Counting and Sampling from Markov Equivalent DAGs Using Clique Trees
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；Sharif University of Technology（谢里夫理工大学）；Georgia Institute of Technology（佐治亚理工学院）；Carnegie Mellon University（卡内基梅隆大学）
AmirEmad Ghassami, Saber Salehkaleybar, Negar Kiyavash, Kun Zhang
A directed acyclic graph (DAG) is the most common graphical model for representing causal relationships among a set of variables. When restricted to using only observational data, the structure of the ground truth DAG is identifiable only up to Markov equivalence, based on conditional independence relations among the variables. Therefore, the number of DAGs equivalent to the ground truth DAG is an indicator of the causal complexity of the underlying structure--roughly speaking, it shows how many interventions or how much additional information is further needed to recover the underlying DAG. In this paper, we propose a new technique for counting the number of DAGs in a Markov equivalence class. Our approach is based on the clique tree representation of chordal graphs. We show that in the case of bounded degree graphs, the proposed algorithm is polynomial time. We further demonstrate that this technique can be utilized for uniform sampling from a Markov equivalence class, which provides a stochastic way to enumerate DAGs in the equivalence class and may be needed for finding the best DAG or for causal inference given the equivalence class as input. We also extend our counting and sampling method to the case where prior knowledge about the underlying DAG is available, and present applications of this extension in causal experiment design and estimating the causal effect of joint interventions.
Causal structure learning with momentum: Sampling distributions over Markov Equivalence Classes of DAGs
Chalmers University of Technology（查尔姆斯理工大学）；University of Gothenburg（哥德堡大学）；Institute for Theoretical Computer Science（理论计算机科学研究所）；University of Lübeck（吕贝克大学）
Graphical models such as Markov random fields (MRFs) that are associated with undirected graphs, and Bayesian networks (BNs) that are associated with directed acyclic graphs, have proven to be a very popular approach for reasoning under uncertainty, prediction problems and causal inference. Parametric MRF likelihoods are well-studied for Gaussian and categorical data. However, in more complicated parametric and semi-parametric settings, likelihoods specified via clique potential functions are generally not known to be congenial {(jointly well-specified)} or non-redundant. Congenial and non-red…
Counting instances of specific subgraphs in a larger graph is an important problem in graph mining. Finding cliques of size k (k-cliques) is one example of this NP-hard problem. Different algorithms for clique counting avoid counting the same clique multiple times by pivoting or ordering the graph. Ordering-based algorithms include an ordering step to direct the edges in the input graph, and a counting step, which is dominated by building node or edge-induced subgraphs. Of the ordering-based algorithms, kClist is the state-of-the art algorithm designed to work on sparse real-world graphs. Desp…
Counting k-cliques in a graph is an important problem in graph analysis with many applications such as community detection and graph partitioning. Counting k-cliques is typically done by traversing search trees starting at each vertex in the graph. Parallelizing k-clique counting has been well-studied on CPUs and many solutions exist. However, there are no performant solutions for k-clique counting on GPUs. Parallelizing k-clique counting on GPUs comes with numerous challenges such as the need for extracting fine-grain multi-level parallelism, sensitivity to load imbalance, and constrained phy…
We show that #SAT is polynomial-time tractable for classes of CNF formulas whose incidence graphs have bounded symmetric clique-width (or bounded clique-width, or bounded rank-width). This result strictly generalizes polynomial-time tractability results for classes of formulas with signed incidence graphs of bounded clique-width and classes of formulas with incidence graphs of bounded modular treewidth, which were the most general results of this kind known so far.
We study the problem of approximately counting cliques and near cliques in a graph, where the access to the graph is only available through crawling its vertices; thus typically seeing only a small portion of it. This model, known as the random walk model or the neighborhood query model has been introduced recently and captures real-life scenarios in which the entire graph is too massive to be stored as a whole or be scanned entirely and sampling vertices independently is non-trivial in it. We introduce DeMEtRIS: Dense Motif Estimation through Random Incident Sampling. This method provides a s…
We present a new parallel algorithm for $k$-clique counting/listing that has polylogarithmic span (parallel time) and is work-efficient (matches the work of the best sequential algorithm) for sparse graphs. Our algorithm is based on computing low out-degree orientations, which we present new linear-work and polylogarithmic-span algorithms for computing in parallel. We also present new parallel algorithms for producing unbiased estimations of clique counts using graph sparsification. Finally, we design two new parallel work-efficient algorithms for approximating the $k$-clique densest subgraph,…
Clique counting is a fundamental task in network analysis, and even the simplest setting of $3$-cliques (triangles) has been the center of much recent research. Getting the count of $k$-cliques for larger $k$ is algorithmically challenging, due to the exponential blowup in the search space of large cliques. But a number of recent applications (especially for community detection or clustering) use larger clique counts. Moreover, one often desires \textit{local} counts, the number of $k$-cliques per vertex/edge. Our main result is Pivoter, an algorithm that exactly counts the number of $k$-cliqu…
We consider the problem of counting $k$-cliques in $s$-uniform Erdos-Renyi hypergraphs $G(n,c,s)$ with edge density $c$, and show that its fine-grained average-case complexity can be based on its worst-case complexity. We prove the following: 1. Dense Erdos-Renyi graphs and hypergraphs: Counting $k$-cliques on $G(n,c,s)$ with $k$ and $c$ constant matches its worst-case time complexity up to a $\mathrm{polylog}(n)$ factor. Assuming randomized ETH, it takes $n^{Ω(k)}$ time to count $k$-cliques in $G(n,c,s)$ if $k$ and $c$ are constant. 2. Sparse Erdos-Renyi graphs and hypergraphs: When $c = Θ(n^…
In this paper, we study new batch-dynamic algorithms for the $k$-clique counting problem, which are dynamic algorithms where the updates are batches of edge insertions and deletions. We study this problem in the parallel setting, where the goal is to obtain algorithms with low (polylogarithmic) depth. Our first result is a new parallel batch-dynamic triangle counting algorithm with $O(Δ\sqrt{Δ+m})$ amortized work and $O(\log^* (Δ+m))$ depth with high probability, and $O(Δ+m)$ space for a batch of $Δ$ edge insertions or deletions. Our second result is an algebraic algorithm based on parallel fa…
We consider the fundamental problem of detecting/counting copies of a fixed pattern graph in a host graph. The recent progress on this problem has not included complete pattern graphs, i.e., cliques (and their complements, i.e., edge-free pattern graphs, in the induced setting). The fastest algorithms for the aforementioned patterns are based on a straightforward reduction to triangle detection/counting. We provide an alternative method of detection/counting copies of fixed size cliques based on a multi-dimensional matrix product. It is at least as time efficient as the triangle method in case…
We tackle the problem of counting the number of $k$-cliques in large-scale graphs, for any constant $k \ge 3$. Clique counting is essential in a variety of applications, among which social network analysis. Due to its computationally intensive nature, we settle for parallel solutions in the MapReduce framework, which has become in the last few years a {\em de facto} standard for batch processing of massive data sets. We give both theoretical and experimental contributions. On the theory side, we design the first exact scalable algorithm for counting (and listing) $k$-cliques. Our algorithm use…
We study the problem of counting the number of {\em isomorphic} copies of a given {\em template} graph, say $H$, in the input {\em base} graph, say $G$. In general, it is believed that polynomial time algorithms that solve this problem exactly are unlikely to exist. So, a lot of work has gone into designing efficient {\em approximation schemes}, especially, when $H$ is a perfect matching. In this work, we present efficient approximation schemes to count $k$-Cliques, $k$-Independent sets and $k$-Clique covers in random graphs. We present {\em fully polynomial time randomized approximation schem…
The AP-LCA problem asks, given an $n$-node directed acyclic graph (DAG), to compute for every pair of vertices $u$ and $v$ in the DAG a lowest common ancestor (LCA) of $u$ and $v$ if one exists. In this paper we study several interesting variants of AP-LCA, providing both algorithms and fine-grained lower bounds for them. The lower bounds we obtain are the first conditional lower bounds for LCA problems higher than $n^{ω-o(1)}$, where $ω$ is the matrix multiplication exponent. Some of our results include: - In any DAG, we can detect all vertex pairs that have at most two LCAs and list all of t…
In this work, we consider the problem of sampling a $k$-clique in a graph from an almost uniform distribution in sublinear time in the general graph query model. Specifically the algorithm should output each $k$-clique with probability $(1\pm ε)/n_k$, where $n_k$ denotes the number of $k$-cliques in the graph and $ε$ is a given approximation parameter. We prove that the query complexity of this problem is \[ Θ^*\left(\max\left\{ \left(\frac{(nα)^{k/2}}{ n_k}\right)^{\frac{1}{k-1}} ,\; \min\left\{nα,\frac{nα^{k-1}}{n_k} \right\}\right\}\right). \] where $n$ is the number of vertices in the grap…
Clique and near-clique counts are important graph properties with applications in graph generation, graph modeling, graph analytics, community detection among others. They are the archetypal examples of dense subgraphs. While there are several different definitions of near-cliques, most of them share the attribute that they are cliques that are missing a small number of edges. Clique counting is itself considered a challenging problem. Counting near-cliques is significantly harder more so since the search space for near-cliques is orders of magnitude larger than that of cliques. We give a form…
Clique counts reveal important properties about the structure of massive graphs, especially social networks. The simple setting of just 3-cliques (triangles) has received much attention from the research community. For larger cliques (even, say 6-cliques) the problem quickly becomes intractable because of combinatorial explosion. Most methods used for triangle counting do not scale for large cliques, and existing algorithms require massive parallelism to be feasible. We present a new randomized algorithm that provably approximates the number of k-cliques, for any constant k. The key insight is…
We study the problem of approximating the number of $k$-cliques in a graph when given query access to the graph. We consider the standard query model for general graphs via (1) degree queries, (2) neighbor queries and (3) pair queries. Let $n$ denote the number of vertices in the graph, $m$ the number of edges, and $C_k$ the number of $k$-cliques. We design an algorithm that outputs a $(1+\varepsilon)$-approximation (with high probability) for $C_k$, whose expected query complexity and running time are $O\left(\frac{n}{C_k^{1/k}}+\frac{m^{k/2}}{C_k}\right)\poly(\log n,1/\varepsilon,k)$. Hence,…
We study tensor networks as a model of arithmetic computation for evaluating multilinear maps. These capture any algorithm based on low border rank tensor decompositions, such as $O(n^{ω+ε})$ time matrix multiplication, and in addition many other algorithms such as $O(n \log n)$ time discrete Fourier transform and $O^*(2^n)$ time for computing the permanent of a matrix. However tensor networks sometimes yield faster algorithms than those that follow from low-rank decompositions. For instance the fastest known $O(n^{(ω+ε)t})$ time algorithms for counting $3t$-cliques can be implemented with ten…
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…
EPG graphs, introduced by Golumbic et al. in 2009, are edge-intersection graphs of paths on an orthogonal grid. The class $B_k$-EPG is the subclass of EPG graphs where the path on the grid associated to each vertex has at most $k$ bends. Epstein et al. showed in 2013 that computing a maximum clique in $B_1$-EPG graphs is polynomial. As remarked in [Heldt et al., 2014], when the number of bends is at least $4$, the class contains $2$-interval graphs for which computing a maximum clique is an NP-hard problem. The complexity status of the Maximum Clique problem remains open for $B_2$ and $B_3$-EP…
Finding large cliques or cliques missing a few edges is a fundamental algorithmic task in the study of real-world graphs, with applications in community detection, pattern recognition, and clustering. A number of effective backtracking-based heuristics for these problems have emerged from recent empirical work in social network analysis. Given the NP-hardness of variants of clique counting, these results raise a challenge for beyond worst-case analysis of these problems. Inspired by the triadic closure of real-world graphs, Fox et al. (SICOMP 2020) introduced the notion of $c$-closed graphs an…
We introduce Tiered Sampling, a novel technique for approximate counting sparse motifs in massive graphs whose edges are observed in a stream. Our technique requires only a single pass on the data and uses a memory of fixed size $M$, which can be magnitudes smaller than the number of edges. Our methods addresses the challenging task of counting sparse motifs - sub-graph patterns that have low probability to appear in a sample of $M$ edges in the graph, which is the maximum amount of data available to the algorithms in each step. To obtain an unbiased and low variance estimate of the count we p…
In the counting Graph Homomorphism problem (#GraphHom) the question is: Given graphs G,H, find the number of homomorphisms from G to H. This problem is generally #P-complete, moreover, Cygan et al. proved that unless the ETH is false there is no algorithm that solves this problem in time O(|V(H)|^{o(|V(G)|)}. This, however, does not rule out the possibility that faster algorithms exist for restricted problems of this kind. Wahlstrom proved that #GraphHom can be solved in plain exponential time, that is, in time k^{|V(G)|+V(H)|}\poly(|V(H)|,|V(G)|) provided H has clique width k. We generalize t…
Directed acyclic graphs (DAGs) can be characterised as directed graphs whose strongly connected components are isolated vertices. Using this restriction on the strong components, we discover that when $m = cn$, where $m$ is the number of directed edges, $n$ is the number of vertices, and $c < 1$, the asymptotic probability that a random digraph is acyclic is an explicit function $p(c)$, such that $p(0) = 1$ and $p(1) = 0$. When $m = n(1 + μn^{-1/3})$, the asymptotic behaviour changes, and the probability that a digraph is acyclic becomes $n^{-1/3} C(μ)$, where $C(μ)$ is an explicit function of…
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 this paper, we design efficient algorithms to approximately count the number of edges of a given $k$-hypergraph, and to sample an approximately uniform random edge. The hypergraph is not given explicitly, and can be accessed only through its colourful independence oracle: The colourful independence oracle returns yes or no depending on whether a given subset of the vertices contains an edge that is colourful with respect to a given vertex-colouring. Our results extend and/or strengthen recent results in the graph oracle literature due to Beame et al. (ITCS 2018), Dell and Lapinskas (STOC 20…
Given query access to an undirected graph $G$, we consider the problem of computing a $(1\pmε)$-approximation of the number of $k$-cliques in $G$. The standard query model for general graphs allows for degree queries, neighbor queries, and pair queries. Let $n$ be the number of vertices, $m$ be the number of edges, and $n_k$ be the number of $k$-cliques. Previous work by Eden, Ron and Seshadhri (STOC 2018) gives an $O^*(\frac{n}{n^{1/k}_k} + \frac{m^{k/2}}{n_k})$-time algorithm for this problem (we use $O^*(\cdot)$ to suppress $\poly(\log n, 1/ε, k^k)$ dependencies). Moreover, this bound is ne…
We consider the problem of counting all $k$-vertex subgraphs in an input graph, for any constant $k$. This problem (denoted sub-cnt$_k$) has been studied extensively in both theory and practice. In a classic result, Chiba and Nishizeki (SICOMP 85) gave linear time algorithms for clique and 4-cycle counting for bounded degeneracy graphs. This is a rich class of sparse graphs that contains, for example, all minor-free families and preferential attachment graphs. The techniques from this result have inspired a number of recent practical algorithms for sub-cnt$_k$. Towards a better understanding o…
Parameterized complexity theory has enabled a refined classification of the difficulty of NP-hard optimization problems on graphs with respect to key structural properties, and so to a better understanding of their true difficulties. More recently, hardness results for problems in P were achieved using reasonable complexity theoretic assumptions such as: Strong Exponential Time Hypothesis (SETH), 3SUM and All-Pairs Shortest-Paths (APSP). According to these assumptions, many graph theoretic problems do not admit truly subquadratic algorithms, nor even truly subcubic algorithms (Williams and Wil…
Over the last two decades, frameworks for distributed-memory parallel computation, such as MapReduce, Hadoop, Spark and Dryad, have gained significant popularity with the growing prevalence of large network datasets. The Massively Parallel Computation (MPC) model is the de-facto standard for studying graph algorithms in these frameworks theoretically. Subgraph counting is one such fundamental problem in analyzing massive graphs, with the main algorithmic challenges centering on designing methods which are both scalable and accurate. Given a graph $G=(V, E)$ with $n$ vertices, $m$ edges and $T$…
Given a graph stream, how can we estimate the number of triangles in it using multiple machines with limited storage? Specifically, how should edges be processed and sampled across the machines for rapid and accurate estimation? The count of triangles (i.e., cliques of size three) has proven useful in numerous applications, including anomaly detection, community detection, and link recommendation. For triangle counting in large and dynamic graphs, recent work has focused largely on streaming algorithms and distributed algorithms but little on their combinations for "the best of both worlds". I…
We consider the problem of sampling and approximately counting an arbitrary given motif $H$ in a graph $G$, where access to $G$ is given via queries: degree, neighbor, and pair, as well as uniform edge sample queries. Previous algorithms for these tasks were based on a decomposition of $H$ into a collection of odd cycles and stars, denoted $\mathcal{D}^*(H)=\{O_{k_1}, \ldots, O_{k_q}, S_{p_1}, \ldots, S_{p_\ell}\}$. These algorithms were shown to be optimal for the case where $H$ is a clique or an odd-length cycle, but no other lower bounds were known. We present a new algorithm for sampling a…
We continue the study of graph classes in which the treewidth can only be large due to the presence of a large clique, and, more specifically, of graph classes with bounded tree-independence number. In [Dallard, Milanič, and Štorgel, Treewidth versus clique number. {II}. Tree-independence number, 2022], it was shown that the Maximum Weight Independent Packing problem, which is a common generalization of the Independent Set and Induced Matching problems, can be solved in polynomial time provided that the input graph is given along with a tree decomposition with bounded independence number. We p…
Pattern counting in graphs is a fundamental primitive for many network analysis tasks, and a number of methods have been developed for scaling subgraph counting to large graphs. Many real-world networks carry a natural notion of strength of connection between nodes, which are often modeled by a weighted graph, but existing scalable graph algorithms for pattern mining are designed for unweighted graphs. Here, we develop a suite of deterministic and random sampling algorithms that enable the fast discovery of the 3-cliques (triangles) with the largest weight in a graph, where weight is measured …
Estimating the number of subgraphs in data streams is a fundamental problem that has received great attention in the past decade. In this paper, we give improved streaming algorithms for approximately counting the number of occurrences of an arbitrary subgraph $H$, denoted $\# H$, when the input graph $G$ is represented as a stream of $m$ edges. To obtain our algorithms, we provide a generic transformation that converts constant-round sublinear-time graph algorithms in the query access model to constant-pass sublinear-space graph streaming algorithms. Using this transformation, we obtain the f…
