Explicit and implicit network connectivity: Analytical formulation and application to transport processes
Connectivity is a fundamental structural feature of a network that determines the outcome of any dynamics that happens on top of it. However, an analytical approach to obtain connection probabilities between nodes associated with to paths of different lengths is still missing. Here, we derive exact expressions for random-walk connectivity probabilities across any range of numbers of steps in a generic temporal, directed, and weighted network. This allows characterizing explicit connectivity realized by causal paths as well as implicit connectivity related to motifs of three nodes and two links called here pitchforks. We directly link such probabilities to the processes of tagging and sampling any quantity exchanged across the network, hence providing a natural framework to assess transport dynamics. Finally, we apply our theoretical framework to study ocean transport features in the Mediterranean Sea. We find that relevant transport structures, such as fluid barriers and corridors, can generate contrasting and counterintuitive connectivity patterns bringing novel insights into how ocean currents drive seascape connectivity. ; T.L. is funded by a Doctoral fellowship obtained through Aix-Marseille University. V.R. and T.L. acknowledge financial support from the European project SEAMoBB (Solutions for sEmi-Automated Monitoring of Benthic Biodiversity), funded by ERA-Net Mar-TERA (id. 145) and managed by the ANR (Grant No. ANR-17-MART-0001-02, P.I.: A. Chenuil). I. H.-C acknowledges the Vicenç Mut contract funded by the Government of the Balearic Island and the European Social Fund (ESF) Operational Programme.
