Network Filtering for Big Data: Triangulated Maximally Filtered Graph

We propose a network-filtering method, the Triangulated Maximally Filtered Graph (TMFG), that provides an approximate solution to the WEIGHTED MAXIMAL PLANAR GRAPH problem. The underlying idea of TMFG consists in building a triangulation that maximizes a score function associated with the amount of information retained by the network.TMFG uses as weights any arbitrary similarity measure to arrange data into a meaningful network structure that can be used for clustering, community detection and modelling. The method is fast, adaptable and scalable to very large datasets; it allows online updating and learning as new data can be inserted and deleted with combinations of local and non-local moves. Further, TMFG permits readjustments of the network in consequence of changes in the strength of the similarity measure. The method is based on local topological moves and can therefore take advantage of parallel and GPUs computing. We discuss how this network-filtering method can be used intuitively and efficiently for big data studies and its significance from an information-theoretic perspective

Sector Neutral Portfolios: Long Memory Motifs Persistence in Market Structure Dynamics

We study soft persistence (existence in subsequent temporal layers of motifs from the initial layer) of motif structures in Triangulated Maximally Filtered Graphs (TMFG) generated from time-varying Kendall correlation matrices computed from stock prices log-returns over rolling windows with exponential smoothing. We observe long-memory processes in these structures in the form of power law decays in the number of persistent motifs. The decays then transition to a plateau regime with a power-law decay with smaller exponent. We demonstrate that identifying persistent motifs allows for forecasting and applications to portfolio diversification. Balanced portfolios are often constructed from the analysis of historic correlations, however not all past correlations are persistently reflected into the future. Sector neutrality has also been a central theme in portfolio diversification and systemic risk. We present an unsupervised technique to identify persistently correlated sets of stocks. These are empirically found to identify sectors driven by strong fundamentals. Applications of these findings are tested in two distinct ways on four different markets, resulting in significant reduction in portfolio volatility. A persistence-based measure for portfolio allocation is proposed and shown to outperform volatility weighting when tested out of sample

Higher-order simplicial synchronization of coupled topological signals

Simplicial complexes capture the underlying network topology and geometry of complex systems ranging from the brain to social networks. Here we show that algebraic topology is a fundamental tool to capture the higher-order dynamics of simplicial complexes. In particular we consider topological signals, i.e., dynamical signals defined on simplices of different dimension, here taken to be nodes and links for simplicity. We show that coupling between signals defined on nodes and links leads to explosive topological synchronization in which phases defined on nodes synchronize simultaneously to phases defined on links at a discontinuous phase transition. We study the model on real connectomes and on simplicial complexes and network models. Finally, we provide a comprehensive theoretical approach that captures this transition on fully connected networks and on random networks treated within the annealed approximation, establishing the conditions for observing a closed hysteresis loop in the large network limit