Endometriosis-associated deep dyspareunia is associated with reduced sexual quality of life, lower self-esteem, and impaired sexual function.The primary objective is to assess the acceptability of a phallus length reducer (brand name: Ohnut [OhnutCo]), which is a buffer worn over the penis or a penetrating object to reduce endometriosis-associated deep dyspareunia, and the feasibility of a definitive randomized controlled trial (RCT). The secondary objective is to obtain estimates of the effectiveness of the buffer. An embedded substudy will explore the acceptability and the preliminary validity and reliability of a vaginal insert for the self-assessment of deep dyspareunia.Ours is an investigator-initiated, 2-arm RCT. We will recruit 40 patient participants with diagnosed endometriosis between the ages of 19 and 49 years, as well as their sexual partners. The participating couples will be randomized in a 1:1 ratio into the experimental arm or the waitlist control arm. The length of the study period will be 10 weeks, during which time all participants will record deep dyspareunia severity following each episode of sexual intercourse. In weeks 1 to 4, all patient participants will record deep dyspareunia severity at each sexual encounter. In weeks 5 to 10, participants in the experimental arm will use the buffer during vaginal penetration; participants in the waitlist control arm will continue engaging in vaginal penetration as usual. Participants will complete questionnaires for assessing measures of anxiety, depression, and sexual function at baseline, at 4 weeks, and at 10 weeks. In the substudy, patient participants will self-assess dyspareunia by using a vaginal insert on 2 occasions, at least 1 week apart. The primary outcomes—the acceptability and feasibility of the buffer—will be assessed with descriptive statistics, and the secondary outcome—phallus length reducer effectiveness—will be assessed by using an analysis of covariance–based approach. For the vaginal insert, we will assess acceptability, test-retest reliability, and convergent validity via correlation analyses comparing the use of the insert to clinical examination in terms of dyspareunia assessment outcomes.Our pilot will provide initial data on the acceptability and effectiveness of the buffer and the feasibility of the study methodology. The results from our study are expected to be submitted for publication by the spring of 2023. As of September 2021, we have consented 31 couples into the study.Our study will provide preliminary evidence for the self-assessment and management of endometriosis-associated deep dyspareunia. The findings will inform the decision to proceed to a definitive RCT.ClinicalTrials.gov NCT04370444; https://clinicaltrials.gov/ct2/show/NCT04370444DERR1-10.2196/39834 Endometriosis is the presence of endometrial-like glands and stromata that abnormally grow outside the uterus . This diEndometriosis-associated deep dyspareunia is multifactorial and can result from the endometriotic lesions themselves and other factors, such as myofascial pelvic pain and central nervous system sensitization ,2,8. A sThe assessment of deep dyspareunia for clinical and research purposes is challenging; pelvic examination in the clinical context may not reliably reflect sexual activities, particularly for patients who have experienced trauma , while qIn our pilot randomized controlled trial (RCT) protocol, the primary objective is to assess the acceptability of a PLR and the feasibility of a definitive RCT. The secondary objective is to obtain estimates of the effectiveness of the PLR, which will be compared to those of a waitlist control for reducing endometriosis-associated deep dyspareunia. An embedded substudy will explore the acceptability, reliability, and preliminary validity if a vaginal insert for the self-assessment of deep dyspareunia.Our study is a pilot study of a parallel-group, 1:1 RCT of the PLR, which will be compared to a waitlist control. An embedded substudy will investigate the use of a vaginal insert   for the This protocol was developed by using the SPIRIT  checklist , the CONEndometriosis and Pelvic Pain Interdisciplinary Cohort  [The study is being conducted at a tertiary care center for endometriosis within a large urban center in western Canada. Our research team manages a prospective patient registry called 2911090) . Baselin2911090) . New or Eligible patient participants will be new or rereferred patients in EPPIC who attended a clinic visit between January 1, 2018, and December 31, 2020; those who consented to being contacted for future research; and those who have a history of surgically confirmed endometriosis. Details of the inclusion and exclusion criteria are listed in Patient participantsInclusion criteria19 to 49 years of ageBirth-assigned femaleMonogamous sexual relationshipSexually active or not currently sexually active due to deep dyspareuniaSelf-reported deep dyspareunia score of ≥4 (out of 10)Sexual partner who consents to participating in the studyWilling to engage in penetrative sex at least once during the duration of the studyExclusion criteriaSelf-reported superficial dyspareunia score of ≥4 Severe anxiety or depression symptoms in the last 2 weeks Confirmed “yes” to question “In the last 2 weeks, have you had intense fear/anxiety in anticipation of, during or as a result of vaginal intercourse?”Current use of a phallus length reducerInability to complete English-language questionnairesPartnersInclusion criteria19 years of age or olderAny sex and genderExclusion criteriaCurrent use of a phallus length reducerInability to complete English-language questionnairesA sample size for a pilot study is often recommended to be at least 12 per group . Given pIndividuals will be selected from a list of potential participants and will be telephoned and emailed about the study by a study coordinator. Interested individuals will be asked a series of screening questions to confirm that they meet the eligibility criteria. An important inclusion criterion listed in The PLR (brand name: Ohnut) is made of a Food and Drug Administration–approved body- and skin-safe polymer blend. The blend is a thermo-set material that does not contain bisphenol A, phthalates, and latex and can be used with either silicone lubrication or water-based lubrication . The PLRThe vaginal insert is made of polyurethane and is stable from −40 to 70 °C. It will be self-inserted by the patient participants to self-assess the tenderness of pelvic structures implicated in deep dyspareunia that are typically assessed by a clinician during a pelvic examination. This is a novel, off-label use of the device.To support the use of the vaginal insert and PLR, the study team developed patient-centered study booklets, which include instructions for the use of each tool . The insThe primary outcomes are the acceptability of the PLR and the feasibility of study methods. Details of the specific outcomes are listed in Outcomes and explanationsResponse: the number of individuals who responded divided by the total number of individuals contactedRefusal: the number of individuals who refuse to participate divided by the number of individuals who respondedScreen fail: the number of individuals who did not meet the inclusion criteria divided by the number of individuals who respondedRecruitment rate: the number of participants enrolled divided by the length of the recruitment period (months)Retention: the number of participants who complete the study divided by the number of participants enrolledAdherence: the number of sexual events completed with the PLR divided by the total number of sexual events reportedPLR acceptability: a 34-item, built-for-purpose questionnaire , with aThe secondary outcome is the effectiveness of the PLR, which will be determined by comparing changes in outcomes reported from week 4 to week 10 by patient participants in the experimental arm to those reported by participants in the control arm. Details of these outcomes are listed in Outcomes and explanationsDeep dyspareunia: an 11-point numeric rating scale, wherein 0 corresponds to no pain and 10 corresponds to the worst pain, will be used to measure deep dyspareunia intensity (using an item previously published by our team ); mean dSexual function: total score on the Female Sexual Function Index, a 19-item self-report measure that captures information on sexual desire, arousal, lubrication, orgasm, satisfaction, and pain over the past 4 weeks Sexual distress: total score on the Female Sexual Distress Scale-Revised, a 13-item self-report measure that captures sex-related personal distress over the past 4 weeks Anxiety: total score on the Generalized Anxiety Disorder-7, a 7-item questionnaire validated for monitoring the severity of general anxiety Depression: total score on the Patient Health Questionnaire-9, a 9-item questionnaire validated for monitoring the severity of depression The substudy will explore the acceptability and the preliminary convergent validity and test-retest reliability of a vaginal insert for the self-assessment of deep dyspareunia. The vaginal insert acceptability substudy outcome is based on an 18-item, built-for-purpose questionnaire adapted from a previously published measure for assessing the acceptability of female condoms  on a resParticipants will use the vaginal insert to palpate various pelvic organs and structures  that have been previously associated with deep dyspareunia , and therandomization to randomly assign consented participants in a 1:1 ratio to 1 of 2 study arms; the experimental PLR arm will receive the PLR, and the waitlist control arm will receive the PLR once the study has concluded. Both arms will receive the vaginal insert for the substudy.The study coordinator will use a REDCap  extension package called Throughout the study, all participants will continue concurrent endometriosis treatments and treatments for other conditions as usual. At the start of the study, both arms will complete baseline demographic, psychological, and sexual function questionnaires. In weeks 1 to 4 of the study, patient participants in both arms will engage in sexual activity as usual and be asked to record a daily diary of all sexual events and the severity of deep dyspareunia (score: range 0-10). Psychological and sexual function questionnaires will be repeated at the end of week 4. In week 5, participants in the experimental arm will receive and begin to use the PLR. Participants in the waitlist control arm will not receive the PLR and will continue to record deep dyspareunia severity at each sexual encounter. At the end of week 10, all patient participants will repeat the psychological and sexual function questionnaires, and participants in the experimental arm will indicate the acceptability of the PLR. At the conclusion of the study period, participants in the waitlist control arm will receive the PLR.For the substudy, during weeks 1 to 4, all patient participants will self-assess deep dyspareunia by using the vaginal insert on weeks 2 and 3 of their menstrual cycle or on another two occasions (1 week apart) in the absence of regular bleeding. At the end of week 4, all patient participants will indicate the acceptability of the vaginal insert. The participant timeline is depicted in The list of potential participants is password protected and stored on a secure server at the study site, and it will only be available to select members of the research team. Upon enrollment to the study, each participant will be given a unique study ID number that does not correlate to any personally identifying information. All reports, data collection sheets, and administrative forms will be identified by this number to maintain confidentiality. This ID number can only be decrypted by a password-protected linking file, which is stored separately on the secure server at the study site; similarly, it will only be available to select members of the research team.Study materials, such as the instruction booklet, the daily diaries, and the PLR, will be mailed to the participants, depending on their study arm allocation. The study coordinator will follow up with each participant to collect the daily diaries of the control and intervention group participants upon the completion of the study. We will not blind the research team to the study arm allocations. Once the participants have completed the study, a research team member will extract the data from the daily diaries and upload the data onto a Microsoft Excel (Microsoft Corporation) sheet. The data will be stored on a secure server at the study site, and they will only be available to select members of the research team. Details of the data management procedures can be found in the study’s research ethics board application, which can be provided upon request. We do not anticipate any adverse events with the use of the PLR, and as such, we will not need a data monitoring committee. If any adverse events occur, the participants can record them in the daily diary, and they will have the contact information of the study team to debrief such cases.Descriptive statistics for demographic data and outcome variables will be presented by using means and SDs for continuous variables that are normally distributed, medians and IQRs for continuous variables that are not normally distributed, and frequency tables for the categorical variables of each study arm and the whole sample. Baseline measures will be compared to assess balance between the two arms, using Fisher exact tests for categorical variables and Kruskal-Wallis tests for continuous variables. With regard to our analytic approach for missing data.For the PLR and vaginal insert, an overall acceptability index will be calculated by summing the acceptability questionnaire item responses and dividing the sum by the total possible score. The index will be characterized by using means and SDs, and individual items will be reported by using medians and IQRs. For the PLR, results that are stratified by participant type (patient or partner) will be reported.t tests, linear regression modeling, and nonparametric approaches for the ANCOVA model.For the experimental and waitlist control arms, the difference in deep dyspareunia scores from week 4 to week 10 between the two arms will be assessed by using an analysis of covariance (ANCOVA)–based approach . We willTenderness at each pelvic anatomic site, as assessed via the vaginal insert test, as well as the highest score across all anatomic locations, will be correlated with the mean deep dyspareunia score, highest deep dyspareunia score, and lowest deep dyspareunia score. The self-assessed tenderness scores will be correlated with self-reported deep dyspareunia scores, using the Spearman rank correlation test. Participants without sexual events during weeks 1 to 4 will be compared to participants with ≥1 sexual event.t test or Mann-Whitney U test.The results of assessments of tenderness (“yes” or “no”) from the most recently performed clinical pelvic examinations, if applicable (these data are collected in EPPIC), will be correlated with self-assessed tenderness, as measured with the vaginal insert (score: range 0-10), by using a 2-tailed To evaluate test-retest reliability, self-assessed tenderness scores for each anatomic site from the first assessment will be correlated with self-assessed tenderness scores from the second assessment to determine if the two sets of vaginal insert data are reproducible over time. This will be done by using scores (0-10) to calculate the intraclass correlation coefficient, as well as by categorizing tenderness (“yes” or “no”) and using the McNemar test.With regard to our analytic approach for missing data, if the maximum amount of missingness in any variable is <5%, we will exclude cases with missing data via pairwise exclusion. Otherwise, if the maximum amount of missingness is >5%, multiple imputation via chained equations will be used to impute the missing values. Results from imputed data sets will be pooled to provide final results.The research presented in this protocol was approved by the Children’s and Women’s Research Ethics Board of the University of British Columbia . The study is also registered on ClinicalTrials.gov . Any subsequent modifications to the study protocol will be reviewed by the Research Ethics Board of the University of British Columbia for approval, submitted to ClinicalTrials.gov, and communicated to the study participants.The results from our study are expected to be submitted for publication by the spring of 2023. As of September 2021, we have recruited 31 couples into the study.Deep dyspareunia affects approximately 50% of people with endometriosis and has significant impacts on their sexual and mental health . One gapPotential feasibility issues pertain to the recruitment rate, retention, and adherence. Both the PLR and vaginal insert will be used in vaginal penetrative activities. The vaginal insert will be used on 2 occasions (in order to test reproducibility), and the PLR will be used during sexual encounters involving vaginal penetration over 6 weeks. Moreover, pain scores will be prospectively recorded for each sexual encounter during the 6 weeks. These intensive procedures and prospective data collection activities may reduce the recruitment rate, retention, and adherence. Acceptability issues may include impacts on the sexual function of the partners and how the placement of the PLR on the penis or a penetrating object affects the sexual response cycles of the participants. We also aim to estimate the potential effectiveness of the PLR in reducing deep dyspareunia by comparing it to a waitlist control.The strengths of the protocol include having a randomized design, taking prospective pain measurements at each sexual encounter, and supplementing the trial by embedding the vaginal insert as a potential novel self-assessment tool. One limitation of the protocol is that it is not placebo controlled; however, it would be difficult to design a placebo for the PLR. Another limitation is that we will be comparing the vaginal insert to clinical examinations that are done at entry into the registry, which may be conducted several months before enrollment into the trial. In a future study, a pelvic examination can also be performed at recruitment. Furthermore, a limitation of the pilot trial is that we will not control for variables like age and BMI, which are potential confounders that may affect the use of the vaginal insert or PLR.In summary, the PLR may be a noninvasive self-management tool for people with endometriosis-associated deep dyspareunia. The results of our pilot RCT will inform a future full-scale RCT of both the PLR and the vaginal insert test. The results from our study will be reported in peer-reviewed journals, in conference presentations, and to the study participants and the public through plain-language knowledge translation strategies. Ultimately, these innovations will contribute to the body of work on the rigorous investigation and evidence-based management of endometriosis-associated deep dyspareunia.


Betweenness centrality is one of the key measures of the node importance in a network. However, it is computationally intractable to calculate the exact betweenness centrality of nodes in large-scale networks. To solve this problem, we present an efficient CBCA  algorithm based on progressive sampling and shortest paths approximation. Our algorithm firstly approximates the shortest paths by generating the network centroids according to the adjacency information entropy of the nodes; then constructs an efficient error estimator using the Monte Carlo Empirical Rademacher averages to determine the sample size which can achieve a balance with accuracy; finally, we present a novel centroid updating strategy based on network density and clustering coefficient, which can effectively reduce the computation burden of updating shortest paths in dynamic networks. The experimental results show that our CBCA algorithm can efficiently output high-quality approximations of the betweenness centrality of a node in large-scale complex networks. Common centrality measures are degree centrality5, betweenness centrality6, closeness centrality7, etc. The betweenness centrality of a node has many applications in various domains, such as identifying critical nodes in transportation networks8, detecting essential proteins in protein networks9, and improving clustering10 and community detection algorithms11.Network analysis16, among which Brandes’ algorithm12 is a representative one that uses the single source shortest paths (SSSP) idea to optimize the computation process. The time complexity of this algorithm is O(nm) for unweighted graphs, where n and m are the numbers of vertices and edges in the graph, respectively.Betweenness centrality (BC) measures the importance of a vertex or an edge based on the shortest paths in a graph . Several exact algorithms for computing Betweenness centrality have been proposed17 due to their increasing size, and research emphasizes the ordering of nodes over exact values. Hence, some sampling-based approximation algorithms have emerged30, which can generate a betweenness centrality approximation with a high probability 31. However, this category of approximation algorithms faces some challenges, such as determining the sample size that can represent the global distribution of parameters with the least samples; selecting samples that can better estimate the parameter distribution; and adapting to changes in network dynamics. Currently, some researchers have proposed many attempts to address these problems. For example, Brandes29 proposed a bp algorithm based on Hoeffding’s inequality and union bound, but this algorithm relies too much on the number of nodes n, resulting in excessive running overhead; then Matteo et al22 proposed an rk algorithm based on Vapnik-Chervonenkis (VC) dimension theory, using VC dimension theory to limit the sample size so that the sample no longer depends on the number of nodes in the graph but on the diameter of the graph . Nonetheless, the VC dimension does not yield the best 20 proposed another ab algorithm based on the Rademacher averages32 and progressive sampling33 to estimate the betweenness centrality of the nodes. Rademacher averages with probability bounds36 and VC dimensions are also often used in pattern mining39. Bergamini et al40 proposed a new fully dynamic adaptive sampling update algorithm based on Matteo’s shortest paths sampling.However, exact algorithms are infeasible for large-scale networks49. These algorithms store the nodes or edges that change by using data structures to avoid re-computing the betweenness centrality from scratch. Hayashi41 proposed a fully dynamic algorithm and a new technique called Two-ball-index and Special-Purpose Reachability Index on the hypersketch sketch of Yoshida26, which improved the dynamic update by several orders of magnitude compared to Berigamini48. Berigamini extensively updated the number of pairwise distances and shortest paths and proposed a faster algorithm to update dependencies40. This algorithm decreased the number of operations performed by dynamic betweenness centrality algorithms, but they have the same worst-case running time as recalculations.At the same time, the network is dynamic, with the addition or deletion of nodes and edges at any time. It would be costly to recalculate the betweenness centrality for each dynamic change. Therefore, many dynamic updating algorithms have been proposedAlthough the above methods propose many improvement strategies for estimating the betweenness centrality, they do not improve the sampling method and consider network characteristics. To address the above problems, we present an efficient approximation algorithm CBCA  based on progressive sampling and shortest paths approximation. The algorithm firstly approximates the shortest paths by using the adjacency information entropy of nodes to generate the network centroid; then constructs Monte Carlo Empirical Rademacher averages based estimator to determine the sample size and achieve a balance between accuracy and efficiency; finally, we propose a new centroid updating strategy according to the network density and clustering coefficients, which effectively reduces the cost of updating the shortest paths in dynamic networks.We present a novel error estimator based on Monte Carlo Empirical Rademacher Averages (MCERA) and progressive sampling methods, which can balance the trade-off between sample size and accuracy. By using MCERA and adjusting the sample size according to the accuracy requirement, our estimator can effectively guarantee that the upper probability bound of the approximation error of the CBCA algorithm is within a predefined range. Experimental results demonstrate that the CBCA algorithm with our estimator is faster and requires smaller samples than existing methods under the same probability condition.We propose a strategy to update the centroid in dynamic networks with frequent BC changes. Our strategy uses a dynamic detection mechanism that monitors the network density and clustering coefficients, which are important features of the network graph. The mechanism updates the centroid when these features exceed a set threshold, without adding or deleting nodes and edges at any time. This reduces the time to reselect high-quality centroids in dynamic networks.50 to select high quality centroids. The experimental results show that our method can cover more than 80% of the shortest paths in networks with scale-free51 and small-world characteristics52.We propose an improved shortest paths approximation method based on network centroids for sampling. Our method can be applied to both undirected and directed graphs. To select the number of centroids, we use different calculation methods of diameter and reduce the computation of dynamic network shortest paths update by distinguishing between undirected and directed graphs. We use the adjacency information entropyThe main contributions and innovations in this paper are as follows:We will describe the definitions and basic ideas in this section.n is the number of all nodes. For any distinct node pair  in the graph, where u and node v, and let u and node v for all shortest paths passing through w. For convenience, we write the shortest paths as SPs.Let b(w) of a node Given a graph 36 are an essential core of statistical learning theory53 and allow measuring the convergence speed of sample means concerning their expectations. More detailed information on the Rademacher averages can be found in36.Rademacher averagesP and a uniform distribution P. Let P to , and let m independent identically distributed samples of P with probability uniform distribution f, which is defined over the samples S, are as followed, respectively:Define a finite field S, we focus on the upper bound of the supremum deviation S is the indicator that can effectively determine the sample-dependent upper bound of the supremum deviation m independent identically distributed  Rademacher random variables that takes the values 35 gives an efficient way to get the apparent sharply probability bound for ERA. For any k-Trials Monto-Carlo Empirical Rademacher Averages  S using k-Trials Monto-Carlo Empirical Rademacher Averages S, which controls the probability bound of k-MCERA to make a better balance between sample size and accuracy. That is because k-MCERA can directly estimate the highest deviation of the function set by data dependence. For each k-MCERA to compute the upper bound of the maximum deviation Now, for a given For anyis a family of functions in the domainPup to  andis a uniform distribution of probabilities over the elements ofP:We have given in Online Appendix A.1, which leverages the properties of the betweenness centrality under large-scale networks and the basic variance formula.Forand the functionwhereis a family of functions fromPto . Letbe amatrix of Rademacher random variables, such that independently and with equal probabilitySbe a sample size ofmdrawn i.i.d. fromP, and take a distributionFor eachdefine:With the probability at leastover the choice ofSandit holds:54 and the symmetry inequality55, as well as the substitution theorem.We put in Online Appendix A.2, using the self-boundary functionk-MCERA outperforms rk and bp. The rk algorithm uses VC dimensional theory to obtain an upper limit on the sample size, which is data-independent and depends on the properties of the graph itself. However, it does not consider the feature of data dependence. The bp algorithm uses Hoeffding’s inequality and union bound, which results in an excessive sample. These two methods lead to a large number of samples to guarantee a high quality approximation, so they are characterized by a sample size that is suboptimal, while k-MCERA can capture the relationship between sample size and accuracy very well.Observing Theorem In this section, we focus on the dynamic changes of complex networks.58, network density and clustering coefficient are closely related to the power law property and small world property of complex networks, which are the key assumptions of our algorithm. Therefore, we choose the network density and clustering coefficient, which are two parameters that can reflect the features of the network. And the network is considered to have been changed measurably when their changes exceed a certain threshold value. The centroid is a critical factor that affects the efficiency of the whole algorithm.We know that dynamic networks change by adding or deleting nodes and edges, but this requires updating and maintenance. However, not all nodes or edges addition and deletion have a significant impact on the network, and we can ignore some minor changes while pursuing approximate estimates. According toWe present a strategy to update the network centroid based on the network density and clustering coefficients to detect the dynamic changes, thus avoiding a large amount of updating time.59: it can be used to characterize the density of interconnected edges of nodes in a network, which is defined as the ratio of the number of variables present in the network with the upper limit of the number of edges that can be accommodated. It is commonly used to measure the intensity of social relationships and the evolutionary trend in online social networks. For n nodes and m edges, the network density is defined as:61: it quantifies how densely nodes form cliques in a graph. There is evidence62 that nodes tend to create tightly bound groups in most real-world networks, especially social networks. This is divided into a global clustering coefficient and a local clustering coefficient. We choose the local clustering coefficient as one of the parameters to reflect the features of the network, because it can capture the local structural changes of nodes and their neighbors in dynamic networks.Network densityL(i) the collection of edges connected to the vertex i. The local clustering coefficient of vertex For mentclass2pt{minimThis section shows our proposed CBCA algorithm, which is an efficient approximation algorithm based on progressive sampling and shortest paths approximation.Progressive sampling is a technique that gradually increases the sample size until a desired accuracy is achieved. It allows us to avoid over-sampling or under-sampling the network, and adapts to the dynamic changes of the network. Shortest paths approximation is a technique that uses the network centroids to estimate the shortest paths between nodes. It allows us to reduce the computational complexity and memory requirement of the algorithm.We firstly introduce the basic process and related results for approximating the shortest paths based on the network centroids, which can efficiently compute the betweenness centrality values after approximating the shortest paths in section “The common shortest paths algorithms do not exploit the properties of complex networks  and small-world property). We leverage these two properties to improve a shortest paths approximation method based on the network centroids. The small-world network has two important properties: high clustering coefficient and short average shortest paths length. The scale-free network characteristic means that few nodes have a high degree and most nodes have a low degree. Then we can naturally select some nodes with high degree or high adjacency information entropy as the centroids, and divide the nodes near or adjacent to the centroids into subgroups. From these subgroups, we use the shortest paths algorithm to calculate the distance between each node and the centroids. This can reduce the search space and time. And it can improve the approximation accuracy. Therefore, the number and quality of the centroids are crucial.t centroids in statistical learning, and we refer the reader to32 for more details on the VC dimension theory:t is the number of centroids with c taking the value D is the diameter of the graph, and We use the VC dimension theory to calculate the number of 22, but c is not the meaning of the coefficients among them in this paper, and they are irrelevant. The reason for employing this equation is that with the data independence of the VC dimension, where the number of centroids does not depend on the number of nodes, but only on the diameter in the graph . It reduces the number of t and speeds up the approximation time. At the same time, the value of c can affect that of t. We need to trade-off between the number of centroids and the ratio The equation is derived from MatteoD. One way to compute the exact value of D is by solving the shortest paths problem between all pairs of nodes. However, this exact calculation of the diameter D is not desirable, because it requires a time complexity of Furthermore, in this Eq. , we needu as the source to all other nodes. We can calculate the diameter D equal to the sum of the two shortest paths from u to two other different nodes w, v.Let 2.D is not necessarily equal to the longest path among the shortest paths between all pairs of nodes, because the edge weights may affect the distance sums. This makes the calculation of the approximate diameter more complicated. We can use the maximum weakly connected component, as an upper bound on the diameter D.For a directed graph with weights, we note that Approximate diameter methods:64. In this paper, we choose a more popular adjacency information entropy formula50 and give a reasonable explanation.We use the adjacency information entropy formula to effectively filter out high-quality centroids based on the small-world feature and the scale-free characteristic of complex network. Commonly used information entropy formulas areG, where For a given G: The node degree of the undirected unweighted graph can be obtained by l is the number of node i and j is the neighbor of node i. i and node j, otherwise it is 0.The undirected unweighted graph G: The node degree of the undirected unweighted graph can be obtained by l is the number of node i, j is the neighbor of node i, and i and node j.The undirected weighted graph The degrees in the directed graph are divided into in degrees and out degrees, which are divided into two cases, directed with weights and directed without weights, as follows.G:G:i.Directed unweighted graph When discussing the influence of graphs, we need to distinguish between directed and undirected graphs. In a directed graph, each node has an incoming degree and an outgoing degree, which indicates the number of edges pointing to and from that node, respectively. In an undirected graph, each node has a degree, which indicates the number of edges connected to that node. Thus, we introduce an influence factor G:G:Directed unweighted graph Adjacency i in the network by considering the probability of node i being selected by its neighbor j with the following probability:Probability t network centroids to achieve the properties of complex networks in this paper, which compound our assumptions:Information entropy formula, the important technical formula is used to carry out how to filter In this section, we describe in detail how to generate subgraphs and how centroids of prime nodes are generated. As an example of an undirected unweighted graph, the same method is used for the other types of graphs.Step 1We use the adjacency list to represent the graph structure, so one Breadth-First Search (BFS) traversal gives us the degrees and weights of all nodes. As shown in Fig. Step 2t. The formula is When performing the BFS traversal, we get the diameter by the approximate diameter calculation method in section "mentclass2pt{minimStep 3The nodes surrounding the centroids can be easily obtained by traversing the neighboring nodes one at a time, as in Fig. Eventually, we get the complete graph with two centroids by the above method.We explain in detail how to perform the shortest paths approximation. According to section “As shown in Fig. Let the relationship of node pairs be written as In this section, we present the CBCA algorithm, which is based on the contributions of section “P mapping to .Our CBCA algorithm takes as input a graph u, v) from P. We define:To improve the computational efficiency, we use:b(w) of each node w in the graph and output an approximate set b(w). Here Our CBCA algorithm is a method based on a progressive algorithm. A progressive algorithm is an algorithm that checks whether a certain stopping condition is satisfied after each iteration. If it is satisfied, the final result is output; if not, the iteration continues. The goal of the CBCA algorithm is to estimate an approximation to the betweenness centrality We now present the CBCA algorithm and next show how to obtain the set of k for the k-MCERA, a user-specified error b(v), and The input parameters of the CBCA algorithm: a graph, a failure probability With probability at leastfor the CBCA algorithm, the outputsuch thatThis is presented in Online Appendix Our CBCA algorithm computes an G) with the diameter method described in section “In line 1, the diameter of the graph is determined by using the procedure getDiameter(t centroids are obtained by following the formula (13) for calculating the centroids described in section “In line 2, the number of required t subgraphs divisions and completes the approximation using the procedure getZone(t).In line 3, the CBCA algorithm obtains the high-quality network centroids based on the adjacency information entropy of section “We now proceed to the second phase of the algorithm.i is the index iteration, while k-MCERA for each iteration  through the domain P.In each iteration of the for loop, our CBCA algorithm performs the following operations: First of all, in line 6, the desired pairs of sample nodes, obeying a uniform distribution, can be easily obtained by using the program getUniformSample. The value of In line 11, the procedure getUpdateWe start with the description of the first phase.rk; and the accuracy rate with those of bp and rk. This is because Silvan is the newest algorithm and consistently outperforms other algorithms in terms of sample size and running time, while the bp algorithm has the highest accuracy rate among the known algorithms. We also tested five artificially generated random networks and networks with power-law property and Small-world property for network feature comparison. Finally, we gave the theoretical analysis and experimental results of dynamic centroids updating. All the above algorithms are guaranteed to satisfy In this section, we successfully implemented the CBCA algorithm and gave experimental evaluation results. We performed the experimental analysis in five aspects: sample size, running time, accuracy, different network characteristics, and dynamic centroids updating. We compared the sample size and running time of the CBCA algorithm with those of the algorithms Silvan and Implementation and environment. This part of the experiment is built on a Linux system, using Ubuntu 18.04 as the main operating system of the experiment platform, with hardware specifications: Intel(R) Core (TM) i9-9900K CPU@3.6GHz processor with 16GB of physical memory. It also has an NVIDIA GeForce RTX 2080Ti graphics processor with 11GB video memory capacity. Software drivers: The 2080Ti graphics processor comes with a 470.103.01 driver with CUDA 11.4 acceleration platform. All code is compiled via GCC 8.Dataset and parameters. We used the publicly available real dataset SNAP from Stanford University. All features of the graphs are given in Table rk. We set the initial sample size k-Monte Carlo trials, referring to Bavarian [18] and MCRapper [44]. It was fixed to 25, which shows that clear bounds can be obtained even using a small number of Monte Carlo trials, but of course, we also experimented with We first show a comparison figure of the required sample sizes on different datasets in Fig. k-MCERA technique, CBCA adopts the shortest paths approximation, which can effectively reduce the time to compute the shortest paths and thus decrease the sample size. In some small graphs, such as ca-GrQc, Silvan’s sample size is smaller than CBCA’s, which is due to CBCA’s consideration of the graph diameter, while Silvan uses an empirical peeling technique to reduce the required sample size. Finally, the CBCA algorithm performs better on large graphs, which is consistent with our experimental objective.Figure rk algorithm. The difference is up to an order of magnitude lower; and at least several times smaller. This is closely related to the different methods they use to reduce the sample sizes. The rk algorithm relies on the VC dimension, which can guarantee a high-quality approximation, but it only considers the diameter of the graph and ignores other features. This leads to an overly conservative estimation of the sample size. However, our CBCA algorithm employs the state-of-the-art k-Monto Carlo trials technique to obtain the maximum correlation index between the nodes with sharp variance-aware probability tail bounds. This can effectively reduce the sample size and provide better guarantees.As shown in Fig. k-Monto Carlo trials techniques.In conclusion, the CBCA algorithm can better obtain the minimum number of samples required to satisfy the high-quality approximation, which illustrates the importance of our shortest paths approximation and we report the running time, sample size, and In Fig. We can see the ratio of time required to achieve approximation in Fig. rk in large graphs and one order of magnitude faster even in small graphs. This is attributed to the improvement of CBCA based on the shortest paths approximation, reducing the running time, as well as the sharply wimpy variance technique for satisfying approximations of high quality, reducing the number of samples.As shown in Fig. we show a comparison of the absolute errors on the four datasets in Fig. k-Monto-Carlo trial, which can capture the probability tail bound with sharply wimpy variance, making a better balance between sample size and accuracy.In this section, we discuss the accuracy of the algorithm that we introduced in section “bp and rk algorithms are smaller. This is because the VC dimensional theory used by rk only considers the diameter of the graph without a more detailed understanding of the other structural underlying distributions of the graph. This results in too many samples, making it better in terms of accuracy guarantees. Moreover, the error of bp is minimal because the algorithm uses Hoeffding’s inequality and union bound. Although both algorithms can satisfy Compared with CBCA, the errors of both 65, WS small-world networks, and BA scale-free networks with 3000, 6000, 9000, 12000, and 15000 nodes .We generate ER random networksOur Shortest Paths Approximation theory is based on two properties of complex networks: the small-world property and the power-law property. To verify this theory, we apply the shortest paths approximation to a random network. The clustering coefficient of random networks is small, while the clustering coefficient of small-world networks is large. Random networks do not have power-law distributions, while scale-free networks have a few high degree nodes and a large number of low degree nodes. These differences affect the effectiveness of the shortest paths approximation.Figure Figure This demonstrates the superiority of the CBCA algorithm’s shortest paths approximation in networks with small-world property and power-law property.According to section “We conduct comparative experiments on different methods or parameters, and based on the evaluation criteria of accuracy and efficiency, we choose:The initial state network is characterized by Threshold qualifier:It can be seen from Table k-Monto Carlo trials technique to trade off the sample size and error to obtain a high-quality approximation of the betweenness centrality of all nodes. The algorithm can also handle dynamic networks with frequent BC changes by using a centroid updating strategy based on network density and clustering coefficients.In this paper, we present a novel betweenness centrality approximation algorithm based on progressive sampling and shortest paths approximation. The algorithm firstly uses the adjacency information entropy to generate network centroids and constructs an efficient shortest paths approximation strategy; then, it uses the Our experimental results show that our algorithm outperforms the baseline algorithm for the same probability in various networks. Our algorithm can efficiently output high-quality approximations of the node betweenness centrality in large-scale complex networks. Our algorithm can also be applied to network analysis and applications, such as identifying the most influential or central nodes in a network, which can help us understand the network structure and function, as well as optimize its performance or resilience. However, our algorithm also has some limitations and challenges, such as relying on the quality of the network centroids, which may not always be optimal or representative. Therefore, we intend to extend our algorithm to address these issues in future work, such as exploring different methods or criteria to select or update network centroids, applying our algorithm to different types of networks, and applying our algorithm to other network analysis tasks.Moreover, the network centroids and shortest paths approximation methods proposed in this paper can also be tried for other centrality measures, such as degree centrality, closeness centrality, etc. These centrality measures can also reflect the importance or influence of network nodes in different aspects, and have many practical applications. We believe that our methods can provide an effective and flexible approximation strategy for other centrality measures, and can adapt to different scales and characteristics of networks. We also believe that the methods proposed in this paper can be used to improve the speed and quality of data analysis and mining, such as finding similar or different points or groups in data faster, evaluating important or abnormal points or edges in data more accurately, etc. These are very important and valuable problems in the field of data science, and have many practical applications.Supplementary Information.


Biological and biomimetic membranes are based on lipid bilayers, which consist of two monolayers or leaflets. To avoid bilayer edges, which form when the hydrophobic core of such a bilayer is exposed to the surrounding aqueous solution, a single bilayer closes up into a unilamellar vesicle, thereby separating an interior from an exterior aqueous compartment. Synthetic nanovesicles with a size below 100 nanometers, traditionally called small unilamellar vesicles, have emerged as potent platforms for the delivery of drugs and vaccines. Cellular nanovesicles of a similar size are released from almost every type of living cell. The nanovesicle morphology has been studied by electron microscopy methods but these methods are limited to a single snapshot of each vesicle. Here, we review recent results of molecular dynamics simulations, by which one can monitor and elucidate the spatio-temporal remodeling of individual bilayers and nanovesicles. We emphasize the new concept of leaflet tensions, which control the bilayers’ stability and instability, the transition rates of lipid flip-flops between the two leaflets, the shape transformations of nanovesicles, the engulfment and endocytosis of condensate droplets and rigid nanoparticles, as well as nanovesicle adhesion and fusion. To actually compute the leaflet tensions, one has to determine the bilayer’s midsurface, which represents the average position of the interface between the two leaflets. Two particularly useful methods to determine this midsurface are based on the density profile of the hydrophobic lipid chains and on the molecular volumes. All biological membranes contain a single lipid bilayer as their basic building block. The importance of lipids was already realized by Langmuir and others at the beginning of the 20th century, using spreading experiments of lipid monolayers. Such a technique was also used by Gorter and Grendel who extracted lipids from red blood cells ,2. They This proposal was confirmed, in the 1950s and 1960s, by imaging cross-sections of biomembranes via electron microscopy, which provided direct evidence that cell membranes are based on single bilayers and showed that these bilayers have a thickness of 4 to 5 nm . ElectroIn order to avoid bilayer edges, at which the hydrophobic core of the bilayer would be exposed to the surrounding aqueous solution, a single bilayer forms a closed unilamellar vesicle. Both synthetic and cellular nanovesicles have been experimentally studied for a long time. Synthetic nanovesicles are assembled from lipid molecules, using a variety of preparation methods ,6, whichindividual nanovesicles.Electron microscopy methods also represent the main experimental approach to study the morphology of nanovesicles ,21,22,23Here, we review recent insights obtained from such molecular dynamics simulations. In fact, the main purpose of this review is to advertize and promote the concept of leaflet tensions, a new concept that has been recently introduced and further developed by our group ,30,31,32We emphasize that one should distinguish the individual leaflet tensions from the bilayer tension, which is equal to the sum of the two leaflet tensions. To avoid membrane rupture, the osmotic conditions have to be adjusted in such a way that both cellular and synthetic membranes experience a relatively low bilayer tension. However, even for vanishing bilayer tension, the individual leaflets can still experience significant leaflet tensions if one leaflet is stretched by a positive leaflet tension whereas the other leaflet is compressed by a negative leaflet tension. In such a situation, the lipid bilayer is subject to a transbilayer stress asymmetry, corresponding to the difference between the two leaflet tensions, which drives the remodeling processes in As far as the simulation methods are concerned, our first molecular dynamics simulations ,36 were Compared to all-atom molecular dynamics, the coarse-grained DPD method used here has several advantages. First, one can study the behavior of relatively large bilayer segments, which are not accessible to all-atom simulations. Second, the molecular models used in DPD are built up from a small number of different beads and, thus, involve only a small number of force parameters for the interaction forces between the beads, see Coarse-grained molecular models bridge the gap between all-atom models and theories of membrane elasticity. The latter theories been very useful to understand and explain the behavior of giant vesicles . If one This review is organized as follows. The next The fluid-elastic behavior of nanovesicles is discussed in The three Lipid bilayers involve many length scales. A single phospholipid molecule has a volume of about 1 nm3 . If 104 Phospholipids are major components of cellular membranes in all living organisms ,56. SyntIn water, lipids assemble into molecular bilayers as shown in symmetric when both leaflets contain the same number of lipids, i.e., asymmetric bilayers are obtained for To set up the simulation studies, planar bilayers are prepared by the assembly of In addition to synthetic liposomes, cellular nanovesicles such as exosomes and extracellular vesicles are released from almost every type of living cell  and are Cellular membranes are maintained in a fluid state which is characterized by fast lateral diffusion of the lipid molecules within one of the bilayer leaflets. The lateral diffusion was first probed by spin-labeled lipids ,63 and sg (FRAP)  and by sg (FRAP) ,68,69,70Each lipid molecule stays within one of the bilayer leaflets until it undergoes a transition or flip-flop to the other leaflet. During such an interleaflet flip-flop, the hydrophilic headgroup of the lipid has to be moved across the hydrophobic core of the bilayer which represents a significant energy barrier. For phospholipids, these barriers are quite large and lead to a wide range of flip-flop rates, with the typical time scale for one flip-flop varying from minutes to days ,77,78,79In their fluid state, lipid bilayers and biomembranes have unusual elastic properties. When we view the bilayers as thin sheets, these sheets can sustain two types of elastic deformations, in-plane stretching and compression as well as out-of-plane bending. Elastic stretching and compression is intimately related to membrane tension whereas bending deformations generate membrane curvature. Next, we will discuss the different notions of membrane tension and curvature without discussing the underlying computational methods, which will be addressed later.bilayer tension which acts to stretch or compress both leaflets of the bilayer simultaneously. When we stretch or compress a molecular bilayer, we will increase or decrease its area. A widely used experimental method to measure the bilayer tension is by micropipette aspiration of giant unilamellar vesicles (GUVs). Using this method, one can obtain the limiting bilayer tension, for which the vesicle membrane ruptures. Typical values for this tension of rupture are a few mN/m [The simplest notion of membrane tension is few mN/m ,84. Becafew mN/m .The lipid bilayer consists of two leaflets which are separated by a leaflet-leaflet interface. This interface is highlighted in both panels of The decomposition of the bilayer tension, On the other hand, if both leaflet tensions, In the following sections, we will consider the stress asymmetry To actually compute the leaflet tensions, we need to identify the spatial regions that are, on average, occupied by the two leaflets and to decompose the overall bilayer tension into two separate contributions from the two leaflets, see During the last eight years, two computational methods were found to be particularly useful for the computation of the midplanes and midsurfaces: the CHAIN protocol based on the density profile of the chain beads ,25,28,30During the assembly of lipids into planar bilayers and nanovesicles, the lipid numbers can be directly controlled as part of the simulation setup. In contrast, the bilayer and leaflet tensions are properties of the lipid assemblies that have to be determined from the analysis of the simulation data and require some computational effort. Thus, compared to the membrane tensions, lipid numbers seem to provide the more obvious control parameters. On the other hand, lipid numbers and membrane tensions are two physical quantities that are fundamentally different as can be understood from the following thought experiments.a and b, that are assembled from Let us consider two vesicles consisting of two lipid bilayers, extensive and intensive variables, respectively. When we combine two systems, extensive variables are added whereas intensive variables are equilibrated. This distinction can be extended to the individual leaflets and to the associated leaflet tensions when we consider the hemifusion of two vesicle bilayers. As a result, we conclude that the lipid numbers of the leaflets represent extensive variables whereas the leaflet tensions represent intensive variables.Using the terminology of thermodynamics , the lipWhen we intend to determine the curvature-elastic properties of a planar bilayer, this bilayer should be prepared in a state of sufficiently low bilayer tension as illustrated in Therefore, in order to characterize a lipid bilayer by its curvature, one has to consider small membrane patches and to average over the molecular conformations within these patches. The minimal lateral size of these patches can be determined from the spectrum of the bilayer’s shape fluctuations and was found to be about 1.5 times the membrane thickness, see In the literature on lipid bilayers, alternative notions of membrane curvature have been proposed. In order to compare these notions with the supramolecular view emphasized here, it is instructive to start from the simple (and popular) view that individual lipid molecules have a certain shape  which leIt is tempting to assume that a phospholipid as in holipids . This inThe assumption that each lipid molecule has a certain, fixed shape is problematic for several reasons. First, most lipids that form an inverted hexagonal phase form other non-bilayer and bilayer phases as well . In thesSecond, thermal noise will affect both the lipid molecule under consideration and its molecular neighborhood. One direct consequence of thermal noise is the lateral diffusion of an individual lipid within the lipid assembly. As a result, the diffusing lipid will encounter different local neighborhoods, in particular when this assembly contains several lipid components. Third, when we consider rigid lipid molecules with a non-cylindrical shape and try to pack them into the two leaflets of a bilayer, the lipids would become “frustated”. It was originally proposed  that sucFor a symmetric bilayer of lipids, the intrinsic curvatures of the lipids in the two leaflets must, on average, cancel out, irrespective of the assumed shape of the lipids. However, bilayer membranes are typically not symmetric because the two leaflets can differ in their molecular composition and can be exposed to different aqueous solutions. This local asymmetry between the two leaflets defines the spontaneous curvature of the bilayers as originally introduced by Helfrich , in closFor planar bilayers with a single lipid component, the reference state with vanishing leaflet tensions represents a symmetric bilayer, for which the preferred or spontaneous curvature must vanish. In contrast, the reference state of bilayers with several lipid components can exhibit a compositional asymmetry even if both leaflet tensions vanish. Therefore, the reference state of a multi-component bilayer can have a significant spontaneous curvature as demonstrated for planar bilayers with two phospholipid components , see SecNanovesicles with a single lipid component also lead to an asymmetric reference state that contains a different number of lipids, urvature .In addition to the number of lipids assembled in the two leaflets, the leaflet tensions are also determined by the overall geometry imposed on the lipid bilayers.a, has the optimal value v, has the optimal value For planar bilayers, the leaflet tensions depend on the lipid numbers 0=3.57d3 . The symBecause the two leaflets of the three planar bilayers in For nanovesicles, the leaflet tensions depend on the lipid numbers This figure displays two spherical nanovesicles, one of which transforms into a stomatocyte with an in-bud, the other into a dumbbell with an out-bud. The spherical nanovesicle in mpressed .The two spherical vesicles in The shape changes of vesicles as displayed in During fission, one membrane compartment is divided up into two daughter compartments. Three examples for a fission process are shown in Figures 22, 31 and 39 below. During fusion, two membrane compartments are combined into a single compartment as illustrated in Figures 33 and 44 further below. In the context of curvature elasticity, the energy change during such a topological transformation is proportional to the Gaussian curvature modulus  surface . The mod≃10−19 J . For the theorem . Thus, dctively, . For theWe now discuss the behavior of planar bilayers in more detail. In this section, we focus on bilayers with a single lipid component. In the next section, we will consider the behavior of bilayers with two and three lipid components.To set up the simulation, a planar bilayer is preassembled by placing lipid molecules onto two adjacent planes, thereby forming the initial states of the lower and upper leaflet of the bilayer. Once such an initial bilayer has been assembled with z, the two lateral coordinates parallel to it by x and y.The simulation box has the shape of a cuboid, see z but are independent of the lateral coordinates x and y. Examples for the density profiles of water (W), headgroup (H), and lipid chain (C) beads are displayed in After a planar bilayer has been assembled, it can be characterized by its density and stress profiles, as illustrated in z-coordinate, which is shifted in such a way that The profiles of the symmetric reference state with tensionless leaflets are depicted in panels a and d of For all bilayers in T=298∘ K . In the z-coordinate. We denote the linear dimension of the simulation box in the z-direction by z-coordinate. The bilayer tension The bilayer tension given by (2)Σ=∫−1z to a smaller interval because the stress profile decays rapidly to zero as we move away from the bilayer, see the examples in In practice, one can reduce the integration over To determine the leaflet tensions z-value, As depicted in ned from ,25,27κEquation , and theκEquationThe numerical coefficient 1/48 in Equation  was deriq for four different values of To corroborate the accuracy of the relationship in Equation , we alsoduced in . The respression ,3617)SSϕulLH=ϕuEquation  with theThe two leaflets of a biomembrane typically differ in their lipid composition. Each lipid molecule stays within one leaflet of the bilayer before it undergoes a transition or flip-flop to the other leaflet. The corresponding flip-flop times are very different for different lipid components and vary over several orders of magnitude. Here, we consider a lipid bilayer with two phospholipid components that do not undergo flip-flops on the accessible time scales and thus may exhibit a wide range of leaflet tensions and stress asymmetries. When we now add a third lipid component that undergoes frequent flip-flops, these flip-flops tend to reduce the stress asymmetry between the leaflets. Furthermore, bilayers with a compositional asymmetry can acquire a significant spontaneous curvature even if both leaflets are tensionless.To comprehend the relaxation of the leaflet tensions, let us consider a planar and tensionless bilayer with Equation . As a coThe relaxation process just described as a steered reshuffling of molecules between the two leaflets can also occur spontaneously if a molecular component is added to the bilayer that undergoes frequent flip-flops between the two leaflets. The latter process has been observed in molecular dynamics simulations for a lipid bilayer that contained the phospholipid POPC and the glycolipid (ganglioside) GM1, both of which did not undergo flip-flops, and, in addition, a model cholesterol, which moved frequently from one leaflet to the other on the time scale of the simulations, as schematically shown in The corresponding simulation data are displayed in z-coordinate in such a way that the midplane is located at m, which is proportional to the first moment of the stress profile, must vanish as well for the reference state of a one-component bilayer. Intuititively, one might expect that the spontaneous curvature vanishes for all bilayers with vanishing leaflet tensions. This expectation agrees with the view, originally proposed by Bancroft for surfactant monolayers in water-oil emulsions [The reference state of a one-component bilayer with vanishing leaflet tension implies that the bilayer is symmetric and that the stress profile satisfies mulsions ,107, thaThe leaflet tensions Equation . On the Equation , and thiThe planar bilayers discussed so far experienced moderate leaflet tensions To avoid bilayer poration and membrane rupture, each bilayer is kept in an OLT state with  zero bilayer tension by adjusting the total number of lipids assembled in the bilayer to the size of the simulation box. We consider such OLT states for one-component bilayers with a total number of 1682 lipids assembled in both leaflets. Such a bilayer is symmetric when each leaflet consists of 841 lipids but becomes asymmetric when the two leaflets contain different number of lipids. When the upper leaflet contains During the simulation time of t.To determine the kinetic rates of lipid flip-flops, we considered several ensembles of bilayers. Each ensemble consisted of more than 120 bilayers that were assembled from the same lipid numbers t, as required for any cumulative distribution function. The associated probability density function The measured cumulative distribution functions the formPcdf(t)=1In addition to flip-flops of the lipid molecules, the bilayers undergo structural instabilities outside of the stability regime in After the initial assembly, the bilayer remained in a metastable state without flip-flops for hundreds of nanoseconds and bulged towards the upper leaflet, see the simulation snapshot after 200 ns in Nanovesicles are closed, bubble-like surfaces with a diameter between 20 and 200 nm, formed by synthetic and cell-derived lipid bilayers. Electron microscopy (EM) studies have shown that these vesicles can attain both spherical and nonspherical shapes. One disadvantage of EM methods is that they provide only a single snapshot of each vesicle. In contrast, molecular dynamics simulations can reveal the spatio-temporal remodeling of each individual nanovesicle. We start with the assembly of spherical vesicles that enclose a certain volume of water and contain a certain total number of lipids. When we reduce their volume, the spherical vesicles transform into a multitude of nonspherical shapes such as oblates and stomatocytes as well as prolates and dumbbells. This surprising polymorphism can be controlled by redistributing a small fraction of lipids between the inner and outer leaflets of the bilayer membranes, which then experience different leaflet tensions.Spherical vesicles were assembled by placing lipid molecules onto two spherical shells corresponding to the two leaflets of the vesicle bilayers. The size of these vesicles was primarily determined by the vesicle volume, that is, by the number of water beads enclosed by the inner leaflet of the membrane. This number was chosen in such a way that the headgroup layers of the inner and outer leaflets had a diameter of about Experimentally, the volume of a vesicle can be changed by osmotic deflation and inflation. In the simulations, the vesicle volume was varied by changing the number r, with r to a small interval because the stress profile For a spherical nanovesicle, the density and stress profiles across the vesicle bilayer depend on the radial coordinate In order to compute the leaflet tensions Equation  accordinThe simplest procedure to determine the midsurface radius ν0≤0.978 . All OLTInterpolating the two sets of data in panel a and b of The Voronoi tesselation for the bead volumes as described in To compute the radial coordinate the form4π3rmid3=given by4π3rmid3=Panels a and b of When we deflate the spherical vesicle with On the other hand, deflating the spherical vesicle with As a third example, let us consider the deflation-induced shape transformation of a spherical vesicle with The deflation of a spherical nanovesicle with The two leaflet tensions  leaflet . Thus, iTo characterize the elastic response of the reference state, it will now be useful to distinguish two types of elastic deformations, corresponding to the black and green data in The OLT states are located on the perpendicular diagonal which is orthogonal to the main diagonal in The green data in The stress asymmetry fined by ,108(25)Equation .The stress asymmetry Equation  vanishesAll OLT states of the vesicle bilayer, corresponding to the black data in M, bounded by a tensionless bilayer with m satisfies the relation [m as given by Equation M, boundeEquation  for planThe vesicle bilayers discussed so far experienced moderate leaflet tensions To illustrate the stability and instability of vesicle bilayers, we consider bilayers that are assembled from a fixed total number of 2875 lipids, with Inspection of When we start from the reference state with tensionless leaflets and reshuffle some lipids between these leaflets, we obtain nonzero leaflet tensions We now focus on the left instability regime in The cumulative distribution functions in PWei(t)=1Equation . For k≠1rocesses ,111.The inset of k is found to be greater than one [In addition, the shape parameter than one  which co−Pcdf(t) ,113. TheIn addition to the flip-flops, the instability regimes in During the first 1300 ns, the bilayer of the vesicle undergoes shape fluctuations that appear to be ‘normal’ until the outer leaflet starts to develop a protrusion by expelling some lipids. This protrusion grows rapidly into a cylindrical micelle, which reaches its maximal extension after 1720 ns as shown in In this section, we consider nanovesicles exposed to an exterior solution of small solutes, which adsorb onto the outer leaflet of the vesicle bilayers. The behavior of these systems depends on several key parameters. First, it depends on the key parameters that determine the vesicle behavior in the absence of solute. For one-component bilayers as considered here, these parameters are the lipid numbers nditions . By defid, and the interaction between two W beads is taken to be the same as the interaction between two S beads [A relatively simple model system for the solution is provided by a binary mixture consisting of water and solute molecules. The mixture is modeled in terms of water (W) and solute (S) beads, both of which represent small molecular groups. For computational simplicity, the two types of beads are taken to have the same diameter,  S beads ,29,31. TThe coordinates for the phase diagram in given by (29)ζ≡ was initially reported by Christine Keating and coworkers . WettingRemodeling of cellular membranes by condensate-membrane interactions has been observed for P-bodies that adhere to the outer nuclear membrane , for lipThe adhesion of a small The partial engulfment of the The planar bilayer displayed in The lateral size The unusual tight-lipped shape of the membrane neck in The three surface segments that meet along the contact line experience three different surface tensions, the interfacial tension surfaces . In addigiven by (30)Σβγmrameters .The unusual tight-lipped shape of the membrane neck, as described in the previous section for the complete engulfment of condensate droplets by planar bilayers, has also been observed for the complete engulfment of rigid nanoparticles. The nanoparticle was constructed using particle beads arranged on an fcc lattice with the number density ρ=3/(d3) . The cloPartial engulfment of such a nanoparticle by a planar bilayer is shown in A variety of simulation methods has been previously used to study the engulfment of solid or rigid nanoparticles by lipid bilayers and nanovesicles. Engulfment of such particles has been investigated by Brownian dynamics , DPD simnegative line tension, and to a tight-lipped shape of the membrane neck, which prevents the fission of this neck. Thus, the non-axisymmetric pathway is similar to the engulfment of a condensate droplet by a symmetric planar bilayer, as shown in positive line tension, and to an axisymmetric membrane neck that undergoes fission, thereby dividing the nanovesicle up into two daughter vesicles. Furthermore, it turns out that the sign of the line tension is determined by the stress asymmetry between the two leaflet tensions: A sufficiently small stress asymmetry We now address the engulfment of a condensate droplet by the bilayer of a nanovesicle. After the droplet has come into contact with the vesicle, the engulfment process is induced by a reduction in the vesicle volume. This process can proceed via two different pathways that involve axisymmetric and non-axisymmetric membrane shapes . The nonEquation  leads toA and volume V [Complete engulfment of a droplet by volume reduction of the vesicle is only possible if the droplet is sufficiently small or the membrane area released by the volume reduction is sufficiently large. When we reduce the volume of a spherical vesicle by volume V ,108.If the droplet is too large or the released membrane area too small, the droplet can only partially be engulfed. An example for partial engulfment is displayed in quations  and 29)ΦS=0.004 Now, let us consider a sufficiently large volume reduction of the nanovesicle so that the condensate droplet can become completely engulfed. When the vesicle bilayer contains Complete engulfment of a condensate droplet by a vesicle bilayer can also proceed via non-axisymmetric shapes. The latter pathway is observed when the vesicle bilayer contains In order to elucidate the mechanism underlying the different pathways for complete engulfment as displayed in The line tension of the contact line between the condensate droplet and the vesicle bilayer can again be computed using the force balance along the contact line as described by Equation . The resInspection of The data displayed in Finally, in this last section, we emphasize that the leaflet tensions also play an important role for the adhesion and fusion of nanovesicles. As a simple example, we consider two identical vesicles, both of which are characterized by tensionless bilayers and by stretched outer leaflets when they come into contact. Depending on the magnitude of the positive outer leaflet tension When the initial stress asymmetry is slightly increased from In this review, we described recent insights into the fluid-elastic behavior of lipid bilayers and nanovesicles as obtained from molecular dynamics simulations. We emphasized that this behavior is primarily controlled by the two leaflet tensions of the bilayer membranes. The position of the undulating leaflet-leaflet interface can be visualized by using different colors for the lipids in the two leaflets as in A global view about the elastic behavior of planar bilayers is obtained from its two-dimensional leaflet tension space in The fluid-elastic behavior of nanovesicles was discussed in A global view about the elastic behavior of nanovesicles is obtained via the two-dimensional leaflet tension space in leaflets  and to sleaflets .Nanovesicles exposed to small solutes that adsorb onto the vesicle bilayers were considered in en necks . For ΦS=on layer . Removalon layer . The mecpositive as follows from a combination of negative as follows again from a combination of In Equation , which iEquation . On the Equation . In thisAll engulfment processes described in Because the flip-flop rate depends on the stress asymmetry between the two leaflet tensions, see In this study, we focused on one-component bilayers for the sake of computational simplicity but our results can be directly generalized to multi-component bilayers as well, provided the bilayer leaflets are fluid and have a laterally uniform composition. Indeed, for leaflets with a uniform molecular composition, the leaflet tensions describe the local stress within these leaflets, irrespective of the number of molecular components. Thus, for a multi-component bilayer, the different components will have different areas per lipid but the average molecular area in each leaflet will again determine the corresponding leaflet tension and vice versa.Multi-component bilayers can also be studied experimentally to measure the cumulative distribution function for the onset of flip-flops . To prepAs discussed in


It has been shown that while feature selection algorithms are able to distinguish between relevant and irrelevant features, they fail to differentiate between relevant and redundant and correlated features. To address this issue, we propose a highly effective approach, called Nested Ensemble Selection (NES), that is based on a combination of filter and wrapper methods. The proposed feature selection algorithm differs from the existing filter-wrapper hybrid methods in its simplicity and efficiency as well as precision. The new algorithm is able to separate the relevant variables from the irrelevant as well as the redundant and correlated features. Furthermore, we provide a robust heuristic for identifying the optimal number of selected features which remains one of the greatest challenges in feature selection. Numerical experiments on synthetic and real-life data demonstrate the effectiveness of the proposed method. The NES algorithm achieves perfect precision on the synthetic data and near optimal accuracy on the real-life data. The proposed method is compared against several popular algorithms including mRMR, Boruta, genetic, recursive feature elimination, Lasso, and Elastic Net. The results show that NES significantly outperforms the benchmarks algorithms especially on multi-class datasets. Thus, there exist many heuristics that attempt to find the optimal subset without going through the exhaustive search. While some feature selection algorithms have shown good results, there remains a significant room for improvement.The aim of feature selection is to reduce the number of features under consideration. It can lead to better interpretability of the model, lower computational load, lower chance of overfitting, and enhanced model accuracy. Ideally, all possible feature subsets should be considered before choosing the best one. However, given It has been shown that feature selection algorithms can often differentiate between the relevant and irrelevant features. On the other hand, the algorithms fail to distinguish the relevant variables from the redundant and correlated variables There exists a trade-off between the precision and speed of feature selection algorithms. While enlarging the search space of feature subsets increases the likelihood of finding the optimal subset, it requires longer computing times. To overcome this challenge, we leverage a key observation that it is possible to quickly separate the relevant features from the irrelevant ones using a filter approach. Filter approaches operate on small search spaces thereby providing a fast method for evaluating feature importances. However, filter methods are not effective at distinguishing between the relevant and redundant and correlated features. Therefore, in the second stage of our algorithm, we employ a wrapper method that searches through the space of all subsets of a fixed size. Since the wrapper method is applied on the filtered features the computing times are exponentially reduced compared to the original set of features. As a result, we obtain an algorithm that performs a detailed search but on a reduced subset of features.1.Achieves perfect precision in identifying the relevant features .2.Provides an effective mechanism for identifying the number of relevant features.3.Applies to both binary and multi-class datasets.One of the key issues in feature selection is determining the number of features to be selected. Although there exist several techniques that attempt to help choose the optimal number of features, it remains a largely open, and perhaps unanswerable, question The paper is structured as follows. In Section 2Feature selection is key component in data science and machine learning applications in multiple fields including gene expression A filter method based on neighborhood multi-granulation rough sets is proposed in Support vector machines (SVM) are one of the popular base models for wrapper methods Recently it has become increasingly popular to use nature inspired optimization techniques such as black widow optimization The proposed feature selection algorithm differs from the existing filter-wrapper hybrid methods in its simplicity and effectiveness. The existing hybrid methods can be divided into 3 main categories: genetic, filter-driven, and fuzzy-based approaches. Genetic feature selection algorithms employ a heuristic to simultaneously optimize both filter and wrapper fitness functions 31.Apply ensemble-based filter method Equation  to obtai2.Apply random forest-based backward sequential search method on the top 20 features until the stopping criterion is achieved to discard the remaining redundant and correlated features.The proposed algorithm is based on the key observation that it is often possible that a feature selection algorithm efficiently discards the irrelevant features using a filter method. Since the filter method is computationally fast, it allows to reduce the feature space in a short amount of time. Afterwards, it remains to discard the redundant and correlated features which are left undetected by the filter method. Thus, the proposed method consists of two main stages:3.1Tree-based ensemble classifiers play a crucial role in the proposed feature selection algorithm. In particular, random forest and extra tree classifiers are used to evaluate individual feature importances during the filtering stage. Random forest is an ensemble classifier that consists of a collection of trees. Each tree is fitted on a bootstrap sample of the data. The final model is constructed by aggregating the predictions of all the base trees 3.2mth region that are from the kth class. Then feature importance is calculated as the total decrease in node impurity that results from splits over that variable, averaged over all trees in the ensemble kth feature, Tree-based classifiers allow us to measure the reduction in impurity from splitting the data on a feature. We employ Gini index to calculate the impurity given by the following equation3.3The proposed feature selection algorithm, called Nested Ensemble Selection (NES), is illustrated in n, we calculate the out-of-bag accuracy of a random forest classifier for all possible subsets of size In the second stage, we perform a more detailed search among the top 20 features that are selected in the first stage. In particular, we perform a backward sequential feature selection. At each iteration, the least important feature is removed from the feature subset. The least important feature is determined based on the out-of-bag accuracy of random tree classifier. Concretely, for a given set of features of size The heuristic for identifying the optimal size of the feature subset is based on the observation that there must be a significant reduction in classification accuracy whenever a relevant feature is removed from the dataset. Conversely, the classification accuracy should change little when an irrelevant or a redundant feature is removed from the subset. The use of backward elimination process provides an added robustness to the proposed approach as it is less likely to miss a relevant feature. Thus, the stopping criterion for the recursive feature elimination is given by a significant drop in the out-of-bag accuracy of a random forest classifier. We employ a visual approach to identify the optimal size of the feature subset. To this end, we consider the graph of the out-of-bag accuracy over the subset size obtained during the sequential search. The point of the sharpest decline in accuracy is considered as the size of the optimal subset. As demonstrated in the numerical experiments, the proposed approach produces consistent results with respect to different types of data. In each of the considered datasets, the graph of the out-of-bag accuracy shows a clear decline which indicates the stoppage of the backward elimination.4In this section, we test the performance of the proposed feature selection algorithm on several synthetic datasets with known relevant features and a real-life dataset. The results show that the NES algorithm correctly identifies all the relevant features in synthetic data. It also selects the top features in the real-life data that produce the same accuracy as the full set of features.4.1We employ 4 synthetic datasets: ORAND, ANDOR, ADDER, and LED-16 that are described in Y is calculated via the following formula:Y is calculated via the following formula:The ORAND dataset contains three relevant features The LED-16 dataset is based on the 16-segment display configuration . which a4.2We benchmark NES against several popular feature selection algorithms: maximum relevance minimum redundancy algorithm (mRMR), Boruta, genetic feature selection, Lasso, Elastic Net, and recursive feature elimination (RFE). mRMR is a highly recognized algorithm that is used both in industry and academia The Boruta algorithm is a well-known wrapper method that compares the importance of the original features to the importance of shadow features Genetic algorithms represent stochastic optimization framework inspired by the processes in evolutionary biology and have gained a large attention in recent years Recursive Feature Selection (RFE) is a popular technique for feature selection that iteratively ranks and selects the most important features. It starts with the set of all features and uses a chosen estimator to evaluate feature importance. The estimator is often chosen to be Support Vector Machines and feature importance is measured by the coefficient value in the estimated model λ. The Lasso is a well known regularization technique that is used in various machine learning models. In classification, it can be used to select relevant features and reduce model complexity by adding a penalty term to the logistic regression cost function Elastic Net is another commonly used regularization technique for feature selection 4.3In the numerical experiments, the proposed NES algorithm is compared to the benchmark methods mRMR, Boruta, Genetic, RFE, Lasso, and Elastic Net. The benchmark methods are employed using mostly their default settings as per the original source. The details of the benchmark methods are provided in Relevant Selected is the number of relevant features selected and Total Selected is the total number of features in the dataset. Note that the datasets contain a redundant copy of each relevant feature, so it is acceptable for an algorithm to select a redundant feature instead of the corresponding relevant feature. However, in case both the relevant feature and its redundant copy are selected, only the relevant is counted. Recall is calculated as follows:Total Relevant is the total number for relevant features in the dataset.Each method is applied to the datasets in The precision and recall score can be used to calculate the balanced F1-score given by the following equation:In case of KDD, the data is first split into train and test subsets. The train set is used to select the relevant features. Then, random forest classifier is trained using the selected features in the train set. Finally, the trained classifier is evaluated on the test set using the selected features. The precision, recall, and F1-score metrics are calculated based on the results of the test set.4.44.4.1We begin our discussion with results using the ORAND dataset. In applying the NES algorithm, first, we calculate the individual feature importances using random forest and extra-trees classifiers based on Equation As shown in The comparison of the features selected by NES and those selected by the benchmark algorithms mRMR, Boruta, Genetic, RFE, Lasso, and Elastic Net is presented in 4.4.2The feature scores for the ANDOR dataset are presented in The accuracy of subsets in the backward sequential search is presented in The comparison of the features selected by NES and those selected by the benchmark algorithms is presented in 4.4.3The feature scores for the ADDER dataset are presented in The second stage of feature selection is illustrated in The comparison of the features selected by NES and those selected by the benchmark algorithms is presented in 4.4.4The LED-16 dataset is significantly different from the previous datasets. In particular, it is a multi-class dataset with 32 different values of the target variable. It has 16 original features. The relevant features have different levels of relevance in that they affect different number of target values. The feature scores for the LED-16 dataset are presented in The results of the backward sequential feature selection are presented in The features selected in the optimal subset are presented in The comparison of the features selected by NES and those selected by the benchmark algorithms is presented in 4.4.5normal and neptune classes. Class membership ranges from 2 (spy) to 87,832  instances. The dataset is split into train and test subsets (70/30). The train set is used to select the important features, while the test set is used to evaluate the accuracy of the trained classifier based on the selected features.The final dataset used in our study is the well-known KDD dataset which simulates network intrusions in a military environment The feature scores for the KDD dataset are presented in Comparison of the features selected by NES and those selected by the benchmark algorithms together with the corresponding accuracy on the test set is presented in 4.5To evaluate the proposed NES algorithm, we conducted numerical experiments on 5 datasets. The proposed algorithm was benchmarked against several popular existing algorithms: mRMR, Boruta, Genetic, RFE, Lasso, and Elastic Net. The precision and recall of the features selection algorithms are presented in As shown in In the LED-16 dataset, NES selects 10 of the 14 relevant features . On the other hand, the feature subset selected by the algorithm has the same test accuracy (0.9417) as the full features set (0.9417). So while NES does not capture all the relevant features it still achieves the same classification accuracy as the full feature set. Note that the algorithm did not select any extraneous variables (precision 1). As mentioned in the previous section, LED-16 dataset contains features of varying importance a so it iIn the last experiment, we employed the KDD dataset where the relevant features are unknown. Since the nature of the features is unknown, we rely on the test accuracy to evaluate the selected feature subset. As shown in Comparison of NES against the benchmark algorithms in The results of F1-score presented in It is important that a distinction is made between the existing and the proposed method. To be sure, there are significant differences between the proposed NES method and other current feature selection techniques. The distinction is two-fold. First, the proposed approach is algorithmically new in its simplicity and the use of double feature importance scoring which reduces the variance of the results. Unlike other ensemble methods that employ multi-stage processes to execute the algorithm, the NES algorithm proposes an uncomplicated, two-stage algorithm that is easy to deploy. Furthermore, the balanced scoring system based on the reduction in impurity using the random forest and extra-trees classifiers provides a stable scoring mechanism. Second, the proposed algorithm is different in its superior performance. The empirical tests  demonstr5In this paper, we proposed a novel ensemble-based feature selection algorithm called Nested Ensemble Selection (NES). The NES algorithm differs from the existing filter-wrapper methods in its simplicity and efficiency as well as near-perfect accuracy. Extensive empirical testing against several popular benchmark algorithms demonstrated the superiority of the proposed method, especially in multi-class datasets.The proposed method was tested on 5 different datasets. The results showed that NES is capable of achieving outstanding and desirable results both in identifying the relevant features as well as classification accuracy . For comAs a future research avenue, extending NES to regression problems can be explored. Since the main components of NES, namely random forest and extra-trees classifiers, do exist for regression tasks it would be logical to research this in the future. The filter-wrapper technique can be used in other application where the complexity of the problem can be reduced.Firuz Kamalov: Conceived and designed the experiments; Wrote the paper; Analyzed and interpreted the data. Hana Sulieman, Sherif Moussa, Murodbek Safaraliev: Contributed reagents, materials, analysis tools or data; Wrote the paper. Jorge Avante Reyes: Performed the experiments; Wrote the paper.The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.