibre and Fibonacci fibre circuits occur in living organisms, notably Escherichia coli; the sixth, the repressilator, is a synthetic GRN. We consider synchronous steady states determined by a fibration symmetry (or balanced colouring) and determine analytic conditions for local bifurcation from such states, which can in principle be either steady-state or Hopf bifurcations. We identify conditions that characterize the first bifurcation, the only one that can be stable near the bifurcation point. We model the state of each gene in terms of two variables: mRNA and protein concentration. We consider all possible 'admissible' models-those compatible with the network structure-and then specialize these general results to simple models based on Hill functions and linear degradation. The results systematically classify using graph symmetries the complexity and dynamics of these circuits, which are relevant to understand the functionality of natural and synthetic cells.

The ultrametric backbone is the union of all minimum spanning forests.

Minimum spanning trees and forests are powerful sparsification techniques that remove cycles from weighted graphs to minimize total edge weight while preserving node reachability, with applications in computer science, network science, and graph theory. Despite their utility and ubiquity, they have several limitations, including that they are only defined for undirected networks, they significantly alter dynamics on networks, and they do not generally preserve important network features such as shortest distances, shortest path distribution, and community structure. In contrast, distance backbones, which are subgraphs formed by all edges that obey a generalized triangle inequality, are well defined in directed and undirected graphs and preserve those and other important network features. The backbone of a graph is defined with respect to a specified path-length operator that aggregates weights along a path to define its length, thereby associating a cost to indirect connections. The backbone is the union of all shortest paths between each pair of nodes according to the specified operator. One such operator, the max function, computes the length of a path as the largest weight of the edges that compose it (a weakest link criterion). It is the only operator that yields an algebraic structure for computing shortest paths that is consistent with De Morgan's laws. Applying this operator yields the ultrametric backbone of a graph in that (semi-triangular) edges whose weights are larger than the length of an indirect path connecting the same nodes (i.e. those that break the generalized triangle inequality based on max as a path-length operator) are removed. We show that the ultrametric backbone is the union of minimum spanning forests in undirected graphs and provides a new generalization of minimum spanning trees to directed graphs that, unlike minimum equivalent graphs and minimum spanning arborescences, preserves all max - min shortest paths and De Morgan's law consistency.

Analysis of induced dynamic biceps EMG signal complexity using Markov transition networks.

Purpose: Surface electromyography (sEMG) is a non-invasive technique to characterize muscle electrical activity. The analysis of sEMG signals under muscle fatigue play a crucial part in the branch of neurorehabilitation, sports medicine, biomechanics, and monitoring neuromuscular pathologies. In this work, a method to transform sEMG signals to complex networks under muscle fatigue conditions using Markov transition field (MTF) is proposed. The importance of normalization to a constant Maximum voluntary contraction (MVC) is also considered. Methods: For this, dynamic signals are recorded using two different experimental protocols one under constant load and another referenced to 50% MVC from Biceps brachii of 50 and 45 healthy subjects respectively. MTF is generated and network graph is constructed from preprocesses signals. Features such as average self-transition probability, average clustering coefficient and modularity are extracted. Results: All the extracted features showed statistical significance for the recorded signals. It is found that during the transition from non-fatigue to fatigue, average clustering coefficient decreases while average self-transition probability and modularity increases. Conclusion: The results indicate higher degree of signal complexity during non-fatigue condition. Thus, the MTF approach may be used to indicate the complexity of sEMG signals. Although both datasets showed same trend in results, sEMG signals under 50% MVC exhibited higher separability for the features. The inter individual variations of the MTF features is found to be more for the signals recorded using constant load. The proposed study can be adopted to study the complex nature of muscles under various neuromuscular conditions.

Resilience of the slow component in timescale-separated synchronized oscillators.

Physiological networks are usually made of a large number of biological oscillators evolving on a multitude of different timescales. Phase oscillators are particularly useful in the modelling of the synchronization dynamics of such systems. If the coupling is strong enough compared to the heterogeneity of the internal parameters, synchronized states might emerge where phase oscillators start to behave coherently. Here, we focus on the case where synchronized oscillators are divided into a fast and a slow component so that the two subsets evolve on separated timescales. We assess the resilience of the slow component by, first, reducing the dynamics of the fast one using Mori-Zwanzig formalism. Second, we evaluate the variance of the phase deviations when the oscillators in the two components are subject to noise with possibly distinct correlation times. From the general expression for the variance, we consider specific network structures and show how the noise transmission between the fast and slow components is affected. Interestingly, we find that oscillators that are among the most robust when there is only a single timescale, might become the most vulnerable when the system undergoes a timescale separation. We also find that layered networks seem to be insensitive to such timescale separations.

A simplex path integral and a simplex renormalization group for high-order interactions.

Modern theories of phase transitions and scale invariance are rooted in path integral formulation and renormalization groups (RGs). Despite the applicability of these approaches in simple systems with only pairwise interactions, they are less effective in complex systems with undecomposable high-order interactions (i.e. interactions among arbitrary sets of units). To precisely characterize the universality of high-order interacting systems, we propose a simplex path integral and a simplex RG (SRG) as the generalizations of classic approaches to arbitrary high-order and heterogeneous interactions. We first formalize the trajectories of units governed by high-order interactions to define path integrals on corresponding simplices based on a high-order propagator. Then, we develop a method to integrate out short-range high-order interactions in the momentum space, accompanied by a coarse graining procedure functioning on the simplex structure generated by high-order interactions. The proposed SRG, equipped with a divide-and-conquer framework, can deal with the absence of ergodicity arising from the sparse distribution of high-order interactions and can renormalize a system with intertwined high-order interactions at thep-order according to its properties at theq-order (p⩽q). The associated scaling relation and its corollaries provide support to differentiate among scale-invariant, weakly scale-invariant, and scale-dependent systems across different orders. We validate our theory in multi-order scale-invariance verification, topological invariance discovery, organizational structure identification, and information bottleneck analysis. These experiments demonstrate the capability of our theory to identify intrinsic statistical and topological properties of high-order interacting systems during system reduction.

Emergent scale-free networks.

Many complex systems-from the Internet to social, biological, and communication networks-are thought to exhibit scale-free structure. However, prevailing explanations require that networks grow over time, an assumption that fails in some real-world settings. Here, we explain how scale-free structure can emerge without growth through network self-organization. Beginning with an arbitrary network, we allow connections to detach from random nodes and then reconnect under a mixture of preferential and random attachment. While the numbers of nodes and edges remain fixed, the degree distribution evolves toward a power-law with an exponent γ = 1 + 1 p that depends only on the proportion p of preferential (rather than random) attachment. Applying our model to several real networks, we infer p directly from data and predict the relationship between network size and degree heterogeneity. Together, these results establish how scale-free structure can arise in networks of constant size and density, with broad implications for the structure and function of complex systems.

Network Models of BACE-1 Inhibitors: Exploring Structural and Biochemical Relationships.

This study investigates the clustering patterns of human ß-secretase 1 (BACE-1) inhibitors using complex network methodologies based on various distance functions, including Euclidean, Tanimoto, Hamming, and Levenshtein distances. Molecular descriptor vectors such as molecular mass, Merck Molecular Force Field (MMFF) energy, Crippen partition coefficient (ClogP), Crippen molar refractivity (MR), eccentricity, Kappa indices, Synthetic Accessibility Score, Topological Polar Surface Area (TPSA), and 2D/3D autocorrelation entropies are employed to capture the diverse properties of these inhibitors. The Euclidean distance network demonstrates the most reliable clustering results, with strong agreement metrics and minimal information loss, indicating its robustness in capturing essential structural and physicochemical properties. Tanimoto and Hamming distance networks yield valuable clustering outcomes, albeit with moderate performance, while the Levenshtein distance network shows significant discrepancies. The analysis of eigenvector centrality across different networks identifies key inhibitors acting as hubs, which are likely critical in biochemical pathways. Community detection results highlight distinct clustering patterns, with well-defined communities providing insights into the functional and structural groupings of BACE-1 inhibitors. The study also conducts non-parametric tests, revealing significant differences in molecular descriptors, validating the clustering methodology. Despite its limitations, including reliance on specific descriptors and computational complexity, this study offers a comprehensive framework for understanding molecular interactions and guiding therapeutic interventions. Future research could integrate additional descriptors, advanced machine learning techniques, and dynamic network analysis to enhance clustering accuracy and applicability.

The Statistics of q-Statistics.

Almost two decades ago, Ernesto P. Borges and Bruce M. Boghosian embarked on the intricate task of composing a manuscript to honor the profound contributions of Constantino Tsallis to the realm of statistical physics, coupled with a concise exploration of q-Statistics. Fast-forward to Constantino Tsallis' illustrious 80th birthday celebration in 2023, where Deniz Eroglu and Ugur Tirnakli delved into Constantino's collaborative network, injecting renewed vitality into the project. With hearts brimming with appreciation for Tsallis' enduring inspiration, Eroglu, Boghosian, Borges, and Tirnakli proudly present this meticulously crafted manuscript as a token of their gratitude.

A complex network approach to analyse pre-trained language models for ancient Chinese.

Ancient Chinese is a splendid treasure within Chinese culture. To facilitate its compilation, pre-trained language models for ancient Chinese are developed. After that, researchers are actively exploring the factors contributing to their success. However, previous work did not study how language models organized the elements of ancient Chinese from a holistic perspective. Hence, we adopt complex networks to explore how language models organize the elements in ancient Chinese system. Specifically, we first analyse the characters' and words' co-occurrence networks in ancient Chinese. Then, we study characters' and words' attention networks, generated by attention heads within SikuBERT from two aspects: static and dynamic network analysis. In the static network analysis, we find that (i) most of attention networks exhibit small-world properties and scale-free behaviour, (ii) over 80% of attention networks exhibit high similarity with the corresponding co-occurrence networks, (iii) there exists a noticeable gap between characters' and words' attention networks across layers, while their fluctuations remain relatively consistent, and (iv) the attention networks generated by SikuBERT tend to be sparser compared with those from Chinese BERT. In dynamic network analysis, we find that the sentence segmentation task does not significantly affect network metrics, while the part-of-speech tagging task makes attention networks sparser.

Beyond network centrality: individual-level behavioral traits for predicting information superspreaders in social media.

Understanding the heterogeneous role of individuals in large-scale information spreading is essential to manage online behavior as well as its potential offline consequences. To this end, most existing studies from diverse research domains focus on the disproportionate role played by highly connected 'hub' individuals. However, we demonstrate here that information superspreaders in online social media are best understood and predicted by simultaneously considering two individual-level behavioral traits: influence and susceptibility. Specifically, we derive a nonlinear network-based algorithm to quantify individuals' influence and susceptibility from multiple spreading event data. By applying the algorithm to large-scale data from Twitter and Weibo, we demonstrate that individuals' estimated influence and susceptibility scores enable predictions of future superspreaders above and beyond network centrality, and reveal new insights into the network positions of the superspreaders.