Number Of Subgraphs Research Articles

An approximation algorithm for K-best enumeration of minimal connected edge dominating sets with cardinality constraints

K-best enumeration, which asks to output k-best solutions without duplication, is a helpful tool in data analysis for many fields. In such fields, graphs typically represent data. Thus subgraph enumeration has been paid much attention to such fields. However, k-best enumeration tends to be intractable since, in many cases, finding one optimum solution is NP-hard. To overcome this difficulty, we combine k-best enumeration with a concept of enumeration algorithms called approximation enumeration algorithms. As a main result, we propose a 4-approximation algorithm for minimal connected edge dominating sets which outputs k minimal solutions with cardinality at most 4⋅OPT‾, where OPT‾ is the cardinality of a minimum solution which is not outputted by the algorithm. Our proposed algorithm runs in O(nm2Δ) delay, where n, m, Δ are the number of vertices, the number of edges, and the maximum degree of an input graph.

Second-order moments of the size of randomly induced subgraphs of given order

For a graph G and a positive integer c, let Mc(G) be the size of a subgraph of G induced by a randomly sampled subset of c vertices. Second-order moments of Mc(G) encode part of the structure of G. We use this fact, coupled to classical moment inequalities, to prove graph theoretical results, to give combinatorial identities, to bound the size of the c-densest subgraph from below and the size of the c-sparsest subgraph from above, and to provide bounds for approximate enumeration of trivial subgraphs.

DuMato: An efficient warp-centric subgraph enumeration system for GPU

Subgraph enumeration is a heavy-computing procedure that lies at the core of Graph Pattern Mining (GPM) algorithms, whose goal is to extract subgraphs from larger graphs according to a given property. Scaling GPM algorithms for GPUs is challenging due to irregularity, high memory demand, and non-trivial choice of enumeration paradigms. In this work we propose a depth-first-search subgraph exploration strategy (DFS-wide) to improve the memory locality and access patterns across different enumeration paradigms. We design a warp-centric workflow to the problem that reduces divergences and ensures that accesses to graph data are coalesced. A weight-based dynamic workload redistribution is also proposed to mitigate load imbalance. We put together these strategies in a system called DuMato, allowing efficient implementations of several GPM algorithms via a common set of GPU primitives. Our experiments show that DuMato's optimizations are effective and that it enables exploring larger subgraphs when compared to state-of-the-art systems.

Exploiting Graphics Processing Units to Speed Up Subgraph Enumeration for Efficient Graph Pattern Mining GraphDuMato

Subgraph selection involves searching an input graph for subgraphs with a certain attribute. Graph pattern mining (GPM) relies on this technique, despite its computing complexity and rapid growth. Problems with uncoalesced access to memory, separation, and strain unbalance make efficient subgraph identification on GPUs a huge problem, given GPUs' success in speeding up operations across multiple industries. It is surprising that these challenges have not received sufficient attention in earlier research. This work presents new methods for effectively building and running subgraph enumeration on GPUs. Optimization of computational resource usage is achieved by combining a warp-centric design with a Depth first search (DFS) style approach. Memory efficiency, execution divergence, and GPU activity parallelization could all be enhanced by integrating these two methods. An affordable load balancing level is also incorporated for the purpose of dispersing work among thread warps, which further decreases GPU idleness. The GraphDuMato system facilitates the utilization of GPM methods with its intuitive application programming interface (API). Testing proves that GraphDuMato can outperform state-of-the-art GPM algorithms on a regular basis and may mine subgraphs with up to twelve trees.

Efficient enumeration of maximal split subgraphs and induced sub-cographs and related classes

In this paper, we are interested in algorithms that take in input an arbitrary graph G, and that enumerate in output all the (inclusion-wise) maximal “subgraphs” of G which fulfil a given property Π. All over this paper, we study several different properties Π, and the notion of subgraph under consideration (induced or not) will vary from a result to another.More precisely, we present efficient algorithms to list all maximal split subgraphs, maximal induced cographs and maximal threshold graphs of a given input graph. All the algorithms presented here run in polynomial delay, and moreover for split graphs it only requires polynomial space. In order to develop an algorithm for maximal split (edge-)subgraphs, we establish a bijection between the maximal split subgraphs and the maximal stable sets of an auxiliary graph. For cographs and threshold graphs, the algorithms rely on a framework recently introduced by Conte & Uno (2019) called Proximity Search. Finally we consider the extension problem, which consists in deciding if there exists a maximal induced subgraph satisfying a property Π that contains a set of prescribed vertices and that avoids another set of vertices. We show that this problem is NP-complete for every non-trivial hereditary property Π. We extend the hardness result to some specific edge version of the extension problem.

TED: Towards Discovering Top-k Edge-Diversified Patterns in a Graph Database

With an exponentially growing number of graphs from disparate repositories, there is a strong need to analyze a graph database containing an extensive collection of small- or medium-sized data graphs (e.g., chemical compounds). Although subgraph enumeration and subgraph mining have been proposed to bring insights into a graph database by a set of subgraph structures, they often end up with similar or homogenous topologies, which is undesirable in many graph applications. To address this limitation, we propose the Top-k Edge-Diversified Patterns Discovery problem to retrieve a set of subgraphs that cover the maximum number of edges in a database. To efficiently process such query, we present a generic and extensible framework called Ted which achieves a guaranteed approximation ratio to the optimal result. Two optimization strategies are further developed to improve the performance. Experimental studies on real-world datasets demonstrate the superiority of Ted to traditional techniques.

Proximity Search for Maximal Subgraph Enumeration

This paper proposes a new general technique for maximal subgraph enumeration which we call proximity search, whose aim is to design efficient enumeration algorithms for problems that could not be solved by existing frameworks. To support this claim and illustrate the technique we include output-polynomial algorithms for several problems for which output-polynomial algorithms were not known, including the enumeration of maximal bipartite subgraphs, maximal -degenerate subgraphs (for bounded ), maximal induced chordal subgraphs, and maximal induced trees. Using known techniques, such as reverse search, the space of all maximal solutions induces an implicit directed graph called “solution graph” or “supergraph,” and solutions are enumerated by traversing it; however, nodes in this graph can have exponential out-degree, thus requiring exponential time to be spent on each solution. The novelty of proximity search is a formalization that allows us to define a better solution graph, and a technique, which we call canonical reconstruction, by which we can exploit the properties of given problems to build such graphs. This results in solution graphs whose nodes have significantly smaller (i.e., polynomial) out-degree with respect to existing approaches, but that remain strongly connected, so that all solutions can be enumerated in polynomial delay by a traversal. A drawback of this approach is the space required to keep track of visited solutions, which can be exponential; we further propose a technique to induce a parent-child relationship among solutions and achieve polynomial space when suitable conditions are met.

BFS-based distributed algorithm for parallel local-directed subgraph enumeration

Abstract Estimating the frequency of subgraphs is of importance for many tasks, including subgraph isomorphism, kernel-based anomaly detection and network structure analysis. While multiple algorithms were proposed for full enumeration or sampling-based estimates, these methods fail in very large graphs. Recent advances in parallelization allow for estimates of total subgraph counts in very large graphs. The task of counting the frequency of each subgraph associated with each vertex also received excellent solutions for undirected graphs. However, there is currently no good solution for very large directed graphs. We here propose VDMC (Vertex specific Distributed Motif Counting)—a fully distributed algorithm to optimally count all the three and four vertices connected directed graphs (network motifs) associated with each vertex of a graph. VDMC counts each motif only once and its efficiency is linear in the number of counted motifs. It is fully parallelized to be efficient in GPU-based computation. VDMC is based on three main elements: (1) Ordering the vertices and only counting motifs containing increasing order vertices; (2) sub-ordering motifs based on the average depth of the tree spanning them via a BFS traversal; and (3) removing isomorphisms only once for the entire graph. We here compare VDMC to analytical estimates of the expected number of motifs in Erdős–Rényi graphs and show its accuracy. VDMC is available as a highly efficient CPU and GPU code with a novel data structure for efficient graph manipulation. We show the efficacy of VDMC on real-world graphs. VDMC allows for the precise analysis of subgraph frequency around each vertex in large graphs and opens the way for the extension of methods until now limited to graphs of thousands of edges to graphs with millions of edges and above. GIT: https://github.com/louzounlab/graph-measures/ PyPI: https://pypi.org/project/graph-measures/

Edge-colored directed subgraph enumeration on the connectome

Following significant advances in image acquisition, synapse detection, and neuronal segmentation in connectomics, researchers have extracted an increasingly diverse set of wiring diagrams from brain tissue. Neuroscientists frequently represent these wiring diagrams as graphs with nodes corresponding to a single neuron and edges indicating synaptic connectivity. The edges can contain “colors” or “labels”, indicating excitatory versus inhibitory connections, among other things. By representing the wiring diagram as a graph, we can begin to identify motifs, the frequently occurring subgraphs that correspond to specific biological functions. Most analyses on these wiring diagrams have focused on hypothesized motifs—those we expect to find. However, one of the goals of connectomics is to identify biologically-significant motifs that we did not previously hypothesize. To identify these structures, we need large-scale subgraph enumeration to find the frequencies of all unique motifs. Exact subgraph enumeration is a computationally expensive task, particularly in the edge-dense wiring diagrams. Furthermore, most existing methods do not differentiate between types of edges which can significantly affect the function of a motif. We propose a parallel, general-purpose subgraph enumeration strategy to count motifs in the connectome. Next, we introduce a divide-and-conquer community-based subgraph enumeration strategy that allows for enumeration per brain region. Lastly, we allow for differentiation of edges by types to better reflect the underlying biological properties of the graph. We demonstrate our results on eleven connectomes and publish for future analyses extensive overviews for the 26 trillion subgraphs enumerated that required approximately 9.25 years of computation time.

Microscopic Structural Analysis of Complex Networks: An Empirical Study Using Motifs

Complex Networks can depict a clear image of real-world systems. A real-world scenario can be represented a graph with interconnected layers - called a multilayer network. Finding motifs can give an idea of the topology of complex systems and help understand the graphs’ dynamics. Looking at motifs as atoms of the network is helpful to analyze the relationship between nodes and between layers. This work suggests a sub-graph enumeration approach to find and count the motifs in a multilayer network. The proposed work has many applications in graph mining, particularly to the structure and dynamics of complex networks.

Towards Efficient Distributed Subgraph Enumeration Via Backtracking-Based Framework

Finding or monitoring subgraph instances that are isomorphic to a given pattern graph in a data graph is a fundamental query operation in many graph analytic applications, such as network motif mining and fraud detection. Existing distributed methods are inefficient in communication. They have to shuffle partial matching results during the distributed multiway join. The partial matching results may be much larger than the data graph itself. To overcome the drawback, we develop the Batch-BENU framework for distributed subgraph enumeration on static data graphs. Batch-BENU executes a group of local search tasks in parallel. Each task enumerates subgraphs around a vertex in the data graph, guided by a backtracking-based execution plan. To handle large-scale data graphs that may exceed the memory capacity of a single machine, Batch-BENU stores the data graph in a distributed database. Each task queries adjacency sets of the data graph on demand, shuffling the data graph instead of partial matching results. To support incremental subgraph enumeration on dynamic data graphs, we propose the Streaming-BENU framework. Streaming-BENU turns the problem of enumerating incremental matching results into enumerating all matching results of incremental pattern graphs at each time step. We implement Batch-BENU and Streaming-BENU with the local database cache and the load balance optimization to improve their efficiency. Extensive experiments show that Batch-BENU and Streaming-BENU can scale to big graphs and complex pattern graphs. They outperform the state-of-the-art distributed methods by up to one and two orders of magnitude, respectively.

Exploiting Reuse for GPU Subgraph Enumeration

Subgraph enumeration is important for many applications such as network motif discovery, community detection, and frequent subgraph mining. To accelerate the execution, recent works utilize graphics processing units (GPUs) to parallelize subgraph enumeration. The performances of these parallel schemes are dominated by the set intersection operations which account for up to 95 percent of the total processing time. (Un)surprisingly, a significant portion (as high as 99 percent) of these operations is actually redundant, i.e., the same set of vertices is repeatedly encountered and evaluated. Therefore, in this article, we seek to salvage and recycle the results of such operations to avoid repeated computation. Our solution consists of two phases. In the first phase, we generate a reusable plan that determines the opportunity for reuse. The plan is based on a novel reuse discovery mechanism that can identify available results to prevent redundant computation. In the second phase, the plan is executed to produce the subgraph enumeration results. This processing is based on a newly designed reusable parallel search strategy that can efficiently maintain and retrieve the results of set intersection operations. Our implementation on GPUs shows that our approach can achieve up to 5 times speedups compared with the state-of-the-art GPU solutions.

Graph Similarity-Based Maximum Stable Subgraph Extraction of Information Topology From a Vehicular Network

Information topology graph has been extensively used in modeling network management, where it is necessary to continually optimize the management of the dynamic network. However, frequent reconfiguration may lead to an increased overhead and a degraded latency. Since the mobility in a vehicular network is restricted by the road, there is an internal relatively stable part embedded in the entire dynamic network. Thus, in this study, this characteristic is first used to reduce reconfiguration, where the maximum stable subgraph is extracted and configuration only needs to be reconsidered for the rest of the network. However, there are three main challenges in the modeling and solving procedure, namely the evaluation of stability of part of the network, the assurance of the coherence in the set of extracted vehicles and the huge computation complexity of the subgraph enumeration. Therefore, in this study, first, the vehicular network is modeled by constructing the time-evolving information topology graph, in which a subgraph similarity is constructed to quantify the stability of part of network. Second, a maximum stable subgraph extraction optimization problem is presented, where the coherence constraint of vehicles is proposed. Finally, a gradual contraction extraction algorithm is proposed to solve the optimization problem with a low complexity. The simulation results reveal that the proposed algorithm can extract a stable subgraph with an approximate maximum size, and has a higher adaptability than the compared ones. Furthermore, it was found that the application of the proposed algorithm in resource allocation can effectively reduce the reconfiguration.

Connected Subgraph Fingerprints: Representing Molecules Using Exhaustive Subgraph Enumeration.

Molecular fingerprints are an efficient and widely used method for similarity-driven virtual screening. Most fingerprint methods can be distinguished by the class of structural features considered. The Connected Subgraph Fingerprint (CSFP) overcomes this limitation and regards all structural features of a compound. This results in a more complete feature space and high adaptive potential to certain application scenarios. The novel descriptor surpasses widely used fingerprint methods in some cases and opens the way for topological search in combinatorial fragment spaces.

FSM: Fast and scalable network motif discovery for exploring higher-order network organizations.

Networks exhibit rich and diverse higher-order organizational structures. Network motifs, which are recurring significant patterns of inter-connections, are recognized as fundamental units to study the higher-order organizations of networks. However, the principle of selecting representative network motifs for local motif based clustering remains largely unexplored. We present a scalable algorithm called FSM for network motif discovery. FSM is advantageous in twofold. First, it accelerates the motif discovery process by effectively reducing the number of times for subgraph isomorphism labeling. Second, FSM adopts multiple heuristic optimizations for subgraph enumeration and classification to further improve its performance. Experimental results on biological networks show that, comparing with the existing network motif discovery algorithm, FSM is more efficient on computational efficiency and memory usage. Furthermore, with the large, frequent, and sparse network motifs discovered by FSM, the higher-order organizational structures of biological networks were successfully revealed, indicating that FSM is suitable to select network representative network motifs for exploring high-order network organizations.

Fast and robust distributed subgraph enumeration

We study the subgraph enumeration problem under distributed settings. Existing solutions either suffer from severe memory crisis or rely on large indexes, which makes them impractical for very large graphs. Most of them follow a synchronous model where the performance is often bottlenecked by the machine with the worst performance. Motivated by this, in this paper, we propose RADS, a Robust Asynchronous Distributed Subgraph enumeration system. RADS first identifies results that can be found using single-machine algorithms. This strategy not only improves the overall performance but also reduces network communication and memory cost. Moreover, RADS employs a novel region-grouped multi-round expand verify & filter framework which does not need to shuffle and exchange the intermediate results, nor does it need to replicate a large part of the data graph in each machine. This feature not only reduces network communication cost and memory usage, but also allows us to adopt simple strategies for memory control and load balancing, making it more robust. Several optimization strategies are also used in RADS to further improve the performance. Our experiments verified the superiority of RADS to state-of-the-art subgraph enumeration approaches.

A linear delay algorithm for enumerating all connected induced subgraphs

BackgroundReal biological and social data is increasingly being represented as graphs. Pattern-mining-based graph learning and analysis techniques report meaningful biological subnetworks that elucidate important interactions among entities. At the backbone of these algorithms is the enumeration of pattern space.ResultsWe propose an efficient algorithm for enumerating all connected induced subgraphs of an undirected graph. Building on this enumeration approach, we propose an algorithm for mining all maximal cohesive subgraphs that integrates vertices’ attributes with subgraph enumeration. To efficiently mine all maximal cohesive subgraphs, we propose two pruning techniques that remove futile search nodes in the enumeration tree.ConclusionsExperiments on synthetic and real graphs show the effectiveness of the proposed algorithm and the pruning techniques. On enumerating all connected induced subgraphs, our algorithm is several times faster than existing approaches. On dense graphs, the proposed approach is at least an order of magnitude faster than the best existing algorithm. Experiments on protein-protein interaction network with cancer gene dysregulation profile show that the reported cohesive subnetworks are biologically interesting.

Enumerating Trillion Subgraphs On Distributed Systems

How can we find patterns from an enormous graph with billions of vertices and edges? The subgraph enumeration, which is to find patterns from a graph, is an important task for graph data analysis with many applications, including analyzing the social network evolution, measuring the significance of motifs in biological networks, observing the dynamics of Internet, and so on. Especially, the triangle enumeration, a special case of the subgraph enumeration, where the pattern is a triangle, has many applications such as identifying suspicious users in social networks, detecting web spams, and finding communities. However, recent networks are so large that most of the previous algorithms fail to process them. Recently, several MapReduce algorithms have been proposed to address such large networks; however, they suffer from the massive shuffled data resulting in a very long processing time. In this article, we propose scalable methods for enumerating trillion subgraphs on distributed systems. We first propose PTE ( Pre-partitioned Triangle Enumeration ), a new distributed algorithm for enumerating triangles in enormous graphs by resolving the structural inefficiency of the previous MapReduce algorithms. PTE enumerates trillions of triangles in a billion scale graph by decreasing three factors: the amount of shuffled data, total work, and network read. We also propose PSE ( Pre-partitioned Subgraph Enumeration ), a generalized version of PTE for enumerating subgraphs that match an arbitrary query graph. Experimental results show that PTE provides 79 times faster performance than recent distributed algorithms on real-world graphs, and succeeds in enumerating more than 3 trillion triangles on the ClueWeb12 graph with 6.3 billion vertices and 72 billion edges. Furthermore, PSE successfully enumerates 265 trillion clique subgraphs with 4 vertices from a subdomain hyperlink network, showing 47 times faster performance than the state of the art distributed subgraph enumeration algorithm.

Multiplex network motifs as building blocks of corporate networks

In corporate networks, firms are connected through links of corporate ownership and shared directors, connecting the control over major economic actors in our economies in meaningful and consequential ways. Most research thus far focused on the connectedness of firms as a result of one particular link type, analyzing node-specific metrics or global network-based methods to gain insights in the modelled corporate system.In this paper, we aim to understand multiplex corporate networks with multiple types of connections, specifically investigating the network’s essential building blocks: multiplex network motifs. Motifs, which are small subgraph patterns occurring at significantly higher frequencies than in similar random networks, have demonstrated their usefulness in understanding the structure of many types of real-world networks. However, detecting motifs in multiplex networks is nontrivial for two reasons. First of all, there are no out-of-the-box subgraph enumeration algorithms for multiplex networks. Second, existing null models to test network motif significance, are unable to incorporate the interlayer dependencies in the multiplex network. We solve these two issues by introducing a layer encoding algorithm that incorporates the multiplex aspect in the subgraph enumeration phase. In addition, we propose a null model that is able to preserve the interlayer connectedness, while taking into account that one of the link types is actually the result of a projection of an underlying bipartite network.The experimental section considers the corporate network of Germany, in which tens of thousands of firms are connected through several hundred thousand links. We demonstrate how incorporating the multiplex aspect in motif detection is able to reveal new insights that could not be obtained by studying only one type of relationship. In a general sense, the motifs reflect known corporate governance practices related to the monitoring of investments and the concentration of ownership. A substantial fraction of the discovered motifs is typical for an industrialized country such as Germany, whereas others seem specific for certain economic sectors. Interestingly, we find that motifs involving financial firms are over-represented amongst the larger and more complex motifs. This demonstrates the prominent role of the financial sector in Germany’s largely industry-oriented corporate network.

PegasusN: A Scalable and Versatile Graph Mining System

How can we find patterns and anomalies in peta-scale graphs? Even recently proposed graph mining systems fail in processing peta-scale graphs. In this work, we propose PegasusN, a scalable and versatile graph mining system that runs on Hadoop and Spark. To handle enormous graphs, PegasusN provides and seamlessly integrates efficient algorithms for various graph mining operations: graph structure analyses, subgraph enumeration, graph generation, and graph visualization. PegasusN quickly processes extra-large graphs that other systems cannot handle.