A Primer on Laplacian Dynamics in Directed Graphs

Chemical Reaction Networks in a Laplacian Framework

The study of the dynamics of chemical reactions, and in particular phenomena such as oscillating reactions, has led to the recognition that many dynamical properties of a chemical reaction can be predicted from graph theoretical properties of a certain directed graph, called a Chemical Reaction Network (CRN). In this graph, the edges represent the reactions and the vertices the reacting combinations of chemical substances. In contrast with the classical treatment, in this work, we heavily rely on a recently developed theory of directed graph Laplacians to simplify the traditional treatment of …

Ultracontractive Properties for Directed Graph Semigroups with Applications to Coupled Oscillators

It is now well known that ultracontractive properties of semigroups with infinitesimal generator given by an undirected graph Laplacian operator can be obtained through an understanding of the geometry of the underlying infinite weighted graph. The aim of this work is to extend these results to semigroups with infinitesimal generator given by a directed graph Laplacian operator, through an analogous inspection of the geometry of the underlying directed graph. In particular, we introduce appropriate nomenclature to discuss the geometry of an infinite directed graph, as well as provide sufficien…

Distributed Averaging Problems over Directed Signed Networks

This paper aims at addressing distributed averaging problems for signed networks in the presence of general directed topologies that are represented by signed digraphs. A new class of improved Laplacian potential functions is proposed by presenting two notions of any signed digraph: induced unsigned digraph and mirror (undirected) signed graph, based on which two distributed averaging protocols are designed using the nearest neighbor rules. It is shown that with any of the designed protocols, signed-average consensus (respectively, state stability) can be achieved if and only if the associated…

Koopman-based spectral clustering of directed and time-evolving graphs

While spectral clustering algorithms for undirected graphs are well established and have been successfully applied to unsupervised machine learning problems ranging from image segmentation and genome sequencing to signal processing and social network analysis, clustering directed graphs remains notoriously difficult. Two of the main challenges are that the eigenvalues and eigenvectors of graph Laplacians associated with directed graphs are in general complex-valued and that there is no universally accepted definition of clusters in directed graphs. We first exploit relationships between the gr…

有向超图上的非马尔可夫热流

Non-Markovian heat flows on directed hypergraphs

该研究提出有向超图拉普拉斯算子的半群框架，拓展经典图热流模型，用谱手术法推导特征值界分析热流长时间行为，揭示有向超图热流的正性与∞-收缩性变化规律，结合多学科方法验证相关现象。

使用时空图拉普拉斯算子对时变网络进行聚类

Clustering Time-Evolving Networks Using the Spatio-Temporal Graph Laplacian

本文提出基于CCA的时空图拉普拉斯算子，将谱聚类推广到动态图，并结合传递算子分析其谱性质，实现对有向和无向时变网络聚类演化的清晰解释。