Algebraic connectivity

About: Algebraic connectivity is a research topic. Over the lifetime, 1392 publications have been published within this topic receiving 60848 citations.

The consensus protocol for multi-agent systems based on the algebraic connectivity

Abstract: The paper studies the consensus protocol of the time invariant multi-agent continuous systems. Based on the direct graph Laplacian matrix theory and it's the second smallest eigenvalue of communication topology, we study the continuous-time integrator agents dynamics system. The issue of stability of the systems is analyzed and the convergence problem of the information states is also discussed. Two examples are presented to show the effectiveness of the results of this paper.

Optimal Network Topology Based on Natural Connectivity and Improved Genetic Algorithm

Abstract: Invulnerability study is an important research content of complex network. On the basis of analysis and comparison of other traditional inadequate measurements of invulnerability, natural connectivity was used as the invulnerability measurement of network topology and an optimal network topology model based on natural connectivity has been proposed. Based on the basic idea of genetic algorithms, we designed an improved algorithm based on genetic algorithm and cloud mode to solve the model. Simulation experiments analyzed the invulnerability changes before and after the network topology optimization and verified the rationality of network topology optimization model and algorithm. Introduction An important purpose of complex network research is to design a good network topology and invulnerability is an important measurement to check the network. Currently, the study of complex network invulnerability is based on graph theory. The graph theory based invulnerability indicators mainly include toughness [1], integrity [2], tenacity [3], algebraic connectivity [4][5], etc., but these indicators have their disadvantages [6][7]. Chvatal initially used tenacity to study Hamiltonicity a graph. Later, it was used to measure the graph invulnerability, but Bauer et al demonstrated that the calculation of toughness was a NP problem. The integrity does not only consider the degree of difficulty of network destruction, but also takes into account of the scale of the largest piece after being destroyed, but the calculation of integrity is also a NP problem. Similarly, tenacity does not solely focus on the degree of difficulty of network destruction, but also takes into account of the scale of the maximum connectivity piece scale and the number of connectivity pieces, but the calculation of tenacity is also a NP problem. Algebraic connectivity is obviously inadequate, and not suitable for large-scale networks. Nature connectivity proposed by Literature [8] focused on proceeding from the internal network topology properties to describe the redundancy of alternative path by the method of closed path amount weighing, with clear physical significance and in concise mathematical form. It can be directly obtained by calculating the characteristic spectrum of the adjacent matrix, with a relatively low calculation complexity. Moreover, the edge adding and removal of natural connectivity is in strict monotonic increase or decrease, which has obvious advantages by comparing with other indicators and can well measure the invulnerability of network topology. Therefore, this paper uses natural connectivity as the invulnerability measurement to establish network topology optimization model, design the algorithm based on genetic algorithm and cloud model, which does not only speed up the search speed of the algorithm, but also reduces the possibility of local optimum. 5th International Conference on Information Engineering for Mechanics and Materials (ICIMM 2015) © 2015. The authors Published by Atlantis Press 1175 Network Topology Invulnerability Optimization Model Network topology representation. Complex network topology can be represented as the Figure G=(V,E)composed of node set V and edge set E, where V is a node set, represented as { } 1 2 , , , N V v v v =  , edge set represented as { } 1 2 , , , W E e e e V V = ⊆ ×  , indicating node connectivity. N V = is used to represent node quantity, whereas W E = represents edge quantity, ( ) ( ) ij N N E G a × = means the adjacent matrix of G, where ij a represents the connection between network node i and j: 1 0 ij i and j is connected a otherwise  =   (1) λ − is the natural connectivity of Figure G, where

MOHCS: Towards Mining Overlapping Highly Connected Subgraphs

Abstract: Many networks in real-life typically contain parts in which some nodes are more highly connected to each other than the other nodes of the network. The collection of such nodes are usually called clusters, communities, cohesive groups or modules. In graph terminology, it is called highly connected graph. In this paper, we first prove some properties related to highly connected graph. Based on these properties, we then redefine the highly connected subgraph which results in an algorithm that determines whether a given graph is highly connected in linear time. Then we present a computationally efficient algorithm, called MOHCS, for mining overlapping highly connected subgraphs. We have evaluated experimentally the performance of MOHCS using real and synthetic data sets from computer-generated graph and yeast protein network. Our results show that MOHCS is effective and reliable in finding overlapping highly connected subgraphs. Keywords-component; Highly connected subgraph, clustering algorithms, minimum cut, minimum degree

Draft: resilient design of complex engineered systems

Abstract: This paper presents a general methodology to calculate the resiliency of complex engineered systems based on complex network theory. Graph spectral theory has been used simultaneously with complex network theory for years. Resiliency is a key driver in how systems are developed to operate in an expected operating environment, and how systems change and respond to the environments in which they operate. This paper uses a popular method for mathematically identifying features in complex networks based on the adjacency matrix; commonly used to represent edge connections between nodes in complex networks. A similar approach can also be used to define the physical connections within complex engineered systems. In conjunction with the adjacency matrix, the degree and Laplacian matrices have eigenvalue and eigenspectrum properties that can be used with complex engineered systems to calculate their design resiliency. One such property of the Laplacian matrix is called the algebraic connectivity. The algebraic connectivity is defined as the second smallest eigenvalue of the Laplacian matrix and is proven to be directly related to the resiliency of a complex network. Our motivation in the present work is to calculate the algebraic connectivity and other graph metrics to predict the resiliency of the system under design.

Some properties of ergodicity coefficients with applications in spectral graph theory

Abstract: The main result is Corollary 2.9 which provides upper bounds on, and even better, approximates the largest non-trivial eigenvalue in absolute value of real constant row-sum matrices by the use of vector norm based ergodicity coefficients Tp. If the constant row-sum matrix is nonsingular, then it is also shown how its smallest non-trivial eigenvalue in absolute value can be bounded by using Tp. In the last section, these two results are applied to bound the spectral radius of the Laplacian matrix as well as the algebraic connectivity of its associated graph. Many other results are obtained. In particular, Theorem 2.15 is a convergence theorem for Tp and Theorem 4.7 compares some ergodicity coefficients to each other.