Node Immunization with Non-backtracking Eigenvalues

Spectral density of the non-backtracking operator

The non-backtracking operator was recently shown to provide a significant improvement when used for spectral clustering of sparse networks. In this paper we analyze its spectral density on large random sparse graphs using a mapping to the correlation functions of a certain interacting quantum disordered system on the graph. On sparse, tree-like graphs, this can be solved efficiently by the cavity method and a belief propagation algorithm. We show that there exists a paramagnetic phase, leading to zero spectral density, that is stable outside a circle of radius $\sqrtρ$, where $ρ$ is the leadin…

Assessing Percolation Threshold Based on High-Order Non-Backtracking Matrices

Percolation threshold of a network is the critical value such that when nodes or edges are randomly selected with probability below the value, the network is fragmented but when the probability is above the value, a giant component connecting large portion of the network would emerge. Assessing the percolation threshold of networks has wide applications in network reliability, information spread, epidemic control, etc. The theoretical approach so far to assess the percolation threshold is mainly based on spectral radius of adjacency matrix or non-backtracking matrix, which is limited to dense …

Graph Distance from the Topological View of Non-backtracking Cycles

Whether comparing networks to each other or to random expectation, measuring dissimilarity is essential to understanding the complex phenomena under study. However, determining the structural dissimilarity between networks is an ill-defined problem, as there is no canonical way to compare two networks. Indeed, many of the existing approaches for network comparison differ in their heuristics, efficiency, interpretability, and theoretical soundness. Thus, having a notion of distance that is built on theoretically robust first principles and that is interpretable with respect to features ubiquito…

Eigenvector localization in real networks and its implications for epidemic spreading

The spectral properties of the adjacency matrix, in particular its largest eigenvalue and the associated principal eigenvector, dominate many structural and dynamical properties of complex networks. Here we focus on the localization properties of the principal eigenvector in real networks. We show that in most cases it is either localized on the star defined by the node with largest degree (hub) and its nearest neighbors, or on the densely connected subgraph defined by the maximum $K$-core in a $K$-core decomposition. The localization of the principal eigenvector is often strongly correlated w…

Fragmenting networks by targeting collective influencers at a mesoscopic level

A practical approach to protecting networks against epidemic processes such as spreading of infectious diseases, malware, and harmful viral information is to remove some influential nodes beforehand to fragment the network into small components. Because determining the optimal order to remove nodes is a computationally hard problem, various approximate algorithms have been proposed to efficiently fragment networks by sequential node removal. Morone and Makse proposed an algorithm employing the non-backtracking matrix of given networks, which outperforms various existing algorithms. In fact, ma…

Spectral Clustering of Graphs with the Bethe Hessian

Spectral clustering is a standard approach to label nodes on a graph by studying the (largest or lowest) eigenvalues of a symmetric real matrix such as e.g. the adjacency or the Laplacian. Recently, it has been argued that using instead a more complicated, non-symmetric and higher dimensional operator, related to the non-backtracking walk on the graph, leads to improved performance in detecting clusters, and even to optimal performance for the stochastic block model. Here, we propose to use instead a simpler object, a symmetric real matrix known as the Bethe Hessian operator, or deformed Lapla…

基于自我中心网络的社会和通信网络中病毒传播过程的结构分析

Structural Analysis of Viral Spreading Processes in Social and Communication Networks Using Egonets

提出基于代数图论和凸优化的数学框架，利用自我中心网络（egonets）约束网络邻接矩阵最大特征值的可能区间，从而分析局部结构对病毒传播动力学的影响。

Pulse strategy for suppressing spreading on networks

In networked spreading models, each node can infect its neighbors and cure spontaneously. The curing is assumed to occur uniformly over time. A pulse immunization/curing strategy is more efficient and broadly applied to suppressing spreading processes. We model the epidemic process by the basic Susceptible-Infected (SI) process with a pulse curing and incorporate the underlying contact network. The mean-field epidemic threshold of the pulse SI model is shown to be $\frac{1}{λ_1}\ln\frac{1}{1-p}$, where $λ_1$ and $p$ are the largest eigenvalue of the adjacency matrix and the fraction of nodes c…

Distinct types of eigenvector localization in networks

The spectral properties of the adjacency matrix provide a trove of information about the structure and function of complex networks. In particular, the largest eigenvalue and its associated principal eigenvector are crucial in the understanding of nodes centrality and the unfolding of dynamical processes. Here we show that two distinct types of localization of the principal eigenvector may occur in heterogeneous networks. For synthetic networks with degree distribution $P(q) \sim q^{-γ}$, localization occurs on the largest hub if $γ>5/2$; for $γ<5/2$ a new type of localization arises on a meso…

图能量作为网络中社区可检测性的度量

Graph energy as a measure of community detectability in networks

本文通过图能量差异研究网络中社区的可检测性阈值，揭示标准图矩阵仍能区分可检测与不可检测社区。