Machine Learning is the study of methods for programming computers to learn. Computers are applied to a wide range of tasks, and for most of these it is relatively easy for programmers to design and implement the necessary software. However, there are many tasks for which this is difficult or impossible. These can be divided into four general categories. First, there are problems for which there exist no human experts. For example, in modern automated manufacturing facilities, there is a need to predict machine failures before they occur by analyzing sensor readings. Because the machines are new, there are no human experts who can be interviewed by a programmer to provide the knowledge necessary to build a computer system. A machine learning system can study recorded data and subsequent machine failures and learn prediction rules. Second, there are problems where human experts exist, but where they are unable to explain their expertise. This is the case in many perceptual tasks, such as speech recognition, hand-writing recognition, and natural language understanding. Virtually all humans exhibit expert-level abilities on these tasks, but none of them can describe the detailed steps that they follow as they perform them. Fortunately, humans can provide machines with examples of the inputs and correct outputs for these tasks, so machine learning algorithms can learn to map the inputs to the outputs. Third, there are problems where phenomena are changing rapidly. In finance, for example, people would like to predict the future behavior of the stock market, of consumer purchases, or of exchange rates. These behaviors change frequently, so that even if a programmer could construct a good predictive computer program, it would need to be rewritten frequently. A learning program can relieve the programmer of this burden by constantly modifying and tuning a set of learned prediction rules. Fourth, there are applications that need to be customized for each computer user separately. Consider, for example, a program to filter unwanted electronic mail messages. Different users will need different filters. It is unreasonable to expect each user to program his or her own rules, and it is infeasible to provide every user with a software engineer to keep the rules up-to-date. A machine learning system can learn which mail messages the user rejects and maintain the filtering rules automatically. Machine learning addresses many of the same research questions as the fields of statistics, data mining, and psychology, but with differences of emphasis. Statistics focuses on understanding the phenomena that have generated the data, often with the goal of testing different hypotheses about those phenomena. Data mining seeks to find patterns in the data that are understandable by people. Psychological studies of human learning aspire to understand the mechanisms underlying the various learning behaviors exhibited by people (concept learning, skill acquisition, strategy change, etc.).

Machine learning

We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

Many practical computing problems concern large graphs. Standard examples include the Web graph and various social networks. The scale of these graphs - in some cases billions of vertices, trillions of edges - poses challenges to their efficient processing. In this paper we present a computational model suitable for this task. Programs are expressed as a sequence of iterations, in each of which a vertex can receive messages sent in the previous iteration, send messages to other vertices, and modify its own state and that of its outgoing edges or mutate graph topology. This vertex-centric approach is flexible enough to express a broad set of algorithms. The model has been designed for efficient, scalable and fault-tolerant implementation on clusters of thousands of commodity computers, and its implied synchronicity makes reasoning about programs easier. Distribution-related details are hidden behind an abstract API. The result is a framework for processing large graphs that is expressive and easy to program.

/pdf/pregel-a-system-for-large-scale-graph-processing-30lgjenxxg.pdf

Pregel: a system for large-scale graph processing

where A represents the magnetic vector potential, is an integral of the hydromagnetic equations. This -integral made it possible to formulate a variational principle for the force-free magnetic fields. The integral expresses the fact that motions cannot transform a given field in an entirely arbitrary different field, if the conductivity of the medium isconsidered infinite. In this paper we shall show that the full set of hydromagnetic equations admit five more integrals, besides the energy integral, if dissipative processes are absent. These integrals, as we shall presently verify, are I2 =fbHvdV, (2)

Proceedings of the National Academy of Sciences

Computational geometry

We present a simple algorithm which maintains the topological order of a directed acyclic graph with n nodes under an online edge insertion sequence in ${\cal O}(n^{2.75})$ time, independent of the number of edges m inserted. For dense DAGs, this is an improvement over the previous best result of ${\cal O}(\min\{m^{\frac{3}{2}} \log{n}, m^{\frac{3}{2}} + n^2 \log{n}\})$ by Katriel and Bodlaender. We also provide an empirical comparison of our algorithm with other algorithms for online topological sorting

An O(n 2.75 ) algorithm for online topological ordering

Near-optimal gossiping algorithms are given for twoand higher dimensional tori. It is assumed that the amount of data each PU contributes is so large that start-up time may be neglected. For twodimensional tori, a previous algorithm achieved optimality in an intricate way, with a time-dependent routing pattern. In all steps of our algorithms, the PUs forward the received packets in the same way.

Time-independent gossiping on full-port tori

We present a simple algorithm which maintains the topological order of a directed acyclic graph (DAG) with n nodes, under an online edge insertion sequence, in O(n2.75) time, independent of the number m of edges inserted. For dense DAGs, this is an improvement over the previous best result of O(min{m3/2 log n, m3/2 + n2 log n}) by Katriel and Bodlaender [2006]. We also provide an empirical comparison of our algorithm with other algorithms for incremental topological sorting.

An O(n2.75) algorithm for incremental topological ordering

LFR is a popular benchmark graph generator used to evaluate community detection algorithms. We present EM-LFR, the first external memory algorithm able to generate massive complex networks following the LFR benchmark. Its most expensive component is the generation of random graphs with prescribed degree sequences which can be divided into two steps: the graphs are first materialized deterministically using the Havel-Hakimi algorithm, and then randomized. Our main contributions are EM-HH and EM-ES, two I/Oefficient external memory algorithms for these two steps. We also propose EM-CM/ES, an alternative sampling scheme using the Configuration Model and rewiring steps to obtain a random simple graph. In an experimental evaluation we demonstrate their performance; our implementation is able to handle graphs with more than 37 billion edges on a single machine, is competitive with a massive parallel distributed algorithm, and is faster than a state-of-theart internal memory implementation even on instances fitting in main memory. EM-LFR’s implementation is capable of generating large graph instances orders of magnitude faster than the original implementation. We give evidence that both implementations yield graphs with matching properties by applying clustering algorithms to generated instances. Similarly, we analyse the evolution of graph properties as EM-ES is executed on networks obtained with EM-CM/ES and find that the alternative approach can accelerate the sampling process. ∗This work was partially supported by the DFG under grants ME 2088/3-2, WA 654/22-2. Parts of this paper were published as [21]. 1 ar X iv :1 60 4. 08 73 8v 3 [ cs .D S] 1 4 Ju n 20 17

Ulrich Meyer

Papers

An O(n 2.75 ) algorithm for online topological ordering

Time-independent gossiping on full-port tori

An O(n2.75) algorithm for incremental topological ordering

I/O-efficient Generation of Massive Graphs Following the LFR Benchmark.

Design, Implementation and Experimental Study of External Memory BFS Algorithms