Global Optimization, Local Adaptation, and the Role of Growth in Distribution Networks

Designing and optimizing different flows in networks is a relevant problem in many contexts. While a number of methods have been proposed in the physics and optimal transport literature for the one-commodity case, we lack similar results for the multi-commodity scenario. In this paper we present a model based on optimal transport theory for finding optimal multi-commodity flow configurations on networks. This model introduces a dynamics that regulates the edge conductivities ...

We present new analytical and numerical results for the elliptic-parabolic system of partial differential equations proposed by Hu and Cai, which models the formation of biological transport networks. The model describes the pressure field using a Darcy's type equation and the dynamics of the conductance network under pressure force effects. Randomness in the material structure is represented by a linear diffusion term and conductance relaxation by an algebraic decay term. Th...

In the realm of biological flow networks, the ability to dynamically adjust to varying demands is paramount. Drawing inspiration from the remarkable adaptability of Physarum polycephalum, we present a novel physical mechanism tailored to optimize flow networks. Central to our approach is the principle that each network component -- specifically, the tubes -- harnesses locally available information to collectively minimize a global cost function. Our findings underscore the sc...

We study network properties of networks evolving in time based on optimal transport principles. These evolve from a structure covering uniformly a continuous space towards an optimal design in terms of optimal transport theory. At convergence, the networks should optimize the way resources are transported through it. As the network structure shapes in time towards optimality, its topological properties also change with it. The question is how do these change as we reach optim...

Numerous networks, such as transportation, distribution and delivery networks optimize their designs in order to increase efficiency and lower costs, improving the stability of its intended functions, etc. Networks that distribute goods, such as electricity, water, gas, telephone and data (Internet), or services as mail, railways and roads are examples of transportation networks. The optimal design fixes network architecture, including clustering, degree distribution, hierarc...

The recent surge in the network modeling of complex systems has set the stage for a new era in the study of fundamental and applied aspects of optimization in collective behavior. This Focus Issue presents an extended view of the state of the art in this field and includes articles from a large variety of domains where optimization manifests itself, including physical, biological, social, and technological networked systems.

In this paper we investigate networks whose evolution is governed by the interaction of a random assembly process and an optimization process. In the first process, new nodes are added one at a time and form connections to randomly selected old nodes. In between node additions, the network is rewired to minimize its pathlength. For timescales, at which neither the assembly nor the optimization processes are dominant, we find a rich variety of complex networks with power law t...

Self-regulation of living tissue as an example of self-organization phenomena in hierarchical systems of biological, ecological, and social nature is under consideration. The characteristic feature of these systems is the absence of any governing center and, thereby, their self-regulation is based on a cooperative interaction of all the elements. The work develops a mathematical theory of a vascular network response to local effects on scales of individual units of peripheral...

We build an evolutionary scenario that explains how some crucial physiological constraints in the arterial network of mammals - i.e. hematocrit, vessels diameters and arterial pressure drops - could have been selected by evolution. We propose that the arterial network evolved while being constrained by its function as an organ. To support this hypothesis, we focus our study on one of the main function of blood network: oxygen supply to the organs. We consider an idealized org...

Many biological, geophysical and technological systems involve the transport of resource over a network. In this paper we present an algorithm for calculating the exact concentration of resource at any point in space or time, given that the resource in the network is lost or delivered out of the network at a given rate, while being subject to advection and diffusion. We consider the implications of advection, diffusion and delivery for simple models of glucose delivery throug...