Understanding the local structure of a graph provides valuable insights about the underlying phenomena from which the graph has originated. Sampling and examining k-subgraphs is a widely used approach to understand the local structure of a graph. In this paper, we study the problem of sampling uniformly k-subgraphs from a given graph. We analyze a few different Markov chain Monte Carlo (MCMC) approaches, and obtain analytical results on their mixing times, which improve significantly the state of the art. In particular, we improve the bound on the mixing times of the standard MCMC approach, and the state-of-the-art MCMC sampling method PSRW, using the canonical-paths argument. In addition, we propose a novel sampling method, which we call recursive subgraph sampling, RSS, and its optimized variant RSS+. The proposed methods, RSS and RSS+, are significantly faster than existing approaches.
We study the following problem: given an integer $k \ge 3$ and a simple graph $G$, sample a connected induced $k$-node subgraph of $G$ uniformly at random. This is a fundamental graph mining primitive with applications in social network analysis, bioinformatics, and more. Surprisingly, no efficient algorithm is known for uniform sampling; the only somewhat efficient algorithms available yield samples that are only approximately uniform, with running times that are unclear or suboptimal. In this work we provide: (i) a near-optimal mixing time bound for a well-known random walk technique, (ii) t…
Counting the frequency of small subgraphs is a fundamental technique in network analysis across various domains, most notably in bioinformatics and social networks. The special case of triangle counting has received much attention. Getting results for 4-vertex patterns is highly challenging, and there are few practical results known that can scale to massive sizes. Indeed, even a highly tuned enumeration code takes more than a day on a graph with millions of edges. Most previous work that runs for truly massive graphs employ clusters and massive parallelization. We provide a sampling algorithm…
Recently [Bhattacharya et al., STOC 2015] provide the first non-trivial algorithm for the densest subgraph problem in the streaming model with additions and deletions to its edges, i.e., for dynamic graph streams. They present a $(0.5-ε)$-approximation algorithm using $\tilde{O}(n)$ space, where factors of $ε$ and $\log(n)$ are suppressed in the $\tilde{O}$ notation. However, the update time of this algorithm is large. To remedy this, they also provide a $(0.25-ε)$-approximation algorithm using $\tilde{O}(n)$ space with update time $\tilde{O}(1)$. In this paper we improve the algorithms by Bha…
We contribute an approach to the problem of locally computing sparse connected subgraphs of dense graphs. In this setting, given an edge in a connected graph $G = (V, E)$, an algorithm locally decides its membership in a sparse connected subgraph $G^* = (V, E^*)$, where $E^* \subseteq E$ and $|E^*| = o(|E|)$. Such an approach to subgraph construction is useful when dealing with massive graphs, where reading in the graph's full network description is impractical. While most prior results in this area require assumptions on $G$ or that $|E'| \le (1+ε)|V|$ for some $ε> 0$, we relax these assumpti…
We present a simple sublinear-time algorithm for sampling an arbitrary subgraph $H$ \emph{exactly uniformly} from a graph $G$ with $m$ edges, to which the algorithm has access by performing the following types of queries: (1) degree queries, (2) neighbor queries, (3) pair queries and (4) edge sampling queries. The query complexity and running time of our algorithm are $\tilde{O}(\min\{m, \frac{m^{ρ(H)}}{\# H}\})$ and $\tilde{O}(\frac{m^{ρ(H)}}{\# H})$, respectively, where $ρ(H)$ is the fractional edge-cover of $H$ and $\# H$ is the number of copies of $H$ in $G$. For any clique on $r$ vertices…
In the subgraph counting problem, we are given a input graph $G(V, E)$ and a target graph $H$; the goal is to estimate the number of occurrences of $H$ in $G$. Our focus here is on designing sublinear-time algorithms for approximately counting occurrences of $H$ in $G$ in the setting where the algorithm is given query access to $G$. This problem has been studied in several recent papers which primarily focused on specific families of graphs $H$ such as triangles, cliques, and stars. However, not much is known about approximate counting of arbitrary graphs $H$. This is in sharp contrast to the …
In the k-arc connected subgraph problem, we are given a directed graph G and an integer k and the goal is the find a subgraph of minimum cost such that there are at least k-arc disjoint paths between any pair of vertices. We give a simple (1 + 1/k)-approximation to the unweighted variant of the problem, where all arcs of G have the same cost. This improves on the 1 + 2/k approximation of Gabow et al. [GGTW09]. Similar to the 2-approximation algorithm for this problem [FJ81], our algorithm simply takes the union of a k in-arborescence and a k out-arborescence. The main difference is in the sele…
We propose Graph Priority Sampling (GPS), a new paradigm for order-based reservoir sampling from massive streams of graph edges. GPS provides a general way to weight edge sampling according to auxiliary and/or size variables so as to accomplish various estimation goals of graph properties. In the context of subgraph counting, we show how edge sampling weights can be chosen so as to minimize the estimation variance of counts of specified sets of subgraphs. In distinction with many prior graph sampling schemes, GPS separates the functions of edge sampling and subgraph estimation. We propose two …
We present a simple algorithm that perfectly samples configurations from the unique Gibbs measure of a spin system on a potentially infinite graph $G$. The sampling algorithm assumes strong spatial mixing together with subexponential growth of $G$. It produces a finite window onto a perfect sample from the Gibbs distribution. The run-time is linear in the size of the window.
We consider the problem of generating uniformly random partitions of the vertex set of a graph such that every piece induces a connected subgraph. For the case where we want to have partitions with linearly many pieces of bounded size, we obtain approximate sampling algorithms based on Glauber dynamics which are fixed-parameter tractable with respect to the bandwidth of $G$, with simple-exponential dependence on the bandwidth. For example, for rectangles of constant or logarithmic width this gives polynomial-time sampling algorithms. More generally, this gives sub-exponential algorithms for bo…
Many real-world networks are prohibitively large for data retrieval, storage and analysis of all of its nodes and links. Understanding the structure and dynamics of these networks entails creating a smaller representative sample of the full graph while preserving its relevant topological properties. In this report, we show that graph sampling algorithms currently proposed in the literature are not able to preserve network properties even with sample sizes containing as many as 20% of the nodes from the original graph. We present a new sampling algorithm, called Tiny Sample Extractor, with a ne…
We introduce a new perfect sampling technique that can be applied to general Gibbs distributions and runs in linear time if the correlation decays faster than the neighborhood growth. In particular, in graphs with sub-exponential neighborhood growth like $\mathbb{Z}^d$, our algorithm achieves linear running time as long as Gibbs sampling is rapidly mixing. As concrete applications, we obtain the currently best perfect samplers for colorings and for monomer-dimer models in such graphs.
In this work we provide a new technique to design fast approximation algorithms for graph problems where the points of the graph lie in a metric space. Specifically, we present a sampling approach for such metric graphs that, using a sublinear number of edge weight queries, provides a {\em linear sampling}, where each edge is (roughly speaking) sampled proportionally to its weight. For several natural problems, such as densest subgraph and max cut among others, we show that by sparsifying the graph using this sampling process, we can run a suitable approximation algorithm on the sparsified gra…
Dense subgraph discovery is an important primitive in graph mining, which has a wide variety of applications in diverse domains. In the densest subgraph problem, given an undirected graph $G=(V,E)$ with an edge-weight vector $w=(w_e)_{e\in E}$, we aim to find $S\subseteq V$ that maximizes the density, i.e., $w(S)/|S|$, where $w(S)$ is the sum of the weights of the edges in the subgraph induced by $S$. Although the densest subgraph problem is one of the most well-studied optimization problems for dense subgraph discovery, there is an implicit strong assumption; it is assumed that the weights of…
We present a perfect marginal sampler of the unique Gibbs measure of a spin system on $\mathbb Z^2$. The algorithm is an adaptation of a previous `lazy depth-first' approach by the authors, but relaxes the requirement of strong spatial mixing to weak. It exploits a classical result in statistical physics relating weak spatial mixing on $\mathbb Z^2$ to strong spatial mixing on squares. When the spin system exhibits weak spatial mixing, the run-time of our sampler is linear in the size of sample. Applications of note are the ferromagnetic Potts model at supercritical temperatures, and the ferro…
Each vertex of an arbitrary simple graph on $n$ vertices chooses $k$ random incident edges. What is the expected number of edges in the original graph that connect different connected components of the sampled subgraph? We prove that the answer is $O(n/k)$, when $k\ge c\log n$, for some large enough $c$. We conjecture that the same holds for smaller values of $k$, possibly for any $k\ge 2$. Such a result is best possible for any $k\ge 2$. As an application, we use this sampling result to obtain a one-way communication protocol with \emph{private} randomness for finding a spanning forest of a g…
Core decomposition is a classic technique for discovering densely connected regions in a graph with large range of applications. Formally, a $k$-core is a maximal subgraph where each vertex has at least $k$ neighbors. A natural extension of a $k$-core is a $(k, h)$-core, where each node must have at least $k$ nodes that can be reached with a path of length $h$. The downside in using $(k, h)$-core decomposition is the significant increase in the computational complexity: whereas the standard core decomposition can be done in $O(m)$ time, the generalization can require $O(n^2m)$ time, where $n$ …
Motivated by the `subgraphs world' view of the ferromagnetic Ising model, we develop a general approach to studying mixing times of Glauber dynamics based on subset expansion expressions for a class of graph polynomials. With a canonical paths argument, we demonstrate that the chains defined within this framework mix rapidly upon graphs of bounded tree-width. This extends known results on rapid mixing for the Tutte polynomial, the adjacency-rank ($R_2$-)polynomial and the interlace polynomial.
Let $H_n$ be a graph on $n$ vertices and let $\ber{H_n}$ denote the complement of $H_n$. Suppose that $Δ= Δ(n)$ is the maximum degree of $\ber{H_n}$. We analyse three algorithms for sampling $d$-regular subgraphs ($d$-factors) of $H_n$. This is equivalent to uniformly sampling $d$-regular graphs which avoid a set $E(\ber{H_n})$ of forbidden edges. Here $d=d(n)$ is a positive integer which may depend on $n$. Two of these algorithms produce a uniformly random $d$-factor of $H_n$ in expected runtime which is linear in $n$ and low-degree polynomial in $d$ and $Δ$. The first algorithm applies when …
Subgraph counting is a fundamental primitive in graph processing, with applications in social network analysis (e.g., estimating the clustering coefficient of a graph), database processing and other areas. The space complexity of subgraph counting has been studied extensively in the literature, but many natural settings are still not well understood. In this paper we revisit the subgraph (and hypergraph) counting problem in the sketching model, where the algorithm's state as it processes a stream of updates to the graph is a linear function of the stream. This model has recently received a lot…
We study the problem of sampling from the power posterior distribution in Bayesian Gaussian mixture models, a robust version of the classical posterior. This power posterior is known to be non-log-concave and multi-modal, which leads to exponential mixing times for some standard MCMC algorithms. We introduce and study the Reflected Metropolis-Hastings Random Walk (RMRW) algorithm for sampling. For symmetric two-component Gaussian mixtures, we prove that its mixing time is bounded as $d^{1.5}(d + \Vert θ_{0} \Vert^2)^{4.5}$ as long as the sample size $n$ is of the order $d (d + \Vert θ_{0} \Ver…
The problem of efficiently sampling from a set of(undirected) graphs with a given degree sequence has many applications. One approach to this problem uses a simple Markov chain, which we call the switch chain, to perform the sampling. The switch chain is known to be rapidly mixing for regular degree sequences. We prove that the switch chain is rapidly mixing for any degree sequence with minimum degree at least 1 and with maximum degree $d_{\max}$ which satisfies $3\leq d_{\max}\leq \frac{1}{4}\, \sqrt{M}$, where $M$ is the sum of the degrees. The mixing time bound obtained is only an order of …
Graph sampling allows mining a small representative subgraph from a big graph. Sampling algorithms deploy different strategies to replicate the properties of a given graph in the sampled graph. In this study, we provide a comprehensive empirical characterization of five graph sampling algorithms on six properties of a graph including degree, clustering coefficient, path length, global clustering coefficient, assortativity, and modularity. We extract samples from fifteen graphs grouped into five categories including collaboration, social, citation, technological, and synthetic graphs. We provid…
We extend the notion of spectral independence (introduced by Anari, Liu, and Oveis Gharan [ALO20]) from the Boolean domain to general discrete domains. This property characterises distributions with limited correlations, and implies that the corresponding Glauber dynamics is rapidly mixing. As a concrete application, we show that Glauber dynamics for sampling proper $q$-colourings mixes in polynomial-time for the family of triangle-free graphs with maximum degree $Δ$ provided $q\ge (α^*+δ)Δ$ where $α^*\approx 1.763$ is the unique solution to $α^*=\exp(1/α^*)$ and $δ>0$ is any constant. This is…
Ge and Stefankovic have recently introduced a novel two-variable graph polynomial. When specialised to a bipartite graphs G and evaluated at the point (1/2,1) this polynomial gives the number of independent sets in the graph. Inspired by this polynomial, they also introduced a Markov chain which, if rapidly mixing, would provide an efficient sampling procedure for independent sets in G. This sampling procedure in turn would imply the existence of efficient approximation algorithms for a number of significant counting problems whose complexity is so far unresolved. The proposed Markov chain is …
