Quantifying the connectivity of a network: the network correlation function method

Deep learning resilience inference for complex networked systems

Resilience, the ability to maintain fundamental functionality amidst failures and errors, is crucial for complex networked systems. Most analytical approaches rely on predefined equations for node activity dynamics and simplifying assumptions on network topology, limiting their applicability to real-world systems. Here, we propose ResInf, a deep learning framework integrating transformers and graph neural networks to infer resilience directly from observational data. ResInf learns representations of node activity dynamics and network topology without simplifying assumptions, enabling accurate resilience inference and low-dimensional visualization. Experimental results show that ResInf significantly outperforms analytical methods, with an F1-score improvement of up to 41.59% over Gao-Barzel-Barabási framework and 14.32% over spectral dimension reduction. It also generalizes to unseen topologies and dynamics and maintains robust performance despite observational disturbances. Our findings suggest that ResInf addresses an important gap in resilience inference for real-world systems, offering a fresh perspective on incorporating data-driven approaches to complex network modeling.

Spreading Dynamics of Capital Flow Transfer in Complex Financial Networks

The financial system, a complex network, operates primarily through the exchange of capital, where the role of information is critical. This study utilizes the transfer entropy method to examine the strength and direction of information flow among different capital flow time series and investigate the community structure within the transfer networks. Moreover, the spreading dynamics of the capital flow transfer networks are observed, and the importance and traveling time of each node are explored. The results imply a dominant role for the food and drink industry within the Chinese market, with increased attention towards the computer industry starting in 2014. The community structure of the capital flow transfer networks significantly differs from those constructed from stock prices, with the main sector predominantly encompassing industry leaders favored by primary funds with robust capital flow connections. The average traveling time from sectors such as food and drink, coal, and utilities to other sectors is the shortest, and the dynamic flow between these sectors displays a significant role. These findings highlight that comprehension of information flow and community structure within the financial system can offer valuable insights into market dynamics and help to identify key sectors and companies.

Distribution equality as an optimal epidemic mitigation strategy

Upon the development of a therapeutic, a successful response to a global pandemic relies on efficient worldwide distribution, a process constrained by our global shipping network. Most existing strategies seek to maximize the outflow of the therapeutics, hence optimizing for rapid dissemination. Here we find that this intuitive approach is, in fact, counterproductive. The reason is that by focusing strictly on the quantity of disseminated therapeutics, these strategies disregard the way in which this quantity distributes across destinations. Most crucially-they overlook the interplay of the therapeutic spreading patterns with those of the pathogens. This results in a discrepancy between supply and demand, that prohibits efficient mitigation even under optimal conditions of superfluous flow. To solve this, we design a dissemination strategy that naturally follows the predicted spreading patterns of the pathogens, optimizing not just for supply volume, but also for its congruency with the anticipated demand. Specifically, we show that epidemics spread relatively uniformly across all destinations, prompting us to introduce an equality constraint into our dissemination that prioritizes supply homogeneity. This strategy may, at times, slow down the supply rate in certain locations, however, thanks to its egalitarian nature, which mimics the flow of the pathogens, it provides a dramatic leap in overall mitigation efficiency, potentially saving more lives with orders of magnitude less resources.

Numerical analysis of first-passage processes in finite Markov chains exhibiting metastability

We describe state-reduction algorithms for the analysis of first-passage processes in discrete- and continuous-time finite Markov chains. We present a formulation of the graph transformation algorithm that allows for the evaluation of exact mean first-passage times, stationary probabilities, and committor probabilities for all nonabsorbing nodes of a Markov chain in a single computation. Calculation of the committor probabilities within the state-reduction formalism is readily generalizable to the first hitting problem for any number of alternative target states. We then show that a state-reduction algorithm can be formulated to compute the expected number of times that each node is visited along a first-passage path. Hence, all properties required to analyze the first-passage path ensemble (FPPE) at both a microscopic and macroscopic level of detail, including the mean and variance of the first-passage time distribution, can be computed using state-reduction methods. In particular, we derive expressions for the probability that a node is visited along a direct transition path, which proceeds without returning to the initial state, considering both the nonequilibrium and equilibrium (steady-state) FPPEs. The reactive visitation probability provides a rigorous metric to quantify the dynamical importance of a node for the productive transition between two endpoint states and thus allows the local states that facilitate the dominant transition mechanisms to be readily identified. The state-reduction procedures remain numerically stable even for Markov chains exhibiting metastability, which can be severely ill-conditioned. The rare event regime is frequently encountered in realistic models of dynamical processes, and our methodology therefore provides valuable tools for the analysis of Markov chains in practical applications. We illustrate our approach with numerical results for a kinetic network representing a structural transition in an atomic cluster.

Resilience centrality in complex networks

Resilience describes a system's ability to adjust its activity to retain the basic functionality when errors or failures occur in components (nodes) of the network. Due to the complexity of a system's structure, different components in the system exhibit diversity in the ability to affect the resilience of the system, bringing us a great challenge to protect the system from collapse. A fundamental problem is therefore to propose a physically insightful centrality index, with which to quantify the resilience contribution of a node in any systems effectively. However, existing centrality indexes are not suitable for the problem because they only consider the network structure of the system and ignore the impact of underlying dynamic characteristics. To break the limits, we derive a new centrality index: resilience centrality from the 1D dynamic equation of systems, with which we can quantify the ability of nodes to affect the resilience of the system accurately. Resilience centrality unveils the long-sought relations between the ability of nodes in a system's resilience and network structure of the system: the capacity is mainly determined by the degree and weighted nearest-neighbor degree of the node, in which weighted nearest-neighbor degree plays a prominent role. Further, we demonstrate that weighted nearest-neighbor degree has a positive impact on resilience centrality, while the effect of the degree depends on a specific parameter, average weighted degree β_{eff}, in the 1D dynamic equation. To test the performance of our approach, we construct four real networks from data, which corresponds to two complex systems with entirely different dynamic characteristics. The simulation results demonstrate the effectiveness of our resilience centrality, providing us theoretical insights into the protection of complex systems from collapse.

The heparin-binding proteome in normal pancreas and murine experimental acute pancreatitis

Acute pancreatitis (AP) is acute inflammation of the pancreas, mainly caused by gallstones and alcohol, driven by changes in communication between cells. Heparin-binding proteins (HBPs) play a central role in health and diseases. Therefore, we used heparin affinity proteomics to identify extracellular HBPs in pancreas and plasma of normal mice and in a caerulein mouse model of AP. Many new extracellular HBPs (360) were discovered in the pancreas, taking the total number of HBPs known to 786. Extracellular pancreas HBPs form highly interconnected protein-protein interaction networks in both normal pancreas (NP) and AP. Thus, HBPs represent an important set of extracellular proteins with significant regulatory potential in the pancreas. HBPs in NP are associated with biological functions such as molecular transport and cellular movement that underlie pancreatic homeostasis. However, in AP HBPs are associated with additional inflammatory processes such as acute phase response signalling, complement activation and mitochondrial dysfunction, which has a central role in the development of AP. Plasma HBPs in AP included known AP biomarkers such as serum amyloid A, as well as emerging targets such as histone H2A. Other HBPs such as alpha 2-HS glycoprotein (AHSG) and histidine-rich glycoprotein (HRG) need further investigation for potential applications in the management of AP. Pancreas HBPs are extracellular and so easily accessible and are potential drug targets in AP, whereas plasma HBPs represent potential biomarkers for AP. Thus, their identification paves the way to determine which HBPs may have potential applications in the management of AP.

Predicting perturbation patterns from the topology of biological networks

High-throughput technologies, offering an unprecedented wealth of quantitative data underlying the makeup of living systems, are changing biology. Notably, the systematic mapping of the relationships between biochemical entities has fueled the rapid development of network biology, offering a suitable framework to describe disease phenotypes and predict potential drug targets. However, our ability to develop accurate dynamical models remains limited, due in part to the limited knowledge of the kinetic parameters underlying these interactions. Here, we explore the degree to which we can make reasonably accurate predictions in the absence of the kinetic parameters. We find that simple dynamically agnostic models are sufficient to recover the strength and sign of the biochemical perturbation patterns observed in 87 biological models for which the underlying kinetics are known. Surprisingly, a simple distance-based model achieves 65% accuracy. We show that this predictive power is robust to topological and kinetic parameter perturbations, and we identify key network properties that can increase up to 80% the recovery rate of the true perturbation patterns. We validate our approach using experimental data on the chemotactic pathway in bacteria, finding that a network model of perturbation spreading predicts with ∼80% accuracy the directionality of gene expression and phenotype changes in knock-out and overproduction experiments. These findings show that the steady advances in mapping out the topology of biochemical interaction networks opens avenues for accurate perturbation spread modeling, with direct implications for medicine and drug development.

Synchronization invariance under network structural transformations

Synchronization processes are ubiquitous despite the many connectivity patterns that complex systems can show. Usually, the emergence of synchrony is a macroscopic observable; however, the microscopic details of the system, as, e.g., the underlying network of interactions, is many times partially or totally unknown. We already know that different interaction structures can give rise to a common functionality, understood as a common macroscopic observable. Building upon this fact, here we propose network transformations that keep the collective behavior of a large system of Kuramoto oscillators invariant. We derive a method based on information theory principles, that allows us to adjust the weights of the structural interactions to map random homogeneous in-degree networks into random heterogeneous networks and vice versa, keeping synchronization values invariant. The results of the proposed transformations reveal an interesting principle; heterogeneous networks can be mapped to homogeneous ones with local information, but the reverse process needs to exploit higher-order information. The formalism provides analytical insight to tackle real complex scenarios when dealing with uncertainty in the measurements of the underlying connectivity structure.

Estimating psychological networks and their accuracy: A tutorial paper

The usage of psychological networks that conceptualize behavior as a complex interplay of psychological and other components has gained increasing popularity in various research fields. While prior publications have tackled the topics of estimating and interpreting such networks, little work has been conducted to check how accurate (i.e., prone to sampling variation) networks are estimated, and how stable (i.e., interpretation remains similar with less observations) inferences from the network structure (such as centrality indices) are. In this tutorial paper, we aim to introduce the reader to this field and tackle the problem of accuracy under sampling variation. We first introduce the current state-of-the-art of network estimation. Second, we provide a rationale why researchers should investigate the accuracy of psychological networks. Third, we describe how bootstrap routines can be used to (A) assess the accuracy of estimated network connections, (B) investigate the stability of centrality indices, and (C) test whether network connections and centrality estimates for different variables differ from each other. We introduce two novel statistical methods: for (B) the correlation stability coefficient, and for (C) the bootstrapped difference test for edge-weights and centrality indices. We conducted and present simulation studies to assess the performance of both methods. Finally, we developed the free R-package bootnet that allows for estimating psychological networks in a generalized framework in addition to the proposed bootstrap methods. We showcase bootnet in a tutorial, accompanied by R syntax, in which we analyze a dataset of 359 women with posttraumatic stress disorder available online.

Dynamic patterns of information flow in complex networks

Although networks are extensively used to visualize information flow in biological, social and technological systems, translating topology into dynamic flow continues to challenge us, as similar networks exhibit fundamentally different flow patterns, driven by different interaction mechanisms. To uncover a network’s actual flow patterns, here we use a perturbative formalism, analytically tracking the contribution of all nodes/paths to the flow of information, exposing the rules that link structure and dynamic information flow for a broad range of nonlinear systems. We find that the diversity of flow patterns can be mapped into a single universal function, characterizing the interplay between the system’s topology and its dynamics, ultimately allowing us to identify the network’s main arteries of information flow. Counter-intuitively, our formalism predicts a family of frequently encountered dynamics where the flow of information avoids the hubs, favoring the network’s peripheral pathways, a striking disparity between structure and dynamics.

Complex networks are a useful tool to investigate spreading processes but topology alone is insufficient to predict information flow. Here the authors propose a measure of information flow and predict its behavior from the interplay between structure and dynamics.

A physical model for dementia

Aging associated brain decline often result in some kind of dementia. Even when this is a complex brain disorder a physical model can be used in order to describe its general behavior. A probabilistic model for the development of dementia is obtained and fitted to some experimental data obtained from the Alzheimer's Disease Neuroimaging Initiative. It is explained how dementia appears as a consequence of aging and why it is irreversible.

Network analysis and juvenile idiopathic arthritis (JIA): a new horizon for the understanding of disease pathogenesis and therapeutic target identification

Juvenile idiopathic arthritis (JIA) is a clinically diverse and genetically complex autoimmune disease. Currently, there is very limited understanding of the potential underlying mechanisms that result in the range of phenotypes which constitute JIA.The elucidation of the functional relevance of genetic associations with phenotypic traits is a fundamental problem that hampers the translation of genetic observations to plausible medical interventions. Genome wide association studies, and subsequent fine-mapping studies in JIA patients, have identified many genetic variants associated with disease. Such approaches rely on 'tag' single nucleotide polymorphisms (SNPs). The associated SNPs are rarely functional variants, so the extrapolation of genetic association data to the identification of biologically meaningful findings can be a protracted undertaking. Integrative genomics aims to bridge the gap between genotype and phenotype.Systems biology, principally through network analysis, is emerging as a valuable way to identify biological pathways of relevance to complex genetic diseases. This review aims to highlight recent findings in systems biology related to JIA in an attempt to assist in the understanding of JIA pathogenesis and therapeutic target identification.

Universal resilience patterns in complex networks

Resilience, a system's ability to adjust its activity to retain its basic functionality when errors, failures and environmental changes occur, is a defining property of many complex systems. Despite widespread consequences for human health, the economy and the environment, events leading to loss of resilience--from cascading failures in technological systems to mass extinctions in ecological networks--are rarely predictable and are often irreversible. These limitations are rooted in a theoretical gap: the current analytical framework of resilience is designed to treat low-dimensional models with a few interacting components, and is unsuitable for multi-dimensional systems consisting of a large number of components that interact through a complex network. Here we bridge this theoretical gap by developing a set of analytical tools with which to identify the natural control and state parameters of a multi-dimensional complex system, helping us derive effective one-dimensional dynamics that accurately predict the system's resilience. The proposed analytical framework allows us systematically to separate the roles of the system's dynamics and topology, collapsing the behaviour of different networks onto a single universal resilience function. The analytical results unveil the network characteristics that can enhance or diminish resilience, offering ways to prevent the collapse of ecological, biological or economic systems, and guiding the design of technological systems resilient to both internal failures and environmental changes.

Constructing minimal models for complex system dynamics

One of the strengths of statistical physics is the ability to reduce macroscopic observations into microscopic models, offering a mechanistic description of a system's dynamics. This paradigm, rooted in Boltzmann's gas theory, has found applications from magnetic phenomena to subcellular processes and epidemic spreading. Yet, each of these advances were the result of decades of meticulous model building and validation, which are impossible to replicate in most complex biological, social or technological systems that lack accurate microscopic models. Here we develop a method to infer the microscopic dynamics of a complex system from observations of its response to external perturbations, allowing us to construct the most general class of nonlinear pairwise dynamics that are guaranteed to recover the observed behaviour. The result, which we test against both numerical and empirical data, is an effective dynamic model that can predict the system's behaviour and provide crucial insights into its inner workings.

Interactions between distant ceRNAs in regulatory networks

Competing endogenous RNAs (ceRNAs) were recently introduced as RNA transcripts that affect each other's expression level through competition for their microRNA (miRNA) coregulators. This stems from the bidirectional effects between miRNAs and their target RNAs, where a change in the expression level of one target affects the level of the miRNA regulator, which in turn affects the level of other targets. By the same logic, miRNAs that share targets compete over binding to their common targets and therefore also exhibit ceRNA-like behavior. Taken together, perturbation effects could propagate in the posttranscriptional regulatory network through a path of coregulated targets and miRNAs that share targets, suggesting the existence of distant ceRNAs. Here we study the prevalence of distant ceRNAs and their effect in cellular networks. Analyzing the network of miRNA-target interactions deciphered experimentally in HEK293 cells, we show that it is a dense, intertwined network, suggesting that many nodes can act as distant ceRNAs of one another. Indeed, using gene expression data from a perturbation experiment, we demonstrate small, yet statistically significant, changes in gene expression caused by distant ceRNAs in that network. We further characterize the magnitude of the propagated perturbation effect and the parameters affecting it by mathematical modeling and simulations. Our results show that the magnitude of the effect depends on the generation and degradation rates of involved miRNAs and targets, their interaction rates, the distance between the ceRNAs and the topology of the network. Although demonstrated for a miRNA-mRNA regulatory network, our results offer what to our knowledge is a new view on various posttranscriptional cellular networks, expanding the concept of ceRNAs and implying possible distant cross talk within the network, with consequences for the interpretation of indirect effects of gene perturbation.

Network analysis: a new approach to study endocrine disorders

Systems biology is the study of the interactions that occur between the components of individual cells - including genes, proteins, transcription factors, small molecules, and metabolites, and their relationships to complex physiological and pathological processes. The application of systems biology to medicine promises rapid advances in both our understanding of disease and the development of novel treatment options. Network biology has emerged as the primary tool for studying systems biology as it utilises the mathematical analysis of the relationships between connected objects in a biological system and allows the integration of varied 'omic' datasets (including genomics, metabolomics, proteomics, etc.). Analysis of network biology generates interactome models to infer and assess function; to understand mechanisms, and to prioritise candidates for further investigation. This review provides an overview of network methods used to support this research and an insight into current applications of network analysis applied to endocrinology. A wide spectrum of endocrine disorders are included ranging from congenital hyperinsulinism in infancy, through childhood developmental and growth disorders, to the development of metabolic diseases in early and late adulthood, such as obesity and obesity-related pathologies. In addition to providing a deeper understanding of diseases processes, network biology is also central to the development of personalised treatment strategies which will integrate pharmacogenomics with systems biology of the individual.

The heparin-binding protein interactome in pancreatic diseases

BACKGROUND

The cellular microenvironment plays an important role in the regulation of homoeostasis and is a source of potential biomarkers and drug targets. In a genome-wide analysis the extracellular proteins that bind to heparin (HBPs) have been shown to form highly modular and interconnected extracellular protein regulatory networks. Using a systems biology approach, we have investigated the role of HBP networks in the normal pancreas and pancreatic digestive diseases.

METHODS

Lists of mRNAs encoding for HBPs associated with the normal pancreas (NP), acute pancreatitis (AP), chronic pancreatitis (CP) and pancreatic ductal adenocarcinoma (PDAC) were obtained using public databases and publications. Networks of the putative protein interactomes derived from mRNA expression data of HBPs were built and analysed using cluster analysis, gene ontology term enrichment and canonical pathways analysis.

RESULTS

The extracellular heparin-binding putative protein interactomes in the pancreas were better connected than their non heparin-binding counterparts, having higher clustering coefficients in the normal pancreas (0.273), acute pancreatitis (0.457), chronic pancreatitis (0.329) and pancreatic ductal adenocarcinoma (0.269). 'Hepatic Fibrosis/Hepatic Stellate Cell Activation' appears to be a significant canonical pathway in pancreatic homoeostasis in health and disease with a large number of important HBPs.

CONCLUSIONS

Our analyses clearly demonstrate that HBPs form disease-specific and highly connected networks that can be explored for potential biomarkers and as collective drug targets via the modification of heparin binding properties.

Network link prediction by global silencing of indirect correlations

Predictions of physical and functional links between cellular components are often based on correlations between experimental measurements, such as gene expression. However, correlations are affected by both direct and indirect paths, confounding our ability to identify true pairwise interactions. Here we exploit the fundamental properties of dynamical correlations in networks to develop a method to silence indirect effects. The method receives as input the observed correlations between node pairs and uses a matrix transformation to turn the correlation matrix into a highly discriminative silenced matrix, which enhances only the terms associated with direct causal links. Against empirical data for Escherichia coli regulatory interactions, the method enhanced the discriminative power of the correlations by twofold, yielding >50% predictive improvement over traditional correlation measures and 6% over mutual information. Overall this silencing method will help translate the abundant correlation data into insights about a system's interactions, with applications ranging from link prediction to inferring the dynamical mechanisms governing biological networks.

Phase clustering in complex networks of delay-coupled oscillators

We study the clusterization of phase oscillators coupled with delay in complex networks. For the case of diffusive oscillators, we formulate the equations relating the topology of the network and the phases and frequencies of the oscillators (functional response). We solve them exactly in directed networks for the case of perfect synchronization. We also compare the reliability of the solution of the linear system for non-linear couplings. Taking advantage of the form of the solution, we propose a frequency adaptation rule to achieve perfect synchronization. We also propose a mean-field theory for uncorrelated random networks that proves to be pretty accurate to predict phase synchronization in real topologies, as for example, the Caenorhabditis elegans or the autonomous systems connectivity.

Structural and functional networks in complex systems with delay

Functional networks of complex systems are obtained from the analysis of the temporal activity of their components, and are often used to infer their unknown underlying connectivity. We obtain the equations relating topology and function in a system of diffusively delay-coupled elements in complex networks. We solve exactly the resulting equations in motifs (directed structures of three nodes) and in directed networks. The mean-field solution for directed uncorrelated networks shows that the clusterization of the activity is dominated by the in-degree of the nodes, and that the locking frequency decreases with increasing average degree. We find that the exponent of a power law degree distribution of the structural topology γ is related to the exponent of the associated functional network as α=(2-γ)(-1) for γ<2.

Mapping from structure to dynamics: a unified view of dynamical processes on networks

Although it is unambiguously agreed that structure plays a fundamental role in shaping the collective dynamics of complex systems, how structure determines dynamics exactly still remains unclear. We investigate a general computational transformation by which we can map the network topology directly to the dynamical patterns emergent on it-independent of the nature of the dynamical processes. Remarkably, we find that many seemingly different dynamical processes on networks, such as coupled oscillators, ensemble neuron firing, epidemic spreading and diffusion can all be understood and unified through this same procedure. Utilizing the inherent multiscale nature of this structure-dynamics transformation, we further define a multiscale complexity measure, which can quantify the functional diversity a general network can support at different organization levels using only its structure. We find that a wide variety of topological features observed in real networks, such as modularity, hierarchy, degree heterogeneity and mixing all result in higher complexity. This result suggests that the demand for functional diversity is driving the structural evolution of physical networks.