Spectrally approximating large graphs with smaller graphs

How does coarsening affect the spectrum of a general graph? We provide conditions such that the principal eigenvalues and eigenspaces of a coarsened and original graph Laplacian matrices are close. The achieved approximation is shown to depend on standard graph-theoretic properties, such as the degree and eigenvalue distributions, as well as on the ratio between the coarsened and actual graph sizes. Our results carry implications for learning methods that utilize coarsening. For the particular case of spectral clustering, they imply that coarse eigenvectors can be used to derive good quality assignments even without refinement---this phenomenon was previously observed, but lacked formal justification.Comment: 22 pages, 10 figure

A Novel Differentiable Loss Function for Unsupervised Graph Neural Networks in Graph Partitioning

In this paper, we explore the graph partitioning problem, a pivotal combina-torial optimization challenge with extensive applications in various fields such as science, technology, and business. Recognized as an NP-hard prob-lem, graph partitioning lacks polynomial-time algorithms for its resolution. Recently, there has been a burgeoning interest in leveraging machine learn-ing, particularly approaches like supervised, unsupervised, and reinforce-ment learning, to tackle such NP-hard problems. However, these methods face significant hurdles: supervised learning is constrained by the necessity of labeled solution instances, which are often computationally impractical to obtain; reinforcement learning grapples with instability in the learning pro-cess; and unsupervised learning contends with the absence of a differentia-ble loss function, a consequence of the discrete nature of most combinatorial optimization problems. Addressing these challenges, our research introduces a novel pipeline employing an unsupervised graph neural network to solve the graph partitioning problem. The core innovation of this study is the for-mulation of a differentiable loss function tailored for this purpose. We rigor-ously evaluate our methodology against contemporary state-of-the-art tech-niques, focusing on metrics: cuts and balance, and our findings reveal that our is competitive with these leading methods.Comment: 2 Tables, 2 Figure

Relaxation-Based Coarsening for Multilevel Hypergraph Partitioning

Multilevel partitioning methods that are inspired by principles of multiscaling are the most powerful practical hypergraph partitioning solvers. Hypergraph partitioning has many applications in disciplines ranging from scientific computing to data science. In this paper we introduce the concept of algebraic distance on hypergraphs and demonstrate its use as an algorithmic component in the coarsening stage of multilevel hypergraph partitioning solvers. The algebraic distance is a vertex distance measure that extends hyperedge weights for capturing the local connectivity of vertices which is critical for hypergraph coarsening schemes. The practical effectiveness of the proposed measure and corresponding coarsening scheme is demonstrated through extensive computational experiments on a diverse set of problems. Finally, we propose a benchmark of hypergraph partitioning problems to compare the quality of other solvers

Scheduling Storms and Streams in the Cloud

Motivated by emerging big streaming data processing paradigms (e.g., Twitter Storm, Streaming MapReduce), we investigate the problem of scheduling graphs over a large cluster of servers. Each graph is a job, where nodes represent compute tasks and edges indicate data-flows between these compute tasks. Jobs (graphs) arrive randomly over time, and upon completion, leave the system. When a job arrives, the scheduler needs to partition the graph and distribute it over the servers to satisfy load balancing and cost considerations. Specifically, neighboring compute tasks in the graph that are mapped to different servers incur load on the network; thus a mapping of the jobs among the servers incurs a cost that is proportional to the number of "broken edges". We propose a low complexity randomized scheduling algorithm that, without service preemptions, stabilizes the system with graph arrivals/departures; more importantly, it allows a smooth trade-off between minimizing average partitioning cost and average queue lengths. Interestingly, to avoid service preemptions, our approach does not rely on a Gibbs sampler; instead, we show that the corresponding limiting invariant measure has an interpretation stemming from a loss system.Comment: 14 page