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This paper is concerned with the identification of important nodes in node-weighted graphs by applying matrix functions, in particular the matrix exponential. Many tools that use an adjacency matrix for a graph have been developed to study the importance of the nodes in unweighted or edge-weighted networks. However, adjacency matrices for node-weighted graphs have not received much attention. The present paper proposes using a line graph associated with a node-weighted graph to construct an edge-weighted graph that can be analyzed with available methods. Both undirected and directed graphs with positive node weights are considered. We show that when the weight of a node increases, the importance of this node in the graph increases as well, provided that the adjacency matrix is suitably scaled. Applications to real-life problems are presented. [ABSTRACT FROM AUTHOR]

Matrix functions play an important role in applied mathematics. In network analysis, in particular, the exponential of the adjacency matrix associated with a network provides valuable information about connectivity, as well as about the relative importance or centrality of nodes. Another popular approach to rank the nodes of a network is to compute the left Perron vector of the adjacency matrix for the network. The present article addresses the problem of evaluating matrix functions, as well as computing an approximation to the left Perron vector, when only some of the columns and/or some of the rows of the adjacency matrix are known. Applications to network analysis are considered, when only some sampled columns and/or rows of the adjacency matrix that defines the network are available. A sampling scheme that takes the connectivity of the network into account is described. Computed examples illustrate the performance of the methods discussed. [ABSTRACT FROM AUTHOR]

This paper is concerned with the identification of important edges in a network, in both their roles as transmitters and receivers of information. We propose a method based on computing the matrix exponential of a matrix associated with a line graph of the given network. Both undirected and directed networks are considered. Edges may be given positive weights. Computed examples illustrate the performance of the proposed method. [ABSTRACT FROM AUTHOR]

Networks in which connections change over time arise in many applications, e.g., when modeling phone calls and flights between airports. This paper discusses new ways to define adjacency matrices associated with this kind of networks. We propose that dynamic networks be modeled with the aid of block upper triangular adjacency matrices. Both modeling and computational aspects are discussed. Several applications to real dynamic networks are presented and illustrate the advantages of the proposed method when compared with an available approach. [ABSTRACT FROM AUTHOR]