Efficient bi-triangle counting for large bipartite networks

2021 ◽  
Vol 14 (6) ◽  
pp. 984-996
Author(s):  
Yixing Yang ◽  
Yixiang Fang ◽  
Maria E. Orlowska ◽  
Wenjie Zhang ◽  
Xuemin Lin
Keyword(s):  
Clustering Coefficient ◽  
Bipartite Network ◽  
Bipartite Networks ◽  
Baseline Method ◽  
Vertex Set ◽  
Triangle Counting ◽  
Baseline Algorithm ◽  
Vertex Sets ◽  
The Relationship ◽  
Sophisticated Algorithm

A bipartite network is a network with two disjoint vertex sets and its edges only exist between vertices from different sets. It has received much interest since it can be used to model the relationship between two different sets of objects in many applications (e.g., the relationship between users and items in E-commerce). In this paper, we study the problem of efficient bi-triangle counting for a large bipartite network, where a bi-triangle is a cycle with three vertices from one vertex set and three vertices from another vertex set. Counting bi-triangles has found many real applications such as computing the transitivity coefficient and clustering coefficient for bipartite networks. To enable efficient bi-triangle counting, we first develop a baseline algorithm relying on the observation that each bi-triangle can be considered as the join of three wedges. Then, we propose a more sophisticated algorithm which regards a bi-triangle as the join of two super-wedges, where a wedge is a path with two edges while a super-wedge is a path with three edges. We further optimize the algorithm by ranking vertices according to their degrees. We have performed extensive experiments on both real and synthetic bipartite networks, where the largest one contains more than one billion edges, and the results show that the proposed solutions are up to five orders of magnitude faster than the baseline method.

EMPIRICAL ANALYSIS OF THE CLUSTERING COEFFICIENT IN THE USER-OBJECT BIPARTITE NETWORKS

2013 ◽  
Vol 24 (08) ◽  
pp. 1350055 ◽  
Author(s):  
JIANGUO LIU ◽  
LEI HOU ◽  
YI-LU ZHANG ◽  
WEN-JUN SONG ◽  
XUE PAN
Keyword(s):  
Null Model ◽  
Clustering Coefficient ◽  
Bipartite Network ◽  
Bipartite Networks ◽  
User Interest ◽  
Time Intervals ◽  
Data Set ◽  
Empirical Results ◽  
Maximum Value ◽  
One Year

The clustering coefficient of the bipartite network, C4, has been widely used to investigate the statistical properties of the user-object systems. In this paper, we empirically analyze the evolution patterns of C4 for a nine year MovieLens data set, where C4 is used to describe the diversity of the user interest. First, we divide the MovieLens data set into fractions according to the time intervals and calculate C4 of each fraction. The empirical results show that, the diversity of the user interest changes periodically with a round of one year, which reaches the smallest value in spring, then increases to the maximum value in autumn and begins to decrease in winter. Furthermore, a null model is proposed to compare with the empirical results, which is constructed in the following way. Each user selects each object with a turnable probability p, and the numbers of users and objects are equal to that of the real MovieLens data set. The comparison result indicates that the user activity has greatly influenced the structure of the user-object bipartite network, and users with the same degree information may have two totally different clustering coefficients. On the other hand, the same clustering coefficient also corresponds to different degrees. Therefore, we need to take the clustering coefficient into consideration together with the degree information when describing the user selection activity.

The configuration model

2018 ◽  
Author(s):  
Mark Newman
Keyword(s):  
Degree Sequence ◽  
Graph Model ◽  
Network Models ◽  
Clustering Coefficient ◽  
Giant Component ◽  
Configuration Model ◽  
Bipartite Networks ◽  
Small Component ◽  
Random Graph Model ◽  
Average Size

A discussion of the most fundamental of network models, the configuration model, which is a random graph model of a network with a specified degree sequence. Following a definition of the model a number of basic properties are derived, including the probability of an edge, the expected number of multiedges, the excess degree distribution, the friendship paradox, and the clustering coefficient. This is followed by derivations of some more advanced properties including the condition for the existence of a giant component, the size of the giant component, the average size of a small component, and the expected diameter. Generating function methods for network models are also introduced and used to perform some more advanced calculations, such as the calculation of the distribution of the number of second neighbors of a node and the complete distribution of sizes of small components. The chapter ends with a brief discussion of extensions of the configuration model to directed networks, bipartite networks, networks with degree correlations, networks with high clustering, and networks with community structure, among other possibilities.

scholarly journals The Non-Commuting, Non-Generating Graph of a Nilpotent Group

10.37236/9802 ◽  
2021 ◽  
Vol 28 (1) ◽  
Author(s):  
Peter Cameron ◽  
Saul Freedman ◽  
Colva Roney-Dougal
Keyword(s):  
Maximal Subgroup ◽  
Nilpotent Group ◽  
Commuting Graph ◽  
Generating Graph ◽  
Central Elements ◽  
Vertex Set ◽  
Infinite Case ◽  
The Difference ◽  
The Relationship

For a nilpotent group $G$, let $\Xi(G)$ be the difference between the complement of the generating graph of $G$ and the commuting graph of $G$, with vertices corresponding to central elements of $G$ removed. That is, $\Xi(G)$ has vertex set $G \setminus Z(G)$, with two vertices adjacent if and only if they do not commute and do not generate $G$. Additionally, let $\Xi^+(G)$ be the subgraph of $\Xi(G)$ induced by its non-isolated vertices. We show that if $\Xi(G)$ has an edge, then $\Xi^+(G)$ is connected with diameter $2$ or $3$, with $\Xi(G) = \Xi^+(G)$ in the diameter $3$ case. In the infinite case, our results apply more generally, to any group with every maximal subgroup normal. When $G$ is finite, we explore the relationship between the structures of $G$ and $\Xi(G)$ in more detail.

scholarly journals A study on small-world characteristics of Chinaäó»s high-speed railway networks

2016 ◽  
Author(s):  
Xin Yuan ◽  
Guo Liu ◽  
Kun Hui Ye
Keyword(s):  
High Speed ◽  
Small World ◽  
Clustering Coefficient ◽  
High Speed Rail ◽  
Attribute Data ◽  
The Social ◽  
The Difference ◽  
Global And Local ◽  
The Relationship ◽  
Space Method

The small-world model provides a useful perspective and method to study the topological structure and intrinsic characteristics of high-speed rail networks (HRNs). In this paper, the P-space method is used to examine global and local HRNs in China, meanwhile the adjacency matrix is developed, then the social network analysis and visualization tool UCINET is used to calculate the spatial and attribute data of HRNs at national and local levels in China. The small-world characteristics of whole HRNs are discussed, three networks which have different properties are determined, and a comparative analysis of the small-world effect is detected. Then, the relationship between the construction of high-speed rail and regional development of China is analysed. The results show that: 1) China's HRNs have small average path length ( L ) and large clustering coefficient (C ), representing a typical small-world network; 2) Local HRNs have a certain correlation with economic development. The reasons for the difference of HRNs with respect to characteristics among regions are eventually discussed.

MINIMUM WEIGHT FEEDBACK VERTEX SETS IN CIRCLE n-GON GRAPHS AND CIRCLE TRAPEZOID GRAPHS

2011 ◽  
Vol 03 (03) ◽  
pp. 323-336 ◽  
Author(s):  
FANICA GAVRIL
Keyword(s):  
Minimum Weight ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Circle Graphs ◽  
Intersection Model ◽  
The Family ◽  
Positive Weights ◽  
Vertex Set ◽  
Vertex Sets ◽  
Trapezoid Graphs

A circle n-gon is the region between n or fewer non-crossing chords of a circle, no chord connecting the arcs between two other chords; the sides of a circle n-gon are either chords or arcs of the circle. A circle n-gon graph is the intersection graph of a family of circle n-gons in a circle. The family of circle trapezoid graphs is exactly the family of circle 2-gon graphs and the family of circle graphs is exactly the family of circle 1-gon graphs. The family of circle n-gon graphs contains the polygon-circle graphs which have an intersection representation by circle polygons, each polygon with at most n chords. We describe a polynomial time algorithm to find a minimum weight feedback vertex set, or equivalently, a maximum weight induced forest, in a circle n-gon graph with positive weights, when its intersection model by n-gon-interval-filaments is given.

Continual representation learning for evolving biomedical bipartite networks

2021 ◽  
Author(s):  
Kishlay Jha ◽  
Guangxu Xun ◽  
Aidong Zhang
Keyword(s):  
Network Structure ◽  
Learning Strategy ◽  
Structure Learning ◽  
Fundamental Problem ◽  
Representation Learning ◽  
Research Area ◽  
Bipartite Network ◽  
Bipartite Networks ◽  
Straightforward Application ◽  
Low Dimensional

Abstract Motivation Many real-world biomedical interactions such as ‘gene-disease’, ‘disease-symptom’ and ‘drug-target’ are modeled as a bipartite network structure. Learning meaningful representations for such networks is a fundamental problem in the research area of Network Representation Learning (NRL). NRL approaches aim to translate the network structure into low-dimensional vector representations that are useful to a variety of biomedical applications. Despite significant advances, the existing approaches still have certain limitations. First, a majority of these approaches do not model the unique topological properties of bipartite networks. Consequently, their straightforward application to the bipartite graphs yields unsatisfactory results. Second, the existing approaches typically learn representations from static networks. This is limiting for the biomedical bipartite networks that evolve at a rapid pace, and thus necessitate the development of approaches that can update the representations in an online fashion. Results In this research, we propose a novel representation learning approach that accurately preserves the intricate bipartite structure, and efficiently updates the node representations. Specifically, we design a customized autoencoder that captures the proximity relationship between nodes participating in the bipartite bicliques (2 × 2 sub-graph), while preserving both the global and local structures. Moreover, the proposed structure-preserving technique is carefully interleaved with the central tenets of continual machine learning to design an incremental learning strategy that updates the node representations in an online manner. Taken together, the proposed approach produces meaningful representations with high fidelity and computational efficiency. Extensive experiments conducted on several biomedical bipartite networks validate the effectiveness and rationality of the proposed approach.

Empirical analysis of the user reputation and clustering property for user-object bipartite networks

2019 ◽  
Vol 30 (05) ◽  
pp. 1950035 ◽  
Author(s):  
Xiao-Lu Liu ◽  
Shu-Wei Jia ◽  
Yan Gu
Keyword(s):  
Standard Deviation ◽  
Null Model ◽  
Clustering Coefficient ◽  
Behavior Patterns ◽  
Bipartite Networks ◽  
Rating Systems ◽  
User Interest ◽  
Collective Behaviors ◽  
User Reputation ◽  
Clustering Property

User reputation is of great significance for online rating systems which can be described by user-object bipartite networks, measuring the user ability of rating accurate assessments of various objects. The clustering coefficients have been widely investigated to analyze the local structural properties of complex networks, analyzing the diversity of user interest. In this paper, we empirically analyze the relation of user reputation and clustering property for the user-object bipartite networks. Grouping by user reputation, the results for the MovieLens dataset show that both the average clustering coefficient and the standard deviation of clustering coefficient decrease with the user reputation, which are different from the results that the average clustering coefficient and the standard deviation of clustering coefficient remain stable regardless of user reputation in the null model, suggesting that the user interest tends to be multiple and the diversity of the user interests is centralized for users with high reputation. Furthermore, we divide users into seven groups according to the user degree and investigate the heterogeneity of rating behavior patterns. The results show that the relation of user reputation and clustering coefficient is obvious for small degree users and weak for large degree users, reflecting an important connection between user degree and collective rating behavior patterns. This work provides a further understanding on the intrinsic association between user collective behaviors and user reputation.

scholarly journals The Simultaneous Strong Resolving Graph and the Simultaneous Strong Metric Dimension of Graph Families

Mathematics ◽  
2020 ◽  
Vol 8 (1) ◽  
pp. 125
Author(s):  
Ismael González Yero
Keyword(s):  
Vertex Cover ◽  
Common Vertex ◽  
Metric Dimension ◽  
Minimum Cardinality ◽  
New Approach ◽  
Strong Metric Dimension ◽  
Vertex Set ◽  
Graph Families ◽  
Vertex Sets ◽  
Np Hardness

We consider in this work a new approach to study the simultaneous strong metric dimension of graphs families, while introducing the simultaneous version of the strong resolving graph. In concordance, we consider here connected graphs G whose vertex sets are represented as V ( G ) , and the following terminology. Two vertices u , v ∈ V ( G ) are strongly resolved by a vertex w ∈ V ( G ) , if there is a shortest w − v path containing u or a shortest w − u containing v. A set A of vertices of the graph G is said to be a strong metric generator for G if every two vertices of G are strongly resolved by some vertex of A. The smallest possible cardinality of any strong metric generator (SSMG) for the graph G is taken as the strong metric dimension of the graph G. Given a family F of graphs defined over a common vertex set V, a set S ⊂ V is an SSMG for F , if such set S is a strong metric generator for every graph G ∈ F . The simultaneous strong metric dimension of F is the minimum cardinality of any strong metric generator for F , and is denoted by Sd s ( F ) . The notion of simultaneous strong resolving graph of a graph family F is introduced in this work, and its usefulness in the study of Sd s ( F ) is described. That is, it is proved that computing Sd s ( F ) is equivalent to computing the vertex cover number of the simultaneous strong resolving graph of F . Several consequences (computational and combinatorial) of such relationship are then deduced. Among them, we remark for instance that we have proved the NP-hardness of computing the simultaneous strong metric dimension of families of paths, which is an improvement (with respect to the increasing difficulty of the problem) on the results known from the literature.

The Behavior Simulation of Manufacturing Services in a Service-Oriented Networked Manufacturing Environment

2017 ◽  
Author(s):  
Jingbo Wang ◽  
Ping Lou ◽  
Xuemei Jiang ◽  
Qin Wei ◽  
YongZhi Qu
Keyword(s):  
Bipartite Network ◽  
Networked Manufacturing ◽  
Efficient Operation ◽  
Agent Based ◽  
Behavior Simulation ◽  
Service Oriented ◽  
Geographically Distributed ◽  
The Cost ◽  
Relationship Of ◽  
The Relationship

In a service-oriented networked manufacturing (SONM) environment, geographically distributed manufacturing resources are encapsulated as various manufacturing services. These manufacturing services release via the Internet and can provide services on the demand of manufacturing tasks. Usually one manufacturing task needs several different services belonged to different organizers to work together. Hence, effective cooperation among services is the foundation of the efficient operation of SONM. In this paper, a bipartite network model is presented to describe the relationship of two different kinds of nodes in SONM, and also is projected as a weighed network for further exploring the behaviors of service nodes. Furthermore, an agent-based model is built for modeling the interactive behaviors of service nodes in a cooperative network and an agent-based simulating system is developed with Repast. The simulation results show that the emergence of cooperative behaviors among service nodes is related to both the cost of cooperation and initial trust of services in the SONM environment.

An Empirical Analysis of Developer Collaboration Network

2013 ◽  
Vol 303-306 ◽  
pp. 2177-2181
Author(s):  
Cheng Xiang Peng
Keyword(s):  
Open Source Software ◽  
Network Theory ◽  
Collaboration Network ◽  
Node Degree ◽  
Bipartite Network ◽  
Bipartite Networks ◽  
Software Projects ◽  
Intrinsic Nature ◽  
Domain Experts ◽  
Social Collaboration

To further verify the uses of bipartite network theory and understand the intrinsic nature in social collaboration network. In this paper, we get the information of open source software projects from Source-Forge web and construct a project management collaboration network by analyzing the data of project and manager. Then, through the ordinary projection two kinds of one-mode network are made and the degree distribution of one-mode network and origin bipartite networks shows a power-law like. Finally we evaluate the node's importance on manager network to acquire the core nodes, namely domain experts, by using the metric of node degree, between and topological potential respectively, and provide some helpful applications.

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