arXiv subjects

Panpan Zhang

Publications and source records attributed to Panpan Zhang.

44 records · Page 3Linked to original sources

arXiv2019

The degree Gini index of several classes of random trees and their poissonized counterparts---an evidence for a duality theory

There is an unproven duality theory hypothesizing that random discrete trees and their poissonized embeddings in continuous time share fundamental properties. We give additional evidence in favor of this theory by showing that several classes of random trees growing in discrete time and their poissonized counterparts have the same limiting degree Gini index. The classes that we consider include binary search trees, binary pyramids and random caterpillars.

math.PR↗

arXiv2019

The Zagreb index of several random models

In this article, we investigate the Zagreb index, a kind of graph-based topological index, of several random networks, including a class of networks extended from random recursive trees, plain-oriented recursive trees, and random caterpillars growing in a preferential attachment manner. We calculate the mean and variance of the Zagreb index for each class. In addition, we prove that the asymptotic distribution of the Zagreb index for the first class is normal, and that the asymptotic distribution of the Zagreb index for the second class is skewed to the right.

math.PR↗

arXiv2018

On several properties of plane-oriented recursive trees

In this paper, several properties of plain-oriented recursive trees (PORTs) are uncovered. Specifically, we investigate the degree profile of a PORT by determining the exact probability mass function of the degree of a node with a fixed label. We compute the expectation and variance of the degree variable via a \Polya\ urn approach. In addition, we look into a topological index, the Zagreb index, of this class of trees. We calculate the exact first two moments of the Zagreb index by using recurrence methods. We also provide several evidence in favor of our conjecture that the Zagreb index of PORTs do not follow a Gaussian law asymptotically. Lastly, we determine the limiting degree distribution in Poissonized PORTs, and show that it is exponential after being properly scaled.

math.PR↗

arXiv2018

Characterizations of asymptotic distributions of continuous-time Pólya processes

We propose an elementary but effective approach to studying a general class of Poissonized tenable and balanced urns on two colors. We characterize the asymptotic behavior of the process via a partial differential equation that governs the process, coupled with the method of moments applied in a bootstrapped manner. We show that the limiting distribution of the process underlying the Bagchi-Pal urn is gamma. We also look into the tenable and balanced processes associated with randomized replacement matrix. Similar results carry over to the process, with minor modifications in the methods of proof, done mutatis mutandis.

math.PR↗

arXiv2018

Distributions in the constant-differentials Pólya process

We study a class of unbalanced constant-differentials Pólya processes on white and blue balls. We show that the number of white balls, the number of blue balls, and the total number of balls, when appropriately scaled, all converge in distribution to a gamma random variables with parameters depending on the differential index and the amount of ball addition at the epochs, but not on the initial conditions. The result is obtained by an analytic approach utilizing partial differential equations.

math.ST↗

arXiv2018

The degree profile and Gini index of random caterpillar trees

In this paper, we investigate the degree profile and Gini index of random caterpillar trees (RCTs). We consider RCTs which evolve in two different manners: uniform and nonuniform. The degrees of the vertices on the central path (i.e., the degree profile) of a uniform RCT follow a multinomial distribution. For nonuniform RCTs, we focus on those growing in the fashion of preferential attachment. We develop methods based on stochastic recurrences to compute the exact expectations and the dispersion matrix of the degree variables. A generalized Pólya urn model is exploited to determine the exact joint distribution of these degree variables. We apply the methods from combinatorics to prove that the asymptotic distribution is Dirichlet. In addition, we propose a new type of Gini index to quantitatively distinguish the evolutionary characteristics of the two classes of RCTs. We present the results via several numerical experiments.

math.ST↗

arXiv2018

Clique-based Method for Social Network Clustering

In this article, we develop a clique-based method for social network clustering. We introduce a new index to evaluate the quality of clustering results, and propose an efficient algorithm based on recursive bipartition to maximize an objective function of the proposed index. The optimization problem is NP-hard, so we approximate the semi-optimal solution via an implicitly restarted Lanczos method. One of the advantages of our algorithm is that the proposed index of each community in the clustering result is guaranteed to be higher than some predetermined threshold, $p$, which is completely controlled by users. We also account for the situation that $p$ is unknown. A statistical procedure of controlling both under-clustering and over-clustering errors simultaneously is carried out to select localized threshold for each subnetwork, such that the community detection accuracy is optimized. Accordingly, we propose a localized clustering algorithm based on binary tree structure. Finally, we exploit the stochastic blockmodels to conduct simulation studies and demonstrate the accuracy and efficiency of our algorithms, both numerically and graphically.

cs.SI↗

arXiv2018

On nodes of small degrees and degree profile in preferential dynamic attachment circuits

We investigate the joint distribution of nodes of small degrees and the degree profile in preferential dynamic attachment circuits. In particular, we study the joint asymptotic distribution of the number of the nodes of outdegree $0$ (terminal nodes) and outdegree $1$ in a very large circuit. The expectation and variance of the number of those two types of nodes are both asymptotically linear with respect to the age of the circuit. We show that the numbers of nodes of outdegree $0$ and $1$ asymptotically follow a two-dimensional Gaussian law via multivariate martingale methods. We also study the exact distribution of the degree of a node, as the circuit ages, via a series of Pólya-Eggenberger urn models with "hiccups" in between. The exact expectation and variance of the degree of nodes are determined by recurrence methods. Phase transitions of these degrees are discussed briefly. This is an extension of the abstract [20].

math.PR↗