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Yong Lin

Publications and source records attributed to Yong Lin.

At least 73 records · Page 4Linked to original sources
arXiv2019

Weighted Path homology of Weighted Digraphs and Persistence

In recent years, A. Grigor'yan, Y. Lin, Y. Muranov and S.T. Yau [6, 7, 8, 9] constructed a path homology theory for digraphs. Later, S. Chowdhury and F. Memoli [3] studied the persistent path homology for directed networks. In this paper, we generalize the path homology theory for digraphs and construct a weighted path homology for weighted digraphs. We study the persistent weighted path homology for weighted digraphs and detect the effects of the weights on the persistent weighted path homology. We prove a persistent version of a Kunneth-type formula for joins of weighted digraphs.

math.AT↗
arXiv2019

Existence of Solutions to Mean Field Equations on Graphs

In this paper, we prove two existence results of solutions to mean field equations $$Δu+e^u=ρδ_0$$ and $$Δu=λe^u(e^u-1)+4 π\sum_{j=1}^{M}{δ_{p_j}}$$ on an arbitrary connected finite graph, where $ρ>0$ and $λ>0$ are constants, $M$ is a positive integer, and $p_1,...,p_M$ are arbitrarily chosen vertices on the graph.

math.AP↗
arXiv2018

Ricci-flat cubic graphs with girth five

We classify all connected, simple, 3-regular graphs with girth at least 5 that are Ricci-flat. We use the definition of Ricci curvature on graphs given in Lin-Lu-Yau, Tohoku Math., 2011, which is a variation of Ollivier, J. Funct. Anal., 2009. A graph is Ricci-flat, if it has vanishing Ricci curvature on all edges. We show, that the only Ricci-flat cubic graphs with girth at least 5 are the Petersen graph, the Triplex and the dodecahedral graph. This will correct the classification in Lin-Lu-Yau, Comm. Anal. Geom., 2014, that misses the Triplex.

math.CO↗
arXiv2018

Li-Yau inequality for unbounded Laplacian on graphs

In this paper, we derive Li-Yau inequality for unbounded Laplacian on complete weighted graphs with the assumption of the curvature-dimension inequality $CDE'(n,K)$, which can be regarded as a notion of curvature on graphs. Furthermore, we obtain some applications of Li-Yau inequality, including Harnack inequality, heat kernel bounds and Cheng's eigenvalue estimate. These are first kind of results on this direction for unbounded Laplacian on graphs.

math.DG↗
arXiv2017

Blow-up problems for nonlinear parabolic equations on locally finite graphs

Let $G=(V,E)$ be a locally finite connected weighted graph, $Δ$ be the usual graph Laplacian. In this paper, we study the blow-up problems for the nonlinear parabolic equation $u_t=Δu + f(u)$ on $G$. The blow-up phenomenons of the equation are discussed in terms of two cases: (i) an initial condition is given; (ii) a Dirichlet boundary condition is given. We prove that if $f$ satisfies appropriate conditions, then the solution of the equation blows up in a finite time.

math.AP↗
arXiv2017

The existence and nonexistence of global solutions for a semilinear heat equation on graphs

Let $G=(V,E)$ be a finite or locally finite connected weighted graph, $Δ$ be the usual graph Laplacian. Using heat kernel estimate, we prove the existence and nonexistence of global solutions for the following semilinear heat equation on $G$ \begin{equation*} \left\{ \begin{array}{lc} u_t=Δu + u^{1+α} &\, \text{in $(0,+\infty)\times V$,}\\ u(0,x)=a(x) &\, \text{in $V$.} \end{array} \right. \end{equation*} We conclude that, for a graph satisfying curvature dimension condition $CDE'(n,0)$ and $V(x,r)\simeq r^m$, if $0 2$, then there is a non-negative global solution $u$ provided that the initial value is small enough. In particular, these results are true on lattice $\mathbb{Z}^m$.

math.AP↗
arXiv2016

On-diagonal lower estimate of heat kernel on graphs

The purpose of this paper is to establish a new continuous-time on-diagonal lower estimate of heat kernel for large time on graphs. To achieve the goal, we first give an upper bound of heat kernel in natural graph metric, and then use this bound and the volume growth condition to show the validity of the on-diagonal lower bound.

math.AP↗
arXiv2016

Yamabe type equations on graphs

Let $G=(V,E)$ be a locally finite graph, $Ω\subset V$ be a bounded domain, $Δ$ be the usual graph Laplacian, and $λ_1(Ω)$ be the first eigenvalue of $-Δ$ with respect to Dirichlet boundary condition. Using the mountain pass theorem due to Ambrosetti-Rabinowitz, we prove that if $α<λ_1(Ω)$, then for any $p>2$, there exists a positive solution to $-Δu-αu=|u|^{p-2}u$ in $Ω^\circ$, $u=0$ on $\partialΩ$, where $Ω^\circ$ and $\partialΩ$ denote the interior and the boundary of $Ω$ respectively. Also we consider similar problems involving the $p$-Laplacian and poly-Laplacian by the same method. Such problems can be viewed as discrete versions of the Yamabe type equations on Euclidean space or compact Riemannian manifolds.

math.AP↗
arXiv2016

Kazdan-Warner equation on graph

Let $G=(V,E)$ be a finite graph and $Δ$ be the usual graph Laplacian. Using the calculus of variations and a method of upper and lower solutions, we give various conditions such that the Kazdan-Warner equation $Δu=c-he^u$ has a solution on $V$, where $c$ is a constant, and $h:V\rightarrow\mathbb{R}$ is a function. We also consider similar equations involving higher order derivatives on graph. Our results can be compared with the original manifold case of Kazdan-Warner (Ann. Math., 1974).

math.AP↗
arXiv2016

Existence of positive solutions to some nonlinear equations on locally finite graphs

Let $G=(V,E)$ be a locally finite graph, whose measure $μ(x)$ have positive lower bound, and $Δ$ be the usual graph Laplacian. Applying the mountain-pass theorem due to Ambrosetti-Rabinowitz, we establish existence results for some nonlinear equations, namely $Δu+hu=f(x,u)$, $x\in V$. In particular, we prove that if $h$ and $f$ satisfy certain assumptions, then the above mentioned equation has strictly positive solutions. Also, we consider existence of positive solutions of the perturbed equation $Δu+hu=f(x,u)+εg$. Similar problems have been extensively studied on the Euclidean space as well as on Riemannian manifolds.

math.AP↗
arXiv2016

Multifractality and Laplace spectrum of horizontal visibility graphs constructed from fractional Brownian motions

Many studies have shown that additional information can be gained on time series by investigating their associated complex networks. In this work, we investigate the multifractal property and Laplace spectrum of the horizontal visibility graphs (HVGs) constructed from fractional Brownian motions. We aim to identify via simulation and curve fitting the form of these properties in terms of the Hurst index $H$. First, we use the sandbox algorithm to study the multifractality of these HVGs. It is found that multifractality exists in these HVGs. We find that the average fractal dimension $\langle D(0)\rangle$ of HVGs approximately satisfies the prominent linear formula $\langle D(0)\rangle = 2 - H$; while the average information dimension $\langle D(1)\rangle$ and average correlation dimension $\langle D(2)\rangle$ are all approximately bi-linear functions of $H$ when $H\ge 0.15$. Then, we calculate the spectrum and energy for the general Laplacian operator and normalized Laplacian operator of these HVGs. We find that, for the general Laplacian operator, the average logarithm of second-smallest eigenvalue $\langle \ln (u_2) \rangle$, the average logarithm of third-smallest eigenvalue $\langle \ln (u_3) \rangle$, and the average logarithm of maximum eigenvalue $\langle \ln (u_n) \rangle$ of these HVGs are approximately linear functions of $H$; while the average Laplacian energy $\langle E_{nL} \rangle$ is approximately a quadratic polynomial function of $H$. For the normalized Laplacian operator, $\langle \ln (u_2) \rangle$ and $\langle \ln (u_3) \rangle$ of these HVGs approximately satisfy linear functions of $H$; while $\langle \ln (u_n) \rangle$ and $\langle E_{nL} \rangle$ are approximately a 4th and cubic polynomial function of $H$ respectively.

cond-mat.stat-mech↗
arXiv2016

Programmable Restoration Granularity in Constraint Programming

In most constraint programming systems, a limited number of search engines is offered while the programming of user-customized search algorithms requires low-level efforts, which complicates the deployment of such algorithms. To alleviate this limitation, concepts such as computation spaces have been developed. Computation spaces provide a coarse-grained restoration mechanism, because they store all information contained in a search tree node. Other granularities are possible, and in this paper we make the case for dynamically adapting the restoration granularity during search. In order to elucidate programmable restoration granularity, we present restoration as an aspect of a constraint programming system, using the model of aspect-oriented programming. A proof-of-concept implementation using Gecode shows promising results.

cs.PL↗
arXiv2016

Recollection: an Alternative Restoration Technique for Constraint Programming Systems

Search is a key service within constraint programming systems, and it demands the restoration of previously accessed states during the exploration of a search tree. Restoration proceeds either bottom-up within the tree to roll back previously performed operations using a trail, or top-down to redo them, starting from a previously stored state and using suitable information stored along the way. In this paper, we elucidate existing restoration techniques using a pair of abstract methods and employ them to present a new technique that we call recollection. The proposed technique stores the variables that were affected by constraint propagation during fix points reasoning steps, and it conducts neither operation roll-back nor recomputation, while consuming much less memory than storing previous visited states. We implemented this idea as a prototype within the Gecode solver. An empirical evaluation reveals that constraint problems with expensive propagation and frequent failures can benefit from recollection with respect to runtime at the expense of a marginal increase in memory consumption, comparing with the most competitive variant of recomputation.

cs.PL↗
arXiv2015

Properties for CD Inequalities with Unbounded Laplacians

The CD equalities were introduced to imply the gradient estimate of laplace operator on graphs. This article is based on the unbounded Laplacians, and finally concludes some equivalent properties of the CD(K,$\infty$)and CD(K,n).

math.CO↗
arXiv2015

Volume doubling, Poincaré inequality and Guassian heat kernel estimate for nonnegative curvature graphs

By studying the heat semigroup, we prove Li-Yau type estimates for bounded and positive solutions of the heat equation on graphs, under the assumption of the curvature-dimension inequality $CDE'(n,0)$, which can be consider as a notion of curvature for graphs. Furthermore, we derive that if a graph has non-negative curvature then it has the volume doubling property, from this we can prove the Gaussian estimate for heat kernel, and then Poincaré inequality and Harnack inequality. As a consequence, we obtain that the dimension of space of harmonic functions on graphs with polynomial growth is finite, which original is a conjecture of Yau on Riemannian manifold proved by Colding and Minicozzi. Under the assumption of positive curvature on graphs, we derive the Bonnet-Myers type theorem that the diameter of graphs is finite and bounded above in terms of the positive curvature by proving some Log Sobolev inequalities.

math.DG↗