Graph Compression Research Articles

Compressed zero-divisor graphs of matrix rings over finite fields

ABSTRACT We extend the notion of the compressed zero-divisor graph to noncommutative rings in a way that still induces a product preserving functor from the category of finite unital rings to the category of directed graphs. For a finite field F, we investigate the properties of , the graph of the matrix ring over F, and give a purely graph-theoretic characterization of this graph when . For we prove that every graph automorphism of is induced by a ring automorphism of . We also show that for finite unital rings R and S, where S is semisimple and has no homomorphic image isomorphic to a field, if , then . In particular, this holds if with .

Universal classes in the category of simple graphs

In this article, we show that the concept of graph compression is important for the study of universal equivalence of simple graphs (not necessarily finite) and graph algebraic structures.

On diameter of the zero-divisor and the compressed zero-divisor graphs of skew Laurent polynomial rings

Let [Formula: see text] be an associative ring with nonzero identity. The zero-divisor graph [Formula: see text] of [Formula: see text] is the (undirected) graph with vertices the nonzero zero-divisors of [Formula: see text], and distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] or [Formula: see text]. Let [Formula: see text] and [Formula: see text] be the set of all right annihilators and the set of all left annihilator of an element [Formula: see text], respectively, and let [Formula: see text]. The relation on [Formula: see text] given by [Formula: see text] if and only if [Formula: see text] is an equivalence relation. The compressed zero-divisor graph [Formula: see text] of [Formula: see text] is the (undirected) graph with vertices the equivalence classes induced by [Formula: see text] other than [Formula: see text] and [Formula: see text], and distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] or [Formula: see text]. The goal of our paper is to study the diameter of zero-divisor and the compressed zero-divisor graph of skew Laurent polynomial rings over noncommutative rings. We give a complete characterization of the possible diameters of [Formula: see text] and [Formula: see text], where the base ring [Formula: see text] is reversible and also has the [Formula: see text]-compatible property.

Domination number of graphs associated with rings

The present work aims to exploit the interplay between the algebraic properties of rings and the graph-theoretic structures of their associated graphs. Let [Formula: see text] be an associative (not necessarily commutative) ring. We focus on the domination number of the zero-divisor graph [Formula: see text], the compressed zero-divisor graph [Formula: see text] and the unit graph [Formula: see text]. We find some relations between the domination number of the zero-divisor graph and that of the compressed zero-divisor graph. Moreover, some relations between the domination number of [Formula: see text] and [Formula: see text], as well as the relations between the domination number of [Formula: see text] and [Formula: see text], are studied.

Regular Decomposition of Large Graphs: Foundation of a Sampling Approach to Stochastic Block Model Fitting

We analyze the performance of regular decomposition, a method for compression of large and dense graphs. This method is inspired by Szemerédi’s regularity lemma (SRL), a generic structural result of large and dense graphs. In our method, stochastic block model (SBM) is used as a model in maximum likelihood fitting to find a regular structure similar to the one predicted by SRL. Another ingredient of our method is Rissanen’s minimum description length principle (MDL). We consider scaling of algorithms to extremely large size of graphs by sampling a small subgraph. We continue our previous work on the subject by proving some experimentally found claims. Our theoretical setting does not assume that the graph is generated from a SBM. The task is to find a SBM that is optimal for modeling the given graph in the sense of MDL. This assumption matches with real-life situations when no random generative model is appropriate. Our aim is to show that regular decomposition is a viable and robust method for large graphs emerging, say, in Big Data area.

Supergraph Search in Graph Databases via Hierarchical Feature-Tree

Supergraph search is a fundamental problem in graph databases that is widely applied in many application scenarios. Given a graph database and a query-graph, supergraph search retrieves all data-graphs contained in the query-graph from the graph database. Most existing solutions for supergraph search follow the pruning-and-verification framework, which prune false answers based on features in the pruning phase and perform subgraph isomorphism testings on the remaining graphs in the verification phase. However, they are not scalable to handle large-sized data-graphs and query-graphs due to three drawbacks. First, they rely on a frequent subgraph mining algorithm to select features which is expensive and cannot generate large features. Second, they require a costly verification phase. Third, they process features in a fixed order without considering their relationships to the query-graph. In this paper, we address the three drawbacks and propose new indexing and query processing algorithms. In indexing, we select features directly from the data-graphs without expensive frequent subgraph mining. The features form a feature-tree that contains all-sized features and both the cost sharing and pruning power of the features are considered. In query processing, we propose a new algorithm, where the order to process features is query-dependent by considering both the cost sharing and the pruning power. We explore two optimization strategies to further improve the algorithm efficiency. The first strategy applies a lightweight graph compression technique and the second strategy optimizes the inclusion of answers. We further introduce how to efficiently maintain the index incrementally when the graph database is updated dynamically. Moreover, we propose an approximation approach to significantly reduce the computational cost for large data-graphs and/or query-graphs while preserving a high result quality. Finally, we conduct extensive performance studies on two real large datasets to demonstrate the efficiency and effectiveness of our algorithms.

Social context-aware trust paths finding for trustworthy service provider selection in social media

Online Social Network (OSN) has been used to enhance service provision and service selection, where trust is one of the most important factors for the decision making of service consumers. Thus, a significant and challenging problem is how to effectively and efficiently find those social trust paths that can yield trustworthy trust evaluation results based on the requirements of a service consumer particularly in contextual OSNs which contains social contexts, like social relationships and social trust between participants, and social positions of participants. In this paper, we propose a new concept called Strong Social Graph (SSG), consisting of participants with strong social connections. We also propose an approach to identify SSGs, and propose a novel index method and a graph compression method for SSG. Then based on the compressed SSG and indices, we propose a new efficient and effective approximation algorithm, called SSG-MCBA by adopting the Monte Carlo method and our optimization search strategies. The experiments conducted onto two real social network datasets illustrate that SSG-MCBA greatly outperforms the state-of-the-art method in both efficiency and effectiveness.

Correct filtering for subgraph isomorphism search in compressed vertex-labeled graphs

We present a revised filtering process that corrects an error in our previous work on subgraph isomorphism search over compressed graphs. We provide additional experiments to test the contribution of the compression itself and the candidate filtering to the performance of four representative backtracking-based subgraph isomorphism search algorithms.

HcRPC: Highly Compact Reachability Preserving Graph Compression With Corrections

Graphs are used in numerous applications to model real-world systems and phenomena. The ever increasing size of graphs makes them difficult to query and analyze. In this paper, we propose HcRPC , a Highly compact Reachability Preserving Graph compression algorithm with Corrections, which is capable of preserving the reachability relations between the nodes in original graph. The highly compressed representation of a given graph consists of a compressed graph and a set of corrections. The original graph is compressed on the basis of equivalence class obtained via the reachability relations between nodes in the original graph. In the compressed graph, each node corresponds to a set of nodes from the original graph with similar ancestors and descendants, and each edge represents linkage between the original nodes in any two node sets. The corrections portion specifies the set of corrections, including equivalent class-node corrections and node-node corrections. MinHash technique is utilized to speed up checking whether equivalence classes are structure-similar and the pair of equivalence classes with high similarity are thus merged to acquire a highly compressed graph. Besides, we develop an algorithm for preserving compressed graph with a set of corrections in response to changes to the original graph. We evaluate our algorithms on real-life graph data sets and the results indicate that graph data sets can be highly compressed while preserving the reachability relations between nodes.

A novel graph compression algorithm for data-intensive scientific networks

A novel graph compression algorithm for data-intensive scientific networks

A novel graph compression algorithm for data-intensive scientific networks

As one of the world's leading scientific and data-intensive computing grids, the worldwide LHC computing grid (WLCG) faces the challenge of improving its computing efficiency and network utilisation. To achieve this goal, WLCG needs an important piece of information: the network topology graphs of participating computing grids. Directly collecting such information from all of the grids, however, would cause high communication overhead and raise many security issues. In this paper, we address these issues by proposing a novel algorithm to compress such a large network topology into a compact, equivalent network topology. We formally define our problem, develop a novel, efficient topology compression algorithm and evaluate its performance using real-world network topologies. Our results show that our algorithm not only achieves a much higher topology compression ratio than state-of-the-art topology transformation algorithms, but also leads to at most 100× reduction in computation time.

StarZIP: Streaming Graph Compression Technique for Data Archiving

The size of a streaming graph is possibly unbounded, and it is updated by a continuous sequence of edges over time. Due to numerous types of real-world interactions, the nature of edge arrival in a streaming graph is dynamic and holds different types of temporal subgraphs, such as stars, bipartite forms, cliques, and chains. The most current techniques find such subgraphs in each snapshot of a dynamic graph and use a dictionary or hash-based summary to compress the graph before applying a stitching technique to demonstrate its temporal behavior. However, it remains difficult to discover those subgraph structures from the continuous stream of edges found in large and rapidly changing dynamic graphs. In this paper, we propose a streaming graph compression algorithm, StarZIP, that uses a new encoding scheme. Our motivational factor is real-world graphs that contain an overwhelmingly large number of stars and a few other structures. We have observed that all subgraph structures can be represented as star-shaped subgraphs. Moreover, the star-shaped representation can easily be arranged in the form of an inverted index, which enables the application of different inverted list encoding techniques for compression. Therefore, we shatter a graph into a uniform representation of stars to compress it. The evaluation of StarZIP on real-world datasets shows that our proposed system reduced the size of a highly dense graph to 60 times less than its original size. Moreover, the experimental results indicate that StarZIP compression is 4 times better than the state-of-the-art techniques.

An information-theoretic framework for the lossy compression of link streams

Graph compression is a data analysis technique that consists in the replacement of parts of a graph by more concise structural patterns in order to reduce its description length. It notably provides interesting exploration tools for the study of real, large-scale, and complex graphs which cannot be grasped at first glance. This article proposes a framework for the compression of temporal graphs, that is for the compression of graphs that evolve with time. This framework first builds on a simple and limited scheme, exploiting structural equivalence for the lossless compression of static graphs, then generalises it to the lossy compression of link streams, a recent formalism for the study of temporal graphs. Such generalisation builds on the natural extension of (bidimensional) relational data by the addition of a third temporal dimension. Moreover, we introduce an information-theoretic measure to quantify and to control the information that is lost during compression, as well as an algebraic characterisation of the space of possible compression patterns to enhance the expressiveness of the initial compression scheme. These contributions lead to the definition of a combinatorial optimisation problem, that is the Lossy Multistream Compression Problem, for which we provide an exact algorithm.

CGAcc: A Compressed Sparse Row Representation-Based BFS Graph Traversal Accelerator on Hybrid Memory Cube

Graph traversal is widely used in map routing, social network analysis, causal discovery and many more applications. Because it is a memory-bound process, graph traversal puts significant pressure on the memory subsystem. Due to poor spatial locality and the increasing size of today’s datasets, graph traversal consumes an ever-larger part of application execution time. One way to mitigate this cost is memory prefetching, which issues requests from the processor to the memory in anticipation of needing certain data. However, traditional prefetching does not work well for graph traversal due to data dependencies, the parallel nature of graphs and the need to move vast amounts of data from memory to the caches. In this paper, we propose a compressed sparse row representation-based graph accelerator on the Hybrid Memory Cube (HMC), called CGAcc. CGAcc combines Compressed Sparse Row (CSR) graph representation with in-memory prefetching and processing to improve the performance of graph traversal. Our approach integrates the prefetching and processing in the logic layer of a 3D stacked Dynamic Random-Access Memory (DRAM) architecture, based on Micron’s HMC. We selected HMC to implement CGAcc because it can provide quite high bandwidth and low access latency. Furthermore, this device has multiple DRAM layers connected to internal logic to control memory access and perform rudimentary computation. Using the CSR representation, CGAcc deploys prefetchers in the HMC to exploit the short transaction latency between the logic and DRAM layers. By doing this, it can also avoid large data movement costs. In the runtime, CGAcc pipelines the prefetching to fetch data from DRAM arrays to improve memory-level parallelism. To further reduce the access latency, several optimized internal caches are also introduced to hold the prefetched data to be Processed In-Memory (PIM). A comprehensive evaluation shows the effectiveness of CGAcc. Experimental results showed that, compared to a conventional HMC main memory equipped with a stream prefetcher, CGAcc achieved an average 3.51× speedup with moderate hardware cost.

An Exponential Lower Bound for Cut Sparsifiers in Planar Graphs

Given an edge-weighted graph G with a set Q of k terminals, a mimicking network is a graph with the same set of terminals that exactly preserves the size of minimum cut between any partition of the terminals. A natural question in the area of graph compression is to provide as small mimicking networks as possible for input graph G being either an arbitrary graph or coming from a specific graph class. We show an exponential lower bound for cut mimicking networks in planar graphs: there are edge-weighted planar graphs with k terminals that require 2^{k-2} edges in any mimicking network. This nearly matches an upper bound of mathcal {O}(k 2^{2k}) of Krauthgamer and Rika (in: Khanna (ed) Proceedings of the twenty-fourth annual ACM-SIAM symposium on discrete algorithms, SODA 2013, New Orleans, 2013) and is in sharp contrast with the upper bounds of mathcal {O}(k^2) and mathcal {O}(k^4) under the assumption that all terminals lie on a single face (Goranci et al., in: Pruhs and Sohler (eds) 25th Annual European symposium on algorithms (ESA 2017), 2017, arXiv:1702.01136; Krauthgamer and Rika in Refined vertex sparsifiers of planar graphs, 2017, arXiv:1702.05951). As a side result we show a tight example for double-exponential upper bounds given by Hagerup et al. (J Comput Syst Sci 57(3):366–375, 1998), Khan and Raghavendra (Inf Process Lett 114(7):365–371, 2014), and Chambers and Eppstein (J Gr Algorithms Appl 17(3):201–220, 2013).

Fast de Bruijn Graph Compaction in Distributed Memory Environments.

De Bruijn graph based genome assembly has gained popularity as short read sequencers become ubiquitous. A core assembly operation is the generation of unitigs, which are sequences corresponding to chains in the graph. Unitigs are used as building blocks for generating longer sequences in many assemblers, and can facilitate graph compression. Chain compaction, by which unitigs are generated, remains a critical computational task. In this paper, we present a distributed memory parallel algorithm for simultaneous compaction of all chains in bi-directed de Bruijn graphs. The key advantages of our algorithm include bounding the chain compaction run-time to logarithmic number of iterations in the length of the longest chain, and ability to differentiate cycles from chains within logarithmic number of iterations in the length of the longest cycle. Our algorithm scales to thousands of computational cores, and can compact a whole genome de Bruijn graph from a human sequence read set in 7.3 seconds using 7680 distributed memory cores, and in 12.9 minutes using 64 shared memory cores. It is 3.7× and 2.0× faster than equivalent steps in the state-of-the-art tools for distributed and shared memory environments, respectively. An implementation of the algorithm is available at https://github.com/ParBLiSS/bruno.

QoS-Guaranteed Wireless Broadcast Scheduling With Network Coding and Rate Adaptation

Network coding is critical to wireless broadcast for real-time applications. Most of the existing approaches make strong assumptions either on application requirements (e.g., single constraint type or the same constraint level) or on data transmission (e.g., ideal channel, fixed transmission rate, or packet length) in order to facilitate formalization and solution. Those assumptions limit the applicability of previous approaches. This work studies quality-of-service-guaranteed broadcast scheduling over wireless networks with network coding and rate selection capabilities, focusing on reducing broadcast completion delay while maximizing the number of packet receptions that satisfy heterogeneous deadline and reliability requirements. To begin with, a multirate graph model is constructed to formulate the optimal broadcast scheduling problem, which is proved to be NP-hard. Then, an adaptive graph compression policy is proposed to reduce the computational burden significantly without sacrificing performance. Furthermore, an approximation framework is presented for each propagation. In the framework, the coding strategy and rate selection can be formulated as a complexity-adjustable clique problem. Finally, a progressive clique search algorithm is designed to make decision on each broadcast. Simulation results demonstrate that compared with typical heuristic algorithms, the proposed algorithm achieves significant performance improvement with lower complexity.

Improving centralized path calculation based on graph compression

Shortest-path calculation on weighted graphs are an essential operation in computer networks. The performance of such algorithms has become a critical challenge in emerging software-defined networks (SDN), since SDN controllers need to centralizedly perform a shortest-path query for every flow, usually on large-scale network. Unfortunately, one of the challenges is that current algorithms will become incalculable as the network size increases. Therefore, inspired by the compression graph in the field of compute visualization, we propose an efficient shortest path algorithm by compressing the original big network graph into a small one, but the important graph properties used to calculate path is reserved. We implement a centralized version of our approach in SDN-enabled network, and the evaluations validate the improvement compared with the well-known algorithms.

MCS-GPM: Multi-Constrained Simulation Based Graph Pattern Matching in Contextual Social Graphs

Graph Pattern Matching (GPM) has been used in lots of areas, like biology, medical science, and physics. With the advent of Online Social Networks (OSNs), recently, GPM has been playing a significant role in social network analysis, which has been widely used in, for example, finding experts, social community mining, and social position detection. Given a query which contains a pattern graph $G_Q$ and a data graph $G_D$ , a GPM algorithm finds those subgraphs, $G_M$ , that match $G_Q$ in $G_D$ . However, the existing GPM methods do not consider the multiple end-to-end constraints of the social contexts, like social relationships, social trust, and social positions on edges in $G_Q$ , which are commonly found in various applications, such as crowdsourcing travel, social network based e-commerce, and study group selection, etc. In this paper, we first conceptually extend Bounded Simulation to Multi-Constrained Simulation (MCS) , and propose a novel NP-Complete Multi-Constrained Graph Pattern Matching (MC-GPM) problem. Then, to address the efficiency issue in large-scale MC-GPM, we propose a new concept called Strong Social Component (SSC), consisting of participants with strong social connections. We also propose an approach to identifying SSCs, and propose a novel index method and a graph compression method for SSC. Moreover, we devise a multithreading heuristic algorithm, called M-HAMC, to bidirectionally search the MC-GPM results in parallel without decompressing graphs. An extensive empirical study over five real-world large-scale social graphs has demonstrated the effectiveness and efficiency of our approach.

A Semantic Approach to Data Reduction for Weighted Graphs and Complex Queries

Graphs are widely used to store complex data nowadays: social networks, recommendation engines, computer networks and bioinformatics, to name a few. With a rapidly growing amount of data on the Internet in recent years, designing scalable systems to process the huge graph data efficiently has become a critical issue. In order to store and process the graph data efficiently in memory/disk, as well as to save time for transferring the data, graph compression techniques are often used. However, most of the existing graph data compression approaches are syntactic, which means they focus on graph structure and reduce it by serialization or redundancy removal. In this paper we focus on a semantic approach, namely query-based graph data reduction, which reduces a graph by preserving only the information relevant to the queries needed by an application. We study several classical graph problems and their applications, and design a suite of graph reduction algorithms to generate reduced graphs in which an application can still compute the same solutions. In addition, we design a synthesis method that can combine existing graph reduction algorithms to generate a reduced graph for a complex graph problem that includes more than one constraint. We also discuss incremental maintenance in order to update a reduced graph without reprocessing the whole graph again when the original graph is modified. We conduct experiments to compare the reduction rate of our algorithms with different sizes and types of data.