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.
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.
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.
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.
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 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 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.
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.
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.
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.
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.
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 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 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.
An Efficient Modified Common Neighbor Approach for Link Prediction in Social ...IOSR Journals
This document discusses link prediction in social networks. It analyzes shortcomings of existing leading link prediction methods like common neighbor. It then proposes a modified common neighbor approach that takes into account both topological network structure and node similarities based on features. The approach generates a weight for each link based on the number of common features between nodes, divided by the total number of features. It then calculates a contribution score for each common neighbor by multiplying the weights of that neighbor's links to the two nodes. Experimental results on co-authorship networks show the modified common neighbor approach outperforms existing methods.
This document outlines steps for analyzing social media text data using semantic network analysis and visualization techniques:
1) Collecting Twitter data using Crimson Hexagon and cleaning the text by removing stop words and punctuation.
2) Performing analyses like frequency analysis, entity detection, topic modeling and sentiment analysis using packages like ConText.
3) Creating semantic networks to show word co-occurrence and visualize relationships between concepts, topics and sentiments.
New prediction method for data spreading in social networks based on machine ...TELKOMNIKA JOURNAL
Information diffusion prediction is the study of the path of dissemination of news, information, or topics in a structured data such as a graph. Research in this area is focused on two goals, tracing the information diffusion path and finding the members that determine future the next path. The major problem of traditional approaches in this area is the use of simple probabilistic methods rather than intelligent methods. Recent years have seen growing interest in the use of machine learning algorithms in this field. Recently, deep learning, which is a branch of machine learning, has been increasingly used in the field of information diffusion prediction. This paper presents a machine learning method based on the graph neural network algorithm, which involves the selection of inactive vertices for activation based on the neighboring vertices that are active in a given scientific topic. Basically, in this method, information diffusion paths are predicted through the activation of inactive vertices byactive vertices. The method is tested on three scientific bibliography datasets: The Digital Bibliography and Library Project (DBLP), Pubmed, and Cora. The method attempts to answer the question that who will be the publisher of thenext article in a specific field of science. The comparison of the proposed method with other methods shows 10% and 5% improved precision in DBL Pand Pubmed datasets, respectively.
I. The document discusses ego networks and how they can be used to study personal networks and relationships. Ego networks combine traditional survey data with network data by collecting information about respondents (egos) and their social ties (alters).
II. Ego network data can be used to examine the effects of network structure and alter characteristics on outcomes of interest. It can also provide insights into diffusion processes within personal networks.
III. While ego network data is useful for studying local network phenomena, global network data is needed to analyze higher-level structural effects, mechanisms of tie formation and diffusion across an entire network. Statistical techniques like randomization and the Quadratic Assignment Procedure are used to analyze ego and global network data
This document discusses how rumors spread quickly through social networks. It simulates a simple rumor spreading process on real-world social networks like Twitter and Orkut as well as theoretical network models. The results show that rumors spread much faster in the structures of actual social networks and preferential attachment networks than in random or complete networks. Specifically, a rumor reaching 45.6 million Twitter users within 8 rounds of communication.
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.
This document summarizes open problems and future directions in the field of social networks and health. It identifies key areas for methodological development including dynamic diffusion models, improved community detection techniques, and understanding triadic network structures. Important theoretical advances involve modeling multiplex and evolving networks over time as well as better understanding social mechanisms linking networks to health. Future data collection should incorporate electronic traces, return to community-based studies, and develop national samples capturing full network contexts.
The document proposes an S-curve network model to describe finite networks with bulk growth. It summarizes that most network models assume infinite growth, but real networks are finite. The model adds new nodes exponentially at each time step based on a logistic curve, with the total number of nodes approaching a carrying capacity. It connects new nodes preferentially to existing high-degree nodes. The model aims to better represent features like the limited growth of real networks like the Chinese IPv4 address network.
01 Introduction to Networks Methods and Measuresdnac
This document provides an introduction to social network analysis. It discusses how networks matter through two fundamental mechanisms: connections and positions. Connections refer to the flow of things through networks, viewing networks as pipes. Positions refer to relational patterns and networks capturing role behavior, viewing networks as roles. The document also covers basic network data structures including nodes, edges, directed/undirected ties, binary/valued ties, and different levels of analysis such as ego networks and complete networks. It provides examples of one-mode and two-mode network data.
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.
This document discusses considerations for collecting social network data through surveys. It addresses research design elements like defining the relevant population boundaries and sampling approaches. For surveys specifically, it covers informed consent, name generator questions to identify social ties, response formats, and balancing depth of network detail collected versus sample size. The key challenges are defining the theoretical population of interest, collecting a sufficiently large and representative network sample, and designing survey questions that accurately capture social ties within time and resource constraints.
Challenging Issues and Similarity Measures for Web Document ClusteringIOSR Journals
This document discusses challenging issues and similarity measures for web document clustering. It begins with an introduction to text mining and document clustering. It then reviews related work on similarity approaches and measures. Some key challenging issues in web document clustering are discussed, such as measuring semantic similarity between words and evaluating cluster validity. Various types of similarity measures are also described, including string-based measures like Jaro-Winkler distance and corpus-based measures like latent semantic analysis. The conclusion states that accurate clustering requires a precise definition of similarity between document pairs and discusses different similarity measures that can be used.
The document analyzes network motifs from biological, social, ecological, and other networks using the FANMOD tool. It finds that:
1) Larger motifs generally contain the most significant 3-node motif as a subgraph, suggesting 3-node motifs are building blocks.
2) Undirected networks commonly share the same significant 3-node motif, an interconnected triangle, indicating similar low-level structure.
3) Directed networks show more distinction between disciplines in motifs, while undirected networks are more similar, especially for small motifs.
This document proposes an approach called "OntoFrac-S" to handle the increasing number of ontologies being created for the semantic web. It suggests using fractals and multi-agent systems to implement the semantic web and link data in a way that accounts for the fractal and self-similar nature of data at different levels. Specifically, it argues that merely integrating local and global ontologies is not sufficient, and that ontologies should be viewed as relative concepts depending on the scale, with each local ontology potentially acting as a global ontology for lower-level sub-ontologies. The approach aims to apply concepts of semantic and ontological relativity using fractals to help build a semantically linked global graph while addressing cross-c
The document discusses a field experiment conducted using a Facebook-like social network platform called MyTito installed at a high school in Siena, Italy. Over three months, 253 of the school's 1,600 students actively used MyTito. The network showed partitioning between classes with no evidence of segregation. Preliminary analysis found the network was not fully connected and stopped at non-reciprocated friendships. Further analysis of message content confirmed when users focused on particular topics. The experiment showed promising early results but would benefit from connecting to other social networks and more in-depth semantic analysis of message texts.
Graph Algorithm to Find Core Periphery Structures using Mutual K-nearest Neig...gerogepatton
Core periphery structures exist naturally in many complex networks in the real-world like social, economic, biological and metabolic networks. Most of the existing research efforts focus on the identification of a meso scale structure called community structure. Core periphery structures are another equally important meso scale property in a graph that can help to gain deeper insights about the relationships between different nodes. In this paper, we provide a definition of core periphery structures suitable for weighted graphs. We further score and categorize these relationships into different types based upon the density difference between the core and periphery nodes. Next, we propose an algorithm called CP-MKNN (Core Periphery-Mutual K Nearest Neighbors) to extract core periphery structures from weighted graphs using a heuristic node affinity measure called Mutual K-nearest neighbors (MKNN). Using synthetic and real-world social and biological networks, we illustrate the effectiveness of developed core periphery structures.
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.
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.
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 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.
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.
An Efficient Modified Common Neighbor Approach for Link Prediction in Social ...IOSR Journals
This document discusses link prediction in social networks. It analyzes shortcomings of existing leading link prediction methods like common neighbor. It then proposes a modified common neighbor approach that takes into account both topological network structure and node similarities based on features. The approach generates a weight for each link based on the number of common features between nodes, divided by the total number of features. It then calculates a contribution score for each common neighbor by multiplying the weights of that neighbor's links to the two nodes. Experimental results on co-authorship networks show the modified common neighbor approach outperforms existing methods.
This document outlines steps for analyzing social media text data using semantic network analysis and visualization techniques:
1) Collecting Twitter data using Crimson Hexagon and cleaning the text by removing stop words and punctuation.
2) Performing analyses like frequency analysis, entity detection, topic modeling and sentiment analysis using packages like ConText.
3) Creating semantic networks to show word co-occurrence and visualize relationships between concepts, topics and sentiments.
New prediction method for data spreading in social networks based on machine ...TELKOMNIKA JOURNAL
Information diffusion prediction is the study of the path of dissemination of news, information, or topics in a structured data such as a graph. Research in this area is focused on two goals, tracing the information diffusion path and finding the members that determine future the next path. The major problem of traditional approaches in this area is the use of simple probabilistic methods rather than intelligent methods. Recent years have seen growing interest in the use of machine learning algorithms in this field. Recently, deep learning, which is a branch of machine learning, has been increasingly used in the field of information diffusion prediction. This paper presents a machine learning method based on the graph neural network algorithm, which involves the selection of inactive vertices for activation based on the neighboring vertices that are active in a given scientific topic. Basically, in this method, information diffusion paths are predicted through the activation of inactive vertices byactive vertices. The method is tested on three scientific bibliography datasets: The Digital Bibliography and Library Project (DBLP), Pubmed, and Cora. The method attempts to answer the question that who will be the publisher of thenext article in a specific field of science. The comparison of the proposed method with other methods shows 10% and 5% improved precision in DBL Pand Pubmed datasets, respectively.
I. The document discusses ego networks and how they can be used to study personal networks and relationships. Ego networks combine traditional survey data with network data by collecting information about respondents (egos) and their social ties (alters).
II. Ego network data can be used to examine the effects of network structure and alter characteristics on outcomes of interest. It can also provide insights into diffusion processes within personal networks.
III. While ego network data is useful for studying local network phenomena, global network data is needed to analyze higher-level structural effects, mechanisms of tie formation and diffusion across an entire network. Statistical techniques like randomization and the Quadratic Assignment Procedure are used to analyze ego and global network data
This document discusses how rumors spread quickly through social networks. It simulates a simple rumor spreading process on real-world social networks like Twitter and Orkut as well as theoretical network models. The results show that rumors spread much faster in the structures of actual social networks and preferential attachment networks than in random or complete networks. Specifically, a rumor reaching 45.6 million Twitter users within 8 rounds of communication.
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.
This document summarizes open problems and future directions in the field of social networks and health. It identifies key areas for methodological development including dynamic diffusion models, improved community detection techniques, and understanding triadic network structures. Important theoretical advances involve modeling multiplex and evolving networks over time as well as better understanding social mechanisms linking networks to health. Future data collection should incorporate electronic traces, return to community-based studies, and develop national samples capturing full network contexts.
The document proposes an S-curve network model to describe finite networks with bulk growth. It summarizes that most network models assume infinite growth, but real networks are finite. The model adds new nodes exponentially at each time step based on a logistic curve, with the total number of nodes approaching a carrying capacity. It connects new nodes preferentially to existing high-degree nodes. The model aims to better represent features like the limited growth of real networks like the Chinese IPv4 address network.
01 Introduction to Networks Methods and Measuresdnac
This document provides an introduction to social network analysis. It discusses how networks matter through two fundamental mechanisms: connections and positions. Connections refer to the flow of things through networks, viewing networks as pipes. Positions refer to relational patterns and networks capturing role behavior, viewing networks as roles. The document also covers basic network data structures including nodes, edges, directed/undirected ties, binary/valued ties, and different levels of analysis such as ego networks and complete networks. It provides examples of one-mode and two-mode network data.
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.
This document discusses considerations for collecting social network data through surveys. It addresses research design elements like defining the relevant population boundaries and sampling approaches. For surveys specifically, it covers informed consent, name generator questions to identify social ties, response formats, and balancing depth of network detail collected versus sample size. The key challenges are defining the theoretical population of interest, collecting a sufficiently large and representative network sample, and designing survey questions that accurately capture social ties within time and resource constraints.
Challenging Issues and Similarity Measures for Web Document ClusteringIOSR Journals
This document discusses challenging issues and similarity measures for web document clustering. It begins with an introduction to text mining and document clustering. It then reviews related work on similarity approaches and measures. Some key challenging issues in web document clustering are discussed, such as measuring semantic similarity between words and evaluating cluster validity. Various types of similarity measures are also described, including string-based measures like Jaro-Winkler distance and corpus-based measures like latent semantic analysis. The conclusion states that accurate clustering requires a precise definition of similarity between document pairs and discusses different similarity measures that can be used.
The document analyzes network motifs from biological, social, ecological, and other networks using the FANMOD tool. It finds that:
1) Larger motifs generally contain the most significant 3-node motif as a subgraph, suggesting 3-node motifs are building blocks.
2) Undirected networks commonly share the same significant 3-node motif, an interconnected triangle, indicating similar low-level structure.
3) Directed networks show more distinction between disciplines in motifs, while undirected networks are more similar, especially for small motifs.
This document proposes an approach called "OntoFrac-S" to handle the increasing number of ontologies being created for the semantic web. It suggests using fractals and multi-agent systems to implement the semantic web and link data in a way that accounts for the fractal and self-similar nature of data at different levels. Specifically, it argues that merely integrating local and global ontologies is not sufficient, and that ontologies should be viewed as relative concepts depending on the scale, with each local ontology potentially acting as a global ontology for lower-level sub-ontologies. The approach aims to apply concepts of semantic and ontological relativity using fractals to help build a semantically linked global graph while addressing cross-c
The document discusses a field experiment conducted using a Facebook-like social network platform called MyTito installed at a high school in Siena, Italy. Over three months, 253 of the school's 1,600 students actively used MyTito. The network showed partitioning between classes with no evidence of segregation. Preliminary analysis found the network was not fully connected and stopped at non-reciprocated friendships. Further analysis of message content confirmed when users focused on particular topics. The experiment showed promising early results but would benefit from connecting to other social networks and more in-depth semantic analysis of message texts.
Graph Algorithm to Find Core Periphery Structures using Mutual K-nearest Neig...gerogepatton
Core periphery structures exist naturally in many complex networks in the real-world like social, economic, biological and metabolic networks. Most of the existing research efforts focus on the identification of a meso scale structure called community structure. Core periphery structures are another equally important meso scale property in a graph that can help to gain deeper insights about the relationships between different nodes. In this paper, we provide a definition of core periphery structures suitable for weighted graphs. We further score and categorize these relationships into different types based upon the density difference between the core and periphery nodes. Next, we propose an algorithm called CP-MKNN (Core Periphery-Mutual K Nearest Neighbors) to extract core periphery structures from weighted graphs using a heuristic node affinity measure called Mutual K-nearest neighbors (MKNN). Using synthetic and real-world social and biological networks, we illustrate the effectiveness of developed core periphery structures.
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.
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.
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 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.
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.
Distribution of maximal clique size of theIJCNCJournal
Our primary objective in this paper is to study the distribution of the maximal clique size of the vertices in complex networks. We define the maximal clique size for a vertex as the maximum size of the clique that the vertex is part of and such a clique need not be the maximum size clique for the entire network. We determine the maximal clique size of the vertices using a modified version of a branch-and-bound based exact algorithm that has been originally proposed to determine the maximum size clique for an entire network graph. We then run this algorithm on two categories of complex networks: One category of networks capture the evolution of small-world networks from regular network (according to the well-known Watts-Strogatz model) and their subsequent evolution to random networks; we show that the distribution of
the maximal clique size of the vertices follows a Poisson-style distribution at different stages of the evolution of the small-world network to a random network; on the other hand, the maximal clique size of the vertices is observed to be in-variant and to be very close to that of the maximum clique size for the entire network graph as the regular network is transformed to a small-world network. The second category
of complex networks studied are real-world networks (ranging from random networks to scale-free networks) and we observe the maximal clique size of the vertices in five of the six real-world networks to follow a Poisson-style distribution. In addition to the above case studies, we also analyze the correlation between the maximal clique size and clustering coefficient as well as analyze the assortativity index of the
vertices with respect to maximal clique size and node degree.
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.
COMMUNICATIONS OF THE ACM November 2004Vol. 47, No. 11 15.docxmonicafrancis71118
COMMUNICATIONS OF THE ACM November 2004/Vol. 47, No. 11 15
N
etworks are hot. The
Internet has made it pos-
sible to observe and mea-
sure linkages
representing relationships of
all kinds. We now recognize
networks everywhere: air
traffic, banking, chemical
bonds, data communications,
ecosystems, finite element
grids, fractals, interstate
highways, journal citations,
material structures, nervous
systems, oil pipelines, orga-
nizational networks, power
grids, social structures, trans-
portation, voice communica-
tion, water supply, Web
URLs, and more.
Several fields are collabo-
rating on the development of
network theory, measurement,
and mapping: mathematics
(graph theory), sociology (net-
works of influence and communi-
cation), computing (Internet), and
business (organizational net-
works). This convergence has pro-
duced useful results for risk
assessment and reduction in com-
plex infrastructure networks,
attacking and defending networks,
protecting against network con-
nectivity failures, operating busi-
nesses, spreading epidemics
(pathogens as well as computer
viruses), and spreading innova-
tion. Here, I will survey the fun-
damental laws of networks that
enable these results.
Defining a Network
A network is usually defined as a
set of nodes and links. The nodes
represent entities such as persons,
machines, molecules, documents,
or businesses; the links represent
relationships between pairs of
entities. A link can be directed
(one-way relationship) or undi-
rected (mutual relationship). A
hop is a transition from one node
to another across a single link
separating them. A path is a series
of hops. Networks are very gen-
eral: they can represent any kind
of relation among entities.
Some common network
topologies (interconnection pat-
terns) have their own names:
clique or island (a connected sub-
network that may be isolated
from other cliques), hierarchical
network (tree structured), hub-
and-spoke network (a special
node, the hub, connected directly
to every other node), and multi-
hub network (several hubs con-
nected directly to many nodes).
Some network topologies are
planned, such as the electric grid,
the interstate highway system, or
Network Laws
M
IC
H
A
EL
S
LO
A
N
Peter J. Denning
Many networks, physical and social, are complex and scale-invariant.
This has important implications from the spread of epidemics and
innovations to protection from attack.
The Profession of IT
16 November 2004/Vol. 47, No. 11 COMMUNICATIONS OF THE ACM
the air traffic system; others are
unplanned. In his seminal papers
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Computational Frameworks for Higher-order Network Data Analysis
1. 1
Computational Frameworks for
Higher-order Network Data Analysis
Austin R. Benson · Cornell University
Texas A&M Institute of Data Science · October 23, 2020
Slides. bit.ly/arb-tamu-20
2. Graph or network data modeling important complex
systems are everywhere.
2
Commerce
nodes are products
edges link co-purchased
products
Communications
nodes are people/accounts
edges show info. exchange
Physical proximity
nodes are people
edges link those that interact
in close proximity
Drug compounds
nodes are substances
edge between substances that
appear in the same drug
3. Network data analysis studies the model to gain insight
and make predictions about these systems.
3
1. Evolution / changes
What new connections will form? (email auto-fill suggestions, rec. systems)
2. Clustering / partitioning / community detection
How to find groups of related nodes? (similar products, protein functions)
3. Spreading and traversing
How does stuff move over the network? (viruses or misinformation)
4. Ranking
Which things are important? (PageRank and its variants)
4. Real-world systems are composed of“higher-order”
interactions that we often reduce to pairwise ones.
4
Commerce
nodes are products
Several products
purchased at once
Communications
nodes are people/accounts
emails often have several
recipients,not just one.
Physical proximity
nodes are people
people gather in groups
Drug compounds
nodes are substances
Drugs are composed of
several substances
6. We can ask the same network analysis questions while
taking into account the higher-order structure.
6
1. Evolution / changes
What new connections will form? (email auto-fill suggestions, rec. systems)
2. Clustering / partitioning / community detection
How to find groups of related nodes? (similar products, protein functions)
3. Spreading and traversing
How does stuff move over the network? (viruses or misinformation)
4. Ranking
Which things are important? (PageRank and its variants)
7. Higher-order Network Data Analysis
7
w/ R. Abebe, M. Schaub,
J. Kleinberg, A. Jadbabaie
1. Temporal evolution of higher-order interactions.
Simplicial Closure and Higher-order Link Prediction,PNAS 2018.
2. Clustering in large networks of higher-order interactions.
Minimizing Localized Ratio Cuts in Hypergraphs,KDD,2020.
3. Diffusions over higher-order interactions in networks.
Random walks on simplicial complexes and the normalized Hodge 1-Laplacian,SIAM Review,2020.
8. We collected many datasets of timestamped simplices,
where each simplex is a subset of nodes.
8
1. Coauthorship in different domains.
2. Emails with multiple recipients.
3. Tags on Q&A forums.
4. Threads on Q&A forums.
5. Contact/proximity measurements.
6. Musical artist collaboration.
7. Substance makeup and
classification codes applied to
drugs the FDA examines.
8. U.S. Congress committee
memberships and bill sponsorship.
9. Combinations of drugs seen in
patients in ER visits. https://math.stackexchange.com/q/80181
bit.ly/sc-holp-data
9. Thinking of higher-order data as a weighted projected
graph with filled-in structures is a convenient viewpoint.
9
1
2
3
4
5
6
7
8
9
2
2
1
1
1
1
1
1
1
1
1
1
1
1
1
t1 : {1, 2, 3, 4}
t2 : {1, 3, 5}
t3 : {1, 6}
t4 : {2, 6}
t5 : {1, 7, 8}
t6 : {3, 9}
t7 : {5, 8}
t8 : {1, 2, 6}
Data.
Projected graph W.
Wij = # of simplices containing nodes i and j.
11. 11
i
j k
i
j k
Warm-up. What’s more common in data?
or
“Open triangle”
each pair has been in a simplex
together but all 3 nodes have
never been in the same simplex
“Closed triangle”
there is some simplex that
contains all 3 nodes
13. Dataset domain separation also occurs at the local level.
13
• Randomly sample 100 egonets per dataset and measure
log of average degree and fraction of open triangles.
• Logistic regression model to predict domain
(coauthorship, tags, threads, email, contact).
• 75% model accuracy vs. 21% with random guessing.
14. 14
How do new simplices form?
Can we predict which simplices will form?
15. Groups of nodes go through trajectories until finally
reaching a“simplicial closure.”
15
t1 : {1, 2, 3, 4}
t2 : {1, 3, 5}
t3 : {1, 6}
t4 : {2, 6}
t5 : {1, 7, 8}
t6 : {3, 9}
t7 : {5, 8}
t8 : {1, 2, 6}
For this talk, we will focus on simplicial closure on 3 nodes.
16. Groups of nodes go through trajectories until finally
reaching a“simplicial closure event.”
16
Substances in marketed drugs recorded in the National Drug Code directory.
We bin weighted edges into “weak” and “strong ties” in the projected graph W.
Wij = # of simplices containing nodes i and j.
• Weak ties. Wij = 1 (one simplex contains i and j)
• Strong ties. Wij > 2 (at least two simplices contain i and j)
17. Simplicial closure depends on structure in projected graph.
17
• First 80% of the data (in time) ⟶ record configurations of triplets not in closed triangle.
• Remainder of data ⟶ find fraction that are now closed triangles.
Increased edge density
increases closure probability.
Increased tie strength
increases closure probability.
Tension between edge
density and tie strength.
Left and middle observations are consistent with theory and empirical studies of social networks.
[Granovetter 73; Kossinets-Watts 06; Backstrom+ 06; Leskovec+ 08]
Closure probability Closure probability Closure probability
18. Simplicial closure on 4 nodes is similar to on 3 nodes,
just“up one dimension.”
18
Increased edge density
increases closure probability.
Increased simplicial tie strength
increases closure probability.
Tension b/w edge density
simplicial tie strength.
Closure probability Closure probability Closure probability
19. We proposed“higher-order link prediction”as a
framework to evaluate models for closure.
19
t1 : {1, 2, 3, 4}
t2 : {1, 3, 5}
t3 : {1, 6}
t4 : {2, 6}
t5 : {1, 7, 8}
t6 : {3, 9}
t7 : {5, 8}
t8 : {1, 2, 6}
Data.
• Observe simplices up to time t.
• Predict which groups of > 2
nodes will appear after time t.
t We predict structure that graph
models would not even consider!
20. 20
Our structural analysis tells us what we should be
looking at for prediction.
1. Edge density matters!
⟶ focus our attention on predicting which open
triangles become closed triangles
(intelligently reduce search space.)
2. Tie strength matters!
⟶ various ways of incorporating this information
i
j k
Wij
Wjk
Wjk
21. 21
For every open triangle,we assign a score function on
first 80% of data based on structural properties.
Score s(i, j, k)…
1. is a function of Wij, Wjk, Wjk
arithmetic mean, harmonic mean, etc.
2. looks at common neighbors of the three nodes.
generalized Jaccard, Adamic-Adar, etc.
3. uses “whole-network” similarity scores on projected graph
sum of PageRank or Katz scores amongst edges
4. is learned from data
logistic regression model with features
i
j k
Wij
Wjk
Wjk
After computing scores, predict that open triangles with
highest scores will be closed triangles in final 20% of data.
i
j k
l
m
x
y
r
z
N(i) = {j, k, l, m, x, y, z}
N(j) = {i, k, l, m, r}
N(k) = {i, j, l, m}<latexit 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scorep(i, j, k)
= (Wp
ij + Wp
jk + Wp
ik)1/p
<latexit 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23. 23
A few lessons learned from applying these ideas.
1. We can predict pretty well on all datasets using some simple method.
→ 4x to 107x better than random w/r/t mean average precision
depending on the dataset/method
(only predicting on open triangles)
2. Thread co-participation and co-tagging on stack exchange are
consistently easy to predict with the harmonic mean.
3. Simple averaging Wij, Wjk, and Wik consistently performs well.
i
j k
Wij
Wjk
Wjk
24. Generalized means of edges weights are often good
predictors of new 3-node simplices appearing.
24
music-rap-genius
NDC-substances
NDC-classes
DAWN
coauth-DBLP
coauth-MAG-geology
coauth-MAG-history
congress-bills
congress-committees
tags-stack-overflow
tags-math-sx
tags-ask-ubuntu
email-Eu
email-Enron
threads-stack-overflow
threads-math-sx
threads-ask-ubuntu
contact-high-school
contact-primary-school
harmonic geometric arithmetic
p
4 3 2 1 0 1 2 3 4
0
20
40
60
80
Relativeperformance
4 3 2 1 0 1 2 3 4
p
2.5
5.0
7.5
10.0
12.5
Relativeperformance
4 3 2 1 0 1 2 3 4
p
1.0
1.5
2.0
2.5
3.0
3.5
Relativeperformance
Good performance from this local information is a deviation from classical link prediction, where
methods that use long paths (e.g., PageRank) perform well [Liben-Nowell & Kleinberg 07].
For structures on k nodes, the subsets of size k-1 contain rich information only when k > 2.
i
j k
Wij
Wjk
Wjk
i
j k
?
scorep(i, j, k)
= (Wp
ij + Wp
jk + Wp
ik)1/p
<latexit 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25. If we only need the top-k weighted triangles,
we have fast algorithms for finding them.
25
scorep(i, j, k)
= (Wp
ij + Wp
jk + Wp
ik)1/p
<latexit 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Simple (incorrect) algorithm.
1. Throw out edges with weight < t.
2. Find triangles in remainder.
i
j k
Wij
Wjk
Wjk
Better (correct) algorithm.
1. Dynamically choose threshold.
2. Careful pruning.
w/ R. Kumar, P. Liu, M. Charikar
26. We often only need the top-k weighted triangles,and
we have fast algorithms for finding them.
26
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dataset # nodes # edges Fast enumeration Fast top-k (k = 1000)
(running time in seconds)
Spotify co-listens 3.6M 1.93B too long 30
MAG co-authorship 173M 544M 596 16
AMINER co-authorship 93M 324M 255 10
Ethereum transactions 38M 103M 91 33
27. Higher-order data is pervasive!
27
• Simplicial Closure and Higher-order Link Prediction.Austin R.Benson,Rediet Abebe,Michael T.Schaub,Ali
Jadbabaie,and Jon Kleinberg. Proc.Natl.Acad.Sci.U.S.A.,2018. github.com/arbenson/ScHoLP-Tutorial
• Retrieving Top Weighted Triangles in Graphs. Raunak Kumar,Paul Liu,Moses Charikar,and Austin R.Benson.
Proc.Of WSDM,2020. github.com/raunakkmr/Retrieving-top-weighted-triangles-in-graphs
1. There are commonalities in temporal evolution. Generative models?
2. There is lots of signal in subsets! Unique to higher-order…
3. Please develop neural embeddings to out-perform our baselines. 😁
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28. 28
w/ Nate Veldt, J. KleinbergHigher-order Network Data Analysis
1. Temporal evolution of higher-order interactions.
Simplicial Closure and Higher-order Link Prediction,PNAS 2018.
2. Clustering in large networks of higher-order interactions.
Minimizing Localized Ratio Cuts in Hypergraphs,KDD,2020.
3. Diffusions over higher-order interactions in networks.
Random walks on simplicial complexes and the normalized Hodge 1-Laplacian,SIAM Review,2020.
29. Graph minimum s-t cuts are fundamental.
29
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]
• Densest subgraph [Goldberg 84; Shang+ 18]
• Graph-based semi-supervised learning algorithms [Blum-Chawla 01]
• Local graph clustering [Andersen-Lang 08; Oreccchia-Zhu 14; Veldt+ 16]
poly-time algorithms!
30. Real-world systems are composed of“higher-order”
interactions that we can model with hypergraphs.
30
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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31. What is a hypergraph minimum s-t cut?
31
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]
edge in a graph size-3 hyperedges
“Only one way to split a triangle”
[Benson+ 16; Li-Milenkovic 17; Yin+ 17]
Must be split 1/1.
32. We model hypergraph cuts with splitting functions.
32
s
t
Given a cut defined by S,
we incur penalty of
at each hyperedge e.
Hypergraph minimum s-t cut problem.
Cardinality-based splitting functions.
S<latexit 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cutH(S) = f (2) + f (1)<latexit 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sha1_base64="JdV0NHpso/GwwYvqd/CeIvys+E4=">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</latexit>
<latexit sha1_base64="6FFH4JtCJ1Fb69WjugRCayyF5vI=">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</latexit>
we(e S)
<latexit sha1_base64="QjrhfsKLxaK/82LxnRurhFCzCik=">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</latexit>
minimizeS⇢V
P
e2E we(e S) ⌘ cutH(S)
subject to s 2 S, t /2 S.
<latexit sha1_base64="vCSQ5hxLftoc4zdzUNdXcsthqGM=">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</latexit>
Non-negativity we(A) 0.
Non-split ignoring we(e) = we(;) = 0.
C-B we(A) = f (min(|A|, |Ae|)).
34. We solve hypergraph cut problems with graph reductions.
34
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.
Quadratic penalty
f(i) = i ( |e| – i )
Linear penalty
f(i) = i
All-or-nothing
f(0) = 0,o/w f(i) = 1
35. b
We made a new gadget for C-B splitting functions.
35
This gadget models min(|A|, |eA|, b).
Theorem [Veldt-Benson-Kleinberg 20a]. Nonnegative linear combinations of the
C-B gadget can model any submodular cardinality-based splitting function.
See also Graph Cuts for Minimizing Robust Higher Order Potentials,Kohli et al.,2008.
<latexit sha1_base64="beQz4cdyY+p8N+9L01TDcNAiwcQ=">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</latexit>
C-B we(A) = f (min(|A|, |eA|)).
(F is submodular on X if F(A B) + F(A [ B) F(A) + F(B) for any A, B ✓ X.)<latexit 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36. 36
Theorem [Veldt-Benson-Kleinberg 20a]. The hypergraph min s-t cut problem
with a cardinality-based splitting function is graph-reducible (via gadgets)
if and only if the splitting function is submodular.
Cardinality-based splitting functions.
s
t
S<latexit 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cutH(S) = f (2) + f (1)<latexit 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Submodularity is key to efficient algorithms.
What happens when the splitting function isn’t submodular?
Can we use some other algorithm?
<latexit sha1_base64="vCSQ5hxLftoc4zdzUNdXcsthqGM=">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</latexit>
Non-negativity we(A) 0.
Non-split ignoring we(e) = we(;) = 0.
C-B we(A) = f (min(|A|, |Ae|)).