What Is in a Simplicial Complex? A Metaplex-Based Approach to Its Structure and Dynamics.

Geometric realization of simplicial complexes makes them a unique representation of complex systems. The existence of local continuous spaces at the simplices level with global discrete connectivity between simplices makes the analysis of dynamical systems on simplicial complexes a challenging problem. In this work, we provide some examples of complex systems in which this representation would be a more appropriate model of real-world phenomena. Here, we generalize the concept of metaplexes to embrace that of geometric simplicial complexes, which also includes the definition of dynamical systems on them. A metaplex is formed by regions of a continuous space of any dimension interconnected by sinks and sources that works controlled by discrete (graph) operators. The definition of simplicial metaplexes given here allows the description of the diffusion dynamics of this system in a way that solves the existing problems with previous models. We make a detailed analysis of the generalities and possible extensions of this model beyond simplicial complexes, e.g., from polytopal and cell complexes to manifold complexes, and apply it to a real-world simplicial complex representing the visual cortex of a macaque.

Network bypasses sustain complexity.

Real-world networks are neither regular nor random, a fact elegantly explained by mechanisms such as the Watts-Strogatz or the Barabási-Albert models, among others. Both mechanisms naturally create shortcuts and hubs, which while enhancing the network's connectivity, also might yield several undesired navigational effects: They tend to be overused during geodesic navigational processes-making the networks fragile-and provide suboptimal routes for diffusive-like navigation. Why, then, networks with complex topologies are ubiquitous? Here, we unveil that these models also entropically generate network bypasses: alternative routes to shortest paths which are topologically longer but easier to navigate. We develop a mathematical theory that elucidates the emergence and consolidation of network bypasses and measure their navigability gain. We apply our theory to a wide range of real-world networks and find that they sustain complexity by different amounts of network bypasses. At the top of this complexity ranking we found the human brain, which points out the importance of these results to understand the plasticity of complex systems.

Topologically induced suppression of explosive synchronization.

Nowadays, explosive synchronization is a well-documented phenomenon consisting in a first-order transition that may coexist with classical synchronization. Typically, explosive synchronization occurs when the network structure is represented by the classical graph Laplacian, and the node frequency and its degree are correlated. Here, we answer the question on whether this phenomenon can be observed in networks when the oscillators are coupled via degree-biased Laplacian operators. We not only observe that this is the case but also that this new representation naturally controls the transition from explosive to standard synchronization in a network. We prove analytically that explosive synchronization emerges when using this theoretical setting in star-like networks. As soon as this star-like network is topologically converted into a network containing cycles, the explosive synchronization gives rise to classical synchronization. Finally, we hypothesize that this mechanism may play a role in switching from normal to explosive states in the brain, where explosive synchronization has been proposed to be related to some pathologies like epilepsy and fibromyalgia.

Nonuniform random graphs on the plane: A scaling study.

We consider random geometric graphs on the plane characterized by a nonuniform density of vertices. In particular, we introduce a graph model where n vertices are independently distributed in the unit disk with positions, in polar coordinates (l,θ), obeying the probability density functions ρ(l) and ρ(θ). Here we choose ρ(l) as a normal distribution with zero mean and variance σ∈(0,∞) and ρ(θ) as a uniform distribution in the interval θ∈[0,2π). Then, two vertices are connected by an edge if their Euclidean distance is less than or equal to the connection radius ℓ. We characterize the topological properties of this random graph model, which depends on the parameter set (n,σ,ℓ), by the use of the average degree 〈k〉 and the number of nonisolated vertices V_{×}, while we approach their spectral properties with two measures on the graph adjacency matrix: the ratio of consecutive eigenvalue spacings r and the Shannon entropy S of eigenvectors. First we propose a heuristic expression for 〈k(n,σ,ℓ)〉. Then, we look for the scaling properties of the normalized average measure 〈X[over ¯]〉 (where X stands for V_{×}, r, and S) over graph ensembles. We demonstrate that the scaling parameter of 〈V_{×}[over ¯]〉=〈V_{×}〉/n is indeed 〈k〉, with 〈V_{×}[over ¯]〉≈1-exp(-〈k〉). Meanwhile, the scaling parameter of both 〈r[over ¯]〉 and 〈S[over ¯]〉 is proportional to n^{-γ}〈k〉 with γ≈0.16.

The world-wide waste web.

Countries globally trade with tons of waste materials every year, some of which are highly hazardous. This trade admits a network representation of the world-wide waste web, with countries as vertices and flows as directed weighted edges. Here we investigate the main properties of this network by tracking 108 categories of wastes interchanged in the period 2001-2019. Although, most of the hazardous waste was traded between developed nations, a disproportionate asymmetry existed in the flow from developed to developing countries. Using a dynamical model, we simulate how waste stress propagates through the network and affects the countries. We identify 28 countries with low Environmental Performance Index that are at high risk of waste congestion. Therefore, they are at threat of improper handling and disposal of hazardous waste. We find evidence of pollution by heavy metals, by volatile organic compounds and/or by persistent organic pollutants, which are used as chemical fingerprints, due to the improper handling of waste in several of these countries.

Cascading from SARS-CoV-2 to Parkinson's Disease through Protein-Protein Interactions.

Extensive extrapulmonary damages in a dozen of organs/systems, including the central nervous system (CNS), are reported in patients of the coronavirus disease 2019 (COVID-19). Three cases of Parkinson's disease (PD) have been reported as a direct consequence of COVID-19. In spite of the scarce data for establishing a definitive link between COVID-19 and PD, some hypotheses have been proposed to explain the cases reported. They, however, do not fit well with the clinical findings reported for COVID-19 patients, in general, and for the PD cases reported, in particular. Given the importance of this potential connection, we present here a molecular-level mechanistic hypothesis that explains well these findings and will serve to explore the potential CNS damage in COVID-19 patients. The model explaining the cascade effects from COVID-19 to CNS is developed by using bioinformatic tools. It includes the post-translational modification of host proteins in the lungs by viral proteins, the transport of modified host proteins via exosomes out the lungs, and the disruption of protein-protein interaction in the CNS by these modified host proteins. Our hypothesis is supported by finding 44 proteins significantly expressed in the CNS which are associated with PD and whose interactions can be perturbed by 24 host proteins significantly expressed in the lungs. These 24 perturbators are found to interact with viral proteins and to form part of the cargoes of exosomes in human tissues. The joint set of perturbators and PD-vulnerable proteins form a tightly connected network with significantly more connections than expected by selecting a random cluster of proteins of similar size from the human proteome. The molecular-level mechanistic hypothesis presented here provides several routes for the cascading of effects from the lungs of COVID-19 patients to PD. In particular, the disruption of autophagy/ubiquitination processes appears as an important mechanism that triggers the generation of large amounts of exosomes containing perturbators in their cargo, which would insult several PD-vulnerable proteins, potentially triggering Parkinsonism in COVID-19 patients.

Loss of structural balance in stock markets.

We use rank correlations as distance functions to establish the interconnectivity between stock returns, building weighted signed networks for the stocks of seven European countries, the US and Japan. We establish the theoretical relationship between the level of balance in a network and stock predictability, studying its evolution from 2005 to the third quarter of 2020. We find a clear balance-unbalance transition for six of the nine countries, following the August 2011 Black Monday in the US, when the Economic Policy Uncertainty index for this country reached its highest monthly level before the COVID-19 crisis. This sudden loss of balance is mainly caused by a reorganization of the market networks triggered by a group of low capitalization stocks belonging to the non-financial sector. After the transition, the stocks of companies in these groups become all negatively correlated between them and with most of the rest of the stocks in the market. The implied change in the network topology is directly related to a decrease in stock predictability, a finding with novel important implications for asset allocation and portfolio hedging strategies.

Communicability distance reveals hidden patterns of Alzheimer's disease.

The communicability distance between pairs of regions in human brain is used as a quantitative proxy for studying Alzheimer's disease. Using this distance, we obtain the shortest communicability path lengths between different regions of brain networks from patients with Alzheimer's disease (AD) and healthy cohorts (HC). We show that the shortest communicability path length is significantly better than the shortest topological path length in distinguishing AD patients from HC. Based on this approach, we identify 399 pairs of brain regions for which there are very significant changes in the shortest communicability path length after AD appears. We find that 42% of these regions interconnect both brain hemispheres, 28% connect regions inside the left hemisphere only, and 20% affect vermis connection with brain hemispheres. These findings clearly agree with the disconnection syndrome hypothesis of AD. Finally, we show that in 76.9% of damaged brain regions the shortest communicability path length drops in AD in relation to HC. This counterintuitive finding indicates that AD transforms the brain network into a more efficient system from the perspective of the transmission of the disease, because it drops the circulability of the disease factor around the brain regions in relation to its transmissibility to other regions.

Protein-driven mechanism of multiorgan damage in COVID-19.

We propose a new plausible mechanism by mean of which SARS-CoV-2 produces extrapulmonary damages in severe COVID-19 patients. The mechanism consist on the existence of vulnerable proteins (VPs), which are (i) mainly expressed outside the lungs; (ii) their perturbations is known to produce human diseases; and (iii) can be perturbed directly or indirectly by SARS-CoV-2 proteins. These VPs are perturbed by other proteins, which are: (i) mainly expressed in the lungs, (ii) are targeted directly by SARS-CoV-2 proteins, (iii) can navigate outside the lungs as cargo of extracellular vesicles (EVs); and (iv) can activate VPs via subdiffusive processes inside the target organ. Using bioinformatic tools and mathematical modeling we identifies 26 VPs and their 38 perturbators, which predict extracellular damages in the immunologic endocrine, cardiovascular, circulatory, lymphatic, musculoskeletal, neurologic, dermatologic, hepatic, gastrointestinal, and metabolic systems, as well as in the eyes. The identification of these VPs and their perturbators allow us to identify 27 existing drugs which are candidates to be repurposed for treating extrapulmonary damage in severe COVID-19 patients. After removal of drugs having undesirable drug-drug interactions we select 7 drugs and one natural product: apabetalone, romidepsin, silmitasertib, ozanezumab, procaine, azacitidine, amlexanox, volociximab, and ellagic acid, whose combinations can palliate the organs and systems found to be damaged by COVID-19. We found that at least 4 drugs are needed to treat all the multiorgan damages, for instance: the combination of romidepsin, silmitasertib, apabetalone and azacitidine.

Fractional diffusion on the human proteome as an alternative to the multi-organ damage of SARS-CoV-2.

The coronavirus 2019 (COVID-19) respiratory disease is caused by the novel coronavirus SARS-CoV-2 (severe acute respiratory syndrome coronavirus 2), which uses the enzyme ACE2 to enter human cells. This disease is characterized by important damage at a multi-organ level, partially due to the abundant expression of ACE2 in practically all human tissues. However, not every organ in which ACE2 is abundant is affected by SARS-CoV-2, which suggests the existence of other multi-organ routes for transmitting the perturbations produced by the virus. We consider here diffusive processes through the protein-protein interaction (PPI) network of proteins targeted by SARS-CoV-2 as an alternative route. We found a subdiffusive regime that allows the propagation of virus perturbations through the PPI network at a significant rate. By following the main subdiffusive routes across the PPI network, we identify proteins mainly expressed in the heart, cerebral cortex, thymus, testis, lymph node, kidney, among others of the organs reported to be affected by COVID-19.

COVID-19 and SARS-CoV-2. Modeling the present, looking at the future.

Since December 2019 the Severe Acute Respiratory Syndrome Coronavirus 2 (SARS-CoV-2) has produced an outbreak of pulmonary disease which has soon become a global pandemic, known as COronaVIrus Disease-19 (COVID-19). The new coronavirus shares about 82% of its genome with the one which produced the 2003 outbreak (SARS CoV-1). Both coronaviruses also share the same cellular receptor, which is the angiotensin-converting enzyme 2 (ACE2) one. In spite of these similarities, the new coronavirus has expanded more widely, more faster and more lethally than the previous one. Many researchers across the disciplines have used diverse modeling tools to analyze the impact of this pandemic at global and local scales. This includes a wide range of approaches - deterministic, data-driven, stochastic, agent-based, and their combinations - to forecast the progression of the epidemic as well as the effects of non-pharmaceutical interventions to stop or mitigate its impact on the world population. The physical complexities of modern society need to be captured by these models. This includes the many ways of social contacts - (multiplex) social contact networks, (multilayers) transport systems, metapopulations, etc. - that may act as a framework for the virus propagation. But modeling not only plays a fundamental role in analyzing and forecasting epidemiological variables, but it also plays an important role in helping to find cures for the disease and in preventing contagion by means of new vaccines. The necessity for answering swiftly and effectively the questions: could existing drugs work against SARS CoV-2? and can new vaccines be developed in time? demands the use of physical modeling of proteins, protein-inhibitors interactions, virtual screening of drugs against virus targets, predicting immunogenicity of small peptides, modeling vaccinomics and vaccine design, to mention just a few. Here, we review these three main areas of modeling research against SARS CoV-2 and COVID-19: (1) epidemiology; (2) drug repurposing; and (3) vaccine design. Therefore, we compile the most relevant existing literature about modeling strategies against the virus to help modelers to navigate this fast-growing literature. We also keep an eye on future outbreaks, where the modelers can find the most relevant strategies used in an emergency situation as the current one to help in fighting future pandemics.

Topological analysis of SARS CoV-2 main protease.

There is an urgent necessity of effective medication against severe acute respiratory syndrome coronavirus 2 (SARS CoV-2), which is producing the COVID-19 pandemic across the world. Its main protease (Mpro) represents an attractive pharmacological target due to its involvement in essential viral functions. The crystal structure of free Mpro shows a large structural resemblance with the main protease of SARS CoV (nowadays known as SARS CoV-1). Here, we report that average SARS CoV-2 Mpro is 1900% more sensitive than SARS CoV-1 Mpro in transmitting tiny structural changes across the whole protein through long-range interactions. The largest sensitivity of Mpro to structural perturbations is located exactly around the catalytic site Cys-145 and coincides with the binding site of strong inhibitors. These findings, based on a simplified representation of the protein as a residue network, may help in designing potent inhibitors of SARS CoV-2 Mpro.

Fractional-Order Susceptible-Infected Model: Definition and Applications to the Study of COVID-19 Main Protease.

We propose a model for the transmission of perturbations across the amino acids of a protein represented as an interaction network. The dynamics consists of a Susceptible-Infected (SI) model based on the Caputo fractional-order derivative. We find an upper bound to the analytical solution of this model which represents the worse-case scenario on the propagation of perturbations across a protein residue network. This upper bound is expressed in terms of Mittag-Leffler functions of the adjacency matrix of the network of inter-amino acids interactions. We then apply this model to the analysis of the propagation of perturbations produced by inhibitors of the main protease of SARS CoV-2. We find that the perturbations produced by strong inhibitors of the protease are propagated far away from the binding site, confirming the long-range nature of intra-protein communication. On the contrary, the weakest inhibitors only transmit their perturbations across a close environment around the binding site. These findings may help to the design of drug candidates against this new coronavirus.

Mathematical modelling for sustainable aphid control in agriculture via intercropping.

Agricultural losses to pests represent an important challenge in a global warming scenario. Intercropping is an alternative farming practice that promotes pest control without the use of chemical pesticides. Here, we develop a mathematical model to study epidemic spreading and control in intercropped agricultural fields as a sustainable pest management tool for agriculture. The model combines the movement of aphids transmitting a virus in an agricultural field, the spatial distribution of plants in the intercropped field and the presence of 'trap crops' in an epidemiological susceptible-infected-removed model. Using this model, we study several intercropping arrangements without and with trap crops and find a new intercropping arrangement that may improve significantly pest management in agricultural fields with respect to the commonly used intercrop systems.

Influence of long-range interactions on quantum interference in molecular conduction. A tight-binding (Hückel) approach.

When certain pairs of atoms in a π-conjugated molecule are connected with nanometer-scale source and drain electrodes, the remarkable quantum interference (QI) effect may arise. In this case, the electron transmission probability is significantly suppressed due to the QI effect. Tight-binding approaches, such as the Hückel molecular orbital (HMO) model, have revealed important features of this quantum phenomenon. However, important deviations from experiments and from more sophisticated calculations are known for a variety of cases. Here, we propose an extension of the HMO method to include non-nearest-neighbor interactions. Such long-range interactions (LRIs) are implemented in the HMO model in the form of a damping function that decays as the topological distance-the number of bonds separating two atoms-gets larger. The proposed model is further developed so that a geometric modification, i.e., the rotation around a single bond, can be taken into account. Our results show that LRI affects both the location of the antiresonance peak due to QI and the intensity of QI, even suppressing it in some cases. These results agree well with what was observed in a Density Functional based Tight-Binding (DFTB) study reported in the literature. These properties can be interpreted on the basis of a graph-theoretic path-counting model as well as the molecular orbital theory. In addition, the geometric LRI model is shown to reproduce the change of transmission as a function of rotation around the single bond separating two benzene rings in biphenyl, in agreement with what was observed in both experiment and DFTB calculation.

Geometrical and spectral study of ß-skeleton graphs.

We perform an extensive numerical analysis of ß-skeleton graphs, a particular type of proximity graphs. In a ß-skeleton graph (BSG) two vertices are connected if a proximity rule, that depends of the parameter ß∈(0,∞), is satisfied. Moreover, for ß>1 there exist two different proximity rules, leading to lune-based and circle-based BSGs. First, by computing the average degree of large ensembles of BSGs we detect differences, which increase with the increase of ß, between lune-based and circle-based BSGs. Then, within a random matrix theory (RMT) approach, we explore spectral and eigenvector properties of random BSGs by the use of the nearest-neighbor energy-level spacing distribution and the entropic eigenvector localization length, respectively. The RMT analysis allows us to conclude that a localization transition occurs at ß=1.

Epidemics on plants: Modeling long-range dispersal on spatially embedded networks.

Here we develop an epidemic model that accounts for long-range dispersal of pathogens between plants. This model generalizes the classical compartmental models-Susceptible-Infected-Susceptible (SIS) and Susceptible-Infected-Recovered (SIR)-to take into account those factors that are key to understand epidemics in real plant populations. These ingredients are the spatial characteristics of the plots and fields in which plants are embedded and the effect of long-range dispersal of pathogens. The spatial characteristics are included through the use of random rectangular graphs which allow to consider the effects of the elongation of plots and fields, while the long-range dispersal is implemented by considering transformations, such as the Mellin and Laplace transforms, of a generalization of the adjacency matrix of the geometric graph. Our results point out that long-range dispersal favors the propagation of pathogens while the elongation of plant plots increases the epidemic threshold and decreases dramatically the number of affected plants. Interestingly, our model is able of reproducing the existence of patchy regions of infected plants and the absence of a clear propagation front centered in the initial infected plants, as it is observed in real plant epidemics.

Quantum Interference, Graphs, Walks, and Polynomials.

In this paper, we explore quantum interference (QI) in molecular conductance from the point of view of graph theory and walks on lattices. By virtue of the Cayley-Hamilton theorem for characteristic polynomials and the Coulson-Rushbrooke pairing theorem for alternant hydrocarbons, it is possible to derive a finite series expansion of the Green's function for electron transmission in terms of the odd powers of the vertex adjacency matrix or Hückel matrix. This means that only odd-length walks on a molecular graph contribute to the conductivity through a molecule. Thus, if there are only even-length walks between two atoms, quantum interference is expected to occur in the electron transport between them. However, even if there are only odd-length walks between two atoms, a situation may come about where the contributions to the QI of some odd-length walks are canceled by others, leading to another class of quantum interference. For nonalternant hydrocarbons, the finite Green's function expansion may include both even and odd powers. Nevertheless, QI can in some circumstances come about for nonalternants from cancellation of odd- and even-length walk terms. We report some progress, but not a complete resolution, of the problem of understanding the coefficients in the expansion of the Green's function in a power series of the adjacency matrix, these coefficients being behind the cancellations that we have mentioned. Furthermore, we introduce a perturbation theory for transmission as well as some potentially useful infinite power series expansions of the Green's function.

Communicability geometry captures traffic flows in cities.

Understanding the structural and dynamical drivers of network flow is an important goal for our complete understanding of complex systems. Particularly challenging is the determination of the routes used by items when flowing through a network. The study of vehicular traffic flow in cities offers a unique opportunity to test theoretical models about network flows and traffic routes using experimental data. Here, we found observational evidence that there is higher vehicular traffic flow through the communicability shortest paths, which assume an 'all-routes' flow, than through the shortest paths in four cities of different sizes, populations and geographical locations. The communicability function is derived here from a coarse-grained theory of traffic on networks accounting for an auxiliary vehicular propagation speed. Finally, we study the vehicular 'all-routes' flow in cities as the perceptual problem of drivers seeing the shortest paths as 'too central to be empty'.