arXiv subjects

Xiaotian Zhou

Publications and source records attributed to Xiaotian Zhou.

23 records · Page 2Linked to original sources

arXiv2023

Maximizing the Smallest Eigenvalue of Grounded Laplacian Matrix

For a connected graph $\mathcal{G}=(V,E)$ with $n$ nodes, $m$ edges, and Laplacian matrix $\boldsymbol{\mathit{L}}$, a grounded Laplacian matrix $\boldsymbol{\mathit{L}}(S)$ of $\mathcal{G}$ is a $(n-k) \times (n-k)$ principal submatrix of $\boldsymbol{\mathit{L}}$, obtained from $\boldsymbol{\mathit{L}}$ by deleting $k$ rows and columns corresponding to $k$ selected nodes forming a set $S\subseteq V$. The smallest eigenvalue $λ(S)$ of $\boldsymbol{\mathit{L}}(S)$ plays a pivotal role in various dynamics defined on $\mathcal{G}$. For example, $λ(S)$ characterizes the convergence rate of leader-follower consensus, as well as the effectiveness of a pinning scheme for the pinning control problem, with larger $λ(S)$ corresponding to smaller convergence time or better effectiveness of a pinning scheme. In this paper, we focus on the problem of optimally selecting a subset $S$ of fixed $k \ll n$ nodes, in order to maximize the smallest eigenvalue $λ(S)$ of the grounded Laplacian matrix $\boldsymbol{\mathit{L}}(S)$. We show that this optimization problem is NP-hard and that the objective function is non-submodular but monotone. Due to the difficulty to obtain the optimal solution, we first propose a na\"ıve heuristic algorithm selecting one optimal node at each time for $k$ iterations. Then we propose a fast heuristic scalable algorithm to approximately solve this problem, using derivative matrix, matrix perturbations, and Laplacian solvers as tools. Our na\"ıve heuristic algorithm takes $\tilde{O}(knm)$ time, while the fast greedy heuristic has a nearly linear time complexity of $\tilde{O}(km)$. We also conduct numerous experiments on different networks sized up to one million nodes, demonstrating the superiority of our algorithm in terms of efficiency and effectiveness.

cs.IT↗

arXiv2022

A Scalable Blockchain-based Smart Contract Model for Decentralized Voltage Stability Using Sharding Technique

Blockchain technologies are one possible avenue for increasing the resilience of the Smart Grid, by decentralizing the monitoring and control of system-level objectives such as voltage stability protection. They furthermore offer benefits in data immutability and traceability, as blockchains are cryptographically secured. However, the performance of blockchain-based systems in real-time grid monitoring and control has never been empirically tested. This study proposes implementing a decentralized voltage stability algorithm using blockchain-based smart contracts, as a testbed for evaluating the performance of blockchains in real-time control. We furthermore investigate sharding mechanisms as a means of improving the system's scalability with fixed computing resources. We implement our models as a proof-of-concept prototype system using Hyperledger Fabric as our blockchain platform, the Matpower library in MATLAB as our power system simulator, and Hyperledger Caliper as our performance evaluation tool. We found that sharding does indeed lead to a substantial improvement in system scalability for this domain, measured by both transaction success rates and transaction latency.

cs.DC↗

arXiv2021

Maximizing Influence of Leaders in Social Networks

The operation of adding edges has been frequently used to the study of opinion dynamics in social networks for various purposes. In this paper, we consider the edge addition problem for the DeGroot model of opinion dynamics in a social network with $n$ nodes and $m$ edges, in the presence of a small number $s \ll n$ of competing leaders with binary opposing opinions 0 or 1. Concretely, we pose and investigate the problem of maximizing the equilibrium overall opinion by creating $k$ new edges in a candidate edge set, where each edge is incident to a 1-valued leader and a follower node. We show that the objective function is monotone and submodular. We then propose a simple greedy algorithm with an approximation factor $(1-\frac{1}{e})$ that approximately solves the problem in $O(n^3)$ time. Moreover, we provide a fast algorithm with a $(1-\frac{1}{e}-ε)$ approximation ratio and $\tilde{O}(mkε^{-2})$ time complexity for any $ε>0$, where $\tilde{O}(\cdot)$ notation suppresses the ${\rm poly} (\log n)$ factors. Extensive experiments demonstrate that our second approximate algorithm is efficient and effective, which scales to large networks with more than a million nodes.

cs.SI↗

arXiv2021

Edge Domination Number and the Number of Minimum Edge Dominating Sets in Pseudofractal Scale-Free Web and Sierpiński Gasket

As a fundamental research object, the minimum edge dominating set (MEDS) problem is of both theoretical and practical interest. However, determining the size of a MEDS and the number of all MEDSs in a general graph is NP-hard, and it thus makes sense to find special graphs for which the MEDS problem can be exactly solved. In this paper, we study analytically the MEDS problem in the pseudofractal scale-free web and the Sierpiński gasket with the same number of vertices and edges. For both graphs, we obtain exact expressions for the edge domination number, as well as recursive solutions to the number of distinct MEDSs. In the pseudofractal scale-free web, the edge domination number is one-ninth of the number of edges, which is three-fifths of the edge domination number of the Sierpiński gasket. Moreover, the number of all MEDSs in the pseudofractal scale-free web is also less than that corresponding to the Sierpiński gasket. We argue that the difference of the size and number of MEDSs between the two studied graphs lies in the scale-free topology.

math.CO↗

arXiv2019

Adaptive Margin Ranking Loss for Knowledge Graph Embeddings via a Correntropy Objective Function

Translation-based embedding models have gained significant attention in link prediction tasks for knowledge graphs. TransE is the primary model among translation-based embeddings and is well-known for its low complexity and high efficiency. Therefore, most of the earlier works have modified the score function of the TransE approach in order to improve the performance of link prediction tasks. Nevertheless, proven theoretically and experimentally, the performance of TransE strongly depends on the loss function. Margin Ranking Loss (MRL) has been one of the earlier loss functions which is widely used for training TransE. However, the scores of positive triples are not necessarily enforced to be sufficiently small to fulfill the translation from head to tail by using relation vector (original assumption of TransE). To tackle this problem, several loss functions have been proposed recently by adding upper bounds and lower bounds to the scores of positive and negative samples. Although highly effective, previously developed models suffer from an expansion in search space for a selection of the hyperparameters (in particular the upper and lower bounds of scores) on which the performance of the translation-based models is highly dependent. In this paper, we propose a new loss function dubbed Adaptive Margin Loss (AML) for training translation-based embedding models. The formulation of the proposed loss function enables an adaptive and automated adjustment of the margin during the learning process. Therefore, instead of obtaining two values (upper bound and lower bound), only the center of a margin needs to be determined. During learning, the margin is expanded automatically until it converges. In our experiments on a set of standard benchmark datasets including Freebase and WordNet, the effectiveness of AML is confirmed for training TransE on link prediction tasks.

cs.CL↗