Infrastructure adaptation and emergence of loops in network routing with time-dependent loads.

Network routing approaches are widely used to study the evolution in time of self-adapting systems. However, few advances have been made for problems where adaptation is governed by time-dependent inputs. In this work we study a dynamical systems where the edge conductivities of a network are regulated by time-varying mass loads injected on nodes. Motivated by empirical observations, we assume that conductivities adapt slowly with respect to the characteristic time of the loads. Furthermore, assuming the loads to be periodic, we derive a dynamics where the evolution of the system is controlled by a matrix obtained with the Fourier coefficients of the input loads. Remarkably, we find a sufficient condition on these coefficients that determines when the resulting network topologies are trees. We show an example of this on the Bordeaux bus network where we tune the input loads to interpolate between loopy and tree topologies. We validate our model on several synthetic networks and provide an expression for long-time solutions of the original conductivities.

Multicommodity routing optimization for engineering networks.

Optimizing passengers routes is crucial to design efficient transportation networks. Recent results show that optimal transport provides an efficient alternative to standard optimization methods. However, it is not yet clear if this formalism has empirical validity on engineering networks. We address this issue by considering different response functions-quantities determining the interaction between passengers-in the dynamics implementing the optimal transport formulation. Particularly, we couple passengers' fluxes by taking their sum or the sum of their squares. The first choice naturally reflects edges occupancy in transportation networks, however the second guarantees convergence to an optimal configuration of flows. Both modeling choices are applied to the Paris metro. We measure the extent of traffic bottlenecks and infrastructure resilience to node removal, showing that the two settings are equivalent in the congested transport regime, but different in the branched one. In the latter, the two formulations differ on how fluxes are distributed, with one function favoring routes consolidation, thus potentially being prone to generate traffic overload. Additionally, we compare our method to Dijkstra's algorithm to show its capacity to efficiently recover shortest-path-like graphs. Finally, we observe that optimal transport networks lie in the Pareto front drawn by the energy dissipated by passengers, and the cost to build the infrastructure.

Arbitrary-order intrinsic virtual element method for elliptic equations on surfaces.

We develop a geometrically intrinsic formulation of the arbitrary-order Virtual Element Method (VEM) on polygonal cells for the numerical solution of elliptic surface partial differential equations (PDEs). The PDE is first written in covariant form using an appropriate local reference system. The knowledge of the local parametrization allows us to consider the two-dimensional VEM scheme, without any explicit approximation of the surface geometry. The theoretical properties of the classical VEM are extended to our framework by taking into consideration the highly anisotropic character of the final discretization. These properties are extensively tested on triangular and polygonal meshes using a manufactured solution. The limitations of the scheme are verified as functions of the regularity of the surface and its approximation.

Network extraction by routing optimization.

Routing optimization is a relevant problem in many contexts. Solving directly this type of optimization problem is often computationally intractable. Recent studies suggest that one can instead turn this problem into one of solving a dynamical system of equations, which can instead be solved efficiently using numerical methods. This results in enabling the acquisition of optimal network topologies from a variety of routing problems. However, the actual extraction of the solution in terms of a final network topology relies on numerical details which can prevent an accurate investigation of their topological properties. In fact, in this context, theoretical results are fully accessible only to an expert audience and ready-to-use implementations for non-experts are rarely available or insufficiently documented. In particular, in this framework, final graph acquisition is a challenging problem in-and-of-itself. Here we introduce a method to extract network topologies from dynamical equations related to routing optimization under various parameters' settings. Our method is made of three steps: first, it extracts an optimal trajectory by solving a dynamical system, then it pre-extracts a network, and finally, it filters out potential redundancies. Remarkably, we propose a principled model to address the filtering in the last step, and give a quantitative interpretation in terms of a transport-related cost function. This principled filtering can be applied to more general problems such as network extraction from images, thus going beyond the scenarios envisioned in the first step. Overall, this novel algorithm allows practitioners to easily extract optimal network topologies by combining basic tools from numerical methods, optimization and network theory. Thus, we provide an alternative to manual graph extraction which allows a grounded extraction from a large variety of optimal topologies. The analysis of these may open up the possibility to gain new insights into the structure and function of optimal networks. We provide an open source implementation of the code online.

Soil-plant interaction monitoring: Small scale example of an apple orchard in Trentino, North-Eastern Italy.

Accurate monitoring and modeling of soil-plant systems are a key unresolved issue that currently limits the development of a comprehensive view of the interactions between soil and atmosphere, with a number of practical consequences including the difficulties in predicting climatic change patterns. This paper presents a case study where time-lapse minimal-invasive 3D micro-electrical tomography (ERT) is used to monitor rhizosphere eco-hydrological processes in an apple orchard in the Trentino region, Northern Italy. In particular we aimed at gaining a better understanding of the soil-vegetation water exchanges in the shallow critical zone, as part of a coordinated effort towards predicting climate-induced changes on the hydrology of Mediterranean basins (EU FP7 CLIMB project). The adopted strategy relied upon the installation of a 3D electrical tomography apparatus consisting of four mini-boreholes carrying 12 electrodes each plus 24 mini-electrodes on the ground surface, arranged in order to image roughly a cubic meter of soil surrounding a single apple tree. The monitoring program was initially tested with repeated measurements over about one year. Subsequently, we performed three controlled irrigation tests under different conditions, in order to evaluate the water redistribution under variable root activities and climatic conditions. Laboratory calibration on soil samples allowed us to translate electrical resistivity variations into moisture content changes, supported also by in-situ TDR measurements. Richards equation modeling was used also to explain the monitoring evidence. The results clearly identified the effect of root water uptake and the corresponding subsoil region where active roots are present, but also marked the need to consider the effects of different water salinity in the water infiltration process. We also gained significant insight about the need to measure quantitatively the plant evapotranspiration in order to close the water balance and separate soil structure effects (primarily, hydraulic conductivity) from water dynamics induced by living plants.

Spanning traceroutes over modular networks and general scaling degree distributions.

We analyze the class of networks characterized by modular structure where a sequence of l Erdös-Renyi random networks of size Nl with random average degrees is joined by links whose structure must remain immaterial. We find that traceroutes spanning the entire macronetwork exhibit scaling degree distributions P(k) approximately k-gamma, where gamma depends on how the degrees of the joined clusters are distributed. We thus suggest that yet another mechanism for the dynamic origin of arbitrary power-law degree distributions observed in natural and artificial networks, many of which belong to the range 2