Extracting Hidden Hierarchies in 3D Distribution Networks

Abstract

Natural and man-made transport webs are frequently dominated by dense sets of nested cycles. The architecture of these networks, as defined by the topology and edge weights, determines how efficiently the networks perform their function. Yet, the set of tools that can characterize such a weighted cycle-rich architecture in a physically relevant, mathematically compact way is sparse. In order to fill this void, we have developed a new algorithm that rests on an abstraction of the physical "tiling" in the case of a twodimensional network to an effective tiling of an abstract surface in 3-space that the network may be thought to sit in. Generically, these abstract surfaces are richer than the flat plane because there are now two families of fundamental units that may aggregate upon cutting weakest links-the plaquettes of the tiling and the longer "topological" cycles associated with the abstract surface itself. Upon sequential removal of the weakest links, as determined by a physically relevant edge weight, such as flow volume or capacity, neighboring plaquettes merge and a new tree graph characterizing this merging process results. The properties of this characteristic tree can provide the physical and topological data required to describe the architecture of the network and to build physical models. The new algorithm can be used for automated phenotypic characterization of any weighted network whose structure is dominated by cycles, such as mammalian vasculature in the organs or the force networks in jammed granular matter.

Resilience of three-dimensional sinusoidal networks in liver tissue

“…The restriction to two-dimensional networks in past research was largely a consequence of the considerable difficulties in imaging three-dimensional networks. Yet, topology suggests fundamental differences between two-dimensional and three-dimensional spatial networks, as it imposes constraints on the distribution of cycles in the network [ 12 ]. Here, we take advantage of recent technological advances in high-resolution imaging of adult mouse liver tissue [ 13 , 14 ], allowing us to study statistical geometry and resilience of three-dimensional sinusoidal networks.…”

Comparing two classes of biological distribution systems using network analysis

“…Understanding how transport in turn affects network development is also important [ 16 , 61 ]. Another intriguing direction would be to further examine the properties of distribution networks that are embedded not in 2-dimensions, but in 3-dimensions [ 26 , 65 ], where relationships between topology and geometric structure or spatial layout may be more complex.…”

On Curiosity: A Fundamental Aspect of Personality, a Practice of Network Growth

“…An example of an indirect constraint is the growth (or otherwise temporal variation) of a tissue that either surrounds the network or serves as a substrate for the network. This sort of indirect constraint has recently been modeled in the context of vasculature networks, which are biological distribution systems that transport nutrients across spatial distances to support the health of an organism (Modes, Magnasco, & Katifori, 2016). Here, the growth of the tissue that the vasculature supports is modeled as one dynamical process, and the growth of the vasculature network itself is modeled as a second dynamical process, coupled to the first (Ronellenfitsch & Katifori, 2016).…”