Graph datasets are frequently constructed by a projection of a bipartite graph, where two nodes are connected in the projection if they share a common neighbor in the bipartite graph; for example, a coauthorship graph is a projection of an author-publication bipartite graph. Analyzing the structure of the projected graph is common, but we do not have a good understanding of the consequences of the projection on such analyses. Here, we propose and analyze a random graph model to study what properties we can expect from the projection step. Our model is based on a Chung-Lu random graph for constructing the bipartite representation, which enables us to rigorously analyze the projected graph. We show that common network properties such as sparsity, heavy-tailed degree distributions, local clustering at nodes, the inverse relationship between node degree, and global transitivity can be explained and analyzed through this simple model. We also develop a fast sampling algorithm for our model, which we show is provably optimal for certain input distributions. Numerical simulations where model parameters come from real-world datasets show that much of the clustering behavior in some datasets can just be explained by the projection step.
The identification of influential nodes in complex network can be very challenging. If the network has a community structure, centrality measures may fail to identify the complete set of influential nodes, as the hubs and other central nodes of the network may lie inside only one community. Here we define a bipartite clustering coefficient that, by taking differently structured clusters into account, can find important nodes across communities.
This paper investigates community detection by modularity maximisation on bipartite networks. In particular we are interested in how the operation of projection, using one node set of the bipartite network to infer connections between nodes in the other set, interacts with community detection. We first define a notion of modularity appropriate for a projected bipartite network and outline an algorithm for maximising it in order to partition the network. Using both real and synthetic networks we compare the communities found by five different algorithms, where each algorithm maximises a differe…
Bipartite projections are used in a wide range of network contexts including politics (bill co-sponsorship), genetics (gene co-expression), economics (executive board co-membership), and innovation (patent co-authorship). However, because bipartite projections are always weighted graphs, which are inherently challenging to analyze and visualize, it is often useful to examine the 'backbone', an unweighted subgraph containing only the most significant edges. In this paper, we introduce the R package backbone for extracting the backbone of weighted bipartite projections, and use bill sponsorship …
This paper introduces a computationally inexpensive method of extracting the backbone of one-mode networks projected from bipartite networks. We show that the edge weights in the one-mode projections are distributed according to a Poisson binomial distribution and that finding the expected weight distribution of a one-mode network projected from a random bipartite network only requires knowledge of the bipartite degree distributions. Being able to extract the backbone of a projection is highly beneficial in filtering out redundant information in large complex networks and narrowing down the in…
Cornell University（康奈尔大学）；The University of Melbourne（墨尔本大学）；Purdue University（普渡大学）
Motivated by applications in community detection and dense subgraph discovery, we consider new clustering objectives in hypergraphs and bipartite graphs. These objectives are parameterized by one or more resolution parameters in order to enable diverse knowledge discovery in complex data. For both hypergraph and bipartite objectives, we identify parameter regimes that are equivalent to existing objectives and share their (polynomial-time) approximation algorithms. We first show that our parameterized hypergraph correlation clustering objective is related to higher-order notions of normalized c…
Despite the abundance of bipartite networked systems, their organizing principles are less studied, compared to unipartite networks. Bipartite networks are often analyzed after projecting them onto one of the two sets of nodes. As a result of the projection, nodes of the same set are linked together if they have at least one neighbor in common in the bipartite network. Even though these projections allow one to study bipartite networks using tools developed for unipartite networks, one-mode projections lead to significant loss of information and artificial inflation of the projected network wi…
Bipartite (two-mode) networks are important in the analysis of social and economic systems as they explicitly show conceptual links between different types of entities. However, applications of such networks often work with a projected (one-mode) version of the original bipartite network. The topology of the projected network, and the dynamics that take place on it, are highly dependent on the degree distributions of the two different node types from the original bipartite structure. To date, the interaction between the degree distributions of bipartite networks and their one-mode projections …
Bipartite networks are a well known strategy to study a variety of phenomena. The commonly used method to deal with this type of network is to project the bipartite data into a unipartite weighted graph and then using a backboning technique to extract only the meaningful edges. Despite the wide availability of different methods both for projection and backboning, we believe that there has been little attention to the effect that the combination of these two processes has on the data and on the resulting network topology. In this paper we study the effect that the possible combinations of proje…
Bipartite networks are currently regarded as providing a major insight into the organization of many real-world systems, unveiling the mechanisms driving the interactions occurring between distinct groups of nodes. One of the most important issues encountered when modeling bipartite networks is devising a way to obtain a (monopartite) projection on the layer of interest, which preserves as much as possible the information encoded into the original bipartite structure. In the present paper we propose an algorithm to obtain statistically-validated projections of bipartite networks, according to …
Monopartite projections of bipartite networks are useful tools for modeling indirect interactions in complex systems. The standard approach to identify significant links is statistical validation using a suitable null network model, such as the popular configuration model (CM) that constrains node degrees and randomizes everything else. However different CM formulations exist, depending on how the constraints are imposed and for which sets of nodes. Here we systematically investigate the application of these formulations in validating the same network, showing that they lead to different resul…
Projections of bipartite or two-mode networks capture co-occurrences, and are used in diverse fields (e.g., ecology, economics, bibliometrics, politics) to represent unipartite networks. A key challenge in analyzing such networks is determining whether an observed number of co-occurrences between two nodes is significant, and therefore whether an edge exists between them. One approach, the fixed degree sequence model (FDSM), evaluates the significance of an edge's weight by comparison to a null model in which the degree sequences of the original bipartite network are fixed. Although the FDSM i…
A survey can be represented by a bipartite network as it has two types of nodes, participants and items in which participants can only interact with items. We introduce an agreement threshold to take a minimal projection of the participants linked by shared responses in order to identify opinion-based groups. We show that in American National Election Studies-data, this can identify polarisation along political attitudes. We also take a projection of attitudes that are linked by how participants respond to them. This can be used to show which attitudes are commonly held together in different c…
Bipartite networks manifest as a stream of edges that represent transactions, e.g., purchases by retail customers. Many machine learning applications employ neighborhood-based measures to characterize the similarity among the nodes, such as the pairwise number of common neighbors (CN) and related metrics. While the number of node pairs that share neighbors is potentially enormous, only a relatively small proportion of them have many common neighbors. This motivates finding a weighted sampling approach to preferentially sample these node pairs. This paper presents a new sampling algorithm that …
We consider spectral clustering algorithms for community detection under a general bipartite stochastic block model (SBM). A modern spectral clustering algorithm consists of three steps: (1) regularization of an appropriate adjacency or Laplacian matrix (2) a form of spectral truncation and (3) a k-means type algorithm in the reduced spectral domain. We focus on the adjacency-based spectral clustering and for the first step, propose a new data-driven regularization that can restore the concentration of the adjacency matrix even for the sparse networks. This result is based on recent work on re…
Bipartite Graph is often a realistic model of complex networks where two different sets of entities are involved and relationship exist only two entities belonging to two different sets. Examples include the user-item relationship of a recommender system, actor-movie relationship of an online movie database systems. One way to compress a bipartite graph is to take unweighted or weighted one mode projection of one side vertices. Properties of this projected network are extremely important in many practical situations (say the selection process of influencing nodes for viral marketing). In this …
A number of real-world networks are, in fact, one-mode projections of bipartite networks comprised of two types of nodes. For institutions engaging in collaboration for technological innovation, the underlying network is bipartite with institutions (agents) linked to the patents they have filed (artifacts), while the projection is the co-patenting network. Projected network topology is highly affected by the underlying bipartite structure, hence a lack of understanding of the bipartite network has consequences for the information that might be drawn from the one-mode co-patenting network. Here…
Bipartite networks are a common type of network data in which there are two types of vertices, and only vertices of different types can be connected. While bipartite networks exhibit community structure like their unipartite counterparts, existing approaches to bipartite community detection have drawbacks, including implicit parameter choices, loss of information through one-mode projections, and lack of interpretability. Here we solve the community detection problem for bipartite networks by formulating a bipartite stochastic block model, which explicitly includes vertex type information and …
Dynamical processes, such as the diffusion of knowledge, opinions, pathogens, "fake news", innovation, and others, are highly dependent on the structure of the social network on which they occur. However, questions on why most social networks present some particular structural features, namely high levels of transitivity and degree assortativity, when compared to other types of networks remain open. First, we argue that every one-mode network can be regarded as a projection of a bipartite network, and show that this is the case using two simple examples solved with the generating functions for…
Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences（中国科学院深圳先进技术研究院）；School of Computer Science, South China Normal University（华南师范大学计算机学院）
In this paper, we propose GPSP, a novel Graph Partition and Space Projection based approach, to learn the representation of a heterogeneous network that consists of multiple types of nodes and links. Concretely, we first partition the heterogeneous network into homogeneous and bipartite subnetworks. Then, the projective relations hidden in bipartite subnetworks are extracted by learning the projective embedding vectors. Finally, we concatenate the projective vectors from bipartite subnetworks with the ones learned from homogeneous subnetworks to form the final representation of the heterogeneo…
We study bipartite community detection in networks, or more generally the network biclustering problem. We present a fast two-stage procedure based on spectral initialization followed by the application of a pseudo-likelihood classifier twice. Under mild regularity conditions, we establish the weak consistency of the procedure (i.e., the convergence of the misclassification rate to zero) under a general bipartite stochastic block model. We show that the procedure is optimal in the sense that it achieves the optimal convergence rate that is achievable by a biclustering oracle, adaptively over t…
Universidad Carlos III de Madrid（马德里卡洛斯三世大学）；Universidad Politécnica de Madrid（马德里理工大学）；Universidad de Zaragoza（萨拉戈萨大学）；Grupo Interdisciplinar de Sistemas Complejos (GISC)（跨学科复杂系统研究组（GISC））；Unidad Mixta Interdisciplinar de Comportamiento y Complejidad Social (UMICCS)（跨学科社会行为与复杂性联合单位（UMICCS））；Grupo de Investigación Ingeniería de Organización y Logística (IOL)（组织与物流工程研究组（IOL））；Instituto de Biocomputación y Física de Sistemas Complejos (BIFI)（生物计算与复杂系统物理研究所（BIFI））；UC3M-Santander Big Data Institute (IBiDat)（马德里卡洛斯三世大学-桑坦德大数据研究所（IBiDat））
When some 'entities' are related by the 'features' they share they are amenable to a bipartite network representation. Plant-pollinator ecological communities, co-authorship of scientific papers, customers and purchases, or answers in a poll, are but a few examples. Analyzing clustering of such entities in the network is a useful tool with applications in many fields, like internet technology, recommender systems, or detection of diseases. The algorithms most widely applied to find clusters in bipartite networks are variants of modularity optimization. Here we provide an hierarchical clusterin…
In this work we are interested in identifying clusters of "positional equivalent" actors, i.e. actors who play a similar role in a system. In particular, we analyze weighted bipartite networks that describes the relationships between actors on one side and features or traits on the other, together with the intensity level to which actors show their features. The main contribution of our work is twofold. First, we develop a methodological approach that takes into account the underlying multivariate dependence among groups of actors. The idea is that positions in a network could be defined on th…
We introduce and study random bipartite networks with hidden variables. Nodes in these networks are characterized by hidden variables which control the appearance of links between node pairs. We derive analytic expressions for the degree distribution, degree correlations, the distribution of the number of common neighbors, and the bipartite clustering coefficient in these networks. We also establish the relationship between degrees of nodes in original bipartite networks and in their unipartite projections. We further demonstrate how hidden variable formalism can be applied to analyze topologi…
Bipartite user-object networks are becoming increasingly popular in representing user interaction data in a web or e-commerce environment. They have certain characteristics and challenges that differentiates them from other bipartite networks. This paper analyzes the properties of five real world user-object networks. In all cases we found a heavy tail object degree distribution with popular objects connecting together a large part of the users causing significant edge inflation in the projected users network. We propose a novel edge weighting strategy based on tf-idf and show that the new sch…
Community detection or clustering is a fundamental task in the analysis of network data. Many real networks have a bipartite structure which makes community detection challenging. In this paper, we consider a model which allows for matched communities in the bipartite setting, in addition to node covariates with information about the matching. We derive a simple fast algorithm for fitting the model based on variational inference ideas and show its effectiveness on both simulated and real data. A variation of the model to allow for degree-correction is also considered, in addition to a novel ap…
Community detection of network flows conventionally assumes one-step dynamics on the links. For sparse networks and interest in large-scale structures, longer timescales may be more appropriate. Oppositely, for large networks and interest in small-scale structures, shorter timescales may be better. However, current methods for analyzing networks at different timescales require expensive and often infeasible network reconstructions. To overcome this problem, we introduce a method that takes advantage of the inner-workings of the map equation and evades the reconstruction step. This makes it pos…
We recently introduced a formalism for the modeling of temporal networks, that we call stream graphs. It emphasizes the streaming nature of data and allows rigorous definitions of many important concepts generalizing classical graphs. This includes in particular size, density, clique, neighborhood, degree, clustering coefficient, and transitivity. In this contribution, we show that, like graphs, stream graphs may be extended to cope with bipartite structures, with node and link weights, or with link directions. We review the main bipartite, weighted or directed graph concepts proposed in the l…
University of Oxford（牛津大学）；Princeton University（普林斯顿大学）；The Alan Turing Institute（阿兰·图灵研究所）；University of Glasgow（格拉斯哥大学）
Clustering is an essential technique for network analysis, with applications in a diverse range of fields. Although spectral clustering is a popular and effective method, it fails to consider higher-order structure and can perform poorly on directed networks. One approach is to capture and cluster higher-order structures using motif adjacency matrices. However, current formulations fail to take edge weights into account, and thus are somewhat limited when weight is a key component of the network under study. We address these shortcomings by exploring motif-based weighted spectral clustering me…
Network science is a powerful tool for analyzing complex systems in fields ranging from sociology to engineering to biology. This paper is focused on generative models of large-scale bipartite graphs, also known as two-way graphs or two-mode networks. We propose two generative models that can be easily tuned to reproduce the characteristics of real-world networks, not just qualitatively, but quantitatively. The characteristics we consider are the degree distributions and the metamorphosis coefficient. The metamorphosis coefficient, a bipartite analogue of the clustering coefficient, is the pro…
Many real-world complex networks are best modeled as bipartite (or 2-mode) graphs, where nodes are divided into two sets with links connecting one side to the other. However, there is currently a lack of methods to analyze properly such graphs as most existing measures and methods are suited to classical graphs. A usual but limited approach consists in deriving 1-mode graphs (called projections) from the underlying bipartite structure, though it causes important loss of information and data storage issues. We introduce here internal links and pairs as a new notion useful for such analysis: it …
Institute of Computer Science, Polish Academy of Sciences（波兰科学院计算机科学研究所）
The purpose of this article is to introduce a new iterative algorithm with properties resembling real life bipartite graphs. The algorithm enables us to generate wide range of random bigraphs, which features are determined by a set of parameters.We adapt the advances of last decade in unipartite complex networks modeling to the bigraph setting. This data structure can be observed in several situations. However, only a few datasets are freely available to test the algorithms (e.g. community detection, influential nodes identification, information retrieval) which operate on such data. Therefore…
A variety of real-world systems can be modeled as bipartite networks. One of the most powerful and simple link prediction methods is Linear-Graph Autoencoder(LGAE) which has promising performance on challenging tasks such as link prediction and node clustering. LGAE relies on simple linear model w.r.t. the adjacency matrix of the graph to learn vector space representations of nodes. In this paper, we consider the case of bipartite link predictions where node attributes are unavailable. When using LGAE, we propose to multiply the reconstructed adjacency matrix with a symmetrically normalized tr…
Performance bounds are given for exploratory co-clustering/ blockmodeling of bipartite graph data, where we assume the rows and columns of the data matrix are samples from an arbitrary population. This is equivalent to assuming that the data is generated from a nonsmooth graphon. It is shown that co-clusters found by any method can be extended to the row and column populations, or equivalently that the estimated blockmodel approximates a blocked version of the generative graphon, with estimation error bounded by $O_P(n^{-1/2})$. Analogous performance bounds are also given for degree-corrected …
Predicting the popularity of items in rating networks is an interesting but challenging problem. This is especially so when an item has first appeared and has received very few ratings. In this paper, we propose a novel approach to predicting the future popularity of new items in rating networks, defining a new bipartite clustering coefficient to predict the popularity of movies and stories in the MovieLens and Digg networks respectively. We show that the clustering behaviour of the first user who rates a new item gives insight into the future popularity of that item. Our method predicts, with…
