Average Vertex Degree Research Articles

Nonlinear Relationship and Its Evolutionary Trace between Average Degree and Average Path Length of Edge Vertices of China Aviation Network Based on Complex Network

Nonlinear Relationship and Its Evolutionary Trace between Average Degree and Average Path Length of Edge Vertices of China Aviation Network Based on Complex Network

Probability Distribution of China Aviation Network Average Degree of Edge Vertices and Its Evolutionary Trace Based on Complex Network

Probability Distribution of China Aviation Network Average Degree of Edge Vertices and Its Evolutionary Trace Based on Complex Network

On the inverse Collatz-Sinogowitz irregularity problem

Abstract The Collatz-Sinogowitz irregularity index is the oldest known numerical measure of graph irregularity. For a simple and connected graph G G of order n n and size m m , it is defined as CS ( G ) = λ 1 − 2 m / n , \hspace{0.1em}\text{CS}\hspace{0.1em}\left(G)={\lambda }_{1}-2m\hspace{0.1em}\text{/}\hspace{0.1em}n, where λ 1 {\lambda }_{1} is the largest eigenvalue of the adjacency matrix of G G , and 2 m / n 2m\hspace{0.1em}\text{/}\hspace{0.1em}n is the average vertex degree of G G . Here, the Collatz-Sinogowitz inverse irregularity problem is studied. For every integer i ≥ 0 i\ge 0 , it is shown that there exists a graph G G such that CS ( G ) = i \hspace{0.1em}\text{CS}\hspace{0.1em}\left(G)=i . Also, for every interval I i = ( i , i + 1 ) {I}_{i}=\left(i,i+1) , it is shown that there are infinitely many graphs whose Collatz-Sinogowitz irregularity lies in I i {I}_{i} .

On the number of independent and $k$-dominating sets in graphs with average vertex degree at most $k$

The following conjecture is formulated: if the average vertex degree in a graph is not greater than a positive integer $k \geqslant 1$, then the number of $k$-dominating sets in this graph does not exceed the number of its independent sets, and these numbers are equal to each other if and only if the graph is $k$-regular. This conjecture is proved for $k \in \{1,2\}$. Bibliography: 10 titles.

Adaptive Vertex Cover of Dynamical Networks With Prospect Theoretic Perspective

Vertex covering a static network has received much attention in the past few years, while vertex covering a dynamical network has rarely been reviewed. In this article, we newly establish a vertex cover game model of dynamical networks with integrating prospect theory, and design an adaptive evolution scheme for network structure and vertices' strategies to achieve the adaptive covering of a dynamical network concurrently. We define a Nash equilibrium state of the network structure and vertices' strategies. Subsequently, we prove that given the reference point in the active region, the designed adaptive evolution scheme can drive the network structure and vertices' strategies to a Nash cover equilibrium (NCE) state with probability one. Intensive examples verify the effectiveness of the designed adaptive evolution scheme to the vertex covering of dynamical networks, presenting that in the NCE states, the less the average number of covered vertices, the less the average clustering coefficient and the ratio of the average degree of covered vertices to the average degree of uncovered vertices.

DynamAP: Architectural Support for Dynamic Graph Traversal on the Automata Processor

Dynamic graph traversals (DGTs) currently are widely used in many important application domains, especially in this big-data era that urgently demands high-performance graph processing and analysis. Unlike static graph traversals, DGTs in real-world application scenarios require not only fast traversal acceleration itself but also, more importantly, a runtime strategy that can effectively accommodate the ever-evolving nature of the graph structure updates followed by a diverse range of graph traversal algorithms . Because of these special features, state-of-the-art designs on conventional compute-centric architectures (e.g., CPU and GPU) struggle to provide sufficient acceleration for DGT processing due to the dominating irregular memory access patterns in graph traversal algorithms and inefficient platform-specific update mechanisms. In this article, we explore the algorithmic features and runtime requirements of real-world DGTs and identify their unique opportunities of acceleration on the recent Micron Automata Processor (AP), an in-situ memory-centric pattern-matching architecture. These features include the natural mapping between traversal algorithms’ path exploration pattern to classic non-deterministic finite automata processing, AP’s architectural and compilation support for DGTs’ evolving traversal operations, and its inherent hardware fitness. However, despite these benefits, enabling highly efficient DGT execution on AP is non-trivial and faces several major challenges. To tackle them, we propose DynamAP , the first AP framework design that enables fast processing for general DGTs. DynamAP is oblivious to periodical traversal algorithm changes and can address the significant overhead caused by frequent graph updates and AP recompilation through our novel hybrid macro designs and associated efficient updating strategies. We evaluate DynamAP against the current DGT designs on a CPU, GPU, and AP with a range of widely adopted DGT algorithms and real-world graphs. For a single update request , our DynamAP achieves an average speedup of 21.3x (up to 39.2x ) over the state-of-the-art implementation on host-AP architecture; an average speedup of 9.2x (up to 14.7x ) and 1.7x (up to 2.8x ) over two highly optimized DGT design frameworks on a 64-GB Intel(R) Xeon CPU and a 32-GB NVIDIA Tesla V100 GPU. DynamAP also maintains high performance and resource utilization for high graph update ratios, and can significantly benefit natural graphs that present a high average vertex degree.

Graphs of low average degree without independent transversals

AbstractAn independent transversal of a graph with a vertex partition is an independent set of intersecting each block of in a single vertex. Wanless and Wood proved that if each block of has size at least and the average degree of vertices in each block is at most , then an independent transversal of exists. We present a construction showing that this result is optimal: for any and sufficiently large , there is a forest with a vertex partition into parts of size at least such that the average degree of vertices in each block is at most , and there is no independent transversal. This unexpectedly shows that methods related to entropy compression such as the Rosenfeld–Wanless–Wood scheme or the Local Cut Lemma are tight for this problem. Further constructions are given for variants of the problem, including the hypergraph version.

On the Normalized Laplacian Spectra of Random Geometric Graphs

In this work, we study the spectrum of the normalized Laplacian and its regularized version for random geometric graphs (RGGs) in various scaling regimes. Two scaling regimes are of special interest, the connectivity and the thermodynamic regime. In the connectivity regime, the average vertex degree grows logarithmically in the graph size or faster. In the thermodynamic regime, the average vertex degree is a constant. We introduce a deterministic geometric graph (DGG) with nodes in a grid and provide an upper bound to the probability that the Hilbert–Schmidt norm of the difference between the normalized Laplacian matrices of the RGG and DGG is greater than a certain threshold in both the connectivity and thermodynamic regime. Using this result, we show that the RGG and DGG normalized Laplacian matrices are asymptotically equivalent with high probability (w.h.p.) in the full range of the connectivity regime. The equivalence is even stronger and holds almost surely when the average vertex degree a_n satisfies the inequality a_n > 24 log (n). Therefore, we use the regular structure of the DGG to show that the limiting eigenvalue distribution of the RGG normalized Laplacian matrix converges to a distribution with a Dirac atomic measure at zero. In the thermodynamic regime, we approximate the eigenvalues of the regularized normalized Laplacian matrix of the RGG by the eigenvalues of the DGG regularized normalized Laplacian and we provide an error bound which is valid w.h.p. and depends upon the average vertex degree.

Edge N-Level Sparse Visibility Graphs: Fast Optimal Any-Angle Pathfinding Using Hierarchical Taut Paths

In the Any-Angle Pathfinding problem, the goal is to find the shortest path between a pair of vertices on a uniform square grid, that is not constrained to any fixed number of possible directions over the grid. Visibility Graphs are a known optimal algorithm for solving the problem with the use of pre-processing. However, Visibility Graphs are known to perform poorly in terms of running time, especially on large, complex maps. In this paper, we introduce two improvements over the Visibility Graph Algorithm to compute optimal paths. Sparse Visibility Graphs (SVGs) are constructed by pruning unnecessary edges from the original Visibility Graph. Edge N-Level Sparse Visibility Graphs (ENLSVGs) is a hierarchical SVG built by iteratively pruning non-taut paths. We also introduce Line-of-Sight Scans, a faster algorithm for building Visibility Graphs over a grid. SVGs run much faster than Visibility Graphs by reducing the average vertex degree. ENLSVGs, a hierarchical algorithm, improves this further, especially on larger maps, with millisecond runtimes even on 6000 x 6000 maps. On large maps, with the use of pre-processing, these algorithms are at least an order of magnitude faster than existing algorithms like Visibility Graphs, Anya and Theta*.

On the density of Cayley graphs of R.Thompson’s group F in symmetric generators

By the density of a finite graph we mean its average vertex degree. For an [Formula: see text]-generated group, the density of its Cayley graph in a given set of generators, is the supremum of densities taken over all its finite subgraphs. It is known that a group with [Formula: see text] generators is amenable if and only if the density of the corresponding Cayley graph equals [Formula: see text]. A famous problem on the amenability of R. Thompson’s group [Formula: see text] is still open. Due to the result of Belk and Brown, it is known that the density of its Cayley graph in the standard set of group generators [Formula: see text], is at least [Formula: see text]. This estimate has not been exceeded so far. For the set of symmetric generators [Formula: see text], where [Formula: see text], the same example only gave an estimate of [Formula: see text]. There was a conjecture that for this generating set equality holds. If so, [Formula: see text] would be non-amenable, and the symmetric generating set would have the doubling property. This would mean that for any finite set [Formula: see text], the inequality [Formula: see text] holds. In this paper, we disprove this conjecture showing that the density of the Cayley graph of [Formula: see text] in symmetric generators [Formula: see text] strictly exceeds [Formula: see text]. Moreover, we show that even larger generating set [Formula: see text] does not have doubling property.

Cliques in high-dimensional random geometric graphs

Random geometric graphs have become now a popular object of research. Defined rather simply, these graphs describe real networks much better than classical Erdős–Rényi graphs due to their ability to produce tightly connected communities. The n vertices of a random geometric graph are points in d-dimensional Euclidean space, and two vertices are adjacent if they are close to each other. Many properties of these graphs have been revealed in the case when d is fixed. However, the case of growing dimension d is practically unexplored. This regime corresponds to a real-life situation when one has a data set of n observations with a significant number of features, a quite common case in data science today. In this paper, we study the clique structure of random geometric graphs when nrightarrow infty, and d rightarrow infty, and average vertex degree grows significantly slower than n. We show that under these conditions, random geometric graphs do not contain cliques of size 4 a. s. if only d gg log ^{1 + epsilon } n. As for the cliques of size 3, we present new bounds on the expected number of triangles in the case log ^2 n ll d ll log ^3 n that improve previously known results. In addition, we provide new numerical results showing that the underlying geometry can be detected using the number of triangles even for small n.

Algorithm for building a website model

The analysis of the structure of the website modeling has been carried out. The models of Internet space representation in the form of semantic networks, frame structures and ontology have been analyzed. The web graph model has been chosen to represent the web resource. The pages of a web resource are connected by hyperlinks, which form the internal structure of the resource. To build a model of a website in the form of a web graph, a method and algorithm for scanning the pages of a web resource have been developed. The web resource scanning is performed by in depth searching with the LIFO (Last In - First Out) method. Links are searched by sorting the lines of the page markup text and extracting links by using regular expressions. Only links to pages within the resource are taken into account in the search process, external links are ignored. The crawling procedure is implemented by using the Scrapy framework and the Python. To account for the presence of additional filters used to select pages with criteria, the rules for selecting URL in HTML code have been strengthened. Web resources are scanned to build their web graphs. Storing information by using a list of edges and an adjacency matrix is used in further work with the obtained web graphs. To visualize the obtained graphs and calculate some metric characteristics, the Gephi software environment and the algorithm for stacking the vertices of the Yifan Hu graph has been used. The graph diameters, the average vertex degree, the average path length, the density factor of the graph are used for analysis of the structural connectivity of the graphs studied. The proposed approach can be applied during the site reengineering procedure.

Constraining cosmology with big data statistics of cosmological graphs

ABSTRACT By utilizing large-scale graph analytic tools implemented in the modern big data platform, apache spark, we investigate the topological structure of gravitational clustering in five different universes produced by cosmological N-body simulations with varying parameters: (1) a WMAP 5-yr compatible ΛCDM cosmology, (2) two different dark energy equation of state variants, and (3) two different cosmic matter density variants. For the big data calculations, we use a custom build of standalone Spark/Hadoop cluster at Korea Institute for Advanced Study and Dataproc Compute Engine in Google Cloud Platform with sample sizes ranging from 7 to 200 million. We find that among the many possible graph-topological measures, three simple ones: (1) the average of number of neighbours (the so-called average vertex degree) α, (2) closed-to-connected triple fraction (the so-called transitivity) $\tau _\Delta$, and (3) the cumulative number density ns ≥ 5 of subgraphs with connected component size s ≥ 5, can effectively discriminate among the five model universes. Since these graph-topological measures are directly related with the usual n-points correlation functions of the cosmic density field, graph-topological statistics powered by big data computational infrastructure opens a new, intuitive, and computationally efficient window into the dark Universe.

Modelling Selected Parameters of Power Grid Network in the South-Eastern Part of Poland: The Case Study

In this paper, a case study is conducted based on the real data obtained from the local Distribution System Operator (DSO) of electrical energy. The analyzed network represents connections and high-voltage switchgears of 110 kV. Selected graph parameters—vertex degree distribution, the average vertex degree, the graph density, network efficiency, the clustering coefficient, the average path length, and the graph diameter were examined, taking into account that in the analysis, some nodes were removed due to the different failures. For each failure, the possible effects on network parameters were tested. As a final result, it was shown that in the analyzed case, the removal of only five nodes could cause a significant (almost four times) fall of graph efficiency. In turn, this means that the whole analyzed network cannot be considered as a fault-tolerant.

Quantitative description and classification of protein structures by a novel robust amino acid network: interaction selective network (ISN)

To quantitatively categorize protein structures, we developed a quantitative coarse-grained model of protein structures with a novel amino acid network, the interaction selective network (ISN), characterized by the links based on interactions in both the main and side chains. We found that the ISN is a novel robust network model to show the higher classification probability in the plots of average vertex degree (k) versus average clustering coefficient (C), both of which are typical network parameters for protein structures, and successfully distinguished between “all-α” and “all-β” proteins. On the other hand, one of the typical conventional networks, the α-carbon network (CAN), was found to be less robust than the ISN, and another typical network, atomic distance network (ADN), failed to distinguish between these two protein structures. Considering that the links in the CAN and ADN are defined by the interactions only between the main chain atoms and by the distance of the closest atom pair between the two amino acid residues, respectively, we can conclude that reflecting structural information from both secondary and tertiary structures in the network parameters improves the quantitative evaluation and robustness in network models, resulting in a quantitative and more robust description of three-dimensional protein structures in the ISN.

DIAMOND: a distributed algorithm for vertex coloring problems and resource allocation

The vertex colouring problem (VCP) and its generalisations have myriad applications in computer networks. To solve the VCP with Δ + 1 colours, numerous distributed algorithms based on LOCAL model have been proposed to reduce time complexity (the number of rounds), where Δ is the maximum vertex degree in the graph. In this paper, the authors present a distributed algorithm based on modified LOCAL model (DIAMOND) that reduces the number of rounds to one. It greedily solves the VCP with at most Δ + 1 colours. Computational results on Geometry (GEOM) graphs show that the number of used colours to colour each instance using DIAMOND is about Δ + 1 / 2 . DIAMOND is easily extended to solve greedily generalised VCPs in only one round. Moreover, they present two efficient resource allocation algorithms using DIAMOND. They allocate more resource to the graph compared with ( Δ + 1 ) -colouring and even to ( d ¯ + 1 ) -colouring algorithms, where d ¯ is the average vertex degree of the graph. They run in two and Δ rounds.

3560 Using Research Performance Progress Report data to Explore CTSI-Stakeholder Engagement through Network Analysis

OBJECTIVES/SPECIFIC AIMS: To develop a social network model of collaborations within and external to the University of Rochester Medical Center (URMC) CTSI using data from the annual Research Performance Progress Report (RPPR) as well as other sources, to provide longitudinal evaluation of the CTSI’s engagement with key stakeholder groups. METHODS/STUDY POPULATION: The annually submitted RPPR follows a specific format with well-defined sections. The Highlights, Milestones and Challenges Report includes areas in which CTSI function leaders provide details about program integration and innovation, including collaborations with other functions or external groups. The Highlights, Milestones and Challenges Report was qualitatively coded to identify function-collaborator dyads. Each entity in the dyad became a node in the network. Nodes were connected by edges named by the dyads. The network included two types of nodes. The first were CTSI internal functions/programs, i.e. the entities that submitted RPPR sections and formed an interconnected sub-network. The second type of nodes were entities external to the CTSI (collaborators, internal or external to the CTSI site). These entities were named by functions submitting RPPR narratives. External nodes with similar meanings were consolidated. Duplicate edges were removed. CTSI-external nodes were grouped into five stakeholder categories: URMC, University of Rochester (UR), community, other CTSA institutions, CTSA consortium. Thus, these nodes were connected to the CTSI internal nodes, but not to each other. A second source of collaboration data was function-reported internal metrics. As part of the internal metric data collection, functions list partners who play a role in improving metric data or who are responsible for providing data. Partners identified in the internal metrics data, but not specified in the RPPR, were added to the network. RESULTS/ANTICIPATED RESULTS: Twenty-three internal CTSI functions submitted an RPPR and represent the CTSI internal nodes. Internal CTSI functions identified 235 collaborations (edges): 125 collaborations with other CTSI internal functions, 57 collaborations with URMC entities, 14 with UR entities, 15 with the external community, 15 with other institutions (CTSA hubs and other universities), and 9 with CTSA consortium entities. Thirty-eight of the collaborations were identified in the internal metrics partners section. In total, the network comprised 104 nodes. Graph density was.022 for full network and.21 for the CTSI internal sub-network. The global clustering coefficient, a measure of connectivity, for the CTSI internal sub-network was.252. DISCUSSION/SIGNIFICANCE OF IMPACT: The RPPR provides an underutilized source of data for annually repeated analyses of internal and external CTSI collaborations and is a way to enhance use of this routinely collected information. Analyses of the network yield metrics for measuring CTSI reach and impact on stakeholder groups over time. For example, measures such as number of nodes representing entities external to CTSI and average vertex degree of the CTSI Internal nodes track aspects of CTSI collaborations. Visualizations using different layouts or highlighting different sub-networks provide a representation of CTSI engagement with the communities of stakeholders as well as insights to relationships between functions, regions of collaboration, and areas of gaps. These data also provide an important new mechanism to engage the CTSI leadership and function leads in understanding how their work contributes to the overall network and synergies they have with each other.

Approximate $\ell_{0}$-penalized estimation of piecewise-constant signals on graphs

We study recovery of piecewise-constant signals on graphs by the estimator minimizing an $l_{0}$-edge-penalized objective. Although exact minimization of this objective may be computationally intractable, we show that the same statistical risk guarantees are achieved by the $\alpha$-expansion algorithm which computes an approximate minimizer in polynomial time. We establish that for graphs with small average vertex degree, these guarantees are minimax rate-optimal over classes of edge-sparse signals. For spatially inhomogeneous graphs, we propose minimization of an edge-weighted objective where each edge is weighted by its effective resistance or another measure of its contribution to the graph’s connectivity. We establish minimax optimality of the resulting estimators over corresponding edge-weighted sparsity classes. We show theoretically that these risk guarantees are not always achieved by the estimator minimizing the $l_{1}$/total-variation relaxation, and empirically that the $l_{0}$-based estimates are more accurate in high signal-to-noise settings.

Limiting eigenvalue distribution of random matrices of Ihara zeta function of long-range percolation graphs

We consider the ensemble of [Formula: see text] real random symmetric matrices [Formula: see text] obtained from the determinant form of the Ihara zeta function associated to random graphs [Formula: see text] of the long-range percolation radius model with the edge probability determined by a function [Formula: see text]. We show that the normalized eigenvalue counting function of [Formula: see text] weakly converges in average as [Formula: see text], [Formula: see text] to a unique measure that depends on the limiting average vertex degree of [Formula: see text] given by [Formula: see text]. This measure converges in the limit of infinite [Formula: see text] to a shift of the Wigner semi-circle distribution. We discuss relations of these results with the properties of the Ihara zeta function and weak versions of the graph theory Riemann Hypothesis.

Signless Laplacian energy of a graph and energy of a line graph

For a simple graph G of order n, size m and with signless Laplacian eigenvalues q1,q2,…,qn, the signless Laplacian energy QE(G) is defined as QE(G)=∑i=1n|qi−d‾|, where d‾=2mn is the average vertex degree of G. We obtain the lower bounds for QE(G), in terms of first Zagreb index M1(G), maximum degree d1, second maximum degree d2, minimum degree dn and second minimum degree dn−1. As a consequence of these bounds, we obtain several bounds for the energy E(L(G)) of the line graph L(G) of graph G in terms of various graph parameters like M1(G), ω (the clique number), n, m, etc., which improve some recently known bounds.