Power-law Graphs Research Articles

Global stability for McKean–Vlasov equations on large networks

Abstract We investigate the mean-field dynamics of stochastic McKean differential equations with heterogeneous particle interactions described by large network structures. To express a wide range of graphs, from dense to sparse structures, we incorporate the recently developed graph limit theory of graphops into the limiting McKean–Vlasov equations. Global stability of the splay steady state is proven via a generalised entropy method, leading to explicit graph structure-dependent decay rates. We highlight the robustness of the entropy approach by extending the results to the closely related Sakaguchi–Kuramoto model with intrinsic frequency distributions. We also present central examples of random graphs, such as power law graphs and the spherical graphop, and analyse the limitations of the applied methodology.

Parallel Maximum Cardinality Matching for General Graphs on GPUs.

The matching problem formulated as Maximum Cardinality Matching in General Graphs (MCMGG) finds the largest matching on graphs without restrictions. The Micali-Vazirani algorithm has the best asymptotic complexity for solving MCMGG when the graphs are sparse. Parallelizing matching in general graphs on the GPU is difficult for multiple reasons. First, the augmenting path procedure is highly recursive, and NVIDIA GPUs use registers to store kernel arguments, which eventually spill into cached device memory, with a performance penalty. Second, extracting parallelism from the matching process requires partitioning the graph to avoid any overlapping augmenting paths. We propose an implementation of the Micali-Vazirani algorithm which identifies bridge edges using thread-parallel breadth-first search, followed by block-parallel path augmentation and blossom contraction. Augmenting path and Union-find methods were implemented as stack-based iterative methods, with a stack allocated in shared memory. Our experimentation shows that compared to the serial implementation, our approach results in up to 15-fold speed-up for very sparse regular graphs, up to 5-fold slowdown for denser regular graphs, and finally a 50-fold slowdown for power-law distributed Kronecker graphs. This implementation has been open-sourced for further research on developing combinatorial graph algorithms on GPUs.

Scalable Approximate Butterfly and Bi-triangle Counting for Large Bipartite Networks

A bipartite graph is a graph that consists of two disjoint sets of vertices and only edges between vertices from different vertex sets. In this paper, we study the counting problems of two common types of em motifs in bipartite graphs: (i) butterflies (2x2 bicliques) and (ii) bi-triangles (length-6 cycles). Unlike most of the existing algorithms that aim to obtain exact counts, our goal is to obtain precise enough estimations of these counts in bipartite graphs, as such estimations are already sufficient and of great usefulness in various applications. While there exist approximate algorithms for butterfly counting, these algorithms are mainly based on the techniques designed for general graphs, and hence, they are less effective on bipartite graphs. Not to mention that there is still a lack of study on approximate bi-triangle counting. Motivated by this, we first propose a novel butterfly counting algorithm, called one-sided weighted sampling, which is tailored for bipartite graphs. The basic idea of this algorithm is to estimate the total butterfly count with the number of butterflies containing two randomly sampled vertices from the same side of the two vertex sets. We prove that our estimation is unbiased, and our technique can be further extended (non-trivially) for bi-triangle count estimation. Theoretical analyses under a power-law random bipartite graph model and extensive experiments on multiple large real datasets demonstrate that our proposed approximate counting algorithms can reach high accuracy, yet achieve up to three orders (resp. four orders) of magnitude speed-up over the state-of-the-art exact butterfly (resp. bi-triangle) counting algorithms. Additionally, we present an approximate clustering coefficient estimation framework for bipartite graphs, which shows a similar speed-up over the exact solutions with less than 1% relative error.

Robustness of clustering coefficients

In analyzing collected network data, measurement error is a common concern. One of the current research interests is to study robustness of network properties to measurement error. In this work, we analytically study the impact of measurement error on the global clustering coefficient and the average clustering coefficient of a network. Two types of common measurement errors are considered: (I) each node is randomly removed with probability β and (II) each edge is randomly removed with probability γ. We analytically derive the limits of the clustering coefficients of an inhomogeneous Erdös-Rényi random graph or a power-law random graph with the above two measurement errors. The limits can be used to quantify robustness of the coefficients: if the limits do not depend on β or γ, the coefficients can be considered as robust. Based on our results, the global clustering coefficient and the average clustering coefficient are robust to random removal of nodes but vulnerable to random removal of edges. Extensive experiments validate our theoretical results.

Computing Graph Edit Distance via Neural Graph Matching

Graph edit distance (GED) computation is a fundamental NP-hard problem in graph theory. Given a graph pair ( G 1 , G 2 ), GED is defined as the minimum number of primitive operations converting G 1 to G 2 . Early studies focus on search-based inexact algorithms such as A*-beam search, and greedy algorithms using bipartite matching due to its NP-hardness. They can obtain a sub-optimal solution by constructing an edit path (the sequence of operations that converts G 1 to G 2 ). Recent studies convert the GED between a given graph pair ( G 1 , G 2 ) into a similarity score in the range (0, 1) by a well designed function. Then machine learning models (mostly based on graph neural networks) are applied to predict the similarity score. They achieve a much higher numerical precision than the sub-optimal solutions found by classical algorithms. However, a major limitation is that these machine learning models cannot generate an edit path. They treat the GED computation as a pure regression task to bypass its intrinsic complexity, but ignore the essential task of converting G 1 to G 2 . This severely limits the interpretability and usability of the solution. In this paper, we propose a novel deep learning framework that solves the GED problem in a two-step manner: 1) The proposed graph neural network GEDGNN is in charge of predicting the GED value and a matching matrix; and 2) A post-processing algorithm based on k -best matching is used to derive k possible node matchings from the matching matrix generated by GEDGNN. The best matching will finally lead to a high-quality edit path. Extensive experiments are conducted on three real graph data sets and synthetic power-law graphs to demonstrate the effectiveness of our framework. Compared to the best result of existing GNN-based models, the mean absolute error (MAE) on GED value prediction decreases by 4.9% ~ 74.3%. Compared to the state-of-the-art searching algorithm Noah, the MAE on GED value based on edit path reduces by 53.6% ~ 88.1%.

A network community detection method with integration of data from multiple layers and node attributes

AbstractMultilayer networks are in the focus of the current complex network study. In such networks, multiple types of links may exist as well as many attributes for nodes. To fully use multilayer—and other types of complex networks in applications, the merging of various data with topological information renders a powerful analysis. First, we suggest a simple way of representing network data in a data matrix where rows correspond to the nodes and columns correspond to the data items. The number of columns is allowed to be arbitrary, so that the data matrix can be easily expanded by adding columns. The data matrix can be chosen according to targets of the analysis and may vary a lot from case to case. Next, we partition the rows of the data matrix into communities using a method which allows maximal compression of the data matrix. For compressing a data matrix, we suggest to extend so-called regular decomposition method for non-square matrices. We illustrate our method for several types of data matrices, in particular, distance matrices, and matrices obtained by augmenting a distance matrix by a column of node degrees, or by concatenating several distance matrices corresponding to layers of a multilayer network. We illustrate our method with synthetic power-law graphs and two real networks: an Internet autonomous systems graph and a world airline graph. We compare the outputs of different community recovery methods on these graphs and discuss how incorporating node degrees as a separate column to the data matrix leads our method to identify community structures well-aligned with tiered hierarchical structures commonly encountered in complex scale-free networks.

DRONE: An Efficient Distributed Subgraph-Centric Framework for Processing Large-Scale Power-law Graphs

Nowadays, the ever-increasing volume of graph-structured data such as social networks, graph databases and knowledge graphs requires to be processed efficiently and scalably. These natural graphs commonly found in the real world have highly skewed power-law degree distribution and are called power-law graphs. The subgraph-centric programming model is a promising approach applied in many state-of-the-art distributed graph computing frameworks. However, the performance of subgraph-centric frameworks is limited when processing large-scale power-law graphs. When deployed to the subgraph-centric framework, existing graph partitioning algorithms are not suitable for power-law graphs. In this paper, we present a novel distributed graph computing framework, DRONE (Distributed gRaph cOmputiNg Engine), which leverages the subgraph-centric model and the vertex-cut graph partitioning strategy. DRONE also supports the fault tolerance mechanism to accommodate the increasing scale of machines with negligible overhead (6.48% on average). We further study the execution workflow of DRONE and propose an efficient and balanced graph partition algorithm (EBV) for DRONE. Experiments show that DRONE reduces the running time on real-world graphs by 25.6%, on average, compared to the state-of-the-art distributed graph computing frameworks. In addition, the EBV graph partition algorithm reduces the replication factor by at least 21.8% than other self-based partition algorithms. Our results indicate that DRONE has excellent potential in processing large-scale power-law graphs.

Epicenter of random epidemic spanning trees on finite graphs

Epidemic source detection is the problem of identifying the network node that originated an epidemic from a partial observation of the epidemic process. The problem finds applications in different contexts, such as detecting the origin of rumors in online social media, and has been studied under various assumptions. Different from prior studies, this work considers an epidemic process on a finite graph that starts on a random node (epidemic source) and terminates when all nodes are infected, yielding a rooted and directed epidemic tree that encodes node infections (i.e., a directed spanning tree of the graph with every edge directed away from the epidemic source). Assuming knowledge of the underlying graph and the undirected spanning tree (i.e., the infection edges but not their directions), can the epidemic source be accurately identified? This work tackles this problem by introducing the epicenter, an efficient estimator for the epidemic source, and thus, the direction of every edge in the epidemic tree. When the underlying graph is vertex-transitive the epicenter can be computed in linear time and it coincides with the well-known distance center of the epidemic tree. Moreover, on a complete graph the epicenter is also the most likely estimator for the source. Finally, the accuracy of the epicenter is evaluated numerically on five different graph models and the performance strongly depends on the graph structure, varying from 31% (on complete graphs) to 13% (on sparse power-law graphs). However, for all graph models considered the epicenter exhibited an accuracy higher than the distance center, being three times more accurate on sparse power-law graphs.

Remarks on power-law random graphs

The theory of graphons is an important tool in understanding properties of large networks. We investigate a power-law random graph model and cast it in the graphon framework. The distinctively different structures of the limit graph are explored in detail in the sub-critical and super-critical regimes. In the sub-critical regime, the graph is empty with high probability, and in the rare event that it is non-empty, it consists of a single edge. Contrarily, in the super-critical regime, a non-trivial random graph exists in the limit, and it serves as an uncovered boundary case between different types of graph convergence.

Minimum and maximum of the average distance in power-law configuration graphs

The article studies configuration graphs with a discrete fixed-parameter power-law distribution of node degrees. Simulation is applied to estimate the average distance (the arithmetic mean of all distances between all pairs of graph vertices), and two characteristics are considered: the minimum and the maximum average distance in the graph. The dependences of these characteristics on the graph size and the vertex degree distribution parameter are obtained and the empirical confidence interval of the average distance in the power-law configuration graph is constructed

Scalable node-disjoint and edge-disjoint multiwavelength routing.

Probabilistic message-passing algorithms are developed for routing transmissions in multiwavelength optical communication networks, under node- and edge-disjoint routing constraints and for various objective functions. Global routing optimization is a hard computational task on its own but is made much more difficult under the node- and edge-disjoint constraints and in the presence of multiple wavelengths, a problem which dominates routing efficiency in real optical communication networks that carry most of the world's internet traffic. The scalable principled method we have developed is exact on trees but provides good approximate solutions on locally treelike graphs. It accommodates a variety of objective functions that correspond to low latency, load balancing, and consolidation of routes and can be easily extended to include heterogeneous signal-to-noise values on edges and a restriction on the available wavelengths per edge. It can be used for routing and managing transmissions on existing topologies as well as for designing and modifying optical communication networks. Additionally, it provides the tool for settling an open and much-debated question on the merit of wavelength-switching nodes and the added capabilities they provide. The methods have been tested on generated networks such as random-regular, Erdős Rényi, and power-law graphs, as well as on optical communication networks in the United Kingdom and United States. They show excellent performance with respect to existing methodology on small networks and have been scaled up to network sizes that are beyond the reach of most existing algorithms.

Accelerating SSSP for Power-Law Graphs.

The single-source shortest path (SSSP) problem is one of the most important and well-studied graph problems widely used in many application domains, such as road navigation, neural image reconstruction, and social network analysis. Although we have known various SSSP algorithms for decades, implementing one for large-scale power-law graphs efficiently is still highly challenging today, because ① a work-efficient SSSP algorithm requires priority-order traversal of graph data, ② the priority queue needs to be scalable both in throughput and capacity, and ③ priority-order traversal requires extensive random memory accesses on graph data. In this paper, we present SPLAG to accelerate SSSP for power-law graphs on FPGAs. SPLAG uses a coarse-grained priority queue (CGPQ) to enable high-throughput priority-order graph traversal with a large frontier. To mitigate the high-volume random accesses, SPLAG employs a customized vertex cache (CVC) to reduce off-chip memory access and improve the throughput to read and update vertex data. Experimental results on various synthetic and real-world datasets show up to a 4.9× speedup over state-of-the-art SSSP accelerators, a 2.6× speedup over 32-thread CPU running at 4.4 GHz, and a 0.9× speedup over an A100 GPU that has 4.1× power budget and 3.4× HBM bandwidth. Such a high performance would place SPLAG in the 14th position of the Graph 500 benchmark for data intensive applications (the highest using a single FPGA) with only a 45 W power budget. SPLAG is written in high-level synthesis C++ and is fully parameterized, which means it can be easily ported to various different FPGAs with different configurations. SPLAG is open-source at https://github.com/UCLA-VAST/splag.

Distinguishing Power-Law Uniform Random Graphs from Inhomogeneous Random Graphs Through Small Subgraphs

We investigate the asymptotic number of induced subgraphs in power-law uniform random graphs. We show that these induced subgraphs appear typically on vertices with specific degrees, which are found by solving an optimization problem. Furthermore, we show that this optimization problem allows to design a linear-time, randomized algorithm that distinguishes between uniform random graphs and random graph models that create graphs with approximately a desired degree sequence: the power-law rank-1 inhomogeneous random graph. This algorithm uses the fact that some specific induced subgraphs appear significantly more often in uniform random graphs than in rank-1 inhomogeneous random graphs.

SimGQ+: Simultaneously evaluating iterative point-to-all and point-to-point graph queries

Graph processing frameworks are typically designed to optimize the evaluation of a single graph query. However, in practice, we often need to respond to multiple graph queries, either from different users or from a single user performing a complex analytics task. Therefore in this paper we develop SimGQ+, a system that optimizes simultaneous evaluation of a group of vertex queries that originate at different source vertices (e.g., multiple shortest path queries originating at different source vertices) and delivers substantial speedups over a conventional framework that evaluates and responds to queries one by one. Our work considers both point-to-all and point-to-point queries. The performance benefits are achieved via batching and sharing. Batching fully utilizes system resources to evaluate a batch of queries and amortizes runtime overheads incurred due to fetching vertices and edge lists, synchronizing threads, and maintaining computation frontiers. Sharing dynamically identifies shared queries that substantially represent subcomputations in the evaluation of different queries in a batch, evaluates the shared queries, and then uses their results to accelerate the evaluation of all queries in the batch. With four input power-law graphs and four graph algorithms SimGQ+ achieves speedups of up to 45.67× with batch sizes of up to 512 queries over the baseline implementation that evaluates the queries one by one using the state of the art Ligra system. Moreover, both batching and sharing contribute substantially to the speedups.

Fast and efficient parallel breadth-first search with power-law graph transformation

In the big data era, graph computing is widely used to exploit the hidden value in real-world graphs in various scenarios such as social networks, knowledge graphs, web searching, and recommendation systems. However, the random memory accesses result in inefficient use of cache and the irregular degree distribution leads to substantial load imbalance. Breadth-First Search (BFS) is frequently utilized as a kernel for many important and complex graph algorithms. In this paper, we describe a preprocessing approach using Reverse Cuthill-Mckee (RCM) algorithm to improve data locality and demonstrate how to achieve an efficient load balancing for BFS. Computations on RCM-reordered graph data are also accelerated with SIMD executions. We evaluate the performance of the graph preprocessing approach on Kronecker graphs of the Graph500 benchmark and real-world graphs. Our BFS implementation on RCM-reordered graph data achieves 326.48 MTEPS/W (mega TEPS per watt) on an ARMv8 system, ranking 2nd on the Green Graph500 list in June 2020 (the 1st rank uses GPU acceleration).

Continual representation learning for node classification in power-law graphs

The recent advent of node embedding techniques enabled a more efficient application of machine learning techniques on graphs. These techniques allow each node of a network to be encoded into an arbitrary low-dimensional vector representation, which can be exploited by statistical learning models. However, the main limitation of these approaches is that the embedding task is solved as an optimization problem on a static snapshot of the graph. In a real scenario, temporal dynamics should be considered with some consequences: new nodes might join the network and get a representation of only these new ones. As a consequence, a new training step over the entire graph is required. Even more, training models with static approaches can have resource-intensive requirements, especially when dealing with large networks. In light of this, a continual feature learning that builds on top of previously already learned knowledge (previous partial embedding of the network) and well-known properties can be a solution to address both limitations efficiently in real scenarios. Our approach is suitable for graphs whose degree distribution is described by a power-law function that is a common property of real systems. This research work presents three main scientific contributions: (a) a continual feature learning meta-algorithm for node embedding, which exploits properties of power-law distribution and spaces alignment techniques; It is suitable with any traditional node embedding techniques that relies on embedding spaces (b) we demonstrate empirically, by performing node labeling tasks, that a lightweight solution to encode new nodes, based on limited knowledge of the embedding of the network hub-nodes, can provide comparable or better performances, with respect to static approaches. (c) Finally, we experimented our algorithm in the temporal graphs domain and we achieved better results in node classification compared with other state of the art techniques.

DTransE: Distributed Translating Embedding for Knowledge Graph

Knowledge graphs play an important role in many applications, such as link prediction and question answering. Translating embedding for knowledge graphs is done with the aim of encoding structured information on entities and their rich relations in a low-dimensional embedding space. TransE is one of the most important methods in translation-based models, and uses translation invariance to implement translating embedding for knowledge graphs. In this line of work, translating embedding models represent the relation as a translation from the head entity to the tail entity and have achieved impressive results. Currently, the TransE model is only developed on single-node machines. Unfortunately, the computing and storage capacities of a single machine can easily reach their limits as knowledge graphs become larger and more complex, which limits the application scope of TransE. In order to solve this problem, we propose a distributed TransE method, known as DTransE, which can utilize distributed computing resources to calculate knowledge graph embeddings. However, building a distributed TransE is complicated and involves challenges of knowledge graph partitioning and computation. To solve these challenges, we provide a high-quality edge partitioning algorithm for the power-law graph by considering the high-degree and low-degree vertices with adaptive weights, which can balance the workload. By using the unactivated Gather-Apply-Scatter model on TransE, the processes periodically exchange messages in a loop. The irregular data distribution among the processes is also optimized to further accelerate communication. As far as we know, this is the first work on a distributed TransE method. We use link prediction to evaluate the DTransE in a distributed environment. Experiments show that, compared to the original TransE method, our proposed DTransE is, on average, 24.5 times faster with a minimum loss of accuracy; compared to the state-of-the-art parallel TransE implementation, DTransE is two times faster on average.

VSIM: Distributed local structural vertex similarity calculation on big graphs

In many graph analytical applications, the local structural vertex similarity calculation is an essential prerequisite for advanced graph mining. The similarity calculation finds out all the similar vertex pairs whose local structural similarity scores (like the number of common neighbors, and the Jaccard index of adjacency sets) are above a given threshold. The real-world applications use a wide range of similarity thresholds. However, the existing distributed methods for the problem only optimize for either high thresholds (> 0.7) or low thresholds (< 0.1). To overcome the drawback, we propose a new distributed vertex similarity calculation framework VSIM that is efficient under a broad range of thresholds. VSIM processes static undirected graphs with local structural similarity scores that measure the similarity between vertices based on the first-order topology information. VSIM generates a similarity calculation task for every vertex in the graph and conducts all the tasks in parallel on a distributed computing platform along with a distributed key-value store. Each task finds vertices similar to a given center vertex with two task execution modes. The two modes optimize for high and low thresholds, respectively. Each task picks the suitable mode adaptively according to cost estimation models. We also propose an efficient implementation for VSIM on Apache Spark, with three optimization techniques to reduce communication costs and balance workloads on power-law graphs. The experimental evaluation shows that VSIM outperforms the state-of-the-art distributed methods by up to 67x speedup. VSIM can achieve near-linear node scalability in low-threshold and small cache scenarios.

Interactive Graph Stream Analytics in Arkouda

Data from emerging applications, such as cybersecurity and social networking, can be abstracted as graphs whose edges are updated sequentially in the form of a stream. The challenging problem of interactive graph stream analytics is the quick response of the queries on terabyte and beyond graph stream data from end users. In this paper, a succinct and efficient double index data structure is designed to build the sketch of a graph stream to meet general queries. A single pass stream model, which includes general sketch building, distributed sketch based analysis algorithms and regression based approximation solution generation, is developed, and a typical graph algorithm—triangle counting—is implemented to evaluate the proposed method. Experimental results on power law and normal distribution graph streams show that our method can generate accurate results (mean relative error less than 4%) with a high performance. All our methods and code have been implemented in an open source framework, Arkouda, and are available from our GitHub repository, Bader-Research. This work provides the large and rapidly growing Python community with a powerful way to handle terabyte and beyond graph stream data using their laptops.

The Power of D-hops in Matching Power-Law Graphs

This paper studies seeded graph matching for power-law graphs. Assume that two edge-correlated graphs are independently edge-sampled from a common parent graph with a power-law degree distribution. A set of correctly matched vertex-pairs is chosen at random and revealed as initial seeds. Our goal is to use the seeds to recover the remaining latent vertex correspondence between the two graphs. Departing from the existing approaches that focus on the use of high-degree seeds in $1$-hop neighborhoods, we develop an efficient algorithm that exploits the low-degree seeds in suitably-defined D-hop neighborhoods. Specifically, we first match a set of vertex-pairs with appropriate degrees (which we refer to as the first slice) based on the number of low-degree seeds in their D-hop neighborhoods. This approach significantly reduces the number of initial seeds needed to trigger a cascading process to match the rest of graphs. Under the Chung-Lu random graph model with n vertices, max degree Θ(√n), and the power-law exponent 2&lt;β&lt;3, we show that as soon as D&gt;4-β/3-β, by optimally choosing the first slice, with high probability our algorithm can correctly match a constant fraction of the true pairs without any error, provided with only Ω((log n)4-β) initial seeds. Our result achieves an exponential reduction in the seed size requirement, as the best previously known result requires n1/2+ε seeds (for any small constant ε^&gt;0). Performance evaluation with synthetic and real data further corroborates the improved performance of our algorithm.