Random trees and random graphs

We show that a random labeled n n -vertex graph almost surely contains isomorphic copies of almost all labeled n n -vertex trees, in two senses. In the first sense, the probability of each edge occurring in the graph diminishes as n n increases, and the set of trees referred to as "almost all" depends on the random graph. In the other sense, the probability of an edge occurring is fixed, and the relevant set of trees is predetermined. The same method applies to show that almost all labeled n n -tournaments contain isomorphic copies of almost all labeled oriented n n -trees.

We present a new technique for proving logarithmic upper bounds for diameters of evolving random graph models, which is based on defining a coupling between random graphs and variants of random recursive trees. The advantage of the technique is three‐fold: it is quite simple and provides short proofs, it is applicable to a broad variety of models including those incorporating preferential attachment, and it provides bounds with small constants. We illustrate this by proving, for the first time, logarithmic upper bounds for the diameters of the following well known models: the forest fire model, the copying model, the PageRank‐based selection model, the Aiello‐Chung‐Lu models, the generalized linear preference model, directed scale‐free graphs, the Cooper‐Frieze model, and random unordered increasingk‐trees. Our results shed light on why the small‐world phenomenon is observed in so many real‐world graphs. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 201–224, 2017

We study a model of random R-enriched trees that is based on weights on the R-structures and allows for a unified treatment of a large family of random discrete structures. We establish distributional limits describing local convergence around fixed and random points in this general context, limit theorems for component sizes when R is a composite class, and a Gromov–Hausdorff scaling limit of random metric spaces patched together from independently drawn metrics on the R-structures. Our main applications treat a selection of examples encompassed by this model. We consider random outerplanar maps sampled according to arbitrary weights assigned to their inner faces, and classify in complete generality distributional limits for both the asymptotic local behaviour near the root-edge and near a uniformly at random drawn vertex. We consider random connected graphs drawn according to weights assigned to their blocks and establish a Benjamini–Schramm limit. We also apply our framework to recover in a probabilistic way a central limit theorem for the size of the largest 2-connected component in random graphs from planar-like classes. We prove Benjamini–Schramm convergence of random k-dimensional trees and establish both scaling limits and local weak limits for random planar maps drawn according to Boltzmann-weights assigned to their 2-connected components.

We show that a random labeled $n$-vertex graph almost surely contains isomorphic copies of almost all labeled $n$-vertex trees, in two senses. In the first sense, the probability of each edge occurring in the graph diminishes as $n$ increases, and the set of trees referred to as "almost all" depends on the random graph. In the other sense, the probability of an edge occurring is fixed, and the relevant set of trees is predetermined. The same method applies to show that almost all labeled $n$-tournaments contain isomorphic copies of almost all labeled oriented $n$-trees.

The subject of this workshop were probabilistic aspects of algorithms for fundamental problems such as sorting, searching, selecting of and within data, random permutations, algorithms based on combinatorial trees or search trees, continuous limits of random trees and random graphs as well as random geometric graphs. The deeper understanding of the complexity of such algorithms and of shape characteristics of large discrete structures require probabilistic models and an asymptotic analysis of random discrete structures. The talks of this workshop focused on probabilistic, combinatorial and analytic techniques to study asymptotic properties of large random combinatorial structures.

ABSTRACTAn independent set of size in a finite undirected graph is a set of vertices of the graph, no two of which are connected by an edge. Let be the number of independent sets of size in the graph and let . In 1987, Alavi, Malde, Schwenk, and Erdős asked if the independent set sequence of a tree is unimodal (the sequence goes up and then down). This problem is still open. In 2006, Levit and Mandrescu showed that the last third of the independent set sequence of a tree is decreasing. We show that the first 46.8% of the independent set sequence of a random tree is increasing with (exponentially) high probability as the number of vertices goes to infinity. So, the question of Alavi, Malde, Schwenk, and Erdős is “four‐fifths true,” with high probability. We also show unimodality of the independent set sequence of Erdős–Rényi random graphs, when the expected degree of a single vertex is large (with [exponentially] high probability as the number of vertices in the graph goes to infinity, except for a small region near the mode). A weaker result is shown for random regular graphs. The structure of independent sets of size as varies is of interest in probability, statistical physics, combinatorics, and computer science.

Student outcomes are of great importance in higher education institutions. Accreditation bodies focus on them as an indicator to measure the performance and effectiveness of the institution. Forecasting students’ academic performance is crucial for every educational establishment seeking to enhance performance and perseverance of its students and reduce the failure rate in the future. The main goal of this study is to predict the performance of undergraduate first-level students in the Computer Department during the years 2016 to 2021 to enhance their performance in future by discovering the best algorithm use to analyze the educational data to identify the students’ academic performance. The secondary data was collected by reviewing the Student Affairs Department at the Faculty of Specific Education at Damietta University, in addition to the Statistics Department at the university. The dataset contained 830 instances after excluding 139 instances of missing values, irrelevant rows, and outliers. The dataset was divided into train (577 instances (70%)), test (253 instances (30%)) and involved six features such year, midterm, practical exam, writing exam, final total degree, and grade. This paper use five machine learning (ML) algorithms which was selected according to the literature review and high accuracy in predicting educational data mining: For the purpose of comparison, a number of different machine learning algorithms, such as Random Forest, Decision Tree, Naive Bayes, Neural Network, and K-Nearest Neighbours, were utilized and evaluated with evaluation metrics such as confusion matrix, accuracy, precision, recall, and F-measure. The Random Forest and Decision Tree classifiers emerged as the top-performing algorithms, accurately categorizing 250 instances when predicting students' performance in the statistics course. This was determined based on the findings of the study. Out of a total of 253 instances that were included in the testing set, they only made three incorrect classifications.

Some mathematical problems involving the asymptotic analysis of rooted random tree graphs and branching patterns for large numbers of vertices are described. The motivation comes from applications to hydrology and geomorphology in which one seeks formulae for the average behavior of various contours (level sets) associated with river networks and which depend on only a few large scale parameters. While this paper is mainly an overview, a new formula is derived in a special case as an illustration. Scaling problems and connections with Aldous’s new theory of “self-similar”continuum random trees are noted.

Every day, millions of people in many institutions communicate with each other on the Internet. The past two decades have witnessed unprecedented levels of Internet use by people around the world. Almost alongside these rapid developments in the internet space, an ever increasing incidence of attacks carried out on the internet has been consistently reported every minute. In such a difficult environment, Anomaly Detection Systems (ADS) play an important role in monitoring and analyzing daily internet activities for security breaches and threats. However, the analytical data routinely generated from computer networks are usually of enormous size and of little use. This creates a major challenge for ADSs, who must examine all the functionality of a certain dataset to identify intrusive patterns. The selection of features is an important factor in modeling anomaly-based intrusion detection systems. An irrelevant characteristic can lead to overfitting which in turn negatively affects the modeling power of classification algorithms. The objective of this study is to analyze and select the most discriminating input characteristics for the construction of efficient and computationally efficient schemes for an ADS. In the first step, a heuristic algorithm called IG-BA is proposed for dimensionality reduction by selecting the optimal subset based on the concept of entropy. Then, the relevant and meaningful features are selected, before implementing Number of Classifiers which includes: (1) An irrelevant feature can lead to overfitting which in turn negatively affects the modeling power of the classification algorithms. Experiment was done on CICIDS-2017 dataset by applying (1) Random Forest (RF), (2) Bayes Network (BN), (3) Naive Bayes (NB), (4) J48 and (5) Random Tree (RT) with results showing better detection precision and faster execution time. The proposed heuristic algorithm outperforms the existing ones as it is more accurate in detection as well as faster. However, Random Forest algorithm emerges as the best classifier for feature selection technique and scores over others by virtue of its accuracy in optimal selection of features.

Genetic variants in multiple genes and loci have been associated with the risk of nonsyndromic cleft lip with or without cleft palate (NSCL ± P). However, the estimation of risk remains challenge, because most of these variants are population-specific rendering the identification of the underlying genetic risk difficult. Herein we examined the use of machine learning network in previously reported single nucleotide polymorphisms (SNPs) to predict risk of NSCL ± P in the Brazilian population. Random forest and neural network methods were applied in 72 SNPs in a case-control sample composed by 722 NSCL ± P and 866 controls for discrimination of NSCL ± P risk. SNP-SNP interactions and functional annotation biological processes associated with the identified NSCL ± P risk genes were verified. Supervised random forest decision trees revealed high scores of importance for the SNPs rs11717284 and rs1875735 in FGF12, rs41268753 in GRHL3, rs2236225 in MTHFD1, rs2274976 in MTHFR, rs2235371 and rs642961 in IRF6, rs17085106 in RHPN2, rs28372960 in TCOF1, rs7078160 in VAX1, rs10762573 and rs2131960 in VCL, and rs227731 in 17q22, with an accuracy of 99% and an error rate of approximately 3% to predict the risk of NSCL ± P. Those same 13 SNPs were considered the most important for the neural network to effectively predict NSCL ± P risk, with an overall accuracy of 94%. Multivariate regression model revealed significant interactions among all SNPs, with an exception of those in FGF12 and MTHFD1. The most significantly biological processes for selected genes were those involved in tissue and epithelium development; neural tube closure; and metabolism of methionine, folate, and homocysteine. Our results provide novel clues for genetic mechanism studies of NSCL ± P and point out for a machine learning model composed by 13 SNPs that is capable of predicting NSCL ± P risk. Although validation is necessary, this genetic panel can be useful in the near future to assist in NSCL ± P genetic counseling.

The escalating prevalence of diabetes has emerged as a burgeoning national concern, necessitating comprehensive investigation and intervention. Leveraging data extracted from the 2021 Behavioral Risk Factor Surveillance System (BRFSS), this study harnesses the power of data mining techniques to discern salient variables within expansive datasets. Subsequently, a series of machine learning models, comprising Random Forest, Decision Tree, Lasso Regression, and Neural Network, were constructed, and evaluated. The principal objective of this research is to facilitate the identification of the optimal predictive model for diabetes. By scrutinizing and contrasting the performance of these diverse models, this study aspires to contribute valuable insights into the field of diabetes prediction, potentially aiding in the development of more accurate and effective diagnostic tools. Consequently, this research and to make a meaningful stride towards mitigating the adverse impacts of the burgeoning diabetes epidemic and underscores the pivotal role of data-driven analytics and machine learning in the realm of public health research and policy formulation.

News is a worldwide resource that is always expanding. Manually categorising texts/data has become tiresome and unproductive as the volume of internet material continues to grow. As a result, there is a compelling need to organise this data systematically and in a personalised manner. This paper presents a case study on identifying MIZO news categories from the news articles collected from the ‘Zonet’ website. Those articles are divided into three categories, namely Tualchhung (local news), Ramchhung (national news) and Rampawn (international news). In this paper, various machine learning techniques, namely random forest, SVM, KNN, decision tree, Naïve Bayes and neural network, have been explored for the classification of Mizo news, and their accuracy is being measured. The result reflects the superiority of the SVM model over the other models.

Heart attacks are one of the foremost causes of death in the world. While doctors can carry out multiple tests to diagnose it, it may go undetected for a long time which can prove fatal. However, it is possible to predict the potential risk of a heart attack using Machine Learning (ML) algorithms. This is done using patients' historical data and has found some success in the literature. However, not all features are equally important in such data which may result in low accuracy of prediction. In this study, we use feature selection techniques to select a subset of features with contribution to improve upon the prediction accuracy of ML models. Inforation theory based models are used for feature selection in combination with popular ML classifiers including Random forest (RF), Decision Trees (DTs), Logistic Regression (LR), k-nearest neighbors (kNN), Gradient Boosting, Bagging, Ensemble, Neural Network (NN) and Support Vector Machines (SVM). The model is tested the real-world and publicly available Framingham dataset. Results show that the proposed approach improves the prediction of heart disease compared to baseline models.

Most supervised classification algorithms produce a soft score (either a probability, a fuzzy degree, a possibility, a cost, etc.) assessing the strength of the association between items and classes. After that, each item is assigned to the class with the highest soft score. In this paper, we show that this last step can be improved through alternative procedures more sensible to the available soft information. To this aim, we propose a general fuzzy bipolar approach that enables learning how to take advantage of the soft information provided by many classification algorithms in order to enhance the generalization power and accuracy of the classifiers. To show the suitability of the proposed approach, we also present some computational experiences for binary classification problems, in which its application to some well-known classifiers as random forest, classification trees and neural networks produces a statistically significant improvement in the performance of the classifiers.

The aim of supervised classification algorithms is to assign objects/items to known classes. Before carrying out the final assignment, many classification algorithms obtain a soft score (probability, fuzzy, possibility, cost...) between each item and each class. In order to improve this final decision, we build a bipolar probabilistic model that considers some extra information about the dissimilarity structure between the classes. We present here some improvements for several supervised classification algorithms such as random forest, decision trees and neural networks for binary classification problems.