📄 Four lectures on probabilistic methods for data science
Roman Vershynin
🔗 https://arxiv.org/pdf/1612.06661v1
📌 ABSTRACT
Methods of high-dimensional probability play a central role in applications for statistics, signal processing theoretical computer science and related fields. These lectures present a sample of particularly useful tools of high-dimensional probability, focusing on the classical and matrix Bernstein's inequality and the uniform matrix deviation inequality. We illustrate these tools with applications for dimension reduction, network analysis, covariance estimation, matrix completion and sparse signal recovery. The lectures are geared towards beginning graduate students who have taken a rigorous course in probability but may not have any experience in data science applications.
Roman Vershynin
🔗 https://arxiv.org/pdf/1612.06661v1
📌 ABSTRACT
Methods of high-dimensional probability play a central role in applications for statistics, signal processing theoretical computer science and related fields. These lectures present a sample of particularly useful tools of high-dimensional probability, focusing on the classical and matrix Bernstein's inequality and the uniform matrix deviation inequality. We illustrate these tools with applications for dimension reduction, network analysis, covariance estimation, matrix completion and sparse signal recovery. The lectures are geared towards beginning graduate students who have taken a rigorous course in probability but may not have any experience in data science applications.
📄 Temporal profiles of avalanches on networks
James P Gleeson, Rick Durrett
🔗 https://arxiv.org/pdf/1612.06477v1
📌 ABSTRACT
An avalanche or cascade occurs when one event causes one or more subsequent events, which in turn may cause further events in a chain reaction. Avalanching dynamics are studied in many disciplines, with a recent focus on average avalanche shapes, i.e., the temporal profiles that characterize the growth and decay of avalanches of fixed duration. At the critical point of the dynamics the average avalanche shapes for different durations can be rescaled so that they collapse onto a single universal curve. We apply Markov branching process theory to derive a simple equation governing the average avalanche shape for cascade dynamics on networks. Analysis of the equation at criticality demonstrates that nonsymmetric average avalanche shapes (as observed in some experiments) occur for certain combinations of dynamics and network topology; specifically, on networks with heavy-tailed degree distributions. We give examples using numerical simulations of models of information spreading, neural dynamics, and threshold models of behaviour adoption.
James P Gleeson, Rick Durrett
🔗 https://arxiv.org/pdf/1612.06477v1
📌 ABSTRACT
An avalanche or cascade occurs when one event causes one or more subsequent events, which in turn may cause further events in a chain reaction. Avalanching dynamics are studied in many disciplines, with a recent focus on average avalanche shapes, i.e., the temporal profiles that characterize the growth and decay of avalanches of fixed duration. At the critical point of the dynamics the average avalanche shapes for different durations can be rescaled so that they collapse onto a single universal curve. We apply Markov branching process theory to derive a simple equation governing the average avalanche shape for cascade dynamics on networks. Analysis of the equation at criticality demonstrates that nonsymmetric average avalanche shapes (as observed in some experiments) occur for certain combinations of dynamics and network topology; specifically, on networks with heavy-tailed degree distributions. We give examples using numerical simulations of models of information spreading, neural dynamics, and threshold models of behaviour adoption.
🍻 Apply now: SFI's Complex Systems Summer School, a 4-week intro to complex behavior in physical & social systems:
🔗 https://www.santafe.edu/engage/learn/schools/sfi-complex-systems-summer-school
🔗 https://www.santafe.edu/engage/learn/schools/sfi-complex-systems-summer-school
www.santafe.edu
Complex Systems Summer School
<p>2021 CSSS will not be held due to the global pandemic.</p>
<p>Please visit this page in mid Oct 2021 to get information and apply for 2022 CSSS.</p>
<p>Please visit this page in mid Oct 2021 to get information and apply for 2022 CSSS.</p>
📄 Scale-free networks emerging from multifractal time series
Marcello A. Budroni, Andrea Baronchelli, Romualdo Pastor-Satorras
🔗 https://arxiv.org/pdf/1612.07070v1
📌 ABSTRACT
Methods connecting dynamical systems and graph theory have attracted increasing interest in the past few years, with applications ranging from a detailed comparison of different kinds of dynamics to the characterisation of empirical data. Here we investigate the effects of the (multi)fractal properties of a time signal, common in sequences arising from chaotic or strange attractors, on the topology of a suitably projected network. Relying on the box counting formalism, we map boxes into the nodes of a network and establish analytic expressions connecting the natural measure of a box with its degree in the graph representation. We single out the conditions yielding to the emergence of a scale-free topology, and validate our findings with extensive numerical simulations.
Marcello A. Budroni, Andrea Baronchelli, Romualdo Pastor-Satorras
🔗 https://arxiv.org/pdf/1612.07070v1
📌 ABSTRACT
Methods connecting dynamical systems and graph theory have attracted increasing interest in the past few years, with applications ranging from a detailed comparison of different kinds of dynamics to the characterisation of empirical data. Here we investigate the effects of the (multi)fractal properties of a time signal, common in sequences arising from chaotic or strange attractors, on the topology of a suitably projected network. Relying on the box counting formalism, we map boxes into the nodes of a network and establish analytic expressions connecting the natural measure of a box with its degree in the graph representation. We single out the conditions yielding to the emergence of a scale-free topology, and validate our findings with extensive numerical simulations.
📄Finding network communities using modularity density
Federico Botta, Charo I. del Genio
🔗 https://arxiv.org/pdf/1612.07297v1
📌 ABSTRACT
Many real-world complex networks exhibit a community structure, in which the modules correspond to actual functional units. Identifying these communities is a key challenge for scientists. A common approach is to search for the network partition that maximizes a quality function. Here, we present a detailed analysis of a recently proposed function, namely modularity density. We show that it does not incur in the drawbacks suffered by traditional modularity, and that it can identify networks without ground-truth community structure, deriving its analytical dependence on link density in generic random graphs. In addition, we show that modularity density allows an easy comparison between networks of different sizes, and we also present some limitations that methods based on modularity density may suffer from. Finally, we introduce an efficient, quadratic community detection algorithm based on modularity density maximization, validating its accuracy against theoretical predictions and on a set of benchmark networks.
Federico Botta, Charo I. del Genio
🔗 https://arxiv.org/pdf/1612.07297v1
📌 ABSTRACT
Many real-world complex networks exhibit a community structure, in which the modules correspond to actual functional units. Identifying these communities is a key challenge for scientists. A common approach is to search for the network partition that maximizes a quality function. Here, we present a detailed analysis of a recently proposed function, namely modularity density. We show that it does not incur in the drawbacks suffered by traditional modularity, and that it can identify networks without ground-truth community structure, deriving its analytical dependence on link density in generic random graphs. In addition, we show that modularity density allows an easy comparison between networks of different sizes, and we also present some limitations that methods based on modularity density may suffer from. Finally, we introduce an efficient, quadratic community detection algorithm based on modularity density maximization, validating its accuracy against theoretical predictions and on a set of benchmark networks.
📄 A Guide to Teaching Data Science
Stephanie C. Hicks, Rafael A. Irizarry
🔗 https://arxiv.org/pdf/1612.07140v1
📌 ABSTRACT:
Demand for data science education is surging and traditional courses offered by statistics departments are not meeting the needs of those seeking this training. This has led to a number of opinion pieces advocating for an update to the Statistics curriculum. The unifying recommendation is that computing should play a more prominent role. We strongly agree with this recommendation, but advocate that the main priority is to bring applications to the forefront as proposed by Nolan and Speed (1999). We also argue that the individuals tasked with developing data science courses should not only have statistical training, but also have experience analyzing data with the main objective of solving real-world problems. Here, we share a set of general principles and offer a detailed guide derived from our successful experience developing and teaching data science courses centered entirely on case studies. We argue for the importance of statistical thinking, as defined by Wild and Pfannkuck (1999) and describe how our approach teaches students three key skills needed to succeed in data science, which we refer to as creating, connecting, and computing. This guide can also be used for statisticians wanting to gain more practical knowledge about data science before embarking on teaching a course.
Stephanie C. Hicks, Rafael A. Irizarry
🔗 https://arxiv.org/pdf/1612.07140v1
📌 ABSTRACT:
Demand for data science education is surging and traditional courses offered by statistics departments are not meeting the needs of those seeking this training. This has led to a number of opinion pieces advocating for an update to the Statistics curriculum. The unifying recommendation is that computing should play a more prominent role. We strongly agree with this recommendation, but advocate that the main priority is to bring applications to the forefront as proposed by Nolan and Speed (1999). We also argue that the individuals tasked with developing data science courses should not only have statistical training, but also have experience analyzing data with the main objective of solving real-world problems. Here, we share a set of general principles and offer a detailed guide derived from our successful experience developing and teaching data science courses centered entirely on case studies. We argue for the importance of statistical thinking, as defined by Wild and Pfannkuck (1999) and describe how our approach teaches students three key skills needed to succeed in data science, which we refer to as creating, connecting, and computing. This guide can also be used for statisticians wanting to gain more practical knowledge about data science before embarking on teaching a course.
🎯 "What are Lyapunov Exponents, and Why are they Interesting?" by Amie Wilkinson
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🗺 Our purpose is to be up-to-date, precise and international.
✔️ Introduce us to your friends and colleagues at all over the globe:
https://telegram.me/ComplexSys
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Complex Systems Studies
What's up in Complexity Science?!
Check out here:
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#complexity #complex_systems #networks #network_science
📨 Contact us: @carimi
Check out here:
@ComplexSys
#complexity #complex_systems #networks #network_science
📨 Contact us: @carimi
📄 Higher-order organization of complex networks
Austin R. Benson, David F. Gleich, Jure Leskovec
🔗 https://arxiv.org/pdf/1612.08447v1
📌 ABSTRACT
Networks are a fundamental tool for understanding and modeling complex systems in physics, biology, neuroscience, engineering, and social science. Many networks are known to exhibit rich, lower-order connectivity patterns that can be captured at the level of individual nodes and edges. However, higher-order organization of complex networks---at the level of small network subgraphs---remains largely unknown. Here we develop a generalized framework for clustering networks based on higher-order connectivity patterns. This framework provides mathematical guarantees on the optimality of obtained clusters and scales to networks with billions of edges. The framework reveals higher-order organization in a number of networks including information propagation units in neuronal networks and hub structure in transportation networks. Results show that networks exhibit rich higher-order organizational structures that are exposed by clustering based on higher-order connectivity patterns.
Austin R. Benson, David F. Gleich, Jure Leskovec
🔗 https://arxiv.org/pdf/1612.08447v1
📌 ABSTRACT
Networks are a fundamental tool for understanding and modeling complex systems in physics, biology, neuroscience, engineering, and social science. Many networks are known to exhibit rich, lower-order connectivity patterns that can be captured at the level of individual nodes and edges. However, higher-order organization of complex networks---at the level of small network subgraphs---remains largely unknown. Here we develop a generalized framework for clustering networks based on higher-order connectivity patterns. This framework provides mathematical guarantees on the optimality of obtained clusters and scales to networks with billions of edges. The framework reveals higher-order organization in a number of networks including information propagation units in neuronal networks and hub structure in transportation networks. Results show that networks exhibit rich higher-order organizational structures that are exposed by clustering based on higher-order connectivity patterns.
📄 Recent history of fractional calculus
http://www.sciencedirect.com/science/article/pii/S1007570410003205
📌Abstract
This survey intends to report some of the major documents and events in the area of fractional calculus that took place since 1974 up to the present date.
#Fractional_calculus
http://www.sciencedirect.com/science/article/pii/S1007570410003205
📌Abstract
This survey intends to report some of the major documents and events in the area of fractional calculus that took place since 1974 up to the present date.
#Fractional_calculus