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In this paper, the leader-follower robust synchronization issue is mainly addressed for reaction-diffusion neural networks (RDNNs) with multiple leaders and external disturbances under directed graphs. Based on the σ modification approach, we propose a novel distributed adaptive controller by adding a term [Formula: see text] to avoid the phenomenon of parameter drift, that is, the adaptive parameters grow to infinity. Meanwhile, different from the adaptive control algorithm proposed in the undirected graph, we introduce a new function χi(t) to provide additional freedom for the design to achieve robust containment when confronted with external disturbances. Further, the robustness of tracking synchronization with one leader is guaranteed by the proposed adaptive controller when the external disturbances concerning L2 norm are bounded. Finally, relevant numerical simulation graphics are displayed separately to verify the correctness of the related theoretical results.

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This paper considers an output feedback consensus control approach for the generic linear multi-agent systems (MASs) under input saturation over a directed graph. A region of stability-based approach has been established for dealing with the input saturation. A conventional Luenberger observer for estimating the states of followers by themselves and an advanced cooperative observer for estimating the state of leader by followers have been applied for an estimated state feedback control. The stability conditions have been derived by considering a three-term-based combined Lyapunov function. Moreover, computationally simple controller and estimator design conditions have been obtained by resorting to a decoupling approach A set of initial conditions has been investigated to achieve the leader-following consensus of MASs under the input saturation constraint. To the best of our knowledge, an output feedback consensus approach, providing a consensus region, for generic linear MASs under input saturation over directed graphs without requiring the exact state of the leader has been explored for the first time. In contrast to the existing methods, the proposed approach considers an output feedback approach (rather than the state feedback), accounts for both linear and nonlinear saturation regions, applies an estimate of the state of the leader through cooperative observer, and is based on a generalized sector condition for the saturation nonlinearity. In addition, it offers a computationally simple design solution owing to the proposed decoupling method. Simulation results are provided to validate the efficacy of the designed protocol for F-18 aircraft and unmanned ground vehicles.

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The emergence of an autocatalytic network from an available set of elements is a fundamental step in early evolutionary processes, such as the origin of metabolism. Given the set of elements, the reactions between them (chemical or otherwise), and with various elements catalysing certain reactions, a Reflexively Autocatalytic F-generated (RAF) set is a subset R[Formula: see text] of reactions that is self-generating from a given food set, and with each reaction in R[Formula: see text] being catalysed from within R[Formula: see text]. RAF theory has been applied to various phenomena in theoretical biology, and a key feature of the approach is that it is possible to efficiently identify and classify RAFs within large systems. This is possible because RAFs can be described as the (nonempty) subsets of the reactions that are the fixed points of an (efficiently computable) interior map that operates on subsets of reactions. Although the main generic results concerning RAFs can be derived using just this property, we show that for systems with at least 12 reactions there are generic results concerning RAFs that cannot be proven using the interior operator property alone.Kindly check and confirm the edit made in the title.I confirm that the edit is fine.

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We study a discrete-time Markov process on triangular arrays of matrices of size d≥1, driven by inverse Wishart random matrices. The components of the right edge evolve as multiplicative random walks on positive definite matrices with one-sided interactions and can be viewed as a d-dimensional generalisation of log-gamma polymer partition functions. We establish intertwining relations to prove that, for suitable initial configurations of the triangular process, the bottom edge has an autonomous Markovian evolution with an explicit transition kernel. We then show that, for a special singular initial configuration, the fixed-time law of the bottom edge is a matrix Whittaker measure, which we define. To achieve this, we perform a Laplace approximation that requires solving a constrained minimisation problem for certain energy functions of matrix arguments on directed graphs.

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Signed directed graphs contain both sign and direction information on their edges, providing richer information about real-world phenomena compared to unsigned or undirected graphs. However, analyzing such graphs is more challenging due to their complexity, and the limited availability of existing methods. Consequently, despite their potential uses, signed directed graphs have received less research attention. In this paper, we propose a novel spectral graph convolution model that effectively captures the underlying patterns in signed directed graphs. To this end, we introduce a complex Hermitian adjacency matrix that can represent both sign and direction of edges using complex numbers. We then define a magnetic Laplacian matrix based on the adjacency matrix, which we use to perform spectral convolution. We demonstrate that the magnetic Laplacian matrix is positive semi-definite (PSD), which guarantees its applicability to spectral methods. Compared to traditional Laplacians, the magnetic Laplacian captures additional edge information, which makes it a more informative tool for graph analysis. By leveraging the information of signed directed edges, our method generates embeddings that are more representative of the underlying graph structure. Furthermore, we showed that the proposed method has wide applicability for various graph types and is the most generalized Laplacian form. We evaluate the effectiveness of the proposed model through extensive experiments on several real-world datasets. The results demonstrate that our method outperforms state-of-the-art techniques in signed directed graph embedding.

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In this paper, a flexible shape generator (FSG) is designed to achieve the divinable transformation process of the time-varying formation, and consider the FSG-based time-varying formation tracking (TVFT) problem of multiple Lagrangian agents with unknown disturbances and directed graphs. A hierarchical control algorithm is newly designed to achieve the control goal without using the prior information of the system model and bounded disturbances, and the specific implementation of the proposed hierarchical algorithms is also provided. By using the Hurwitz criterion and adaptive system theory, the sufficient conditions are derived and the stability analysis show that the formation tracking errors of the considered system are uniform ultimate bounded. Several simulation examples are performed on five two-degree-of-freedom mechanical arms to show the effectiveness of theoretical results.

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CONTEXT: Ecological Risk Assessments (ERAs) are important tools for supporting evidence-based decision making. However, most ERA frameworks rarely consider complex ecological feedbacks, which limit their capacity to evaluate risks at community and ecosystem levels of organisation. METHOD: We used qualitative mathematical modelling to add additional perspectives to previously conducted ERAs for the rehabilitation of the Ranger uranium mine (Northern Territory, Australia) and support an assessment of the cumulative risks from the mine site. Using expert elicitation workshops, separate qualitative models and scenarios were developed for aquatic and terrestrial systems. The models developed in the workshops were used to construct Bayes Nets that predicted whole-of-ecosystem outcomes after components were perturbed. RESULTS: The terrestrial model considered the effect of fire and weeds on established native vegetation that will be important for the successful rehabilitation of Ranger. It predicted that a combined intervention that suppresses both weeds and fire intensity gave similar response predictions as for weed control alone, except for lower levels of certainty to tall grasses and fire intensity in models with immature trees or tall grasses. However, this had ambiguous predictions for short grasses and forbs, and tall grasses in models representing mature vegetation. The aquatic model considered the effects of magnesium (Mg), a key solute in current and predicted mine runoff and groundwater egress, which is known to adversely affect many aquatic species. The aquatic models provided support that attached algae and phytoplankton assemblages are the key trophic base for food webs. It predicted that shifts in phytoplankton abundance arising from increase in Mg to receiving waters, may result in cascading effects through the food-chain. CONCLUSION: The qualitative modelling approach was flexible and capable of modelling both gradual (i.e. decadal) processes in the mine-site restoration and the comparatively more rapid (seasonal) processes of the aquatic ecosystem. The modelling also provides a useful decision tool for identifying important ecosystem sub-systems for further research efforts.

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We initiate the parameterized complexity study of minimum t-spanner problems on directed graphs. For a positive integer t, a multiplicative t-spanner of a (directed) graph G is a spanning subgraph H such that the distance between any two vertices in H is at most t times the distance between these vertices in G, that is, H keeps the distances in G up to the distortion (or stretch) factor t. An additive t-spanner is defined as a spanning subgraph that keeps the distances up to the additive distortion parameter t, that is, the distances in H and G differ by at most t. The task of Directed Multiplicative Spanner is, given a directed graph G with m arcs and positive integers t and k, decide whether G has a multiplicative t-spanner with at most m - k arcs. Similarly, Directed Additive Spanner asks whether G has an additive t-spanner with at most m - k arcs. We show that (i) Directed Multiplicative Spanner admits a polynomial kernel of size O ( k 4 t 5 ) and can be solved in randomized ( 4 t ) k · n O ( 1 ) time, (ii) the weighted variant of Directed Multiplicative Spanner can be solved in k 2 k · n O ( 1 ) time on directed acyclic graphs, (iii) Directed Additive Spanner is W [ 1 ] -hard when parameterized by k for every fixed t ≥ 1 even when the input graphs are restricted to be directed acyclic graphs. The latter claim contrasts with the recent result of Kobayashi from STACS 2020 that the problem for undirected graphs is FPT when parameterized by t and k.

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A binary state on a graph means an assignment of binary values to its vertices. A time-dependent sequence of binary states is referred to as binary dynamics. We describe a method for the classification of binary dynamics of digraphs, using particular choices of closed neighbourhoods. Our motivation and application comes from neuroscience, where a directed graph is an abstraction of neurons and their connections, and where the simplification of large amounts of data is key to any computation. We present a topological/graph theoretic method for extracting information out of binary dynamics on a graph, based on a selection of a relatively small number of vertices and their neighbourhoods. We consider existing and introduce new real-valued functions on closed neighbourhoods, comparing them by their ability to accurately classify different binary dynamics. We describe a classification algorithm that uses two parameters and sets up a machine learning pipeline. We demonstrate the effectiveness of the method on simulated activity on a digital reconstruction of cortical tissue of a rat, and on a nonbiological random graph with similar density.

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In this work, we present the release of a novel easy to use software package called DGM or Directed-Graph-Mapping. DGM can automatically analyze any type of arrhythmia to find reentry or focal sources if the measurements are synchronized in time. Currently, DGM requires the local activation times (LAT) and the spatial coordinates of the measured electrodes. However, there is no requirement for any spatial organization of the electrodes, allowing to analyze clinical, experimental or computational data. DGM creates directed networks of the activation, which are analyzed with fast algorithms to search for reentry (cycles in the network) and focal sources (nodes with outgoing arrows). DGM has been mainly optimized to analyze atrial tachycardia, but we also discuss other applications of DGM demonstrating its wide applicability. The goal is to release a free software package which can allow researchers to save time in the analysis of cardiac data. An academic license is attached to the software, allowing only non-commercial use of the software. All updates of the software, user and installation guide will be published on a dedicated website www.dgmapping.com . Graphical Abstract Direct-Graph-Mapping is a method to automatically analyze a given arrhythmia by converting measured data of the electrodes in a directed network. DGM requires the local activation times (LAT) and the spatial coordinates of the measured electrodes. There is no requirement for any spatial organization of the electrodes, allowing to analyze clinical, experimental or computational data (see left). An example could be the LATs and coordinates from a CARTO file. DGM creates a directed network of the activation by (1) determining the neighbors of each node, 2 (2) allowing a directed arrow between two neighbors if propagation of the electrical wave is possible, (3) repeating this process for all nodes, (4) if necessary, redistributing the nodes more uniformly and repeating step (1)-(3). Two possible steps are (5) to visualize the wavefront by creating an average graph or (6) find the cycles in the network which represent the reentry loops. Focal sources are nodes with only outgoing arrows.

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How cooperation emerges in human societies is both an evolutionary enigma and a practical problem with tangible implications for societal health. Population structure has long been recognized as a catalyst for cooperation because local interactions facilitate reciprocity. Analysis of population structure typically assumes bidirectional social interactions. But human social interactions are often unidirectional-where one individual has the opportunity to contribute altruistically to another, but not conversely-as the result of organizational hierarchies, social stratification, popularity effects, and endogenous mechanisms of network growth. Here we expand the theory of cooperation in structured populations to account for both uni- and bidirectional social interactions. Even though unidirectional interactions remove the opportunity for reciprocity, we find that cooperation can nonetheless be favored in directed social networks and that cooperation is provably maximized for networks with an intermediate proportion of unidirectional interactions, as observed in many empirical settings. We also identify two simple structural motifs that allow efficient modification of interaction directions to promote cooperation by orders of magnitude. We discuss how our results relate to the concepts of generalized and indirect reciprocity.

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This work presents a novel strategy for classifying neurons, represented by nodes of a directed graph, based on their circuitry (edge connectivity). We assume a stochastic block model (SBM) in which neurons belong together if they connect to neurons of other groups according to the same probability distributions. Following adjacency spectral embedding of the SBM graph, we derive the number of classes and assign each neuron to a class with a Gaussian mixture model-based expectation maximization (EM) clustering algorithm. To improve accuracy, we introduce a simple variation using random hierarchical agglomerative clustering to initialize the EM algorithm and picking the best solution over multiple EM restarts. We test this procedure on a large (≈212-215 neurons), sparse, biologically inspired connectome with eight neuron classes. The simulation results demonstrate that the proposed approach is broadly stable to the choice of embedding dimension, and scales extremely well as the number of neurons in the network increases. Clustering accuracy is robust to variations in model parameters and highly tolerant to simulated experimental noise, achieving perfect classifications with up to 40% of swapped edges. Thus, this approach may be useful to analyze and interpret large-scale brain connectomics data in terms of underlying cellular components.

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Given the recent outbreak of Sars-CoV-2, several countries started to seek different strategies to control contamination and minimize fatalities, which are usually the primary objectives for all strategies. Secondary objectives are related to economic factors, therefore ensuring that society would be able is to keep its essential activities and avoid supply disruptions. This paper presents an application of anonymized mobile phone users' location data to estimate population flow amongst cities with an origin-destination matrix. The work includes a clustering analysis of cities, which may enable policymakers (and epidemiologists) to develop public policies giving the appropriate consideration for each set of cities within a Province or State. Risk measures are included to analyze the severity of the spread among the clusters, which can be ranked. Then, intelligence can be obtained from the analysis, and some clusters could be isolated to avoid contagion while keeping their economic activities. Therefore, this analysis is reproducible for other states of Brazil and other countries and can be adapted for districts within a city, especially considering the possibility of a second wave COVID-19 pandemic.

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Alzheimer's disease has been extensively studied using undirected graphs to represent the correlations of BOLD signals in different anatomical regions through functional magnetic resonance imaging (fMRI). However, there has been relatively little analysis of this kind of data using directed graphs, which potentially offer the potential to capture asymmetries in the interactions between different anatomical brain regions. The detection of these asymmetries is relevant to detect the disease in an early stage. For this reason, in this paper, we analyze data extracted from fMRI images using the net4Lap algorithm to infer a directed graph from the available BOLD signals, and then seek to determine asymmetries between the left and right hemispheres of the brain using a directed version of the Return Random Walk (RRW). Experimental evaluation of this method reveals that it leads to the identification of anatomical brain regions known to be implicated in the early development of Alzheimer's disease in clinical studies.

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BACKGROUND: Metabolic networks reflect the relationships between metabolites (biomolecules) and the enzymes (proteins), and are of particular interest since they describe all chemical reactions of an organism. The metabolic networks are constructed from the genome sequence of an organism, and the graphs can be used to study fluxes through the reactions, or to relate the graph structure to environmental characteristics and phenotypes. About ten years ago, Takemoto et al. (2007) stated that the structure of prokaryotic metabolic networks represented as undirected graphs, is correlated to their living environment. Although metabolic networks are naturally directed graphs, they are still usually analysed as undirected graphs. RESULTS: We implemented a pipeline to reconstruct metabolic networks from genome data and confirmed some of the results of Takemoto et al. (2007) with today data using up-to-date databases. However, Takemoto et al. (2007) used only a fraction of all available enzymes from the genome and taking into account all the enzymes we fail to reproduce the main results. Therefore, we introduce three robust measures on directed representations of graphs, which lead to similar results regardless of the method of network reconstruction. We show that the size of the largest strongly connected component, the flow hierarchy and the Laplacian spectrum are strongly correlated to the environmental conditions. CONCLUSIONS: We found a significant negative correlation between the size of the largest strongly connected component (a cycle) and the optimal growth temperature of the considered prokaryotes. This relationship holds true for the spectrum, high temperature being associated with lower eigenvalues. The hierarchy flow shows a negative correlation with optimal growth temperature. This suggests that the dynamical properties of the network are dependant on environmental factors.

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The protein sequence-structure gap results from the contrast between rapid, low-cost deep sequencing, and slow, expensive experimental structure determination techniques. Comparative homology modelling may have the potential to close this gap by predicting protein structure in target sequences using existing experimentally solved structures as templates. This paper presents the first use of force-directed graphs for the visualization of sequence space in two dimensions, and applies them to the choice of suitable RNA-dependent RNA polymerase (RdRP) target-template pairs within human-infective RNA virus genera. Measures of centrality in protein sequence space for each genus were also derived and used to identify centroid nearest-neighbour sequences (CNNs) potentially useful for production of homology models most representative of their genera. Homology modelling was then carried out for target-template pairs in different species, different genera and different families, and model quality assessed using several metrics. Reconstructed ancestral RdRP sequences for individual genera were also used as templates for the production of ancestral RdRP homology models. High quality ancestral RdRP models were consistently produced, as were good quality models for target-template pairs in the same genus. Homology modelling between genera in the same family produced mixed results and inter-family modelling was unreliable. We present a protocol for the production of optimal RdRP homology models for use in further experiments, e.g. docking to discover novel anti-viral compounds. (219 words).

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This article presents a simple and easily implementable Bayesian approach to model and quantify uncertainty in small descriptive social networks. While statistical methods for analyzing networks have seen burgeoning activity over the last decade or so, ranging from social sciences to genetics, such methods usually involve sophisticated stochastic models whose estimation requires substantial structure and information in the networks. At the other end of the analytic spectrum, there are purely descriptive methods based upon quantities and axioms in computational graph theory. In social networks, popular descriptive measures include, but are not limited to, the so called Krackhardt's axioms. Another approach, recently gaining attention, is the use of PageRank algorithms. While these descriptive approaches provide insight into networks with limited information, including small networks, there is, as yet, little research detailing a statistical approach for small networks. This article aims to contribute at the interface of Bayesian statistical inference and social network analysis by offering practicing social scientists a relatively straightforward Bayesian approach to account for uncertainty while conducting descriptive social network analysis. The emphasis is on computational feasibility and easy implementation using existing R packages, such as sna and rjags, that are available from the Comprehensive R Archive Network (https://cran.r-project.org/). We analyze a network comprising 18 websites from the US and UK to discern transnational identities, previously analyzed using descriptive graph theory with no uncertainty quantification, using fully Bayesian model-based inference.

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We consider the problem of solving consensus using deterministic algorithms in a synchronous dynamic network with unreliable, directional point-to-point links, which are under the control of a message adversary. In contrast to the large body of existing work that focuses on message adversaries that pick the communication graphs from a predefined set of candidate graphs arbitrarily, we consider message adversaries that also allow to express eventual properties, like stable periods that occur only eventually. Such message adversaries can model systems that exhibit erratic boot-up phases or recover after repeatedly occurring, massive transient faults. We precisely determine how much eventual stability is necessary and sufficient, and provide an optimal consensus algorithm. Unlike in the case of longer stability periods, where standard algorithms can be adapted for solving consensus, different algorithmic techniques are needed in the case of short-lived stability.

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A special type of social networks is the so-called affiliation network, consisting of two modes of vertices: actors and events. Up to now, in the undirected case, the closeness of actors in such networks has been measured by their jointly-attended events. Indirect contacts and attenuated and directed links are of minor interest in affiliation networks. These flaws make a veritable estimation of, e.g., possible message transfers amongst actors questionable. In this contribution, first, we discuss these matters from a graph-theoretical point of view. Second, so as to avoid the identified weaknesses, we propose an up-and-coming entropy-based approach for modeling such networks in their generic structure, replacing directed (attenuated) links by conditionals: if-then. In this framework, the contribution of actors and events to a reliable message transfer from one actor to another-even via intermediaries-is then calculated applying the principle of maximum entropy. The usefulness of this new approach is demonstrated by the analysis of an affiliation network called "corporate directors".

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We consider the problem of measuring the similarity between two graphs using continuous-time quantum walks and comparing their time-evolution by means of the quantum Jensen-Shannon divergence. Contrary to previous works that focused solely on undirected graphs, here we consider the case of both directed and undirected graphs. We also consider the use of alternative Hamiltonians as well as the possibility of integrating additional node-level topological information into the proposed framework. We set up a graph classification task and we provide empirical evidence that: (1) our similarity measure can effectively incorporate the edge directionality information, leading to a significant improvement in classification accuracy; (2) the choice of the quantum walk Hamiltonian does not have a significant effect on the classification accuracy; (3) the addition of node-level topological information improves the classification accuracy in some but not all cases. We also theoretically prove that under certain constraints, the proposed similarity measure is positive definite and thus a valid kernel measure. Finally, we describe a fully quantum procedure to compute the kernel.