A queueing network-based distributed Laplacian solver for directed graphs

Highlights

• First distributed Laplacian solver analyzed in terms of the underlying directed graph.

• New approach to solve Laplacians using the theory of queueing networks, Markov chains.

• Works for systems $x^{T}L=b^T$ where only one coordinate of b is negative like electrical flow.

• Uses equivalence of Laplacian system to steady-state equation of a stochastic process.

• Running time of $\tilde{O}(n)$ for random digraphs.

Abstract

We present a distributed algorithm for solving Laplacian systems on strongly connected directed graphs, the first that can be analyzed in terms of the underlying graph's parameters. Our distributed solver works for a large and important class of Laplacian systems that we call “one-sink” Laplacian systems, which includes the important electrical flow computation problem. Specifically, given $D_{\text{out}}$ as the diagonal out-degree matrix and $\overrightarrow{A}$ as the adjacency matrix of the directed graph with $L=D_{\text{out}}-\overrightarrow{A}$, our solver can produce solutions for systems of the form ${\mathit{x}}^{T}L={\mathit{b}}^T$, where exactly one of the coordinates of b is negative. Our solver takes $\tilde{O}(t_{\text{hit}}{d}_{\max }^{\text{in}})$ time (where $\tilde{O}$ hides $\text{poly}\log n$ factors) to produce an approximate solution where $t_{\text{hit}}$ is the worst-case hitting time of the random walk on the graph, which is $\mathrm{\Theta }(n)$ for a large set of important graphs and $d_{\max }^{\text{in}}$ is the maximum in-degree of the graph.

Introduction

Solving a Laplacian system of equations forms a core routine for a number of fields, including network analysis, computer vision, operations research, and machine learning. Following Spielman and Teng's [1] pioneering work proposing a quasi-linear time algorithm for solving undirected Laplacians, a powerful collection of algorithmic tools have been developed for the undirected case. On the other hand, similar developments have remained elusive for the directed graphs. In a recent line of work, Cohen et al. [2], [3] extensively studied directed Laplacians and gave a new class of nearly-linear time algorithms for such asymmetric linear systems. However, all these solvers are centralized. Recently, Tang and Mei [4] studied a consensus-based distributed algorithm for the directed Laplacians, which they show converges to a solution at a geometric rate. However, they do not characterize this rate in terms of the underlying graph parameters. Thus, to the best of our knowledge currently there is no such algorithm for directed graphs. In this work, we endeavor to fill that gap by describing a simple distributed algorithm to solve a large and useful class of Laplacian systems that includes the electrical flow computation, which in turn is used as a primitive for computing max flows [5], random spanning trees [6], graph sparsification [7], and computation of current flow closeness centrality [8]. Also, our method is a completely new approach for solving Laplacian systems based on the theory of queueing networks, multidimensional Markov chains, and random walks.

In particular, we first formulate a stochastic problem that captures the properties of Laplacian systems of the form ${\mathit{x}}^{T}L={\mathit{b}}^T$ with a constraint that only one element in b is negative, where $L=D_{\text{out}}-\overrightarrow{A}$ such that $D_{\text{out}}$ is the diagonal out-degree matrix and $\overrightarrow{A}$ is the adjacency matrix of the directed graph. We call such systems “one-sink” Laplacian systems since the stochastic process can be viewed as a network in which some nodes are generating data, and there is a single sink that collects all the packets that it receives. We call our stochastic process the “Data Collection Process” and show that this process has an equivalence to the one-sink Laplacian system at stationarity. The main challenge while handling the directed graphs is the computation of the stationary distribution of its corresponding Markov chain. Our proposed solver works by quickly approximating the stationary distribution of a multidimensional Markov chain induced by the queueing network of the Data Collection Process. We show that when this multidimensional chain is ergodic the vector whose vth coordinate is proportional to the probability at stationarity of the queue at v being non-empty is a solution to ${\mathit{x}}^{T}L={\mathit{b}}^T$. We estimate this solution in a natural way by statistically determining the proportion of time slots for which the queues are empty. A fast-mixing property of the multidimensional chain allows us to achieve the running time result.

Furthermore, we consider a synchronous model of communication such that in each round all nodes in parallel can send a message of size $O(\log n)$ to one of its neighbors and perform local computation on messages received in previous rounds from its neighbors. Under this model, our solver takes $\tilde{O}(t_{\text{hit}})$ distributed rounds to solve one-sink Laplacian systems. However, since under this model each node can get at most $d_{\max }^{\text{in}}$ messages from its neighbors, where $d_{\max }^{\text{in}}$ is the maximum in-degree of the graph, so simulating each round in worst-case would require $d_{\max }^{\text{in}}$ time. In Table 1, we present the running time of our solver for various graphs where we add a factor of $d_{\max }^{\text{in}}$ to the distributed round complexity to account for multiple reception by a node in each round. This will not affect the constant degree graphs while giving an extra $O(n)$ for graphs whose in-degree grows with the number of vertices.