Effect of localization on the stability of mutualistic ecological networks

Virus communities rather than bacterial communities contribute more on nutrient pool in polluted aquatic environment

The degradation of animal carcasses can lead to rapid waste release (e.g., pathogenic bacteria, viruses, prions, or parasites) and also result in nutrient accumulation in the surrounding environment. However, how viral profile responds and influences nutrient pool (carbon (C), nitrogen (N), phosphorus (P) and sulfur (S)) in polluted water caused by animal carcass decomposition had not been explored. Here, we combined metagenomic analysis, 16S rRNA gene sequencing and water physicochemical assessment to explore the response of viral communities under different temperatures (23 °C, 26 °C, 29 °C, 32 °C, and 35 °C) in water polluted by cadaver, as well as compare the contribution of viral/bacterial communities on water nutrient pool. We found that a total of 15,240 viral species were classified and mainly consisted of Siphoviridae. Both temperature and carrion reduced the viral diversity and abundance. Only a small portion of the viruses (∼8.8 %) had significant negative correlations with temperature, while most were not sensitive. Our results revealed that the viruses had lager contribution on nutrient pool than bacteria. Besides, viral-related functional genes involved in C, N, P and S cycling. These functional genes declined during carcass decomposition and covered part of the central nutrient cycle metabolism (including carbon sugar transformation, denitrification, P mineralization and extracelluar sulfate transfer, etc.). Our result implies that human regulation of virus communities may be more important than bacterial communities in regulating and managing polluted water quality and nutrition.

Optimization hardness constrains ecological transients

Living systems operate far from equilibrium, yet few general frameworks provide global bounds on biological transients. In high-dimensional biological networks like ecosystems, long transients arise from the separate timescales of interactions within versus among subcommunities. Here, we use tools from computational complexity theory to frame equilibration in complex ecosystems as the process of solving an analogue optimization problem. We show that functional redundancies among species in an ecosystem produce difficult, ill-conditioned problems, which physically manifest as transient chaos. We find that the recent success of dimensionality reduction methods in describing ecological dynamics arises due to preconditioning, in which fast relaxation decouples from slow solving timescales. In evolutionary simulations, we show that selection for steady-state species diversity produces ill-conditioning, an effect quantifiable using scaling relations originally derived for numerical analysis of complex optimization problems. Our results demonstrate the physical toll of computational constraints on biological dynamics.

Understanding pesticide-induced tipping in plant-pollinator networks across geographical scales: Prioritizing richness and modularity over nestedness

Mutually beneficial interactions between plants and pollinators are crucial for biodiversity, ecosystem stability, and crop production. A threat to a mutualistic network is the occurrence of a tipping point at which the species abundances collapse to a near zero level. In modern agriculture, there is widespread use of pesticides. What are the effects of extensive pesticide use on mutualistic networks? We develop a plant-pollinator-pesticide model and study its dynamics using 123 mutualistic networks across the globe. We demonstrate that pesticide exposure can lead to a tipping point. Furthermore, while the network characteristics such as richness and modularity exhibit a strong association with pesticide-induced tipping, nestedness shows a weak association. A surprising finding is that the mutualistic networks in the African continent are less pesticide tolerant than those in Europe. We articulate and test a pragmatic intervention strategy through targeted management of pesticide levels within specific plant species to delay or avert the tipping point. Our study provides quantitative insights into the phenomenon of pesticide-induced tipping for safeguarding mutualistic networks that are fundamental to agriculture and ecosystems.

Quantifying the Resilience of Coal Energy Supply in China Toward Carbon Neutrality

Facing the challenge of achieving the goal of carbon neutrality, China is decoupling the currently close dependence of its economy on coal use. The energy supply and demand decarbonization has substantial influence on the resilience of the coal supply. However, a general understanding of the precise impact of energy decarbonization on the resilience of the coal energy supply is still lacking. Here, from the perspective of network science, we propose a theoretical framework to explore the resilience of the coal market of China. We show that the processes of increasing the connectivity and the competition between the coal enterprises, which are widely believed to improve the resilience of the coal market, can undermine the sustainability of the coal supply. Moreover, our results reveal that the policy of closing small-sized coal mines may not only reduce the safety accidents in the coal production but also improve the resilience of the coal market network. Using our model, we also suggest a few practical policies for minimizing the systemic risk of the coal energy supply.

Unraveling the mesoscale organization induced by network-driven processes

Complex systems are characterized by emergent patterns created by the nontrivial interplay between dynamical processes and the networks of interactions on which these processes unfold. Topological or dynamical descriptors alone are not enough to fully embrace this interplay in all its complexity, and many times one has to resort to dynamics-specific approaches that limit a comprehension of general principles. To address this challenge, we employ a metric-that we name Jacobian distance-which captures the spatiotemporal spreading of perturbations, enabling us to uncover the latent geometry inherent in network-driven processes. We compute the Jacobian distance for a broad set of nonlinear dynamical models on synthetic and real-world networks of high interest for applications from biological to ecological and social contexts. We show, analytically and computationally, that the process-driven latent geometry of a complex network is sensitive to both the specific features of the dynamics and the topological properties of the network. This translates into potential mismatches between the functional and the topological mesoscale organization, which we explain by means of the spectrum of the Jacobian matrix. Finally, we demonstrate that the Jacobian distance offers a clear advantage with respect to traditional methods when studying human brain networks. In particular, we show that it outperforms classical network communication models in explaining functional communities from structural data, therefore highlighting its potential in linking structure and function in the brain.

The development of the biological soil crust regulates the fungal distribution and the stability of fungal networks

The heterogeneous composition of fungi plays an indispensable role in the foundation of the multifunctionalities of ecosystems within drylands. The precise mechanisms that govern fluctuations in soil fungal assemblages in dryland ecosystems remain incompletely elucidated. In this study, biological soil crusts (biocrusts) at different successional stages in the Gurbantunggut Desert were used as substrates to examine the characteristics and driving factors that influence fungal abundance and community dynamics during biocrust development using qPCR and high-throughput sequencing of the ITS2 region. The findings showed that the physicochemical properties changed significantly with the development of biocrusts. In particular, total nitrogen increased 4.8 times, along with notable increases in ammonium, total phosphorus (2.1 times) and soil organic carbon (6.5 times). Initially, there was a rise in fungal abundance, which was subsequently followed by a decline as the biocrust developed, with the highest abundance detected in lichen crust (2.66 × 107 copies/g soil) and the lowest in bare sand (7.98 × 106 copies/g soil). Ascomycetes and Basidiomycetes emerged as dominant phyla, collectively forming 85% of the fungal community. As the biocrust developed, noticeable alterations occurred in fungal community compositions, resulting from changes in the relative proportions of Dothideomycetes, Lecanoromycetes and unclassified ascomycetes. Nitrogen, phosphorus, organic carbon content, and pH of biocrusts were identified as direct or indirect regulators of fungal abundance and community structure. The complexity of fungal networks increased as biocrusts developed as revealed by network analysis, but reduced in the stability of fungal communities within algal and lichen crusts. Keystone species within the fungal community also underwent changes as biocrust developed. These results suggested that shifts in interspecies relationships among fungi could further contribute to the variation in fungal communities during the development of biocrusts.

Land abandonment transforms soil microbiome stability and functional profiles in apple orchards of the Chinese Losses Plateau

Land abandonment is considered an effective strategy for ecological restoration on a global scale. However, few studies have focused on how environmental heterogeneity associated with the age of land abandonment affects the assembly and potential functions of the soil microbial community. In the present study, we investigated the community assembly of soil bacteria and fungi as well as the stability of soil networks and their potential functions in the chronosequence of abandoned apple orchards. We elucidated that the Shannon diversity of bacteria and the richness of fungi increased as land abandonment progressed. In addition, land abandonment destabilized the microbial network stability but increased network complexity. Soil available nitrogen, total carbon, and moisture are the potentially important factors in shaping the soil microbial assembly. Importantly, we showed that the microbial community diversity and functional diversity presented a synchronization effect in response to the different stages of land abandonment. Furthermore, specific bacterial taxa related to carbon fixation, dissimilatory nitrate reduction, and organic phosphorus mineralization were significantly enriched during the early abandonment stage. Collectively, these results indicate that land abandonment significantly transformed soil microbiome assembly and functional adaptation during the restoration process. These findings provide valuable insights into the influence of ecological restoration on soil microbiome and ecosystem functions in arable areas.

Eigenvector localization in hypergraphs: Pairwise versus higher-order links

Localization behaviors of Laplacian eigenvectors of complex networks furnish an explanation to various dynamical phenomena of the corresponding complex systems. We numerically examine roles of higher-order and pairwise links in driving eigenvector localization of hypergraphs Laplacians. We find that pairwise interactions can engender localization of eigenvectors corresponding to small eigenvalues for some cases, whereas higher-order interactions, even being much much less than the pairwise links, keep steering localization of the eigenvectors corresponding to larger eigenvalues for all the cases considered here. These results will be advantageous to comprehend dynamical phenomena, such as diffusion, and random walks on a range of real-world complex systems having higher-order interactions in better manner.

Coexistence in diverse communities with higher-order interactions

A central assumption in most ecological models is that the interactions in a community operate only between pairs of species. However, two species may interactively affect the growth of a focal species. Although interactions among three or more species, called higher-order interactions, have the potential to modify our theoretical understanding of coexistence, ecologists lack clear expectations for how these interactions shape community structure. Here we analytically predict and numerically confirm how the variability and strength of higher-order interactions affect species coexistence. We found that as higher-order interaction strengths became more variable across species, fewer species could coexist, echoing the behavior of pairwise models. If interspecific higher-order interactions became too harmful relative to self-regulation, coexistence in diverse communities was destabilized, but coexistence was also lost when these interactions were too weak and mutualistic higher-order effects became prevalent. This behavior depended on the functional form of the interactions as the destabilizing effects of the mutualistic higher-order interactions were ameliorated when their strength saturated with species' densities. Last, we showed that more species-rich communities structured by higher-order interactions lose species more readily than their species-poor counterparts, generalizing classic results for community stability. Our work provides needed theoretical expectations for how higher-order interactions impact species coexistence in diverse communities.

Seasonal structure of interactions enhances multidimensional stability of mutualistic networks

Community ecologists have made great advances in understanding how natural communities can be both diverse and stable by studying communities as interaction networks. However, focus has been on interaction networks aggregated over time, neglecting the consequences of the seasonal organization of interactions (hereafter 'seasonal structure') for community stability. Here, we extended previous theoretical findings on the topic in two ways: (i) by integrating empirical seasonal structure of 11 plant–hummingbird communities into dynamic models, and (ii) by tackling multiple facets of network stability together. We show that, in a competition context, seasonal structure enhances community stability by allowing diverse and resilient communities while preserving their robustness to species extinctions. The positive effects of empirical seasonal structure on network stability vanished when using randomized seasonal structures, suggesting that eco-evolutionary dynamics produce stabilizing seasonal structures. We also show that the effects of seasonal structure on community stability are mainly mediated by changes in network structure and productivity, suggesting that the seasonal structure of a community is an important and yet neglected aspect in the diversity–stability and diversity–productivity debates.

Complexity-stability trade-off in empirical microbial ecosystems

May's stability theory, which holds that large ecosystems can be stable up to a critical level of complexity, a product of the number of resident species and the intensity of their interactions, has been a central paradigm in theoretical ecology. So far, however, empirically demonstrating this theory in real ecological systems has been a long-standing challenge with inconsistent results. Especially, it is unknown whether this theory is pertinent in the rich and complex communities of natural microbiomes, mainly due to the challenge of reliably reconstructing such large ecological interaction networks. Here we introduce a computational framework for estimating an ecosystem's complexity without relying on a priori knowledge of its underlying interaction network. By applying this method to human-associated microbial communities from different body sites and sponge-associated microbial communities from different geographical locations, we found that in both cases the communities display a pronounced trade-off between the number of species and their effective connectance. These results suggest that natural microbiomes are shaped by stability constraints, which limit their complexity.

Eigenvalue ratio statistics of complex networks: Disorder versus randomness

The distribution of the ratios of consecutive eigenvalue spacings of random matrices has emerged as an important tool to study spectral properties of many-body systems. This article numerically investigates the eigenvalue ratios distribution of various model networks, namely, small-world, Erdős-Rényi random, and (dis)assortative random having a diagonal disorder in the corresponding adjacency matrices. Without any diagonal disorder, the eigenvalues ratio distribution of these model networks depict Gaussian orthogonal ensemble (GOE) statistics. Upon adding diagonal disorder, there exists a gradual transition from the GOE to Poisson statistics depending upon the strength of the disorder. The critical disorder (w_{c}) required to procure the Poisson statistics increases with the randomness in the network architecture. We relate w_{c} with the time taken by maximum entropy random walker to reach the steady state. These analyses will be helpful to understand the role of eigenvalues other than the principal one for various network dynamics such as transient behavior.

Comparison of the Responses of Soil Fungal Community to Straw, Inorganic Fertilizer, and Compost in a Farmland in the Loess Plateau

The Loess Plateau is located in the arid and semi-arid regions in northern China. The ecosystem is particularly sensitive to natural and anthropogenic disturbances. Fungi can produce extracellular enzymes, decompose a variety of organic matter, and regulate carbon and nutrient balance. We studied the changes of soil fungal community compositions in response to straw, inorganic fertilizer, and compost in a typical farmland in the Loess Plateau. Our results demonstrated that the addition of straw significantly reduces the Shannon index of the fungal community, in addition, the participation of straw significantly affects the composition of the fungal community. Functional prediction based on FUNGuild showed that straw significantly reduced the relative abundance of saprotrophs, pathotrophs, symbiotrophs, lichenized, ectomycorrhizal, and plant pathogens. Although fertilization practices destroyed the co-occurrence pattern among the fungal species, the addition of straw alleviated this affect. No significant effect of straw, compost, and inorganic fertilizers on the co-occurrence pattern among species in the soil fungal community was observed. Compared with compost and inorganic fertilizer, the addition of straw shaped the community composition by changing the relative abundance of fungal functional taxa. Thus, in the fragile Loess Plateau environment, over-fertilizing or non-order-fertilizing may destroy the co-occurrence pattern of the fungal communities and Loess Plateau ecosystem. IMPORTANCE Determining the response of soil fungi in sensitive ecosystems to external environmental disturbances is an important, yet little-known, topic in microbial ecology. In this study, we evaluated the impact of traditional fertilization management practices on the composition, co-occurrence pattern, and functional groups of fungal communities in loessial soil. Our results show that in the fragile Loess Plateau environment, fertilizer management changed the composition of the fungal community and disrupted the co-occurrence pattern between fungi. The application of straw alleviates the destroying of the co-occurrence pattern. The current research emphasizes the necessity of rational fertilization of farmland in loessial soil.

Role of mitochondrial genetic interactions in determining adaptation to high altitude human population

Physiological and haplogroup studies performed to understand high-altitude adaptation in humans are limited to individual genes and polymorphic sites. Due to stochastic evolutionary forces, the frequency of a polymorphism is affected by changes in the frequency of a near-by polymorphism on the same DNA sample making them connected in terms of evolution. Here, first, we provide a method to model these mitochondrial polymorphisms as "co-mutation networks" for three high-altitude populations, Tibetan, Ethiopian and Andean. Then, by transforming these co-mutation networks into weighted and undirected gene-gene interaction (GGI) networks, we were able to identify functionally enriched genetic interactions of CYB and CO3 genes in Tibetan and Andean populations, while NADH dehydrogenase genes in the Ethiopian population playing a significant role in high altitude adaptation. These co-mutation based genetic networks provide insights into the role of different set of genes in high-altitude adaptation in human sub-populations.

Accuracy of a one-dimensional reduction of dynamical systems on networks

Resilience is an ability of a system with which the system can adjust its activity to maintain its functionality when it is perturbed. To study resilience of dynamics on networks, Gao et al. [Nature (London) 530, 307 (2016)0028-083610.1038/nature16948] proposed a theoretical framework to reduce dynamical systems on networks, which are high dimensional in general, to one-dimensional dynamical systems. The accuracy of this one-dimensional reduction relies on three approximations in addition to the assumption that the network has a negligible degree correlation. In the present study, we analyze the accuracy of the one-dimensional reduction assuming networks without degree correlation. We do so mainly through examining the validity of the individual assumptions underlying the method. Across five dynamical system models, we find that the accuracy of the one-dimensional reduction hinges on the spread of the equilibrium value of the state variable across the nodes in most cases. Specifically, the one-dimensional reduction tends to be accurate when the dispersion of the node's state is small. We also find that the correlation between the node's state and the node's degree, which is common for various dynamical systems on networks, is unrelated to the accuracy of the one-dimensional reduction.

Spatial variation in species' roles in host-helminth networks

Species interactions may vary considerably across space as a result of spatial and environmental gradients. With respect to host-parasite interactions, this suggests that host and parasite species may play different functional roles across the different networks they occur in. Using a global occurrence database of helminth parasites, we examine the conservation of species' roles using data on host-helminth interactions from 299 geopolitical locations. Defining species' roles in a two-dimensional space which captures the tendency of species to be more densely linked within species subgroups than between subgroups, we quantified species' roles in two ways, which captured if and which species' roles are conserved by treating species' utilization of this two-dimensional space as continuous, while also classifying species into categorical roles. Both approaches failed to detect the conservation of species' roles for a single species out of over 38 000 host and helminth parasite species. Together, our findings suggest that species' roles in host-helminth networks may not be conserved, pointing to the potential role of spatial and environmental gradients, as well as the importance of the context of the local host and helminth parasite community. This article is part of the theme issue 'Infectious disease macroecology: parasite diversity and dynamics across the globe'.

The impact of individual variation on abrupt collapses in mutualistic networks

Individual variation is central to species involved in complex interactions with others in an ecological system. Such ecological systems could exhibit tipping points in response to changes in the environment, consequently leading to abrupt transitions to alternative, often less desirable states. However, little is known about how individual trait variation could influence the timing and occurrence of abrupt transitions. Using 101 empirical mutualistic networks, I model the eco‐evolutionary dynamics of such networks in response to gradual changes in strength of co‐evolutionary interactions. Results indicated that individual variation facilitates the timing of transition in such networks, albeit slightly. In addition, individual variation significantly increases the occurrence of large abrupt transitions. Furthermore, topological network features also positively influence the occurrence of such abrupt transitions. These findings argue for understanding tipping points using an eco‐evolutionary perspective to better forecast abrupt transitions in ecological systems.

Fluctuation spectra of large random dynamical systems reveal hidden structure in ecological networks

Understanding the relationship between complexity and stability in large dynamical systems-such as ecosystems-remains a key open question in complexity theory which has inspired a rich body of work developed over more than fifty years. The vast majority of this theory addresses asymptotic linear stability around equilibrium points, but the idea of 'stability' in fact has other uses in the empirical ecological literature. The important notion of 'temporal stability' describes the character of fluctuations in population dynamics, driven by intrinsic or extrinsic noise. Here we apply tools from random matrix theory to the problem of temporal stability, deriving analytical predictions for the fluctuation spectra of complex ecological networks. We show that different network structures leave distinct signatures in the spectrum of fluctuations, and demonstrate the application of our theory to the analysis of ecological time-series data of plankton abundances.

Dimensionality reduction of complex dynamical systems

One of the outstanding problems in complexity science and engineering is the study of high-dimensional networked systems and of their susceptibility to transitions to undesired states as a result of changes in external drivers or in the structural properties. Because of the incredibly large number of parameters controlling the state of such complex systems and the heterogeneity of its components, the study of their dynamics is extremely difficult. Here we propose an analytical framework for collapsing complex N-dimensional networked systems into an S+1-dimensional manifold as a function of S effective control parameters with S << N. We test our approach on a variety of real-world complex problems showing how this new framework can approximate the system's response to changes and correctly identify the regions in the parameter space corresponding to the system's transitions. Our work offers an analytical method to evaluate optimal strategies in the design or management of networked systems.

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We analytically collapse N-dimensional networked dynamics in low-dimensional manifolds

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We test this approach on a variety of real-world complex problems

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We accurately predict the system's response to changes in parameter values

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We identify regions in parameter space corresponding to system's critical transitions

Information limitation and the dynamics of coupled ecological systems

The dynamics of large ecological systems result from vast numbers of interactions between individual organisms. Here, we develop mathematical theory to show that the rate of such interactions is inherently limited by the ability of organisms to gain information about one another. This phenomenon, which we call 'information limitation', is likely to be widespread in real ecological systems and can dictate both the rates of ecological interactions and long-run dynamics of interacting populations. We show how information limitation leads to sigmoid interaction rate functions that can stabilize antagonistic interactions and destabilize mutualistic ones; as a species or type becomes rare, information on its whereabouts also becomes rare, weakening coupling with consumers, pathogens and mutualists. This can facilitate persistence of consumer-resource systems, alter the course of pathogen infections within a host and enhance the rates of oceanic productivity and carbon export. Our findings may shed light on phenomena in many living systems where information drives interactions.

Optimal Microbiome Networks: Macroecology and Criticality

The human microbiome is an extremely complex ecosystem considering the number of bacterial species, their interactions, and its variability over space and time. Here, we untangle the complexity of the human microbiome for the Irritable Bowel Syndrome (IBS) that is the most prevalent functional gastrointestinal disorder in human populations. Based on a novel information theoretic network inference model, we detected potential species interaction networks that are functionally and structurally different for healthy and unhealthy individuals. Healthy networks are characterized by a neutral symmetrical pattern of species interactions and scale-free topology versus random unhealthy networks. We detected an inverse scaling relationship between species total outgoing information flow, meaningful of node interactivity, and relative species abundance (RSA). The top ten interacting species are also the least relatively abundant for the healthy microbiome and the most detrimental. These findings support the idea about the diminishing role of network hubs and how these should be defined considering the total outgoing information flow rather than the node degree. Macroecologically, the healthy microbiome is characterized by the highest Pareto total species diversity growth rate, the lowest species turnover, and the smallest variability of RSA for all species. This result challenges current views that posit a universal association between healthy states and the highest absolute species diversity in ecosystems. Additionally, we show how the transitory microbiome is unstable and microbiome criticality is not necessarily at the phase transition between healthy and unhealthy states. We stress the importance of considering portfolios of interacting pairs versus single node dynamics when characterizing the microbiome and of ranking these pairs in terms of their interactions (i.e., species collective behavior) that shape transition from healthy to unhealthy states. The macroecological characterization of the microbiome is useful for public health and disease diagnosis and etiognosis, while species-specific analyses can detect beneficial species leading to personalized design of pre- and probiotic treatments and microbiome engineering.

Divergent responses of bacterial activity, structure, and co-occurrence patterns to long-term unbalanced fertilization without nitrogen, phosphorus, or potassium in a cultivated vertisol

Unbalanced fertilization lacking nitrogen (N), phosphorus (P), or potassium (K) is a worldwide phenomenon; however, whether they affect bacterial community composition and intraspecific interactions in a similar pattern and how they affect bacterial activity are not systematically compared. Soils under different kinds of unbalanced fertilization in a 21-year field experiment were collected to investigate the variation in dehydrogenase activity (DHA), bacterial community diversity, structure, composition, and possible interactions. Compared to the balanced fertilization of NPK, the DHA from unbalanced fertilization of NP, PK, and NK was 8.70, 11.59, and 14.17% lower, respectively, and from the unfertilized treatment (Nil) was 13.41% lower; however, the Shannon index from NP, PK, and Nil was 4.48-7.21% higher and from NK was 3.95% lower. Based on principal coordinate analyses (PCoA), bacterial community structure was separated by N application or not along PCo1 and was further separated by P application or not along PCo2, indicating a more influence by N deficiency. Moreover, the structure was mainly determined by soil pH, soil organic carbon (SOC), and total phosphorus (TP). The network complexity using co-occurrence analysis followed the order NP > NPK > PK > NK > Nil, indicating a more influence by P deficiency on intraspecific interactions. Structural equation modeling (SEM) revealed that the reduced DHA in NP was mainly regulated by the decreased SOC and increased Shannon index, in PK by the decreased SOC and increased Shannon index and pH, and in NK by the decreased SOC and TP and increased PCo2. The significantly lower abundance of Bacteroidetes and Chitinophagaceae in NK may also contribute to the reduced DHA. Our results imply that N deficiency had the greatest impact on bacterial community structure and composition, P deficiency had the greatest impact on network construction and bacterial activity, and K deficiency has minimal effect. Our results also suggest that main factors regulating the variation in soil functions may vary among different nutrient deficiencies.

Reconciling cooperation, biodiversity and stability in complex ecological communities

Empirical evidences show that ecosystems with high biodiversity can persist in time even in the presence of few types of resources and are more stable than low biodiverse communities. This evidence is contrasted by the conventional mathematical modeling, which predicts that the presence of many species and/or cooperative interactions are detrimental for ecological stability and persistence. Here we propose a modelling framework for population dynamics, which also include indirect cooperative interactions mediated by other species (e.g. habitat modification). We show that in the large system size limit, any number of species can coexist and stability increases as the number of species grows, if mediated cooperation is present, even in presence of exploitative or harmful interactions (e.g. antibiotics). Our theoretical approach thus shows that appropriate models of mediated cooperation naturally lead to a solution of the long-standing question about complexity-stability paradox and on how highly biodiverse communities can coexist.

On the stability of multilayer Boolean networks under targeted immunization

In this paper, we study targeted immunization in a multilayer Boolean network model for genetic regulatory networks. Given a specific set of nodes immune to perturbations, we find that the stability of a multilayer Boolean network is determined by the largest eigenvalue of the weighted non-backtracking matrix of corresponding aggregated network. Aimed to minimize this largest eigenvalue, we developed the metric of multilayer collective influence (MCI) to quantify the impact of immunizing individual nodes on the stability of the system. Compared with other competing heuristics, immunizing nodes with high MCI scores can stabilize an unstable multilayer network with higher efficiency on both synthetic and real-world networks. Moreover, despite that coupling nodes can exert direct influence across multiple layers, they are found to exhibit less importance as measured by the MCI score. Our work reveals the mechanism of maintaining the stability of multilayer Boolean networks and provides an efficient targeted immunization strategy, which can be potentially applied to the location of pathogenesis of diseases and the development of targeted therapy.

Network spectra for drug-target identification in complex diseases: new guns against old foes

The fundamental understanding of altered complex molecular interactions in a diseased condition is the key to its cure. The overall functioning of these molecules is kind of jugglers play in the cell orchestra and to anticipate these relationships among the molecules is one of the greatest challenges in modern biology and medicine. Network science turned out to be providing a successful and simple platform to understand complex interactions among healthy and diseased tissues. Furthermore, much information about the structure and dynamics of a network is concealed in the eigenvalues of its adjacency matrix. In this review, we illustrate rapid advancements in the field of network science in combination with spectral graph theory that enables us to uncover the complexities of various diseases. Interpretations laid by network science approach have solicited insights into molecular relationships and have reported novel drug targets and biomarkers in various complex diseases.

How to Invade an Ecological Network

Invasion science is in a state of paradox, having low predictability despite strong, identifiable covariates of invasion performance. We propose shifting the foundation metaphor of biological invasions from a linear filtering scheme to one that invokes complex adaptive networks. We link invasion performance and invasibility directly to the loss of network stability and indirectly to network topology through constraints from the emergence of the stability criterion in complex systems. We propose the wind vane of an invaded network - the major axis of its adjacency matrix - which reveals how species respond dynamically to invasions. We suggest that invasion ecology should steer away from comparative macroecological studies, to rather explore the ecological network centred on the focal species.

Effect of population abundances on the stability of large random ecosystems

Random matrix theory successfully connects the structure of interactions of large ecological communities to their ability to respond to perturbations. One of the most debated aspects of this approach is that so far studies have neglected the role of population abundances on stability. While species abundances are well studied and empirically accessible, studies on stability have so far failed to incorporate this information. Here we tackle this question by explicitly including population abundances in a random matrix framework. We derive an analytical formula that describes the spectrum of a large community matrix for arbitrary feasible species abundance distributions. The emerging picture is remarkably simple: while population abundances affect the rate to return to equilibrium after a perturbation, the stability of large ecosystems is uniquely determined by the interaction matrix. We confirm this result by showing that the likelihood of having a feasible and unstable solution in the Lotka-Volterra system of equations decreases exponentially with the number of species for stable interaction matrices.

Warnings and caveats in brain controllability

A recent article by Gu et al. (Nat. Commun. 6, 2015) proposed to characterize brain networks, quantified using anatomical diffusion imaging, in terms of their "controllability", drawing on concepts and methods of control theory. They reported that brain activity is controllable from a single node, and that the topology of brain networks provides an explanation for the types of control roles that different regions play in the brain. In this work, we first briefly review the framework of control theory applied to complex networks. We then show contrasting results on brain controllability through the analysis of five different datasets and numerical simulations. We find that brain networks are not controllable (in a statistical significant way) by one single region. Additionally, we show that random null models, with no biological resemblance to brain network architecture, produce the same type of relationship observed by Gu et al. between the average/modal controllability and weighted degree. Finally, we find that resting state networks defined with fMRI cannot be attributed specific control roles. In summary, our study highlights some warning and caveats in the brain controllability framework.

Epidemicity thresholds for water-borne and water-related diseases

Determining the conditions that favor pathogen establishment in a host community is key to disease control and eradication. However, focusing on long-term dynamics alone may lead to an underestimation of the threats imposed by outbreaks triggered by short-term transient phenomena. Achieving an effective epidemiological response thus requires to look at different timescales, each of which may be endowed with specific management objectives. In this work we aim to determine epidemicity thresholds for some prototypical examples of water-borne and water-related diseases, a diverse family of infections transmitted either directly through water infested with pathogens or by vectors whose lifecycles are closely associated with water. From a technical perspective, while conditions for endemicity are determined via stability analysis, epidemicity thresholds are defined through generalized reactivity analysis, a recently proposed method that allows the study of the short-term instability properties of ecological systems. Understanding the drivers of water-borne and water-related disease dynamics over timescales that may be relevant to epidemic and/or endemic transmission is a challenge of the utmost importance, as large portions of the developing world are still struggling with the burden imposed by these infections.

Disentangling the co-structure of multilayer interaction networks: degree distribution and module composition in two-layer bipartite networks

Species establish different interactions (e.g. antagonistic, mutualistic) with multiple species, forming multilayer ecological networks. Disentangling network co-structure in multilayer networks is crucial to predict how biodiversity loss may affect the persistence of multispecies assemblages. Existing methods to analyse multilayer networks often fail to consider network co-structure. We present a new method to evaluate the modular co-structure of multilayer networks through the assessment of species degree co-distribution and network module composition. We focus on modular structure because of its high prevalence among ecological networks. We apply our method to two Lepidoptera-plant networks, one describing caterpillar-plant herbivory interactions and one representing adult Lepidoptera nectaring on flowers, thereby possibly pollinating them. More than 50% of the species established either herbivory or visitation interactions, but not both. These species were over-represented among plants and lepidopterans, and were present in most modules in both networks. Similarity in module composition between networks was high but not different from random expectations. Our method clearly delineates the importance of interpreting multilayer module composition similarity in the light of the constraints imposed by network structure to predict the potential indirect effects of species loss through interconnected modular networks.

Explorability and the origin of network sparsity in living systems

The increasing volume of ecologically and biologically relevant data has revealed a wide collection of emergent patterns in living systems. Analysing different data sets, ranging from metabolic gene-regulatory to species interaction networks, we find that these networks are sparse, i.e. the percentage of the active interactions scales inversely proportional to the system size. To explain the origin of this puzzling common characteristic, we introduce the new concept of explorability: a measure of the ability of an interacting system to adapt to newly intervening changes. We show that sparsity is an emergent property resulting from optimising both explorability and dynamical robustness, i.e. the capacity of the system to remain stable after perturbations of the underlying dynamics. Networks with higher connectivities lead to an incremental difficulty to find better values for both the explorability and dynamical robustness, associated with the fine-tuning of the newly added interactions. A relevant characteristic of our solution is its scale invariance, i.e., it remains optimal when several communities are assembled together. Connectivity is also a key ingredient in determining ecosystem stability and our proposed solution contributes to solving May's celebrated complexity-stability paradox.

Optimized evolution of networks for principal eigenvector localization

Network science is increasingly being developed to get new insights about behavior and properties of complex systems represented in terms of nodes and interactions. One useful approach is investigating the localization properties of eigenvectors having diverse applications including disease-spreading phenomena in underlying networks. In this work, we evolve an initial random network with an edge rewiring optimization technique considering the inverse participation ratio as a fitness function. The evolution process yields a network having a localized principal eigenvector. We analyze various properties of the optimized networks and those obtained at the intermediate stage. Our investigations reveal the existence of a few special structural features of such optimized networks, for instance, the presence of a set of edges which are necessary for localization, and rewiring only one of them leads to complete delocalization of the principal eigenvector. Furthermore, we report that principal eigenvector localization is not a consequence of changes in a single network property and, preferably, requires the collective influence of various distinct structural as well as spectral features.

Super-Resolution Community Detection for Layer-Aggregated Multilayer Networks

Applied network science often involves preprocessing network data before applying a network-analysis method, and there is typically a theoretical disconnect between these steps. For example, it is common to aggregate time-varying network data into windows prior to analysis, and the trade-offs of this preprocessing are not well understood. Focusing on the problem of detecting small communities in multilayer networks, we study the effects of layer aggregation by developing random-matrix theory for modularity matrices associated with layer-aggregated networks with N nodes and L layers, which are drawn from an ensemble of Erdős-Rényi networks with communities planted in subsets of layers. We study phase transitions in which eigenvectors localize onto communities (allowing their detection) and which occur for a given community provided its size surpasses a detectability limit K* . When layers are aggregated via a summation, we obtain [Formula: see text], where T is the number of layers across which the community persists. Interestingly, if T is allowed to vary with L, then summation-based layer aggregation enhances small-community detection even if the community persists across a vanishing fraction of layers, provided that T/L decays more slowly than 𝒪(L-1/2). Moreover, we find that thresholding the summation can, in some cases, cause K* to decay exponentially, decreasing by orders of magnitude in a phenomenon we call super-resolution community detection. In other words, layer aggregation with thresholding is a nonlinear data filter enabling detection of communities that are otherwise too small to detect. Importantly, different thresholds generally enhance the detectability of communities having different properties, illustrating that community detection can be obscured if one analyzes network data using a single threshold.

Collapse of resilience patterns in generalized Lotka-Volterra dynamics and beyond

Recently, a theoretical framework aimed at separating the roles of dynamics and topology in multidimensional systems has been developed [Gao et al., Nature (London) 530, 307 (2016)10.1038/nature16948]. The validity of their method is assumed to hold depending on two main hypotheses: (i) The network determined by the the interaction between pairs of nodes has negligible degree correlations; (ii) the node activities are uniform across nodes on both the drift and the pairwise interaction functions. Moreover, the authors consider only positive (mutualistic) interactions. Here we show the conditions proposed by Gao and collaborators [Nature (London) 530, 307 (2016)10.1038/nature16948] are neither sufficient nor necessary to guarantee that their method works in general and validity of their results are not independent of the model chosen within the class of dynamics they considered. Indeed we find that a new condition poses effective limitations to their framework and we provide quantitative predictions of the quality of the one-dimensional collapse as a function of the properties of interaction networks and stable dynamics using results from random matrix theory. We also find that multidimensional reduction may work also for an interaction matrix with a mixture of positive and negative signs, opening up an application of the framework to food webs, neuronal networks, and social and economic interactions.

Feasibility and coexistence of large ecological communities

The role of species interactions in controlling the interplay between the stability of ecosystems and their biodiversity is still not well understood. The ability of ecological communities to recover after small perturbations of the species abundances (local asymptotic stability) has been well studied, whereas the likelihood of a community to persist when the conditions change (structural stability) has received much less attention. Our goal is to understand the effects of diversity, interaction strengths and ecological network structure on the volume of parameter space leading to feasible equilibria. We develop a geometrical framework to study the range of conditions necessary for feasible coexistence. We show that feasibility is determined by few quantities describing the interactions, yielding a nontrivial complexity–feasibility relationship. Analysing more than 100 empirical networks, we show that the range of coexistence conditions in mutualistic systems can be analytically predicted. Finally, we characterize the geometric shape of the feasibility domain, thereby identifying the direction of perturbations that are more likely to cause extinctions.

A central question in theoretical ecology is how diverse species can coexist in communities, and how that coexistence depends on network properties. Here, Grilli et al. quantify the extent of feasible coexistence of empirical networks, showing that it is smaller for trophic than mutualism networks.

A Data Driven Network Approach to Rank Countries Production Diversity and Food Specialization

The easy access to large data sets has allowed for leveraging methodology in network physics and complexity science to disentangle patterns and processes directly from the data, leading to key insights in the behavior of systems. Here we use country specific food production data to study binary and weighted topological properties of the bipartite country-food production matrix. This country-food production matrix can be: 1) transformed into overlap matrices which embed information regarding shared production of products among countries, and or shared countries for individual products, 2) identify subsets of countries which produce similar commodities or subsets of commodities shared by a given country allowing for visualization of correlations in large networks, and 3) used to rank country fitness (the ability to produce a diverse array of products weighted on the type of food commodities) and food specialization (quantified on the number of countries producing a specific food product weighted on their fitness). Our results show that, on average, countries with high fitness produce both low and high specializion food commodities, whereas nations with low fitness tend to produce a small basket of diverse food products, typically comprised of low specializion food commodities.

Symmetric and Asymmetric Tendencies in Stable Complex Systems

A commonly used approach to study stability in a complex system is by analyzing the Jacobian matrix at an equilibrium point of a dynamical system. The equilibrium point is stable if all eigenvalues have negative real parts. Here, by obtaining eigenvalue bounds of the Jacobian, we show that stable complex systems will favor mutualistic and competitive relationships that are asymmetrical (non-reciprocative) and trophic relationships that are symmetrical (reciprocative). Additionally, we define a measure called the interdependence diversity that quantifies how distributed the dependencies are between the dynamical variables in the system. We find that increasing interdependence diversity has a destabilizing effect on the equilibrium point, and the effect is greater for trophic relationships than for mutualistic and competitive relationships. These predictions are consistent with empirical observations in ecology. More importantly, our findings suggest stabilization algorithms that can apply very generally to a variety of complex systems.