1,720,980 research outputs found
The structured distance to singularity of a symmetric tridiagonal Toeplitz matrix
Full text linkThis paper is concerned with the distance of a symmetric tridiagonal Toeplitz
matrix to the variety of similarly structured singular matrices, and with
determining the closest matrix to in this variety. Explicit formulas are
presented, that exploit the analysis of the sensitivity of the spectrum of
with respect to structure-preserving perturbations of its entries.Comment: 16 pages, 5 Figure
Edge importance in complex networks
Full text linkComplex networks are made up of vertices and edges. The latter connect the vertices. There are several ways to measure the importance of the vertices, e.g., by counting the number of edges that start or end at each vertex, or by using the subgraph centrality of the vertices. It is more difficult to assess the importance of the edges. One approach is to consider the line graph associated with the given network and determine the importance of the vertices of the line graph, but this is fairly complicated except for small networks. This paper compares two approaches to estimate the importance of edges of medium-sized to large networks. One approach computes partial derivatives of the total communicability of the weights of the edges, where a partial derivative of large magnitude indicates that the corresponding edge may be important. Our second approach computes the Perron sensitivity of the edges. A high sensitivity signals that the edge may be important. The performance of these methods and some computational aspects are discussed. Applications of interest include to determine whether a network can be replaced by a network with fewer edges with about the same communicability
Communication in multiplex transportation networks
No full textComplex networks are made up of vertices and edges. The edges, which may be directed or undirected, are equipped with positive weights. Modeling complex systems that consist of different types of objects leads to multilayer networks, in which vertices in distinct layers represent different kinds of objects. Multiplex networks are special vertex-aligned multilayer networks, in which vertices in distinct layers are identified with each other and inter-layer edges connect each vertex with its copy in other layers and have a fixed weight γ>0 associated with the ease of communication between layers. This paper discusses two different approaches to analyze communication in a multiplex. One approach focuses on the multiplex global efficiency by using the multiplex path length matrix, the other approach considers the multiplex total communicability. The sensitivity of both the multiplex global efficiency and the multiplex total communicability to structural perturbations in the network is investigated to help to identify intra-layer edges that should be strengthened to enhance communicability
Enhancing multiplex global efficiency
No full textModeling complex systems that consist of different types of objects leads to multilayer networks, in which vertices are connected by both inter-layer and intra-layer edges. In this paper, we investigate multiplex networks, in which vertices in different layers are identified with each other, and the only inter-layer edges are those that connect a vertex with its copy in other layers. Let the third-order adjacency tensor A∈ RN×N×L and the parameter γ≥ 0 , which is associated with the ease of communication between layers, represent a multiplex network with N vertices and L layers. To measure the ease of communication in a multiplex network, we focus on the average inverse geodesic length, which we refer to as the multiplex global efficiency eA(γ) by means of the multiplex path length matrix P∈ RN×N . This paper generalizes the approach proposed in [15] for single-layer networks. We describe an algorithm based on min-plus matrix multiplication to construct P, as well as variants PK that only take into account multiplex paths made up of at most K intra-layer edges. These matrices are applied to detect redundant edges and to determine non-decreasing lower bounds eAK(γ) for eA(γ) , for K= 1 , 2 , ⋯ , N- 2 . Finally, the sensitivity of eAK(γ) to changes of the entries of the adjacency tensor A is investigated to determine edges that should be strengthened to enhance the multiplex global efficiency the most
Network analysis with the aid of the path length matrix
Full text linkLet a network be represented by a simple graph G with n vertices. A common approach to investigate properties of a network is to use the adjacency matrix A=[aij]i,j=1n∈Rn×n associated with the graph G , where aij> 0 if there is an edge pointing from vertex vi to vertex vj , and aij= 0 otherwise. Both A and its positive integer powers reveal important properties of the graph. This paper proposes to study properties of a graph G by also using the path length matrix for the graph. The (ij) th entry of the path length matrix is the length of the shortest path from vertex vi to vertex vj ; if there is no path between these vertices, then the value of the entry is ∞ . Powers of the path length matrix are formed by using min-plus matrix multiplication and are important for exhibiting properties of G . We show how several known measures of communication such as closeness centrality, harmonic centrality, and eccentricity are related to the path length matrix, and we introduce new measures of communication, such as the harmonic K-centrality and global K-efficiency, where only (short) paths made up of at most K edges are taken into account. The sensitivity of the global K-efficiency to changes of the entries of the adjacency matrix also is considered
Estimating and increasing the structural robustness of a network
Full text linkThe capability of a network to cope with threats and survive attacks is referred to as its robustness. This article discusses one kind of robustness, commonly denoted structural robustness, which increases when the spectral radius of the adjacency matrix associated with the network decreases. We discuss computational techniques for identifying edges, whose removal may significantly reduce the spectral radius. Nonsymmetric adjacency matrices are studied with the aid of their pseudospectra. In particular, we consider nonsymmetric adjacency matrices that arise when people seek to avoid being infected by Covid-19 by wearing facial masks of different qualities
An iterative method for computing the eigenvalues of second kind Fredholm operators and applications
No full text
- …
