In this paper we study variants of the widely used spectral clustering that partitions a graph into k clusters by (1) embedding the vertices of a graph into a low-dimensional space using the bottom eigenvectors of the Laplacian matrix, and (2) grouping the embedded points into k clusters via k-means algorithms. We show that, for a wide class of graphs, spectral clustering gives a good approximation of the optimal clustering. While this approach was proposed in the early 1990s and has comprehensive applications, prior to our work similar results were known only for graphs generated from stochastic models.We also give a nearly-linear time algorithm for partitioning well-clustered graphs based on computing a matrix exponential and approximate nearest neighbor data structures.

For any undirected and weighted graph G = (V, E, w) with n vertices and m edges, we call a sparse subgraph H of G, with proper reweighting of the edges, a (1 + ε)-spectral sparsifier ifholds for any x ∈ R n , where L G and L H are the respective Laplacian matrices of G and H. Noticing that Ω(m) time is needed for any algorithm to construct a spectral sparsifier and a spectral sparsifier of G requires Ω(n) edges, a natural question is to investigate, for any constant ε, if a (1 + ε)-spectral sparsifier of G with O(n) edges can be constructed in O(m) time, where the O notation suppresses polylogarithmic factors. All previous constructions on spectral sparsification [ST11, SS11, BSS12, Zou12, ALO15, LS15] require either super-linear number of edges or m 1+Ω(1) time.In this work we answer this question affirmatively by presenting an algorithm that, for any undirected graph G and ε > 0, outputs a (1 + ε)-spectral sparsifier of G with O(n/ε 2 ) edges in O(m/ε O(1) ) time. Our algorithm is based on three novel techniques: (1) a new potential function which is much easier to compute yet has similar guarantees as the potential functions used in previous references; (2) an efficient reduction from a two-sided spectral sparsifier to a one-sided spectral sparsifier; (3) constructing a one-sided spectral sparsifier by a semi-definite program.

We present the first almost-linear time algorithm for constructing linear-sized spectral sparsification for graphs. This improves all previous constructions of linear-sized spectral sparsification, which requires Ω(n 2 ) time [BSS12, Zou12, AZLO15]. A key ingredient in our algorithm is a novel combination of two techniques used in literature for constructing spectral sparsification: Random sampling by effective resistance [SS11], and adaptive construction based on barrier functions [BSS12, AZLO15].

We consider the problem of balancing load items (tokens) in networks. Starting with an arbitrary load distribution, we allow nodes to exchange tokens with their neighbors in each round. The goal is to achieve a distribution where all nodes have nearly the same number of tokens.For the continuous case where tokens are arbitrarily divisible, most load balancing schemes correspond to Markov chains, whose convergence is fairly well-understood in terms of their spectral gap. However, in many applications, load items cannot be divided arbitrarily, and we need to deal with the discrete case where the load is composed of indivisible tokens. This discretization entails a non-linear behavior due to its rounding errors, which makes this analysis much harder than in the continuous case.We investigate several randomized protocols for different communication models in the discrete case. As our main result, we prove that for any regular network in the matching model, all nodes have the same load up to an additive constant in (asymptotically) the same number of rounds as required in the continuous case. This generalizes and tightens the previous best result, which only holds for expander graphs [17], and demonstrates that there is almost no difference between the discrete and continuous cases. Our results also provide a positive answer to the question of how well discrete load balancing can be approximated by (continuous) Markov chains, which has been posed by many researchers, e.g., [23,29,32]. * A preliminary version of this work appeared in