Learning embeddings for multiplex networks using triplet loss

AbstractLearning low-dimensional representations of graphs has facilitated the use of traditional machine learning techniques to solving classic network analysis tasks such as link prediction, node classification, community detection, etc. However, to date, the vast majority of these learning tasks are focused on traditional single-layer/unimodal networks and largely ignore the case of multiplex networks. A multiplex network is a suitable structure to model multi-dimensional real-world complex systems. It consists of multiple layers where each layer represents a different relationship among the network nodes. In this work, we propose MUNEM, a novel approach for learning a low-dimensional representation of a multiplex network using a triplet loss objective function. In our approach, we preserve the global structure of each layer, while at the same time fusing knowledge among different layers during the learning process. We evaluate the effectiveness of our proposed method by testing and comparing on real-world multiplex networks from different domains, such as collaboration network, protein-protein interaction network, online social network. Finally, in order to deliberately examine the effect of our model’s parameters we conduct extensive experiments on synthetic multiplex networks.

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Nonlinear Structural Fusion for Multiplex Network

Complexity ◽

10.1155/2020/7041564 ◽

2020 ◽

Vol 2020 ◽

pp. 1-17

Author(s):

Nianwen Ning ◽

Feiyu Long ◽

Chunchun Wang ◽

Youjie Zhang ◽

Yilin Yang ◽

...

Keyword(s):

Network Analysis ◽

Real World ◽

Structural Information ◽

Single Layer ◽

Multiplex Networks ◽

Multiplex Network ◽

Low Weight ◽

Fusion Methods ◽

Low Dimensional ◽

Single Layer Network

Many real-world complex systems have multiple types of relations between their components, and they are popularly modeled as multiplex networks with each type of relation as one layer. Since the fusion analysis of multiplex networks can provide a comprehensive insight, the structural information fusion of multiplex networks has become a crucial issue. However, most of these existing data fusion methods are inappropriate for researchers to apply to complex network analysis directly. The feature-based fusion methods ignore the sharing and complementarity of interlayer structural information. To tackle this problem, we propose a multiplex network structural fusion (MNSF) model, which can construct a network with comprehensive information. It is composed of two modules: the network feature extraction (NFE) module and the network structural fusion (NSF) module. (1) In NFE, MNSF first extracts a low-dimensional vector representation of a node from each layer. Then, we construct a node similarity network based on embedding matrices and K-D tree algorithm. (2) In NSF, we present a nonlinear enhanced iterative fusion (EIF) strategy. EIF can strengthen high-weight edges presented in one (i.e., complementary information) or more (i.e., shared information) networks and weaken low-weight edges (i.e., redundant information). The retention of low-weight edges shared by all layers depends on the tightness of connections of their K-order proximity. The usage of higher-order proximity in EIF alleviates the dependence on the quality of node embedding. Besides, the fused network can be easily exploited by traditional single-layer network analysis methods. Experiments on real-world networks demonstrate that MNSF outperforms the state-of-the-art methods in tasks link prediction and shared community detection.

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Link prediction via layer relevance of multiplex networks

International Journal of Modern Physics C ◽

10.1142/s0129183117501017 ◽

2017 ◽

Vol 28 (08) ◽

pp. 1750101 ◽

Cited By ~ 7

Author(s):

Yabing Yao ◽

Ruisheng Zhang ◽

Fan Yang ◽

Yongna Yuan ◽

Qingshuang Sun ◽

...

Keyword(s):

Structural Properties ◽

Link Prediction ◽

Structural Information ◽

Similarity Index ◽

Single Layer ◽

Structural Features ◽

Prediction Performance ◽

Multiplex Networks ◽

Multiplex Network ◽

Node Similarity

In complex networks, the existing link prediction methods primarily focus on the internal structural information derived from single-layer networks. However, the role of interlayer information is hardly recognized in multiplex networks, which provide more diverse structural features than single-layer networks. Actually, the structural properties and functions of one layer can affect that of other layers in multiplex networks. In this paper, the effect of interlayer structural properties on the link prediction performance is investigated in multiplex networks. By utilizing the intralayer and interlayer information, we propose a novel “Node Similarity Index” based on “Layer Relevance” (NSILR) of multiplex network for link prediction. The performance of NSILR index is validated on each layer of seven multiplex networks in real-world systems. Experimental results show that the NSILR index can significantly improve the prediction performance compared with the traditional methods, which only consider the intralayer information. Furthermore, the more relevant the layers are, the higher the performance is enhanced.

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Crawling the Community Structure of Multiplex Networks

Proceedings of the AAAI Conference on Artificial Intelligence ◽

10.1609/aaai.v33i01.3301168 ◽

2019 ◽

Vol 33 ◽

pp. 168-175 ◽

Cited By ~ 3

Author(s):

Ricky Laishram ◽

Jeremy D. Wendt ◽

Sucheta Soundarajan

Keyword(s):

Community Structure ◽

Real World ◽

Multiplex Networks ◽

Data Collector ◽

Multiplex Network ◽

Multiple Levels ◽

Novel Algorithm ◽

Collection Costs

We examine the problem of crawling the community structure of a multiplex network containing multiple layers of edge relationships. While there has been a great deal of work examining community structure in general, and some work on the problem of sampling a network to preserve its community structure, to the best of our knowledge, this is the first work to consider this problem on multiplex networks. We consider the specific case in which the layers of a multiplex network have different query (collection) costs and reliabilities; and a data collector is interested in identifying the community structure of the most expensive layer. We propose MultiComSample (MCS), a novel algorithm for crawling a multiplex network. MCS uses multiple levels of multi-armed bandits to determine the best layers, communities and node roles for selecting nodes to query. We test MCS against six baseline algorithms on real-world multiplex networks, and achieved large gains in performance. For example, after consuming a budget equivalent to sampling 20% of the nodes in the expensive layer, we observe that MCS outperforms the best baseline by up to 49%.

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The Multiplex Dependency Structure of Financial Markets

Complexity ◽

10.1155/2017/9586064 ◽

2017 ◽

Vol 2017 ◽

pp. 1-13 ◽

Cited By ~ 26

Author(s):

Nicolò Musmeci ◽

Vincenzo Nicosia ◽

Tomaso Aste ◽

Tiziana Di Matteo ◽

Vito Latora

Keyword(s):

Financial Markets ◽

Financial Time Series ◽

Single Layer ◽

Separate Analysis ◽

Dependency Structure ◽

Multiplex Networks ◽

Multiplex Network ◽

Different Types ◽

Partial Correlations ◽

Filtering Procedure

We propose here a multiplex network approach to investigate simultaneously different types of dependency in complex datasets. In particular, we consider multiplex networks made of four layers corresponding, respectively, to linear, nonlinear, tail, and partial correlations among a set of financial time series. We construct the sparse graph on each layer using a standard network filtering procedure, and we then analyse the structural properties of the obtained multiplex networks. The study of the time evolution of the multiplex constructed from financial data uncovers important changes in intrinsically multiplex properties of the network, and such changes are associated with periods of financial stress. We observe that some features are unique to the multiplex structure and would not be visible otherwise by the separate analysis of the single-layer networks corresponding to each dependency measure.

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Link prediction in real-world multiplex networks via layer reconstruction method

Royal Society Open Science ◽

10.1098/rsos.191928 ◽

2020 ◽

Vol 7 (7) ◽

pp. 191928

Author(s):

Amir Mahdi Abdolhosseini-Qomi ◽

Seyed Hossein Jafari ◽

Amirheckmat Taghizadeh ◽

Naser Yazdani ◽

Masoud Asadpour ◽

...

Keyword(s):

Real World ◽

Link Prediction ◽

Large Body ◽

Single Layer ◽

Structural Features ◽

Prediction Performance ◽

Reconstruction Method ◽

Multiplex Networks ◽

Technological Complex ◽

Different Types

Networks are invaluable tools to study real biological, social and technological complex systems in which connected elements form a purposeful phenomenon. A higher resolution image of these systems shows that the connection types do not confine to one but to a variety of types. Multiplex networks encode this complexity with a set of nodes which are connected in different layers via different types of links. A large body of research on link prediction problem is devoted to finding missing links in single-layer (simplex) networks. In recent years, the problem of link prediction in multiplex networks has gained the attention of researchers from different scientific communities. Although most of these studies suggest that prediction performance can be enhanced by using the information contained in different layers of the network, the exact source of this enhancement remains obscure. Here, it is shown that similarity w.r.t. structural features (eigenvectors) is a major source of enhancements for link prediction task in multiplex networks using the proposed layer reconstruction method and experiments on real-world multiplex networks from different disciplines. Moreover, we characterize how low values of similarity w.r.t. structural features result in cases where improving prediction performance is substantially hard.

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Multiplex Networks of the Guarantee Market: Evidence from China

Complexity ◽

10.1155/2017/9781890 ◽

2017 ◽

Vol 2017 ◽

pp. 1-7 ◽

Cited By ~ 4

Author(s):

Shouwei Li ◽

Shihang Wen

Keyword(s):

Betweenness Centrality ◽

Single Layer ◽

Eigenvector Centrality ◽

Scale Free ◽

Multiplex Networks ◽

Multiplex Network ◽

Different Types ◽

Single Layer Network

We investigate a multiplex network of the guarantee market with three layers corresponding to different types of guarantee relationships in China. We find that three single-layer networks all have the scale-free property and are of disassortative nature. A single-layer network is not quite representative of another single-layer network. The result of the betweenness centrality shows that central companies in one layer are not necessarily central in another layer. And the eigenvector centrality has the same result of the betweenness centrality except the top central company.

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Sampling Graphlets of Multiplex Networks: A Restricted Random Walk Approach

ACM Transactions on the Web ◽

10.1145/3456291 ◽

2021 ◽

Vol 15 (4) ◽

pp. 1-31

Author(s):

Simiao Jiao ◽

Zihui Xue ◽

Xiaowei Chen ◽

Yuedong Xu

Keyword(s):

Random Walk ◽

Real World ◽

High Efficiency ◽

Large Network ◽

Induced Subgraph ◽

Multiplex Networks ◽

Multiplex Network ◽

Random Node ◽

Edge Sampling ◽

And Function

Graphlets are induced subgraph patterns that are crucial to the understanding of the structure and function of a large network. A lot of effort has been devoted to calculating graphlet statistics where random walk-based approaches are commonly used to access restricted graphs through the available application programming interfaces (APIs). However, most of them merely consider individual networks while overlooking the strong coupling between different networks. In this article, we estimate the graphlet concentration in multiplex networks with real-world applications. An inter-layer edge connects two nodes in different layers if they actually belong to the same node. The access to a multiplex network is restrictive in the sense that the upper layer allows random walk sampling, whereas the nodes of lower layers can be accessed only through the inter-layer edges and only support random node or edge sampling. To cope with this new challenge, we define a suit of two-layer graphlets and propose novel random walk sampling algorithms to estimate the proportion of all the three-node graphlets. An analytical bound on the sampling steps is proved to guarantee the convergence of our unbiased estimator. We further generalize our algorithm to explore the tradeoff between the estimated accuracy of different graphlets when the sample budget is split into different layers. Experimental evaluation on real-world and synthetic multiplex networks demonstrates the accuracy and high efficiency of our unbiased estimators.

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Inverse Percolation to Quantify Robustness in Multiplex Networks

Complexity ◽

10.1155/2020/8796360 ◽

2020 ◽

Vol 2020 ◽

pp. 1-11

Author(s):

Edwin Montes-Orozco ◽

Roman-Anselmo Mora-Gutiérrez ◽

Bibiana Obregón-Quintana ◽

Sergio-G. de-los-Cobos-Silva ◽

Eric A. Rincón-García ◽

...

Keyword(s):

Mathematical Model ◽

Social Networks ◽

Single Layer ◽

Multiplex Networks ◽

Vertex Separator ◽

Multiplex Network ◽

The World ◽

Structural Measure ◽

The Mathematical Model

Inverse percolation is known as the problem of finding the minimum set of nodes whose elimination of their links causes the rupture of the network. Inverse percolation has been widely used in various studies of single-layer networks. However, the use and generalization of multiplex networks have been little considered. In this work, we propose a methodology based on inverse percolation to quantify the robustness of multiplex networks. Specifically, we present a modified version of the mathematical model for the multiplex-vertex separator problem (m-VSP). By solving the m-VSP, we can find nodes that cause the rupture of the mutually connected giant component (MCGC) and the large viable cluster (LVC) when their links are removed from the network. The methodology presented in this work was tested in a set of benchmark networks, and as case study, we present an analysis using a set of multiplex social networks modeled with information about the main characteristics of the best universities in the world and the universities in Mexico. The results show that the methodology presented in this work can work in different models and types of 2- and 3-layer multiplex networks without dividing the entire multiplex network into single-layer as some techniques described in the specific literature. Furthermore, thanks to the fact that the technique does not require the calculation of some structural measure or centrality metric, and it is easy to scale for networks of different sizes.

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Multiplex Network Embedding Model with High-Order Node Dependence

Complexity ◽

10.1155/2021/6644111 ◽

2021 ◽

Vol 2021 ◽

pp. 1-18

Author(s):

Nianwen Ning ◽

Qiuyue Li ◽

Kai Zhao ◽

Bin Wu

Keyword(s):

Real World ◽

Information Diffusion ◽

Structural Information ◽

High Order ◽

Network Embedding ◽

Multiplex Networks ◽

Cross Domain ◽

Multiplex Network ◽

Node Attributes ◽

Two Phases

Multiplex networks have been widely used in information diffusion, social networks, transport, and biology multiomics. They contain multiple types of relations between nodes, in which each type of the relation is intuitively modeled as one layer. In the real world, the formation of a type of relations may only depend on some attribute elements of nodes. Most existing multiplex network embedding methods only focus on intralayer and interlayer structural information while neglecting this dependence between node attributes and the topology of each layer. Attributes that are irrelevant to the network structure could affect the embedding quality of multiplex networks. To address this problem, we propose a novel multiplex network embedding model with high-order node dependence, called HMNE. HMNE simultaneously considers three properties: (1) intralayer high-order proximity of nodes, (2) interlayer dependence in respect of nodes, and (3) the dependence between node attributes and the topology of each layer. In the intralayer embedding phase, we present a symmetric graph convolution-deconvolution model to embed high-order proximity information as the intralayer embedding of nodes in an unsupervised manner. In the interlayer embedding phase, we estimate the local structural complementarity of nodes as an embedding constraint of interlayer dependence. Through these two phases, we can achieve the disentangled representation of node attributes, which can be treated as fined-grained semantic dependence on the topology of each layer. In the restructure phase of node attributes, we perform a linear fusion of attribute disentangled representations for each node as a reconstruction of original attributes. Extensive experiments have been conducted on six real-world networks. The experimental results demonstrate that the proposed model outperforms the state-of-the-art methods in cross-domain link prediction and shared community detection tasks.

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Principled approach to the selection of the embedding dimension of networks

Nature Communications ◽

10.1038/s41467-021-23795-5 ◽

2021 ◽

Vol 12 (1) ◽

Author(s):

Weiwei Gu ◽

Aditya Tandon ◽

Yong-Yeol Ahn ◽

Filippo Radicchi

Keyword(s):

Real World ◽

Structural Information ◽

General Purpose ◽

Embedding Dimension ◽

Network Embedding ◽

Machine Learning Technique ◽

Learning Technique ◽

Low Dimensional ◽

Large Corpus ◽

Selection Of

AbstractNetwork embedding is a general-purpose machine learning technique that encodes network structure in vector spaces with tunable dimension. Choosing an appropriate embedding dimension – small enough to be efficient and large enough to be effective – is challenging but necessary to generate embeddings applicable to a multitude of tasks. Existing strategies for the selection of the embedding dimension rely on performance maximization in downstream tasks. Here, we propose a principled method such that all structural information of a network is parsimoniously encoded. The method is validated on various embedding algorithms and a large corpus of real-world networks. The embedding dimension selected by our method in real-world networks suggest that efficient encoding in low-dimensional spaces is usually possible.

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