Multilayer networks: new methods and applications

We will discuss community detection in multi-layer and dynamical networks. First, we will describe fundamental information theoretic thresholds for community recovery in these models. Next, we will introduce algorithms based on Approximate Message Passing (AMP) which attain the information theoretic threshold. These AMP algorithms require knowledge of the underlying model parameters. We will use an empirical bayes approach to learn these model parameters---this will yield a fully data driven algorithm for community recovery. Time permitting, we will discuss some applications of this empirical bayes approach to dynamical networks.

 

The modeling of transportation systems as multilayer networks presents significant challenges due to the complex interactions between different modes of transit. In this framework, each layer corresponds to a specific transportation mode, with nodes representing stations or stops and links denoting available routes. The node sets between layers may differ, with inter-layer connections representing feasible transfers, including walking distances. Our focus is on analyzing the topological structure of these multilayer networks and estimating the impact of inter-layer links on the convenience and efficiency of the overall network. We propose a multilayer network model to capture these complex associations and assess the feasibility of introducing new routes and stations. By evaluating both intra- and inter-layer connectivity, we aim to identify strategies for improving transit efficiency. This model offers valuable insights for future urban public transit planning. 

Network data is ubiquitous in various scientific disciplines, including sociology, economics, and neuroscience. Latent space models are often employed in network data analysis, but the geometric effect of latent space curvature remains a significant, unresolved issue. In this work, we propose a hyperbolic network latent space model with a learnable curvature parameter. We theoretically justify that learning the optimal curvature is essential to minimizing the embedding error across all hyperbolic embedding methods beyond network latent space models. A maximum-likelihood estimation strategy, employing manifold gradient optimization, is developed, and we establish the consistency and convergence rates for the maximum-likelihood estimators, both of which are technically challenging due to the non-linearity and non-convexity of the hyperbolic distance metric. We further demonstrate the geometric effect of latent space curvature and the superior performance of the proposed model through extensive simulation studies and an application using a Facebook friendship network. 

Multilayer networks continue to gain significant attention in many areas of study, particularly, due to their high utility in modeling interdependent systems such as critical infrastructures, human brain connectome, and socio-environmental ecosystems. However, the clustering of multilayer networks, especially, using the information on higher order interactions of the system entities, yet remains in its infancy. In turn, higher order connectivity is often the key in such multilayer network applications as developing optimal partitioning of critical infrastructures in order to isolate unhealthy system components under cyber-physical threats and simultaneous identification of multiple brain regions affected by trauma or mental illness.

In this talk we introduce the concepts of Topological Data Analysis (TDA) to studies of complex multilayer networks and propose a new topological approach for network clustering. The key rationale is to group nodes based not on pairwise connectivity patterns or relationships between observations recorded at two individual nodes, but based on how similar in shape their local neighborhoods are at various resolution scales. Since shapes of local node neighborhoods are quantified using the topological summary, termed persistence diagrams, we refer to the new approach as Clustering using Persistence Diagrams (CPD). CPD systematically accounts for the important heterogeneous higher-order properties of node interactions within and in-between network layers and integrates information from the node neighbors. We illustrate the utility of CPD in application to to an emerging problem of societal importance - vulnerability zoning of residential properties to weather- and climate-induced risks in the context of house insurance claim dynamics.
 

The network model for brain structural-functional connectivity coupling is challenging because both structural and functional brain connectivity (SC and FC) exhibit organized and complex network structures. We represent SC and FC as graphs, where nodes denote brain regions and edges represent connections. The node sets for SC and FC networks may differ. Our focus is on assessing which sets of SC edges are associated with sets of FC edges and understanding the underlying network topological structure. We propose a triple-layer network model to capture the complex associations and identify dense SC-FC subgraph pairs. In a dense SC-FC subgraph pair, we maximize the proportion of SC-FC related edges in both SC and FC subgraphs while ensuring that most related SC-FC edges are covered by SC-FC subgraph pairs. We conduct extensive simulations and apply the proposed method to data from the Human Connectome Project and UK Biobank.