In network science, one of the significant and challenging subjects is the detection of communities. Modularity [1] is a measure of community structure that compares connectivity in the network with the expected connectivity in a graph sampled from a random null model. Its optimisation is a common approach to tackle the community detection problem. We present a new method for modularity maximisation, which is based on the observation that modularity can be expressed in terms of total variation on the graph and signless total variation on the null model. The resulting algorithm is of Merriman–Bence–Osher (MBO) type. Different from earlier methods of this type, the new method can easily accommodate different choices of the null model. Besides theoretical investigations of the method, we include in this paper numerical comparisons with other community detection methods, among which the MBO-type methods of Hu et al. [2] and Boyd et al. [3], and the Leiden algorithm [4].

We consider the community detection problem in sparse random hypergraphs under the non-uniform hypergraph stochastic block model (HSBM), a general model of random networks with community structure and higher-order interactions. When the random hypergraph has bounded expected degrees, we provide a spectral algorithm that outputs a partition with at least a $\gamma$ fraction of the vertices classified correctly, where $\gamma \in (0.5,1)$ depends on the signal-to-noise ratio (SNR) of the model. When the SNR grows slowly as the number of vertices goes to infinity, our algorithm achieves weak consistency, which improves the previous results in Ghoshdastidar and Dukkipati ((2017) Ann. Stat. 45(1) 289–315.) for non-uniform HSBMs.

Our spectral algorithm consists of three major steps: (1) Hyperedge selection: select hyperedges of certain sizes to provide the maximal signal-to-noise ratio for the induced sub-hypergraph; (2) Spectral partition: construct a regularised adjacency matrix and obtain an approximate partition based on singular vectors; (3) Correction and merging: incorporate the hyperedge information from adjacency tensors to upgrade the error rate guarantee. The theoretical analysis of our algorithm relies on the concentration and regularisation of the adjacency matrix for sparse non-uniform random hypergraphs, which can be of independent interest.

We study the community detection problem on a Gaussian mixture model, in which vertices are divided into $k\geq 2$ distinct communities. The major difference in our model is that the intensities for Gaussian perturbations are different for different entries in the observation matrix, and we do not assume that every community has the same number of vertices. We explicitly find the necessary and sufficient conditions for the exact recovery of the maximum likelihood estimation, which can give a sharp phase transition for the exact recovery even though the Gaussian perturbations are not identically distributed; see Section 7. Applications include the community detection on hypergraphs.

Suicide is a leading cause of death in the United States, particularly among adolescents. In recent years, suicidal ideation, attempts, and fatalities have increased. Systems maps can effectively represent complex issues such as suicide, thus providing decision-support tools for policymakers to identify and evaluate interventions. While network science has served to examine systems maps in fields such as obesity, there is limited research at the intersection of suicidology and network science. In this paper, we apply network science to a large causal map of adverse childhood experiences (ACEs) and suicide to address this gap. The National Center for Injury Prevention and Control (NCIPC) within the Centers for Disease Control and Prevention recently created a causal map that encapsulates ACEs and adolescent suicide in 361 concept nodes and 946 directed relationships. In this study, we examine this map and three similar models through three related questions: (Q1) how do existing network-based models of suicide differ in terms of node- and network-level characteristics? (Q2) Using the NCIPC model as a unifying framework, how do current suicide intervention strategies align with prevailing theories of suicide? (Q3) How can the use of network science on the NCIPC model guide suicide interventions?

Researchers theorize that many real-world networks exhibit community structure where within-community edges are more likely than between-community edges. While numerous methods exist to cluster nodes into different communities, less work has addressed this question: given some network, does it exhibit statistically meaningful community structure? We answer this question in a principled manner by framing it as a statistical hypothesis test in terms of a general and model-agnostic community structure parameter. Leveraging this parameter, we propose a simple and interpretable test statistic used to formulate two separate hypothesis testing frameworks. The first is an asymptotic test against a baseline value of the parameter while the second tests against a baseline model using bootstrap-based thresholds. We prove theoretical properties of these tests and demonstrate how the proposed method yields rich insights into real-world datasets.

Most community detection methods focus on clustering actors with common features in a network. However, clustering edges offers a more intuitive way to understand the network structure in many real-life applications. Among the existing methods for network edge clustering, the majority are algorithmic, with the exception of the latent space edge clustering (LSEC) model proposed by Sewell (Journal of Computational and Graphical Statistics, 30(2), 390–405, 2021). LSEC was shown to have good performance in simulation and real-life data analysis, but fitting this model requires prior knowledge of the number of clusters and latent dimensions, which are often unknown to researchers. Within a Bayesian framework, we propose an extension to the LSEC model using a sparse finite mixture prior that supports automated selection of the number of clusters. We refer to our proposed approach as the automated LSEC or aLSEC. We develop a variational Bayes generalized expectation-maximization approach and a Hamiltonian Monte Carlo-within Gibbs algorithm for estimation. Our simulation study showed that aLSEC reduced run time by 10 to over 100 times compared to LSEC. Like LSEC, aLSEC maintains a computational cost that grows linearly with the number of actors in a network, making it scalable to large sparse networks. We developed the R package aLSEC which implements the proposed methodology.

We explored the transmission mechanisms of corporate fraud and its punishments within social network communities. Using fraud triangle theory and trust triangle theory, we hypothesize four transmitting channels of how fraud commission and detection are affected by peers’ fraud and punishment. Based on Chinese listed corporations from 2008 to 2018, we first construct and detect interlocked social network communities with a community-detecting algorithm, and then examine hypotheses using a bivariate probit model with partial observability. Our findings indicate that peer-concealing and -hinting effects exist within social network communities. The peer-concealing effect decreases the likelihood of being detected when committing fraud, for those with more and closer fraudulent peers. The peer-hinting effect increases the likelihood of being detected when committing fraud, for those with more and closer punished peers. There is no evidence to support peer-contagion and vicarious-punishment effects. Thus, an improved understanding of the transmission mechanism of corporate fraud commission and detection within communities is provided to prevent and detect corporate fraud. In addition, stakeholders and regulators should be aware of the deviant subculture and social distancing in social network communities.

Multilayer networks are in the focus of the current complex network study. In such networks, multiple types of links may exist as well as many attributes for nodes. To fully use multilayer—and other types of complex networks in applications, the merging of various data with topological information renders a powerful analysis. First, we suggest a simple way of representing network data in a data matrix where rows correspond to the nodes and columns correspond to the data items. The number of columns is allowed to be arbitrary, so that the data matrix can be easily expanded by adding columns. The data matrix can be chosen according to targets of the analysis and may vary a lot from case to case. Next, we partition the rows of the data matrix into communities using a method which allows maximal compression of the data matrix. For compressing a data matrix, we suggest to extend so-called regular decomposition method for non-square matrices. We illustrate our method for several types of data matrices, in particular, distance matrices, and matrices obtained by augmenting a distance matrix by a column of node degrees, or by concatenating several distance matrices corresponding to layers of a multilayer network. We illustrate our method with synthetic power-law graphs and two real networks: an Internet autonomous systems graph and a world airline graph. We compare the outputs of different community recovery methods on these graphs and discuss how incorporating node degrees as a separate column to the data matrix leads our method to identify community structures well-aligned with tiered hierarchical structures commonly encountered in complex scale-free networks.

Dense networks with weighted connections often exhibit a community-like structure, where although most nodes are connected to each other, different patterns of edge weights may emerge depending on each node’s community membership. We propose a new framework for generating and estimating dense weighted networks with potentially different connectivity patterns across different communities. The proposed model relies on a particular class of functions which map individual node characteristics to the edges connecting those nodes, allowing for flexibility while requiring a small number of parameters relative to the number of edges. By leveraging the estimation techniques, we also develop a bootstrap methodology for generating new networks on the same set of vertices, which may be useful in circumstances where multiple data sets cannot be collected. Performance of these methods is analyzed in theory, simulations, and real data.

We make the first steps towards generalising the theory of stochastic block models, in the sparse regime, towards a model where the discrete community structure is replaced by an underlying geometry. We consider a geometric random graph over a homogeneous metric space where the probability of two vertices to be connected is an arbitrary function of the distance. We give sufficient conditions under which the locations can be recovered (up to an isomorphism of the space) in the sparse regime. Moreover, we define a geometric counterpart of the model of flow of information on trees, due to Mossel and Peres, in which one considers a branching random walk on a sphere and the goal is to recover the location of the root based on the locations of leaves. We give some sufficient conditions for percolation and for non-percolation of information in this model.

We propose a new methodology for inferring political actors’ latent memberships in communities of collective activity that drive their observable interactions. Unlike existing methods, the proposed Bipartite Link Community Model (biLCM) (1) applies to two groups of actors, (2) takes into account that actors may be members of more than one community, and (3) allows a pair of actors to interact in more than one way. We apply this method to characterize legislative communities of special interest groups and politicians in the 113th U.S. Congress. Previous empirical studies of interest group politics have been limited by the difficulty of observing the ties between interest groups and politicians directly. We therefore first construct an original dataset that connects the politicians who sponsor congressional bills with the interest groups that lobby on those bills based on more than two million textual descriptions of lobbying activities. We then use the biLCM to make quantitative measurements of actors’ community memberships ranging from narrow targeted interactions according to industry interests and jurisdictional committee membership to broad multifaceted connections across multiple policy domains.

Isolation is a concept originally conceived in the context of clique enumeration in static networks, mostly used to model communities that do not have much contact to the outside world. Herein, a clique is considered isolated if it has few edges connecting it to the rest of the graph. Motivated by recent work on enumerating cliques in temporal networks, we transform the isolation concept to the temporal setting. We discover that the addition of the time dimension leads to six distinct natural isolation concepts. Our main contribution is the development of parameterized enumeration algorithms for five of these six isolation types for clique enumeration, employing the parameter “degree of isolation.” In a nutshell, this means that the more isolated these cliques are, the faster we can find them. On the empirical side, we implemented and tested these algorithms on (temporal) social network data, obtaining encouraging results.