Synthetic lethality (SL) is a promising gold mine for the discovery of anti-cancer drug targets. Wet-lab screening of SL pairs is afflicted with high cost, batch-effect, and off-target problems. Current computational methods for SL prediction include gene knock-out simulation, knowledge-based data mining and machine learning methods. Most of the existing methods tend to assume that SL pairs are independent of each other, without taking into account the shared biological mechanisms underlying the SL pairs. Although several methods have incorporated genomic and proteomic data to aid SL prediction, these methods involve manual feature engineering that heavily relies on domain knowledge.Here, we propose a novel graph neural network (GNN)-based model, named KG4SL, by incorporating knowledge graph (KG) message-passing into SL prediction. The KG was constructed using 11 kinds of entities including genes, compounds, diseases, biological processes and 24 kinds of relationships that could be pertinent to SL. The integration of KG can help harness the independence issue and circumvent manual feature engineering by conducting message-passing on the KG. Our model outperformed all the state-of-the-art baselines in area under the curve, area under precision-recall curve and F1. Extensive experiments, including the comparison of our model with an unsupervised TransE model, a vanilla graph convolutional network model, and their combination, demonstrated the significant impact of incorporating KG into GNN for SL prediction.https://github.com/JieZheng-ShanghaiTech/KG4SL.: KG4SL is freely available at Bioinformatics online.Supplementary data are available at  Complex biological systems cannot be composed by a large number of genes acting independently, but rely on the interactions between genes which can be further classified into enhancing and suppressive effects . The supin silico knockouts using metabolic network models. A spectrum of computational methods has been proposed for SL prediction. These methods can be categorized into three classes. The first class involves simulating 2MF  might latently invalidate the assumption of independency. For instance, poly (adenosine diphosphate-ribose) polymerase 1 (PARP1) and breast cancer 1 (BRCA1) are a famous SL gene pair, leading to the first clinically approved SL-based cancer drug, PARP inhibitor . PARP1 a2MF and GRSMF] have injected some genomic and proteomic data to facilitate the SL prediction, and the results of these studies have underscored the significance of integrating additional information. Meanwhile, GNN-based methods can also encode such information as input features. However, these methods extracted features manually based on domain knowledge and some features might be left out. Therefore, to attain a more comprehensive set of features to improve the performance of SL prediction, we need a new method capable of automatic knowledge integration and feature extraction.A subset of existing methods [e.g. SVM, random forests (RFs), SLE are defined as triplets Knowledge graphs (KGs) are a type of multi-relational graph, where nodes and edges have different types. A KG is denoted by wo nodes . By incoHere, we propose a novel KGNN-based method for SL prediction, named KG4SL, which utilizes KG MP as a back-end. We approach the independency issue by injecting various factors including biological processes, diseases, compounds etc. that could be pertinent to SL, into our KG. Our model comprises three parts. In the first part, we derive a gene-specific subgraph from the original KG for each gene. In the second part, we conduct MP on the gene-specific subgraph, to automatically associate genes with factors that could be decisive in identifying an SL pair. In the third part, we define a decoder to reconstruct gene–gene similarity in a supervised fashion. To the best of our knowledge, this is the first framework to integrate KG with GNN for SL prediction. We compared our model with 10 state-of-the-art methods for SL prediction, and our model outperformed all the baselines in area under ROC curve (AUC), area under precision-recall curve (AUPR) and F1. Another contribution of our work is that we studied the impact of KG, which suggests that introducing a KG combined with MP process in GNN can significantly improve the performance of SL prediction.In this section, we first introduce the data and the problem of SL prediction. Then, we present the details of the proposed KG4SL model.http://synlethdb.sist.shanghaitech.edu.cn/v2/#/; SynLethDB ,  and . And the other 8 relationships are associated with drug and compounds. Besides, 7 out of 11 kinds of entities are directly related to genes, i.e. pathway, cellular component, biological process, molecular function, disease, compound and anatomy. They are in the format of . These entities can be reached from genes in one hop, whereas the other three kinds of entities  can be reached from genes in two hops. After removing isolated nodes, the final graph of SynLethKG contains 54 012 nodes and 2 231 921 edges as shown in The 24 relationships in n is the number of genes involved in the SL pairs. In this SL matrix S, an entry ie and gene je, and 0 otherwise. Note that gene pairs with entry value 0 are unknown pairs, some of which could be potential SL pairs not yet discovered. The KG SynLethKG is denoted by eV and a set of relationships rV. Each edge in the KG is defined as a triplet r between head entity h and tail entity t, where Formally, the SL data can be modeled as a matrix S and the KG G, the problem we aim to solve is to predict the SL relationship between gene ie and gene je. To achieve this goal, we propose a GNN-based model to learn a scoring function ie and gene je is an SL pair, where W, A and b denote the learnable parameters in function F.Given the SL matrix The overall framework of KG4SL is laid out in 2.3.1Given an SL-related gene, we first construct a weighted subgraph from the KG. Identifying relevant nodes and determining the edge weights are two key operations to construct the gene-specific weighted subgraph.e is the central node/entity and N(e) is the set of neighbors of e (i.e. entities directly connected to e). In SynLethKG, the size of N(e) varies greatly among the entities. For example, network hubs may have thousands of relations, whereas some nodes are less studied and thus have a limited number of neighbors. In this work, we sample a fixed number of k neighbors for each entity to characterize its local structure and we repeat this process for H hops . An example of 2-hop subgraph with neighbor sampling size k = 4 in each hop can be seen in Assume that ie, je), the weight for an edge i’es subgraph is computed by e is one of the entities in the subgraph of ie, and je and relation g is an inner product function. Here, je.In a gene-specific subgraph, we can assign different weights for edges to describe the importance of the relations. For an SL pair ; hyper-parameters α,d, k, h and epoch;Output:  1: Initialization:eW;2: entity embedding matrix rW;3: relation embedding matrix 4: step ← 0;5: while step < epoch do6:   for  doGene-specific Subgraph(ie);7:    Gene-specific Subgraph(je);8:    9:    for m do10:     11:     for h = 1,2,…,H do12:      for  do13:       14:       15:      end for16:     end for17:     18:    end for19:    Compute the predicted probability 20:    Compute the loss 21:   end for22:   step ← step + 1;23:  end whileIn this section, we first introduce the state-of-the-art baseline methods and their implementation details, and then we compare our model with the baselines, followed by an analysis of the influence of the KG. The KG4SL model was implemented with Python 3.6 and Tensorflow 1.15.0. We adopt AUC, AUPR and F1 as the evaluation metrics.3.1.1We compare KG4SL with the following baselines:2MFSL  (rk (GCN)  is the mGraphSAGE , RW-based baselines  and GNN-based baselines . We evaluated DeepWalk, Node2Vec, LINE using CogDL  into the GNN model. Many existing methods view each SL pair as an independent sample from a latent representation, but this assumption is an oversimplification in the biological context of cancer cells. KG can bring additional information such as biological processes which could be crucial for discovering new SL pairs. By injecting the KG, our model was capable of harnessing the aforementioned issue, without manual feature engineering. Extensive experiments have been conducted to examine the impact of KG. The results show that both KG and MP in GNN are essential for boosting the model performance. Therefore, KG4SL represents a breakthrough in applying supervised machine learning to SL prediction.Our future work will focus on the following directions. First, we plan to further improve our model, using the strategy of contrastive learning, which is a self-supervised approach to learn graph representation. Secondly, the degrees of some nodes in a KG might be very large, while sampling a fixed-sized neighborhood may not fully capture the neighborhood topological structures of the nodes. To address this issue, it will be a highly desirable future work to contrast multiple receptive fields for a given node, to learn a more robust and enriched embedding. Moreover, inspired by the recently proposed KG attention network . By insp


The authors have been informed of some errors that occurred in the published paper. The errors are listed as:•In the M&M section in the main text, the word “performed” has been changed to “recruited”.•Some references have been modified in the text.•The type of randomization has been corrected as the random number table.•The Mann-Whitney test was added in the statistical analysis section.•The statistical test which has been used in Table I was added as a table subtitle.•The study results were re-analyzed, and some p-values were modified in the text and table, especially in Table IV.The authors wish to apologize for these errors.http://journals.ssu.ac.ir/ijrmnew/article-1-813-en.html&sw=(http://doi.org/10.29252/ijrm.15.4.231).The online version of the article has been updated on September 4, 2021 and can be found at


Procedia-Social and Behavioral Sciences. Spain is the country with the highest volume of production. It is concluded that research on ELHI use does not have an established line of research, due to its recent creation and the lack of related research. The bibliometric analysis specifies that the research is oriented towards knowing the level of acceptance and application of the pedagogical method in the teaching and learning processes.E-learning is conceived as a purely virtual training approach. Different learning styles have been proliferated in recent years, especially now, due to the impact of COVID-19 in the educational field. The aim of this study is to discover the evolution of e-learning in higher education (ELHI) in scientific literature indexed on the Web of Science. Co-word analysis and bibliometric analysis was performed. A total matrix of 1261 documents was analyzed through SciMAT software. The results revealed that studies on ELHI are written in English and presented by conference papers. The main source of publication for the conferences is EDULEARN proceedings, while the journal source is  Information and communication technologies (hereinafter ICT) are deeply rooted, widely accepted, and developed in today’s society . The fieIn this sense, new technological means have made it possible to generate new learning environments and methodologies ,5. In adAs a consequence of technological development in the field of education, the digital competence of teachers and students is presented as a relevant factor for the development of good practices with ICTs ,9,10. ThE-learning can be found within the broad spectrum of innovative learning methodologies. This training approach is conceived as a virtual instruction, carried out through content management platforms to carry out the teaching and learning process in a digital environment . This trThere are several advantages that e-learning has over other, more traditional teaching methods . E-learnThis type of teaching allows greater communicative interaction between educational agents, by being able to emit messages through the platforms established by educational centers . In addiDespite the potentialities reflected, e-learning also has limitations or disadvantages . This inAfter analyzing the literature on e-learning, it was observed that there are no studies that have analyzed the construct in a comprehensive manner. This is, from a bibliometric perspective, based on the scientific mapping of published studies. This problem, which has arisen in the impact literature, is intended to minimize with this work, presenting the importance and evolution of the concept of e-learning, specifically in higher education spaces. This research also offers the themes that e-learning runs through, as well as the way forward in coming years, revealing trends that are on the rise.Due to the importance of the subject today, the purpose of this work is to know the evolution of e-learning in the higher education stage in the scientific documents registered in the Web of Science (WoS) database.In this research, the concepts “e-learning” and “higher education” (ELHI) in WoS are studied. The analysis of this work was developed through scientific mapping ,35,36,37The purpose of this research is to reveal the evolution of ELHI through the analysis of WoS export documents. In addition, it focuses on knowing how scientific production performs and evolves, as well as specifying the most influential topics and authors.2 index [The research methodology used is bibliometrics. Bibliometrics allows to analyze different bibliographic characteristics in documents and the analysis of diverse features of scientific activity. This kind of analysis is important to scientific publications as a tool that allows qualifying the quality of the knowledge-generating process and the impact of this process on the scientific background . Specifi2 index . This isn ≥ 13); knowledge area (n ≥ 89); type of document (n ≥ 70); institutions (n ≥ 12); authors (n ≥ 5); source of origin (n ≥ 34); countries (n ≥ 74); citation (the four most cited references ≥ 100). These actions reported 1305 publications. The cut-off values for the criteria are established with those results whose nodes are formed by more than one document or value.The research procedure has been carried out in several stages: (1) selection of the database to be analyzed (WoS), considered the most important scientific database; (2) choice of keywords under study (“e-learning” and “higher education”); (3) construction of the search equation (“e-learning” (TITLE) AND “higher education” (SUBJECT)). Several inclusion criteria were also applied: year of publication ; language  that have been prepared under the criterion of documentary similarity.All presented analysis was carried out with the citation reports tool of WOS and SciMAT software.The set of documents related to ELHI in Web of Science is 1261. The start of production dates back to the year 2000. From that date until 2010, the growth in the volume of documents was high (being increasing and constant throughout the period). In 2011 the scientific production decreased compared to the previous year, but in 2012 and 2013, it increased again. From that same year until 2017, the production was constant (concerning quantity). However, in 2018 and 2019, production fell to levels close to 2008. Peak production occurred in 2015 .The language, mainly used by researchers, is English. All other languages are far removed from it .The studies on ELHI are collected mainly in the knowledge area Education Educational Research, being far from the other areas .Both papers and research articles are the main types of documents used by researchers to show their findings. The number of papers stand out, confirming the fact that this field of study is of a recent appearance in the scientific community .The Universitat Oberta de Catalunya is the leading institution in ELHI-related studies. The fact that the most prolific universities are Spanish stands out .The author with the highest level of scientific production on ELHI is Pelet, J.E. The rest of the authors are at a close distance, which shows the variety and number of authors who study the field under analysis .The main sources collecting research on ELHI are two books of conference proceedings, with EDULEARN as the main reference. This fact coincides with the data offered previously on the source of production. In this case, the main journal that publishes studies on this subject is Procedia Social and Behavioral Science .Two countries are the reference countries in this field of study in scientific production. Spain is in first place, closely followed by England .In ELHI’s field of study, the most widely cited work has 234 citations . The secThe evolution of keywords shows the number of keywords collected in a given period. In addition, it also shows the keywords that go out and come in a certain period. In this study, the level of keyword matching between contiguous periods is low, and it increased from its beginnings until 2019. This shows that the line of investigation is not established, although it is beginning to be established .The academic performance presents the most valuable subjects in each of the established time periods. These values are based on bibliometric indicators of various kinds . Academic performance is shown according to established time periods. In the first interval (2000–2011), the subject with the highest bibliometric value is “education”. In the second interval (2012–2015), “e-learning” is the most important. In the third interval (2016–2019), it is “extension” and “students” .Thematic diagrams provide data on the relevance of the different topics in a given time period. The coordinate axis shows the level of centrality and density. The former analyzes the relational strength of external links. The second analyzes the relational strength of internal links. The study of the three diagrams shows that no single theme is repeated as the driving force in the three periods. Only “satisfaction” stands out, which appears in the second and third periods .In the first interval (2000–2011), the driving themes are “perceived-usefulness”, which is related to “impact”, “technology-acceptance-model”, and “user-acceptance”; “e-readiness”, which is based on “measuring-e-readiness”, “cultural-aspects”, and “IT”; and “on-line” which is oriented towards “qualitative-differences”, “knowledge-work”, “teaching-quality”, “technology”, “satisfaction”, “conceptions”, “teaching/learning-strategies”, and “face-to-face”. In this period, although ELHI is analyzed, it seems that the research interest is more oriented towards distance learning than e-learning.In the second interval (2012–2015), the driving themes are “used-acceptance”, which is related to “perceived-ease”, “extension”, “national-culture”, “information-technology”, “model”, “behavioral-intention”, “self-efficacy” and “technology-acceptance”; and “satisfaction”, which focuses on “information-systems-success”, “continuance-intention”, “instructor”, “critical-success-factors”, “web”, “management”, and “web-based-e-learning-system”. In this period, the evaluation and acceptance of the e-learning method, mainly oriented to technological resources, takes on an important nuance.In the third and final interval (2016–2019), the driving themes are “DeLone” which is related to “McLean-model”, “information-systems-success”, “attitudes”, “e-learning-system”, and “management-systems”; “extension”, which is oriented towards “unified-theory”, “determinants”, “students-behavioral-intention”, “information-technology”, “technology-acceptance-model”, “user-acceptance”, “usage” and “variables”; “intention” which is based on “acceptance”, “information-systems-continuance”, “trust”, “computer”, “model”, “adoption”, “self-efficacy” and “perceived-ease”; “satisfaction”, which relates to “e-learning-success”, “students-perspective”, “public-administration-education”, “quality-of-e-learning-system”, “critical-success-factors”, “performance”, “context” and “use”; and “behavioral-intention” which focuses on “technology-acceptance”, “TAM”, “system-characteristics”, “perspective”, “preservice-teachers”, “structural-equation-modeling”, “university-students”, and “perceived-usefulness”. In this period, the e-learning method is more relevant, the behavior of the subjects when faced with the pedagogical method and the relationship between the people involved in the pedagogical act. Furthermore, in this period, the themes “MOOC”, “gamification”, “blended-learning”, and “Moodle” must be taken into account, given that their location in the diagram places them as unknown themes. In other words, the topics may become the next driving force, or they may disappear from the scientists’ lines of research.The scientific evolution provides information on the relationship between the themes of the recognized contiguous periods. To establish these relationships, the Jaccard index has been taken into account. The relationships can be of two types: continuous line and discontinuous line. The thickness of the lines shows the strength of the relationship between the themes.The scientific evolution of ELHI presents a conceptual gap. This is due to the fact that there is not one theme that is repeated in the three periods. Even so, two lines of research can be glimpsed, on the one hand that established by “model-user/acceptance-extension/behavioral-intention” and—on the other—that of “students-e/learning-students”. In other words, studies in this field of study are mainly oriented towards the acceptance of the pedagogical model and in the students themselves. Furthermore, as can be seen in The author considered as the driving force in the ELHI study is Gullu, F. , becauseThe ELHI field of study is relatively young, as it began in 2000 according to WoS. From that date to 2019, the level of production was irregular, with a great deal of incidence by the scientific community from 2000 to 2010, and then remaining constant over time. Since 2018, there was a downward trend in production, although with the health crisis caused by the COVID-19, it will probably rise again, given that it is currently the model used to develop all types of education, both university and non-university levels.The studies on ELHI are written in English and presented in conference papers. The area of knowledge where research is based on is Education Educational Research. The most important institution in this field of study is the Universitat Oberta de Catalunya , with Spanish institutions being the most productive. Pelet, J.E. is the author with the largest number of productions, although Gullu, F. stands out as the most relevant in this field of study. The main source of publication for the conferences is EDULEARN proceedings, while the journal source is Procedia Social and Behavioral Sciences. Spain has the largest scientific production on ELHI. The most cited manuscript is , with 23If we look at the co-word analysis, we can see that this field of study does not have a solid line of investigation recognized, given that the level of coincidence of key words between the established periods is low. Even so, it can be seen that a line of research is beginning to be established in this field of study, given that, over time, the number of key words coinciding between periods is increasing.Academic performance reiterates and confirms this fact, given that there is not one theme that repeats in all three periods, not even in two. This analysis shows that the topics dealt with in this field of research are oriented towards education, online teaching, and students.With regard to the most relevant topics in this field of study, it is shown that there is an evolution over time, with varying interest and focus on the part of the researchers. In the first period, interest is focused more on distance learning than on online learning itself, given that it uses terms that are not typical of the e-learning method. In the second period, this fact changes radically, given that interest is focused on the assessment and acceptance of e-learning by those involved in this teaching process. In the third period, the line of the second period is maintained a little, although the researchers extend their field of interest to the intention of those involved in the training process.Over the next few years, the themes of “MOOC”, “gamification”, “blended-learning” and “Moodle” should be kept in mind when researching the use of e-learning in higher education, and could be the next driving force.The thematic evolution once again confirms what was indicated previously—that is to say, there is no solid and stable line of research in this field. Moreover, there is a conceptual research gap related to ELHI. Even so, there are two lines that have focused more interest on the part of the scientific community. These timelines are “model-user/acceptance-extension/behavioral-intention” and “students-e/learning-students”. In other words, studies in this field of study are mainly oriented towards the acceptance of the pedagogical model and in the students themselves.It can be concluded that research on the use of ELHI does not have an established line of investigation. This bibliometric study indicates that the subject is quite new in the Web of Science database. That is to say, studies on this line of research have begun to be registered recently. The bibliometric analysis specifies that the research is oriented towards knowing the level of acceptance and application of the pedagogical method in the teaching and learning processes. This information offered here can guide researchers and professors in their work. The prospective of this research consists of offering the scientific community new trends in this field of study. In addition, the aim is to show the educational community the latest trends on this method of teaching and learning.As for the limitations of this research, we must say that there are several. Firstly, generating the database in WoS requires a great deal of effort for researchers. This is because the entire scientific production must be read to determine whether it follows the inclusion criteria set out in the PRISMA  protocol. Secondly, the time distribution has been carried out, taking into account the criterion of equity. An attempt has been made to collect an even number of manuscripts in each of the established time periods. Thirdly and finally, the criteria for inclusion have been established at the discretion of the researchers, which have sought to present results that are in line with each other in terms of size and relevance. In this case, the data shown in this study should be considered with caution. The change in the inclusion criteria may slightly change the data shown in this research. Future lines of research include analyzing the e-learning method in other educational stages, such as compulsory secondary education, baccalaureate, or vocational training.


D. The networks need not be binary, so vertices in the networks may have more than two parents and/or more than two children. In general, in order to make the resulting network acyclic, additional arcs not in D must also be contracted. This paper shows how to choose D so that the resulting acyclic network is “pre-normal”. As a result, removal of all redundant arcs yields a normal network. The set D can be selected based only on the geometry of the network, giving a well-defined normal phylogenetic network depending only on the given network. There are CSD maps relating most of the networks. The resulting network can be visualized as a “wired lift” in the original network, which appears as the original network with each arc drawn in one of three ways.As phylogenetic networks grow increasingly complicated, systematic methods for simplifying them to reveal properties will become more useful. This paper considers how to modify acyclic phylogenetic networks into other acyclic networks by contracting specific arcs that include a set  A phylogenetic tree is a directed tree whose vertices represent biological species, whose leaves typically correspond to known extant species, and whose branchings indicate speciation events, usually by genetic mutation. As such, internal vertices have in-degree one and out-degree at least two (except for the root with in-degree zero). In the last decades it has become clear that other events such as hybridization and lateral gene transfer are also important in evolution, even though they are not easily modeled using phylogenetic trees One measure of the complexity of an acyclic X-network and Suppose N is a rooted tree, then (a) If N is normal, then (b) If N is regular, then (c) If N is tree-child, then v is unbounded.(d) If n indicates that normal networks are potentially a more tractable network type than regular or tree-child. Also indicative of their tractable nature is the fact  N, from which properties of N. The wired lift is not a subnetwork of N in the usual sense.If entclass1pt{minima Willson  show thaN, even though there is usually no CSD map from N to N with each arc drawn in one of three different ways. Thus we can visualize the resulting normal network by looking at a redrawn diagram of N. The current author thinks such visualizations can provide a tool for better understanding complicated networks.Section Section X-network N, to find a related, uniquely determined, normal X-network, which I will denote N. The fast program PhyloSketch  is regarded as directed from u to v, so we call u a parent of v and v a child of u. We assume N is a simple graph: there are no loops ; and there is at most one arc  for Let corresponding undirected graphUnd(N) edges and are not directed. In this paper, N will always refer to a directed graph unless otherwise specified.If in-degree of a vertex v in N, denoted indeg(v) or indeg , is the number of arcs , i.e. the number of parents of v. The out-degree of a vertex v, denoted outdeg(v) , is the number of arcs  , i.e., the number of children of v.The v may have We shall not assume that our directed graphs are binary. Thus a vertex leaf is a vertex root is a vertex v is hybrid or reticulate if u of v is a tree-child if v, u) is the only arc coming into u. A vertex v is trivial if A u and v are vertices, a path or, for emphasis, a directed path from u to v is a sequence of vertices i, length of the path is the number n of arcs. Note that the arcs are uniquely determined by the vertices in the list since N is a simple graph. No two successive vertices can be the same since there are no loops. We say the path contains arc segment.  is a simple directed graph; Let X-networkN is an X-directed graph An V is a finite set (the set of nodes or vertices).(N1) A (the set of arcs) is a finite set of ordered pairs  with u and v distinct members of V.(N2) root, is a node with in-degree 0.(N3) (N4) The map (N5) Each leaf is a vertex with in-degree 1 and hence has a unique parent.(N6) The image of (N7) v.(N8) For each v to some leaf.(N9) For each acyclicX-network is an X-network that also satisfiesAn N has no cycles.(N10) phylogeneticX-network to be an acyclic X-network that contains no trivial vertices.Following Steel , we defiThese assumptions are not intended to be the minimal possible; rather, they tell the properties we will utilize the most.p(x) or p. The arc of form x-arc. If x is not specified, any such arc will be called an X-arc.If N is an X-network. By (N4) and (N6) we may identify X with the set of leaves.Suppose u to v then we write X-network is acyclic, then u and v we have If there is a directed path from X-networks or two X-directed graphs X-isomorphic iff there exists a map Two f is one-to-one and surjective.(i) (ii) (iii) (iv) N and In this situation, cl (or cl(v) when N is understood) for cluster of v. Note that v, cl(v) is nonempty.For each It is immediate that if N.Let N be an X-labeled graph. An arc  is redundant or a short-cut if there exists a path a, b). Thus, there is no lengthening or a lengthening path of . Examples will be seen in several figures later, such as Fig. Let We shall have need of the following result:N is an acyclic X-network. Suppose there is a directed path in N from a to b. Then, a directed path in N from a to b of maximal length contains no redundant arc.Suppose P in N of maximal length k. If the result is false, we may assume that for some P by replacing the segment Since the vertex set is finite and there are no cycles, there is an upper bound to the length of a path. Suppose X-networks which will be of interest:There are several types of X-network N is tree-child  is the power set of X;(1) the cluster map N has no redundant arcs; and(2) (3) X-network N is normal  N contains no redundant arc.(2) Sometimes there are small differences in the definition of a network. In Baroni et al.  and WillN and X-networks. One interesting way to compare them is their Robinson-Foulds distanceCl(N) and X-networks X, it is a metric on the collection of regular X-networks  denote the result of contracting every arc  such that N, for some p(x) is hybrid and has out-degree one. Then in S(N) the arc N will have been contracted, and S(N) will not satisfy (N5). Thus in S(N) a leaf can be hybrid. Moreover, any trivial vertices will have been suppressed.If The following result shows that two normal networks N is a normal X-network, then S(N) is a regular X-network and (1) If X-networks and (2) Suppose X-network N, if  is an arc and N is normal then S(N) remains normal and hence is a regular network (Willson (1) For any  Willson .X-networks. (2) If X-networks, it can be used to compare how well various networks of various types “approximate” a given network. For example, if N is a complicated acyclic X-network and T and X-networks that are rooted trees, then T might be a better approximation to N than The X-network N into a normal X-network In this paper we will be “simplifying” an acyclic X-directed graphs. A connected surjective digraph (CSD) map (Willson Let  Willson  \documen(C1) digraph map).(C2) For each arc (C3) For each (C4) u, v in V such that (C5) For each N. Thus in the undirected graph Und(N) of N, if W and edge set (C6) For each N and Note that if X-networks. A CSD-map leaf-preserving if for each Let V such that (C7) x-arc x-arc (C8) the If  Willson  that theX-network” refers to what in this paper is an X-directed graph satisfying (N1), (N2), (N3), (N4), (N6), (N7), (N8), and (N9). Thus the networks in Willson  which contains D and is strongly closed. Hence, we are able to define X-network that results from contracting the arcs of D and also the other arcs needed for acyclicity. The sections after this one will rely on the iterated use of this construction. The fundamental problem studied in this paper is, roughly, how to choose D so that we can find a normal network from Here is a summary of this fundamental section: The basic tool used in this paper is that of successively contracting arcs in an X-network. Suppose V. Let [v] denote the equivalence class of V under Let quotient digraphAs in Willson  the quot(1) (2) (3) For each u] and [v] be in (4) Let [We will denote this quotient digraph by connected if each equivalence class [v] is connected in N. An equivalence class [v] is convex if, whenever N with both i, convex if each equivalence class [v] is convex.The equivalence relation root-preserving if the equivalence class leaf-preserving providedThe relation (1) if u such that (2) for any X-networks. A CSD-map leaf-preserving if for each Let V such that (1) x-arc x-arc (2) the The following result is similar to Theorem X-network. Let V, and let Let X-network.(1) (2) The natural map a to b in N, then there is a path from First we prove (1). (N1) and (N2) are immediate. (N4) is immediate since u] and hence an arc from some N.Each N there is a path in N from v to v] to Let v] is a leaf of v is not a leaf of N, then there is a path in N from v to v] is a leaf, it follows Suppose [N from Suppose in u] has in-degree 0. By (N8) there is a path from u, say u] has in-degree 0, it follows that u] has in-degree 0, it follows For (N7), suppose . (C2) is immediate since if v] is connected as well.We now prove (2). (C1) is immediate since every vertex of (C7) restates that N is acyclic.We do not claim that projection from N to We will refer to the map X-network. Let D be a subset of A. Define a relation V by saying vertices a and b satisfy Let D be a subset of A. A path D-path provided that for each i, Droot-preserving if whenever D-path, then every path from a is a D-path.Let X-network. Let D be a subset of A. ThenLet (1) D contains no X-arc.(2) D is root-preserving.(3) v the trivial path at v shows D is a subset of A, it is clear that each equivalence class is connected. This proves (1).(1) For any vertex v under Write D contains no X-arc, then for each (2) If D is root-preserving. We must show that the equivalence class N such that (3) Assume that v] if D is understood) denote the equivalence class of v under equivalence relation determined by D. It is clearly the smallest equivalence relation u, v) satisfying Let X-network. Let D be a subset of A. Assume D contains no X-arc and D is root-preserving. Then the quotient digraph X-network. Moreover, the projection Let By Theorem D contains no X-arc and is root-preserving, we will write quotient X-network of N under D and refer to its formation as contracting or merging the arcs of D. In general, N is acyclic.Henceforth if It is not hard to program a computer to find X-network N. Let D contains no X-arc. Note the root Figure X-network. A subset closed if it satisfies the following: Suppose i satisfying a, b) is an arc and Let D is not closed since In Fig. X-network. Let D is closed then each equivalence class of Let D is closed. Let v] is convex. Write N where Suppose D is not closed.Figure u, v) is merged, the number of vertices drops by one, reducing the resolution. In Fig. D contained 4 arcs, and the number of vertices dropped from 13 in (a) to 9 in (b). On the other hand, if Figure X-network. A subset strongly closed if it satisfies the following: Suppose there are vertices V with k such that K.Let X-network. Let D is strongly closed, then D is closed. Hence D is root-preserving.Let D is strongly closed. Suppose there is a sequence (not necessarily a path) a to b. Then D is strongly closed, for D is closed.Suppose D is root-preserving. By Theorem D that is closed but not strongly closed. Indeed, Figure D is strongly closed, then The next result shows that, if X-network. Assume X-arc. If D is strongly closed, then Suppose D is strongly closed. By Theorem D is root-preserving. By Theorem X-network. Suppose D is strongly closed, each arc Write D we can construct a uniquely determined strongly closed set K that contains D.The following theorem shows that from a given X-network and Let (i) K is strongly closed, and(ii) (iii) for every strongly closed K is the unique minimal strongly closed subset of A containing D.Thus A. Let We define a sequence V such that n by If A is a finite set, the chain K contains C that contains C must contain K. By construction D is a set of arcs in the X-network N, the strong closureD is the smallest set K of arcs that contains D and is strongly closed. By Theorem 3.7K is uniquely determined.If The next theorem is the main result of this section.X-network. Assume X-arc. Let K(D) be the strong closure of D. Let Let X-network.(1) v] of (2) Each equivalence class  to [b]. Then in N there is a g-path from a to b.(4) Suppose there is a path in Write N. Since i, either (1). Let (2) (3) Since (4) Suppose X-network. Let b is not a leafD is strongly closed and contains no X-arcs.Let D contains no X-arcs since such were specifically excluded. Note that if i, either X-arc in D, no It is immediate that D is strongly closed, suppose there are vertices V such that D. Then To see that X-network. Let b is not a leafLet X-network.(1) (2) For every b is a leaf.(3) For every arc (4) p(x) for some (5) No vertex of (6) (1) follows from Theorem N from v to (2) By Lemma D it follows that Since b is not a leaf. If For (3) suppose u, v) is an arc of v is not a leaf. There exists b is not a leaf since For (4) suppose (c. Then For (5) write For (6), note v is trivial if X-network v with unique child c. When this happens, p(x) is trivial, it has a unique parent u(x). By Theorem u(x) satisfies p(x) can be merged with u(x) and hence removed. We state this as a theorem:Recall that a vertex X-network and p(x) for Suppose X-network.(1) (2) There is a leaf-preserving CSD map (3) X-network.(4) (5) E contains no X-arcs. It is easy to see that E is strongly closed. Hence, X-network, proving part of (1).It is immediate that N to u(x) did not have out-degree one in u(x)] does not have out-degree one in u(x)] is not trivial. Thus, There are leaf-preserving CSD maps A very similar network was described in Willson  by a difX-network N, and Fig. N and much simpler than N. Vertices in N with the same cluster.Figure X-network N all its redundant arcs to form Our goal in this paper is the construction of normal networks which by definition contain no redundant arcs. A crucial step will be removing from an N is an X-network, let N by removing all redundant arcs. More precisely, if A by removing all arcs redundant in N.If X-network. Then X-network.Suppose The proof is straight-forward and is omitted.f may not induce a CSD map from X-network M. Let M has no redundant arcs, while N has two redundant arcs, indicated by dashes. The projection map N without the dashed arcs, and If N to In the same figure, one sees easily that there is no CSD map from N is an acyclic X-network. For each Suppose v to N. By Theorem v to N of maximal length contains no redundant arc, hence lies in v to N from v to x, proving Suppose X-networks. Suppose wired lift of N is described. It provides a method for visualizing N. In this section we modify and generalize the notion so that connected map. This will let us obtain wired lifts from a process that includes both CSD maps and removing redundant arcs.Let  Willson  a wired X-networks, and let f is a connected map Let (K1) (K2) (K3) for all (K4) for each (K5) for each f is a connected map while in general it is not a CSD map.It is immediate that a CSD map is connected, but a connected map need not be CSD. Suppose V. A  wired lift of f (or of N) is a pair Let (W1) For each arc u, v) representspre-arc of (W2) For every arc representative arcs since each represents an arc of Call the members of V. Thus for all Note that the collection of all backwards mapg is a map f is a CSD map, for each u, v) exists, and Suppose There are several situations that give rise to wired lifts. We describe three of them in the next theorem. A fourth will be given in Theorem X-networks, and let Let (1) Let (2) Suppose (3) Let f can be regarded as a map (1) and (2) are immediate from the definitions since a CSD map is connected. For (3), note that N. An example is shown below in Fig. N except that each arc may be wide solid, thin solid, or thin dashed. Suppose u, v) a wide solid arrow if N and each vertex of g-paths using allowed steps, which we will now describe.Given a connected map X-networks, with f. If u and v are in V, we say there is an allowed step from u to v if either Let X-networks, with f. Let V defined by Let N there is a sequence of vertices (1) In N such that, for i such that (2) There is a sequence of vertices Suppose there is a sequence of type (1). If Conversely, given a sequence of type (2), if the allowed step from generalized path or g-path from a to b in We will call a sequence of either type a X-networks. Let f.Let N (of type (2)). Then (1) Suppose N from a to b.(2) Suppose If there is an allowed step from Conversely suppose In Willson  there waN; thus if we ignore thickness and dashing and include all arcs with their indicated directions, whether thin, wide, or dashed, the diagram exhibits N. A vertex of f(10) of f(15) has inverse image Figure f(16) to f(1). The corresponding path in N from 16 to 1. Similarly the g-path 21,17,4 shows that in The dashed arcs are not permitted on g-paths, and wide solid arcs must be followed in their direction. Thin solid arcs can be followed in either direction. Thus, 16, 10, 8, 9, 15, 1 is a g-path showing that X-networks. Suppose f is nevertheless a connected map and there is a wired lift of f. Consequently, we are able to visualize simplifications of Suppose X-network. Write Suppose for (i) (ii) (1) f.(2) f is well defined since To see (1), note that For (2), to prove (W1) suppose For (W2), given any arc f on the diagram of In the situation of Theorem u, v) if (1) a thin solid arc  if (2) a thin dashed arc  if (3) a wide solid arc  an obstacle if (1) v is not a leaf, and (2) every child of the vertex v in v may be regarded as an “obstacle” to N being pre-normal.Let We must ignore redundant arcs when deciding our strategies concerning which arcs to merge. To make these decisions we need to have a notion of in-degree and out-degree that does not count redundant arcs.v is a vertex of an X-network N. The non-redundant in-degree of v, denoted nrindeg(v), is the number of non-redundant arcs ; hence it is the number of parents of v by non-redundant arcs. If non-redundant out-degree of v, denoted nroutdeg(v), is the number of non-redundant arcs , hence the number of children of v by non-redundant arcs. If v is not a leaf, then by Theorem Suppose v of N is nonr-hybrid if c is a nonr-child of v if  is a non-redundant arc; we also say v is a nonr-parent of c. A nonr-child c of v is a nonr-tree-child of v if nonr-path if no arc A vertex v is a pre-normal obstacle iff (1) v is not a leaf, and (2) every nonr-child of v is nonr-hybrid.It is immediate that X-network with redundant arcs. Note that 8 is an obstacle since both its children 12 and 13 are nonr-hybrids. But 9 is not an obstacle since the only nonr-parent of 10 is 9 and Figure X-network N is obstacle-free if it contains no pre-normal obstacle.An N is an acyclic X-network that is obstacle-free. Then X-network and N is pre-normal.Suppose v that is not a leaf, there is a non-redundant arc  with c is a tree-child of v. Since N is pre-normal. By hypothesis, for every vertex X-network is always pre-normal, but a pre-normal network need not be tree-child.Theorem X-network N, when we seek a normal network M, we know by Lemma 2.2 that S(M) will be regular; by (1) in the definition of regular network, S(M) is SCD. We are therefore seeking a network that is very close to being an SCD network, and it is plausible to start with the very general SCD network Theorem N be an acyclic X-network. Suppose c is an obstacle. An allowable 1-fold parent chain of c is a path d is via a redundant arc). An obstacle c is of type 1 if c has an allowable 1-fold parent chain. If c has type 1 and Now we show that there are different types of pre-normal obstacles. Let c is an obstacle and allowable k-fold parent chain for c is a nonr-path d distinct from c is of type k ifSuppose c is not of type (a) c has an allowable k-fold parent chain.(b) k-fold parent chain writeIn this situation, for this N be an acyclic X-network. Then every pre-normal obstacle c has a unique type.Let It is clear that the type, if it exists, is unique.c which has maximal length k. Write this path as d other than k-fold parent chain of c, so c has type at most k. If, instead, q of d other c has type Consider a path from c is not a leaf hence must have a nonr-child e. Since c is an obstacle, e has a nonr-parent q must start c to q, hence a nonr-path from c to q to e, showing that  is redundant, a contradiction. Hence some such c has an allowable If no such The following result shows a simple way to remove a type 1 obstacle:N is an acyclic X-network and c is a type 1 obstacle with allowable 1-fold parent chain p, c, where p has nonr-tree-child Suppose c is an obstacle, it is not a leaf, so D contains no X-arc. Moreover, D is strongly closed since  is not redundant, and X-network. Note p, c], d). If q is any parent of d in N other than p, then  is redundant since d is a nonr-tree-child of p. Hence by Theorem d. Thus the lengthening must include the nonr-parent p of d and there is a path in N with non-redundant arcs from q to p to d. Since q, d) is redundant in Since k. The next result assumes for simplicity that D is strongly closed.The result above often generalizes to obstacles of type N be an acyclic X-network with pre-normal obstacle c of type k. Suppose k-fold parent chain, where D is strongly closed. Form d and Let D contains no X-arcs since c is not a leaf. By Lemma d becomes a nonr-tree-child of d becomes a nonr-tree-child of d. Note that The following lemma shows that, often, once an obstacle is removed, it does not reappear when subsequent arcs are merged.p, c) is a non-redundant arc in the acyclic X-network N and p and c. Let a, b) is a non-redundant arc in N and b is a nonr-tree-child of a  (2) There is a leaf-preserving CSD map (3) N is normal, then (4) If E contains no X-arc because a leaf does not have out-degree one. Since E is clearly strongly closed, X-network. (1) and (2) follow as in Theorem u is trivial, then Note that N is tree-child, N by removing one trivial vertex t with parent q and child c. In q obviously still has a tree-child, the same one as in N. We must show that q has a tree-child in t has a tree-child in N which must be c so c has no other nonr-parent than t in N. Hence q in c which has no other nonr-parent and is therefore a tree-child. This proves For (4) we first show that since N by the removal of a single trivial vertex t with parent q and child c. The only possible redundant arc in q, c). If it is redundant, there is a path in q to c other than the arc, hence a path in N from q to c not through t. Such a path of maximal length by the proof of Theorem c has a nonr-parent besides t. This contradicts that N was normal, since t in N has no tree-child. For (4) we must also prove that X-network N, to compute a uniquely determined normal X-network. We first compute a uniquely determined pre-normal acyclic X-network pre-normalization of N. The computation uses the procedure PRENORM described below. Briefly, if N is not already a pre-normal acyclic X-network, we compute c we compute its type k and find all the allowable k-fold parent chains for c. Let D(c) be the union of c. Let D be the union of the D(c) for all the obstacles c. We then compute We now show how, given an Here is a more detailed description of the computation of height of r. Thus N is acyclic and pre-normal; otherwise, r is 1 more than the number of times the procedure passes through steps 5 , 6, 7. The height r is a crude indicator of the complexity of the calculation. The examples in this paper satisfy The network output by the procedure will be denoted The next theorem shows that the procedure works.X-network.Let N. ThenApply procedure PRENORM to X-network (1) The procedure terminates and outputs an acyclic N.(2) (3) The projection N that contains no dashed arcs.(4) Let N and X-network which contains no trivial vertices. If it contains no obstacles, it is pre-normal by Theorem c. For each obstacle c of type k and each allowable k-fold parent chain c the set X-arc since the obstacle c cannot be a leaf. Hence D(c) contains no X-arc, so D contains no X-arc. By Theorem X-network, which we expect will be pre-normal by Lemmas If the procedure returns D is nonempty, so more arcs are contracted and N is finite, the procedure must terminate. This proves (1).Each time step 5 is utilized on The procedure never involves a choice, such as favoring some obstacles or some allowable parent chains over others. Hence (2) is true.We wish to see (3). If For (4), since X-network, we remove the redundant arcs to form normalization of N. The next theorem records its basic properties.Since N is an X-network. Let Suppose X-network containing no trivial vertices, hence a phylogenetic X-network.(1) N.(2) The definition of (3) The projection N to (4) The composition N which may contain dashed arcs.(5) There is a wired lift of (6) X-network by Theorem N. Then (3) follows from Theorem For (1) note There is an interesting variant of the procedure PRENORM. Define the procedure VARIANT PRENORM to be exactly like PRENORM except that step (5c) is replaced byk-fold parent chain c, and let  The procedure terminates and outputs an acyclic (2) The projection N that contains no dashed arcs.(3) There is a wired lift of Let X-network containing no trivial vertices, hence a phylogenetic X-network.(4) (5) The projection (6) The composition N.(7) There is a wired lift of a variant prenormalization of N and is denoted C records the particular choice of allowable k-fold parent chain for c made in step  c of type k there is more than one allowable k-fold parent chain, then (2) If for some obstacle u and v are vertices of (3) If (4) u and v are vertices of (5) If (6) If (7) If c of type k, E contains exactly one allowable k-fold parent chain for c, while D contains all allowable k-fold parent chains for c. Then (2) is obvious and (3) follows from (1). For (4), go through the proof of Theorem K(E), it necessarily must be added also to K(D). Then (5) follows from (4). For (6), the vertices of (1) is true since for a given obstacle A calculation of Other variants making other choices of parent chains are possible as well.N in Fig. N has four more distinct clusters than the tree This example continues Example N in Fig. N is already SCD. Consider D is strongly closed, and we find v, N contains the arc .The projection map f is a connected map. Figure N. The thin solid arcs show immediately that there were identifications Let mentclasspt{minimaN contains vertices with clusters N. In this example, It is easy to compute that X-network N with a single obstacle 16 of type 3. N is seen by changing all thin solid or dashed arcs to wide solid. N is easily verified to be already SCD, so N is 11, 14, 17, 16; another is 13, 14, 17, 16. The chain 13,14,16 is not an allowable parent chain because  is redundant. To find Figure N. When these arcs are removed from u, v) such that As maps of vertex sets The wired lift of This section contains two examples from real biological data.N be their graph. A wired lift of N is seen if each arc in Fig. N, which contains 31 vertices (of which eight are hybrid), and 38 arcs. We find Glémin et al.  study pen et al.  to indicN for the angiosperm genus Viola in their Fig. D contains 4 arcs. u, v) can map to the same arc Marcussen et al.  exhibit N than s et al.  in theirD has 3 rather than 4 arcs, leading to one more vertex in N as well as does If we use instead VARIANT PRENORM, there are two possible normal networks N by Theorem Both such Both variant normalizations satisfy Further comments concerning this example are given in Sect. Comparison ofandX-network. It is interesting to contrast X-networks depending only on the geometry of N. Both allow vertices of N to have in-degree greater than 2 or out-degree greater than 2, and both apply quite generally.s et al. . Both arN. A vertex v is visible if there exists v. This set of visible vertices forms the initial vertex set of N, as is done in Francis et al. , where u and v are distinct visible vertices of N, precisely when N and there is no third visible vertex w such that u could be present because of directed paths in N from u to u to The arcs of N is for the Viola genus of Marcussen et al.  are visible and hence they do not appear in N, and (c) there is no other path from 22 to 18 containing a third visible vertex. Similarly N. Thus these three distinct arcs in N involving 13 and 14.Consider again Fig. n et al. . The diaN that appear in g-paths; the thin solid arcs tell what arcs must be merged to obtain the normal network g-paths lets us understand In contrast, the arcs of Software The author has written software using Xcode which implements the calculation of Future work One can ask whether there are other classes of networks besides normal networks for which a similar construction could be used to simplify a network N into one of this other class. Suppose, given an X-network N, we sought a tree-child X-network C. Since a tree child network may contain redundant arcs, we should like a construction that depends only on the geometry of N and yields a CSD map


Scientific Reports 10.1038/s41598-021-89417-8, published online 12 May 2021Correction to: The original version of this Article contained errors.In the PDF version, Table 1 was incorrectly sized resulting in the loss of the final two columns of data: “DIF” and “BMI”. The HTML version was unaffected.F statistics, where F  and F  were incorrectly given as F =  and F =  respectively.In addition, there was a repeated error in the The original Article has been corrected.The conclusions of the Article remain unchanged.


Meniscoid lesions have been reported in patients with chronic ankle injuries, especially in soccer athletes, and such lesions cause soft-tissue impingement and pain. To our knowledge, we are the first to report a meniscoid lesion in the ankle joint presenting as a long-term sequela of avascular necrosis of the talus that developed in childhood.In this paper, we describe a 55-year-old Caucasian male patient who presented with a 1-year history of intermittent locking, “giving way,” weight-bearing pain, and swelling over the anterior aspect of the ankle joint. Imaging showed a rare case of avascular necrosis of the talus associated with an unstable plica-like lesion that was removed arthroscopically after unsuccessful conservative treatment.We demonstrate that unstable meniscoid lesions of the ankle joint can be treated successfully with arthroscopic debridement. We obtained satisfactory short-term clinical results at the 2-year follow-up, even though advanced osteoarthritis was present. Avascular necrosis (AVN) of the talus is a challenging condition to treat, and is usually associated with major disabilities in patients. Fracture of the talus is the most common cause of AVN, accounting for up to 75% of cases . AnatomiA 55-year-old Caucasian male presented for consultation with a 1-year history of intermittent locking, “giving away,” weight-bearing pain, and swelling over the anterior aspect of his right ankle that had recently been aggravated. The patient was a sedentary worker and non-athlete. He stated that he had experienced right ankle trauma at 8 years old, following an accidental fall from a height that was never investigated or treated. His pain and swelling had disappeared after several months, and the patient never had any further symptoms until recently.Physical examination revealed pain over the anterior aspect of the ankle joint. No snap was audible, and no lateral instability was exhibited. Provocative tests of the flexor hallucis longus were painless, dorsiflexion was slightly limited and painful, and forced plantar flexion was full and painless. Moreover, we noted pain with a single leg squat. There was no proprioception impairment, postural control difficulty, or strength deficit. Shortening of the right inferior limb was measured to be approximately 1 cm.Plain radiographs Fig. a and b sAfter 6 months of conservative treatment, including rehabilitation, pain medication, and intraarticular corticoid injection, the patient still had the same complaints. Surgical arthroscopy was chosen for debridement. An ambulatory arthroscopic intervention was performed under general anesthesia. A tourniquet was applied for hemostasis. Two portals were used: anterolateral and anteromedial. At the time of surgery, no instability was seen during anesthesia. Upon arthroscopic examination through the anterior medial portal, an unstable band of white, meniscus-like tissue accompanied by an unstable 1.5 cm ossicle Fig. a–d was fThe follow-up radiograph Fig. a and cliet al. in 1950 [et al. [et al. [et al. [et al. [Meniscoid lesions in the ankle were first reported in the literature by Wolin  in 1950 . Other s in 1950 . Acute i in 1950 . We have in 1950  identifi in 1950 . McCarro [et al.  advocate [et al.  that chi [et al.  and has  [et al. , in the  [et al. , 14, whe [et al. .In this case report, we present a case of an unstable preexisting meniscoid lesion accompanied by an unstable ossicle as the long-term sequela of AVN of the talus. This case initially developed in childhood and became more symptomatic in adulthood. Both lesions led to fibrous and osseous impingements that were removed arthroscopically. The treatment was a success, with the patient complaining of only very rare pain and swelling at the 2-year follow-up. This result supports the role of arthroscopy in symptomatic meniscoid lesions in the ankle joint as a first line of surgical treatment. However, for later end-stage ankle arthritis, more invasive measures such as tibiotalar fusion may be needed.


Scientific Reports 10.1038/s41598-018-24414-y, published online 23 April 2018Correction to: This Article contains an error in Figure 1A. As a result of a mistake in figure assembly, the band for p-MLKL is incorrect. The correct Figure This change does not affect the conclusions of the Article.


One of the major approaches in such a study is inverse quantitative structure activity/property relationship (inverse QSAR/QSPR) analysis, which is to infer chemical structures from given chemical activities/properties. Recently, a novel two-phase framework has been proposed for inverse QSAR/QSPR, where in the first phase an artificial neural network (ANN) is used to construct a prediction function. In the second phase, a mixed integer linear program (MILP) formulated on the trained ANN and a graph search algorithm are used to infer desired chemical structures. The framework has been applied to the case of chemical compounds with cycle index up to 2 so far. The computational results conducted on instances with  In computational molecular biology, various types of data have been utilized, which include sequences, gene expression patterns, and protein structures. Graph structured data have also been extensively utilized, which include metabolic pathways, protein-protein interaction networks, gene regulatory networks, and chemical graphs. Much attention has recently been paid to the analysis of chemical graphs due to its potential applications to computer-aided drug design. One of the major approaches to computer-aided drug design is quantitative structure activity/property relationship (QSAR/QSPR) analysis, the purpose of which is to derive quantitative relationships between chemical structures and their activities/properties. Furthermore, inverse QSAR/QSPR has been extensively studied , 2, the descriptors and correspond to feature vectors in machine learning. Using these chemical descriptors, various heuristic and statistical methods have been developed for finding optimal or nearly optimal graph structures under given objective functions  denote the set of integers i with Let graph stands for a simple undirected graph, where an edge joining two vertices u and v is denoted by uvH are denoted by V(H) and E(H), respectively. Let V of vertices and a set E of edges. For a vertex v in H is denoted by degreev is defined to be distanceu and v in H. The diameterH is defined to be the maximum distance between two vertices in H; i.e., P.A Centers of trees For a tree T with an even  diameter d, the center is defined to be the vertex v  is designated as the root. Let T be a rooted tree, where for two adjacent vertices u and v, vertex u is called the parent of v if u is closer to the root than v is. The heightv in T is defined to be the maximum length of a path from v to a leaf u in the descendants of v, where v in T. Figure mentclass2pt{minimDegree-bounded trees For positive integers a, b and c with T denote the rooted tree such that the number of children of the root is a, the number of children of each non-root internal vertex is b and the distance from the root to each leaf is c. We see that the number of vertices in T is T is T, we denote the vertices by T, let j such that j such that i, j) of vertices T. We call proper if the next conditions hold: T such that For each pair of vertices T with Each subtree Note that a proper set Branch-height in trees In this paper, we introduce “branch-height” of a tree as a new measure to the “agglomeration degree” of trees. We specify a non-negative integer k, called a branch-parameter to define branch-height. First we regard T as a rooted tree by choosing the center of T as the root. Figure T.leafk-branch: A non-root vertex v in T such that A non-leafk-branch: A non-root vertex v in T such that v has at least two children, and for each child u of v it holds that k-branch a k-branch. Figure k-branches of the rooted tree A entclass1pt{minimak-branch-path: A path P in T that joins two vertices u and u and k-branch and P does not contain the root or a k-branch as an internal vertex.A k-branch-subtree of T: The subtree of T that consists of the edges in all k-branch-paths of T. We call a vertex  in T a k-internal vertex  if it is contained in the k-branch-subtree of T and a k-external vertex  otherwise. Let k-internal and k-external vertices  in T.The k-branch-tree of T: The rooted tree obtained from the k-branch-subtree of T by replacing each k-branch-path with a single edge. Figure k-branches and non-leaf k-branches are leaves and branching points in the k-branch-tree.The mentclass2pt{minimk-fringe-tree: One of the connected components that consists of the edges not in the k-branch-subtree. Each k-fringe-tree v in the k-branch-subtree, where v. Note that the height of any k-fringe-tree is at most k. Figure k-fringe-trees of the rooted tree A entclass1pt{minimak-branch-leaf numberk-branches in T. For the trees The mentclass2pt{minimk-branch heightT: The maximum number of k-branches along a path from the root to a leaf of T; i.e., k-branch-tree The mentclass2pt{minimk-branch height for chemical cyclic graphs . The core of a chemical cyclic graph G is defined to be the induced subgraph G that consists of vertices in a cycle or the vertices in a path joining two cycles. A vertex in the core (not in the core) is called a core vertex . The edges not in the core of a chemical cyclic graph G form a collection of trees T, which we call a non-core tree. Each non-core tree contains exactly one core vertex and is regarded as a tree rooted at the core vertex. The k-branch height of a chemical cyclic graph G is defined to be the maximum of k-branch heights over all non-core trees. We observe that most chemical graphs G with at most 50 non-hydrogen atoms satisfy Even though this paper deals exclusively with acyclic graphs, we formally introduce the For convenient reference, we summarize the graph-related notation used throughout this paper in Table We represent the graph structure of a chemical compound as a graph with labels on vertices and multiplicity on edges in a hydrogen-suppressed model. Let adjacency-configuration of the atom pair. Choose a set We introduce a total order < over the elements in We use a hydrogen-suppressed model because hydrogen atoms can be added at the final stage.v and a bond-multiplicity m is assigned to an edge e, respectively. For a notational convenience, we denote the sum of bond-multiplicities of edges incident to a vertex A tuple chemical graph over (i)H is connected;(ii)(iii)A chemical graph H is an acyclic graph. Similarly for other types of graphs for H.Let bond-configuration of an edge G to be a tuple u and v of e. Let We define the In summary, we give the notation on modeling chemical compounds used throughout this paper in Table feature vectorf(G) that consists of the following 11 kinds of descriptors. We choose an integer In our method, we use only graph-theoretical descriptors for defining a feature vector, which facilitates our design of an algorithm for constructing graphs. Given a chemical acyclic graph General chemical graph descriptorsn(G): the number |V| of vertices.H divided by G.G.Descriptors for vertices of certain degreei in H, where the bond-multiplicity of edges incident to a vertex v is ignored in the degree of v.Descriptors for branch-leaf number and branch-heightG.G.Descriptors for vertex labelsDescriptors for the number of bondsm.Descriptors for adjacency-configurationsG.Descriptors for bond-configurationsG.Note thatK of descriptors in our feature vector K descriptors are not independent in the sense that some descriptors depend on the combination of other descriptors. For example, descriptor a(G) the observed value of the property G. As the first phase, we solve (I) Prediction Problem with the following three steps.We review the framework that solves the inverse QSAR/QSPR by using MILPs , 21, whiPhase 1.Stage 1: Let K. We call f(G) the feature vector of f(G) a descriptor of G.Stage 2: Introduce a feature function a(G) for many chemical graphs in Stage 3: Construct a prediction function j-th descriptor j-th descriptor, In this paper, we use the range-based method to define an applicability domain (AD)  to our iIn the second phase, we try to find a vector agamochi , Chiewvaagamochi  showed tLetbe an ANN with a piecewise-linear activation function for an input vectordenote the number of nodes in the architecture anddenote the total number of break-points over all activation functions. Then there is an MILPthat consists of variable vectors, and an auxiliary variable vectorfor some integerand a setofconstraints on these variables such that:if and only if there is a vectorfeasible to, 21) Let Let21]) See Appendix “mentclass2pt{minimadmissible if there is a chemical graph q. For the class q so thatA feasible solution A vector Afterwards, for the classes of chemical graphs with cycle index 1 and 2, Ito et al. [A vector t}f(G)=x . Let \dom et al.  introduco et al.  and Zhu o et al.  presenteInverse Problem for the inverse QSAR/QSPR by treating the following inference problems.As the second phase, we solve (II) (II-a) Inference of VectorsInput: A real Output: Vectors (II-b) Inference of GraphsInput: A vector Output: All graphs The second phase consists of the next two steps.Phase 2.The second requirement may be replaced with inequalities Stage 4:  Formulate Problem (II-a) as the above MILP Stage 5:  To solve Problem (II-b), enumerate all (or a specified number) of graphs In practical applications, there would be many criteria that a target chemical compound needs to satisfy rather than a single chemical property G such thatG is at most 4;The maximum degree in k-branch height k; andThe k-fringe-tree in G is bounded.The size of each The reason why we restrict ourselves to the graphs in k-branch height and the size of 2-fringe-trees. According to this, over 55%  of acyclic chemical compounds with up to 100 non-hydrogen atoms in PubChem have the maximum degree 3 ; and nearly 87%  of acyclic chemical compounds with up to 50 non-hydrogen atoms in PubChem have the 2-branch height at most 1 . This implies that k-branch height and the k-branch-leaf number for a branch-parameter k.In this paper, we choose a branch-parameter H such thatH is at most 3 when The maximum degree of a vertex in n(H) of vertices in H is The number H is The diameter The The  holds.To design Stage 4 for our class More formally, given specified integers G so that the number G with Design of Stage 5, i.e., generating chemical graphs x 0 to 2 –24. All To conclude the description of the target graph class to be inferred by the inverse QSAR/QSPR framework developed in this paper, we summarize the global parameters in Table G in the class In this section, we describe an idea of formulating an MILP H is as follows. See a rooted tree T including the root T as the H.From the tree T and replace each of the edges H.Next, we choose some edges in the tree k as the H. The resulting tree is a required rooted tree H.Finally, we append to the tree Our new idea of constructing an acyclic graph T of In our MILP, we prepare a binary variable for each of the vertices and edges in e with a path F of directed edges between the vertices in F in our MILP. When an edge To represent a replacement of an edge F and the set of rooted trees at the vertices in scheme graphWe call the graph that consists of all the above graphs H with H in Fig. H in Fig. Figure In this paper, we obtain the following result.Letbe a set of chemical elements,be a set of adjacency-configurations, where, andGiven non-negative integers,,,,and, there is an MILPthat consists of variable vectorsandfor an integerand a setof constraints onxandgwith sizesuch that:is feasible toif and only ifforms a chemical acyclic graphsuch thatandNote that our MILP requires only See Appendices n(G) is larger than around 20.Previous methods of inferring chemical graphs –19 use aTreat acyclic graphs with a certain limited structure that frequently appears among chemical compounds registered in the chemical database; andG from the process of computing the vectors.Instead of manipulating acyclic graphs directly, first compute the frequency vectors In (a), we choose a branch-parameter G that have a small 2-branch number such as G with This section introduces a new dynamic programming method for designing an algorithm in Stage 5. We consider the following aspects: mentclass2pt{minimT to be a vector that consists of the frequency of each chemical element T. We are given a vector G to be inferred.We design a method in (b) based on the mechanism of dynamic programming in the following way. Define a frequency vector We first construct a set m, as illustrated in Fig. m to obtain the frequency vector m.We next combine two chemical rooted trees G with p trees in m is equal to the given vector G with m.Our algorithm for generating a chemical acyclic graph G with With a slight modification, the algorithm can generate a chemical acyclic graph G with Appendix Kow), boiling point (Bp) and heat of combustion (Hc). We executed the experiments on a PC with Two Intel Xeon CPUs E5-1660 v3 @3.00GHz, 32 GB of RAM running under OS: Ubuntu 14.04.6 LTS. We show 2D drawings of some of the inferred chemical graphs, where ChemDoodle version 10.2.0 was used for constructing the drawings.We implemented our method of Stages 1 to 5 for inferring chemical acyclic graphs and conducted experiments to evaluate the computational efficiency for three chemical properties Results on Phase 1.  We implemented Stages 1, 2, and 3, in Phase 1 as follows.Kow, Bp, HcStage 1. We set a graph class Kow, Bp and Hc;n(G) of non-hydrogen atoms over the compounds G in G in G in a(G) for G in Table f that consists of the descriptors defined in “Stage 2. We used a feature function K: the number of descriptors for the chemical compounds in data set Activation: the choice of activation function;a, b, 1) consists of an input layer with a nodes, a hidden layer with b nodes and an output layer with a single node, where a is equal to the number K of descriptors;Architecture: (L-time: the average time (in seconds) to construct ANNs for each trial;test test From Table Stage 3. We used scikit-learn version 0.21.6 with Python 3.7.4 to construct ANNs $$i\in 1,$$\end{do$$i\in 1,$$\end{doFor each chemical property Results on Phase 2.  We implemented Stages 4 and 5 in Phase 2 as follows.Set the 2-leaf-branch number Fix the instance size Set the diameter Set the maximum degree Kow, Bp, HcFor each instance size Based on the above setting, we generated six instances for each instance size Stage 4. In this step, we solve the MILP IP-time: the time (sec.) to an MILP instance to find vectors We observe that most of the MILP instances with Tables g et al.  could soa–c illustrate some chemical acyclic graphs G with Figure 7a–c illustrate some chemical acyclic graphs G with Figure 8Stage 5. In this stage, we execute our new graph search algorithms for generating target graphs h in Step 2 and an execution of Steps 1 and 3 for h in Steps 2 and 3 and We introduce a time limit of 10 minutes for each iteration When no feasible vector pair is found by the graph search algorithms, we output the target graph G-LB: a lower bound on the number of all target graphs G such that G has been found from the vector G-time: the running time (sec.) to execute Stage 5 for a given feature vector Previously, an instance of chemical acyclic graphs with size Tables m et al. . For them et al.  and Zhu m et al.  was arouAn additional experiment We also conducted some additional experiment to demonstrate that our MILP-based method is flexible to control conditions on inference of chemical graphs. In Stage 3, we constructed an ANN Kow, Bp, HcKow, Bp and Hc, it is possible to infer a chemical acyclic graph G that satisfies a target value G illustrated in Fig. In this paper, we introduced a new measure, branch-height of a tree, and showed that many chemical compounds in the chemical database have a simple structure where the number of 2-branches is small. Based on this, we proposed a new method of applying the framework for inverse QSAR/QSPR –19 to thWe implemented our new method and conducted some experiments on chemical properties such as octanol/water partition coefficient, boiling point and heat of combustion.The resulting method improved the performance so that chemical graphs with around 50 non-hydrogen atoms and around diameter 30 can be inferred. Since there are many acyclic chemical compounds having large diameters, this is a significant improvement.It is left as a future work to design MILPs and graph search algorithms based on the new idea of the paper for classes of graphs with a higher rank. Recently, a method for inferring a chemical cyclic graph with any rank has been designed by Akutsu and Nagamochi  based on


J. Appl. Cryst. (2021), 54, 317–332]Peterson, Olds, McDonnell & Page Peterson, Olds, McDonnell & Page [ are corrected and clarified, respectively.Errors and ambiguities in the article by  Illustrated formalisms for total scattering data: a guide for new practitioners  and ρ(r)] that are defined identically by Keen  ; the powers of ten, although implicitly present, could be ignored in practice. They are now more commonly tabulated in fm , 1(f), 2(e), 2(f), 3 [GK(r) and T(r)] and 4 [GK(r) and T(r)] of Peterson et al. , GPDF(r) or ρ(r)] are unaffected as the scattering length units cancel out.Neutron scattering lengths were typically given in units of 10 al. 2021. This ha al. 2021; the funQ limit of total scattering structure factors is related in a complex way to thermodynamic functions and fluctuations  most ofKeen 2001 and by MKeen 2001 and is f al. 2021]. The fi al. 2021. It is uKeen 2001 and Pete al. 2021. Since tKeen 2001], whilstKeen 2001. These l al. 2021, which a al. 2021 with verS(Q), FPDF(Q) and GPDF(r) etc., which are divided through by 〈bcoh〉2], are useful when comparing with models and calculations, the functions FK(Q), GK(r) etc., which are not scaled by 〈bcoh〉−2, permit a much more direct and unambiguous assessment of absolute data normalization when correcting experimental data. FPDF(Q) and GPDF(r) are more challenging in this regard as their respective Q- and r-dependent asymptotes to the origin make determination of the low-Q and low-r trends less obvious ‘by eye’. S(Q) should also be used cautiously; although it is unitless this hides the fact that scattering factors are incorporated within the function, and even though many S(Q) have a low-Q limit that is close to zero this does not mean that zero is the limiting value by definition.The above discussion also highlights another important point. Although the scaled functions . Typically when correcting data, a ‘good’ low-Q limit is often much harder to achieve than a ‘good’ high-Q limit. This is especially the case for time-of-flight neutron diffractometers where data corrections are usually more challenging in the lower-Q regime. The low-Q limit of S(Q) should approximately equal c)]. The low-r levels of GK(r) = −〈bcoh〉2 and gPDF(r) = 0 . Here a back-transform correction has been applied to direct these PDFs to their theoretical values for r < 1 Å; encouragingly these values are maintained to much higher r, including in the gaps between the first few low-r peaks. For completeness, plots of FPDF(Q) and GPDF(r) are shown in Figs. 2e) and 2f), respectively.As a worked example of this, we show in Fig. 2et al.  will not be symmetric and centred in gPDF(r) . Symmetet al. (2021et al. (2021et al. (2021We have worked together on this corrigendum to try to ensure that these corrections to the Peterson  al. 2021 paper ar al. 2021 are not  al. 2021, thus un al. 2021, but it  al. 2021. We as a al. 2021 and Pete al. 2021 with thi


The Laurent–Stieltjes constants of Let  Let Dirichlet L-functionStieltjes constants or generalized Euler constants. In the particular case when L-function Riemann zeta functionL-functions. This gives rise to the widely used name “Stieltjes constants” for these coefficients.Let L-functions have also been studied by many authors. We introduce some of their results in the following section.The asymptotic behavior of : Briggs , Mitrovi: Briggs , Israilo: Briggs , Matsuokmentclasspt{minima: Briggs , Knessl : Briggs , Adell [entclass1pt{minima: Briggs ). The stL-functions. We consider functions in a class larger than the Selberg class. We first introduce the Selberg class In this paper, we are interested in investigating the Stieltjes constants of more general f be an arithmetic function, and denote by f. We say the Dirichlet series Let p:All functions L-functions K. The sum of the parameters degree of the L-function Notable examples of functions in refer to , Chap. 6generalized Laurent–Stieltjes constants or the Laurent–Stieltjes constants of the extended Selberg class. In this paper, we study these coefficients and give an upper bound of From now on we keep our focus on Our main theorem is stated as follows.Q be the positive real number appearing in condition n witha satisfies Let L-functions have many applications not only in analytic number theory, but also in algebraic number theory and even fields outside of number theory. They can be used to determine zero-free regions for Finally we remark that the Laurent–Stieltjes constants of zeta and sis, see .L-functions. She gave the following upper bound for q and for every The first explicit upper bound for y Briggs , which wy Briggs  and Isray Briggs . In 1985mentclasspt{minimad author  extendedhowed in  that thiK be a number field and Dedekind zeta functionK, K, h(K) the class number of K, R(K) the regulator of K, K and D(K) the discriminant of the extension K. The second author [L-functions, has been studied so far. For more details see for example [L-functions in the extended Selberg class.Finally, let tion. In  they ared author  studied  example , 20. ThiIn order to prove our main result, we first show a proposition and two necessary lemmas. Recall the notation used when we defined mentclass2pt{minimLet For the standard case when we assume entclass1pt{minimaNow without entclass1pt{miniman be an integer with Let D is the positively oriented rectangular path passing through the vertices A and T are sufficiently large numbers. Let us now divide D into the line segments By Cauchy’s formula, we can writementclass2pt{minimBy Lemma Next we consider the integral over Thus, for fined in . This comentclass2pt{minimFor mentclass2pt{minimPut λm. From , we havea be a real number satisfying Now we are ready to prove our main theorem. We again putBy Proposition


Nature Plants 10.1038/s41477-021-01014-9, published online 15 November 2021.Correction to: In the version of this article initially published, there were errors in the Fig. 1 color key. In the key, “Cortex” should have been by the blue bar, rather than yellow, and “Endodermis” by the yellow bar, rather than blue. The errors have been corrected in the html and PDF versions of the article.