Mean species responses predict effects of environmental change on coexistence.

Environmental change research is plagued by the curse of dimensionality: the number of communities at risk and the number of environmental drivers are both large. This raises the pressing question if a general understanding of ecological effects is achievable. Here, we show evidence that this is indeed possible. Using theoretical and simulation-based evidence for bi- and tritrophic communities, we show that environmental change effects on coexistence are proportional to mean species responses and depend on how trophic levels on average interact prior to environmental change. We then benchmark our findings using relevant cases of environmental change, showing that means of temperature optima and of species sensitivities to pollution predict concomitant effects on coexistence. Finally, we demonstrate how to apply our theory to the analysis of field data, finding support for effects of land use change on coexistence in natural invertebrate communities.

Global Topological Synchronization on Simplicial and Cell Complexes.

Topological signals, i.e., dynamical variables defined on nodes, links, triangles, etc. of higher-order networks, are attracting increasing attention. However, the investigation of their collective phenomena is only at its infancy. Here we combine topology and nonlinear dynamics to determine the conditions for global synchronization of topological signals defined on simplicial or cell complexes. On simplicial complexes we show that topological obstruction impedes odd dimensional signals to globally synchronize. On the other hand, we show that cell complexes can overcome topological obstruction and in some structures signals of any dimension can achieve global synchronization.

Self-segregation in heterogeneous metapopulation landscapes.

Complex interactions are at the root of the population dynamics of many natural systems, particularly for being responsible for the allocation of species and individuals across apposite niches of the ecological landscapes. On the other side, the randomness that unavoidably characterises complex systems has increasingly challenged the niche paradigm providing alternative neutral theoretical models. We introduce a network-inspired metapopulation individual-based model (IBM), hereby named self-segregation, where the density of individuals in the hosting patches (local habitats) drives the individuals spatial assembling while still constrained by nodes' saturation. In particular, we prove that the core-periphery structure of the networked landscape triggers the spontaneous emergence of vacant habitat patches, which segregate the population in multistable patterns of isolated (sub)communities separated by empty patches. Furthermore, a quantisation effect in the number of vacant patches is observed once the total system mass varies continuously, emphasising thus a striking feature of the robustness of population stationary distributions. Notably, our model reproduces the patch vacancy found in the fragmented habitat of the Glanville fritillary butterfly Melitaea cinxia, an endemic species of the Åland islands. We argue that such spontaneous breaking of the natural habitat supports the concept of the highly contentious (Grinnellian) niche vacancy and also suggests a new mechanism for the endogeneous habitat fragmentation and consequently the peripatric speciation.

Synchronization Dynamics in Non-Normal Networks: The Trade-Off for Optimality.

Synchronization is an important behavior that characterizes many natural and human made systems that are composed by several interacting units. It can be found in a broad spectrum of applications, ranging from neuroscience to power-grids, to mention a few. Such systems synchronize because of the complex set of coupling they exhibit, with the latter being modeled by complex networks. The dynamical behavior of the system and the topology of the underlying network are strongly intertwined, raising the question of the optimal architecture that makes synchronization robust. The Master Stability Function (MSF) has been proposed and extensively studied as a generic framework for tackling synchronization problems. Using this method, it has been shown that, for a class of models, synchronization in strongly directed networks is robust to external perturbations. Recent findings indicate that many real-world networks are strongly directed, being potential candidates for optimal synchronization. Moreover, many empirical networks are also strongly non-normal. Inspired by this latter fact in this work, we address the role of the non-normality in the synchronization dynamics by pointing out that standard techniques, such as the MSF, may fail to predict the stability of synchronized states. We demonstrate that, due to a transient growth that is induced by the structure's non-normality, the system might lose synchronization, contrary to the spectral prediction. These results lead to a trade-off between non-normality and directedness that should be properly considered when designing an optimal network, enhancing the robustness of synchronization.

Patterns of non-normality in networked systems.

Several mechanisms have been proposed to explain the spontaneous generation of self-organised patterns, hypothesised to play a role in the formation of many of the magnificent patterns observed in Nature. In several cases of interest, the system under scrutiny displays a homogeneous equilibrium, which is destabilised via a symmetry breaking instability which reflects the specificity of the problem being inspected. The Turing instability is among the most celebrated paradigms for pattern formation. In its original form, the diffusion constants of the two mobile species need to be quite different from each other for the instability to develop. Unfortunately, this condition limits the applicability of the theory. To overcome this impediment, and with the ambitious long term goal to eventually reconcile theory and experiments, we here propose an alternative mechanism for promoting the onset of pattern. To this end a multi-species reactive model is studied, assuming a generalized transport on a discrete and directed network-like support: the instability is triggered by the non-normality of the embedding network. The non-normal character of the dynamics instigates a short time amplification of the imposed perturbation, thus making the system unstable for a choice of parameters that would yield stability under the conventional scenario. In other words, non-normality promotes the emergence of patterns in cases where a classical linear analysis would not predict them. The importance of our result relies also on the fact that non-normal networks are pervasively found, motivating the general interest of the mechanism discussed here.

A minimally invasive neurostimulation method for controlling abnormal synchronisation in the neuronal activity.

Many collective phenomena in Nature emerge from the -partial- synchronisation of the units comprising a system. In the case of the brain, this self-organised process allows groups of neurons to fire in highly intricate partially synchronised patterns and eventually lead to high level cognitive outputs and control over the human body. However, when the synchronisation patterns are altered and hypersynchronisation occurs, undesirable effects can occur. This is particularly striking and well documented in the case of epileptic seizures and tremors in neurodegenerative diseases such as Parkinson's disease. In this paper, we propose an innovative, minimally invasive, control method that can effectively desynchronise misfiring brain regions and thus mitigate and even eliminate the symptoms of the diseases. The control strategy, grounded in the Hamiltonian control theory, is applied to ensembles of neurons modelled via the Kuramoto or the Stuart-Landau models and allows for heterogeneous coupling among the interacting unities. The theory has been complemented with dedicated numerical simulations performed using the small-world Newman-Watts network and the random Erdos-Rényi network. Finally the method has been compared with the gold-standard Proportional-Differential Feedback control technique. Our method is shown to achieve equivalent levels of desynchronisation using lesser control strength and/or fewer controllers, being thus minimally invasive.

Resilience for stochastic systems interacting via a quasi-degenerate network.

A stochastic reaction-diffusion model is studied on a networked support. In each patch of the network, two species are assumed to interact following a non-normal reaction scheme. When the interaction unit is replicated on a directed linear lattice, noise gets amplified via a self-consistent process, which we trace back to the degenerate spectrum of the embedding support. The same phenomenon holds when the system is bound to explore a quasidegenerate network. In this case, the eigenvalues of the Laplacian operator, which governs species diffusion, accumulate over a limited portion of the complex plane. The larger the network, the more pronounced the amplification. Beyond a critical network size, a system deemed deterministically stable, hence resilient, can develop seemingly regular patterns in the concentration amount. Non-normality and quasidegenerate networks may, therefore, amplify the inherent stochasticity and so contribute to altering the perception of resilience, as quantified via conventional deterministic methods.

Hopping in the Crowd to Unveil Network Topology.

We introduce a nonlinear operator to model diffusion on a complex undirected network under crowded conditions. We show that the asymptotic distribution of diffusing agents is a nonlinear function of the nodes' degree and saturates to a constant value for sufficiently large connectivities, at variance with standard diffusion in the absence of excluded-volume effects. Building on this observation, we define and solve an inverse problem, aimed at reconstructing the a priori unknown connectivity distribution. The method gathers all the necessary information by repeating a limited number of independent measurements of the asymptotic density at a single node, which can be chosen randomly. The technique is successfully tested against both synthetic and real data and is also shown to estimate with great accuracy the total number of nodes.

Theory of Turing Patterns on Time Varying Networks.

The process of pattern formation for a multispecies model anchored on a time varying network is studied. A nonhomogeneous perturbation superposed to an homogeneous stable fixed point can be amplified following the Turing mechanism of instability, solely instigated by the network dynamics. By properly tuning the frequency of the imposed network evolution, one can make the examined system behave as its averaged counterpart, over a finite time window. This is the key observation to derive a closed analytical prediction for the onset of the instability in the time dependent framework. Continuously and piecewise constant periodic time varying networks are analyzed, setting the framework for the proposed approach. The extension to nonperiodic settings is also discussed.

Multi-objective genetic algorithm for the optimization of a flat-plate solar thermal collector.

We present a multi-objective genetic algorithm we developed for the optimization of a flat-plate solar thermal collector. This collector consists of a waffle-shaped Al substrate with NiCrOx cermet and SnO(2) anti-reflection conformal coatings. Optimal geometrical parameters are determined in order to (i) maximize the solar absorptance α and (ii) minimize the thermal emittance ε. The multi-objective genetic algorithm eventually provides a whole set of Pareto-optimal solutions for the optimization of α and ε, which turn out to be competitive with record values found in the literature. In particular, a solution that enables α = 97.8% and ε = 4.8% was found.

Speed-accuracy trade-offs in best-of-n collective decision making through heterogeneous mean-field modeling.

To succeed in their objectives, groups of individuals must be able to make quick and accurate collective decisions on the best option among a set of alternatives with different qualities. Group-living animals aim to do that all the time. Plants and fungi are thought to do so too. Swarms of autonomous robots can also be programed to make best-of-n decisions for solving tasks collaboratively. Ultimately, humans critically need it and so many times they should be better at it! Thanks to their mathematical tractability, simple models like the voter model and the local majority rule model have proven useful to describe the dynamics of such collective decision-making processes. To reach a consensus, individuals change their opinion by interacting with neighbors in their social network. At least among animals and robots, options with a better quality are exchanged more often and therefore spread faster than lower-quality options, leading to the collective selection of the best option. With our work, we study the impact of individuals making errors in pooling others' opinions caused, for example, by the need to reduce the cognitive load. Our analysis is grounded on the introduction of a model that generalizes the two existing models (local majority rule and voter model), showing a speed-accuracy trade-off regulated by the cognitive effort of individuals. We also investigate the impact of the interaction network topology on the collective dynamics. To do so, we extend our model and, by using the heterogeneous mean-field approach, we show the presence of another speed-accuracy trade-off regulated by network connectivity. An interesting result is that reduced network connectivity corresponds to an increase in collective decision accuracy.

Phase chimera states on nonlocal hyperrings.

Chimera states are dynamical states where regions of synchronous trajectories coexist with incoherent ones. A significant amount of research has been devoted to studying chimera states in systems of identical oscillators, nonlocally coupled through pairwise interactions. Nevertheless, there is increasing evidence, also supported by available data, that complex systems are composed of multiple units experiencing many-body interactions that can be modeled by using higher-order structures beyond the paradigm of classic pairwise networks. In this work we investigate whether phase chimera states appear in this framework, by focusing on a topology solely involving many-body, nonlocal, and nonregular interactions, hereby named nonlocal d-hyperring, (d+1) being the order of the interactions. We present the theory by using the paradigmatic Stuart-Landau oscillators as node dynamics, and we show that phase chimera states emerge in a variety of structures and with different coupling functions. For comparison, we show that, when higher-order interactions are "flattened" to pairwise ones, the chimera behavior is weaker and more elusive.

Altered dynamical integration/segregation balance during anesthesia-induced loss of consciousness.

In recent years, brain imaging studies have begun to shed light on the neural correlates of physiologically-reversible altered states of consciousness such as deep sleep, anesthesia, and psychedelic experiences. The emerging consensus is that normal waking consciousness requires the exploration of a dynamical repertoire enabling both global integration i.e., long-distance interactions between brain regions, and segregation, i.e., local processing in functionally specialized clusters. Altered states of consciousness have notably been characterized by a tipping of the integration/segregation balance away from this equilibrium. Historically, functional MRI (fMRI) has been the modality of choice for such investigations. However, fMRI does not enable characterization of the integration/segregation balance at sub-second temporal resolution. Here, we investigated global brain spatiotemporal patterns in electrocorticography (ECoG) data of a monkey (Macaca fuscata) under either ketamine or propofol general anesthesia. We first studied the effects of these anesthetics from the perspective of band-specific synchronization across the entire ECoG array, treating individual channels as oscillators. We further aimed to determine whether synchrony within spatially localized clusters of oscillators was differently affected by the drugs in comparison to synchronization over spatially distributed subsets of ECoG channels, thereby quantifying changes in integration/segregation balance on physiologically-relevant time scales. The findings reflect global brain dynamics characterized by a loss of long-range integration in multiple frequency bands under both ketamine and propofol anesthesia, most pronounced in the beta (13-30 Hz) and low-gamma bands (30-80 Hz), and with strongly preserved local synchrony in all bands.

Diffusion-driven instability of topological signals coupled by the Dirac operator.

The study of reaction-diffusion systems on networks is of paramount relevance for the understanding of nonlinear processes in systems where the topology is intrinsically discrete, such as the brain. Until now, reaction-diffusion systems have been studied only when species are defined on the nodes of a network. However, in a number of real systems including, e.g., the brain and the climate, dynamical variables are not only defined on nodes but also on links, faces, and higher-dimensional cells of simplicial or cell complexes, leading to topological signals. In this work, we study reaction-diffusion processes of topological signals coupled through the Dirac operator. The Dirac operator allows topological signals of different dimension to interact or cross-diffuse as it projects the topological signals defined on simplices or cells of a given dimension to simplices or cells of one dimension up or one dimension down. By focusing on the framework involving nodes and links, we establish the conditions for the emergence of Turing patterns and we show that the latter are never localized only on nodes or only on links of the network. Moreover, when the topological signals display a Turing pattern their projection does as well. We validate the theory hereby developed on a benchmark network model and on square lattices with periodic boundary conditions.

Machine learning in spectral domain.

Deep neural networks are usually trained in the space of the nodes, by adjusting the weights of existing links via suitable optimization protocols. We here propose a radically new approach which anchors the learning process to reciprocal space. Specifically, the training acts on the spectral domain and seeks to modify the eigenvalues and eigenvectors of transfer operators in direct space. The proposed method is ductile and can be tailored to return either linear or non-linear classifiers. Adjusting the eigenvalues, when freezing the eigenvectors entries, yields performances that are superior to those attained with standard methods restricted to operate with an identical number of free parameters. To recover a feed-forward architecture in direct space, we have postulated a nested indentation of the eigenvectors. Different non-orthogonal basis could be employed to export the spectral learning to other frameworks, as e.g. reservoir computing.

Training of sparse and dense deep neural networks: Fewer parameters, same performance.

Deep neural networks can be trained in reciprocal space by acting on the eigenvalues and eigenvectors of suitable transfer operators in direct space. Adjusting the eigenvalues while freezing the eigenvectors yields a substantial compression of the parameter space. This latter scales by definition with the number of computing neurons. The classification scores as measured by the displayed accuracy are, however, inferior to those attained when the learning is carried in direct space for an identical architecture and by employing the full set of trainable parameters (with a quadratic dependence on the size of neighbor layers). In this paper, we propose a variant of the spectral learning method as in Giambagli et al. [Nat. Commun. 12, 1330 (2021)2041-172310.1038/s41467-021-21481-0], which leverages on two sets of eigenvalues for each mapping between adjacent layers. The eigenvalues act as veritable knobs which can be freely tuned so as to (1) enhance, or alternatively silence, the contribution of the input nodes and (2) modulate the excitability of the receiving nodes with a mechanism which we interpret as the artificial analog of the homeostatic plasticity. The number of trainable parameters is still a linear function of the network size, but the performance of the trained device gets much closer to those obtained via conventional algorithms, these latter requiring, however, a considerably heavier computational cost. The residual gap between conventional and spectral trainings can be eventually filled by employing a suitable decomposition for the nontrivial block of the eigenvectors matrix. Each spectral parameter reflects back on the whole set of internode weights, an attribute which we effectively exploit to yield sparse networks with stunning classification abilities as compared to their homologs trained with conventional means.

Turing patterns in a network-reduced FitzHugh-Nagumo model.

Reduction of a two-component FitzHugh-Nagumo model to a single-component model with long-range connection is considered on general networks. The reduced model describes a single chemical species reacting on the nodes and diffusing across the links with weighted long-range connections, which can be interpreted as a class of networked dynamical systems on a multigraph with local and nonlocal Laplace matrices that self-consistently emerge from the adiabatic elimination. We study the conditions for the instability of homogeneous states in the original and reduced models and show that Turing patterns can emerge in both models. We also consider generality of the adiabatic elimination for a wider class of slow-fast systems and discuss the peculiarity of the FitzHugh-Nagumo model.

Random walks on hypergraphs.

In the past 20 years network science has proven its strength in modeling many real-world interacting systems as generic agents, the nodes, connected by pairwise edges. Nevertheless, in many relevant cases, interactions are not pairwise but involve larger sets of nodes at a time. These systems are thus better described in the framework of hypergraphs, whose hyperedges effectively account for multibody interactions. Here we propose and study a class of random walks defined on such higher-order structures and grounded on a microscopic physical model where multibody proximity is associated with highly probable exchanges among agents belonging to the same hyperedge. We provide an analytical characterization of the process, deriving a general solution for the stationary distribution of the walkers. The dynamics is ultimately driven by a generalized random-walk Laplace operator that reduces to the standard random-walk Laplacian when all the hyperedges have size 2 and are thus meant to describe pairwise couplings. We illustrate our results on synthetic models for which we have full control of the high-order structures and on real-world networks where higher-order interactions are at play. As the first application of the method, we compare the behavior of random walkers on hypergraphs to that of traditional random walkers on the corresponding projected networks, drawing interesting conclusions on node rankings in collaboration networks. As the second application, we show how information derived from the random walk on hypergraphs can be successfully used for classification tasks involving objects with several features, each one represented by a hyperedge. Taken together, our work contributes to unraveling the effect of higher-order interactions on diffusive processes in higher-order networks, shedding light on mechanisms at the heart of biased information spreading in complex networked systems.

Topological resilience in non-normal networked systems.

The network of interactions in complex systems strongly influences their resilience and the system capability to resist external perturbations or structural damages and to promptly recover thereafter. The phenomenon manifests itself in different domains, e.g., parasitic species invasion in ecosystems or cascade failures in human-made networks. Understanding the topological features of the networks that affect the resilience phenomenon remains a challenging goal for the design of robust complex systems. We hereby introduce the concept of non-normal networks, namely networks whose adjacency matrices are non-normal, propose a generating model, and show that such a feature can drastically change the global dynamics through an amplification of the system response to exogenous disturbances and eventually impact the system resilience. This early stage transient period can induce the formation of inhomogeneous patterns, even in systems involving a single diffusing agent, providing thus a new kind of dynamical instability complementary to the Turing one. We provide, first, an illustrative application of this result to ecology by proposing a mechanism to mute the Allee effect and, second, we propose a model of virus spreading in a population of commuters moving using a non-normal transport network, the London Tube.

Structure and dynamical behavior of non-normal networks.

We analyze a collection of empirical networks in a wide spectrum of disciplines and show that strong non-normality is ubiquitous in network science. Dynamical processes evolving on non-normal networks exhibit a peculiar behavior, as initial small disturbances may undergo a transient phase and be strongly amplified in linearly stable systems. In addition, eigenvalues may become extremely sensible to noise and have a diminished physical meaning. We identify structural properties of networks that are associated with non-normality and propose simple models to generate networks with a tunable level of non-normality. We also show the potential use of a variety of metrics capturing different aspects of non-normality and propose their potential use in the context of the stability of complex ecosystems.