Graph Compression Research Articles

Eigenvalues of zero-divisor graphs of finite commutative rings

We investigate eigenvalues of the zero-divisor graph Gamma (R) of finite commutative rings R and study the interplay between these eigenvalues, the ring-theoretic properties of R and the graph-theoretic properties of Gamma (R). The graph Gamma (R) is defined as the graph with vertex set consisting of all nonzero zero-divisors of R and adjacent vertices x, y whenever xy = 0. We provide formulas for the nullity of Gamma (R), i.e., the multiplicity of the eigenvalue 0 of Gamma (R). Moreover, we precisely determine the spectra of Gamma ({mathbb {Z}}_p times {mathbb {Z}}_p times {mathbb {Z}}_p) and Gamma ({mathbb {Z}}_p times {mathbb {Z}}_p times {mathbb {Z}}_p times {mathbb {Z}}_p) for a prime number p. We introduce a graph product times _{Gamma } with the property that Gamma (R) cong Gamma (R_1) times _{Gamma } cdots times _{Gamma } Gamma (R_r) whenever R cong R_1 times cdots times R_r. With this product, we find relations between the number of vertices of the zero-divisor graph Gamma (R), the compressed zero-divisor graph, the structure of the ring R and the eigenvalues of Gamma (R).

A community detection algorithm based on graph compression for large-scale social networks

Uncovering the underlying community structure of a social network is an important task in social network analysis. To solve this problem, many community detection algorithms for the full topology of an original social network have been proposed. However, these algorithms are not very effective at analyzing large-scale social networks. To overcome this deficiency, this paper proposes a community detection algorithm based on graph compression. Specifically, a compressed graph is first obtained by iteratively merging vertices with a degree of 1 or 2 into their neighbors with a higher degree. Then, two indices, i.e., the density and quality of vertices, are defined to evaluate the probability of vertices as community seeds. By considering these two measures together, in a compressed social network, the number of communities and the corresponding initial community seeds are determined simultaneously. After obtaining the community structure of the compressed social network via seed expansion, the community results are propagated to the original social network. Extensive experiments conducted on various social networks have demonstrated the superiority of our proposal compared to several existing state-of-the-art community detection algorithms.

On the compressed essential graph of a module over a commutative ring

Let [Formula: see text] be a commutative ring and [Formula: see text] be an [Formula: see text]-module. The compressed essential graph of [Formula: see text], denoted by [Formula: see text] is a simple undirected graph associated to [Formula: see text] whose vertices are classes of torsion elements of [Formula: see text] and two distinct classes [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] is an essential ideal of [Formula: see text]. In this paper, we study diameter and girth of [Formula: see text] and we characterize all modules for which the compressed essential graph is connected. Moreover, it is proved that [Formula: see text], whenever [Formula: see text] is Noetherian and [Formula: see text] is a finitely generated multiplication module with [Formula: see text].

Compressed graph representation for scalable molecular graph generation

Recently, deep learning has been successfully applied to molecular graph generation. Nevertheless, mitigating the computational complexity, which increases with the number of nodes in a graph, has been a major challenge. This has hindered the application of deep learning-based molecular graph generation to large molecules with many heavy atoms. In this study, we present a molecular graph compression method to alleviate the complexity while maintaining the capability of generating chemically valid and diverse molecular graphs. We designate six small substructural patterns that are prevalent between two atoms in real-world molecules. These relevant substructures in a molecular graph are then converted to edges by regarding them as additional edge features along with the bond types. This reduces the number of nodes significantly without any information loss. Consequently, a generative model can be constructed in a more efficient and scalable manner with large molecules on a compressed graph representation. We demonstrate the effectiveness of the proposed method for molecules with up to 88 heavy atoms using the GuacaMol benchmark.

A Hybrid Update Strategy for I/O-Efficient Out-of-Core Graph Processing

In recent years, a number of out-of-core graph processing systems have been proposed to process graphs with billions of edges on just one commodity computer, due to their high cost efficiency. To obtain a better performance, these systems adopt a full I/O model that scans all edges during the computation to avoid the inefficiency of random I/Os. Although this model ensures good I/O access locality, it leads to a large number of useless edges to be loaded when running graph algorithms that only access a small portion of edges in each iteration. An intuitive method to solve this I/O inefficiency problem is the on-demand I/O model that only accesses the active edges. However, this method only works well for the graph algorithms with very few active edges, since the I/O cost will grow rapidly as the number of active edges increases due to the increasing amount of random I/Os. In this article, we present HUS-Graph, an efficient out-of-core graph processing system to address the above I/O issues and achieve a good balance between I/O traffic and I/O access locality. HUS-Graph adopts a hybrid update strategy including two update models, Row-oriented Push (ROP) and Column-oriented Pull (COP). It supports switching between ROP and COP adaptively, for the graph algorithms that have different computation and I/O features. For traversal-based algorithms, HUS-Graph also provides an immediate propagation-based vertex update scheme to accelerate the vertex state propagation and convergence speed. Furthermore, HUS-Graph adopts a locality-optimized dual-block representation to organize graph data and an I/O-based performance prediction method to enable the system to dynamically select the optimal update model between ROP and COP. To save the disk space and further reduce I/O traffic, HUS-Graph implements a space-efficient storage format by combining several graph compression methods. Extensive experimental results show that HUS-Graph outperforms two existing out-of-core systems GraphChi and GridGraph by 1.2x-52.8x.

Interpretable multi-scale graph descriptors via structural compression

Graph representations that preserve relevant topological information allow the use of a rich machine learning toolset for data-driven network analytics. Some notable graph representations in the literature are fruitful in their respective applications but they either lack interpretability or are unable to effectively encode a graph’s structure at both local and global scale. In this work, we propose the Higher-Order Structure Descriptor (HOSD): an interpretable graph descriptor that captures information about the patterns in a graph at multiple scales. Scaling is achieved using a novel graph compression technique that reveals successive higher-order structures. The proposed descriptor is invariant to node permutations due to its graph-theoretic nature. We analyze the HOSD algorithm for time complexity and also prove the NP-completeness of three interesting graph compression problems. A faster version, HOSD-Lite, is also presented to approximate HOSD on dense graphs. We showcase the interpretability of our model by discussing structural patterns found within real-world datasets using HOSD. HOSD and HOSD-Lite are evaluated on benchmark datasets for applicability to classification problems; results demonstrate that a simple random forest setup based on our representations competes well with the current state-of-the-art graph embeddings.

Compression of Dynamic Graphs Generated by a Duplication Model

We continue building up the information theory of non-sequential data structures such as trees, sets, and graphs. In this paper, we consider dynamic graphs generated by a full duplication model in which a new vertex selects an existing vertex and copies all of its neighbors. We ask how many bits are needed to describe the labeled and unlabeled versions of such graphs. We first estimate entropies of both versions and then present asymptotically optimal compression algorithms up to two bits. Interestingly, for the full duplication model the labeled version needs Theta (n) bits while its unlabeled version (structure) can be described by Theta (log n) bits due to significant amount of symmetry (i.e. large average size of the automorphism group of sample graphs).

C-Blondel: An Efficient Louvain-Based Dynamic Community Detection Algorithm

One of the most interesting topics in the scope of social network analysis is dynamic community detection, keeping track of communities’ evolutions in a dynamic network. This article introduces a new Louvain-based dynamic community detection algorithm relied on the derived knowledge of the previous steps of the network evolution. The algorithm builds a compressed graph, where its supernodes represent the detected communities of the previous step and its superedges show the edges among the supernodes. The algorithm not only constructs the compressed graph with low computational complexity but also detects the communities through the integration of the Louvain algorithm into the graph. The efficiency of the proposed algorithms is widely investigated in this article. By doing so, several evaluations have been performed over three standard real-world data sets, namely Enron Email, Cit-HepTh, and Facebook data sets. The obtained results indicate the superiority of the proposed algorithm with respect to the execution time as an efficiency metric. Likewise, the results show the modularity of the proposed algorithm as another effectiveness metric compared with the other well-known related algorithms.

PACC: Large scale connected component computation on Hadoop and Spark.

A connected component in a graph is a set of nodes linked to each other by paths. The problem of finding connected components has been applied to diverse graph analysis tasks such as graph partitioning, graph compression, and pattern recognition. Several distributed algorithms have been proposed to find connected components in enormous graphs. Ironically, the distributed algorithms do not scale enough due to unnecessary data IO & processing, massive intermediate data, numerous rounds of computations, and load balancing issues. In this paper, we propose a fast and scalable distributed algorithm PACC (Partition-Aware Connected Components) for connected component computation based on three key techniques: two-step processing of partitioning & computation, edge filtering, and sketching. PACC considerably shrinks the size of intermediate data, the size of input graph, and the number of rounds without suffering from load balancing issues. PACC performs 2.9 to 10.7 times faster on real-world graphs compared to the state-of-the-art MapReduce and Spark algorithms.

Traversing large graphs on GPUs with unified memory

Due to the limited capacity of GPU memory, the majority of prior work on graph applications on GPUs has been restricted to graphs of modest sizes that fit in memory. Recent hardware and software advances make it possible to address much larger host memory transparently as a part of a feature known as unified virtual memory. While accessing host memory over an interconnect is understandably slower, the problem space has not been sufficiently explored in the context of a challenging workload with low computational intensity and an irregular data access pattern such as graph traversal. We analyse the performance of breadth first search (BFS) for several large graphs in the context of unified memory and identify the key factors that contribute to slowdowns. Next, we propose a lightweight offline graph reordering algorithm, HALO (Harmonic Locality Ordering), that can be used as a pre-processing step for static graphs. HALO yields speedups of 1.5x-1.9x over baseline in subsequent traversals. Our method specifically aims to cover large directed real world graphs in addition to undirected graphs whereas prior methods only account for the latter. Additionally, we demonstrate ties between the locality ordering problem and graph compression and show that prior methods from graph compression such as recursive graph bisection can be suitably adapted to this problem.

Reachability preserving compression for dynamic graph

Reachability preserving compression generates small graphs that preserve the information only relevant to reachability queries, and the compressed graph can answer any reachability query without decompression. Existing reachability preserving compression algorithms either require a long compression time or include redundant data in the compressed graph. In this paper, we propose a novel edge sorting alogrithm for fast, non-redudant reachability preserving compression. On the other hand, we found that the current incremental reachability compression algorithms for dynamic graphs may return incorrect results in some cases. Therefore, we propose two novel incremental reachability compression algorithms. An algorithm called incremental reachability preserving compression with optimum compression ratio, which generates an update compressed graph that is exactly the same as the graph computed by recompression. The other algorithm called fast incremental reachability preserving compression, which can update the compressed graph in a short time. Extensive experiments on real datasets show the efficiency and the effectiveness of our methods.

Test dense subgraphs in sparse uniform hypergraph

In network data analysis, one of the fundamental data mining tasks is to find dense subgraphs, which has wide applications in biology, finance, graph compression and so on. In practice, given network data, it’s unknown whether a dense subgraph exists or not. Statistically, this problem can be formulated as a hypothesis testing problem: under the null distribution, there is no dense subgraph while under the alternative hypothesis, a dense subgraph exists. In the graph regime, this problem has been well studied. However, in the hypergraph case, to our knowledge, it’s still open. In this paper, we study the problem in the context of uniform hypergraphs. Specifically, we propose a test statistic, derive its asymptotic distributions under the null and alternative hypothesis, and show the power may approach one when the sample size goes to infinity. Then we use simulation and real data to evaluate the finite sample performance of our test.

Path optimization of box-covering based routing to minimize average packet delay in software defined network

A routing algorithm plays a major role in Data communication across an inter-network. Software-Defined Networking (SDN) is a new era of computer networking that separates Data plane from Control plane which controls, changes and manages the network behavior dynamically via open interfaces with the help of SDN Controllers. In SDN, the networking devices route packets in the form of flows with the help of flow rules given by the controllers and the flow rules are stored in the corresponding flow tables of the networking devices. The Box-Covering (BC) algorithm is an existing algorithm used in SDN for achieving renormalization of networks by calculating the fractal dimension of networks and covering the network with the minimum possible number of boxes and Dijkstra’s algorithm is used for calculating shortest paths. The proposed work focuses on Path Optimization in the Box-Covering-Based Routing (BCR) algorithm, which is an existing algorithm used in large-scale SDN. In the proposed work, a Link-Weight system has been used for bounding the Link Utilization of the shortest path between the Source and the Destination to minimize the Average Packet Delay in the Network. The results show that the proposed work reduces the Average Packet Delay compared to the existing algorithms such as Dijkstra’s algorithm, Graph Compression algorithm and BCR algorithm and also improves the performance of the network.

Graph Compression Storage Based on Spatial Cluster Entity Optimization

Graph storage technology is confronted with an enormous challenge as far as the compact and complex graph-structure data. This phenomenon is derived from social networks with spatially intensive data. Since a hot event can cause the generation of a network cluster, which consists of a massive duplicate associated entities in the social networks, the space utilization and processing speed of graph data is obstructed. Therefore, it is necessary to design a graph storage mechanism specifically for the above data. In this paper, we propose a G raph compression S torage engine based on spatial C luster entity O ptimization ( GSCO ), which improves the native graph storage model through the proposed the many-to-one mapping structure and a H eat E volution E limination algorithm ( H2E ). Firstly, we define the spatial cluster entity formally and confirm the compressed storage objects. Then, we introduce the many-to-one relationship to transfer the mapping structure between the node and property. It compresses the data to raise the space utilization of the graph database. Finally, we propose the H2E algorithm that allows the representative nodes to be anchored an extended period in memory according to the heat evolution acceleration. It increases the hit rate and throughput and reduces the I/O operation by deleting the redundancy of data. Extensive experiments results show that the proposed GSCO storage model is better than Neo4j for reading and writing data in spatial clustering entity. It significantly promotes the effectiveness of graph operation, including the data loading, the common queries, and the clustering test.

Zuckerli: A New Compressed Representation for Graphs

Zuckerli is a scalable compression system meant for large real-world graphs. Graphs are notoriously challenging structures to store efficiently due to their linked nature, which makes it hard to separate them into smaller, compact components. Therefore, effective compression is crucial when dealing with large graphs, which can have billions of nodes and edges. Furthermore, a good compression system should give the user fast and reasonably flexible access to parts of the compressed data without requiring full decompression, which may be unfeasible on their system. Compared to WebGraph, the de-facto standard in compressing real-world graphs, Zuckerli improves multiple aspects by using advanced compression techniques and novel heuristic graph algorithms. Zuckerli can produce both a compressed representation for storage and one which allows fast direct access to the adjacency lists of the compressed graph without decompressing the entire graph. We validate its effectiveness on real-world graphs with up to a billion nodes and 90 billion edges, conducting an extensive experimental evaluation of both compression density and decompression performance. We show that Zuckerli-compressed graphs are 10% to 29% smaller, and more than 20% in most cases, with a resource usage for decompression comparable to that of WebGraph.

Universal Tree Source Coding Using Grammar-Based Compression

The problem of universal source coding for binary trees is considered. Zhang, Yang, and Kieffer derived upper bounds on the average-case redundancy of codes based on directed acyclic graph (DAG) compression for binary tree sources with certain properties. In this paper, a natural class of binary tree sources is presented such that the demanded properties are fulfilled. Moreover, for both subclasses considered in the paper of Zhang, Yang, and Kieffer, their result is improved by deriving bounds on the maximal pointwise redundancy (or worst-case redundancy) instead of the average-case redundancy. Finally, using context-free tree grammars instead of DAGs, upper bounds on the maximal pointwise redundancy for certain binary tree sources are derived. This yields universal codes for new classes of binary tree sources.

Building a morpho-semantic knowledge graph for Arabic information retrieval

In this paper, we propose to build a morpho-semantic knowledge graph from Arabic vocalized corpora. Our work focuses on classical Arabic as it has not been deeply investigated in related works. We use a tool suite which allows analyzing and disambiguating Arabic texts, taking into account short diacritics to reduce ambiguities. At the morphological level, we combine Ghwanmeh stemmer and MADAMIRA which are adapted to extract a multi-level lexicon from Arabic vocalized corpora. At the semantic level, we infer semantic dependencies between tokens by exploiting contextual knowledge extracted by a concordancer. Both morphological and semantic links are represented through compressed graphs, which are accessed through lazy methods. These graphs are mined using a measure inspired from BM25 to compute one-to-many similarity. Indeed, we propose to evaluate the morpho-semantic Knowledge Graph in the context of Arabic Information Retrieval (IR). Several scenarios of document indexing and query expansion are assessed. That is, we vary indexing units for Arabic IR based on different levels of morphological knowledge, a challenging issue which is not yet resolved in previous research. We also experiment several combinations of morpho-semantic query expansion. This permits to validate our resource and to study its impact on IR based on state-of-the art evaluation metrics.

On the compressed essential graph of a commutative ring

Let $R$ be a commutative ring. In this paper, we introduce and study the compressed essential graph of $R$, $EG_E(R)$. The compressed essential graph of $R$ is a graph whose vertices are equivalence classes of non-zero zero-divisors of $R$ and two distinct vertices $[x]$ and $[y]$ are adjacent if and only if $\ann(xy)$ is an essential ideal of $R$. It is shown if $R$ is reduced, then $EG_E(R)=\Gamma_E(R)$, where $\Gamma_E(R)$ denotes the compressed zero-divisor graph of $R$. Furthermore, for a non-reduced Noetherian ring $R$ with $3<|EG_E(R)|<\infty $, it is shown that $EG_E(R)=\Gamma_E(R)$ if and only if \begin{itemize} \item[(i)] $\Nil(R)=\ann(Z(R))$. \item[(ii)] Every non-zero element of $\Nil(R)$ is irreducible in $Z(R)$. \end{itemize}

Modular Decomposition-Based Graph Compression for Fast Reachability Detection

Fast reachability detection is one of the key problems in graph applications. Most of the existing works focus on creating an index and answering reachability based on that index. For these approaches, the index construction time and index size can become a concern for large graphs. More recently query-preserving graph compression has been proposed, and searching reachability over the compressed graph has been shown to be able to significantly improve query performance as well as reducing the index size. In this paper, we introduce a multilevel compression scheme for DAGs, which builds on existing compression schemes, but can further reduce the graph size for many real-world graphs. We propose an algorithm to answer reachability queries using the compressed graph. Extensive experiments with four existing state-of-the-art reachability algorithms and 12 real-world datasets demonstrate that our approach outperforms the existing methods. Experiments with synthetic datasets ensure the scalability of this approach. We also provide a discussion on possible compression for k-reachability.

Nearest Embedded and Embedding Self-Nested Trees

Self-nested trees present a systematic form of redundancy in their subtrees and thus achieve optimal compression rates by directed acrylic graph (DAG) compression. A method for quantifying the degree of self-similarity of plants through self-nested trees was introduced by Godin and Ferraro in 2010. The procedure consists of computing a self-nested approximation, called the nearest embedding self-nested tree, that both embeds the plant and is the closest to it. In this paper, we propose a new algorithm that computes the nearest embedding self-nested tree with a smaller overall complexity, but also the nearest embedded self-nested tree. We show from simulations that the latter is mostly the closest to the initial data, which suggests that this better approximation should be used as a privileged measure of the degree of self-similarity of plants.