skip to main content

This content will become publicly available on November 6, 2024

Title: Singular Value Approximation and Sparsifying Random Walks on Directed Graphs
In this paper, we introduce a new, spectral notion of approximation between directed graphs, which we call singular value (SV) approximation. SV-approximation is stronger than previous notions of spectral approximation considered in the literature, including spectral approximation of Laplacians for undirected graphs [ST04], standard approximation for directed graphs [CKP + 17], and unit-circle (UC) approximation for directed graphs [AKM + 20]. Further, SV approximation enjoys several useful properties not possessed by previous notions of approximation, e.g., it is preserved under products of randomwalk matrices and bounded matrices. We provide a nearly linear-time algorithm for SV-sparsifying (and hence UC-sparsifying) Eulerian directed graphs, as well as ℓ-step random walks on such graphs, for any ℓ≤poly(n). Combined with the Eulerian scaling algorithms of [CKK + 18], given an arbitrary (not necessarily Eulerian) directed graph and a set S of vertices, we can approximate the stationary probability mass of the (S,Sc) cut in an ℓ-step random walk to within a multiplicative error of 1/polylog(n) and an additive error of 1/poly(n) in nearly linear time. As a starting point for these results, we provide a simple black-box reduction from SV-sparsifying Eulerian directed graphs to SV-sparsifying undirected graphs; such a directed-to-undirected reduction was not known for previous notions of spectral approximation.  more » « less
Award ID(s):
Author(s) / Creator(s):
; ; ;
Publisher / Repository:
64th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2023
Date Published:
Journal Name:
64th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2023
Page Range / eLocation ID:
Medium: X
Santa Cruz, California
Sponsoring Org:
National Science Foundation
More Like this
  1. We introduce a notion called entropic independence that is an entropic analog of spectral notions of high-dimensional expansion. Informally, entropic independence of a background distribution $\mu$ on $k$-sized subsets of a ground set of elements says that for any (possibly randomly chosen) set $S$, the relative entropy of a single element of $S$ drawn uniformly at random carries at most $O(1/k)$ fraction of the relative entropy of $S$. Entropic independence is the analog of the notion of spectral independence, if one replaces variance by entropy. We use entropic independence to derive tight mixing time bounds, overcoming the lossy nature of spectral analysis of Markov chains on exponential-sized state spaces. In our main technical result, we show a general way of deriving entropy contraction, a.k.a. modified log-Sobolev inequalities, for down-up random walks from spectral notions. We show that spectral independence of a distribution under arbitrary external fields automatically implies entropic independence. We furthermore extend our theory to the case where spectral independence does not hold under arbitrary external fields. To do this, we introduce a framework for obtaining tight mixing time bounds for Markov chains based on what we call restricted modified log-Sobolev inequalities, which guarantee entropy contraction not for all distributions, but for those in a sufficiently large neighborhood of the stationary distribution. To derive our results, we relate entropic independence to properties of polynomials: $\mu$ is entropically independent exactly when a transformed version of the generating polynomial of $\mu$ is upper bounded by its linear tangent; this property is implied by concavity of the said transformation, which was shown by prior work to be locally equivalent to spectral independence. We apply our results to obtain (1) tight modified log-Sobolev inequalities and mixing times for multi-step down-up walks on fractionally log-concave distributions, (2) the tight mixing time of $O(n\log n)$ for Glauber dynamics on Ising models whose interaction matrix has eigenspectrum lying within an interval of length smaller than $1$, improving upon the prior quadratic dependence on $n$, and (3) nearly-linear time $\widetilde O_{\delta}(n)$ samplers for the hardcore and Ising models on $n$-node graphs that have $\delta$-relative gap to the tree-uniqueness threshold. In the last application, our bound on the running time does not depend on the maximum degree $\Delta$ of the graph, and is therefore optimal even for high-degree graphs, and in fact, is sublinear in the size of the graph for high-degree graphs. 
    more » « less
  2. null (Ed.)
    The problem of sparsifying a graph or a hypergraph while approximately preserving its cut structure has been extensively studied and has many applications. In a seminal work, Benczúr and Karger (1996) showed that given any n-vertex undirected weighted graph G and a parameter ε ∈ (0,1), there is a near-linear time algorithm that outputs a weighted subgraph G' of G of size Õ(n/ε²) such that the weight of every cut in G is preserved to within a (1 ± ε)-factor in G'. The graph G' is referred to as a (1 ± ε)-approximate cut sparsifier of G. Subsequent recent work has obtained a similar result for the more general problem of hypergraph cut sparsifiers. However, all known sparsification algorithms require Ω(n + m) time where n denotes the number of vertices and m denotes the number of hyperedges in the hypergraph. Since m can be exponentially large in n, a natural question is if it is possible to create a hypergraph cut sparsifier in time polynomial in n, independent of the number of edges. We resolve this question in the affirmative, giving the first sublinear time algorithm for this problem, given appropriate query access to the hypergraph. Specifically, we design an algorithm that constructs a (1 ± ε)-approximate cut sparsifier of a hypergraph H(V,E) in polynomial time in n, independent of the number of hyperedges, when given access to the hypergraph using the following two queries: 1) given any cut (S, ̄S), return the size |δ_E(S)| (cut value queries); and 2) given any cut (S, ̄S), return a uniformly at random edge crossing the cut (cut edge sample queries). Our algorithm outputs a sparsifier with Õ(n/ε²) edges, which is essentially optimal. We then extend our results to show that cut value and cut edge sample queries can also be used to construct hypergraph spectral sparsifiers in poly(n) time, independent of the number of hyperedges. We complement the algorithmic results above by showing that any algorithm that has access to only one of the above two types of queries can not give a hypergraph cut sparsifier in time that is polynomial in n. Finally, we show that our algorithmic results also hold if we replace the cut edge sample queries with a pair neighbor sample query that for any pair of vertices, returns a random edge incident on them. In contrast, we show that having access only to cut value queries and queries that return a random edge incident on a given single vertex, is not sufficient. 
    more » « less
  3. null (Ed.)
    Consider an algorithm performing a computation on a huge random object (for example a random graph or a "long" random walk). Is it necessary to generate the entire object prior to the computation, or is it possible to provide query access to the object and sample it incrementally "on-the-fly" (as requested by the algorithm)? Such an implementation should emulate the random object by answering queries in a manner consistent with an instance of the random object sampled from the true distribution (or close to it). This paradigm is useful when the algorithm is sub-linear and thus, sampling the entire object up front would ruin its efficiency. Our first set of results focus on undirected graphs with independent edge probabilities, i.e. each edge is chosen as an independent Bernoulli random variable. We provide a general implementation for this model under certain assumptions. Then, we use this to obtain the first efficient local implementations for the Erdös-Rényi G(n,p) model for all values of p, and the Stochastic Block model. As in previous local-access implementations for random graphs, we support Vertex-Pair and Next-Neighbor queries. In addition, we introduce a new Random-Neighbor query. Next, we give the first local-access implementation for All-Neighbors queries in the (sparse and directed) Kleinberg’s Small-World model. Our implementations require no pre-processing time, and answer each query using O(poly(log n)) time, random bits, and additional space. Next, we show how to implement random Catalan objects, specifically focusing on Dyck paths (balanced random walks on the integer line that are always non-negative). Here, we support Height queries to find the location of the walk, and First-Return queries to find the time when the walk returns to a specified location. This in turn can be used to implement Next-Neighbor queries on random rooted ordered trees, and Matching-Bracket queries on random well bracketed expressions (the Dyck language). Finally, we introduce two features to define a new model that: (1) allows multiple independent (and even simultaneous) instantiations of the same implementation, to be consistent with each other without the need for communication, (2) allows us to generate a richer class of random objects that do not have a succinct description. Specifically, we study uniformly random valid q-colorings of an input graph G with maximum degree Δ. This is in contrast to prior work in the area, where the relevant random objects are defined as a distribution with O(1) parameters (for example, n and p in the G(n,p) model). The distribution over valid colorings is instead specified via a "huge" input (the underlying graph G), that is far too large to be read by a sub-linear time algorithm. Instead, our implementation accesses G through local neighborhood probes, and is able to answer queries to the color of any given vertex in sub-linear time for q ≥ 9Δ, in a manner that is consistent with a specific random valid coloring of G. Furthermore, the implementation is memory-less, and can maintain consistency with non-communicating copies of itself. 
    more » « less
  4. We present an algorithm that, with high probability, generates a random spanning tree from an edge-weighted undirected graph in \Otil(n^{5/3 }m^{1/3}) time\footnote{The \Otil(\cdot) notation hides \poly(\log n) factors}. The tree is sampled from a distribution where the probability of each tree is proportional to the product of its edge weights. This improves upon the previous best algorithm due to Colbourn et al. that runs in matrix multiplication time, O(n^\omega). For the special case of unweighted graphs, this improves upon the best previously known running time of \tilde{O}(\min\{n^{\omega},m\sqrt{n},m^{4/3}\}) for m >> n^{7/4} (Colbourn et al. '96, Kelner-Madry '09, Madry et al. '15). The effective resistance metric is essential to our algorithm, as in the work of Madry et al., but we eschew determinant-based and random walk-based techniques used by previous algorithms. Instead, our algorithm is based on Gaussian elimination, and the fact that effective resistance is preserved in the graph resulting from eliminating a subset of vertices (called a Schur complement). As part of our algorithm, we show how to compute \eps-approximate effective resistances for a set SS of vertex pairs via approximate Schur complements in \Otil(m+(n + |S|)\eps^{-2}) time, without using the Johnson-Lindenstrauss lemma which requires \Otil( \min\{(m + |S|)\eps^{-2}, m+n\eps^{-4} +|S|\eps^{-2}\}) time. We combine this approximation procedure with an error correction procedure for handing edges where our estimate isn't sufficiently accurate. 
    more » « less
  5. We show variants of spectral sparsification routines can preserve the total spanning tree counts of graphs, which by Kirchhoff's matrix-tree theorem, is equivalent to determinant of a graph Laplacian minor, or equivalently, of any SDDM matrix. Our analyses utilizes this combinatorial connection to bridge between statistical leverage scores / effective resistances and the analysis of random graphs by [Janson, Combinatorics, Probability and Computing `94]. This leads to a routine that in quadratic time, sparsifies a graph down to about $n^{1.5}$ edges in ways that preserve both the determinant and the distribution of spanning trees (provided the sparsified graph is viewed as a random object). Extending this algorithm to work with Schur complements and approximate Choleksy factorizations leads to algorithms for counting and sampling spanning trees which are nearly optimal for dense graphs. We give an algorithm that computes a $(1 \pm \delta)$ approximation to the determinant of any SDDM matrix with constant probability in about $n^2 \delta^{-2}$ time. This is the first routine for graphs that outperforms general-purpose routines for computing determinants of arbitrary matrices. We also give an algorithm that generates in about $n^2 \delta^{-2}$ time a spanning tree of a weighted undirected graph from a distribution with total variation distance of $\delta$ from the $w$-uniform distribution . 
    more » « less