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1. Root and community inference on the latent growth process of a network

   https://doi.org/10.1093/jrsssb/qkad102  

   Crane, Harry; Xu, Min (September 2023, Journal of the Royal Statistical Society Series B: Statistical Methodology)

   Abstract Many statistical models for networks overlook the fact that most real-world networks are formed through a growth process. To address this, we introduce the Preferential Attachment Plus Erdős–Rényi model, where we let a random network G be the union of a preferential attachment (PA) tree T and additional Erdős–Rényi (ER) random edges. The PA tree captures the underlying growth process of a network where vertices/edges are added sequentially, while the ER component can be regarded as noise. Given only one snapshot of the final network G, we study the problem of constructing confidence sets for the root node of the unobserved growth process; the root node can be patient zero in an infection network or the source of fake news in a social network. We propose inference algorithms based on Gibbs sampling that scales to networks with millions of nodes and provide theoretical analysis showing that the size of the confidence set is small if the noise level of the ER edges is not too large. We also propose variations of the model in which multiple growth processes occur simultaneously, reflecting the growth of multiple communities; we use these models to provide a new approach to community detection. 

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2. A Unified Spectral Clustering Approach for Detecting Community Structure in Multilayer Networks

   https://doi.org/10.3390/sym15071368  

   Al-sharoa, Esraa; Aviyente, Selin (July 2023, Symmetry)

   Networks offer a compact representation of complex systems such as social, communication, and biological systems. Traditional network models are often inadequate to capture the diverse nature of contemporary networks, which may exhibit temporal variation and multiple types of interactions between entities. Multilayer networks (MLNs) provide a more comprehensive representation by allowing interactions between nodes to be represented by different types of links, each reflecting a distinct type of interaction. Community detection reveals meaningful structure and provides a better understanding of the overall functioning of networks. Current approaches to multilayer community detection are either limited to community detection over the aggregated network or are extensions of single-layer community detection methods with simplifying assumptions such as a common community structure across layers. Moreover, most of the existing methods are limited to multiplex networks with no inter-layer edges. In this paper, we introduce a spectral-clustering-based community detection method for two-layer MLNs. The problem of detecting the community structure is formulated as an optimization problem where the normalized cut for each layer is minimized simultaneously with the normalized cut for the bipartite network along with regularization terms that ensure the consistency of the within- and across-layer community structures. The proposed method is evaluated on both synthetic and real networks and compared to state-of-the-art methods. MLNs. The problem of detecting the community structure is formulated as an optimization problem where the normalized cut for each layer is minimized simultaneously with the normalized cut for the bipartite network along with regularization terms that ensure the consistency of the intra- and inter-layer community structures. The proposed method is evaluated on both synthetic and real networks and compared to state-of-the-art methods. 

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3. On clustering network-valued data

   Mukherjee, Soumendu Sundar; Sarkar, Purnamrita; Lin, Lizhen (January 2017, Advances in neural information processing systems)

   Community detection, which focuses on clustering nodes or detecting communities in (mostly) a single network, is a problem of considerable practical interest and has received a great deal of attention in the research community. While being able to cluster within a network is important, there are emerging needs to be able to \emph{cluster multiple networks}. This is largely motivated by the routine collection of network data that are generated from potentially different populations. These networks may or may not have node correspondence. When node correspondence is present, we cluster networks by summarizing a network by its graphon estimate, whereas when node correspondence is not present, we propose a novel solution for clustering such networks by associating a computationally feasible feature vector to each network based on trace of powers of the adjacency matrix. We illustrate our methods using both simulated and real data sets, and theoretical justifications are provided in terms of consistency. 

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4. The Multivariate Community Hawkes model for dependent relational events in continuous-time networks

   Soliman, Hadeel; Zhao, Lingfei; Huang, Zhipeng; Paul, Subhadeep; Xu, Kevin S. (January 2022, Proceedings of Machine Learning Research)

   The stochastic block model (SBM) is one of the most widely used generative models for network data. Many continuous-time dynamic network models are built upon the same assumption as the SBM: edges or events between all pairs of nodes are conditionally independent given the block or community memberships, which prevents them from reproducing higher-order motifs such as triangles that are commonly observed in real networks. We propose the multivariate community Hawkes (MULCH) model, an extremely flexible community-based model for continuous-time networks that introduces dependence between node pairs using structured multivariate Hawkes processes. We fit the model using a spectral clustering and likelihood-based local refinement procedure. We find that our proposed MULCH model is far more accurate than existing models both for predictive and generative tasks. 

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5. Heterformer: Transformer-based Deep Node Representation Learning on Heterogeneous Text-Rich Networks

   https://doi.org/10.1145/3580305.3599376  

   Jin, Bowen; Zhang, Yu; Zhu, Qi; Han, Jiawei (August 2023, ACM)

   Proc. 2023 ACM SIGKDD Int. Conf. on Knowledge Discovery and Data Mining (Ed.)

   Representation learning on networks aims to derive a meaningful vector representation for each node, thereby facilitating downstream tasks such as link prediction, node classification, and node clustering. In heterogeneous text-rich networks, this task is more challenging due to (1) presence or absence of text: Some nodes are associated with rich textual information, while others are not; (2) diversity of types: Nodes and edges of multiple types form a heterogeneous network structure. As pretrained language models (PLMs) have demonstrated their effectiveness in obtaining widely generalizable text representations, a substantial amount of effort has been made to incorporate PLMs into representation learning on text-rich networks. However, few of them can jointly consider heterogeneous structure (network) information as well as rich textual semantic information of each node effectively. In this paper, we propose Heterformer, a Heterogeneous Network-Empowered Transformer that performs contextualized text encoding and heterogeneous structure encoding in a unified model. Specifically, we inject heterogeneous structure information into each Transformer layer when encoding node texts. Meanwhile, Heterformer is capable of characterizing node/edge type heterogeneity and encoding nodes with or without texts. We conduct comprehensive experiments on three tasks (i.e., link prediction, node classification, and node clustering) on three large-scale datasets from different domains, where Heterformer outperforms competitive baselines significantly and consistently. 

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