We consider the problem of estimating the differences between two causal directed acyclic graph (DAG) models given i.i.d. samples from each model. This is of interest for example in genomics, where large-scale gene expression data is becoming available under different cellular contexts, for different cell types, or disease states. Changes in the structure or edge weights of the underlying causal graphs reflect alterations in the gene regulatory networks and provide important insights into the emergence of a particular phenotype. While the individual networks are usually very large, containing high-degree hub nodes and thus difficult to learn, the overall change between two related networks can be sparse. We here provide the first provably consistent method for directly estimating the differences in a pair of causal DAGs without separately learning two possibly large and dense DAG models and computing their difference. Our two-step algorithm first uses invariance tests between regression coefficients of the two data sets to estimate the skeleton of the difference graph and then orients some of the edges using invariance tests between regression residual variances. We demonstrate the properties of our method through a simulation study and apply it to the analysis of gene expression data from ovarian cancer and during T-cell activation.
A simpler task is learning differences of undirected graphical models. Let @math and @math denote the precision matrices corresponding to @math and @math . The support of @math consists of the edges in the undirected graph (UG) models corresponding to @math . We define the difference-UG ( D-UG ) by @math , with @math if and only if @math for @math . Two recent methods that directly learn the difference of two UG models are KLIEP @cite_37 and DPM @cite_34 ; for a review and comparison of these methods see @cite_19 . These methods can be used as a first step towards estimating the D-DAG @math : under genericity assumptions, the formulae for @math in imply that if @math then @math . Hence, the skeleton of @math is a subgraph of @math , i.e., @math . In the following section we present our algorithm showing how to obtain @math and determine some of the edge directions in @math . We end this section with a piece of notation needed for introducing our algorithm; we define the to be @math .
@cite_19 @cite_37 @cite_34
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We present two new approximation algorithms for the problem of finding a k-node connected spanning subgraph (directed or undirected) of minimum cost. The best known approximation guarantees for this problem were @math for both directed and undirected graphs, and @math for undirected graphs with @math , where @math is the number of nodes in the input graph. Our first algorithm has approximation ratio @math , which is @math except for very large values of @math , namely, @math . This algorithm is based on a new result on @math -connected @math -critical graphs, which is of independent interest in the context of graph theory. Our second algorithm uses the primal-dual method and has approximation ratio @math for all values of @math . Combining these two gives an algorithm with approximation ratio @math , which asymptotically improves the best known approximation guarantee for directed graphs for all values of @math , and for undirected graphs for @math . Moreover, this is the first algorithm that has an approximation guarantee better than @math for all values of @math . Our approximation ratio also provides an upper bound on the integrality gap of the standard LP-relaxation. For an unweighted graph @math , @math is a subgraph if @math , and @math is a Steiner graph if @math , and for any pair of vertices @math , the distance between them in @math (denoted @math ) is at least the distance between them in @math (denoted @math ). In this paper we introduce the notion of distance preserver. A subgraph (resp., Steiner graph) @math of a graph @math is a subgraph (resp., Steiner) @math -preserver of @math if for every pair of vertices @math with @math , @math . We show that any graph (resp., digraph) has a subgraph @math -preserver with at most @math edges (resp., arcs), and there are graphs and digraphs for which any undirected Steiner @math -preserver contains @math edges. However, we show that if one allows a directed Steiner (diSteiner) @math -preserver, then these bounds can be improved. Specifically, we show that for any graph or digraph there exists a diSteiner @math -preserver with @math arcs, and that this result is tight up to a constant factor. We also study @math -preserving distance labeling schemes, that are labeling schemes that guarantee precise calculation of distances between pairs of vertices that are at a distance of at least @math one from another. We show that there exists a @math -preserving labeling scheme with labels of size @math , and that labels of size @math are required for any @math -preserving labeling scheme. Let k ≥ 2 be an integer. We show that any undirected and unweighted graph G = (V, E) on n vertices has a subgraph G' = (V, E') with O(kn1+1 k) edges such that for any two vertices u, v ∈ V, if Δ G (u, v) = d, then Δ G' (u, v) = d+O(d1-1 k-1). Furthermore, we show that such subgraphs can be constructed in O(mn1 k) time, where m and n are the number of edges and vertices in the original graph. We also show that it is possible to construct a weighted graph G* = (V, E*) with O(kn1+1 (2k-1)) edges such that for every u, v ∈ V, if Δ G (u, v) = d, then Δ ≤ Δ G* (u, v) = d + O(d1-1 k-1). These are the first such results with additive error terms of the form o(d), i.e., additive error terms that are sublinear in the distance being approximated. Let G=(V,E) be an unweighted undirected graph on n vertices. A simple argument shows that computing all distances in G with an additive one-sided error of at most 1 is as hard as Boolean matrix multiplication. Building on recent work of [SIAM J. Comput., 28 (1999), pp. 1167--1181], we describe an @math -time algorithm APASP2 for computing all distances in G with an additive one-sided error of at most 2. Algorithm APASP2 is simple, easy to implement, and faster than the fastest known matrix-multiplication algorithm. Furthermore, for every even k>2, we describe an @math -time algorithm APASPk for computing all distances in G with an additive one-sided error of at most k. We also give an @math -time algorithm @math for producing stretch 3 estimated distances in an unweighted and undirected graph on n vertices. No constant stretch factor was previously achieved in @math time. We say that a weighted graph F=(V,E') k-emulates an unweighted graph G=(V,E) if for every @math we have @math . We show that every unweighted graph on n vertices has a 2-emulator with @math edges and a 4-emulator with @math edges. These results are asymptotically tight. Finally, we show that any weighted undirected graph on n vertices has a 3-spanner with @math edges and that such a 3-spanner can be built in @math time. We also describe an @math -time algorithm for estimating all distances in a weighted undirected graph on n vertices with a stretch factor of at most 3.