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Xing Shi Cai

Publications and source records attributed to Xing Shi Cai.

16 recordsLinked to original sources
arXiv2025

Minimum stationary values of sparse random directed graphs

We consider the stationary distribution of the simple random walk on the directed configuration model with bounded degrees. Provided that the minimum out-degree is at least $2$, with high probability (whp) there is a unique stationary distribution. We show that the minimum positive stationary value is whp $n^{-(1+C+o(1))}$ for some constant $C \ge 0$ determined by the degree distribution. In particular, $C$ is the competing combination of two factors: (1) the contribution of atypically "thin" in-neighbourhoods, controlled by subcritical branching processes; and (2) the contribution of atypically "light" trajectories, controlled by large deviation rate functions. Additionally, our proof implies that whp the hitting and the cover time are both $n^{1+C+o(1)}$. Our results complement those of Caputo and Quattropani who showed that if the minimum in-degree is at least 2, stationary values have logarithmic fluctuations around $n^{-1}$.

math.PR↗
arXiv2021

The fluctuations of the giant cluster for percolation on random split trees

A split tree of cardinality $n$ is constructed by distributing $n$ "balls" in a subset of vertices of an infinite tree which encompasses many types of random trees such as $m$-ary search trees, quad trees, median-of-$(2k+1)$ trees, fringe-balanced trees, digital search trees and random simplex trees. In this work, we study Bernoulli bond percolation on arbitrary split trees of large but finite cardinality $n$. We show for appropriate percolation regimes that depend on the cardinality $n$ of the split tree that there exists a unique giant cluster, the fluctuations of the size of the giant cluster as $n \rightarrow \infty$ are described by an infinitely divisible distribution that belongs to the class of stable Cauchy laws. This work generalizes the results for the random $m$-ary recursive trees in Berzunza (2015). Our approach is based on a remarkable decomposition of the size of the giant percolation cluster as a sum of essentially independent random variables which may be useful for studying percolation on other trees with logarithmic height; for instance in this work we study also the case of regular trees.

math.PR↗
arXiv2021

Rankings in directed configuration models with heavy tailed in-degrees

We consider the extremal values of the stationary distribution of sparse directed random graphs with given degree sequences and their relation to the extremal values of the in-degree sequence. The graphs are generated by the directed configuration model. Under the assumption of bounded $(2+η)$-moments on the in-degrees and of bounded out-degrees, we obtain tight comparisons between the maximum value of the stationary distribution and the maximum in-degree. Under the further assumption that the order statistics of the in-degrees have a power-law behavior, we show that the extremal values of the stationary distribution also have a power-law behavior with the same index. In the same setting, we prove that these results extend to the PageRank scores of the random digraph, thus confirming a version of the so-called power-law hypothesis. Along the way, we establish several facts about the model, including the mixing time cutoff and the characterization of the typical values of the stationary distribution, which were previously obtained under the assumption of bounded in-degrees.

math.PR↗
arXiv2020

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 converges 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.

math.PR↗
arXiv2020

The diameter of the directed configuration model

We show that the diameter of the directed configuration model with $n$ vertices rescaled by $\log n$ converges in probability to a constant. Our assumptions are the convergence of the in- and out-degree of a uniform random vertex in distribution, first and second moment. Our result extends previous results on the diameter of the model and applies to many other random directed graphs.

math.PR↗
arXiv2019

Cutting resilient networks -- complete binary trees

In our previous work, we introduced the random $k$-cut number for rooted graphs. In this paper, we show that the distribution of the $k$-cut number in complete binary trees of size $n$, after rescaling, is asymptotically a periodic function of $\lg n - \lg \lg n$. Thus there are different limit distributions for different subsequences, where these limits are similar to weakly 1-stable distributions. This generalizes the result for the case $k = 1$, i.e., the traditional cutting model, by Janson.

math.PR↗
arXiv2019

K-cut on paths and some trees

We define the (random) $k$-cut number of a rooted graph to model the difficulty of the destruction of a resilient network. The process is as the cut model of Meir and Moon except now a node must be cut $k$ times before it is destroyed. The first order terms of the expectation and variance of $\mathcal{X}_{n}$, the $k$-cut number of a path of length $n$, are proved. We also show that $\mathcal{X}_{n}$, after rescaling, converges in distribution to a limit $\mathcal{B}_{k}$, which has a complicated representation. The paper then briefly discusses the $k$-cut number of some trees and general graphs. We conclude by some analytic results which may be of interest.

math.PR↗
arXiv2018

Inversions in split trees and conditional Galton--Watson trees

We study $I(T)$, the number of inversions in a tree $T$ with its vertices labeled uniformly at random, which is a generalization of inversions in permutations. We first show that the cumulants of $I(T)$ have explicit formulas involving the $k$-total common ancestors of $T$ (an extension of the total path length). Then we consider $X_n$, the normalized version of $I(T_n)$, for a sequence of trees $T_n$. For fixed $T_{n}$'s, we prove a sufficient condition for $X_n$ to converge in distribution. As an application, we identify the limit of $X_n$ for complete $b$-ary trees. For $T_n$ being split trees, we show that $X_n$ converges to the unique solution of a distributional equation. Finally, when $T_n$'s are conditional Galton--Watson trees, we show that $X_n$ converges to a random variable defined in terms of Brownian excursions. By exploiting the connection between inversions and the total path length, we are able to give results that are stronger and much broader compared to previous work by Panholzer and Seitz.

math.PR↗
arXiv2018

A note on the asymptotic expansion of the Lerch's transcendent

In a previous paper by Ferreira and López [Journal of Mathematical Analysis and Applications, 298(1), 2004], the authors derived an asymptotic expansion of the Lerch's transcendent $Φ(z,s,a)$ for large $\vert a\vert$, valid for $\mathrm{Re}(a)>0$, $\mathrm{Re}(s)>0$ and $z\in\mathbb{C}\setminus[1,\infty)$. In this paper we study the special case $z\ge 1$ not covered in the previous result, deriving a complete asymptotic expansion of the Lerch's transcendent $Φ(z,s,a)$ for $z > 1$ and $\mathrm{Re}(s)>0$ as $\mathrm{Re}(a)$ goes to infinity. We also show that when $a$ is a positive integer, this expansion is convergent for $\mathrm{Re}(z) \ge 1$. As a corollary, we get a full asymptotic expansion for the sum $\sum_{n=1}^{m} z^{n}/n^{s}$ for fixed $z >1 $ as $m \to \infty$. Some numerical results show the accuracy of the approximation.

math.CV↗
arXiv2018

Non-fringe subtrees in conditioned Galton--Watson trees

We study $S(\mathcal T_{n})$, the number of subtrees in a conditioned Galton--Watson tree of size $n$. With two very different methods, we show that $\log(S(\mathcal T_{n}))$ has a Central Limit Law and that the moments of $S(\mathcal T_{n})$ are of exponential scale.

math.CO↗
arXiv2016

The graph structure of a deterministic automaton chosen at random: full version

A deterministic finite automaton (DFA) of $n$ states over a $k$-letter alphabet can be seen as a digraph with $n$ vertices which all have exactly $k$ labeled out-arcs ($k$-out digraph). In 1973 Grusho first proved that with high probability (whp) in a random $k$-out digraph there is a strongly connected component (SCC) of linear size that is reachable from all vertices, i.e., a giant. He also proved that the size of the giant follows a central limit law. We show that whp the part outside the giant contains at most a few short cycles and mostly consists of overlapping tree-like structures. Thus the directed acyclic graph (DAG) of a random $k$-out digraph is almost the same as the digraph with the giant contracted into one vertex. These findings lead to a new, concise and self-contained proof of Grusho's theorem. This work also contains some other results including the structure outside the giant, the phase transition phenomenon in strong connectivity, the typical distance, and an extension to simple digraphs.

math.PR↗
arXiv2016

Acyclic edge colourings of graphs with large girth

An edge colouring of a graph $G$ is called acyclic if it is proper and every cycle contains at least three colours. We show that for every $\varepsilon>0$, there exists a $g=g(\varepsilon)$ such that if $G$ has girth at least $g$ then $G$ admits an acyclic edge colouring with at most $(1+\varepsilon)Δ$ colours.

math.CO↗
arXiv2016

A study of large fringe and non-fringe subtrees in conditional Galton-Watson trees

We study the conditions for families of subtrees to exist with high probability (whp) in a Galton-Walton tree of size $n$. We first give a Poisson approximation of fringe subtree counts, which yields the height of the maximal complete $r$-ary fringe subtree. Then we determine the maximal $K_n$ such that every tree of size at most $K_n$ appears as fringe subtree whp. Finally, we study non-fringe subtree counts and determine the height of the maximal complete $r$-ary non-fringe subtree.

math.PR↗
arXiv2015

The Analysis of Kademlia for random IDs

Kademlia is the de facto standard searching algorithm for P2P (peer-to-peer) networks on the Internet. In our earlier work, we introduced two slightly different models for Kademlia and studied how many steps it takes to search for a target node by using Kademlia's searching algorithm. The first model, in which nodes of the network are labelled with deterministic IDs, had been discussed in that paper. The second one, in which nodes are labelled with random IDs, which we call the Random ID Model, was only briefly mentioned. Refined results with detailed proofs for this model are given in this paper. Our analysis shows that with high probability it takes about $c \log n$ steps to locate any node, where $n$ is the total number of nodes in the network and $c$ is a constant that does not depend on $n$.

cs.DS↗
arXiv2013

A Probabilistic Analysis of Kademlia Networks

Kademlia is currently the most widely used searching algorithm in P2P (peer-to-peer) networks. This work studies an essential question about Kademlia from a mathematical perspective: how long does it take to locate a node in the network? To answer it, we introduce a random graph K and study how many steps are needed to locate a given vertex in K using Kademlia's algorithm, which we call the routing time. Two slightly different versions of K are studied. In the first one, vertices of K are labelled with fixed IDs. In the second one, vertices are assumed to have randomly selected IDs. In both cases, we show that the routing time is about c*log(n), where n is the number of nodes in the network and c is an explicitly described constant.

cs.NI↗