Ecology

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The world’s population is rapidly urbanizing, particularly along coastlines, where population density is now three times higher than the global average1,2. According to the National Oceanic and Atmospheric Administration (NOAA), almost 40% of the United States (U.S.) population resides in coastal zones with population density being over five times greater in coastal shoreline counties than the national average. As a result, human encroachment on coastal ecosystems is significantly modifying natural landscapes and reducing intact coastal habitat. Along densely populated coastlines, residential development often involves unsustainable land-use planning and armoring of shorelines, where natural habitats such as saltmarshes, mangroves, seagrasses and oyster reefs, are replaced with artificial structures, including vertical bulkheads, seawalls, boat ramps, and other gray infrastructures3. In areas with dense residential development between 50–90% of shorelines can be armored, whereby, on average, 14% of all U.S. shorelines have been modified from their natural conditions and replaced with artificial structures3. This transition represents an extensive loss of natural coastal habitats and the critical ecosystem services they provide.As more ecologically harmful infrastructure is developed to meet the demands of human population growth, urbanization concurrently alters ecosystem services and functions by negatively impacting biodiversity, ecological conditions and environmental quality, specifically through a decrease in native habitat, increased water pollution, and creation of impervious surfaces4. Urbanization may also lead to less resilient and adaptable coastal communities against natural hazards and climate change threats, such as sea level rise and hurricanes. This is because in urban areas, ecosystem functioning is reduced and associated services are lost, resulting in increasing risk of shoreline erosion, saltwater intrusion, storm surges, and coastal flooding2,5.These human-environment interactions in coastal ecosystems can lead to, and at the same time be derived by, decisions that will shape the future structure, function, and sustainability of coastal ecosystems6. These social decisions (e.g., large-scale policies or individual level choices) can have long-lasting consequences for both the environment and society, especially as coastal development increases. Decisions that modify and change the biophysical nature of the environment (e.g., waterfront residents’ decision to use artificial structures for storm protection and shoreline stabilization) impact its ecological functionality7. At the same time, these alterations may change the degree of connectivity that individual humans have to their environments, which might extend to broader societies’ ecological knowledge8,9.Few studies provide evidence that the removal and lack of natural environments in urbanized environments reduces individuals’ environmental connectedness and ecological knowledge, and subsequently lowers pro-environmental behaviors10,11,12. This is of critical importance since a general lack of environmental connectedness, and in particular, a lack of ecological knowledge is a phenomenon often used to explain the non-appreciation of, and deleterious behaviors toward, the natural environment, even though many studies theorize these relationships opposed to empirically test them (e.g., see refs. 13,14 and the discussion in ref. 15 about “nature-deficit disorder”). Furthermore, if there exists a general lack of ecological knowledge, social decisions at the individual level that reflect these limited perceptions (e.g., utilitarian land-use decisions12 or waterfront homeowners’ preference to install a bulkhead) can often cascade to larger societal impacts through domino-effects, where individual decisions trigger similar, reactive decisions by neighbors leading to broader societal patterns16. For example, Gittman et al.17 found that one of the stronger predictors of an individual decision to have an armored shoreline was presence of armoring on a neighboring parcel. When considered across a community scale, such societal patterns can alter natural coastal habitats significantly.In this current study, we investigate the relationship between residents’ knowledge, or mental models, of human-environment interactions, their self-reported pro-environmental behavior, and how these perceptions and behaviors are associated with urbanization. A mental model is the cognitive internal representation of a system in the external world that articulates causal relationships among system components (i.e., abstract concepts)18,19. Mental models that represent causal knowledge can be graphically obtained through cognitive mapping techniques in the form of directed graphs, which are networks in which nodes represent concepts (i.e., system components) and graph edges (arrows) represent the causal relationships between the concepts20. We combine methods from social science, data science, and network science to conduct an analysis using mental models of coastal residents along an urbanization gradient to better understand the interconnections among urbanization, people’s knowledge of human-environment interactions, and their pro-environmental behavior.We surveyed residents across eight coastal states in the northeast U.S., including Maine, New Hampshire, Massachusetts, Rhode Island, Connecticut, New York, New Jersey, and Delaware. We used a fuzzy cognitive mapping (FCM) approach21 to elicit mental models of coastal ecosystems with a focus on environmental connectedness, ecosystem health, human wellbeing, climate, and sustainable coasts (see Methods). Here, we propose a concept, Urbanized Knowledge Syndrome (UKS), which represents recurring patterns in urban dwellers’ mental models about natural ecosystems – their internal understanding of how humans and environment interact. Here, syndrome should not be interpreted as a set of medical signs and symptoms which are associated with a particular disease or disorder. These recurring patterns include (1) diminished systems thinking (e.g., complexity of mental models decreases as degree of urban development increases) and (2) the erosion of cognitive diversity (i.e., diversity of mental models among residents decreases as degree of urban development increases). These patterns demonstrate a type of thinking that is simplified to some extent or otherwise limited or focused on fewer social-ecological relationships than exist in reality.Systems thinking – a holistic view that considers factors and interactions and how they result in a possible outcome – is an important skillset that helps people better understand complex systems and adapt to changes22. Individuals with higher degrees of systems thinking are more likely to consider interdependencies, identify leverage points to intervene within the system and produce desired outcomes23, better anticipate system function and emergence of patterns of behavior24, and avoid unintended consequences18. As such, systems thinking may help coastal residents develop mental models that enable more nuanced reasoning about diverse causal pathways between humans and natural coastal ecosystems25,26,27,28, which may lead to behaviors that are driven by more predominant cognition of complex feedbacks, trade-offs, and reciprocal interdependencies between humans and nature. In contrast, bounded systems thinking (or linear thinking) may lead humans to develop limited cognition of their surrounding world, reduce their ability to accurately and adequately perceive the complexity of the environment they inhabit and interact with22, and thus may give rise to counterproductive behaviors and decisions27,29. For example, a simple causal relationship might be that seawall construction increases coastal protection as a form of structural defense to control shoreline erosion; whereas a more complex relationship might be that seawalls lead to alterations in hydrodynamic processes, which reduces erosion locally and accelerates coastal erosion downstream30, and at the same time, shoreline armoring can also lead to losses of natural coastal habitats and their critical ecosystem functions3.While cities are beneficial to human development, working as engines of socioeconomic change, cultural transformation, and technological innovation, their psychological influences on people and how these influences drive urban residents’ perceptions and behavior must be noted. Firstly, the salience of ecosystem services is limited for inhabitants of more urbanized areas, as compared to rural areas. Exposure to nature provides multiple opportunities for cognitive development which increases the potential for stewardship of the environment and for a stronger recognition of ecosystem functions13. Urban residents, however, are more routinely exposed to built environment and gray infrastructure, such as armored shorelines and artificial structures along coastlines, as opposed to natural environment, and thus their local experience of, and connection to, ecosystem services can be limited31.Secondly, urbanization generally comes with complex technology and commerce, allowing individuals to meet their needs quickly and through many choices with less appreciation of, and first hand experience with, provisioning ecosystem services (e.g., food comes from many grocery stores not a farm or garden; fish comes over a counter not across a dock or the end of a spear; and potable water comes from a pipeline not a spring or well). This may cause the development of a wider gap in human perceptions of benefits received from natural ecosystems32, fostering the emergence of societies that are increasingly disconnected and seemingly independent from ecosystem services31.Finally, residents of urbanized areas may be exposed to a set of social norms, information, and perspectives that encourage anthropocentric values and thinking including human exemptionalism (“the tendency to see human systems as exempt from the constraints of natural environment”33) and human exceptionalism (“the tendency to see humans as biologically unique and discontinuous with the rest of the animal world”34), therefore limiting their understanding of the importance and substantiality of reciprocal interdependencies between humans and natural environment13,34. These urbanization aspects may spark what we call ‘limits to systems thinking’ in the social-ecological realm.Therefore, we hypothesize (H1) that in more urbanized areas, mental models are predominantly characterized by linear thinking of coastal ecosystems, as opposed to systems thinking, where components are connected mostly by simple causal patterns. This class of mental models is associated with limited cognition of synergies and trade-offs, emergence of global patterns from local relationships, reciprocal interdependencies, and feedback loops between humans and natural ecosystems, which may lead to a gap in residents’ perception of nonlinear complex structures. To test our hypothesis, we analyze the structure of causal relationships using the network structure and graph-theoretic metrics of cognitive maps (i.e., graphical representations of mental models). We use cluster analysis to identify predominant classes of mental models about coastal ecosystems. Distinct clusters of mental models represent archetypal cognitions that individuals develop to perceive human-environment interdependencies13,27. We then use network analyses to measure the complexity of causal structures in cognitive maps and determine the overall degree of systems thinking in each cluster (see Methods). Finally, we investigate the association between urbanization and the degree of systems thinking across those clusters.The second important feature that helps systems adapt to changes is diversity, ranging from ecosystems35 to economic systems36. There is also evidence that these same relationships between diversity and adaptability hold true for cultural knowledge systems, governance systems, and among diverse communities and social institutions that function more effectively as resilient collectives28,37,38.In contrast, as cultural homogenization theories explain, survival in cities depends on fitting in and adopting practices that are considered socially normal by the dominant culture39. Although cities are magnets for people from all corners of the world with seemingly more diverse composition of race and ethnicity compared to rural areas40, assimilation of diverse values, beliefs, cultural knowledge, and social norms into a universal, governing culture—sometimes referred to as “cultural colonialism” or “cultural normalization” – is a major component of urban societies41. This cultural normalization among urban dwellers is exacerbated by dominant exposure to the universal language and education system, greater access to the Internet, social media and news outlets, and market-driven policies and global standardizations for laws and finance41.In addition, an important characteristic of urbanization is the centralization of the population into cities, “where neighborhoods in different regions have similar patterns of roads, residential lots, commercial areas, and aquatic features”42. Such physical and environmental homogenization across urban areas, which is visually evident, is influenced by monocentric land-management and policies, economic pressures for land development and use, engineering necessities, codes and standards, and preferences for particular aesthetics and recreations. Prior studies have shown that this homogenization extends to ecological structure, meaning that across urbanized areas, similar built environment and landscape structures can lead to homogenized ecological characteristics, function, and the range of ecosystem services they can supply42,43.Here, we argue that homogenization in cultural, physical, and ecological systems also extends to residents’ perceptions and understanding of human-environment interactions. We, therefore, hypothesize (H2) that increased urbanization is associated with more homogenized mental models of coastal ecosystems. To test our second hypothesis, we measure the structural dissimilarity of individuals’ mental models (i.e., cognitive maps) using some of the widely used methods for comparing graphs44. We measure the mean of pairwise cognitive distances (i.e., a quantitative metric that represents the mean of graph dissimilarity between any two individual cognitive maps) and compare this metric across clusters of mental models, and thus, explore the correspondence between urbanization and mental model homogenization (i.e., testing the hypothesis that urbanization is associated with more similar mental models in terms of causal structures represented in cognitive maps) (see Methods). More

Farmers at a Great Green Wall site in Niger. Researchers have found that the project is not always benefiting the most vulnerable people.Credit: Boureima Hama/AFP/Getty

It’s now 15 years since the African Union gave its blessing to Africa’s Great Green Wall, one of the world’s most ambitious ecological-restoration schemes. The project is intended to combat desertification across the width of Africa, and spans some 8,000 kilometres, from Senegal to Djibouti. Its ambition is staggering: it aims to restore 100 million hectares of degraded land by 2030, capturing 250 million tonnes of carbon dioxide and creating 10 million jobs in the process. But it continues to struggle.An assessment two years ago by independent experts commissioned by the United Nations stated that somewhere between 4% and 20% of the restoration target had been achieved (go.nature.com/39zqgkr). That figure has not changed, according to the latest edition of Global Land Outlook (go.nature.com/3kdjtw5) from the UN Convention to Combat Desertification (UNCCD), out last week. Equally concerning is the fact that funding for the project continues to lag. Africa’s governments and international donors need to find around US$30 billion to reach the 100-million-hectare target. So far, $19 billion has been raised.A pandemic — and now a cost-of-living crisis — has placed demands on all governments, and that means countries might be expected to reduce their green-wall commitments. But the project continues to be weighed down by other difficulties, including the complex system through which it is funded and governed, as well as how its success is measured. These problems can and must be fixed, otherwise it will struggle to achieve its goals.One potential solution — improved metrics — comes from an analysis published last year by Matthew Turner at the University of Wisconsin–Madison and his colleagues (M. D. Turner et al. Land Use Policy 111, 105750; 2021). The researchers explored limitations in the Great Green Wall project metrics by assessing the impact of World Bank funding from 2006 to 2020. As their work indicates, definitions of success depend on which measure is used.In Niger, for example, green-wall projects could be said to be succeeding if measured by the area of eroded soil that has been recovered or by the number of trees that have been planted. But the authors report that these gains were not necessarily benefiting the most vulnerable people. In places, women were being excluded from employment in green-wall projects, and in some cases, local administrations looked to privatize restored land that might instead have been owned by everyone in a community.Broader problems with metrics are highlighted in the UN’s latest land-degradation report. This estimates that nearly half of the land that has been pledged for restoration worldwide will be planted predominantly with fast-growing trees and plants. This will provide only a fraction of the ecosystem services produced by forests that are allowed to naturally regenerate, including significantly less carbon storage, groundwater recharge and wildlife habitat.The Great Green Wall project also needs more predictable funding and more transparent governance. The project was conceived by Africa’s leaders for the benefit of the continent’s people, on the basis of warnings from scientists about the risks of desertification and land degradation. The original idea was not brought to Africa by international donors, as is often the case in international science-based development projects. But it still relies on donor financing, and lots of it — and that brings other problems, among them coordination challenges.The project is the responsibility of an organization set up by the African Union called the Pan African Agency of the Great Green Wall, based in Nouakchott, Mauritania. But some donors, such as the European Union and the World Bank, are not providing most of their Great Green Wall funding through this agency. Instead, they often deal directly with individual governments, because this gives them more control over how their money is spent. It is unfair to expect the Pan African Agency to coordinate a raft of donors doing one-on-one deals with individual countries. Bypassing the Pan African Agency also creates a problem for transparency, because it makes it harder for the African Union to determine precisely who is funding what.In January 2021, at an international biodiversity summit hosted by France, Emmanuel Macron, the French president, announced that the Great Green Wall would receive an extra $14 billion in funding for 5 years. He also said that a new body, called the Great Green Wall Accelerator, based in Bonn, Germany, would be responsible for pulling together funding pledges and tracking progress against targets. This is well-intentioned, but the accelerator needs to coordinate its work with the Pan African Agency. It is not yet clear how this will happen.A potentially more transformative solution was proposed two years ago by a group of UN-appointed experts. They recommended that a single trust fund be set up that all donors could contribute to and through which they could decide funding priorities together. Regrettably, this has not happened, and observers say it is not likely to happen in the current climate.This month, the international community will come together in Abidjan, Côte d’Ivoire, for the 15th conference of the parties to the UNCCD. The green wall’s funders and participating countries will all be there. If a single trust fund is off the table, they must work together to find a better way to coordinate their green-wall project activities. It is also essential that they study the findings of Turner and colleagues’ review. Along with a focus on existing metrics, the Great Green Wall needs evaluation criteria that take better account of the needs of all people in participating countries, particularly the most vulnerable. More

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Overall strategyAn admixture analysis aims to estimate the admixture proportions (or ancestries), Q, of each sampled individual in a given number of K source populations (Pritchard et al. 2000), and the characteristic allele frequencies, P, at each locus of each inferred source population. Even though Q is frequently of the primary interest, P must be estimated simultaneously because we have genotype data only and Q is highly dependent on P which actually defines the source populations. For N individuals from K source populations genotyped at L loci with a total number of A alleles, the numbers of independent variables in Q and P are VQ = (K − 1)N and VP = (A − L)K, respectively. The high dimensionality of an admixture analysis, with V = VQ + VP = (K − 1)N + (A − L)K variables, not only incurs a large computational burden, but also poses a high risk of non-convergence (to the global maximum) for any algorithm, especially when either Q or P is expected to be poorly estimated in difficult situations such as a small sample (say, a couple) of individuals from each source population or low differentiation.I propose a two-step procedure with corresponding algorithms to reduce the risk of non-convergence, to speed up the computation, and to make more accurate inferences of both Q and P. In the first step, I assume a mixture model (Pritchard et al. 2000; Falush et al. 2003) that individuals in a sample can come from different source populations, but each individual’s genome comes exclusively from a single population. Under this simplified probabilistic model, I conduct a clustering analysis to obtain estimates of both individual memberships and allele frequencies of each cluster by a global maximisation algorithm, simulated annealing, with extra care (details below) of convergence. In the absence of admixture and with sufficient information for complete recovery of population structure, the estimated individual memberships and allele frequencies of the clusters are expected to be equivalent to Q (with element qik = 1 and qil = 0 if individual i is inferred to be in cluster k where l ≠ k) and P, respectively. Otherwise, they are expected to be good approximations of Q and P, because an admixed individual i with the highest ancestral proportion from a population would be expected to be assigned (exclusively) to that population. In the second step, I assume an admixture model (Pritchard et al. 2000; Falush et al. 2003) to refine estimates of Q and P, using an EM algorithm and the start parameter (Q and P) values obtained from the clustering analysis. Because the starting values are already close to the truth, the algorithm is fast and has a much-reduced risk of converging to a local maximum than the original EM algorithms (Tang et al. 2005; Alexander et al. 2009).Clustering analysisI assume N diploid individuals are sampled from K source populations. The origin of a sampled individual from the K source populations is unknown, which is the primary interest of structure analysis. However, if it is (partially) known, this information can be used to supervise (help) the clustering analysis of other sampled individuals of unknown origins. Each individual’s genome comes exclusively from one of the K unknown source populations (i.e., mixture model, no admixture). I assume each individual is genotyped at L loci, with a diploid genotype {xil1, xil2} for individual i (=1, 2, …, N) at locus l (=1, 2, …, L). The task of the clustering analysis is to sort the N individuals with genotype data X = {xila:i = 1, 2, …, N; l = 1, 2, …, L; a = 1, 2} into K clusters, with each representing a source population. No assumption is made about the evolutionary relationships of the populations, which, when summarized by F statistics, are estimated from the same genotype data in both clustering and admixture analyses.Suppose, in a given clustering configuration Ω = {Ω1, Ω2, …, ΩK}, cluster k (=1, 2, …, K), Ωk, contains a set of Nk (with Nk  > 0 and (mathop {sum}nolimits_{k = 1}^K {N_k equiv N})) individuals, denoted by Ωk = {ωk1, ωk2, …, ωkNk} where ωkj is the index of the jth individual in cluster k. The genotype data of the Nk individuals in cluster k is Xk = {xila: i ∈ Ωk; l = 1, 2, …, L; a = 1, 2}. The log-likelihood of Ωk is then the log probability of observing Xk given Ωk$${{{mathcal{L}}}}_kleft( {{{{mathbf{Omega }}}}_k} right) = {{{mathrm{LogP}}}}left( {{{{mathbf{X}}}}_kleft| {{{{mathbf{Omega }}}}_k} right.} right) = mathop {sum}limits_{l = 1}^L {mathop {sum}limits_{j = 1}^{J_l} {c_{klj}{{{mathrm{Log}}}}left( {p_{klj}} right)} }$$
(1)
where cklj and pklj are the count of copies and the frequency, respectively, of allele j at locus l in cluster k, and Jl is the number of alleles at locus l. Given Ωk, cklj is counted from genotype data Xk, and allele frequency pklj is estimated by$$p_{klj} = left( {p_{lj} + c_{klj}} right)/mathop {sum}limits_{m = 1}^{J_l} {left( {p_{lm} + c_{klm}} right)}$$
(2)
where plj is the frequency of allele j at locus l in the entire population represented by the K clusters. plj is calculated by$$p_{lj} = mathop {sum}limits_{k = 1}^K {c_{klj}} /mathop {sum}limits_{m = 1}^{J_l} {mathop {sum}limits_{k = 1}^K {c_{klm}} } = c_{lj}/mathop {sum}limits_{m = 1}^{J_l} {c_{lm}}$$
(3)
where (c_{lm} = mathop {sum}nolimits_{k = 1}^K {c_{klm}}) is the count of allele m (=1, 2, …, Jl) at locus l in the entire sample of individuals.Under the mixture model above, clusters are only weakly dependent (with the extent of dependency decreasing with an increasing value of K) and the total log-likelihood of the clustering configuration, Ω = {Ω1, Ω2, …, ΩK}, is thus$${{{mathcal{L}}}}left( {{{mathbf{Omega }}}} right) = mathop {sum}limits_{k = 1}^K {{{{mathcal{L}}}}_kleft( {{{{mathbf{Omega }}}}_k} right)} ,$$
(4)
where ({{{mathcal{L}}}}_kleft( {{{{mathbf{Omega }}}}_k} right)) is calculated by (1).It is worth noting that allele frequencies, P, are modelled as hidden or nuisance variables and are estimated as a by-product of maximising (4) for estimates of Ω. Yet, careful modelling of P proves important for estimating Ω, as the two are highly dependent. Bayesian admixture methods assume allele frequencies pkl = {pkl1, pkl2, …, (p_{klj_l})} in a Dirichlet distribution (e.g., Foreman et al. 1997; Rannala and Mountain 1997; Pritchard et al. 2000), ({{{mathcal{D}}}}left( {lambda _1,lambda _2, ldots ,lambda _{J_l}} right)). For any population k, the uncorrelated (Pritchard et al. 2000) and correlated (Falush et al. 2003) allele frequency model assumes λj = 1 and (lambda_j=p_{ol_j}F_K/(1-F_k)), respectively, for j = 1, 2, …, Jl. In the latter model, p0lj is the frequency of allele j at locus l in the ancestral population (common to the K derived populations), and Fk is the differentiation of population k from the ancestral population. In contrast, likelihood admixture methods (e.g., Tang et al. 2005; Alexander et al. 2009; Frichot et al. 2014) and non-model based clustering methods (e.g., K-means method, Jombart et al. 2010) do not use any prior, which is equivalent to assuming plj ≡ 0 for j = 1, 2, …, Jl in Eq. (2). However, properly modelling prior allele frequencies, as carefully considered in Bayesian methods (Pritchard et al. 2000; Falush et al. 2003), becomes important in situations where allele frequencies are not well defined or tricky to estimate, such as when few individuals are sampled from a source population or when rare alleles are present. The frequentist estimator (2) is in spirit similar to the Bayesian correlated allele frequency model (Falush et al. 2003), and leads to accurate results in various situations to be shown in this study. I have also tried alternatives such as plj ≡ 1/Jl (which is similar to the uncorrelated allele frequency model of Pritchard et al. 2000) or plj ≡ 0 (which is equivalent to the treatment in previous likelihood admixture analysis or non-model based clustering analysis) in replacement of (2), but none works as well as (2) and could yield much less accurate results in difficult situations (below).Scaling for unbalanced samplingBayesian methods of STRUCTURE’s admixture model assume an individual i’s ancestry, qi = {qi1, qi2, …, qiK}, follows a prior Dirichlet probability distribution ({{{mathbf{q}}}}_isim {{{mathbf{{{{mathcal{D}}}}}}}}left( {alpha _1,alpha _2, ldots ,alpha _K} right)) (Pritchard et al. 2000; Falush et al. 2003). By default, α1 = α2 = ··· = αK = α, which essentially assumes that an individual has its ancestry originating from each of the assumed K populations at an equal prior probability of 1/K. To model unequal sample sizes such that an individual comes from a more intensively sampled population at a higher prior probability, STRUCTURE also has applied an alternative prior, α1 ≠ α2 ≠ ··· ≠ αK. It is shown that, when sampling intensity is heavily unbalanced among populations, the default prior could lead to the split of a large cluster and the merge of small clusters, while the alternative prior yields much more accurate results (Wang 2017). These priors have a large impact on admixture analysis; applying the default prior to data of highly unbalanced samples leads to inaccurate Q estimates even when many informative markers are used (Wang 2017).Unfortunately, current non-model based or likelihood-based admixture analysis methods do not utilise these or other priors for handling unbalanced sampling. As a result, they can give inaccurate admixture estimates, just like STRUCTURE under the default ancestry prior model, for data from highly unbalanced sampling. To reduce the cluster split and merge problems, herein I propose the following method to scale the likelihood of a cluster by the size, the number of individual members, of the cluster.The original log-likelihood of cluster k, ({{{mathcal{L}}}}_kleft( {{{{mathbf{Omega }}}}_k} right)), is calculated by (1). It is then scaled by the cluster size, Nk, as$${{{mathcal{L}}}}_{Sk}left( {{{{mathbf{Omega }}}}_k} right) = {{{mathcal{L}}}}_kleft( {{{{mathbf{Omega }}}}_k} right)/left( {1 + e^{sN_k/left( {8N} right)}} right),$$
(5)
where s is the scaling factor taking values 1, 2, 3 for weak, medium and strong scaling, respectively. This scaling scheme encourages large clusters and discourages small clusters. Although (5) is not an analytically derived but an empirical equation and is thus not guaranteed to be optimal, extensive simulations (some shown below) verify that the scaling scheme works very well for data from highly unbalanced sampling, yielding accurate clustering analysis results and thus similarly or more accurate admixture estimates than STRUCTURE under its alternative ancestry model. The most appropriate scaling level (1, 2 or 3) for a particular dataset depends on how unbalanced the sampling is, how much differentiated the populations are, and how much informative the markers are. For example, a low scaling level, s = 1, is appropriate when many markers are genotyped for a set of lowly differentiated (low FST) populations. Usually, we do not know these factors in analysing the data. Therefore, when the data are suspected to be unbalanced in sampling among populations, they are better analysed comparatively with different levels of scaling (0, 1, 2, and 3). When the applied level of scaling is too low, large populations tend to be split and small populations tend to be merged. When the applied level of scaling is too high, small populations tend to be merged among themselves or with a large population. With the help of some internal information such as consistency of replicate runs at the same scaling level and the same K value and some external information such as sampling locations in examining the admixture estimates, the appropriate scaling level can be determined.Simulated annealing algorithmA likelihood function with many variables, such as (4), is difficult to maximise for estimates of the variables. Traditional methods, such as derivative based Newton-Raphson algorithm (e.g., Tang et al. 2005) and non-derivative based EM algorithm (Dempster et al. 1977; Tang et al. 2005; Alexander et al. 2009), may converge to a local rather than the global maximum for a large scale problem with ridges and plateaus (Gaffe et al. 1994). Although trying multiple replicate runs with different starting values and choosing the run with the highest likelihood could reduce the risk of landing on a local maximum, a global maximum cannot be guaranteed regardless of the number of runs. The Bayesian approach as implemented in STRUCTURE (Pritchard et al. 2000) has a similar problem, as different replicate runs of the same data with the same parameter and model choices but different random number seeds may yield different admixture estimates and likelihood values (Tang et al. 2005; below).Simulated annealing (SA) was developed to optimise very large and complex systems (Kirkpatrick et al. 1983). Using the Metropolis algorithm (Metropolis et al. 1953) from statistical mechanics, SA can find the global maximum by searching both downhill and uphill and by traversing deep valleys on the likelihood surface to avoid getting stuck on a local maximum (Kirkpatrick et al. 1983; Goffe et al. 1994). It has been proved to be highly powerful in pedigree reconstruction (Wang 2004; Wang and Santure 2009) from genotype data, which is probably more difficult than population structure reconstruction (i.e., clustering analysis) because the genetic structure (i.e., sibship) of the former is, in general, more numerous, more complicated with hierarchy, and smaller (thus more elusive and more difficult to define) than that in the latter. Herein I propose a SA algorithm for a population clustering analysis, as detailed in Supplementary Appendix 1.Admixture analysisUnder the mixture model, the above clustering analysis partitions the N sampled individuals into a predefined K clusters, each representing a source population. The properties (e.g., genetic diversity) of and the relationships (e.g., FST) among these populations can be learnt from the inferred clusters. However, the clustering results are accurate only when the mixture model is valid. For a sample containing a substantial proportion of highly admixed individuals (i.e., who have recent ancestors from multiple source populations), the clustering results are just approximations. In such a case, the admixture model is more appropriate and can be used to refine the mixture analysis results by inferring the admixture proportions (or ancestry coefficients) of each sampled individual.Under the admixture model (Pritchard et al. 2000), an individual i’s ancestry (or admixture proportions) can be characterised by a vector qi = {qi1, qi2, …, qiK}, where qik is the proportion of its genome coming from (contributed by) source population k. Equivalently, qik can also be taken as the probability that an allele sampled at random from individual i comes from source population k. Obviously, we have qik ≥ 0 and (mathop {sum}nolimits_{k = 1}^K {q_{ik} equiv 1}). The overall admixture extent of individual i can be measured by (M_i = 1 – mathop {sum}nolimits_{k = 1}^K {q_{ik}^2}), the probability that the two alleles at a randomly drawn locus come from different source populations. Individual i is purebred and admixed when Mi = 0 and Mi  > 0, respectively. An F1 and F2 hybrid individual i is expected to have Mi = 0.5 and Mi = 0.625, respectively.The task of an admixture analysis is to infer qi for each individual i, denoted by Q = {q1, q2, …, qN}. The log-likelihood function is$${{{mathcal{L}}}}left( {{{{mathbf{Q}}}},{{{mathbf{P}}}}left| {{{mathbf{X}}}} right.} right) = mathop {sum}limits_{i = 1}^N {mathop {sum}limits_{l = 1}^L {mathop {sum}limits_{a = 1}^2 {{{{mathrm{Log}}}}left( {mathop {sum}limits_{k = 1}^K {q_{ik}p_{klx_{ila}}} } right)} } }$$
(6)
Note (6) is essentially the same as those proposed in previous studies (e.g., Tang et al. 2005; Alexander et al. 2009). It assumes independence of individuals conditional on the genetic structure defined by Q, and independence of alleles both within and between loci. The former can be violated when the data have genetic structure in addition to the subpopulation structure defined by Q, such as the presence of familial structure (Rodríguez‐Ramilo and Wang 2012) or inbreeding (Gao et al. 2007) within a subpopulation. The assumption of independence among loci is violated for markers in linkage disequilibrium. It, as well as the assumption of independence between paternal and maternal alleles within a locus, is also violated due to admixture (Tang et al. 2005) or inbreeding (Gao et al. 2007). However, (6) is a good approximation and works well in general even when these assumptions are violated, as checked by extensive simulations.If P were known, it would be trivial to estimate Q from X. Unfortunately, usually, the only information we have is genotype data X, from which we must infer K, Q and P jointly. Herein I modify the EM algorithm of Tang et al. (2005) to solve (6) for maximum likelihood estimates of Q and P given K, as detailed in Supplementary Appendix 2.Despite essentially the same likelihood function, my EM algorithm differs from that of Tang et al. (2005) in three aspects. First, I use the clustering results of mixture model as initial values of Q. Even in the worst scenario of many highly admixed individuals included in a sample, the clustering results should still be much closer to the true Q than a random guess, as used in previous likelihood methods (Tang et al. 2005; Alexander et al. 2009). It is possible (and indeed it has been trialled) to use the results of a faster non-model based clustering method, such as K-means method, in place of those of the likelihood-based clustering method with simulated annealing algorithm as described above. However, such non-model based methods are less reliable and less accurate, especially in difficult situations (below). Second, rather than updating Q and P in alternation, I update Q to asymptotic convergence under a given P. I then update P using the converged Q. This two-step iteration process is repeated until the convergence of both Q and P is reached. Third, the allele frequencies for a specific individual i are calculated by excluding the genotypes of the individual, which are then used in the EM procedure for iteratively updating qi.Optimal KThe above-described clustering analysis and admixture analysis are conducted by assuming a given number of source populations, K. Apparently, different genetic structures would be inferred from the same genotype data if different K values are assumed. In some cases, a reasonable K value is roughly known. For example, individuals might be sampled from K known discrete locations (say, lakes), and the purposes of a structure analysis are to confirm that populations from different locations are indeed differentiated and thus distinguishable, to identify migrants between the locations, and to find out the patterns of genetic differentiations (e.g., whether isolation by distance applies or not). In many other cases, however, we may have no idea of the most likely K value. For example, individuals might be sampled from the same breeding or feeding ground and we wish to know how many populations are using the same ground, and to learn the properties of these populations from the individuals sampled and assigned to them. In such a situation of hidden genetic structure, we need first to identify the most likely one or more K values, and then investigate the corresponding structure/admixture.Estimating the most likely K value from genotype data is difficult (Pritchard et al. 2000). Although many methods have been proposed and applied (see review by Wang 2019), they are all ad hoc to some extent and may be inaccurate in difficult situations such as highly unbalanced sampling from different populations and low differentiation (Wang 2019). Herein I propose two ad hoc estimators of K that can be calculated from the clustering analysis presented in this study. They have a satisfactory accuracy as checked by many test datasets, simulated and empirical.The first estimator is based on the second order rate of change of the estimated log-likelihood as a function of K in a clustering analysis, DLK2. This estimator is similar in spirit to the ∆K method of Evanno et al. (2005), but does not use the mean and standard deviation of log-likelihood values among replicate runs (for a given K value) because the standard deviation (the denominator of ∆K) is frequently zero thanks to the convergence of our clustering analysis by the simulated annealing algorithm.The second estimator, denoted by FSTIS, is based on Wright (1984)’s F-statistics. The best K should produce the strongest population structure, with high differentiation (measured by FST) of each inferred cluster and low deviation from Hardy-Weinberg equilibrium (measured by FIS) within each inferred cluster. Details of how to calculate the two estimators are in Supplementary Appendix 3.SimulationsTo evaluate the accuracy, robustness, and computational efficiency of the new methods implemented in PopCluster in comparison with other methods, I simulated and analysed data with different population structures and sampling intensities. The simulation procedure described below is implemented in the software package PopCluster.Simulation 1, small samplesA population becomes difficult to define genetically when few individuals from it are sampled and included in an admixture analysis. However, a small sample of individuals can be common in practice when, for example, archaeological samples (usually few) are used in studying ancient population structure or in studying the relationship between ancient and current populations (e.g., Lazaridis et al. 2014). In a mixed stock analysis (Smouse et al. 1990) or a wildlife forensic analysis of source populations, there might also be few sampled individuals representing a rare population. To investigate the impact of sample sizes on an admixture analysis, I simulated 10 populations in an island model with FST = 0.05. Nk (=2, 3, …, 10 and 20) individuals were sampled from each of the 10 populations, or 1 individual was sampled from each of the first five populations and 2 individuals were sampled from each of the last five populations (the case Nk = 1.5, Table 1). Other simulation parameters are summarized in Table 1.Table 1 Simulation parameters.Full size tableSimulation 2, many populationsAdmixture becomes increasingly difficult to infer with an increasing K, the number of assumed populations, because the dimensions of both Q and P increase linearly with K. This contrasts with the number of individuals, N, and the number of loci, L, which determines the dimensions of Q and P only, respectively. Therefore, the scale of an admixture analysis, in terms of the number of parameters to be estimated, is predominantly determined by K rather than N or L. I simulated data with a widely variable number of populations (K = [6, 100]) to see if the structure can be accurately reconstructed by using relatively highly informative markers (parameters in Table 1), especially when K is large which is rarely considered in previous simulation studies.Simulation 3, spatial admixture modelThe spatial admixture model resembles isolation by distance where population structure changes gradually as a function of geographic location. Under this model, populations are not discrete as assumed by admixture models and have no recognisable boundaries, posing challenges to an admixture analysis. To simulate the spatially gradual changes in genetic structure, I assume source populations 1, 2, …, K are equally spaced in that order along a line (say, a river in reality). Sampled individuals 1, 2, …, N are also equally spaced in that order on the same line. The admixture proportions of individual i, qi = {qi1, qi2, …, qiK}, being the proportional genetic contributions to i from source populations k, are a function of the individual’s proximity to these K source populations. Formally, we have$$q_{ik} = frac{{q_{ik}^ ast }}{{mathop {sum}nolimits_{k = 1}^K {q_{ik}^ ast } }}$$
(7)
where$$q_{ik}^ ast = left[ {1 – left( {frac{{i – 1}}{{N – 1}} – frac{{k – 1}}{{K – 1}}} right)^2} right]^S$$and parameter S is used to regulate the admixture extent of the N sampled individuals. Under this spatial admixture model, an individual i’s admixture (qi) is determined by its location, or the distances from the K source populations. The 1st and the last sampled individuals (i = 1, N) always have the least admixture, measured by (M_i = 1 – mathop {sum}nolimits_{k = 1}^K {q_{ik}^2}). q11 (=qNK) is always the largest among the qik values for i = 1, 2, …, N and k = 1, 2, …, K. Given a desired value of q11 and K, the scaler parameter S can be solved from the above equations. Given K, N and S, qi of an individual i can then be calculated from the above equations. In this study, I simulated and analysed samples generated with parameters K = 5, N = 500, L = 10000 SNPs, and q11 varying between 0.5 and 1.0 (Table 1).Simulation 4, low differentiationPopulation structure analysis becomes increasingly difficult with a decreasing differentiation, usually measured by FST, among subpopulations. Fortunately, with genomic data of many SNPs, it is still possible to detect weak and subtle population structures (Patterson et al. 2006) as demonstrated in human fine-structure analysis (e.g., Leslie et al. 2015). I simulated data with varying weak population structures (low FST, Table 1) and otherwise ideal populational (only 3 equally differentiated subpopulations) and sampling conditions (i.e., a large sample of individuals per subpopulation, and many SNPs). The number of SNPs used in analyse was L = 1000/FST such that in principle the population structures should be inferred with roughly equal power and accuracy. Because L is large for low FST, STRUCTURE analysis was abandoned due to computational difficulties.Simulation 5, unbalanced samplingSamples of individuals from different source populations are rarely identical in size in practice. Frequently, different source populations are represented by different numbers of individuals in a sample. The impact of unbalanced sampling and how to mitigate it in applying STRUCTURE have been investigated (e.g., Puechmaille 2016; Wang 2017). Similar problems exist for other admixture or clustering analysis methods but have not been studied yet. The same population structure and unbalanced sampling schemes (see parameters in Table 1) used in Wang (2017) were used to simulate data, which were then analysed by various methods to understand their robustness to unbalanced sampling.Simulation 6, computational efficiencySamples from a variable number of populations (Table 1) were analysed by the four programs on a linux cluster to compare their computational efficiencies. Each program uses a single core (no parallelisation) of a processor (Intel Xeon Gold 6248 2.5 GHz) for a maximal allowed time of 48 or 72 (when K = 1024 only) hours. Default parameter settings are used for all four programs. For STRUCTURE, both burn-in and run lengths were set to 104, although much higher burn-in is required for convergence when K is large (say K  > 20). The running time for STRUCTURE is thus conservative, especially when K is not small.Further simulations were conducted to investigate the effects of high admixture and the presence of familial relationships and inbreeding on the relative performance of different admixture analysis methods, as detailed in Supplementary Appendix 4.In all simulations except for the spatial admixture model, I assumed a population with K discrete subpopulations in Wright’s (1931) island model in equilibrium among mutation, drift and migration. For a locus l (=1, 2, …, L) with Jl alleles, allele frequencies of the ancestral population, p0l = {p0l1, p0l2, …, (p_{0lJ_l})}, were drawn from a uniform Dirichlet distribution, ({{{mathcal{D}}}}left( {lambda _1,lambda _2, ldots ,lambda _{J_l}} right)) where λj = 1 for j = 1, 2, …, Jl. Given p0l, allele frequencies of subpopulation k (=1, 2, …, K), pkl = {pkl1, pkl2, …, (p_{klJ_l})}, were drawn from a uniform Dirichlet distribution, ({{{mathcal{D}}}}left( {lambda _1,lambda _2, ldots ,lambda _{J_l}} right)), where (lambda _j = ( {frac{1}{{F_{ST}}} – 1} )p_{0lj}) for j = 1, 2, …, Jl (Nicholson et al. 2002; Falush et al. 2003). Given pkl and the admixture proportion qi of individual i, two alleles at locus l were drawn independently to form the individual’s genotype. The multilocus genotype of an individual was obtained by combining single locus genotypes sampled independently, assuming linkage equilibrium. Nk individuals were drawn at random from population k (= 1, 2, …, K), which were then pooled and subjected to a structure analysis.For the spatial population and sampling model, allele frequencies at a locus l, p0l and pkl, are generated as before, assuming FST = 0.05 among K = 5 subpopulations. A number of N = 500 individuals, equally spaced on the line between source populations 1 and 5, are sampled. The admixture proportion of individual i, qi, is determined by its location, calculated by Eq. (7). Given pkl and qi, the multilocus genotype of individual i is simulated as described above.For each parameter combination, 100 replicate datasets were simulated, analysed and assessed for estimation accuracy. Each dataset was analysed for admixture by different methods (see below for details) with an assumed K as used in simulations. I did not consider estimating the optimal K by analysing a simulated dataset in a range of possible K values. This is because, like previous studies (e.g., Pritchard et al. 2000; Alexander et al. 2009), I am more concerned with admixture inference under a given K, which is important of itself and forms the basis for inferring the optimal K as well. This is also because it is almost impossible computationally to estimate the optimal K for so many replicate datasets and so many parameter combinations in a large-scale simulation study like the present one, even when using large computer clusters. The optimal K was estimated for several empirical datasets (below).Measurement of accuracyInference accuracy could be assessed by comparing, for each individual i, the agreement between simulated ancestry coefficients, qi, and estimated ancestry coefficients, (widehat {{{mathbf{q}}}}_i), obtained by an admixture analysis assuming the true/simulated subpopulation number K. Because the reconstructed populations are labelled arbitrarily (Pritchard et al. 2000), no meaningful results can be gained by comparing qi and (widehat {{{mathbf{q}}}}_i) directly, however. It is possible to relabel the reconstructed populations and find the labelling scheme that has the maximum agreement between qi and (widehat {{{mathbf{q}}}}_i) as the measurement of accuracy. However, there are K! possible labelling schemes, making the approach difficult to calculate when K is large (say, K > 50).The labelling becomes irrelevant when pairs of individuals are considered for the co-assignment probabilities (or coancestry) (Dawson and Belkhir 2001). I calculate and use the average difference between simulated and estimated coancestry for pairs of sampled individuals to measure the average assignment error, AAE (Wang 2017),$$AAE = left( {frac{1}{{Nleft( {N – 1} right)/2}}mathop {sum}limits_{i = 1}^N {mathop {sum}limits_{j = 1 + 1}^N {left( {mathop {sum}limits_{k = 1}^K {q_{ik}q_{jk}} – mathop {sum}limits_{k = 1}^K {widehat q_{ik}widehat q_{jk}} } right)^2} } } right)^{1/2}.$$
(8)
The minimum value of AAE is 0, when ancestry (admixture) is inferred perfectly. The maximum value is 1, when there are no admixed individuals in the sample, individuals from the same source population are always assigned to different populations and individuals from different source populations are always assigned to the same population. It is worth noting that the minimum AAE value of 0 is always possible for any population structure. However, the maximum value varies and can be much smaller than 1, depending on the actual underlying population structure. With an increasing K value or increasing admixture (i.e., qik→1/K for any individual i), the maximum value of AAE tends to decrease. For this reason, AAE cannot be compared fairly between different genetic structures (e.g., different K values, different actual Q for a given K, or different sizes of subsamples from the source populations) for measuring the relative inference qualities. However, it can always be used to compare the accuracy of different inference methods for a given simulated genetic structure and a given sample.Analysis of real datasetsAn ant datasetIt was originally used in a study of the mating system of an ant species, Leptothorax acervorum (Hammond et al. 2001). Ten sampled colonies, A, B, C, D, E, F, G, H, I, and J, contribute respectively 9, 7, 47, 45, 45, 45, 45, 45, 44, and 45 diploid workers to a sample of 377 individuals. For this species, we know that each colony is headed by a single diploid queen mated with a single haploid male. Therefore, workers from the same colony are full-sibs and workers from different colonies are non-sibs. Each sampled worker was genotyped at up to 6 microsatellite loci, which have 3 to 22 alleles per locus observed in the 377 individuals. This dataset was analysed to reconstruct the genetic structure of the sample, which actually is the family structure. ADMIXTURE and sNMF cannot handle multiallelic marker data and therefore only STRUCTURE and PopCluster are used for analysing this dataset.For STRUCTURE, I used the default parameter settings, except for the burning-in and run lengths which were both set to 105 to reduce the risk of non-convergence. Two analyses were conducted. First, optimal K values were determined using three estimators (Wang 2019) calculated from STRUCTURE outputs, and using the DLK2 estimator of PopCluster. For this K estimation purpose, 20 replicate runs for each possible K value in the range [1, 15] were conducted by both STRUCTURE and PopCluster. Second, assuming K = 10, a number of 100 replicate runs (each with a distinctive seed for the random number generator) were conducted by both STRUCTURE and PopCluster to investigate their convergence.An Arctic charr datasetShikano et al. (2015) sampled 328 Arctic charr individuals from 6 locations in northern Fennoscandia: two lakes (Galggojavri and Gallajavri) and one pond (Leenanlampi) in the Skibotn watercourse drain into the Atlantic Ocean and three lakes (Somasjärvi, Urtas-Riimmajärvi and Kilpisjärvi) in the Tornio-Muoniojoki watercourse drain into the Baltic Sea. Individuals were genotyped at 15 microsatellite loci to study the genetic structure and demography. The data were again analysed by STRUCTURE and PopCluster but not by ADMIXTURE and sNMF because the markers are multiallelic. I conducted two separate analyses of the genotype data. First, I estimated the most likely K value by each program, making 20 replicate runs with each K value in the range [1, 10]. Second, I investigated the convergence of each program by conducting 100 replicate runs of the data at K = 6. STRUCTURE analyses were run with default parameter settings except for both burn-in and run lengths being 105.A human SNP datasetUsing FRAPPE (Tang et al. 2005), Li et al. (2008) studied the world-wide human population structure represented by 938 individuals sampled from 51 populations of the Human Genome Diversity Panel (HGDP). Each individual was genotyped at 650000 common SNP loci. The data were expanded to include genotypes of 1043 individuals at 644258 SNPs, available from http://www.cephb.fr/en/hgdp_panel.php#basedonnees. In this study, the expanded data were comparatively analysed by PopCluster, ADMIXTURE, and sNMF, assuming K = 7 clusters (regions) as in the original study (Li et al. 2008). STRUCTURE was too slow to analyse this big dataset and thus it was abandoned.The human 1000 genomes phase I datasetThe dataset (Abecasis et al. 2012), available from https://www.internationalgenome.org/data/, has 1092 human individuals sampled from 14 populations across all continents, with each individual having 38 million SNP genotypes. After removing monomorphic loci (note, no pruning was applied regarding missing data, minor allele frequency and linkage disequilibrium, in contrast to other studies), genotypes at a number of L = 38035992 SNPs were analysed by PopCluster and sNMF, assuming K = 9 clusters (regions). Both STRUCTURE and ADMIXTURE were too slow to analyse this huge dataset and thus were abandoned. No attempts are made to find the optimal K for this dataset as done for the ant and Arctic charr datasets, because too much computational time is required for PopCluster or sNMF to analyse the data with a number of replicate runs at each of a number of K values even when using a large cluster, and there might be multiple K values that explain the data equally well (at different spatial and time scales). For a better understanding of the world-wide human population genetic structure, the data should be analysed at least with one replicate under each of a number of possible K values, say K = [5, 12], to reveal and compare the genetic structure. This study analysed the data at a single K = 9 for the purpose of demonstrating the capacity of different methods, and comparing the admixture estimates of PopCluster and sNMF at this particular value of K. Because of the incompleteness of the analysis, the biological interpretations of the results should be taken with caution.Comparative analyses by different softwareI compared the accuracy and computational time of STRUCTURE (Pritchard et al. 2000; Falush et al. 2003), ADMIXTURE (Alexander et al. 2009), sNMF (Frichot et al. 2014) and PopCluster in analysing both simulated and empirical datasets described above. Quite a few other model-based methods implemented in various software exist. I choose STRUCTURE and ADMIXTURE because they are the most popular model-based admixture analysis methods used for small and large datasets, respectively. I also choose sNMF because it is a very fast model-based method that works for huge datasets for which other methods, such as ADMIXTURE, fail to run or take unrealistically too much time to run.STRUCTURE can handle both diallelic (such as SNPs) and multiallelic (such as microsatellites) markers, but runs too slowly to analyse large datasets with many markers, many individuals, or many populations. It was therefore used to analyse all simulated and empirical datasets with no more than 10000 loci. The default parameter setting was used for most datasets, with a burn-in length of 104 and a run length of 104 iterations. For better convergence, the burn-in and run lengths were increased to 105 iterations for analyses involving a large number of simulated populations (say, when K ≥ 10) or for analyses of empirical datasets. For unbalanced sampling, the alternative ancestry model instead of the default model was used by setting POPALPHAS = 1.Both ADMIXTURE and sNMF were developed specifically for diallelic markers and could not analyse multiallelic marker data. In this study, they were used to analyse SNP data only. For the human 1000 genome phase I data, however, ADMIXTURE could not complete the analysis within a realistic period of time (72 h, the maximum allowed in the linux cluster used for the analysis) even when the maximal number of parallel threads were used. Therefore, only sNMF and PopCluster were used to analyse this dataset.To understand the relative computational efficiency and how much speedup can be gained by parallelisation, ADMIXTURE, sNMF and PopCluster were used to analyse the HGDP dataset and the 1000 genome dataset, by using a variable number of parallel threads on a linux cluster with many nodes, each having 32 cores. 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