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The purpose of this paper is to analyze the degree index and clustering index in random graphs. The degree index in our setup is a certain measure of degree irregularity whose basic properties are well studied in the literature, and the corresponding theoretical analysis in a random graph setup turns out to be tractable. On the other hand, the clustering index, based on a similar reasoning, is first introduced in this manuscript. Computing exact expressions for the expected clustering index turns out to be more challenging even in the case of Erd\H{o}s-R\'enyi graphs, and our results are on obtaining relevant upper bounds. These are also complemented with observations based on Monte Carlo simulations. Besides the Erd\H{o}s-R\'enyi case, we also do simulation-based analysis for random regular graphs, the Barab\'asi-Albert model and the Watts-Strogatz model., Comment: 21 pages, 6 figures

The purpose of this paper is to give an overview of the time series forecasting problem based on similarity of trajectories. Various methodologies are introduced and studied, and detailed discussions on hyperparameter optimization, outlier handling and distance measures are provided. The suggested new approaches involve variations in both the selection of similar trajectories and assembling the candidate forecasts. After forming a general framework, an experimental study is conducted to compare the methods that use similar trajectories along with some other standard models (such as ARIMA and Random Forest) from the literature. Lastly, the forecasting setting is extended to interval forecasts, and the prediction intervals resulting from the similar trajectories approach are compared with the existing models from the literature, such as historical simulation and quantile regression. Throughout the paper, the experimentations and comparisons are conducted via the time series of traffic flow from the California PEMS dataset., Comment: 32 pages

The edge flipping is a non-reversible Markov chain on a given connected graph, which is defined by Chung and Graham. In the same paper, its eigenvalues and stationary distributions for some classes of graphs are identified. We further study its spectral properties to show a lower bound for the rate of convergence in the case of regular graphs. Moreover, we show that a cutoff occurs at 1/4 n log n for the edge flipping on the complete graph by a coupling argument, Comment: This manuscript has been withdrawn as a duplicate. The article is available at: arXiv:2201.03315v2

We introduce a cover time problem for random walks on dynamic graphs in which the graph expands in time and the walker moves at random times. Time to cover all nodes and number of returns to original states are analyzed in resulting model.

We obtain an explicit formula for the variance of the number of $k$-peaks in a uniformly random permutation. This is then used to obtain an asymptotic formula for the variance of the length of longest $k$-alternating subsequence in random permutations. Also a central limit is proved for the latter statistic., Comment: Typos and errors corrected

For a given permutation $\pi_n$ in $S_n$, a random permutation graph is formed by including an edge between two vertices $i$ and $j$ if and only if $(i - j) (\pi_n(i) - \pi_n (j)) < 0$. In this paper, we study various statistics of random permutation graphs. In particular, the degree of a given node, the number of nodes with a given degree, the number of isolated vertices, and the number of cliques are analyzed. Further, explicit formulas for the probabilities of having a given number of connected components and isolated vertices are obtained., Comment: 21 pages, 1 figure

The purpose of this paper is to study a statistic that is used to compare the similarity between two strings, which is first introduced by Michael Steele in 1982. It was proposed as an alternative to the length of the longest common subsequences, for which the variance problem is still open. Our results include moment asymptotics and distributional asymptotics for Steele's statistic and a variation of it in random words and random permutations., Comment: 17 pages

The purpose of this paper is to analyze certain statistics of a recently introduced non-uniform random tree model, biased recursive trees. This model is based on constructing a random tree by establishing a correspondence with non-uniform permutations, biased riffle shuffles. The statistics that are treated include the number of nodes with a given number of descendants, the depth of the tree, and the number of branches. The model yields the uniform recursive trees as a certain limit, some new results for the uniform case are obtained as well., Comment: 27 pages, partially based on Hiesmayr's master's thesis

A uniform recursive tree on $n$ vertices is a random tree where each possible $(n-1)!$ labeled recursive rooted tree is selected with equal probability. In this paper we introduce and study weighted trees, a non-uniform recursive tree model departing from the recently introduced Hoppe trees. This class generalizes both uniform recursive trees and Hoppe trees. The generalization provides diversity among the nodes, making the model more flexible for applications. We also analyze the number of leaves, the height, the depth, the number of branches, and the size of the largest branch in these weighted trees., Comment: 16 pages. Based on master's thesis of Hiesmayr, arxiv:1710.01402

Let $X=(X_1,\ldots,X_n)$ be a vector of i.i.d. random variables where $X_i$'s take values over $\mathbb{N}$. The purpose of this paper is to study the number of weakly increasing subsequences of $X$ of a given length $k$, and the number of all weakly increasing subsequences of $X$. For the former, it is shown that a central limit theorem holds. Also, the first two moments of each of those two random variables are analyzed, their asymptotics are investigated, and results are related to the case of similar statistics in uniformly random permutations. We conclude the paper with applications on a similarity measure of Steele, and on increasing subsequences of riffle shuffles., Comment: Final version

In this paper we study the inverse of so-called unfair permutations, and explore various properties of them. Our investigation begins with comparing this class of permutations with uniformly random permutations, and showing that they behave very much alike for locally dependent random variables. As an example of a globally dependent statistic we use the number of inversions, and show that this statistic satisfies a central limit theorem after proper centering and scaling. A secondary example of a globally dependent statistic to be studied will be the number of fixed points. Finally, we introduce two different generalizations of inverse-unfair permutations., Comment: Final version after a major revision. An error in Proof of Theorem 4.1 is fixed

Let $\mathbf{X}^{(1)}_{n},\ldots,\mathbf{X}^{(m)}_{n}$, where $\mathbf{X}^{(i)}_{n}=(X^{(i)}_{1},\ldots,X^{(i)}_{n})$, $i=1,\ldots,m$, be $m$ independent sequences of independent and identically distributed random variables taking their values in a finite alphabet $\mathcal{A}$. Let the score function $S$, defined on $\mathcal{A}^{m}$, be non-negative, bounded, permutation-invariant, and satisfy a bounded differences condition. Under a variance lower-bound assumption, a central limit theorem is proved for the optimal alignments score of the $m$ random words., Comment: 30 pages

Let $(X_i)_{i \geq 1}$ and $(Y_i)_{i\geq1}$ be two independent sequences of independent identically distributed random variables taking their values in a common finite alphabet and having the same law. Let $LC_n$ be the length of the longest common subsequences of the two random words $X_1\cdots X_n$ and $Y_1\cdots Y_n$. Under a lower bound assumption on the order of its variance, $LC_n$ is shown to satisfy a central limit theorem. This is in contrast to the limiting distribution of the length of the longest common subsequences in two independent uniform random permutations of $\{1, \dots, n\}$, which is shown to be the Tracy-Widom distribution., Comment: Revised arguments, main result unchanged; Corrected typos; Added references

Threshold-type counts based on multivariate occupancy models with log concave marginals admit bounded size biased couplings under weak conditions, leading to new concentration of measure results for random graphs, germ-grain models in stochastic geometry, multinomial allocation models and multivariate hypergeometric sampling. The work generalizes and improves upon previous results in a number of directions., Comment: 35 pages. Version 2 is the result of a major revision, including corrections and significant clarifications in particular as regards Lemmas 2.6 and 2.7, and the introduction and use of Property (B,*) in Definition 2.2. Version 3 added some references and removed Theorem 1.1