MPA network design based on graph network theory and emergent properties of larval dispersal

Despite the recognised effectiveness of networks of Marine Protected Areas (MPAs) as a biodiversity conservation instrument, nowadays MPA network design frequently disregards the importance of connectivity patterns. In the case of sedentary marine populations, connectivity stems not only from the stochastic nature of the physical environment that affects early-life stages dispersal, but also from the spawning stock attributes that affect the reproductive output (e.g., passive eggs and larvae) and its survivorship. Early-life stages are virtually impossible to track in the ocean. Therefore, numerical ocean current simulations coupled to egg and larval Lagrangian transport models remain the most common approach for the assessment of marine larval connectivity. Inferred larval connectivity may be different depending on the type of connectivity considered; consequently, the prioritisation of sites for marine populations' conservation might also differ. Here, we introduce a framework for evaluating and designing MPA networks based on the identification of connectivity hotspots using graph theoretic analysis. We use as a case of study a network of open-access areas and MPAs, off Mallorca Island (Spain), and test its effectiveness for the protection of the painted comber Serranus scriba. Outputs from network analysis are used to: (1) identify critical areas for improving overall larval connectivity; (2) assess the impact of species' biological parameters in network connectivity; and (3) explore alternative MPA configurations to improve average network connectivity. Results demonstrate the potential of graph theory to identify non-trivial egg/larval dispersal patterns and emerging collective properties of the MPA network which are relevant for increasing protection efficiency.Comment: 8 figures, 3 tables, 1 Supplementary material (including 4 table; 3 figures and supplementary methods

A Method Based on Total Variation for Network Modularity Optimization using the MBO Scheme

The study of network structure is pervasive in sociology, biology, computer science, and many other disciplines. One of the most important areas of network science is the algorithmic detection of cohesive groups of nodes called "communities". One popular approach to find communities is to maximize a quality function known as {\em modularity} to achieve some sort of optimal clustering of nodes. In this paper, we interpret the modularity function from a novel perspective: we reformulate modularity optimization as a minimization problem of an energy functional that consists of a total variation term and an $\ell_2$ balance term. By employing numerical techniques from image processing and $\ell_1$ compressive sensing -- such as convex splitting and the Merriman-Bence-Osher (MBO) scheme -- we develop a variational algorithm for the minimization problem. We present our computational results using both synthetic benchmark networks and real data.Comment: 23 page