Small Subgraphs Research Articles

Threshold functions for small subgraphs in simple graphs and multigraphs

We revisit the problem of counting the number of copies of a fixed graph in a random graph or multigraph, for various models of random (multi)graphs. For our proofs we introduce the notion of patchworks to describe the possible overlappings of copies of subgraphs. Furthermore, the proofs are based on analytic combinatorics to carry out asymptotic computations. The flexibility of our approach allows us to tackle a wide range of problems. We obtain the asymptotic number and the limiting distribution of the number of subgraphs which are isomorphic to a graph from a given set of graphs. The results apply to multigraphs as well as to (multi)graphs with degree constraints. One application is to scale-free multigraphs, where the degree distribution follows a power law, for which we show how to obtain the asymptotic number of copies of a given subgraph and give as an illustration the expected number of small cycles.

Estimating the number of connected components in a graph via subgraph sampling

Learning properties of large graphs from samples has been an important problem in statistical network analysis since the early work of Goodman (Ann. Math. Stat. 20 (1949) 572–579) and Frank (Scand. J. Stat. 5 (1978) 177–188). We revisit a problem formulated by Frank (Scand. J. Stat. 5 (1978) 177–188) of estimating the number of connected components in a large graph based on the subgraph sampling model, in which we randomly sample a subset of the vertices and observe the induced subgraph. The key question is whether accurate estimation is achievable in the sublinear regime where only a vanishing fraction of the vertices are sampled. We show that it is impossible if the parent graph is allowed to contain high-degree vertices or long induced cycles. For the class of chordal graphs, where induced cycles of length four or above are forbidden, we characterize the optimal sample complexity within constant factors and construct linear-time estimators that provably achieve these bounds. This significantly expands the scope of previous results which have focused on unbiased estimators and special classes of graphs such as forests or cliques. Both the construction and the analysis of the proposed methodology rely on combinatorial properties of chordal graphs and identities of induced subgraph counts. They, in turn, also play a key role in proving minimax lower bounds based on construction of random instances of graphs with matching structures of small subgraphs.

Guided sampling for large graphs

Large real-world graphs claim lots of resources in terms of memory and computational power to study them and this makes their full analysis extremely challenging. In order to understand the structure and properties of these graphs, we intend to extract a small representative subgraph from a big graph while preserving its topology and characteristics. In this work, we aim at producing good samples with sample size as low as 0.1% while maintaining the structure and some of the key properties of a network. We exploit the fact that average values of degree and clustering coefficient of a graph can be estimated accurately and efficiently. We use the estimated values to guide the sampling process and extract tiny samples that preserve the properties of the graph and closely approximate their distributions in the original graph. The distinguishing feature of our work is that we apply traversal based sampling that utilizes only the local information of nodes as opposed to the global information of the network and this makes our approach a practical choice for crawling online networks. We evaluate the effectiveness of our sampling technique using real-world datasets and show that it surpasses the existing methods.

List all Maximal Cliques of an Undirected Graph: A Parallable Algorithm

Listing all the maximal cliques of an undirected graph is considered as a NP-complete problem, even if the existing fastest algorithm for listing all the maximal cliques also needs an exponential time complexity, and these algorithms are not able to be reconstructed into parallel processing algorithms, therefore, which are also unsuited to solve all maximal cliques of a complex undirected graph with a large number of maximal cliques or vertices. This paper aims to develop a parallable algorithm for listing all maximal cliques of the large size complex undirected graph. Firstly we gave some definitions, including adjacent sub-graph, suspended sub-graph and so on, then adopted a partition and recursive strategy to partition equivalently a parent-graph into smaller sub-graphs, so we can get all maximal cliques of parent-graph by recursively listing all maximal cliques of sub-graphs. we have developed a single-thread and a multi-thread program of the algorithm by using Java language. Lastly, we compared the performance of our algorithm in a benchmark data set DIMACS with the existing main algorithms. The results show that our single-thread program is basically equivalent performance with existing algorithms, but the multi-thread program has better performance than the existing algorithms. Because our method is to divide the complex problem into some smaller sub-problems and achieve the solution of the original problem by solving these sub-problems, our algorithm is easy to be implemented by using multi-thread, and also easy to be expanded to the parallel algorithm, which has better feasibility and practicability in solving the large scale complex graph.

Fastest Path Query Answering using Time-Dependent Hop-Labeling in Road Network

Finding the fastest path in the time-dependent road network is time consuming because its problem complexity is $\Omega(T(|V|\log|V| + |E|))$ , where $T$ is the size of the result's time-dependent function, $|V|$ and $|E|$ are the number of vertices and edges. There are three kinds of fastest path problems: SSFP (Single-Staring Time Fastest Path) that has a fixed departure time, ISFP (Interval-Staring Time Fastest Path) that selects the best departure time from an interval, and FPP (Fastest Path Profile) that returns the travel time of the entire time domain. In this paper, we aim to answer these three queries in time-dependent road network faster by extending the 2-hop labeling approach, which is fast in answering shortest distance query in the static graph. However, it is hard to construct index for SSFP and ISFP because there are $T$ and $|T|^2$ possible time points and intervals, where $T$ is the time domain. Therefore, we first propose the time-dependent hop-labeling for FPP, then provide the specific optimizations for SSFP and ISFP query answering. Moreover, it is both time and space consuming to build an index in a large time-dependent graph, so we partition road network into smaller sub-graphs and build indexes within and between the partitions. Furthermore, we propose an online approximation technique AT-Dijkstra and a bottom-up compression method to further reduce the label size, save construction time and speedup query answering. Experiments on real world road network show that our approach outperforms the state-of-art fastest path index approaches and can speed up the query answering by hundreds of times.

Bounds on the Spectral Radius of Digraphs from Subgraph Counts

The spectral radius of a directed graph is a metric that can only be computed when the structure of the network is completely known. However, in many practical scenarios, it is not possible to exactly retrieve the whole structure of the network; hence, the exact value of the spectral radius is not computable. Even in these scenarios, it is typically possible to extract local structural properties of a network using, for example, graph crawlers. In this paper, we develop a novel measure-theoretic framework to upper and lower bound the spectral radius of a directed graph using local structural information, in particular, using the counts of a collection of small subgraphs or motifs. Our framework is based on recent results relating the multivariate moment problem with semidefinite programming. Using these results, we develop a hierarchy of (small) semidefinite programs whose solutions provide upper and lower bounds on the spectral radius of a directed graph using, solely, subgraph and motif counts. We numerically validate the quality of our bounds using both random and real-world directed graphs.

Graph simulation on large scale temporal graphs

Graph simulation is one of the most important queries in graph pattern matching, and it is being increasingly used in various applications, e.g., protein interaction networks, software plagiarism detection. Most previous studies mainly focused on the simulation problem on static graphs, which neglected the temporal factors in daily life. In this paper, we investigate a novel problem, namely, temporal bounded simulation on temporal graphs. Specifically, given a pattern query Q with bound constraint and a temporal graph G, we aim to find out the matches Q(G) of Q on G, such that Q(G) satisfies temporal reachability and bound constraint of Q. To tackle this problem efficiently, we propose a simulation matching framework, which consists of three phases, namely pattern segmentation, temporal bounded simulation of pattern segments, and result integration. We first divide the Q into simple small subgraphs (segments). We construct a temporal reachable tree of G. Based on the tree, we propose a basic matching algorithm for the pattern segments. Furthermore, we propose two optimization algorithms by leveraging multi-layer partition strategy to accelerate query processing. After that, we integrate the simulation results obtained from the pattern segments according to the constraints of temporal reachability and the bound of query pattern. Finally, we use real temporal and synthetic datasets to empirically verify the efficiency and effectiveness of our solutions for temporal bounded simulation matching.

The ropelengths of knots are almost linear in terms of their crossing numbers

For a knot or link [Formula: see text], let [Formula: see text] be the ropelength of [Formula: see text] and [Formula: see text] be the crossing number of [Formula: see text]. In this paper, we show that there exists a constant [Formula: see text] such that [Formula: see text] for any [Formula: see text], i.e. the upper bound of the ropelength of any knot is almost linear in terms of its minimum crossing number. This result is a significant improvement over the best known upper bound established previously, which is of the form [Formula: see text]. The proof is based on a divide-and-conquer approach on 4-regular plane graphs: a 4-regular plane graph of [Formula: see text] is first repeatedly subdivided into many small subgraphs and then reconstructed from these small subgraphs on the cubic lattice with its topology preserved with a total length of the order [Formula: see text]. The result then follows since a knot can be recovered from a graph that is topologically equivalent to a regular projection of it (which is a 4-regular plane graph).

Top-k relevant semantic place retrieval on spatiotemporal RDF data

RDF data are traditionally accessed using structured query languages, such as SPARQL. However, this requires users to understand the language as well as the RDF schema. Keyword search on RDF data aims at relieving users from these requirements; users only input a set of keywords, and the goal is to find small RDF subgraphs that contain all keywords. At the same time, popular RDF knowledge bases also include spatial and temporal semantics, which opens the road to spatiotemporal-based search operations. In this work, we propose and study novel keyword-based search queries with spatial semantics on RDF data, namely kSP queries. The objective of the kSP query is to find RDF subgraphs which contain the query keywords and are rooted at spatial entities close to the query location. To add temporal semantics to the kSP query, we propose the kSPT query that uses two ways to incorporate temporal information. One way is considering the temporal differences between the keyword-matched vertices and the query timestamp. The other way is using a temporal range to filter keyword-matched vertices. The novelty of kSP and kSPT queries is that they are spatiotemporal-aware and that they do not rely on the use of structured query languages. We design an efficient approach containing two pruning techniques and a data preprocessing technique for the processing of kSP queries. The proposed approach is extended and improved with four optimizations to evaluate kSPT queries. Extensive empirical studies on two real datasets demonstrate the superior and robust performance of our proposals compared to baseline methods.

On Computing Entity Relatedness in Wikipedia, with Applications

Many text mining tasks, such as clustering, classification, retrieval, and named entity linking, benefit from a measure of relatedness between entities in a knowledge graph. We present a thorough study of all entity relatedness measures in recent literature based on Wikipedia as the knowledge graph. To facilitate this study, we introduce a new dataset with human judgments of entity relatedness. No clear dominance is seen between measures based on textual similarity and graph proximity. Some of the better measures involve expensive global graph computations. We propose a new, space-efficient, computationally lightweight, two-stage framework for relatedness computation. In the first stage, a small weighted subgraph is dynamically grown around the two query entities; in the second stage, relatedness is derived based on computations on this subgraph. Our system shows better agreement with human judgment than existing proposals both on the new dataset and on an established one. Our framework also shows improvements with respect to the state-of-the-art on three different extrinsic evaluations in the domains of ranking entity pairs, entity linking, and synonym extraction.

Exploring GPU-Accelerated Routing for FPGAs

Field Programmable Gate Arrays (FPGAs) are reconfigurable architectures able to provide a good balance between energy efficiency and flexibility with respect to CPUs and ASICs. The main drawback in using FPGAs, however, is their timing-consuming routing process, significantly hindering the designer productivity. An emerging solution to this problem is to accelerate the routing by parallelization. Existing attempts of parallelizing the FPGA routing either do not fully exploit the parallelism or suffer from an excessive quality loss. Massive parallelism using GPUs has the potential to solve this issue but faces non-trivial challenges. To cope with these challenges, this paper explores GPU-accelerated routing approach for FPGAs. We leverage the idea of problem size reduction by limiting the single-net routing in a small subgraph rather than in an entire graph, further enabling the GPU-friendly shortest path algorithm to be used in FPGA routing. We maintain the convergence after problem size reduction by using the dynamic expansion of the routing resource subgraph, where the routing region of subgraph will be progressively expanded to find a feasible solution to each net. In addition, we are based on a GPU platform to explore the fine-grained single-net parallel routing in three ways and propose a hybrid approach to combine the static and dynamic parallelization for better speedup in FPGA routing. To explore the coarse-grained multi-net parallelization, We propose a dynamic programming-based partitioning algorithm to parallelize the routing of multiple nets while generating the equivalent routing results as the original single-net routing. Experimental results show that our proposed approach can provide an average of about 21.53× speedup on a single GPU with a tolerable loss in the routing quality and maintain a scalable speedup on large-scale routing resource graphs. To our knowledge, this is the first work to demonstrate the effectiveness of GPU-accelerated routing for FPGAs.

Variational principle for scale-free network motifs

For scale-free networks with degrees following a power law with an exponent τ ∈ (2, 3), the structures of motifs (small subgraphs) are not yet well understood. We introduce a method designed to identify the dominant structure of any given motif as the solution of an optimization problem. The unique optimizer describes the degrees of the vertices that together span the most likely motif, resulting in explicit asymptotic formulas for the motif count and its fluctuations. We then classify all motifs into two categories: motifs with small and large fluctuations.

F98. Counts of Small Subgraphs Within the Resting Functional Connectome are Parsimonious, Stable and Individualized Features in Healthy as Well as Disordered Mood

F98. Counts of Small Subgraphs Within the Resting Functional Connectome are Parsimonious, Stable and Individualized Features in Healthy as Well as Disordered Mood

A safety analysis of roundabouts and turbo roundabouts based on Petri nets

Objective: In this work, a roundabout and a turbo roundabout model are compared and previous modeling with continuous Petri nets and safety are analyzed through indicators of complexity. Petri nets are a graphic and mathematical representation that allow a faithful modeling of urban systems.Method: The methodology has been designed for the transformation of a real system to small subgraphs that represent the maneuvers in roundabouts, approximated as roads and lanes of incorporation. Places within the roundabout have been located and defined as continuous places from their influence and visibility toward adjacent conditions. The transitions have been modeled by time and inhibitory arcs, which represent priorities and areas where drivers must pay attention. The created networks represent a faithful model of vehicle flow trajectories in the roundabouts.Results: The methodology is applied to the same real road intersection. The case study is a recent transformation from roundabout to turbo roundabout. The roundabout network complexity is corroborated by a greater number of entries and exits that lead to each roundabout place (reflected in the maneuvers that can be performed) and a greater number of inhibiting arcs. In most of the turbo roundabout places, the driver’s only option is reduced to occupying next place. The possibility of choosing between several places supposes a greater trajectory intersection and an increased time for decision making. The only situation where the complexity is the same between both systems is when a vehicle accesses the inner lane of the roundabout from the left lane on a single-lane road. The main maneuvers causing accidents have been modeled and their solution in a turbo roundabout is presented.Conclusions: The reduced complexity of the turbo roundabout is due to the strict limitations in lane changes, turning turbo roundabouts into a safer model: A lower number of possible movements that can be performed by drivers and a smaller number of trajectories with collision risk. Petri nets have proven to be perfectly applicable to the representation of traffic circular systems (such as roundabouts and turbo roundabouts) and to measure the complexity and security of the system.

Connectivity problems on heterogeneous graphs

BackgroundNetwork connectivity problems are abundant in computational biology research, where graphs are used to represent a range of phenomena: from physical interactions between molecules to more abstract relationships such as gene co-expression. One common challenge in studying biological networks is the need to extract meaningful, small subgraphs out of large databases of potential interactions. A useful abstraction for this task turned out to be the Steiner Network problems: given a reference “database” graph, find a parsimonious subgraph that satisfies a given set of connectivity demands. While this formulation proved useful in a number of instances, the next challenge is to account for the fact that the reference graph may not be static. This can happen for instance, when studying protein measurements in single cells or at different time points, whereby different subsets of conditions can have different protein milieu.Results and discussionWe introduce the condition Steiner Network problem in which we concomitantly consider a set of distinct biological conditions. Each condition is associated with a set of connectivity demands, as well as a set of edges that are assumed to be present in that condition. The goal of this problem is to find a minimal subgraph that satisfies all the demands through paths that are present in the respective condition. We show that introducing multiple conditions as an additional factor makes this problem much harder to approximate. Specifically, we prove that for C conditions, this new problem is NP-hard to approximate to a factor of C - epsilon , for every C ge 2 and epsilon > 0, and that this bound is tight. Moving beyond the worst case, we explore a special set of instances where the reference graph grows monotonically between conditions, and show that this problem admits substantially improved approximation algorithms. We also developed an integer linear programming solver for the general problem and demonstrate its ability to reach optimality with instances from the human protein interaction network.ConclusionOur results demonstrate that in contrast to most connectivity problems studied in computational biology, accounting for multiplicity of biological conditions adds considerable complexity, which we propose to address with a new solver. Importantly, our results extend to several network connectivity problems that are commonly used in computational biology, such as Prize-Collecting Steiner Tree, and provide insight into the theoretical guarantees for their applications in a multiple condition setting.

Functional geometry of protein interactomes.

Protein-protein interactions (PPIs) are usually modeled as networks. These networks have extensively been studied using graphlets, small induced subgraphs capturing the local wiring patterns around nodes in networks. They revealed that proteins involved in similar functions tend to be similarly wired. However, such simple models can only represent pairwise relationships and cannot fully capture the higher-order organization of protein interactomes, including protein complexes. To model the multi-scale organization of these complex biological systems, we utilize simplicial complexes from computational geometry. The question is how to mine these new representations of protein interactomes to reveal additional biological information. To address this, we define simplets, a generalization of graphlets to simplicial complexes. By using simplets, we define a sensitive measure of similarity between simplicial complex representations that allows for clustering them according to their data types better than clustering them by using other state-of-the-art measures, e.g. spectral distance, or facet distribution distance. We model human and baker's yeast protein interactomes as simplicial complexes that capture PPIs and protein complexes as simplices. On these models, we show that our newly introduced simplet-based methods cluster proteins by function better than the clustering methods that use the standard PPI networks, uncovering the new underlying functional organization of the cell. We demonstrate the existence of the functional geometry in the protein interactome data and the superiority of our simplet-based methods to effectively mine for new biological information hidden in the complexity of the higher-order organization of protein interactomes. Codes and datasets are freely available at http://www0.cs.ucl.ac.uk/staff/natasa/Simplets/. Supplementary data are available at Bioinformatics online.

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.

Local Properties in Colored Graphs, Distinct Distances, and Difference Sets

We study Extremal Combinatorics problems where local properties are used to derive global properties. That is, we consider a given configuration where every small piece of the configuration satisfies some restriction, and use this local property to derive global properties of the entire configuration. We study one such Ramsey problem of Erdős and Shelah, where the configurations are complete graphs with colored edges and every small induced subgraph contains many distinct colors. Our bounds for this Ramsey problem show that the known probabilistic construction is tight in various cases. We study one Discrete Geometry variant, also by Erdős, where we have a set of points in the plane such that every small subset spans many distinct distances. Finally, we consider an Additive Combinatorics problem, where we are given sets of real numbers such that every small subset has a large difference set. We derive new bounds for all of the above problems. Our proof technique is based on introducing a variant of additive energy, which is based on edge colors in graphs.

Network motifs for translator stylometry identification.

Despite the extensive literature investigating stylometry analysis in authorship attribution research, translator stylometry is an understudied research area. The identification of translator stylometry contributes to many fields including education, intellectual property rights and forensic linguistics. In a two stage process, this paper first evaluates the use of existing lexical measures for the translator stylometry problem. Similar to previous research we found that using vocabulary richness in its traditional form as it has been used in the literature could not identify translator stylometry. This encouraged us to design an approach with the aim of identifying the distinctive patterns of a translator by employing network-motifs. Networks motifs are small sub-graphs which aim at capturing the local structure of a complex network. The proposed approach achieved an average accuracy of 83% in three-way classification. These results demonstrate that classic tools based on lexical features can be used for identifying translator stylometry if they get augmented with appropriate non-parametric scaling. Moreover, the use of complex network analysis and network motifs mining provided made it possible to design features that can solve translator stylometry analysis problems.

Spare Pose Graph Decomposition and Optimization for SLAM

The graph optimization has become the mainstream technology to solve the problems of SLAM (simultaneous localization and mapping). The pose graph in the graph based SLAM is consisted with a series of nodes and edges that connect the adjacent or related poses. With the widespread use of mobile robots, the scale of pose graph has rapidly increased. Therefore, optimizing a large-scale pose graph is the bottleneck of application of graph based SLAM. In this paper, we propose an optimization method basing on the decomposition of pose graph, of which we have noticed the sparsity. With the extraction of the Single-chain and the Parallel-chain, the pose graph is decomposed into many small subgraphs. Compared with directly processing the original graph, the speed of calculation is accelerated by separately optimizing the subgraph, which is because the computational complexity is increasing exponentially with the increase of the graph’s scale. This method we proposed is very suitable for the current multi-threaded framework adopted in the mainstream SLAM, which separately calculate the subgraph decomposed by our method, rather than the original optimization requiring a large block of time in once may cause CPU obstruction. At the end of the paper, our algorithm is validated with the open source dataset of the mobile robot, of which the result illustrates our algorithm can reduce the one-time resource consumption and the time consumption of the calculation with the same map-constructing accuracy.