Various graph algorithms have been developed with multiple random walks, the movement of several independent random walkers on a graph. Designing an efficient graph algorithm based on multiple random walks requires investigating multiple random walks theoretically to attain a deep understanding of their characteristics. The first meeting time is one of the important metrics for multiple random walks. The first meeting time on a graph is defined by the time it takes for multiple random walkers to meet at the same node in a graph. This time is closely related to the rendezvous problem, a fundamental problem in computer science. The first meeting time of multiple random walks has been analyzed previously, but many of these analyses focused on regular graphs. In this paper, we analyze the first meeting time of multiple random walks in arbitrary graphs and clarify the effects of graph structures on expected values. First, we derive the spectral formula of the expected first meeting time on the basis of spectral graph theory. Then, we examine the principal component of the expected first meeting time using the derived spectral formula. The clarified principal component reveals that (a) the expected first meeting time is almost dominated by $n/(1+d_{ m std}^2/d_{ mavg}^2)$ and (b) the expected first meeting time is independent of the starting nodes of random walkers, where n is the number of nodes of the graph. davg and dstd are the average and the standard deviation of weighted node degrees, respectively. Characteristic (a) is useful for understanding the effect of the graph structure on the first meeting time. According to the revealed effect of graph structures, the variance of the coefficient dstd/davg (degree heterogeneity) for weighted degrees facilitates the meeting of random walkers.

Spectral graph theory provides an algebraic approach to investigate the characteristics of weighted networks using the eigenvalues and eigenvectors of a matrix (e.g., normalized Laplacian matrix) that represents the structure of the network. However, it is difficult to accurately represent the structures of large-scale and complex networks (e.g., social network) as a matrix. This difficulty can be avoided if there is a universality, such that the eigenvalues are independent of the detailed structure in large-scale and complex network. In this paper, we clarify Wigner's Semicircle Law for weighted networks as such a universality. The law indicates that the eigenvalues of the normalized Laplacian matrix of weighted networks can be calculated from a few network statistics (the average degree, average link weight, and square average link weight) when the weighted networks satisfy a sufficient condition of the node degrees and the link weights.

Spectral graph theory, based on the adjacency matrix or the Laplacian matrix that represents the network topology and link weights, provides a useful approach for analyzing network structure. However, in large scale and complex social networks, since it is difficult to completely know the network topology and link weights, we cannot determine the components of these matrices directly. To solve this problem, we propose a method for indirectly determining the Laplacian matrix by estimating its eigenvalues and eigenvectors using the resonance of oscillation dynamics on networks.

Spectral graph theory gives an algebraic approach to the analysis of the dynamics of a network by using the matrix that represents the network structure. However, it is not easy for social networks to apply the spectral graph theory because the matrix elements cannot be given exactly to represent the structure of a social network. The matrix element should be set on the basis of the relationship between persons, but the relationship cannot be quantified accurately from obtainable data (e.g., call history and chat history). To get around this problem, we utilize the universality of random matrices with the feature of social networks. As such a random matrix, we use the normalized Laplacian matrix for a network where link weights are randomly given. In this paper, we first clarify that the universality (i.e., the Wigner semicircle law) of the normalized Laplacian matrix appears in the eigenvalue frequency distribution regardless of the link weight distribution. Then, we analyze the information propagation speed by using the spectral graph theory and the universality of the normalized Laplacian matrix. As a result, we show that the worst-case speed of the information propagation changes up to twice if the structure (i.e., relationship among people) of a social network changes.

This paper proposes an oscillation model for analyzing the dynamics of activity propagation across social media networks. In order to analyze such dynamics, we generally need to model asymmetric interactions between nodes. In matrix-based network models, asymmetric interaction is frequently modeled by a directed graph expressed as an asymmetric matrix. Unfortunately, the dynamics of an asymmetric matrix-based model is difficult to analyze. This paper, first of all, discusses a symmetric matrix-based model that can describe some types of link asymmetry, and then proposes an oscillation model on networks. Next, the proposed oscillation model is generalized to arbitrary link asymmetry. We describe the outlines of four important research topics derived from the proposed oscillation model. First, we show that the oscillation energy of each node gives a generalized notion of node centrality. Second, we introduce a framework that uses resonance to estimate the natural frequency of networks. Natural frequency is important information for recognizing network structure. Third, by generalizing the oscillation model on directed networks, we create a dynamical model that can describe flaming on social media networks. Finally, we show the fundamental equation of oscillation on networks, which provides an important breakthrough for generalizing the spectral graph theory applicable to directed graphs.