Signal Processing on Higher-Order Networks: Livin' on the Edge ... and Beyond

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. key words: spectral graph theory, coupled oscillators, node centrality, resonance, flaming, quantum theory

https://www.jstage.jst.go.jp/article/transcom/E101.B/1/E101.B_2017EBN0001/_pdf

Oscillation Model for Describing Network Dynamics Caused by Asymmetric Node Interaction

We introduce a novel harmonic analysis for functions defined on the vertices of a strongly connected directed graph of which the random walk operator is the cornerstone. As a first step, we consider the set of eigenvectors of the random walk operator as a non-orthogonal Fourier-type basis for functions over directed graphs. We found a frequency interpretation by linking the variation of the eigenvectors of the random walk operator obtained from their Dirichlet energy to the real part of their associated eigenvalues. From this Fourier basis, we can proceed further and build multi-scale analyses on directed graphs. We propose both a redundant wavelet transform and a decimated wavelet transform by extending the diffusion wavelets framework by Coifman and Maggioni for directed graphs. The development of our harmonic analysis on directed graphs thus leads us to consider both semi-supervised learning problems and signal modeling problems on graphs applied to directed graphs highlighting the efficiency of our framework.

Harmonic analysis on directed graphs and applications: from Fourier analysis to wavelets

This paper proposes an oscillation model for describing the flaming phenomena in on-line social networks, and discusses countermeasures to flaming based on the proposed oscillation model. The most significant feature of the proposed model is that the cause of flaming can be explained by the structure of the network. Based on the proposed model, we can suppress the generation of flaming by controlling link weights of a part of the network. This passive solution to flaming is important for the stable operation of social media networks.

https://dl.acm.org/doi/pdf/10.1145/3110025.3120982

Dynamical Model of Flaming Phenomena in On-Line Social Networks

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. key words: Laplacian matrix, spectral graph theory, resonance

Network Resonance Method: Estimating Network Structure from the Resonance of Oscillation Dynamics

Graph signal processing is a useful tool for representing, analyzing, and processing the signal lying on a graph, and has attracted attention in several fields including data mining and machine learning. A key to construct the graph signal processing is the graph Fourier transform, which is defined by using eigenvectors of the graph Laplacian of an undirected graph. The orthonormality of eigenvectors gives the graph Fourier transform algebraically desirable properties, and thus the graph signal processing for undirected graphs has been well developed. However, since eigenvectors of the graph Laplacian of a directed graph are generally not orthonormal, it is difficult to simply extend the graph signal processing to directed graphs. In this paper, we present a general framework for extending the graph signal processing to directed graphs. To this end, we introduce the Hermitian Laplacian which is a complex matrix obtained from an extension of the graph Laplacian. The Hermitian Laplacian is defined so as to preserve the edge directionality and Hermitian property and enables the graph signal processing to be straightforwardly extended to directed graphs. Furthermore, the Hermitian Laplacian guarantees some desirable properties, such as non-negative real eigenvalues and the unitarity of the Fourier transform. Finally, experimental results for representation learning and signal denoising of/on directed graphs show the effectiveness of our framework.

Graph Signal Processing for Directed Graphs Based on the Hermitian Laplacian

Sybils are users created for carrying out nefarious actions in online social networks (OSNs) and threaten the security of OSNs. Therefore, Sybil detection is an urgent security task, and various detection methods have been proposed. Existing Sybil detection methods are based on the relationship (i.e., graph structure) of users in OSNs. Structure-based methods can be classified into two categories: Random Walk (RW)-based and Belief Propagation (BP)-based. However, although almost all methods have been experimentally evaluated in terms of their performance and robustness to noise, the theoretical understanding of them is insufficient. In this paper, we interpret the Sybil detection problem from the viewpoint of graph signal processing and provide a framework to formulate RW- and BPbased methods as low-pass filtering. This framework enables us to theoretically compare RW- and BP-based methods and explain why BP-based methods perform well for scale-free graphs, unlike RW-based methods. Furthermore, by this framework, we relate RW- and BP-based methods and Graph Neural Networks (GNNs) and discuss the difference among these methods. Finally, we evaluate the validity of this framework through numerical experiments.

Sybil Detection as Graph Filtering

Spectral graph theory gives a useful approach to analyzing network structure based on the adjacency matrix or the Laplacian matrix that represents the network topology and link weights. However, in large scale and complex social networks, since it is difficult to know the network topology and link weights, we cannot determine the components of these matrices directly. To solve this problem, we consider a method for indirectly determining a Laplacian matrix from its eigenvalues and eigenvectors. As the first step, our prior study proposed a method for estimating eigenvalues of a Laplacian matrix by using the resonance of oscillation dynamics on networks with no a priori information about the network structure, and showed the effectiveness of this method. In this paper, we propose a method for estimating the eigenvectors of a Laplacian matrix by once again using the resonance of oscillation dynamics on networks.

Method for Estimating the Eigenvectors of a Scaled Laplacian Matrix Using the Resonance of Oscillation Dynamics on Networks

Eigenvalues of the Laplacian matrix play an important role in characterizing structural and dynamical properties of networks In the procedure for calculating eigenvalues of the Laplacian matrix, we need to get the Laplacian matrix that represents structures of the network Since the actual structure of networks and the strength of links are difficult to know, it is difficult to determine elements of the Laplacian matrix To solve this problem, our previous study proposed a concept of the network resonance method, which is for estimating eigenvalues of the scaled Laplacian matrix using resonance of oscillation dynamics on networks This method does not need a priori information about the network structure In this research, we investigate feasibility of the network resonance method, and show that the method can estimate eigenvalues of the scaled Laplacian matrix of the entire network through observations of oscillation dynamics even if observable nodes are restricted to a part of network

Satoshi Furutani

Papers

Graph Signal Processing for Directed Graphs Based on the Hermitian Laplacian

Network Resonance Method: Estimating Network Structure from the Resonance of Oscillation Dynamics

Sybil Detection as Graph Filtering

Method for Estimating the Eigenvectors of a Scaled Laplacian Matrix Using the Resonance of Oscillation Dynamics on Networks

Proposal of the Network Resonance Method for Estimating Eigenvalues of the Scaled Laplacian Matrix