Components Of Graph Research Articles

Limits of random tree-like discrete structures

We study a model of random R-enriched trees that is based on weights on the R-structures and allows for a unified treatment of a large family of random discrete structures. We establish distributional limits describing local convergence around fixed and random points in this general context, limit theorems for component sizes when R is a composite class, and a Gromov–Hausdorff scaling limit of random metric spaces patched together from independently drawn metrics on the R-structures. Our main applications treat a selection of examples encompassed by this model. We consider random outerplanar maps sampled according to arbitrary weights assigned to their inner faces, and classify in complete generality distributional limits for both the asymptotic local behaviour near the root-edge and near a uniformly at random drawn vertex. We consider random connected graphs drawn according to weights assigned to their blocks and establish a Benjamini–Schramm limit. We also apply our framework to recover in a probabilistic way a central limit theorem for the size of the largest 2-connected component in random graphs from planar-like classes. We prove Benjamini–Schramm convergence of random k-dimensional trees and establish both scaling limits and local weak limits for random planar maps drawn according to Boltzmann-weights assigned to their 2-connected components.

Using combinatorics to compute fluid routing alternatives in a hydrocarbon production network

Typically, hydrocarbon production networks have several fluid routing alternatives that are applied by opening and closing on–off valves. This usually sends the wells’ fluids through a specific pipeline, pump or compressor, or to a particular separator, among other requirements. This paper presents a general methodology to compute all fluid routing configurations of a production network using a graph representation of it. The particular implementation case discussed in this paper involves interacting with a preexisting steady-state computational model of the production network. The method starts by extracting from the model the name list of elements in the network and their type. Equipment (wells, separators, junctions, pumps, compressors, valves, etc.) is tagged as nodes and pipes (flowlines, connectors) are tagged as edges. The nodes are further classified by type: sources (wells), internals (e.g., junctions) or sinks (separators). The start and end element of each edge is recorded. This process yields a network connectivity list. A depth-first search is executed from each source to each sink. The search keeps track of the edges that must be active in each path and honors (if any) pre-specified edge directions. All paths for one source node in one component are combined to form all feasible edge combinations for that source node, and these combinations are again combined for all the source nodes in each component. This is repeated for all graph components. The unique combinations are stored and reported at the end. The method has been tried in a production network with seven wells representing a typical subsea production system in the North Sea where the wells have the option to produce (through two flowlines) to two separators on the platform. The production network model was available in a commercial software; thus, there was no access to the code or the underlying equations. The model was controlled from an external computational routine using automation. The graph was extracted from the model, all operating configurations of the network were computed (2187), and then each one was applied (by enabling or disabling flowlines) and evaluated in the commercial software. This allowed to identify routing configurations that provided maximum total oil production or maximum total gas production. There were only 306 configurations that yielded a total oil production close (within 10%) to the maximum recorded oil production. The input data of the production system model are given in the appendix for verification and benchmarking by a third party. Details about the implementation are provided.

Local commensurability graphs of solvable groups

ABSTRACTThe commensurability index between two subgroups A,B of a group G is [A:A∩B][B:A∩B]. This gives a notion of distance among finite index subgroups of G, which is encoded in the p-local commensurability graphs of G. We show that for any metabelian group, any component of the p-local commensurabilty graph of G has diameter bounded above by 4. However, no universal upper bound on diameters of components exists for the class of finite solvable groups. In the appendix we give a complete classification of components for upper triangular matrix groups in GL(2,𝔽q).

Studying the use and effect of graph decomposition in qualitative spatial and temporal reasoning

AbstractWe survey the use and effect of decomposition-based techniques in qualitative spatial and temporal constraint-based reasoning, and clarify the notions of a tree decomposition, a chordal graph, and a partitioning graph, and their implication with a particular constraint property that has been extensively used in the literature, namely, patchwork. As a consequence, we prove that a recently proposed decomposition-based approach that was presented in the study by Nikolaou and Koubarakis for checking the satisfiability of qualitative spatial constraint networks lacks soundness. Therefore, the approach becomes quite controversial as it does not seem to offer any technical advance at all, while results of an experimental evaluation of it in a following work presented in the study by Sioutis become questionable. Finally, we present a particular tree decomposition that is based on the biconnected components of the constraint graph of a given large network, and show that it allows for cost-free utilization of parallelism for a qualitative constraint language that has patchwork for satisfiable atomic networks.

Branching graphs for finite unitary groups in nondefining characteristic

ABSTRACTWe show that the modular branching rule (in the sense of Harish-Chandra) on unipotent modules for finite unitary groups is piecewise described by particular connected components of the crystal graph of well-chosen Fock spaces, under favorable conditions. Besides, we give the combinatorial formula to pass from one to the other in the case of modules arising from cuspidal modules of defect 0. This partly proves a recent conjecture of Jacon and the authors.

Resonance graphs of fullerenes

A fullerene G is a 3 -regular plane graph consisting only of pentagonal and hexagonal faces. The resonance graph R ( G ) of G reflects the structure of its perfect matchings. The Zhang-Zhang polynomial of a fullerene is a counting polynomial of resonant structures called Clar covers. The cube polynomial is a counting polynomial of induced hypercubes in a graph. In the present paper we show that the resonance graph of every fullerene is bipartite and each connected component has girth 4 or is a path. Also, the equivalence of the Zhang-Zhang polynomial of a fullerene and the cube polynomial of its resonance graph is established. Furthermore, it is shown that every subgraph of the resonance graph isomorphic to a hypercube is an induced subgraph in the resonance graph. For benzenoid systems and tubulenes each connected component of the resonance graph is the covering graph of a distributive lattice; for fullerenes this is not true, as we show with an example.

Using landscape graphs to delineate ecologically functional areas

Landscape graphs are widely used to model connectivity and to support decision-making in conservation planning. Compartmentalization methods applied to such graphs aim to define clusters of highly interconnected patches. Recent studies show that compartmentalization based on modularity is suitable, but it applies to non-weighted graphs whereas most landscape graphs involve weighted nodes and links. We propose to adapt modularity computation to weighted landscape graphs and to validate the relevance of the resulting compartments using demographic or genetic data about the patches. A weighted adjacency matrix was designed to express potential fluxes, associating patch capacities and inter-patch distances. Eight weighting scenarios were compared. The statistical evaluation of each compartmentalization was based on Wilks’ Lambda. These methods were performed on a grassland network where patches are documented by annual densities of water voles in the Jura massif (France). The scenarios in which patch capacity is assigned a small weight led to the more relevant results, giving high modularity values and low Wilks’ Lambda values. When considering a fixed number of compartments, we found a significant negative correlation between these two criteria. Comparison showed that compartments are ecologically more valid than graph components. The method proposed is suitable for designing ecologically functional areas from weighted landscape graphs. Maximum modularity values can serve as a guide for setting the parameters of the adjacency matrix.

Continuum limit of critical inhomogeneous random graphs

Motivated by applications, the last few years have witnessed tremendous interest in understanding the structure as well as the behavior of dynamics for inhomogeneous random graph models. In this study we analyze the maximal components at criticality of one famous class of such models, the rank-one inhomogeneous random graph model. Viewing these components as measured random metric spaces, under finite moment assumptions for the weight distribution, we show that the components in the critical scaling window with distances scaled by $n^{-1/3}$ converge in the Gromov-Haussdorf-Prokhorov metric to rescaled versions of the limit objects identified for the Erd\H{o}s-R\'enyi random graph components at criticality Addario-Berry, Broutin and Goldschmidt (2012). A key step is the construction of connected components of the random graph through an appropriate tilt of a famous class of random trees called $\mathbf{p}$-trees (studied previously by Aldous, Miermont and Pitman (2004) and by Camarri and Pitman (2000)). This is the first step in rigorously understanding the scaling limits of objects such as the Minimal spanning tree and other strong disorder models from statistical physics (see Braunstein et al., 2003) for such graph models. By asymptotic equivalence (Janson, 2010), the same results are true for the Chung-Lu model and the Britton-Deijfen-Lof model. A crucial ingredient of the proof of independent interest is tail bounds for the height of $\mathbf{p}$-trees. The techniques developed in this paper form the main technical bedrock for proving continuum scaling limits in the critical regime for a wide array of other random graph models (Bhamidi, Broutin, Sen and Wang, 2014) including the configuration model and inhomogeneous random graphs with general kernels which were introduced by Bollobas, Janson and Riordan (2007).

BinGold: Towards robust binary analysis by extracting the semantics of binary code as semantic flow graphs (SFGs)

Binary analysis is useful in many practical applications, such as the detection of malware or vulnerable software components. However, our survey of the literature shows that most existing binary analysis tools and frameworks rely on assumptions about specific compilers and compilation settings. It is well known that techniques such as refactoring and light obfuscation can significantly alter the structure of code, even for simple programs. Applying such techniques or changing the compiler and compilation settings can significantly affect the accuracy of available binary analysis tools, which severely limits their practicability, especially when applied to malware. To address these issues, we propose a novel technique that extracts the semantics of binary code in terms of both data and control flow. Our technique allows more robust binary analysis because the extracted semantics of the binary code is generally immune from light obfuscation, refactoring, and varying the compilers or compilation settings. Specifically, we apply data-flow analysis to extract the semantic flow of the registers as well as the semantic components of the control flow graph, which are then synthesized into a novel representation called the semantic flow graph (SFG). Subsequently, various properties, such as reflexive, symmetric, antisymmetric, and transitive relations, are extracted from the SFG and applied to binary analysis. We implement our system in a tool called BinGold and evaluate it against thirty binary code applications. Our evaluation shows that BinGold successfully determines the similarity between binaries, yielding results that are highly robust against light obfuscation and refactoring. In addition, we demonstrate the application of BinGold to two important binary analysis tasks: binary code authorship attribution, and the detection of clone components across program executables. The promising results suggest that BinGold can be used to enhance existing techniques, making them more robust and practical.

Correlative pattern based data aggregation mechanism for WSN

Environment monitoring is one of the typical application scenarios of the wireless sensor networks. As an energy limited system, most of the energy consumption is for the data transmission. As a well-known principle, the difference among the physical parameters of adjacent nodes is approximate a constant. Eliminating these data to be transmitted will lead to remarkable energy saving. A correlative pattern based data aggregation mechanism following this principle is proposed in this paper, which is named the Correlative Pattern based Data Aggregation (CPDA). CPDA mines the correlations of every adjacent nodes pair, and generates a correlation graph of the network, then builds an aggregation routing tree for each connected component of correlation graph based on the shortest path methodology. Following the CPDA algorithm, a node’s sensed data will be suppressed when the data and the children’s match the restriction that is defined by CPDA. When the aggregated data arrive at the Sink node, all the data can be recovered. The recovery error will be limited within a specified small error threshold based on the reversed mechanism. The simulations based on the data set of Berkeley lab show that CPDA has excellent performance in aggregation degree and average error. Further more, a real established temperature sensing experiment also gives the same conclusion.

Module-based visualization of large-scale graph network data

The efficient visualization of dynamic network structures has become a dominant problem in many big data applications, such as large network analytics, traffic management, resource allocation graphs, logistics, social networks, and large document repositories. In this paper, we present a large-graph visualization system called ModuleGraph. ModuleGraph is a scalable representation of graph structures by treating a graph as a set of modules. The main objectives are: (1) to detect graph patterns in the visualization of large-graph data, and (2) to emphasize the interconnecting structures to detect potential interactions between local modules. Our first contribution is a hybrid modularity measure. This measure partitions the cohesion of the graph at various levels of details. We aggregate clusters of nodes and edges into several modules to reduce the overlap between graph components on a 2D display. Our second contribution is a k-clustering method that can flexibly detect the local patterns or substructures in modules. Patterns of modules are preserved by the ModuleGraph system to avoid information loss, while sub-graphs are clustered as a single node. Our experiments show that this method can efficiently support large-scale social and spatial network visualization. Graphical Abstract text

Heuristics for a hub location‐routing problem

We investigate a variant of the many‐to‐many hub location‐routing problem which consists in partitioning the set of nodes of a graph into routes containing exactly one hub each, and determining an extra route interconnecting all hubs. A variable neighborhood descent with neighborhood structures based on remove/add, swap and exchange moves nested with routing and location operations is used as a local search procedure in a multistart algorithm. We also consider a sequential version of this local search in the multistart. In addition, a biased random‐key genetic algorithm working with a local search routine, which also considers routing and location operations, is applied to the problem. To compare the heuristic solutions, we develop an integer programming formulation which is solved with a branch‐and‐cut algorithm. Capacity and path elimination constraints are added in a cutting plane fashion. The separation algorithms are based on the computation of min‐cut trees and on the connected components of a support graph. Computational experiments were conducted on several benchmark instances of routing problems and show that the heuristics are effective on medium to large‐sized instances, while the branch‐and‐cut algorithm solves small to medium sized problems to optimality. These algorithms were also compared with a commercial hybrid solver showing that the heuristics are quite competitive. © 2016 Wiley Periodicals, Inc. NETWORKS, Vol. 68(1), 54–90 2016

Codimension one connectedness of the graph of associated varieties

Let $ \pi $ be an irreducible Harish-Chandra $ (\mathfrak{g}, K) $-module, and denote its associated variety by $ \mathcal{AV}(\pi) $. If $ \mathcal{AV}(\pi) $ is reducible, then each irreducible component must contain codimension one boundary component. Thus we are interested in the codimension one adjacency of nilpotent orbits for a symmetric pair $ (G, K) $. We define the notion of orbit graph and associated graph for $ \pi $, and study its structure for classical symmetric pairs; number of vertices, edges, connected components, etc. As a result, we prove that the orbit graph is connected for even nilpotent orbits. Finally, for indefinite unitary group $ U(p, q) $, we prove that for each connected component of the orbit graph $ \Gamma_K(\mathcal{O}^G_\lambda) $ thus defined, there is an irreducible Harish-Chandra module $ \pi $ whose associated graph is exactly equal to the connected component.

On the number of connected components of divisibility graph for certain simple groups

The divisibility graph $mathscr{D}(G)$ for a finite group $G$ is a graph with vertex set $cs(G)setminus{1}$‎ ‎where $cs(G)$ is the set of conjugacy class sizes of $G$‎. ‎Two vertices $a$ and $b$ are adjacent whenever $a$ divides‎ ‎$b$ or $b$ divides $a$‎. ‎In this paper we will find the number of connected components of $mathscr{D}(G)$ where $G$ is a‎ ‎simple Zassenhaus group or an sporadic simple group‎.

Extending Partial Representations of Proper and Unit Interval Graphs

The recently introduced problem of extending partial interval representations asks, for an interval graph with some intervals pre-drawn by the input, whether the partial representation can be extended to a representation of the entire graph. In this paper, we give a linear-time algorithm for extending proper interval representations and an almost quadratic-time algorithm for extending unit interval representations. We also introduce the more general problem of bounded representations of unit interval graphs, where the input constrains the positions of some intervals by lower and upper bounds. We show that this problem is NP-complete for disconnected input graphs and give a polynomial-time algorithm for the special class of instances, where the ordering of the connected components of the input graph along the real line is prescribed. This includes the case of partial representation extension. The hardness result sharply contrasts the recent polynomial-time algorithm for bounded representations of proper interval graphs (Balko et al. in 2013). So unless $${\textsf {P}} = {\textsf {NP}}$$P=NP, proper and unit interval representations have vastly different structure. This explains why partial representation extension problems for these different types of representations require substantially different techniques.

Efficient Multicore Algorithms For Identifying Biconnected Components

In this paper we design and implement an algorithm for finding the biconnected components of a given graph. Our algorithm is based on experimental evidence that finding the bridges of a graph is usually easier and faster in the parallel setting. We use this property to first decompose the graph into independent and maximal 2-edge-connected subgraphs. To identify the articulation points in these 2-edge connected subgraphs, we again convert this into a problem of finding the bridges on an auxiliary graph. It is interesting to note that during the conversion process, the size of the graph may increase. However, we show that this small increase in size and the run time is offset by the consideration that finding bridges is easier in a parallel setting. We implement our algorithm on an Intel i7 980X CPU running 12 threads. We show that our algorithm is on average 2.45x faster than the best known current algorithms implemented on the same platform. Finally, we extend our approach to dense graphs by applying the sparsification technique suggested by Cong and Bader in [7].Â

The analysis of split graphs in social networks based on the k-cardinality assignment problem

In terms of social networks, split graphs correspond to the variety of interpersonal and intergroup relations. In this paper we analyse the interaction between the cliques (socially strong and trusty groups) and the independent sets (fragmented and non-connected groups of people) as the basic components of any split graph. Based on the Semi-Lagrangean relaxation for the k-cardinality assignment problem, we show the way of minimizing the socially risky interactions between the cliques and the independent sets within the social network.

Network analysis of functional brain connectivity in borderline personality disorder using resting-state fMRI.

Borderline personality disorder (BPD) is associated with symptoms such as affect dysregulation, impaired sense of self, and self-harm behaviors. Neuroimaging research on BPD has revealed structural and functional abnormalities in specific brain regions and connections. However, little is known about the topological organizations of brain networks in BPD. We collected resting-state functional magnetic resonance imaging (fMRI) data from 20 patients with BPD and 10 healthy controls, and constructed frequency-specific functional brain networks by correlating wavelet-filtered fMRI signals from 82 cortical and subcortical regions. We employed graph-theory based complex network analysis to investigate the topological properties of the brain networks, and employed network-based statistic to identify functional dysconnections in patients. In the 0.03–0.06 Hz frequency band, compared to controls, patients with BPD showed significantly larger measures of global network topology, including the size of largest connected graph component, clustering coefficient, small-worldness, and local efficiency, indicating increased local cliquishness of the functional brain network. Compared to controls, patients showed lower nodal centrality at several hub nodes but greater centrality at several non-hub nodes in the network. Furthermore, an interconnected subnetwork in 0.03–0.06 Hz frequency band was identified that showed significantly lower connectivity in patients. The links in the subnetwork were mainly long-distance connections between regions located at different lobes; and the mean connectivity of this subnetwork was negatively correlated with the increased global topology measures. Lastly, the key network measures showed high correlations with several clinical symptom scores, and classified BPD patients against healthy controls with high accuracy based on linear discriminant analysis. The abnormal topological properties and connectivity found in this study may add new knowledge to the current understanding of functional brain networks in BPD. However, due to limitation of small sample sizes, the results of the current study should be viewed as exploratory and need to be validated on large samples in future works.

Simple groups with m-regular first prime graph component

Let G be a finite simple group and GK(G) be the prime graph of G. The connected component of GK(G) whose vertex set contains 2 is denoted by $\pi_1(G)$. In this paper, our purpose is to classify the finite simple groups G such that $\pi_1(G)$ is regular. We prove that $\pi_1(G)$ is regular if and only if all the connected components of GK(G) are cliques.

Giant Component in Random Multipartite Graphs with given Degree Sequences

We study the problem of the existence of a giant component in a random multipartite graph. We consider a random multipartite graph with p parts generated according to a given degree sequence [Formula: see text] which denotes the number of vertices in part i of the multipartite graph with degree given by the vector d in an n-node graph. We assume that the empirical distribution of the degree sequence converges to a limiting probability distribution. Under certain mild regularity assumptions, we characterize the conditions under which, with high probability, there exists a component of linear size. The characterization involves checking whether the Perron-Frobenius norm of the matrix of means of a certain associated edge-biased distribution is greater than unity. We also specify the size of the giant component when it exists. We use the exploration process of Molloy and Reed Molloy and Reed (1995) to analyze the size of components in the random graph. The main challenges arise due to the multidimensionality of the random processes involved which prevents us from directly applying the techniques from the standard unipartite case. In this paper we use techniques from the theory of multidimensional Galton-Watson processes along with Lyapunov function technique to overcome the challenges.