Turing patterns mediated by network topology in homogeneous active systems

The relationship between clustering and networked Turing patterns

Networked Turing patterns often manifest as groups of nodes distributed on either side of the homogeneous equilibrium, exhibiting high and low density. These pattern formations are significantly influenced by network topological characteristics, such as the average degree. However, the impact of clustering on them remains inadequately understood. Here, we investigate the relationship between clustering and networked Turing patterns using classical prey-predator models. Our findings reveal that when nodes of high and low density are completely distributed on both sides of the homogeneous equilibrium, there is a linear decay in Turing patterns as global clustering coefficients increase, given a fixed node size and average degree; otherwise, this linear decay may not always hold due to the presence of high-density nodes considered as low-density nodes. This discovery provides a qualitative assessment of how clustering coefficients impact the formation of Turing patterns and may contribute to understanding why using refuges in ecosystems could enhance the stability of prey-predator systems. The results link network topological structures with the stability of prey-predator systems, offering new insights into predicting and controlling pattern formations in real-world systems from a network perspective.

A time independent least squares algorithm for parameter identification of Turing patterns in reaction-diffusion systems

Turing patterns arising from reaction-diffusion systems such as epidemic, ecology or chemical reaction models are an important dynamic property. Parameter identification of Turing patterns in spatial continuous and networked reaction-diffusion systems is an interesting and challenging inverse problem. The existing algorithms require huge account operations and resources. These drawbacks are amplified when apply them to reaction-diffusion systems on large-scale complex networks. To overcome these shortcomings, we present a new least squares algorithm which is rooted in the fact that Turing patterns are the stationary solutions of reaction-diffusion systems. The new algorithm is time independent, it translates the parameter identification problem into a low dimensional optimization problem even a low order linear algebra equations. The numerical simulations demonstrate that our algorithm has good effectiveness, robustness as well as performance.

Connectome-based prediction of functional impairment in experimental stroke models

Experimental rat models of stroke and hemorrhage are important tools to investigate cerebrovascular disease pathophysiology mechanisms, yet how significant patterns of functional impairment induced in various models of stroke are related to changes in connectivity at the level of neuronal populations and mesoscopic parcellations of rat brains remain unresolved. To address this gap in knowledge, we employed two middle cerebral artery occlusion models and one intracerebral hemorrhage model with variant extent and location of neuronal dysfunction. Motor and spatial memory function was assessed and the level of hippocampal activation via Fos immunohistochemistry. Contribution of connectivity change to functional impairment was analyzed for connection similarities, graph distances and spatial distances as well as the importance of regions in terms of network architecture based on the neuroVIISAS rat connectome. We found that functional impairment correlated with not only the extent but also the locations of the injury among the models. In addition, via coactivation analysis in dynamic rat brain models, we found that lesioned regions led to stronger coactivations with motor function and spatial learning regions than with other unaffected regions of the connectome. Dynamic modeling with the weighted bilateral connectome detected changes in signal propagation in the remote hippocampus in all 3 stroke types, predicting the extent of hippocampal hypoactivation and impairment in spatial learning and memory function. Our study provides a comprehensive analytical framework in predictive identification of remote regions not directly altered by stroke events and their functional implication.

Complex Network Evolution Model Based on Turing Pattern Dynamics

Complex network models are helpful to explain the evolution rules of network structures, and also are the foundations of understanding and controlling complex networks. The existing studies (e.g., scale-free model, small-world model) are insufficient to uncover the internal mechanisms of the emergence and evolution of communities in networks. To overcome the above limitation, in consideration of the fact that a network can be regarded as a pattern composed of communities, we introduce Turing pattern dynamic as theory support to construct the network evolution model. Specifically, we develop a Reaction-Diffusion model according to Q-Learning technology (RDQL), in which each node regarded as an intelligent agent makes a behavior choice to update its relationships, based on the utility and behavioral strategy at every time step. Extensive experiments indicate that our model not only reveals how communities form and evolve, but also can generate networks with the properties of scale-free, small-world and assortativity. The effectiveness of the RDQL model has also been verified by its application in real networks. Furthermore, the depth analysis of the RDQL model provides a conclusion that the proportion of exploration and exploitation behaviors of nodes is the only factor affecting the formation of communities. The proposed RDQL model has potential to be the basic theoretical tool for studying network stability and dynamics.

Turing patterns in simplicial complexes

The spontaneous emergence of patterns in nature, such as stripes and spots, can be mathematically explained by reaction-diffusion systems. These patterns are often referred as Turing patterns to honor the seminal work of Alan Turing in the early 1950s. With the coming of age of network science, and with its related departure from diffusive nearest-neighbor interactions to long-range links between nodes, additional layers of complexity behind pattern formation have been discovered, including irregular spatiotemporal patterns. Here we investigate the formation of Turing patterns in simplicial complexes, where links no longer connect just pairs of nodes but can connect three or more nodes. Such higher-order interactions are emerging as a new frontier in network science, in particular describing group interaction in various sociological and biological systems, so understanding pattern formation under these conditions is of the utmost importance. We show that a canonical reaction-diffusion system defined over a simplicial complex yields Turing patterns that fundamentally differ from patterns observed in traditional networks. For example, we observe a stable distribution of Turing patterns where the fraction of nodes with reactant concentrations above the equilibrium point is exponentially related to the average degree of 2-simplexes, and we uncover parameter regions where Turing patterns will emerge only under higher-order interactions, but not under pairwise interactions.

Pattern mechanism in stochastic SIR networks with ER connectivity

The diffusion of the susceptible and infected is a vital factor in spreading infectious diseases. However, the previous SIR networks cannot explain the dynamical mechanism of periodic behavior and endemic diseases. Here, we incorporate the diffusion and network effect into the SIR model and describes the mechanism of periodic behavior and endemic diseases through wavenumber and saddle-node bifurcation. We also introduce the standard network structured entropy (NSE) and demonstrate diffusion effect could induce the saddle-node bifurcation and Turing instability. Then we reveal the mechanism of the periodic outbreak and endemic diseases by the mean-field method. We provide the Turing instability condition through wavenumber in this network-organized SIR model. In the end, the data from COVID-19 authenticated the theoretical results.

The Consensus Problem in Polities of Agents with Dissimilar Cognitive Architectures

Agents interacting with their environments, machine or otherwise, arrive at decisions based on their incomplete access to data and their particular cognitive architecture, including data sampling frequency and memory storage limitations. In particular, the same data streams, sampled and stored differently, may cause agents to arrive at different conclusions and to take different actions. This phenomenon has a drastic impact on polities-populations of agents predicated on the sharing of information. We show that, even under ideal conditions, polities consisting of epistemic agents with heterogeneous cognitive architectures might not achieve consensus concerning what conclusions to draw from datastreams. Transfer entropy applied to a toy model of a polity is analyzed to showcase this effect when the dynamics of the environment is known. As an illustration where the dynamics is not known, we examine empirical data streams relevant to climate and show the consensus problem manifest.

The qualitative and quantitative relationships between pattern formation and average degree in networked reaction-diffusion systems

The Turing pattern is an important dynamic behavior characteristic of activator-inhibitor systems. Differentiating from traditional assumption of activator-inhibitor interactions in a spatially continuous domain, a Turing pattern in networked reaction-diffusion systems has received much attention during the past few decades. In spite of its great progress, it still fails to evaluate the precise influences of network topology on pattern formation. To this end, we try to promote the research on this important and interesting issue from the point of view of average degree-a critical topological feature of networks. We first qualitatively analyze the influence of average degree on pattern formation. Then, a quantitative relationship between pattern formation and average degree, the exponential decay of pattern formation, is proposed via nonlinear regression. The finding holds true for several activator-inhibitor systems including biology model, ecology model, and chemistry model. The significance of this study lies that the exponential decay not only quantitatively depicts the influence of average degree on pattern formation, but also provides the possibility for predicting and controlling pattern formation.

Turing pattern induced by the directed ER network and delay

Infectious diseases generally spread along with the asymmetry of social network propagation because the asymmetry of urban development and the prevention strategies often affect the direction of the movement. But the spreading mechanism of the epidemic remains to explore in the directed network. In this paper, the main effect of the directed network and delay on the dynamic behaviors of the epidemic is investigated. The algebraic expressions of Turing instability are given to show the role of the directed network in the spread of the epidemic, which overcomes the drawback that undirected networks cannot lead to the outbreaks of infectious diseases. Then, Hopf bifurcation is analyzed to illustrate the dynamic mechanism of the periodic outbreak, which is consistent with the transmission of COVID-19. Also, the discrepancy ratio between the imported and the exported is proposed to explain the importance of quarantine policies and the spread mechanism. Finally, the theoretical results are verified by numerical simulation.

Assessing the risk of pandemic outbreaks across municipalities with mathematical descriptors based on age and mobility restrictions

By March 14th 2022, Spain is suffering the sixth wave of the COVID-19 pandemic. All the previous waves have been intimately related to the degree of imposed mobility restrictions and its consequent release. Certain factors explain the incidence of the virus across regions revealing the weak locations that probably require some medical reinforcements. The most relevant ones relate with mobility restrictions by age and administrative competence, i.e., spatial constrains. In this work, we aim to find a mathematical descriptor that could identify the critical communities that are more likely to suffer pandemic outbreaks and, at the same time, to estimate the impact of different mobility restrictions. We analyze the incidence of the virus in combination with mobility flows during the so-called second wave (roughly from August 1st to November 30th, 2020) using a SEIR compartmental model. After that, we derive a mathematical descriptor based on linear stability theory that quantifies the potential impact of becoming a hotspot. Once the model is validated, we consider different confinement scenarios and containment protocols aimed to control the virus spreading. The main findings from our simulations suggest that the confinement of the economically non-active individuals may result in a significant reduction of risk, whose effects are equivalent to the confinement of the total population. This study is conducted across the totality of municipalities in Spain.

A sampling-guided unsupervised learning method to capture percolation in complex networks

The use of machine learning methods in classical and quantum systems has led to novel techniques to classify ordered and disordered phases, as well as uncover transition points in critical phenomena. Efforts to extend these methods to dynamical processes in complex networks is a field of active research. Network-percolation, a measure of resilience and robustness to structural failures, as well as a proxy for spreading processes, has numerous applications in social, technological, and infrastructural systems. A particular challenge is to identify the existence of a percolation cluster in a network in the face of noisy data. Here, we consider bond-percolation, and introduce a sampling approach that leverages the core-periphery structure of such networks at a microscopic scale, using onion decomposition, a refined version of the k-core. By selecting subsets of nodes in a particular layer of the onion spectrum that follow similar trajectories in the percolation process, percolating phases can be distinguished from non-percolating ones through an unsupervised clustering method. Accuracy in the initial step is essential for extracting samples with information-rich content, that are subsequently used to predict the critical transition point through the confusion scheme, a recently introduced learning method. The method circumvents the difficulty of missing data or noisy measurements, as it allows for sampling nodes from both the core and periphery, as well as intermediate layers. We validate the effectiveness of our sampling strategy on a spectrum of synthetic network topologies, as well as on two real-word case studies: the integration time of the US domestic airport network, and the identification of the epidemic cluster of COVID-19 outbreaks in three major US states. The method proposed here allows for identifying phase transitions in empirical time-varying networks.

Optimal control of networked reaction-diffusion systems

Patterns in nature are fascinating both aesthetically and scientifically. Alan Turing's celebrated reaction-diffusion model of pattern formation from the 1950s has been extended to an astounding diversity of applications: from cancer medicine, via nanoparticle fabrication, to computer architecture. Recently, several authors have studied pattern formation in underlying networks, but thus far, controlling a reaction-diffusion system in a network to obtain a particular pattern has remained elusive. We present a solution to this problem in the form of an analytical framework and numerical algorithm for optimal control of Turing patterns in networks. We demonstrate our method's effectiveness and discuss factors that affect its performance. We also pave the way for multidisciplinary applications of our framework beyond reaction-diffusion models.

Robust controlled formation of Turing patterns in three-component systems

Over the past few decades, formation of Turing patterns in reaction-diffusion systems has been shown to be the underlying process in several examples of biological morphogenesis, confirming Alan Turing's hypothesis, put forward in 1952. However, theoretical studies suggest that Turing patterns formation via classical "short-range activation and long-range inhibition" concept in general can happen within only narrow parameter ranges. This feature seemingly contradicts the accuracy and reproducibility of biological morphogenesis given the stochasticity of biochemical processes and the influence of environmental perturbations. Moreover, it represents a major hurdle to synthetic engineering of Turing patterns. In this work it is shown that this problem can be overcome in some systems under certain sets of interactions between their elements, one of which is immobile and therefore corresponding to a cell-autonomous factor. In such systems Turing patterns formation can be guaranteed by a simple universal control under any values of kinetic parameters and diffusion coefficients of mobile elements. This concept is illustrated by analysis and simulations of a specific three-component system, characterized in absence of diffusion by a presence of codimension two pitchfork-Hopf bifurcation.

Predictable topological sensitivity of Turing patterns on graphs

Reaction-diffusion systems implemented as dynamical processes on networks have recently renewed the interest in their self-organized collective patterns known as Turing patterns. We investigate the influence of network topology on the emerging patterns and their diversity, defined as the variety of stationary states observed with random initial conditions and the same dynamics. We show that a seemingly minor change, the removal or rewiring of a single link, can prompt dramatic changes in pattern diversity. The determinants of such critical occurrences are explored through an extensive and systematic set of numerical experiments. We identify situations where the topological sensitivity of the attractor landscape can be predicted without a full simulation of the dynamical equations, from the spectrum of the graph Laplacian and the linearized dynamics. Unexpectedly, the main determinant appears to be the degeneracy of the eigenvalues or the growth rate and not the number of unstable modes.

Risk evaluation at municipality level of a COVID-19 outbreak incorporating relevant geographic data: the study case of Galicia

The COVID-19 pandemic was an inevitable outcome of a globalized world in which a highly infective disease is able to reach every country in a matter of weeks. While lockdowns and strong mobility restrictions have proven to be efficient to contain the exponential transmission of the virus, its pervasiveness has made it impossible for economies to maintain this kind of measures in time. Understanding precisely how the spread of the virus occurs from a territorial perspective is crucial not only to prevent further infections but also to help with policy design regarding human mobility. From the large spatial differences in the behavior of the virus spread we can unveil which areas have been more vulnerable to it and why, and with this information try to assess the risk that each community has to suffer a future outbreak of infection. In this work we have analyzed the geographical distribution of the cumulative incidence during the first wave of the pandemic in the region of Galicia (north western part of Spain), and developed a mathematical approach that assigns a risk factor for each of the different municipalities that compose the region. This risk factor is independent of the actual evolution of the pandemic and incorporates geographic and demographic information. The comparison with empirical information from the first pandemic wave demonstrates the validity of the method. Our results can potentially be used to design appropriate preventive policies that help to contain the virus.

A theory of pattern formation for reaction–diffusion systems on temporal networks

Networks have become ubiquitous in the modern scientific literature, with recent work directed at understanding ‘temporal networks’—those networks having structure or topology which evolves over time. One area of active interest is pattern formation from reaction–diffusion systems, which themselves evolve over temporal networks. We derive analytical conditions for the onset of diffusive spatial and spatio-temporal pattern formation on undirected temporal networks through the Turing and Benjamin–Feir mechanisms, with the resulting pattern selection process depending strongly on the evolution of both global diffusion rates and the local structure of the underlying network. Both instability criteria are then extended to the case where the reaction–diffusion system is non-autonomous, which allows us to study pattern formation from time-varying base states. The theory we present is illustrated through a variety of numerical simulations which highlight the role of the time evolution of network topology, diffusion mechanisms and non-autonomous reaction kinetics on pattern formation or suppression. A fundamental finding is that Turing and Benjamin–Feir instabilities are generically transient rather than eternal, with dynamics on temporal networks able to transition between distinct patterns or spatio-temporal states. One may exploit this feature to generate new patterns, or even suppress undesirable patterns, over a given time interval.

Turing instability in the reaction-diffusion network

It is an established fact that a positive wave number plays an essential role in Turing instability. However, the impact of a negative wave number on Turing instability remains unclear. Here, we investigate the effect of the weights and nodes on Turing instability in the FitzHugh-Nagumo model, and theoretical results reveal genesis of Turing instability due to a negative wave number through the stability analysis and mean-field method. We obtain the Turing instability region in the continuous media system and provide the relationship between degree and eigenvalue of the network matrix by the Gershgorin circle theorem. Furthermore, the Turing instability condition about nodes and the weights is provided in the network-organized system. Additionally, we found chaotic behavior because of interactions between I Turing instability and II Turing instability. Besides, we apply this above analysis to explaining the mechanism of the signal conduction of the inhibitory neuron. We find a moderate coupling strength and corresponding number of links are necessary to the signal conduction.

Cross-diffusion-induced patterns in an SIR epidemic model on complex networks

Infectious diseases are a major threat to global health. Spatial patterns revealed by epidemic models governed by reaction-diffusion systems can serve as a potential trend indicator of disease spread; thus, they have received wide attention. To characterize important features of disease spread, there are two important factors that cannot be ignored in the reaction-diffusion systems. One is that a susceptible individual has an ability to recognize the infected ones and keep away from them. The other is that populations are usually organized as networks instead of being continuously distributed in space. Consequently, it is essential to study patterns generated by epidemic models with self- and cross-diffusion on complex networks. Here, with the help of a linear analysis method, we study Turing instability induced by cross-diffusion for a network organized SIR epidemic model and explore Turing patterns on several different networks. Furthermore, the influences of cross-diffusion and network structure on patterns are also investigated.