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Fan Chung

Publications and source records attributed to Fan Chung.

At least 19 recordsLinked to original sources
arXiv2026

Optimal Transport on Graphs and Stochastically Evolving Trees

We give an effective algorithm for determining the transportation distance between two given probability density functions defined on the vertices of a graph $G=(V,E)$ by analyzing an associated polytope. The vertices of the polytope correspond to feasible flows on spanning trees in $G$, and the $1$-skeleton of the polytope is a projection of the spanning tree state graph associated with the Glauber dynamics on $G$. The optimal value of this transportation problem, known as the $1$-Wasserstein distance, can be computed by tracing the transportation cost along the vertices of this polytope. We show that a local minimum of the transportation cost is also a global minimum, and this leads to a steepest descent algorithm for solving the transportation problem. If the probability density functions take discrete values in $\delta \mathbb{Z}$ for some $\delta>0$, then the optimal transport cost can be reached in at most $\frac{|V|-1}{\delta}$ steps. As an application, we give an efficient algorithm for computing the Ollivier--Ricci curvature of a graph.

math.CO↗
arXiv2025

Stochastically Evolving Graphs via Edit Semigroups

We investigate a randomly evolving process of subgraphs in an underlying host graph using the spectral theory of semigroups related to the Tsetlin library and hyperplane arrangements. Starting with some initial subgraph, at each iteration, we apply a randomly selected edit to the current subgraph. Such edits vary in nature from simple edits consisting of adding or deleting an edge, or compound edits which can affect several edges at once. This evolving process generates a random walk on the set of all possible subgraphs of the host graph. We show that the eigenvalues of this random walk can be naturally indexed by subsets of edges of the host graph. We also provide, in the case of simple edits, a closed-form formula for the eigenvectors of the transition probability matrix and a sharp bound for the rate of convergence of this random walk. We consider extensions to the case of compound edits; examples of this model include the previously studied Moran forest model and a dynamic random intersection graph model. Evolving graphs arise in a variety of fields ranging from deep learning and graph neural networks to epidemic modeling and social networks. Our random evolving process serves as a general stochastic model for sampling random subgraphs from a given graph.

math.CO↗
arXiv2022

Quasi-Random Influences of Boolean Functions

We examine a hierarchy of equivalence classes of quasi-random properties of Boolean Functions. In particular, we prove an equivalence between a number of properties including balanced influences, spectral discrepancy, local strong regularity, homomorphism enumerations of colored or weighted graphs and hypergraphs associated with Boolean functions as well as the $k$th-order strict avalanche criterion amongst others. We further construct families of quasi-random boolean functions which exhibit the properties of our equivalence theorem and separate the levels of our hierarchy.

math.CO↗
arXiv2022

Fan-complete Ramsey numbers

For graphs $G$ and $H$, we consider Ramsey numbers $r(G,H)$ with tight lower bounds, namely, $r(G,H) \geq (\chi(G)-1)(|H|-1)+1,$ where $\chi(G)$ denotes the chromatic number of $G$ and $|H|$ denotes the number of vertices in $H$. We say $H$ is $G$-good if the equality holds. Let $G+H$ be the join graph obtained from graphs $G$ and $H$ by adding all edges between the disjoint vertex sets of $G$ and $H$. Let $nH$ denote the union graph of $n$ disjoint copies of $H$. We show that $K_1+nH$ is $K_p$-good if $n$ is sufficiently large. In particular, the fan-graph $F_n=K_1 + n K_2$ is $K_p$-good if $n\geq 27p^2$, improving previous tower-type lower bounds for $n$ due to Li and Rousseau (1996). Moreover, we give a stronger lower bound inequality for Ramsey number $r(G, K_1+F)$ for the case of $G=K_p(a_1, a_2, \dots, a_p)$, the complete $p$-partite graph with $a_1=1$ and $a_i \leq a_{i+1}$. In particular, using a stability-supersaturation lemma by Fox, He and Wigderson (2021), we show that for any fixed graph $H$, \begin{align*} r(G,K_1+nH) = \left\{ \begin{array}{ll} (p-1)(n |H|+a_2-1)+1 & \textrm{if $n|H|+a_2-1$ is even or $a_2-1$ is even,}\\ (p-1)(n |H|+a_2-2)+1 & \textrm{otherwise,} \end{array} \right. \end{align*} where $G=K_p(1,a_2, \dots, a_p)$ with $a_i$'s satisfying some mild conditions and $n$ is sufficiently large. The special case of $H=K_1$ gives an answer to Burr's question (1981) about the discrepancy of $r(G, K_{1,n})$ from $G$-goodness for sufficiently large $n$. All bounds of $n$ we obtain are not of tower-types.

math.CO↗
arXiv2021

Forest formulas of discrete Green's functions

The discrete Green's functions are the pseudoinverse (or the inverse) of the Laplacian (or its variations) of a graph. In this paper, we will give combinatorial interpretations of Green's functions in terms of enumerating trees and forests in a graph that will be used to derive further formulas for several graph invariants. For example, we show that the trace of the Green's function $\mathbf{G}$ associated with the combinatorial Laplacian of a connected simple graph $\Gamma$ on $n$ vertices satisfies $\text{Tr}(\mathbf{G})=\sum_{\lambda_i \neq 0} \frac 1 {\lambda_i}= \frac{1}{n\tau}|\mathbb{F}^*_2|$, where $\lambda_i$ denotes the eigenvalues of the combinatorial Laplacian, $\tau$ denotes the number of spanning trees and $\mathbb{F}^*_2$ denotes the set of rooted spanning $2$-forests in $\Gamma$. We will prove forest formulas for discrete Green's functions for directed and weighted graphs and apply them to study random walks on graphs and digraphs. We derive a forest expression of the hitting time for digraphs, which gives combinatorial proofs to old and new results about hitting times, traces of discrete Green's functions, and other related quantities.

math.CO↗
arXiv2019

Slow Fibonacci Walks

For a positive integer $n$, we study the number of steps to reach $n$ by a {\it Fibonacci walk} for some starting pair $a_1$ and $a_2$ satisfying the recurrence of $a_{k+2}=a_{k+1}+a_k$. The problem of slow Fibonacci walks, first suggested by Richard Stanley, is to determine the maximum number $s(n)$ of steps for such a Fibonacci walk ending at $n$. Stanley conjectured that for most $n$, there is a slow Fibonacci walk reaching $n = a_s$ with the property that $a_{s+1}$ is the integer closest to $\phi n$ where $\phi=(1+\sqrt{5})/2$. We prove that this is true for only a positive fraction of $n$. We give explicit formulas for the choice of the starting pairs and the determination of $s(n)$ by giving a characterization theorem. We also derive a number of density results concerning the distribution of down and up cases (that is, those $n$ with $a_{s+1}=\lfloor \phi n\rfloor$ or $\lceil \phi n \rceil$, respectively), as well as for more general `paradoxical' cases.

math.NT↗
arXiv2018

The maximum relaxation time of a random walk

We show the minimum spectral gap of the normalized Laplacian over all simple, connected graphs on $n$ vertices is $(1+o(1))\tfrac{54}{n^3}$. This minimum is achieved asymptotically by a double kite graph. Consequently, this leads to sharp upper bounds for the maximum relaxation time of a random walk, settling a conjecture of Aldous and Fill. We also improve an eigenvalue-diameter inequality by giving a new lower bound for the spectral gap of the normalized Laplacian. This eigenvalue lower bound is asymptotically best possible.

math.CO↗
arXiv2016

Extreme values of the stationary distribution of random walks on directed graphs

We examine the stationary distribution of random walks on directed graphs. In particular, we focus on the {\em principal ratio}, which is the ratio of maximum to minimum values of vertices in the stationary distribution. We give an upper bound for this ratio over all strongly connected graphs on $n$ vertices. We characterize all graphs achieving the upper bound and we give explicit constructions for these extremal graphs. Additionally, we show that under certain conditions, the principal ratio is tightly bounded. We also provide counterexamples to show the principal ratio cannot be tightly bounded under weaker conditions.

math.CO↗
arXiv2015

Distributed Algorithms for Finding Local Clusters Using Heat Kernel Pagerank

A distributed algorithm performs local computations on pieces of input and communicates the results through given communication links. When processing a massive graph in a distributed algorithm, local outputs must be configured as a solution to a graph problem without shared memory and with few rounds of communication. In this paper we consider the problem of computing a local cluster in a massive graph in the distributed setting. Computing local clusters are of certain application-specific interests, such as detecting communities in social networks or groups of interacting proteins in biological networks. When the graph models the computer network itself, detecting local clusters can help to prevent communication bottlenecks. We give a distributed algorithm that computes a local cluster in time that depends only logarithmically on the size of the graph in the CONGEST model. In particular, when the value of the optimal local cluster is known, the algorithm runs in time entirely independent of the size of the graph and depends only on error bounds for approximation. We also show that the local cluster problem can be computed in the k-machine distributed model in sublinear time. The speedup of our local cluster algorithms is mainly due to the use of our distributed algorithm for heat kernel pagerank.

cs.DC↗
arXiv2015

Finding Consensus in Multi-Agent Networks Using Heat Kernel Pagerank

We present a new and efficient algorithm for determining a consensus value for a network of agents. Different from existing algorithms, our algorithm evaluates the consensus value for very large networks using heat kernel pagerank. We consider two frameworks for the consensus problem, a weighted average consensus among all agents, and consensus in a leader-following formation. Using a heat kernel pagerank approximation, we give consensus algorithms that run in time sublinear in the size of the network, and provide quantitative analysis of the tradeoff between performance guarantees and error estimates.

eess.SY↗
arXiv2015

Juggling card sequences

Juggling patterns can be described by a sequence of cards which keep track of the relative order of the balls at each step. This interpretation has many algebraic and combinatorial properties, with connections to Stirling numbers, Dyck paths, Narayana numbers, boson normal ordering, arc-labeled digraphs, and more. Some of these connections are investigated with a particular focus on enumerating juggling patterns satisfying certain ordering constraints, including where the number of crossings is fixed.

math.CO↗
arXiv2015

Computing Heat Kernel Pagerank and a Local Clustering Algorithm

Heat kernel pagerank is a variation of Personalized PageRank given in an exponential formulation. In this work, we present a sublinear time algorithm for approximating the heat kernel pagerank of a graph. The algorithm works by simulating random walks of bounded length and runs in time $O\big(\frac{\log(\epsilon^{-1})\log n}{\epsilon^3\log\log(\epsilon^{-1})}\big)$, assuming performing a random walk step and sampling from a distribution with bounded support take constant time. The quantitative ranking of vertices obtained with heat kernel pagerank can be used for local clustering algorithms. We present an efficient local clustering algorithm that finds cuts by performing a sweep over a heat kernel pagerank vector, using the heat kernel pagerank approximation algorithm as a subroutine. Specifically, we show that for a subset $S$ of Cheeger ratio $\phi$, many vertices in $S$ may serve as seeds for a heat kernel pagerank vector which will find a cut of conductance $O(\sqrt{\phi})$.

cs.DS↗
arXiv2015

Solving Local Linear Systems with Boundary Conditions Using Heat Kernel Pagerank

We present an efficient algorithm for solving local linear systems with a boundary condition using the Green's function of a connected induced subgraph related to the system. We introduce the method of using the Dirichlet heat kernel pagerank vector to approximate local solutions to linear systems in the graph Laplacian satisfying given boundary conditions over a particular subset of vertices. With an efficient algorithm for approximating Dirichlet heat kernel pagerank, our local linear solver algorithm computes an approximate local solution with multiplicative and additive error $\epsilon$ by performing $O(\epsilon^{-5}s^3\log(s^3\epsilon^{-1})\log n)$ random walk steps, where $n$ is the number of vertices in the full graph and $s$ is the size of the local system on the induced subgraph.

cs.DS↗
arXiv2014

Decomposition of random graphs into complete bipartite graphs

We consider the problem of partitioning the edge set of a graph $G$ into the minimum number $\tau(G)$ of edge-disjoint complete bipartite subgraphs. We show that for a random graph $G$ in $G(n,p)$, for $p$ is a constant no greater than $1/2$, almost surely $\tau(G)$ is between $n- c(\ln_{1/p} n)^{3+\epsilon}$ and $n - 2\ln_{1/(1-p)} n$ for any positive constants $c$ and $\epsilon$.

math.CO↗
arXiv2013

From Quasirandom graphs to Graph Limits and Graphlets

We generalize the notion of quasirandom which concerns a class of equivalent properties that random graphs satisfy. We show that the convergence of a graph sequence under the spectral distance is equivalent to the convergence using the (normalized) cut distance. The resulting graph limit is called graphlets. We then consider several families of graphlets and, in particular, we characterize graphlets with low ranks for both dense and sparse graphs.

math.CO↗
arXiv2010

Descent polynomials for permutations with bounded drop size

Motivated by juggling sequences and bubble sort, we examine permutations on the set {1,2,...,n} with d descents and maximum drop size k. We give explicit formulas for enumerating such permutations for given integers k and d. We also derive the related generating functions and prove unimodality and symmetry of the coefficients.

math.CO↗
arXiv2006

The Workshop on Internet Topology (WIT) Report

Internet topology analysis has recently experienced a surge of interest in computer science, physics, and the mathematical sciences. However, researchers from these different disciplines tend to approach the same problem from different angles. As a result, the field of Internet topology analysis and modeling must untangle sets of inconsistent findings, conflicting claims, and contradicting statements. On May 10-12, 2006, CAIDA hosted the Workshop on Internet topology (WIT). By bringing together a group of researchers spanning the areas of computer science, physics, and the mathematical sciences, the workshop aimed to improve communication across these scientific disciplines, enable interdisciplinary crossfertilization, identify commonalities in the different approaches, promote synergy where it exists, and utilize the richness that results from exploring similar problems from multiple perspectives. This report describes the findings of the workshop, outlines a set of relevant open research problems identified by participants, and concludes with recommendations that can benefit all scientific communities interested in Internet topology research.

cs.NI↗