Higher-order motif structures and multi-vertex interactions are becoming increasingly important in studies of functionalities and evolution patterns of complex networks. To elucidate the role of higher-order structures in community detection over networks, we introduce a Superimposed Stochastic Block Model (SupSBM). The model is based on a random graph framework in which certain higher-order structures or subgraphs are generated through an independent hyperedge generation process and then replaced with graphs superimposed with edges generated by an inhomogeneous random graph model. Consequently, the model introduces dependencies between edges which allow for capturing more realistic network phenomena, namely strong local clustering in a sparse network, short average path length, and community structure. We then proceed to rigorously analyze the performance of a recently proposed higher-order spectral clustering method on the SupSBM. In particular, we prove non-asymptotic upper bounds on the misclustering error of higher-order spectral community detection for a SupSBM setting in which triangles are superimposed with undirected edges. We assess the model fit of the proposed model and compare it with existing random graph models in terms of observed properties of real network data obtained from diverse domains by sampling networks from the fitted models and a nonparametric network cross-validation approach.

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.

Networks and temporal point processes serve as fundamental building blocks for modeling complex dynamic relational data in various domains. We propose the latent space Hawkes (LSH) model, a novel generative model for continuous-time networks of relational events, using a latent space representation for nodes. We model relational events between nodes using mutually exciting Hawkes processes with baseline intensities dependent upon the distances between the nodes in the latent space and sender and receiver specific effects. We demonstrate that our proposed LSH model can replicate many features observed in real temporal networks including reciprocity and transitivity, while also achieving superior prediction accuracy and providing more interpretable fits than existing models.

In many application settings involving networks, such as messages between users of an on-line social network or transactions between traders in financial markets, the observed data consist of timestamped relational events, which form a continuous-time network. We propose the Community Hawkes Independent Pairs (CHIP) generative model for such networks. We show that applying spectral clustering to an aggregated adjacency matrix constructed from the CHIP model provides consistent community detection for a growing number of nodes and time duration. We also develop consistent and computationally efficient estimators for the model parameters. We demonstrate that our proposed CHIP model and estimation procedure scales to large networks with tens of thousands of nodes and provides superior fits than existing continuous-time network models on several real networks.