Networks beyond pairwise interactions: Structure and dynamics

This article reviews and presents various solved and open problems in the development, analysis, and control of epidemic models. The proper modeling and analysis of spreading processes has been a long-standing area of research among many different fields, including mathematical biology, physics, computer science, engineering, economics, and the social sciences. One of the earliest epidemic models conceived was by Daniel Bernoulli in 1760, which was motivated by studying the spread of smallpox [1]. In addition to Bernoulli, there were many different researchers also working on mathematical epidemic models around this time [2]. These initial models were quite simplistic, and the further development and study of such models dates back to the 1900s [3]-[6], where still-simple models were studied to provide insight into how various diseases can spread through a population. In recent years, there has been a resurgence of interest in these problems as the concept of "networks" becomes increasingly prevalent in modeling many different aspects of the world today. A more comprehensive review of the history of mathematical epidemiology can be found in [7] and [8].

Analysis and Control of Epidemics: A Survey of Spreading Processes on Complex Networks

Google's PageRank method was developed to evaluate the importance of web-pages via their link structure. The mathematics of PageRank, however, are entirely general and apply to any graph or network in any domain. Thus, PageRank is now regularly used in bibliometrics, social and information network analysis, and for link prediction and recommendation. It's even used for systems analysis of road networks, as well as biology, chemistry, neuroscience, and physics. We'll see the mathematics and ideas that unite these diverse applications.

PageRank Beyond the Web

We analyze the performance of spectral clustering for community extraction in stochastic block models. We show that, under mild conditions, spectral clustering applied to the adjacency matrix of the network can consistently recover hidden communities even when the order of the maximum expected degree is as small as $\log n$, with $n$ the number of nodes. This result applies to some popular polynomial time spectral clustering algorithms and is further extended to degree corrected stochastic block models using a spherical $k$-median spectral clustering method. A key component of our analysis is a combinatorial bound on the spectrum of binary random matrices, which is sharper than the conventional matrix Bernstein inequality and may be of independent interest.

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Consistency of spectral clustering in stochastic block models

Consider the problem of estimating the entries of a large matrix, when the observed entries are noisy versions of a small random fraction of the original entries. This problem has received widespread attention in recent times, especially after the pioneering works of Emmanuel Candes and collaborators. This paper introduces a simple estimation procedure, called Universal Singular Value Thresholding (USVT), that works for any matrix that has "a little bit of structure." Surprisingly, this simple estimator achieves the minimax error rate up to a constant factor. The method is applied to solve problems related to low rank matrix estimation, blockmodels, distance matrix completion, latent space models, positive definite matrix completion, graphon estimation and generalized Bradley--Terry models for pairwise comparison.

Matrix estimation by Universal Singular Value Thresholding

In this paper, we examine a spectral clustering algorithm for similarity graphs drawn from a simple random graph model, where nodes are allowed to have varying degrees, and we provide theoretical bounds on its performance. The random graph model we study is the Extended Planted Partition (EPP) model, a variant of the classical planted partition model. The standard approach to spectral clustering of graphs is to compute the bottom k singular vectors or eigenvectors of a suitable graph Laplacian, project the nodes of the graph onto these vectors, and then use an iterative clustering algorithm on the projected nodes. However a challenge with applying this approach to graphs generated from the EPP model is that unnormalized Laplacians do not work, and normalized Laplacians do not concentrate well when the graph has a number of low degree nodes. We resolve this issue by introducing the notion of a degree-corrected graph Laplacian. For graphs with many low degree nodes, degree correction has a regularizing eect on the Laplacian. Our spectral clustering algorithm projects the nodes in the graph onto the bottom k right singular vectors of the degree-corrected random-walk Laplacian, and clusters the nodes in this subspace. We show guarantees on the performance of this algorithm, demonstrating that it outputs the correct partition under a wide range of parameter values. Unlike some previous work, our algorithm does not require access to any generative parameters of the model.

http://www.math.ucsd.edu/~fan/wp/planted.pdf

Spectral Clustering of Graphs with General Degrees in the Extended Planted Partition Model

We give an analysis of a variant of the contact process on finite graphs, allowing for nonuniform cure rates, modeling antidote distribution. We examine an inoculation scheme using PageRank vectors that quantify the correlations among vertices in the contact graph. We show that for a contact graph on n nodes we can select a set H of nodes to inoculate such that with probability at least 1 − 2e, any infection from any starting infected set of s nodes will die out in c log s+cˈ time, where c and cˈ depend only on the probabilistic error bound e and the infection rate, and the size of H depends only on s, e, and the topology around the initially infected nodes, independent of the size of the whole graph.

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Distributing Antidote Using PageRank Vectors

We give algorithms for finding graph clusters and drawing graphs, highlighting local community structure within the context of a larger network. For a given graph G, we use the personalized PageRank vectors to determine a set of clusters, by optimizing the jumping parameter α subject to several cluster variance measures in order to capture the graph structure according to PageRank. We then give a graph visualization algorithm for the clusters using PageRank-based coordinates. Several drawings of real-world data are given, illustrating the partition and local community structure.

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Finding and Visualizing Graph Clusters Using PageRank Optimization

Motivated by numerous models of representing trust and distrust within a network ranking system, we examine a quantitative vertex ranking with consideration of the influence of a subset of nodes. We propose and analyze a general ranking metric, called Dirichlet PageRank, which gives a ranking of vertices in a subset S of nodes subject to some specified conditions on the vertex boundary of S. In addition to the usual Dirichlet boundary condition (which disregards the influence of nodes outside of S), we consider general boundary conditions allowing the presence of negative (distrustful) nodes or edges. We give an efficient approximation algorithm for computing Dirichlet PageRank vectors. Furthermore, we give several algorithms for solving various trust-based ranking problems using Dirichlet PageRank with general boundary conditions.

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Dirichlet PageRank and Ranking Algorithms Based on Trust and Distrust

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Alexander Tsiatas

Papers

Spectral Clustering of Graphs with General Degrees in the Extended Planted Partition Model

Distributing Antidote Using PageRank Vectors

Finding and Visualizing Graph Clusters Using PageRank Optimization

Dirichlet PageRank and Ranking Algorithms Based on Trust and Distrust

Finding and Visualizing Graph Clusters Using PageRank Optimization