StationRank: Aggregate dynamics of the Swiss railway

Increasing availability and quality of actual, as opposed to scheduled, open transport data offers new possibilities for capturing the spatiotemporal dynamics of the railway and other networks of social infrastructure. One way to describe such complex phenomena is in terms of stochastic processes. At its core, a stochastic model is domain-agnostic and algorithms discussed here have been successfully used in other applications, including Google's PageRank citation ranking. Our key assumption is that train routes constitute meaningful sequences analogous to sentences of literary text. A corpus of routes is thus susceptible to the same analytic tool-set as a corpus of sentences. With our experiment in Switzerland, we introduce a method for building Markov Chains from aggregated daily streams of railway traffic data. The stationary distributions under normal and perturbed conditions are used to define systemic risk measures with non-evident,valuable information about railway infrastructure

A Tutorial on the Spectral Theory of Markov Chains

Markov chains are a class of probabilistic models that have achieved widespread application in the quantitative sciences. This is in part due to their versatility, but is compounded by the ease with which they can be probed analytically. This tutorial provides an in-depth introduction to Markov chains, and explores their connection to graphs and random walks. We utilize tools from linear algebra and graph theory to describe the transition matrices of different types of Markov chains, with a particular focus on exploring properties of the eigenvalues and eigenvectors corresponding to these matrices. The results presented are relevant to a number of methods in machine learning and data mining, which we describe at various stages. Rather than being a novel academic study in its own right, this text presents a collection of known results, together with some new concepts. Moreover, the tutorial focuses on offering intuition to readers rather than formal understanding, and only assumes basic exposure to concepts from linear algebra and probability theory. It is therefore accessible to students and researchers from a wide variety of disciplines

Markov Chains and Spectral Clustering

Part 3: ModelingInternational audienceThe importance of Markov chains in modeling diverse systems, including biological, physical, social and economic systems, has long been known and is well documented. More recently, Markov chains have proven to be effective when applied to internet search engines such as Google’s PageRank model [7], and in data mining applications wherein data trends are sought. It is with this type of Markov chain application that we focus our research efforts. Our starting point is the work of Fiedler who in the early 70’s developed a spectral partitioning method to obtain the minimum cut on an undirected graph (symmetric system). The vector that results from the spectral decomposition, called the Fiedler vector, allows the nodes of the graph to be partitioned into two subsets. At the same time that Fiedler proposed his spectral approach, Stewart proposed a method based on the dominant eigenvectors of a Markov chain — a method which was more broadly applicable to nonsymmetric systems. Enlightened by these, somewhat orthogonal, results and combining them together, we show that spectral partitioning can be viewed in the framework of state clustering on Markov chains. Our research results to date are two-fold. First, we prove that the second eigenvector of the signless Laplacian provides a heuristic solution to the NP-complete state clustering problem which is the dual of the minimum cut problem. Second, we propose two clustering techniques for Markov chains based on two different clustering measures