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.
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.
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.
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 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.
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.
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.
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.
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.
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.
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.
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 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.
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.
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.
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.
The study about the analysis of responsiveness pair clustering tosocial netwo...acijjournal
In this study, regional (cities, towns and villages
) data and tweet data are obtained from Twitter, an
d
extract information of "purchase information (Where
and what bought)" from the tweet data by
morphological analysis and rule-based dependency an
alysis. Then, the "The regional information" and th
e
"Theinformation of purchase history (Where and wha
t bought information)" are captured as bipartite
graph, and Responsiveness Pair Clustering analysis
(a clustering using correspondence analysis as
similarity measure) is conducted. In this study, si
nce it was found to be difficult to analyze a netwo
rk such
as bipartite graph having limitations in links by u
sing modularity Q, responsiveness is used instead o
f
modularity Q as similarity measure. As a result of
this analysis, "regional information cluster" which
refers
to similar "Theinformation of purchase history" nod
es group is generated. Finally, similar regions are
visualized by mapping the regional information clus
ter on the map. This visualization system is expect
ed to
contribute as an analytical tool for customers’ pur
chasing behaviour and so on.
This document discusses predicting new friendships in social networks using temporal information. It describes research on predicting new links in social networks over time using supervised learning models trained on temporal features from past network interactions. The researchers used anonymized Facebook data over 28 months to train decision tree and neural network classifiers to predict new relationships, finding models using temporal information performed better than those without it.
International Journal of Computational Engineering Research (IJCER) is dedicated to protecting personal information and will make every reasonable effort to handle collected information appropriately. All information collected, as well as related requests, will be handled as carefully and efficiently as possible in accordance with IJCER standards for integrity and objectivity.
The document presents an overview of searching in metric spaces. It discusses how similarity searching is needed for unstructured data like text, images, and audio, where exact matching is not possible. It describes how similarity is modeled using a distance function between objects in a metric space. The document surveys existing solutions from different fields that address proximity searching in metric spaces and vector spaces. It aims to provide a unified framework to analyze and categorize existing algorithms.
Scalable Local Community Detection with Mapreduce for Large NetworksIJDKP
Community detection from complex information networks draws much attention from both academia and
industry since it has many real-world applications. However, scalability of community detection algorithms
over very large networks has been a major challenge. Real-world graph structures are often complicated
accompanied with extremely large sizes. In this paper, we propose a MapReduce version called 3MA that
parallelizes a local community identification method which uses the $M$ metric. Then we adopt an
iterative expansion approach to find all the communities in the graph. Empirical results show that for large
networks in the order of millions of nodes, the parallel version of the algorithm outperforms the traditional
sequential approach to detect communities using the M-measure. The result shows that for local community
detection, when the data is too big for the original M metric-based sequential iterative expension approach
to handle, our MapReduce version 3MA can finish in a reasonable time.
This document presents a novel approach to anomaly detection in link mining based on applying mutual information. It adapts the CRISP-DM methodology for link mining and applies it to a case study using co-citation data. The methodology includes data description, preprocessing, transformation, exploration, modeling through graph mapping and hierarchical clustering, and evaluation. Mutual information is used to interpret the semantics of anomalies identified in clusters. The case study identifies collective and community anomalies and confirms mutual information can validate clustering results by showing strong links within clusters but independence between objects in one cluster.
The document discusses a link mining methodology adapted from the CRISP-DM process to incorporate anomaly detection using mutual information. It applies this methodology in a case study of co-citation data. The methodology involves data description, preprocessing, transformation, exploration, modeling, and evaluation. Hierarchical clustering identified 5 clusters, with cluster 1 showing strong links and cluster 5 weak links. Mutual information validated the results, showing cluster 5 had the lowest mutual information, indicating independent variables. The case study demonstrated the approach can interpret anomalies semantically and be used with real-world data volumes and inconsistencies.
COLOCATION MINING IN UNCERTAIN DATA SETS: A PROBABILISTIC APPROACHIJCI JOURNAL
In this paper we investigate colocation mining problem in the context of uncertain data. Uncertain data is a
partially complete data. Many of the real world data is Uncertain, for example, Demographic data, Sensor
networks data, GIS data etc.,. Handling such data is a challenge for knowledge discovery particularly in
colocation mining. One straightforward method is to find the Probabilistic Prevalent colocations (PPCs).
This method tries to find all colocations that are to be generated from a random world. For this we first
apply an approximation error to find all the PPCs which reduce the computations. Next find all the
possible worlds and split them into two different worlds and compute the prevalence probability. These
worlds are used to compare with a minimum probability threshold to decide whether it is Probabilistic
Prevalent colocation (PPCs) or not. The experimental results on the selected data set show the significant
improvement in computational time in comparison to some of the existing methods used in colocation
mining.
K-means and bayesian networks to determine building damage levelsTELKOMNIKA JOURNAL
Many troubles in life require decision-making with convoluted processes because they are caused by uncertainty about the process of relationships that appear in the system. This problem leads to the creation of a model called the Bayesian Network. Bayesian Network is a Bayesian supported development supported by computing advancements. The Bayesian network has also been developed in various fields. At this time, information can implement Bayesian Networks in determining the extent of damage to buildings using individual building data. In practice, there is mixed data which is a combination of continuous and discrete variables. Therefore, to simplify the study it is assumed that all variables are discrete in order to solve practical problems in the implementation of theory. Discretization method used is the K-Means clustering because the percentage of validity obtained by this method is greater than the binning method.
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
Recent Trends in Incremental Clustering: A ReviewIOSRjournaljce
This document provides a review of recent trends in incremental clustering algorithms. It discusses clustering methods based on both similarity measures and those not based on similarity measures. Specific incremental clustering algorithms covered include single-pass clustering, k-nearest neighbors clustering, suffix tree clustering, incremental DBSCAN, and ICIB (incremental clustering based on information bottleneck theory). The document also reviews various techniques for clustering, including particle swarm optimization, ant colony optimization, and genetic algorithms. Applications of genetic algorithm based clustering are discussed.
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.
Statistical models for networks aim to compare observed networks to random graphs in order to assess statistical significance. Simple random graphs are commonly used as a baseline null model but are unrealistic. More developed null models condition on key network structures like degree distribution or mixing patterns to generate more reasonable random graphs for comparison. Network inference problems evaluate whether an observed network exhibits random or non-random properties relative to an appropriate null model.
IJERD (www.ijerd.com) International Journal of Engineering Research and Devel...IJERD Editor
This document discusses distance similarity measures that can be used for data mining classification and clustering techniques. It proposes a novel distance similarity measure called "Supervised & Unsupervised learning" that uses Euclidean distance similarity to partition training data into clusters. It then builds decision trees on each cluster to improve classification performance. The document also discusses using these measures for other applications like image processing, where k-means clustering can be used to segment images into clusters of similar pixel intensities. In conclusion, it states these similarity measures can help analyze complex datasets for business analysis purposes.
Introduction to Topological Data AnalysisMason Porter
Here are slides for my 3/14/21 talk on an introduction to topological data analysis.
This is the first talk in our Short Course on topological data analysis at the 2021 American Physical Society (APS) March Meeting: https://march.aps.org/program/dsoft/gsnp-short-course-introduction-to-topological-data-analysis/
Concurrent Inference of Topic Models and Distributed Vector RepresentationsParang Saraf
Abstract: Topic modeling techniques have been widely used to uncover dominant themes hidden inside an unstructured document collection. Though these techniques first originated in the probabilistic analysis of word distributions, many deep learning approaches have been adopted recently. In this paper, we propose a novel neural network based architecture that produces distributed representation of topics to capture topical themes in a dataset. Unlike many state-of-the-art techniques for generating distributed representation of words and documents that directly use neighboring words for training, we leverage the outcome of a sophisticated deep neural network to estimate the topic labels of each document. The networks, for topic modeling and generation of distributed representations, are trained concurrently in a cascaded style with better runtime without sacrificing the quality of the topics. Empirical studies reported in the paper show that the distributed representations of topics represent intuitive themes using smaller dimensions than conventional topic modeling approaches.
For more information, please visit: http://people.cs.vt.edu/parang/ or contact parang at firstname at cs vt edu
INTELLIGENT SOCIAL NETWORKS MODEL BASED ON SEMANTIC TAG RANKINGIJwest
The document presents a new model for intelligent social networks based on semantic tag ranking. It uses a multi-agent system approach with agents performing indexing and ranking. For indexing, it uses an enhanced Latent Dirichlet Allocation (E-LDA) model that optimizes LDA parameters. Tags above a threshold from E-LDA output are ranked using Tag Rank. Simulation results showed improvements in indexing and ranking over conventional methods. The model introduces semantics to social networks to improve search and link recommendation.
Slides: Concurrent Inference of Topic Models and Distributed Vector Represent...Parang Saraf
Abstract: Topic modeling techniques have been widely used to uncover dominant themes hidden inside an unstructured document collection. Though these techniques first originated in the probabilistic analysis of word distributions, many deep learning approaches have been adopted recently. In this paper, we propose a novel neural network based architecture that produces distributed representation of topics to capture topical themes in a dataset. Unlike many state-of-the-art techniques for generating distributed representation of words and documents that directly use neighboring words for training, we leverage the outcome of a sophisticated deep neural network to estimate the topic labels of each document. The networks, for topic modeling and generation of distributed representations, are trained concurrently in a cascaded style with better runtime without sacrificing the quality of the topics. Empirical studies reported in the paper show that the distributed representations of topics represent intuitive themes using smaller dimensions than conventional topic modeling approaches.
For more information, please visit: http://people.cs.vt.edu/parang/ or contact parang at firstname at cs vt edu
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.
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.
The study about the analysis of responsiveness pair clustering tosocial netwo...acijjournal
In this study, regional (cities, towns and villages
) data and tweet data are obtained from Twitter, an
d
extract information of "purchase information (Where
and what bought)" from the tweet data by
morphological analysis and rule-based dependency an
alysis. Then, the "The regional information" and th
e
"Theinformation of purchase history (Where and wha
t bought information)" are captured as bipartite
graph, and Responsiveness Pair Clustering analysis
(a clustering using correspondence analysis as
similarity measure) is conducted. In this study, si
nce it was found to be difficult to analyze a netwo
rk such
as bipartite graph having limitations in links by u
sing modularity Q, responsiveness is used instead o
f
modularity Q as similarity measure. As a result of
this analysis, "regional information cluster" which
refers
to similar "Theinformation of purchase history" nod
es group is generated. Finally, similar regions are
visualized by mapping the regional information clus
ter on the map. This visualization system is expect
ed to
contribute as an analytical tool for customers’ pur
chasing behaviour and so on.
This document discusses predicting new friendships in social networks using temporal information. It describes research on predicting new links in social networks over time using supervised learning models trained on temporal features from past network interactions. The researchers used anonymized Facebook data over 28 months to train decision tree and neural network classifiers to predict new relationships, finding models using temporal information performed better than those without it.
International Journal of Computational Engineering Research (IJCER) is dedicated to protecting personal information and will make every reasonable effort to handle collected information appropriately. All information collected, as well as related requests, will be handled as carefully and efficiently as possible in accordance with IJCER standards for integrity and objectivity.
The document presents an overview of searching in metric spaces. It discusses how similarity searching is needed for unstructured data like text, images, and audio, where exact matching is not possible. It describes how similarity is modeled using a distance function between objects in a metric space. The document surveys existing solutions from different fields that address proximity searching in metric spaces and vector spaces. It aims to provide a unified framework to analyze and categorize existing algorithms.
Scalable Local Community Detection with Mapreduce for Large NetworksIJDKP
Community detection from complex information networks draws much attention from both academia and
industry since it has many real-world applications. However, scalability of community detection algorithms
over very large networks has been a major challenge. Real-world graph structures are often complicated
accompanied with extremely large sizes. In this paper, we propose a MapReduce version called 3MA that
parallelizes a local community identification method which uses the $M$ metric. Then we adopt an
iterative expansion approach to find all the communities in the graph. Empirical results show that for large
networks in the order of millions of nodes, the parallel version of the algorithm outperforms the traditional
sequential approach to detect communities using the M-measure. The result shows that for local community
detection, when the data is too big for the original M metric-based sequential iterative expension approach
to handle, our MapReduce version 3MA can finish in a reasonable time.
This document presents a novel approach to anomaly detection in link mining based on applying mutual information. It adapts the CRISP-DM methodology for link mining and applies it to a case study using co-citation data. The methodology includes data description, preprocessing, transformation, exploration, modeling through graph mapping and hierarchical clustering, and evaluation. Mutual information is used to interpret the semantics of anomalies identified in clusters. The case study identifies collective and community anomalies and confirms mutual information can validate clustering results by showing strong links within clusters but independence between objects in one cluster.
The document discusses a link mining methodology adapted from the CRISP-DM process to incorporate anomaly detection using mutual information. It applies this methodology in a case study of co-citation data. The methodology involves data description, preprocessing, transformation, exploration, modeling, and evaluation. Hierarchical clustering identified 5 clusters, with cluster 1 showing strong links and cluster 5 weak links. Mutual information validated the results, showing cluster 5 had the lowest mutual information, indicating independent variables. The case study demonstrated the approach can interpret anomalies semantically and be used with real-world data volumes and inconsistencies.
COLOCATION MINING IN UNCERTAIN DATA SETS: A PROBABILISTIC APPROACHIJCI JOURNAL
In this paper we investigate colocation mining problem in the context of uncertain data. Uncertain data is a
partially complete data. Many of the real world data is Uncertain, for example, Demographic data, Sensor
networks data, GIS data etc.,. Handling such data is a challenge for knowledge discovery particularly in
colocation mining. One straightforward method is to find the Probabilistic Prevalent colocations (PPCs).
This method tries to find all colocations that are to be generated from a random world. For this we first
apply an approximation error to find all the PPCs which reduce the computations. Next find all the
possible worlds and split them into two different worlds and compute the prevalence probability. These
worlds are used to compare with a minimum probability threshold to decide whether it is Probabilistic
Prevalent colocation (PPCs) or not. The experimental results on the selected data set show the significant
improvement in computational time in comparison to some of the existing methods used in colocation
mining.
K-means and bayesian networks to determine building damage levelsTELKOMNIKA JOURNAL
Many troubles in life require decision-making with convoluted processes because they are caused by uncertainty about the process of relationships that appear in the system. This problem leads to the creation of a model called the Bayesian Network. Bayesian Network is a Bayesian supported development supported by computing advancements. The Bayesian network has also been developed in various fields. At this time, information can implement Bayesian Networks in determining the extent of damage to buildings using individual building data. In practice, there is mixed data which is a combination of continuous and discrete variables. Therefore, to simplify the study it is assumed that all variables are discrete in order to solve practical problems in the implementation of theory. Discretization method used is the K-Means clustering because the percentage of validity obtained by this method is greater than the binning method.
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
Recent Trends in Incremental Clustering: A ReviewIOSRjournaljce
This document provides a review of recent trends in incremental clustering algorithms. It discusses clustering methods based on both similarity measures and those not based on similarity measures. Specific incremental clustering algorithms covered include single-pass clustering, k-nearest neighbors clustering, suffix tree clustering, incremental DBSCAN, and ICIB (incremental clustering based on information bottleneck theory). The document also reviews various techniques for clustering, including particle swarm optimization, ant colony optimization, and genetic algorithms. Applications of genetic algorithm based clustering are discussed.
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.
Statistical models for networks aim to compare observed networks to random graphs in order to assess statistical significance. Simple random graphs are commonly used as a baseline null model but are unrealistic. More developed null models condition on key network structures like degree distribution or mixing patterns to generate more reasonable random graphs for comparison. Network inference problems evaluate whether an observed network exhibits random or non-random properties relative to an appropriate null model.
IJERD (www.ijerd.com) International Journal of Engineering Research and Devel...IJERD Editor
This document discusses distance similarity measures that can be used for data mining classification and clustering techniques. It proposes a novel distance similarity measure called "Supervised & Unsupervised learning" that uses Euclidean distance similarity to partition training data into clusters. It then builds decision trees on each cluster to improve classification performance. The document also discusses using these measures for other applications like image processing, where k-means clustering can be used to segment images into clusters of similar pixel intensities. In conclusion, it states these similarity measures can help analyze complex datasets for business analysis purposes.
Introduction to Topological Data AnalysisMason Porter
Here are slides for my 3/14/21 talk on an introduction to topological data analysis.
This is the first talk in our Short Course on topological data analysis at the 2021 American Physical Society (APS) March Meeting: https://march.aps.org/program/dsoft/gsnp-short-course-introduction-to-topological-data-analysis/
Concurrent Inference of Topic Models and Distributed Vector RepresentationsParang Saraf
Abstract: Topic modeling techniques have been widely used to uncover dominant themes hidden inside an unstructured document collection. Though these techniques first originated in the probabilistic analysis of word distributions, many deep learning approaches have been adopted recently. In this paper, we propose a novel neural network based architecture that produces distributed representation of topics to capture topical themes in a dataset. Unlike many state-of-the-art techniques for generating distributed representation of words and documents that directly use neighboring words for training, we leverage the outcome of a sophisticated deep neural network to estimate the topic labels of each document. The networks, for topic modeling and generation of distributed representations, are trained concurrently in a cascaded style with better runtime without sacrificing the quality of the topics. Empirical studies reported in the paper show that the distributed representations of topics represent intuitive themes using smaller dimensions than conventional topic modeling approaches.
For more information, please visit: http://people.cs.vt.edu/parang/ or contact parang at firstname at cs vt edu
INTELLIGENT SOCIAL NETWORKS MODEL BASED ON SEMANTIC TAG RANKINGIJwest
The document presents a new model for intelligent social networks based on semantic tag ranking. It uses a multi-agent system approach with agents performing indexing and ranking. For indexing, it uses an enhanced Latent Dirichlet Allocation (E-LDA) model that optimizes LDA parameters. Tags above a threshold from E-LDA output are ranked using Tag Rank. Simulation results showed improvements in indexing and ranking over conventional methods. The model introduces semantics to social networks to improve search and link recommendation.
Slides: Concurrent Inference of Topic Models and Distributed Vector Represent...Parang Saraf
Abstract: Topic modeling techniques have been widely used to uncover dominant themes hidden inside an unstructured document collection. Though these techniques first originated in the probabilistic analysis of word distributions, many deep learning approaches have been adopted recently. In this paper, we propose a novel neural network based architecture that produces distributed representation of topics to capture topical themes in a dataset. Unlike many state-of-the-art techniques for generating distributed representation of words and documents that directly use neighboring words for training, we leverage the outcome of a sophisticated deep neural network to estimate the topic labels of each document. The networks, for topic modeling and generation of distributed representations, are trained concurrently in a cascaded style with better runtime without sacrificing the quality of the topics. Empirical studies reported in the paper show that the distributed representations of topics represent intuitive themes using smaller dimensions than conventional topic modeling approaches.
For more information, please visit: http://people.cs.vt.edu/parang/ or contact parang at firstname at cs vt edu
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.
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.
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.
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.
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.
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.
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.
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.
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.
Tensor Eigenvectors and Stochastic ProcessesAustin Benson
This document provides an overview of a tutorial on tensor eigenvectors and stochastic processes. The tutorial consists of 6 acts that cover basic tensor notation and operations, motivating applications involving tensors, a review of stochastic processes and Markov chains, introducing spacey random walks as stochastic processes, the theory of spacey random walks, and applications of spacey random walks. The goal is to interpret tensor objects from a stochastic perspective using concepts like random walks on graphs and higher-order Markov chains.
Simplicial closure and higher-order link prediction LA/OPTAustin Benson
- The speaker proposes a framework called "higher-order link prediction" to evaluate models of higher-order network data. This extends classical link prediction to predict new groups of nodes that will form simplices.
- Analysis of datasets shows that many have many "open triangles" of nodes connected by edges but not in a simplex. A simple probabilistic model can account for variation in open triangles.
- Simplicial closure probability depends on edge density and tie strength between nodes, both for 3 and 4-node groups.
- For higher-order link prediction, the speaker evaluates score functions based on edge weights, structural properties, whole-network similarities, and machine learning to predict which open triangles will close.
The document summarizes research on spacey random walks, which are a type of stochastic process that can model higher-order Markov chains. Key points:
1. Spacey random walks generalize higher-order Markov chains by forgetting history but pretending to remember a random previous state, with the stationary distribution given by a tensor eigenvector of the transition tensor.
2. This connects higher-order Markov chains to tensor eigenvectors and provides a stochastic interpretation of tensor eigenvectors as stationary distributions.
3. The dynamics of spacey random walks can be modeled as an ordinary differential equation, allowing tensor eigenvectors to be computed by numerically integrating the dynamical system.
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.
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 May 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.
1. 1
Joint work with
Junteng Jia (Cornell),
Michael T.Schaub (MIT),&
Santiago Segarra (Rice)
Graph-based semi-supervised
learning for edge flows
Austin R. Benson · Cornell University
NetSci HONS
May 28, 2019
Slides. bit.ly/arb-HONS-19
3. Two major questions in semi-supervised learning.
3
1. Interpolation. Given the measurements, how do I
interpolate to locations where I don’t know have data.
2. Active learning. Where are the best locations to make
my measurements, knowing step 1?
4. Background. Classical graph-based semi-supervised
learning interpolates from labels on a few vertices.
4
Key idea.
My label is similar to the
labels of my connections.
minimize
labels x
X
(i,j)2E
(xi xj)2
subject to x matches given labels<latexit 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5. In the higher-order case of edge flows,we have a
different type of objective.
5
Key idea (“divergence-free”).
Net flow into a node should be
similar to net flow out of a node.
6. An edge flow represents net flow along an edge.
6
• As an alternating function: F(i, j) = -F(j, i)
• For the linear algebra, first orient each edge i → j if i < j.
Then vector f gives flows on these oriented edges.
• If fi,j > 0, if net flow aligns with orientation
• If fi,j < 0, net flow is opposite of orientation.
1
2
3 4
5
6
7
f1,3 > 0 f5,7 < 0
7. In the higher-order case of edge flows,we have a
different type of objective.
7
Key idea (“divergence-free”).
Net flow into a node should be
similar to net flow out of a node.
minimize
flows f
X
i
2
4
X
j>i,(i,j)2E
fij
X
k<i,(k,i)2E
fki
3
5
2
subject to f matches labels<latexit 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8. There is a close relationship between node-based SSL
and edge-based SSLobjective functions.
8
X
(i,j)2E
(xi xj)2
= xT
Lx = xT
BBT
x = kBT
xk2
2
<latexit 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sha1_base64="ySnVyFu/bHlEXTQVOztKcP12UE4=">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</latexit>
X
i
2
4
X
j>i,(i,j)2E
fij
X
k<i,(k,i)2E
fki
3
5
2
= fT
BT
Bf = kBfk2
<latexit sha1_base64="8Wots+j4DG5+yjap6jJuU1BS0Yo=">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</latexit><latexit 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sha1_base64="DzXqcCT2r5jJ8wnUxXYU8xa/QFc=">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</latexit>
Bk,(i,j) =
8
><
>:
1 k = i, i < j
1 k = j, i < j
0 otherwise<latexit 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One catch.
• Having labels in node case gives unique answer.
• Having labels in edge case is under-constrained.
9. We add regularization to get a nice sparse linear least
squares problem.
9
minimize
flows f
kBfk2
2 + kfk2
2
subject to f matches labels<latexit 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sha1_base64="FX1ilveIMUE873uv0A2QVMsGL2A=">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</latexit>
• We use iterative solvers
LSQR or LSMR to compute
the solution efficiently.
10. Key Idea
My label is similar
to the labels of my
connections.
Objective
Net in flow =
net out flow
at all nodes.
minimize
vertex values x
kBT
xk2
2
subject to x matches labels<latexit 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sha1_base64="loGbjI0bxqRY8lMfhZLYiEbFdFU=">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</latexit><latexit sha1_base64="loGbjI0bxqRY8lMfhZLYiEbFdFU=">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</latexit>
minimize
edge flows f
kBfk2
2 + kfk2
2
subject to f matches labels<latexit 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12. Why do we do so poorly on Chicago?
12
ftruth = y z, y 2 R, z 2 C<latexit 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sha1_base64="9yOl/N+Z/nATZtvcMOokoBxa4rs=">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</latexit>
Our divergence-free assumption says that ftruth ⇡ z.<latexit 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sha1_base64="xvKzv/3dvMkp8PffS78cmxHikj4=">AAAHdnicfVVdb9s2FFW7Lem0r3R9HDCwS40Nhe3YKbIkAwoYWFGsWItmc9IWiIyMkq4kwqSkkVeJHEI/ar9m2Nv2L/a4S9tZLCebAFsUec89vIeHZFhKYXAw+OPO3ffe/2Bj896H/kcff/LpZ1v3P39jikpHcBIVstDvQm5AihxOUKCEd6UGrkIJb8Pp92787TloI4r8GGclTBRPc5GIiCN1nW39+LrSLBYUkkIeQS/RAIwbU6nSBTDDZ4ZhxpE9Cs6TMxsg1GhRV5g1DQt4WeqiZsH55aM+O9vaHvQH84fdbAyXjW1v+Ryd3d94EMRFVCnIMZLEejoclDixXKOIJDR+UBkoeTTlKZxSM+cKzMTOq25Yh3pilhSafjmyea+/CqE8ms9aWSzysJJc1+3esCimNGIa329zYnIwsSIvKyR1FpRJJRkWzIlJwmmIUM5YmxfF9LKbiwgSzaMuV0ZxzLqlcPPs4vSyl2peZl3FpxCBlNddi1k5uBSh5nrmSiguTDekzKkuqjw23ZIjgs4N4VGLumsyXoLpJgK7EZeR+44dppQFKq6n5r+y9hUgp8G5chLQHlcJws8QN1ZD/PBg8DCUxLsagRmk5JC8sfOXi7nIBMJaTCgraKz7X4nwOyxDLM13Oztkor5Byg11lPE8hX5UqJ1fKzDOdGZn+O3e4e7hjgHlnBmSV1XvQmDWc0X0RN4LyeKg53FP9rcXLz9wgnLaAk4fP0hlEXIZ0GfgYCPITaVhFBeSDDCiDRAVMTwNNEheX2ELmnzbRKfHw4l1C+cM0Frlo+Mxz524GnK4oAIUz2MbJFwJOYsh4ZXExgYmuWq3TWIS54rG76ySGVpBiJ8O+ofdSAkiJVtIsjwRYG0Sl6JdJOUOcqxdqtECbM3jU9pre5NmvahnQJtMw3imwkI+p5LsIotp7OtXLxubOwolGqsaK2i6wRjwtmDqiNch4RKy5HCAcRXScmLllvR2gnWG8fNXTpIrguNhSz4b1o018prEBS/Q9gVFOg24LDPeXE/1lxdrqsepBBFlvYX2t43QQhs6Xtrng3JpVldZjUWqiClYuMqls0GobLDob27YQr2kQzm+DbEcaNoUj4M65PqUzBdkYVHb4Nz9d/wg05UEloFIM6TTdX+vRNZhxxkd2xFWXDKC+cGUTohBf3cP6g67ejrsGV0onI55FgJe0P51sYzImJnL6C+oOj5j8wS9QX8IqnOFHmeFJnVEnjK6GchUTEKCzIgYHGKlru1h828SugCe/G8SPa9knqVxKtA1Mly/NG423uz2hzS9n3a3RwfLC+We94X3lfeNN/T2vZH3g3fknXiR95v3u/en99fG35tfbnY2v16E3r2zxDzwWs/m4B+9paDJ</latexit><latexit sha1_base64="xvKzv/3dvMkp8PffS78cmxHikj4=">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</latexit>
Does this actually hold in our data?