River deltas are intricate landscapes with complex channel networks that self-organize to deliver water, sediment, and nutrients from the apex to the delta top and eventually to the coastal zone. The natural balance of material and energy fluxes, which maintains a stable hydrologic, geomorphologic, and ecological state of a river delta, is often disrupted by external perturbations causing topological and dynamical changes in the delta structure and function. A formal quantitative framework for studying delta channel network connectivity and transport dynamics and their response to change is lacking. Here we present such a framework based on spectral graph theory and demonstrate its value in computing delta's steady state fluxes and identifying upstream (contributing) and downstream (nourishment) areas and fluxes from any point in the network. We use this framework to construct vulnerability maps that quantify the relative change of sediment and water delivery to the shoreline outlets in response to possible perturbations in hundreds of upstream links. The framework is applied to the Wax Lake delta in the Louisiana coast of the U.S. and the Niger delta in West Africa. In a companion paper, we present a comprehensive suite of metrics that quantify topologic and dynamic complexity of delta channel networks and, via application to seven deltas in diverse environments, demonstrate their potential to reveal delta morphodynamics and relate to notions of vulnerability and robustness.

Deltas are landforms that deliver water, sediment and nutrient fluxes from upstream rivers to the deltaic surface and eventually to oceans or inland water bodies via multiple pathways. Despite their importance, quantitative frameworks for their analysis lack behind those available for tributary networks. In a companion paper, delta channel networks were conceptualized as directed graphs and spectral graph theory was used to design a quantitative framework for exploring delta connectivity and flux dynamics. Here we use this framework to introduce a suite of graph-theoretic and entropy-based metrics, to quantify two components of a delta's complexity: (1) Topologic, imposed by the network connectivity and (2) Dynamic, dictated by the flux partitioning and distribution. The metrics are aimed to facilitate comparing, contrasting, and establishing connections between deltaic structure, process, and form. We illustrate the proposed analysis using seven deltas in diverse morphodynamic environments and of various degrees of channel complexity. By projecting deltas into a topo-dynamic space whose coordinates are given by topologic and dynamic delta complexity metrics, we show that this space provides a basis for delta comparison and physical insight into their dynamic behavior. The examined metrics are demonstrated to relate to the intuitive notion of vulnerability, measured by the impact of upstream flux changes to the shoreline flux, and reveal that complexity and vulnerability are inversely related. Finally, a spatially explicit metric, akin to a delta width function, is introduced to classify shapes of different delta types.

SignificanceRiver deltas are critically important Earthscapes at the land–water interface, supporting dense populations and diverse ecosystems while also providing disproportionately large food and energy resources. Deltas exhibit complex channel networks that dictate how water, sediment, and nutrients are spread over the delta surface. By adapting concepts from information theory, we show that a range of field and numerically generated deltas obey an optimality principle that suggests that deltas self-organize to increase the diversity of sediment transport pathways across the delta channels to the shoreline. We suggest that optimal delta configurations are also more resilient because the same mechanism that diversifies the delivery of fluxes to the shoreline also enhances the dampening of possible perturbations.

We study the dynamics of diffusion processes acting on directed multiplex networks, i.e., coupled multilayer networks where at least one layer consists of a directed graph. We reveal that directed multiplex networks may exhibit a faster diffusion at an intermediate degree of coupling than when the two layers are fully coupled. We use three simple multiplex examples and a real-world topology to illustrate the characteristics of the directed dynamics that give rise to a regime in which an optimal coupling exists. Given the ubiquity of both directed and multilayer networks in nature, our results could have important implications for the dynamics of multilevel complex systems towards optimality. PACS numbers: 89.75.Hc, 89.20.a, 89.75.Kd Multiplex networks are coupled multilayer networks where each layer consists of the same set of nodes but possibly different topologies and layers interact with each other only via counterpart nodes in each layer [1-3]. Multiplex networks have been shown useful for the study of diverse processes including social networks [4][5][6], transportation networks [7], and biochemical networks [8,9], among others. Recent studies have shown that the coupling of the layers in a multiplex network can result in emergent structural [10] and dynamical behavior such as enhanced diffusion (superdiffusion) [11], increased resilience to random failure [12], and emergence of critical points in the dynamics of coupled spreading processes [4,6]. These richer dynamics arise as a direct consequence of the emergence of more paths between every pair of nodes brought about by layer switching via an interlayer link.Most of the theory for multiplex networks has been developed when all layers consist of undirected networks [1]. However, more often than not, real social, biological and natural networks are structurally directed. Additionally, even if the underlying topology is not directed, the functional and dynamical connectivity of undirected networks is often directed due to gradients or the directionality in the flow of mass or information. These processes include geophysical processes on tributary river networks [13][14][15][16] and river delta channel networks [17][18][19][20][21][22], food webs [23][24][25], gene regulation networks [23,26] and social dynamics [27] to name a few. In this paper, we study diffusion processes on directed multiplex networks, defined here as multiplex networks wherein the connectivity of at least one of the layers forms a directed graph. We document a non-monotonic increase in the rate of convergence to the steady state as a function of the degree of coupling between layers. We uncover that due to the directionality of (at least one of) the layers in a directed multiplex network an optimal coupling regime can emerge where transport processes are enhanced and diffusion is faster than when layers are fully coupled. Let x(t) represent the N × 1 vector of concentration associated with the N nodes of a network at time t (throughout, vectors are thought of as column vectors). The d...