Hypergraph Cuts with General Splitting FunctionsAustin Benson
The document discusses joint work on hypergraph cuts with Cornell researchers Nate Veldt and Jon Kleinberg. It presents at the SIAM MDS 2020 conference on Pattern Analysis for Networks and Network Generalizations. The work aims to generalize graph minimum s-t cuts to hypergraphs by minimizing a cut function subject to constraints.
Spectral embeddings and evolving networksAustin Benson
Spectral embeddings provide a fundamental approach for many machine learning tasks but are challenging for dynamic networks that change over time. The authors develop improved methods for maintaining spectral embeddings on evolving networks by (1) incrementally updating embeddings with small perturbations, (2) measuring convergence to speed computations, and (3) proposing a new dynamic graph model to test algorithms. They formalize these ideas with perturbation theory and algorithms that warm-start iterative solvers using the previous embedding.
This document summarizes a research paper that proposes using a genetic algorithm to solve the NP-hard graph partitioning problem. The paper aims to partition graphs to minimize the number of cuts between partitions. It describes representing graph partitions as chromosomes that are evolved over generations using genetic operators like crossover and mutation. An algorithm is presented that initially partitions a graph randomly and then applies genetic operators to iteratively improve the partitioning solution by reducing cut size, considered the fitness function. The algorithm was tested on sample graphs and able to find partitioning solutions with a minimum cut size of 10 and average cut size reduced from 14 to 11 over 100 generations.
This document discusses network analysis and measures of centrality and communicability in networks. It provides mathematical definitions and formulas for quantifying properties like betweenness centrality, clustering coefficient, communicability between nodes, and the number of walks and routes connecting nodes in a network. Examples of applying these metrics to real-world networks like social and biological networks are also mentioned.
IJCER (www.ijceronline.com) International Journal of computational Engineerin...ijceronline
This document summarizes a research paper that proposes a distributed Canny edge detection algorithm with the following key points:
1. The algorithm divides an input image into overlapping blocks that can be processed independently and in parallel to reduce memory requirements, latency, and increase throughput compared to the original Canny algorithm.
2. A novel method is proposed for calculating hysteresis thresholds based on an 8-bin non-uniform quantized gradient magnitude histogram to reduce computational complexity compared to previous methods.
3. An FPGA architecture is presented for implementing the proposed distributed Canny algorithm, along with simulation results demonstrating it can process an image 16 times faster than the original Canny algorithm with no loss in performance.
Random spatial network models for core-periphery structure.Austin Benson
The document proposes a random spatial network model for generating networks with core-periphery structure. The model assigns each node u a weight e^θu and the probability of an edge between nodes u and v is proportional to e^θu + e^θv. This leads to dense connections between high-weight core nodes and sparser connections between core and low-weight peripheral nodes.
Random spatial network models for core-periphery structureAustin Benson
The document proposes a random spatial network model for generating networks with core-periphery structure. The model assigns each node u a weight e^θu and the probability of an edge between nodes u and v is proportional to e^θu + e^θv. This generates networks where high-weight nodes in the "core" have many connections and low-weight "periphery" nodes have few connections.
Hypergraph Cuts with General Splitting FunctionsAustin Benson
The document discusses joint work on hypergraph cuts with Cornell researchers Nate Veldt and Jon Kleinberg. It presents at the SIAM MDS 2020 conference on Pattern Analysis for Networks and Network Generalizations. The work aims to generalize graph minimum s-t cuts to hypergraphs by minimizing a cut function subject to constraints.
Spectral embeddings and evolving networksAustin Benson
Spectral embeddings provide a fundamental approach for many machine learning tasks but are challenging for dynamic networks that change over time. The authors develop improved methods for maintaining spectral embeddings on evolving networks by (1) incrementally updating embeddings with small perturbations, (2) measuring convergence to speed computations, and (3) proposing a new dynamic graph model to test algorithms. They formalize these ideas with perturbation theory and algorithms that warm-start iterative solvers using the previous embedding.
This document summarizes a research paper that proposes using a genetic algorithm to solve the NP-hard graph partitioning problem. The paper aims to partition graphs to minimize the number of cuts between partitions. It describes representing graph partitions as chromosomes that are evolved over generations using genetic operators like crossover and mutation. An algorithm is presented that initially partitions a graph randomly and then applies genetic operators to iteratively improve the partitioning solution by reducing cut size, considered the fitness function. The algorithm was tested on sample graphs and able to find partitioning solutions with a minimum cut size of 10 and average cut size reduced from 14 to 11 over 100 generations.
This document discusses network analysis and measures of centrality and communicability in networks. It provides mathematical definitions and formulas for quantifying properties like betweenness centrality, clustering coefficient, communicability between nodes, and the number of walks and routes connecting nodes in a network. Examples of applying these metrics to real-world networks like social and biological networks are also mentioned.
IJCER (www.ijceronline.com) International Journal of computational Engineerin...ijceronline
This document summarizes a research paper that proposes a distributed Canny edge detection algorithm with the following key points:
1. The algorithm divides an input image into overlapping blocks that can be processed independently and in parallel to reduce memory requirements, latency, and increase throughput compared to the original Canny algorithm.
2. A novel method is proposed for calculating hysteresis thresholds based on an 8-bin non-uniform quantized gradient magnitude histogram to reduce computational complexity compared to previous methods.
3. An FPGA architecture is presented for implementing the proposed distributed Canny algorithm, along with simulation results demonstrating it can process an image 16 times faster than the original Canny algorithm with no loss in performance.
Random spatial network models for core-periphery structure.Austin Benson
The document proposes a random spatial network model for generating networks with core-periphery structure. The model assigns each node u a weight e^θu and the probability of an edge between nodes u and v is proportional to e^θu + e^θv. This leads to dense connections between high-weight core nodes and sparser connections between core and low-weight peripheral nodes.
Random spatial network models for core-periphery structureAustin Benson
The document proposes a random spatial network model for generating networks with core-periphery structure. The model assigns each node u a weight e^θu and the probability of an edge between nodes u and v is proportional to e^θu + e^θv. This generates networks where high-weight nodes in the "core" have many connections and low-weight "periphery" nodes have few connections.
Line Detection is computationally more intense than humans often would
expect. A graphics processing unit (GPU) can meet this need with substantial computational power, but the classic algorithmic approaches to line detection are often of a serial nature
and/or
utilize statistical sampling that cannot provide deterministic detection guarantuees.
Our talk presents a line detection algorithm that is able to detect lines of any angle, throughout the image. It is as parallel as the number of given image pixels multiplied by the
number of potential line angle bins. In contrast to the Hough transform, it is able to locate start and end of found line segments as well. Its redundant image accesses and bilinear
interpolations needed for
the multi-angle edge detection are managed by the texture cache, conserving DRAM memory bandwidth and computational complexity.
It is based on local edge detection filtering to fill small line angle candidates, followed by the inference of line primitives by a segmented scan, all happening in a data-parallel
fashion.
The output is a 2D array of line segments, providing the length of all line segments that originate from a given 2D position and a given line angle bin. This line segment map can then
be used to either infer higher-level vector symbols built from line primitives, again in a data-parallel fashion, using either GPU atomics or a data compaction algorithm in stream
fashion such as HistoPyramids. We exemplify this with the detection of parallel lines and quadriliterals.
While the algorithm's implementation benefits from atomics and shared memory, the basic algorithmic implementation is so simple that it can even be implemented on OpenGL ES 2.0 hardware
such as mobile phones.
Through a WebGL implementation, the line detection can even be applied to HTML5-based
camera input, providing a platform portable approach to low-level computer vision, and, in continuation, augmented reality and symbol detection on mobile phones.
https://www.geofront.eu/demos/lines
1) Plastic analysis was performed using the lower-bound theorem and equilibrium method to determine the collapse load of a W30x99 beam with continuous lateral support.
2) The working load was first determined by calculating the yield moment My. Once yielding occurred, the plastic moment capacity Mp was used.
3) Equilibrium of internal and external moments was satisfied at the collapse mechanism to determine the ultimate load. The uniqueness theorem confirmed this was the collapse load.
This document discusses machine learning applications and provides examples. It begins with an overview of machine learning algorithms being used in parallel to combine results from individual classifiers and extract all possible information from datasets. It then provides examples of mobile marketplaces using machine learning for fraud detection, personalization, and other applications. It concludes by discussing how machine learning can be incorporated into design to make use of visual, aural, corporal, and environmental inputs.
The document discusses various machine learning applications including:
1) Using multiple machine learning algorithms in parallel to combine results from individual classifiers and extract more information from datasets.
2) A mobile marketplace that uses machine learning for fraud detection, search, recommendations, personalization, and other applications by processing transaction data in data lakes and streaming it through batch and real-time layers.
3) How machine learning can be incorporated into design by learning from visual, aural, corporal, and environmental inputs to discover new opportunities.
Recent Developments in Computational Methods for the Analysis of Ducted Prope...João Baltazar
This paper presents an overview of the recent developments at IST and MARIN in applying computational methods for the hydrodynamic analysis of ducted propellers. The developments focus on the propeller performance prediction in open water conditions using Boundary Element Methods and Reynolds-averaged Navier-Stokes solvers. The paper starts with an estimation of the numerical errors involved in both methods. Then, the different viscous mechanisms involved in the ducted propeller flow are discussed and numerical procedures for the potential flow solution proposed. Finally, the numerical predictions are compared with experimental measurements.
Remote sensing data from satellite with high temporal resolution typically have lower spatial resolution, with one pixel often spanning over a square kilometer. The signal recorded by such satellite at a pixel is typically a mixture of reflectance from different types of land covers within
the pixel, resulting in a mixed pixel. In this talk we introduce a couple of parametric and nonparametric statistical approaches to deal with the un-mixing problem which integrate information from multiple sources, and present some preliminary results applying the methodology to data
from the SMOS (Soil Moisture and Ocean Salinity) mission and the OCO-2 (Orbiting Carbon Observatory 2) mission, which motivated this research.
IRJET- Comparative Result of Displacement and Stress for Tapered Beam L/D=...IRJET Journal
This document compares the results of a finite element analysis of a cantilever tapered deep beam using different methods. A cantilever prismatic deep beam with an L/D ratio of 1.25 was analyzed using a finite element program, ANSYS 2D, and ANSYS 3D. The deflection, flexural stresses, and shear stresses obtained from each method are compared. In general, the finite element methods provided more accurate results for the deflection and stress distribution compared to classical beam theory. Specifically, finite element analysis showed that the neutral axis is not linear and moves downward for deep beams under loading. The results from the different finite element methods showed close agreement with each other.
The document discusses the connections between cosmology and fundamental physics, noting that cosmology is interconnected with theories of inflation, dark matter, quantum gravity, and more. It outlines strategies for improving models of inflation, large scale structure, and the late universe through both top-down and bottom-up approaches, emphasizing the use of principles like symmetries, effective field theory, and causality. Key observational probes discussed include the CMB, large scale structure, and gravitational waves.
1. This document discusses tension members and their design strength. Tension members are structural elements that are primarily subjected to tensile forces such as those in trusses, suspension bridges, and cable-stayed bridges.
2. The design strength of a tension member is based on either its gross section resisting yielding, or its net section resisting fracture. Allowable stresses are reduced using strength reduction factors to obtain the design strength.
3. Examples are provided to calculate the design strength of given tension members based on their material properties and dimensions. The effective net area is considered to account for things like bolt holes. Combinations of loads are also checked to ensure the design strength is not exceeded.
Computational Frameworks for Higher-order Network Data AnalysisAustin Benson
1. The document discusses computational frameworks for analyzing higher-order network data, where interactions can involve more than two nodes. Real-world systems often involve higher-order interactions that are reduced to pairwise connections.
2. The author presents several datasets involving higher-order interactions and shows that predicting the formation of new higher-order connections is similar to link prediction but considers groups of nodes rather than individual links. Structural properties like edge density and tie strength influence the likelihood of simplicial closure.
3. Models are proposed to score open simplices based on structural features and predict which will transition to closed simplices. Accounting for higher-order structure provides new insights beyond traditional network analysis of pairwise connections.
Higher-order link prediction and other hypergraph modelingAustin Benson
Higher-order link prediction and other hypergraph modeling can better model real-world systems composed of higher-order interactions that are often reduced to pairwise ones. Hypergraphs allow the modeling of interactions between more than two nodes, like groups of people collaborating, multiple recipients of emails, students gathering in groups, and drug compounds made of several substances.
Simplicial closure & higher-order link predictionAustin Benson
The document discusses higher-order link prediction in networks. It summarizes previous work representing higher-order interactions as tensors, hypergraphs, etc. It then proposes evaluating models of higher-order data using "higher-order link prediction" to predict which groups of more than two nodes will interact based on past data. The authors analyze dynamics of triadic closure in several real-world networks and propose methods to predict closure based on structural properties like edge weights.
Three hypergraph eigenvector centralitiesAustin Benson
Three hypergraph eigenvector centralities are proposed to measure the importance of nodes in complex systems modeled as hypergraphs. Hypergraphs generalize graphs by allowing edges to connect any number of nodes. The proposed centralities are adaptations of the standard graph eigenvector centrality to hypergraphs. They measure a node's centrality based on 1) the centralities of its neighbors, 2) being positive values, and 3) being the principal eigenvector of the hypergraph adjacency matrix.
The document discusses different models for how social networks grow over time, including preferential attachment and fitness models. It proposes using discrete choice theory as a way to model network growth, which allows incorporating covariates and flexible modeling. The approach is statistically rigorous and allows easy incorporation of new models and effects compared to traditional static network models.
Link prediction in networks with core-fringe structureAustin Benson
1. The document discusses link prediction in networks with a core-fringe structure. It examines how including connections from fringe nodes affects the performance of link prediction algorithms on the core nodes.
2. An experiment was conducted where a link prediction algorithm was run multiple times, each time including more fringe nodes and connections in order to measure the effect on link prediction accuracy for the core nodes.
3. The results showed that including more information from the fringe helped improve the link prediction performance on the core nodes.
This document discusses research on modeling and predicting higher-order interactions in networks beyond pairwise connections. The researchers collected datasets containing time-stamped groups or "simplices" of nodes and analyzed properties like triangle closure. They propose "higher-order link prediction" to predict which new simplices will form based on structural features like edge weights between nodes. Scoring functions were tested and averages of edge weights often performed well, differing from classical link prediction methods.
This document summarizes a talk about higher-order link prediction in networks. It discusses organizational principles of systems with higher-order interactions, how they evolve over time through simplicial closure events, and how insights can be used to create effective higher-order link prediction methods. Key points include that simplicial closure depends on the structure and strength of ties in the projected graph, and this closure process is similar for 3 and 4 nodes.
Simplicial closure & higher-order link predictionAustin Benson
This document discusses higher-order link prediction and simplicial closure as ways to analyze and model higher-order interactions in network data. It summarizes that networks can be viewed as weighted projected graphs where simplices "fill in" structures, and that new simplices and closed triangles tend to form through trajectories of nodes reaching "simplicial closure events". It proposes evaluating models of higher-order structure through higher-order link prediction, predicting the formation of new simplices.
More Related Content
Similar to Hypergraph Cuts with General Splitting Functions
Line Detection is computationally more intense than humans often would
expect. A graphics processing unit (GPU) can meet this need with substantial computational power, but the classic algorithmic approaches to line detection are often of a serial nature
and/or
utilize statistical sampling that cannot provide deterministic detection guarantuees.
Our talk presents a line detection algorithm that is able to detect lines of any angle, throughout the image. It is as parallel as the number of given image pixels multiplied by the
number of potential line angle bins. In contrast to the Hough transform, it is able to locate start and end of found line segments as well. Its redundant image accesses and bilinear
interpolations needed for
the multi-angle edge detection are managed by the texture cache, conserving DRAM memory bandwidth and computational complexity.
It is based on local edge detection filtering to fill small line angle candidates, followed by the inference of line primitives by a segmented scan, all happening in a data-parallel
fashion.
The output is a 2D array of line segments, providing the length of all line segments that originate from a given 2D position and a given line angle bin. This line segment map can then
be used to either infer higher-level vector symbols built from line primitives, again in a data-parallel fashion, using either GPU atomics or a data compaction algorithm in stream
fashion such as HistoPyramids. We exemplify this with the detection of parallel lines and quadriliterals.
While the algorithm's implementation benefits from atomics and shared memory, the basic algorithmic implementation is so simple that it can even be implemented on OpenGL ES 2.0 hardware
such as mobile phones.
Through a WebGL implementation, the line detection can even be applied to HTML5-based
camera input, providing a platform portable approach to low-level computer vision, and, in continuation, augmented reality and symbol detection on mobile phones.
https://www.geofront.eu/demos/lines
1) Plastic analysis was performed using the lower-bound theorem and equilibrium method to determine the collapse load of a W30x99 beam with continuous lateral support.
2) The working load was first determined by calculating the yield moment My. Once yielding occurred, the plastic moment capacity Mp was used.
3) Equilibrium of internal and external moments was satisfied at the collapse mechanism to determine the ultimate load. The uniqueness theorem confirmed this was the collapse load.
This document discusses machine learning applications and provides examples. It begins with an overview of machine learning algorithms being used in parallel to combine results from individual classifiers and extract all possible information from datasets. It then provides examples of mobile marketplaces using machine learning for fraud detection, personalization, and other applications. It concludes by discussing how machine learning can be incorporated into design to make use of visual, aural, corporal, and environmental inputs.
The document discusses various machine learning applications including:
1) Using multiple machine learning algorithms in parallel to combine results from individual classifiers and extract more information from datasets.
2) A mobile marketplace that uses machine learning for fraud detection, search, recommendations, personalization, and other applications by processing transaction data in data lakes and streaming it through batch and real-time layers.
3) How machine learning can be incorporated into design by learning from visual, aural, corporal, and environmental inputs to discover new opportunities.
Recent Developments in Computational Methods for the Analysis of Ducted Prope...João Baltazar
This paper presents an overview of the recent developments at IST and MARIN in applying computational methods for the hydrodynamic analysis of ducted propellers. The developments focus on the propeller performance prediction in open water conditions using Boundary Element Methods and Reynolds-averaged Navier-Stokes solvers. The paper starts with an estimation of the numerical errors involved in both methods. Then, the different viscous mechanisms involved in the ducted propeller flow are discussed and numerical procedures for the potential flow solution proposed. Finally, the numerical predictions are compared with experimental measurements.
Remote sensing data from satellite with high temporal resolution typically have lower spatial resolution, with one pixel often spanning over a square kilometer. The signal recorded by such satellite at a pixel is typically a mixture of reflectance from different types of land covers within
the pixel, resulting in a mixed pixel. In this talk we introduce a couple of parametric and nonparametric statistical approaches to deal with the un-mixing problem which integrate information from multiple sources, and present some preliminary results applying the methodology to data
from the SMOS (Soil Moisture and Ocean Salinity) mission and the OCO-2 (Orbiting Carbon Observatory 2) mission, which motivated this research.
IRJET- Comparative Result of Displacement and Stress for Tapered Beam L/D=...IRJET Journal
This document compares the results of a finite element analysis of a cantilever tapered deep beam using different methods. A cantilever prismatic deep beam with an L/D ratio of 1.25 was analyzed using a finite element program, ANSYS 2D, and ANSYS 3D. The deflection, flexural stresses, and shear stresses obtained from each method are compared. In general, the finite element methods provided more accurate results for the deflection and stress distribution compared to classical beam theory. Specifically, finite element analysis showed that the neutral axis is not linear and moves downward for deep beams under loading. The results from the different finite element methods showed close agreement with each other.
The document discusses the connections between cosmology and fundamental physics, noting that cosmology is interconnected with theories of inflation, dark matter, quantum gravity, and more. It outlines strategies for improving models of inflation, large scale structure, and the late universe through both top-down and bottom-up approaches, emphasizing the use of principles like symmetries, effective field theory, and causality. Key observational probes discussed include the CMB, large scale structure, and gravitational waves.
1. This document discusses tension members and their design strength. Tension members are structural elements that are primarily subjected to tensile forces such as those in trusses, suspension bridges, and cable-stayed bridges.
2. The design strength of a tension member is based on either its gross section resisting yielding, or its net section resisting fracture. Allowable stresses are reduced using strength reduction factors to obtain the design strength.
3. Examples are provided to calculate the design strength of given tension members based on their material properties and dimensions. The effective net area is considered to account for things like bolt holes. Combinations of loads are also checked to ensure the design strength is not exceeded.
Computational Frameworks for Higher-order Network Data AnalysisAustin Benson
1. The document discusses computational frameworks for analyzing higher-order network data, where interactions can involve more than two nodes. Real-world systems often involve higher-order interactions that are reduced to pairwise connections.
2. The author presents several datasets involving higher-order interactions and shows that predicting the formation of new higher-order connections is similar to link prediction but considers groups of nodes rather than individual links. Structural properties like edge density and tie strength influence the likelihood of simplicial closure.
3. Models are proposed to score open simplices based on structural features and predict which will transition to closed simplices. Accounting for higher-order structure provides new insights beyond traditional network analysis of pairwise connections.
Higher-order link prediction and other hypergraph modelingAustin Benson
Higher-order link prediction and other hypergraph modeling can better model real-world systems composed of higher-order interactions that are often reduced to pairwise ones. Hypergraphs allow the modeling of interactions between more than two nodes, like groups of people collaborating, multiple recipients of emails, students gathering in groups, and drug compounds made of several substances.
Simplicial closure & higher-order link predictionAustin Benson
The document discusses higher-order link prediction in networks. It summarizes previous work representing higher-order interactions as tensors, hypergraphs, etc. It then proposes evaluating models of higher-order data using "higher-order link prediction" to predict which groups of more than two nodes will interact based on past data. The authors analyze dynamics of triadic closure in several real-world networks and propose methods to predict closure based on structural properties like edge weights.
Three hypergraph eigenvector centralitiesAustin Benson
Three hypergraph eigenvector centralities are proposed to measure the importance of nodes in complex systems modeled as hypergraphs. Hypergraphs generalize graphs by allowing edges to connect any number of nodes. The proposed centralities are adaptations of the standard graph eigenvector centrality to hypergraphs. They measure a node's centrality based on 1) the centralities of its neighbors, 2) being positive values, and 3) being the principal eigenvector of the hypergraph adjacency matrix.
The document discusses different models for how social networks grow over time, including preferential attachment and fitness models. It proposes using discrete choice theory as a way to model network growth, which allows incorporating covariates and flexible modeling. The approach is statistically rigorous and allows easy incorporation of new models and effects compared to traditional static network models.
Link prediction in networks with core-fringe structureAustin Benson
1. The document discusses link prediction in networks with a core-fringe structure. It examines how including connections from fringe nodes affects the performance of link prediction algorithms on the core nodes.
2. An experiment was conducted where a link prediction algorithm was run multiple times, each time including more fringe nodes and connections in order to measure the effect on link prediction accuracy for the core nodes.
3. The results showed that including more information from the fringe helped improve the link prediction performance on the core nodes.
This document discusses research on modeling and predicting higher-order interactions in networks beyond pairwise connections. The researchers collected datasets containing time-stamped groups or "simplices" of nodes and analyzed properties like triangle closure. They propose "higher-order link prediction" to predict which new simplices will form based on structural features like edge weights between nodes. Scoring functions were tested and averages of edge weights often performed well, differing from classical link prediction methods.
This document summarizes a talk about higher-order link prediction in networks. It discusses organizational principles of systems with higher-order interactions, how they evolve over time through simplicial closure events, and how insights can be used to create effective higher-order link prediction methods. Key points include that simplicial closure depends on the structure and strength of ties in the projected graph, and this closure process is similar for 3 and 4 nodes.
Simplicial closure & higher-order link predictionAustin Benson
This document discusses higher-order link prediction and simplicial closure as ways to analyze and model higher-order interactions in network data. It summarizes that networks can be viewed as weighted projected graphs where simplices "fill in" structures, and that new simplices and closed triangles tend to form through trajectories of nodes reaching "simplicial closure events". It proposes evaluating models of higher-order structure through higher-order link prediction, predicting the formation of new simplices.
Simplicial closure and simplicial diffusionsAustin Benson
This document summarizes research on modeling higher-order interactions in network data using simplicial complexes. It finds that most real-world network datasets exhibit a mixture of closed and open triangles, with the fraction varying by domain. A simple probabilistic model can account for this variation. The document proposes that groups of nodes go through trajectories of interactions until reaching a "simplicial closure event" where a new simplex is formed, analogous to triangle closure. It evaluates models' ability to predict such closures using a framework of "higher-order link prediction". Key indicators of closure are edge density and tie strength between nodes.
Sampling methods for counting temporal motifsAustin Benson
The document summarizes research on developing scalable algorithms for counting temporal network motifs in real-time from high-throughput temporal network data streams. It discusses existing methods being insufficient and the problem of not having algorithms that can analyze modern temporal network datasets at fine time scales and high frequencies. It also briefly introduces the idea of using parallel sampling to speed up motif counting algorithms and enable analysis of very large temporal networks.
The document discusses three perspectives on predicting sets of items rather than single items. It describes how sets are common in data such as team formations, medical codes, and online purchases. It then discusses three specific approaches to set prediction: (1) predicting which sets an individual will interact with based on their history, (2) modeling sequences of sets using a generative model, and (3) understanding characteristics of set-based data like subsets and repeats. Applications include prediction, analysis, and simulation.
1. The document discusses sequences of sets, which is a common data structure where each item in a sequence can be a set of elements rather than a single element. Examples of data that can be modeled as sequences of sets include email recipients, tags on questions, academic coauthors, and contacts.
2. The authors provide a generative model to capture important characteristics of sequences of sets, such as subsets and supersets of prior sets being common and recency bias in repeat behavior.
3. Applications of the model include predicting new sets, anomaly detection, understanding user behaviors, and simulation.
Simplicial closure and higher-order link prediction --- SIAMNS18Austin Benson
The document discusses higher-order link prediction, which aims to predict the formation of new groups or "simplices" containing more than two nodes, based on structural properties in timestamped simplex data from various domains. It finds that predicting the closure of open triangles (where a pair of nodes have interacted but not with the third) performs well, and that simply averaging the edge weights in a triangle is often a good predictor. Predicting new structures in communication, collaboration and proximity networks can provide insights beyond classical link prediction.
Simplicial closure and higher-order link prediction (SIAMNS18)Austin Benson
This document summarizes research on modeling and predicting the formation of higher-order relationships or interactions between nodes in network datasets. It introduces the concept of "simplicial closure" to describe how groups of nodes interact over time until forming a simplex or higher-order relationship. The researchers propose "higher-order link prediction" as a framework to evaluate models for predicting the formation of new simplices. They test various methods for scoring open triangles based on edge weights and other structural properties to predict which will become closed triangles. The results show these approaches can significantly outperform random prediction, with simply averaging edge weights often performing well.
Simplicial closure and higher-order link predictionAustin Benson
This document summarizes research on simplicial closure and higher-order link prediction in network science. It finds that groups of nodes often interact through complex trajectories before reaching "simplicial closure" where all nodes are jointly present in a simplex. Predicting these closed simplices is framed as a higher-order link prediction problem. Various score functions are proposed based on edge weights, node neighborhoods, and similarity measures. Scores combining local edge weight information consistently perform well, outperforming classical link prediction approaches. The results provide insights into higher-order structure and a framework for evaluating models of complex relational data.
Higher-order clustering coefficients generalize the clustering coefficient to capture clustering with respect to larger cliques (denser subgraphs) beyond triangles. The speaker defines higher-order clustering coefficients as the fraction of (r-1)-cliques paired with an adjacent edge that induce an r-clique. These coefficients reveal that real-world networks exhibit clustering to different orders and provide additional insights into network structure compared to only considering triangles. The coefficients also vary across networks such as neural, social, and collaboration networks in ways not explained by random graph models.
New perspectives on measuring network clusteringAustin Benson
This document summarizes a talk on mining and modeling network data given at SIAM DM'18. The talk introduces two new classes of network clustering measures: higher-order clustering coefficients and closure coefficients. Higher-order clustering coefficients measure the probability that cliques of different orders close to form larger cliques. Closure coefficients measure the probability that the friend of a friend becomes a friend. These new measures provide insights into real-world network structure beyond what can be seen from traditional clustering coefficients. They also have applications in data mining tasks like community detection, anomaly detection, and predictive modeling.
06-18-2024-Princeton Meetup-Introduction to MilvusTimothy Spann
06-18-2024-Princeton Meetup-Introduction to Milvus
tim.spann@zilliz.com
https://www.linkedin.com/in/timothyspann/
https://x.com/paasdev
https://github.com/tspannhw
https://github.com/milvus-io/milvus
Get Milvused!
https://milvus.io/
Read my Newsletter every week!
https://github.com/tspannhw/FLiPStackWeekly/blob/main/142-17June2024.md
For more cool Unstructured Data, AI and Vector Database videos check out the Milvus vector database videos here
https://www.youtube.com/@MilvusVectorDatabase/videos
Unstructured Data Meetups -
https://www.meetup.com/unstructured-data-meetup-new-york/
https://lu.ma/calendar/manage/cal-VNT79trvj0jS8S7
https://www.meetup.com/pro/unstructureddata/
https://zilliz.com/community/unstructured-data-meetup
https://zilliz.com/event
Twitter/X: https://x.com/milvusio https://x.com/paasdev
LinkedIn: https://www.linkedin.com/company/zilliz/ https://www.linkedin.com/in/timothyspann/
GitHub: https://github.com/milvus-io/milvus https://github.com/tspannhw
Invitation to join Discord: https://discord.com/invite/FjCMmaJng6
Blogs: https://milvusio.medium.com/ https://www.opensourcevectordb.cloud/ https://medium.com/@tspann
Expand LLMs' knowledge by incorporating external data sources into LLMs and your AI applications.
Discover the cutting-edge telemetry solution implemented for Alan Wake 2 by Remedy Entertainment in collaboration with AWS. This comprehensive presentation dives into our objectives, detailing how we utilized advanced analytics to drive gameplay improvements and player engagement.
Key highlights include:
Primary Goals: Implementing gameplay and technical telemetry to capture detailed player behavior and game performance data, fostering data-driven decision-making.
Tech Stack: Leveraging AWS services such as EKS for hosting, WAF for security, Karpenter for instance optimization, S3 for data storage, and OpenTelemetry Collector for data collection. EventBridge and Lambda were used for data compression, while Glue ETL and Athena facilitated data transformation and preparation.
Data Utilization: Transforming raw data into actionable insights with technologies like Glue ETL (PySpark scripts), Glue Crawler, and Athena, culminating in detailed visualizations with Tableau.
Achievements: Successfully managing 700 million to 1 billion events per month at a cost-effective rate, with significant savings compared to commercial solutions. This approach has enabled simplified scaling and substantial improvements in game design, reducing player churn through targeted adjustments.
Community Engagement: Enhanced ability to engage with player communities by leveraging precise data insights, despite having a small community management team.
This presentation is an invaluable resource for professionals in game development, data analytics, and cloud computing, offering insights into how telemetry and analytics can revolutionize player experience and game performance optimization.
We are pleased to share with you the latest VCOSA statistical report on the cotton and yarn industry for the month of March 2024.
Starting from January 2024, the full weekly and monthly reports will only be available for free to VCOSA members. To access the complete weekly report with figures, charts, and detailed analysis of the cotton fiber market in the past week, interested parties are kindly requested to contact VCOSA to subscribe to the newsletter.
Discovering Digital Process Twins for What-if Analysis: a Process Mining Appr...Marlon Dumas
This webinar discusses the limitations of traditional approaches for business process simulation based on had-crafted model with restrictive assumptions. It shows how process mining techniques can be assembled together to discover high-fidelity digital twins of end-to-end processes from event data.
1. 1
Joint work with
Nate Veldt & Jon Kleinberg (Cornell)
Hypergraph Cuts with General Splitting Functions
Austin R. Benson · Cornell University
Applied and Computational Discrete Algorithms Minisymposium
SIAM Annual · July 6, 2020
Slides. bit.ly/arb-ACDA-AN20
2. Graph minimum s-t cuts are fundamental.
2
minimizeS⇢V cut(S)
subject to s 2 S, t /2 S.<latexit 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1 3
2 4
5
6
7
8
s
t
• Maximum flow / min s-t cut [Ford,Fulkerson,Dantzig 1950s]
• Computer vision [Bokykov-Kolmogorov 01; Kolmogorov-Zabih 04]
• Densest subgraph [Goldberg 84; Shang+ 18]
• First graph-based semi-supervised learning algorithms [Blum-Chawla 01]
• Local graph clustering [Andersen-Lang 08; Oreccchia-Zhu 14; Veldt+ 16]
Also see any undergraduate algorithms class
poly-time algorithms!
3. Real-world systems are composed of“higher-order”
interactions that we can model with hypergraphs.
3
Physical proximity
• nodes are students
• hyperedges are students
in the same class
Drug compounds
• nodes are substances
• hyperedges are substances
combined in a drug
linear-algebra discrete-mathematics
math-software
combinatorics
category-theory
logic
terminology
algebraic-graph-theory
combinatorial-designs
hypergraphs
graph-theory
cayley-graphs
group-theory
finite-groups
Categorical information
• nodes are tags
• hyperedges are groups of tags (e.g.,for the
same question on mathoverflow.com)
Networks beyond pairwise interactions: structure and dynamics. Battiston et al., 2020.
The why, how, and when of representations for complex systems. Torres et al., 2020.
4. Real-world systems are composed of“higher-order”
interactions that we can model with hypergraphs.
4
H = (V, E), edge e 2 E is a subset of V (e ⇢ V)<latexit 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1 2
3
4
5
V = {1, 2, 3, 4, 5}
E = {{1, 2, 3}, {2, 4, 5}}<latexit 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5. 5
1. What is a hypergraph minimum s-t cut?
2. If we know what they are, can we find them efficiently?
3. If we can find them efficiently, what can we use them for?
We should have a foundation for
hypergraph minimum s-t cuts,but…
6. What is a hypergraph minimum s-t cut?
6
s
t
Should we treat the 2/2 split
differently from the 1/3 split?
Historically, no. [Lawler 73,Ihler+ 93]
More recently, yes.
[Li-Milenkovic 17,Veldt-Benson-Kleinberg 20]
1 3
2 4
5
6
7
8
s
t
There is only one way to
split an edge (1/1).
7. We model hypergraph cuts with splitting functions.
7
s
t
Non-negativity we(U) 0 for all U ⇢ e.
Symmetry we(U) = we(eU) for all U ⇢ e.
Non-split ignoring we(e) = we(;) = 0.<latexit 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Splitting function for separating edge e into U and U e.
For each edge e, we have a function we with
minimizeS⇢V
P
e2E we(e S) ⌘ cutH(S)
subject to s 2 S, t /2 S.<latexit 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Hypergraph minimum s-t cut problem.
1. Anonymity. A node’s identity doesn’t affect the function.
2. Heterogeneity. Same splitting function at each edge.
Cardinality-based splitting functions.
S<latexit 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cutH(S) = f (2) + f (1)<latexit 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we(U) = f (min(|U|, |Ue|))<latexit 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9. Cardinality-based splitting functions are easy to specify.
9
Cardinality-based splitting functions.
minimizeS⇢V
P
e2E we(e S) ⌘ cutH(S)
subject to s 2 S, t /2 S.<latexit 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sha1_base64="dNi2W8uQiA5FM9ge7UsagYj+LZU=">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</latexit>
s
t
One extra scaling DOF, so set w1 = 1. Specify w2, ... , wbr/2c.<latexit 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sha1_base64="SMjjx0KffHfUKRIVd6aJj9NDt0M=">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</latexit><latexit sha1_base64="SMjjx0KffHfUKRIVd6aJj9NDt0M=">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</latexit>
Non-negativity we(U) 0 for all U ⇢ e.
Non-split ignoring we(e) = we(;) = 0.
C-B we(U) = f (min(|U|, |Ue|)).<latexit 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cutH(S) = f (2) + f (1) = w2 + 1<latexit 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Only need to specify f(1), f(2), …, f(⌊r / 2⌋), where r = max hyperedge size.
Just scalars. f(i) = wi.
10. Cardinality-based splitting functions are easy to specify.
10
Just need to specify w2, ... , wbr/2c and assume w1 = 1.<latexit 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r = 2 (graphs) r = 3 (3-uniform hypergraph)
“Only one way to split a triangle”
[Benson+ 16; Li-Milenkovic 17; Yin+ 17]
s
t
s
t
s
t
r = 4 w2 = 0.5 solution w2 = 1.5 solution w3 = 1.5 solution
11. 1.0 1.25 1.5 1.75 2.0
fusion- systems
topological- stacks
graph- invariants
adjacency- matrix
signed- graph
gorenstein
cohen- macaulay
topological- k- theory
difference- sets
pushforward
regular- rings
graph- connectivity
block- matrices
directed- graphs
eulerian- path
central- extensions
group- extensions
semidirect- product
wreath- product
graded- algebras
supergeometry
geometric- complexity
soliton- theory
matrix- congruences
teichmueller- theory
superalgebra
string- theory
riemann- surfaces
group- cohomology
dglas
celestial- mechanics
s- seed = symplectic- linear- algebra
t- seed = bernoulli- numbers
Different weights lead to different min cuts in practice.
11
1.00 1.25 1.50 1.75 2.00
0.7
0.8
0.9
1.0
JaccardSimilarity
12. 12
1. What is a hypergraph minimum s-t cut?
2. If we know what they are, can we find them efficiently?
3. If we can find them efficiently, what can we use them for?
We should have a foundation for
hypergraph minimum s-t cuts,but…
13. We solve hypergraph cut problems with graph reductions.
13
1/21/2
1/2
1
1
1
1
∞
∞ ∞
∞
∞∞
Gadgets (expansions) model a hyperedge with a small graph.
clique expansion star expansion Lawler gadget [1973]hyperedge
In a graph reduction, we first replace all hyperedges with graph gadgets...
s
t
s
t
s
t
s
t
… then solve the (min s-t cut) problem exactly on the graph,
and finally convert the solution to a hypergraph solution.
14. s
t
s
t
s
t
s
t
Existing gadgets model cardinality-based splitting functions.
14
1/21/2
1/2
1
1
1
1
∞
∞ ∞
∞
∞∞
clique expansion star expansion Lawler gadget [1973]hyperedge
Quadratic penalty
wi = i ( k – i )
k = hyperedge size
Linear penalty
wi = i
All-or-nothing
wi = 1
15. s
t
Existing gadgets model cardinality-based splitting functions.
15
1
∞
∞ ∞
∞
∞∞s
t
1
∞
∞ ∞
∞
∞∞with s
with t
with t
must go
with s
must go
with t
⟶ penalty = 1
1
∞
∞ ∞
∞
∞∞with s
with s
with s
must go
with s
must go
with s
⟶ penalty = 0
Directed min
s-t graph cut