This paper provides an in-depth study of the fundamental problems of finding small subgraphs in distributed dynamic networks. While some problems are trivially easy to handle, such as detecting a triangle that emerges after an edge insertion, we show that, perhaps somewhat surprisingly, other problems exhibit a wide range of complexities in terms of the trade-offs between their round and bandwidth complexities. In the case of triangles, which are only affected by the topology of the immediate neighborhood, some end results are: The bandwidth complexity of @math -round dynamic triangle detection or listing is @math . The bandwidth complexity of @math -round dynamic triangle membership listing is @math for node edge deletions, @math for edge insertions, and @math for node insertions. The bandwidth complexity of @math -round dynamic triangle membership detection is @math for node edge deletions, @math for edge insertions, and @math for node insertions. Most of our upper and lower bounds are . Additionally, we provide almost always tight upper and lower bounds for larger cliques.
In the CONGEST model and in related models, where the nodes synchronously communicate with @math bits, much research is devoted to subgraph detection, e.g., @cite_4 @cite_26 @cite_3 @cite_1 @cite_19 @cite_12 @cite_30 @cite_0 .
@cite_30 @cite_26 @cite_4 @cite_1 @cite_3 @cite_0 @cite_19 @cite_12
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We present simple deterministic algorithms for subgraph finding and enumeration in the broadcast CONGEST model of distributed computation: -- For any constant @math , detecting @math -paths and trees on @math nodes can be done in constantly many rounds, and @math -cycles in @math rounds. -- On @math -degenerate graphs, cliques and @math -cycles can be enumerated in @math rounds, and @math -cycles in @math rounds. In many cases, these bounds are tight up to logarithmic factors. Moreover, we show that the algorithms for @math -degenerate graphs can be improved to optimal complexity @math and @math , respectively, in the supported CONGEST model, which can be seen as an intermediate model between CONGEST and the congested clique. Given a simple graph G=(V,E) and a set of sources S ⊆ V, denote for each node ν e V by Lν(∞) the lexicographically ordered list of distance source pairs (d(s,v),s), where s ∈ S. For integers d,k ∈ N∪ ∞ , we consider the source detection, or (S,d,k)-detection task, requiring each node v to learn the first k entries of Lν(∞) (if for all of them d(s,v) ≤ d) or all entries (d(s,v),s) ∈ Lν(∞) satisfying that d(s,v) ≤ d (otherwise). Solutions to this problem provide natural generalizations of concurrent breadth-first search (BFS) tree constructions. For example, the special case of k=∞ requires each source s ∈ S to build a complete BFS tree rooted at s, whereas the special case of d=∞ and S=V requires constructing a partial BFS tree comprising at least k nodes from every node in V. In this work, we give a simple, near-optimal solution for the source detection task in the CONGEST model, where messages contain at most O(log n) bits, running in d+k rounds. We demonstrate its utility for various routing problems, exact and approximate diameter computation, and spanner construction. For those problems, we obtain algorithms in the CONGEST model that are faster and in some cases much simpler than previous solutions. In the standard CONGEST model for distributed network computing, it is known that "global" tasks such as minimum spanning tree, diameter, and all-pairs shortest paths, consume large bandwidth, for their running-time is @math rounds in @math -node networks with constant diameter. Surprisingly, "local" tasks such as detecting the presence of a 4-cycle as a subgraph also requires @math rounds, even using randomized algorithms, and the best known upper bound for detecting the presence of a 3-cycle is @math rounds. The objective of this paper is to better understand the landscape of such subgraph detection tasks. We show that, in contrast to , which are hard to detect in the CONGEST model, there exists a deterministic algorithm for detecting the presence of a subgraph isomorphic to @math running in a number of rounds, for every tree @math . Our algorithm provides a distributed implementation of a combinatorial technique due to Erd o for sparsening the set of partial solutions kept by the nodes at each round. Our result has important consequences to , i.e., to randomized algorithms whose aim is to distinguish between graphs satisfying a property, and graphs far from satisfying that property. In particular, we get that, for every graph pattern @math composed of an edge and a tree connected in an arbitrary manner, there exists a distributed testing algorithm for @math -freeness, performing in a constant number of rounds. Although the class of graph patterns @math formed by a tree and an edge connected arbitrarily may look artificial, all previous results of the literature concerning testing @math -freeness for classical patterns such as cycles and cliques can be viewed as direct consequences of our result, while our algorithm enables testing more complex patterns. We study the computation power of the congested clique, a model of distributed computation where n players communicate with each other over a complete network in order to compute some function of their inputs. The number of bits that can be sent on any edge in a round is bounded by a parameter b We consider two versions of the model: in the first, the players communicate by unicast, allowing them to send a different message on each of their links in one round; in the second, the players communicate by broadcast, sending one message to all their neighbors. It is known that the unicast version of the model is quite powerful; to date, no lower bounds for this model are known. In this paper we provide a partial explanation by showing that the unicast congested clique can simulate powerful classes of bounded-depth circuits, implying that even slightly super-constant lower bounds for the congested clique would give new lower bounds in circuit complexity. Moreover, under a widely-believed conjecture on matrix multiplication, the triangle detection problem, studied in [8], can be solved in O(ne) time for any e > 0. The broadcast version of the congested clique is the well-known multi-party shared-blackboard model of communication complexity (with number-in-hand input). This version is more amenable to lower bounds, and in this paper we show that the subgraph detection problem studied in [8] requires polynomially many rounds for several classes of subgraphs. We also give upper bounds for the subgraph detection problem, and relate the hardness of triangle detection in the broadcast congested clique to the communication complexity of set disjointness in the 3-party number-on-forehead model. Triangle-free graphs play a central role in graph theory, and triangle detection (or triangle finding) as well as triangle enumeration (triangle listing) play central roles in the field of graph algorithms. In distributed computing, algorithms with sublinear round complexity for triangle finding and listing have recently been developed in the powerful CONGEST clique model, where communication is allowed between any two nodes of the network. In this paper we present the first algorithms with sublinear complexity for triangle finding and triangle listing in the standard CONGEST model, where the communication topology is the same as the topology of the network. More precisely, we give randomized algorithms for triangle finding and listing with round complexity O(n2 3(log n)2 3) and O(n3 4log n), respectively, where n denotes the number of nodes of the network. We also show a lower bound Ω(n1 3 log n) on the round complexity of triangle listing, which also holds for the CONGEST clique model. In this work, we use algebraic methods for studying distance computation and subgraph detection tasks in the congested clique model. Specifically, we adapt parallel matrix multiplication implementations to the congested clique, obtaining an @math round matrix multiplication algorithm, where @math is the exponent of matrix multiplication. In conjunction with known techniques from centralised algorithmics, this gives significant improvements over previous best upper bounds in the congested clique model. The highlight results include: -- triangle and 4-cycle counting in @math rounds, improving upon the @math triangle detection algorithm of [DISC 2012], -- a @math -approximation of all-pairs shortest paths in @math rounds, improving upon the @math -round @math -approximation algorithm of Nanongkai [STOC 2014], and -- computing the girth in @math rounds, which is the first non-trivial solution in this model. In addition, we present a novel constant-round combinatorial algorithm for detecting 4-cycles. Numerous graph mining applications rely on detecting subgraphs which are large near-cliques. Since formulations that are geared towards finding large near-cliques are hard and frequently inapproximable due to connections with the Maximum Clique problem, the poly-time solvable densest subgraph problem which maximizes the average degree over all possible subgraphs "lies at the core of large scale data mining" [10]. However, frequently the densest subgraph problem fails in detecting large near-cliques in networks. In this work, we introduce the k-clique densest subgraph problem, k ≥ 2. This generalizes the well studied densest subgraph problem which is obtained as a special case for k=2. For k=3 we obtain a novel formulation which we refer to as the triangle densest subgraph problem: given a graph G(V,E), find a subset of vertices S* such that τ(S*)=max limitsS ⊆ V t(S) |S|, where t(S) is the number of triangles induced by the set S. On the theory side, we prove that for any k constant, there exist an exact polynomial time algorithm for the k-clique densest subgraph problem . Furthermore, we propose an efficient 1 k-approximation algorithm which generalizes the greedy peeling algorithm of Asahiro and Charikar [8,18] for k=2. Finally, we show how to implement efficiently this peeling framework on MapReduce for any k ≥ 3, generalizing the work of Bahmani, Kumar and Vassilvitskii for the case k=2 [10]. On the empirical side, our two main findings are that (i) the triangle densest subgraph is consistently closer to being a large near-clique compared to the densest subgraph and (ii) the peeling approximation algorithms for both k=2 and k=3 achieve on real-world networks approximation ratios closer to 1 rather than the pessimistic 1 k guarantee. An interesting consequence of our work is that triangle counting, a well-studied computational problem in the context of social network analysis can be used to detect large near-cliques. Finally, we evaluate our proposed method on a popular graph mining application. This work presents a novel modeling and analysis framework for graph sequences which addresses the challenge of detecting and contextualizing anomalies in labeled, streaming graph data. We introduce a generalization of the BTER model of by adding flexibility to community structure, and use this model to perform multi-scale graph anomaly detection. Specifically, probability models describing coarse subgraphs are built by aggregating node probabilities, and these related hierarchical models simultaneously detect deviations from expectation. This technique provides insight into the graphs' structure and helps contextualized detected event. For evaluation, two new hierarchical detectors are tested against a baseline Gaussian method on a synthetic graph sequence with seeded anomalies. We demonstrate that in a labeled setting with community structure, our graph statistics-based approach outperforms both a distribution-based detector and the baseline, accurately detecting anomalies at the node, subgraph, and graph levels. When working with network datasets, the theoretical framework of detection theory for Euclidean vector spaces no longer applies. Nevertheless, it is desirable to determine the detectability of small, anomalous graphs embedded into background networks with known statistical properties. Casting the problem of subgraph detection in a signal processing context, this article provides a framework and empirical results that elucidate a "detection theory" for graph-valued data. Its focus is the detection of anomalies in unweighted, undirected graphs through L1 properties of the eigenvectors of the graph's so-called modularity matrix. This metric is observed to have relatively low variance for certain categories of randomly-generated graphs, and to reveal the presence of an anomalous subgraph with reasonable reliability when the anomaly is not well-correlated with stronger portions of the background graph. An analysis of subgraphs in real network datasets confirms the efficacy of this approach. A cohesive subgraph is a primary vehicle for massive graph analysis, and a newly introduced cohesive subgraph, k-truss, which is motivated by a natural observation of social cohesion, has attracted more and more attention. However, the existing parallel solutions to identify the k-truss are inefficient for very large graphs, as they still suffer from huge communication cost and large number of iterations during the computation. In this paper, we propose a novel parallel and efficient truss detection algorithm, called PeTa. The PeTa produces a triangle complete subgraph (TC-subgraph) for every computing node. Based on the TC-subgraphs, PeTa can detect the local k-truss in parallel within a few iterations. We theoretically prove, within this new paradigm, the communication cost of PeTa is bounded by three times of the number of triangles, the total computation complexity of PeTa is the same order as the best known serial algorithm and the number of iterations for a given partition scheme is minimized as well. Furthermore, we present a subgraph-oriented model to efficiently express PeTa in parallel graph computing systems. The results of comprehensive experiments demonstrate, compared with the existing solutions, PeTa saves 2X to 19X in communication cost, reduces 80 to 95 number of iterations and improves the overall performance by 80 across various real-world graphs. Densest subgraph computation has emerged as an important primitive in a wide range of data analysis tasks such as community and event detection. Social media such as Facebook and Twitter are highly dynamic with new friendship links and tweets being generated incessantly, calling for efficient algorithms that can handle very large and highly dynamic input data. While either scalable or dynamic algorithms for finding densest subgraphs have been proposed, a viable and satisfactory solution for addressing both the dynamic aspect of the input data and its large size is still missing. We study the densest subgraph problem in the the dynamic graph model, for which we present the first scalable algorithm with provable guarantees. In our model, edges are added adversarially while they are removed uniformly at random from the current graph. We show that at any point in time we are able to maintain a 2(1+e)-approximation of a current densest subgraph, while requiring O(polylog(n+r)) amortized cost per update (with high probability), where r is the total number of update operations executed and n is the maximum number of nodes in the graph. In contrast, a naive algorithm that recomputes a dense subgraph every time the graph changes requires Omega(m) work per update, where m is the number of edges in the current graph. Our theoretical analysis is complemented with an extensive experimental evaluation on large real-world graphs showing that (approximate) densest subgraphs can be maintained efficiently within hundred of microseconds per update. Subgraph isomorphism can be determined by means of a brute-force tree-search enumeration procedure. In this paper a new algorithm is introduced that attains efficiency by inferentially eliminating successor nodes in the tree search. To assess the time actually taken by the new algorithm, subgraph isomorphism, clique detection, graph isomorphism, and directed graph isomorphism experiments have been carried out with random and with various nonrandom graphs. A parallel asynchronous logic-in-memory implementation of a vital part of the algorithm is also described, although this hardware has not actually been built. The hardware implementation would allow very rapid determination of isomorphism. Detecting and counting the number of copies of certain subgraphs (also known as network motifs or graphlets) is motivated by applications in a variety of areas ranging from biology to the study of the World Wide Web. Several polynomial-time algorithms have been suggested for counting or detecting the number of occurrences of certain network motifs. However, a need for more efficient algorithms arises when the input graph is very large, as is indeed the case in many applications of motif counting. In this paper we design sublinear-time algorithms for approximating the number of copies of certain constant-size subgraphs in a graph G. That is, our algorithms do not read the whole graph, but rather query parts of the graph. Specifically, we consider algorithms that may query the degree of any vertex of their choice and may ask for any neighbor of any vertex of their choice. The main focus of this work is on the basic problem of counting the number of length-2 paths and more generally on counting the number of... Mining discriminative subgraph patterns from graph data has attracted great interest in recent years. It has a wide variety of applications in disease diagnosis, neuroimaging, etc. Most research on subgraph mining focuses on the graph representation alone. However, in many real-world applications, the side information is available along with the graph data. For example, for neurological disorder identification, in addition to the brain networks derived from neuroimaging data, hundreds of clinical, immunologic, serologic and cognitive measures may also be documented for each subject. These measures compose multiple side views encoding a tremendous amount of supplemental information for diagnostic purposes, yet are often ignored. In this paper, we study the problem of discriminative subgraph selection using multiple side views and propose a novel solution to find an optimal set of subgraph features for graph classification by exploring a plurality of side views. We derive a feature evaluation criterion, named gSide, to estimate the usefulness of subgraph patterns based upon side views. Then we develop a branch-and-bound algorithm, called gMSV, to efficiently search for optimal subgraph features by integrating the subgraph mining process and the procedure of discriminative feature selection. Empirical studies on graph classification tasks for neurological disorders using brain networks demonstrate that subgraph patterns selected by the multi-side-view guided subgraph selection approach can effectively boost graph classification performances and are relevant to disease diagnosis. Consider the following network communication setup, originating in a sensor networking application we refer to as the "sensor reachback" problem. We have a directed graph G=(V,E), where V= v sub 0 v sub 1 ...v sub n and E spl sube V spl times V. If (v sub i ,v sub j ) spl isin E, then node i can send messages to node j over a discrete memoryless channel (DMC) (X sub ij ,p sub ij (y|x),Y sub ij ), of capacity C sub ij . The channels are independent. Each node v sub i gets to observe a source of information U sub i (i=0...M), with joint distribution p(U sub 0 U sub 1 ...U sub M ). Our goal is to solve an incast problem in G: nodes exchange messages with their neighbors, and after a finite number of communication rounds, one of the M+1 nodes (v sub 0 by convention) must have received enough information to reproduce the entire field of observations (U sub 0 U sub 1 ...U sub M ), with arbitrarily small probability of error. In this paper, we prove that such perfect reconstruction is possible if and only if H(U sub s | U sub S(c) ) < spl Sigma sub i spl isin S,j spl isin S(c) for all S spl sube 0...M ,S spl ne O,0 spl isin S(c). Our main finding is that in this setup, a general source channel separation theorem holds, and that Shannon information behaves as a classical network flow, identical in nature to the flow of water in pipes. At first glance, it might seem surprising that separation holds in a fairly general network situation like the one we study. A closer look, however, reveals that the reason for this is that our model allows only for independent point-to-point channels between pairs of nodes, and not multiple-access and or broadcast channels, for which separation is well known not to hold. This "information as flow" view provides an algorithmic interpretation for our results, among which perhaps the most important one is the optimality of implementing codes using a layered protocol stack. In the subgraph-freeness problem, we are given a constant-sized graph H, and wish to de- termine whether the network graph contains H as a subgraph or not. Until now, the only lower bounds on subgraph-freeness known for the CONGEST model were for cycles of length greater than 3; here we extend and generalize the cycle lower bound, and obtain polynomial lower bounds for subgraph-freeness in the CONGEST model for two classes of subgraphs. The first class contains any graph obtained by starting from a 2-connected graph H for which we already know a lower bound, and replacing the vertices of H by arbitrary connected graphs. We show that the lower bound on H carries over to the new graph. The second class is constructed by starting from a cycle Ck of length k ≥ 4, and constructing a graph H from Ck by replacing each edge i, (i + 1) mod k of the cycle with a connected graph Hi, subject to some constraints on the graphs H_ 0 , . . . , H_ k−1 . In this case we obtain a polynomial lower bound for the new graph H , depending on the size of the shortest cycle in H passing through the vertices of the original k-cycle. The ability to find and count subgraphs of a given network is an important non trivial task with multidisciplinary applicability. Discovering network motifs or computing graphlet signatures are two examples of methodologies that at their core rely precisely on the subgraph counting problem. Here we present the g-trie, a data-structure specifically designed for discovering subgraph frequencies. We produce a tree that encapsulates the structure of the entire graph set, taking advantage of common topologies in the same way a prefix tree takes advantage of common prefixes. This avoids redundancy in the representation of the graphs, thus allowing for both memory and computation time savings. We introduce a specialized canonical labeling designed to highlight common substructures and annotate the g-trie with a set of conditional rules that break symmetries, avoiding repetitions in the computation. We introduce a novel algorithm that takes as input a set of small graphs and is able to efficiently find and count them as induced subgraphs of a larger network. We perform an extensive empirical evaluation of our algorithms, focusing on efficiency and scalability on a set of diversified complex networks. Results show that g-tries are able to clearly outperform previously existing algorithms by at least one order of magnitude. The subgraph isomorphism problem involves deciding whether a copy of a pattern graph occurs inside a larger target graph. The non-induced version allows extra edges in the target, whilst the induced version does not. Although both variants are NP-complete, algorithms inspired by constraint programming can operate comfortably on many real-world problem instances with thousands of vertices. However, they cannot handle arbitrary instances of this size. We show how to generate " really hard " random instances for subgraph isomorphism problems, which are computationally challenging with a couple of hundred vertices in the target, and only twenty pattern vertices. For the non-induced version of the problem, these instances lie on a satisfiable unsatisfiable phase transition, whose location we can predict; for the induced variant, much richer behaviour is observed, and constrained-ness gives a better measure of difficulty than does proximity to a phase transition. These results have practical consequences: we explain why the widely researched " filter verify " indexing technique used in graph databases is founded upon a misunderstanding of the empirical hardness of NP-complete problems, and cannot be beneficial when paired with any reasonable subgraph isomorphism algorithm. In this paper, we study the problem of subgraph matching that extracts all subgraph isomorphic embeddings of a query graph q in a large data graph G. The existing algorithms for subgraph matching follow Ullmann's backtracking approach; that is, iteratively map query vertices to data vertices by following a matching order of query vertices. It has been shown that the matching order of query vertices is a very important aspect to the efficiency of a subgraph matching algorithm. Recently, many advanced techniques, such as enforcing connectivity and merging similar vertices in query or data graphs, have been proposed to provide an effective matching order with the aim to reduce unpromising intermediate results especially the ones caused by redundant Cartesian products. In this paper, for the first time we address the issue of unpromising results by Cartesian products from "dissimilar" vertices. We propose a new framework by postponing the Cartesian products based on the structure of a query to minimize the redundant Cartesian products. Our second contribution is proposing a new path-based auxiliary data structure, with the size O(|E(G)| x |V(q)|), to generate a matching order and conduct subgraph matching, which significantly reduces the exponential size O(|V(G)||V(q)|-1) of the existing path-based auxiliary data structure, where V (G) and E (G) are the vertex and edge sets of a data graph G, respectively, and V (q) is the vertex set of a query @math . Extensive empirical studies on real and synthetic graphs demonstrate that our techniques outperform the state-of-the-art algorithms by up to @math orders of magnitude. Let G = (V,E) be an n-vertex graph and M_d a d-vertex graph, for some constant d. Is M_d a subgraph of G? We consider this problem in a model where all n processes are connected to all other processes, and each message contains up to O(log n) bits. A simple deterministic algorithm that requires O(n^((d-2) d) log n) communication rounds is presented. For the special case that M_d is a triangle, we present a probabilistic algorithm that requires an expected O(ceil(n^(1 3) (t^(2 3) + 1))) rounds of communication, where t is the number of triangles in the graph, and O(min n^(1 3) log^(2 3) n (t^(2 3) + 1), n^(1 3) ) with high probability. We also present deterministic algorithms specially suited for sparse graphs. In any graph of maximum degree Delta, we can test for arbitrary subgraphs of diameter D in O(ceil(Delta^(D+1) n)) rounds. For triangles, we devise an algorithm featuring a round complexity of O(A^2 n + log_(2+n A^2) n), where A denotes the arboricity of G.