Recursive Tree Research Articles

Exact calculations of network coherence in weighted ring-trees networks and recursive trees

In this paper, we study noisy consensus dynamics in two families of weighted ring-trees networks and recursive trees with a controlled initial state. Based on the topological structures, we obtain exact expressions for the first- and second-order network coherence as a function of the involved parameters and provide the scalings of network coherence regarding network size. We then show that the weights dominate the consensus behaviors and the scalings. Finally, we make a comparison of the network coherence between the ring-trees networks and the recursive trees with the same number of nodes and show that the consensus of ring-trees networks is better than the trees since the initial state in the ring-trees networks is a ring.

DOP60 Simplified rules to identify bio-naïve patients with Crohn’s Disease with higher likelihood of clinical remission when initiating vedolizumab versus anti-TNFα therapies: Analysis of EVOLVE study data

Abstract Background Previously, we identified subsets of biologic-naïve patients (pts) with Crohn’s disease (CD) from the EVOLVE study who had higher rates of clinical remission (CR) [Fig 1] when initiating vedolizumab (VDZ) vs anti-TNFα treatment, using prediction models based on multiple baseline characteristics.1,2 To aid use in practice, we investigated whether these subsets could be identified using simpler rules based on fewer baseline characteristics. Methods Using data from EVOLVE, we used recursive partitioning and regression tree (RPART) classification to predict membership in the previously identified higher CR subsets for VDZ. The RPART algorithm repeatedly splits data, based on baseline predictors (demographics, prior treatments, clinical characteristics at treatment initiation, Charlson comorbidity index, prior extraintestinal manifestations [EIMs], and prior healthcare resource use). At each split, the predictor and its value as chosen by the algorithm to split the data were those that maximized the number of pts classified correctly. Simplified rules were developed from the resulting RPART decision trees. Analyses of the treatment effect of VDZ vs anti-TNFα were conducted in the subsets of pts identified by each rule. Results Pts with data on CR and candidate predictors were included (VDZ [n=195]; anti-TNFα [n=245]). Three simplified rules (A, B, & C) were identified (Table 1). Pt characteristics included in the rules (exacerbation ongoing at treatment initiation, no emergency department/emergency room (ED/ER) visits prior to treatment initiation, no fistulae at most recent assessment prior to treatment initiation, pre-initiation disease behaviour) were among the main predictors of CR in VDZ pts identified previously. Pts identified by Rule A comprised 32% of the EVOLVE population, and were those who 1) had an exacerbation ongoing at index, 2) did not have ED/ER visits prior to initiation and 3) had pre-initiation disease behaviour classified as other than stricturing with/without perianal disease. Among these pts, median time to CR for VDZ and anti-TNFα pts were 6.7 and 18.1 months, respectively (unadjusted log-rank p<0.001), and the adjusted hazard ratio (HR) of CR for VDZ vs anti-TNFα was 2.9 (95% CI: 1.7, 5.0). Rules B & C identified larger subsets in which VDZ vs anti-TNFα treatment differences were smaller but still statistically significant. Conclusion Simple rules were developed to identify biologic-naïve, CD pts in whom VDZ initiation appeared to have a larger effect on CR relative to anti-TNFα initiation. Validation of these rules in other data sources is important to confirm these findings; if validated, these simplified rules can inform targeting of treatment and optimization of outcomes for pts with CD treated with VDZ.

Dependence structure estimation using Copula Recursive Trees

We construct the COpula Recursive Tree (CORT) estimator: a flexible, consistent, piecewise linear estimator of a copula, leveraging the patchwork copula formalization and various piecewise constant density estimators. While the patchwork structure imposes a grid, the CORT estimator is data-driven and constructs the (possibly irregular) grid recursively from the data, minimizing a chosen distance on the copula space. The addition of the copula constraints makes usual density estimators unusable, whereas the CORT estimator is only concerned with dependence and guarantees the uniformity of margins. Refinements such as localized dimension reduction and bagging are developed, analyzed, and tested through simulated data.

DNA Methylation Markers for Detection of Cholangiocarcinoma: Discovery, Validation, and Clinical Testing in Biliary Brushings and Plasma.

Cholangiocarcinoma (CCA) has poor prognosis due to late‐stage, symptomatic presentation. Altered DNA methylation markers may improve diagnosis of CCA. Reduced‐representation bisulfite sequencing was performed on DNA extracted from frozen CCA tissues and matched to adjacent benign biliary epithelia or liver parenchyma. Methylated DNA markers (MDMs) identified from sequenced differentially methylated regions were selected for biological validation on DNA from independent formalin‐fixed, paraffin‐embedded CCA tumors and adjacent hepatobiliary control tissues using methylation‐specific polymerase chain reaction. Selected MDMs were then blindly assayed on DNA extracted from independent archival biliary brushing specimens, including 12 perihilar cholangiocarcinoma, 4 distal cholangiocarcinoma cases, and 18 controls. Next, MDMs were blindly assayed on plasma DNA from patients with extrahepatic CCA (eCCA), including 54 perihilar CCA and 5 distal CCA cases and 95 healthy and 22 primary sclerosing cholangitis controls, balanced for age and sex. From more than 3,600 MDMs discovered in frozen tissues, 39 were tested in independent samples. In the clinical pilot of 16 MDMs on cytology brushings, methylated EMX1 (empty spiracles homeobox 1) had an area under the curve (AUC) of 0.98 (95% confidence interval [CI], 0.95‐1.0). In the clinical pilot on plasma, a cross‐validated recursive partitioning tree prediction model from nine MDMs was accurate for de novo eCCA (AUC, 0.88 [0.81‐0.95]) but not for primary sclerosing cholangitis–associated eCCA (AUC, 0.54 [0.35‐0.73]). Conclusion: Next‐generation DNA sequencing yielded highly discriminant methylation markers for CCA. Confirmation of these findings in independent tissues, cytology brushings, and plasma supports further development of DNA methylation to augment diagnosis of CCA.

Predicting in-Hospital Mortality of Patients with COVID-19 Using Machine Learning Techniques.

The present work aims to identify the predictors of COVID-19 in-hospital mortality testing a set of Machine Learning Techniques (MLTs), comparing their ability to predict the outcome of interest. The model with the best performance will be used to identify in-hospital mortality predictors and to build an in-hospital mortality prediction tool. The study involved patients with COVID-19, proved by PCR test, admitted to the “Ospedali Riuniti Padova Sud” COVID-19 referral center in the Veneto region, Italy. The algorithms considered were the Recursive Partition Tree (RPART), the Support Vector Machine (SVM), the Gradient Boosting Machine (GBM), and Random Forest. The resampled performances were reported for each MLT, considering the sensitivity, specificity, and the Receiving Operative Characteristic (ROC) curve measures. The study enrolled 341 patients. The median age was 74 years, and the male gender was the most prevalent. The Random Forest algorithm outperformed the other MLTs in predicting in-hospital mortality, with a ROC of 0.84 (95% C.I. 0.78–0.9). Age, together with vital signs (oxygen saturation and the quick SOFA) and lab parameters (creatinine, AST, lymphocytes, platelets, and hemoglobin), were found to be the strongest predictors of in-hospital mortality. The present work provides insights for the prediction of in-hospital mortality of COVID-19 patients using a machine-learning algorithm.

Tree limits and limits of random trees

AbstractWe explore the tree limits recently defined by Elek and Tardos. In particular, we find tree limits for many classes of random trees. We give general theorems for three classes of conditional Galton–Watson trees and simply generated trees, for split trees and generalized split trees (as defined here), and for trees defined by a continuous-time branching process. These general results include, for example, random labelled trees, ordered trees, random recursive trees, preferential attachment trees, and binary search trees.

Metric dimension of critical Galton–Watson trees and linear preferential attachment trees

The metric dimension of a graph G is the minimal size of a subset R of vertices of G that, upon reporting their graph distance from a distinguished (source) vertex v⋆, enable unique identification of the source vertex v⋆ among all possible vertices of G. In this paper we show a Law of Large Numbers (LLN) for the metric dimension of some classes of trees: critical Galton–Watson trees conditioned to have size n, and growing general linear preferential attachment trees. The former class includes uniform random trees, the latter class includes Yule-trees (also called random recursive trees), m-ary increasing trees, binary search trees, and positive linear preferential attachment trees. In all these cases, we are able to identify the limiting constant in the LLN explicitly. Our result relies on the insight that the metric dimension can be related to subtree properties, and hence we can make use of the powerful fringe-tree literature developed by Aldous and Janson et al.

MSO 0-1 law for recursive random trees

We prove the monadic second-order 0-1 law for two recursive tree models: uniform attachment tree and preferential attachment tree. We also show that the first order 0-1 law does not hold for non-tree uniform attachment models.

Profile of Random Exponential Recursive Trees

We introduce the random exponential recursive tree in which at each point of discrete time every node recruits a child (new leaf) with probability p, or fails to do so with probability 1 − p. We study the distribution of the size of these trees and the average level composition, often called the profile. We also study the size and profile of an exponential version of the plane-oriented recursive tree (PORT), wherein every insertion positions in the “gaps” between the edges recruits a child (new leaf) with probability p, or fails to do so with probability 1 − p. We use martingales in conjunction with distributional equations to establish strong laws for the size of both exponential flavors; in both cases, the limit laws are characterized by their moments. Via generating functions, we get an exact expression for the average expectation of the number of nodes at each level. Asymptotic analysis reveals that the most populous level is $\frac {p}{p+1} n$ in exponential recursive trees, and is $\frac {p}{2p+1} n$ in exponential PORTs.

The $k$-Cut Model in Deterministic and Random Trees

The \(k\)-cut number of rooted graphs was introduced by Cai et al. as a generalization of the classical cutting model by Meir and Moon. In this paper, we show that all moments of the \(k\)-cut number of conditioned Galton-Watson trees converge after proper rescaling, which implies convergence in distribution to the same limit law regardless of the offspring distribution of the trees. This extends the result of Janson. Using the same method, we also show that the \(k\)-cut number of various random or deterministic trees of logarithmic height converges in probability to a constant after rescaling, such as random split-trees, uniform random recursive trees, and scale-free random trees.

A note on the independence number, domination number and related parameters of random binary search trees and random recursive trees

We identify the mean growth of the independence number of random binary search trees and random recursive trees and show normal fluctuations around their means. Similarly we also show normal limit laws for the domination number and variations of it for these two cases of random tree models. Our results are an application of a recent general theorem of Holmgren and Janson on fringe trees in these two random tree models.

Geometry of weighted recursive and affine preferential attachment trees

We study two models of growing recursive trees. For both models, the tree initially contains a single vertex u1 and at each time n≥2 a new vertex un is added to the tree and its parent is chosen randomly according to some rule. In the weighted recursive tree, we choose the parent uk of un among {u1,u2,…,un−1} with probability proportional to wk, where (wn)n≥1 is some deterministic sequence that we fix beforehand. In the affine preferential attachment tree with fitnesses, the probability of choosing any uk is proportional to ak+deg+(uk), where deg+(uk) denotes its current number of children, and the sequence of fitnesses (an)n≥1 is deterministic and chosen as a parameter of the model. We show that for any sequence (an)n≥1, the corresponding preferential attachment tree has the same distribution as some weighted recursive tree with a random sequence of weights (with some explicit distribution). We then prove almost sure scaling limit convergences for some statistics associated with weighted recursive trees as time goes to infinity, such as degree sequence, height, profile and also the weak convergence of some measures carried on the tree. Thanks to the connection between the two models, these results also apply to affine preferential attachment trees.

HYPER HEURISTIC EVOLUTIONARY APPROACH FOR CONSTRUCTING DECISION TREE CLASSIFIERS

Decision tree models have earned a special status in predictive modeling since these are considered comprehensible for human analysis and insight. Classification and Regression Tree (CART) algorithm is one of the renowned decision tree induction algorithms to address the classification as well as regression problems. Finding optimal values for the hyper parameters of a decision tree construction algorithm is a challenging issue. While making an effective decision tree classifier with high accuracy and comprehensibility, we need to address the question of setting optimal values for its hyper parameters like the maximum size of the tree, the minimum number of instances required in a node for inducing a split, node splitting criterion and the amount of pruning. The hyper parameter setting influences the performance of the decision tree model. As researchers, we know that no single setting of hyper parameters works equally well for different datasets. A particular setting that gives an optimal decision tree for one dataset may produce a sub-optimal decision tree model for another dataset. In this paper, we present a hyper heuristic approach for tuning the hyper parameters of Recursive and Partition Trees (rpart), which is a typical implementation of CART in statistical and data analytics package R. We employ an evolutionary algorithm as hyper heuristic for tuning the hyper parameters of the decision tree classifier. The approach is named as Hyper heuristic Evolutionary Approach with Recursive and Partition Trees (HEARpart). The proposed approach is validated on 30 datasets. It is statistically proved that HEARpart performs significantly better than WEKA’s J48 algorithm in terms of error rate, F-measure, and tree size. Further, the suggested hyper heuristic algorithm constructs significantly comprehensible models as compared to WEKA’s J48, CART and other similar decision tree construction strategies. The results show that the accuracy achieved by the hyper heuristic approach is slightly less as compared to the other comparative approaches.

Site Frequency Spectrum of the Bolthausen-Sznitman Coalescent

We derive explicit formulas for the two first moments of he site frequency spectrum $(SFS_{n,b})_{1\leq b\leq n-1}$ of the Bolthausen-Sznitman coalescent along with some precise and efficient approximations, even for small sample sizes $n$. These results provide new $L_2$-asymptotics for some values of $b=o(n)$. We also study the length of internal branches carrying $b>n/2$ individuals. In this case we obtain the distribution function and a convergence in law. Our results rely on the random recursive tree construction of the Bolthausen-Sznitman coalescent.

Deadlock Avoidance Algorithms for Recursion-Tree Modeled Requests in Parallel Executions

We present an extension of the bankers algorithm to resolve deadlock for programs whose resource-request graph can be modeled as a recursion tree for parallel execution. Our algorithm implements the bankers logic, with the key difference being that some properties of the tree are fully exploited to improve the resource utilization and safety check in deadlock avoidance. For an n-node tree modeled program making requests to m types of resources, our recursion-tree based algorithm can obtain a time complexity of O(mn loglogn) on average in safety check while reducing the conservativeness in resource utilization. We reap these benefits by proposing a concept of the resource critical tree and leverage it to localize the maximum claim associated with each node in the tree. To tackle the case when the tree model is not statically known, we relax the definition of a local maximum claim by sacrificing some resource utilization. With this trade-off, the algorithm can resolve the deadlock and achieve more efficient safety checks within time of O(m loglogn). Our empirical studies on a two-dimensional integration problem on sparse grids show that the proposed algorithms can reduce resource utilization conservativeness and improve avoidance performance by minimizing the number of safety checks.

On Bollobás‐Riordan random pairing model of preferential attachment graph

AbstractBollobás‐Riordan random pairing model of a preferential attachment graph is studied. Let {Wj}j ≤ mn + 1 be the process of sums of independent exponentials with mean 1. We prove that the degrees of the first vertices are jointly, and uniformly, asymptotic to , and that with high probability (whp) the smallest of these degrees is , at least. Next we bound the probability that there exists a pair of large vertex sets without connecting edges, and apply the bound to several special cases. We propose to measure an influence of a vertex v by the size of a maximal recursive tree (max‐tree) rooted at v. We show that whp the set of the first vertices does not contain a max‐tree, and the largest max‐tree has size of order n. We prove that, for m > 1, . We show that the distribution of scaled size of a generic max‐tree in converges to a mixture of two beta distributions.

Predictive Value of Midsagittal Tissue Bridges on Functional Recovery After Spinal Cord Injury

Background The majority of patients with spinal cord injury (SCI) have anatomically incomplete lesions and present with preserved tissue bridges, yet their outcomes vary. Objective To assess the predictive value of the anatomical location (ventral/dorsal) and width of preserved midsagittal tissue bridges for American Spinal Injury Association (ASIA) Impairment Scale (AIS) grade conversion and SCI patient stratification into recovery-specific subgroups. Methods This retrospective longitudinal study includes 70 patients (56 men, age: 52.36 ± 18.58 years) with subacute (ie, 1 month) SCI (45 tetraplegics, 25 paraplegics), 1-month neuroimaging data, and 1-month and 12-month clinical data. One-month midsagittal T2-weighted scans were used to determine the location and width of tissue bridges. Their associations with functional outcomes were assessed using partial correlation and unbiased recursive partitioning conditional inference tree (URP-CTREE). Results Fifty-seven (81.4%) of 70 patients had tissue bridges (2.53 ± 2.04 mm) at 1-month post-SCI. Larger ventral (P = .001, r = 0.511) and dorsal (P < .001, r = 0.546) tissue bridges were associated with higher AIS conversion rates 12 months post-SCI (n = 39). URP-CTREE analysis identified 1-month ventral tissue bridges as predictors of 12-month total motor scores (0.4 mm cutoff, P = .008), recovery of upper extremity motor scores at 12 months (1.82 mm cutoff, P = .002), 12-month pin-prick scores (1.4 mm cutoff, P = .018), and dorsal tissue bridges at 1 month as predictors of 12-month Spinal Cord Independence Measure scores (0.5 mm cutoff, P = .003). Conclusions Midsagittal tissue bridges add predictive value to baseline clinical measures for post-SCI recovery. Based on tissue bridges’ width, patients can be classified into subgroups of clinical recovery profiles. Midsagittal tissue bridges provide means to optimize patient stratification in clinical trials.

Network Trees: A Method for Recursively Partitioning Covariance Structures.

In many areas of psychology, correlation-based network approaches (i.e., psychometric networks) have become a popular tool. In this paper, we propose an approach that recursively splits the sample based on covariates in order to detect significant differences in the structure of the covariance or correlation matrix. Psychometric networks or other correlation-based models (e.g., factor models) can be subsequently estimated from the resultant splits. We adapt model-based recursive partitioning and conditional inference tree approaches for finding covariate splits in a recursive manner. The empirical power of these approaches is studied in several simulation conditions. Examples are given using real-life data from personality and clinical research.

A non-increasing tree growth process for recursive trees and applications

AbstractWe introduce a non-increasing tree growth process $((T_n,{\sigma}_n),\, n\ge 1)$ , where Tn is a rooted labelled tree on n vertices and σn is a permutation of the vertex labels. The construction of (Tn, σn) from (Tn−1, σn−1) involves rewiring a random (possibly empty) subset of edges in Tn−1 towards the newly added vertex; as a consequence Tn−1 ⊄ Tn with positive probability. The key feature of the process is that the shape of Tn has the same law as that of a random recursive tree, while the degree distribution of any given vertex is not monotone in the process.We present two applications. First, while couplings between Kingman’s coalescent and random recursive trees were known for any fixed n, this new process provides a non-standard coupling of all finite Kingman’s coalescents. Second, we use the new process and the Chen–Stein method to extend the well-understood properties of degree distribution of random recursive trees to extremal-range cases. Namely, we obtain convergence rates on the number of vertices with degree at least $c\ln n$ , c ∈ (1, 2), in trees with n vertices. Further avenues of research are discussed.

Trees grown under young-age preferential attachment

AbstractWe introduce a class of non-uniform random recursive trees grown with an attachment preference for young age. Via the Chen–Stein method of Poisson approximation, we find that the outdegree of a node is characterized in the limit by ‘perturbed’ Poisson laws, and the perturbation diminishes as the node index increases. As the perturbation is attenuated, a pure Poisson limit ultimately emerges in later phases. Moreover, we derive asymptotics for the proportion of leaves and show that the limiting fraction is less than one half. Finally, we study the insertion depth in a random tree in this class. For the insertion depth, we find the exact probability distribution, involving Stirling numbers, and consequently we find the exact and asymptotic mean and variance. Under appropriate normalization, we derive a concentration law and a limiting normal distribution. Some of these results contrast with their counterparts in the uniform attachment model, and some are similar.