Understanding the Design of Physical Networks
Physical networks, such as the human vascular system and brain, differ from typical networks studied in network science because of their tangible nodes and links. These are made of material resources that limit their layout. The importance of these material elements has been recognized in many fields. For instance, early researchers suggested that the laws conserving the volume of the 'wire' could explain the design of neurons. Others applied volume minimization principles to vascular networks, coming up with the branching principles known as Murray's law.
These days, wiring optimization is used to explain the form and layout of a variety of physical systems. These range from the distribution of neuronal branch sizes and lengths, to the structure and flow in transport networks, and even the wiring of the internet.
Optimal Wiring Hypothesis
The optimal wiring hypothesis is the foundation of wiring economy approaches. It envisions physical networks as a set of connected one-dimensional wires, with their total length minimized. The optimal wiring, in this case, is accurately predicted by the Steiner graph. However, the lack of high-quality data on physical networks has limited the systematic testing of the Steiner predictions. At best, evidence of their validity has been mixed.
However, there have been significant improvements in data availability in recent years. Advances in microscopy and three-dimensional reconstruction techniques have given access to the detailed three-dimensional structure of physical networks. This ranges from high-resolution layouts of brain connectomes to vascular networks or the structure of coral trees.
Challenges of the Optimal Wiring Hypothesis
Despite these advancements, there have been systematic deviations from both the Steiner predictions and volume optimization. These deviations are rooted in the hypothesis that estimates the cost of physical networks as the sum of their link lengths or as simple cylinders. The reality is, the links of real physical networks are inherently three-dimensional. This suggests that their true material cost must also consider surface constraints.
Building on previous analyses that introduced volumetric constraints, we successfully account for the local surface morphology. This ensures that, when links intersect, they blend together continuously and smoothly, free of singularities. This is dictated by the physical nature of their material structure.
Unraveling the Secrets of Physical Networks
To achieve this, we map the local tree structure of physical networks into two-dimensional manifolds. This results in a numerically challenging surface and volume minimization problem. However, we discover a formal mapping between surface minimization and high-dimensional Feynman diagrams. This allows us to use a well-developed string-theoretical toolset to predict the basic characteristics of minimal surfaces.
We find that surface minimization can not only account for the empirically observed discrepancies from the Steiner predictions but offers testable predictions on the degree distribution and the angle asymmetry of physical networks. This provides a crucial insight into the design principles of physical networks.
Conclusion
In conclusion, while we see the signature of the Steiner theorem and volume optimization in the prevalence of k = 3 nodes, the optimal wiring hypothesis is unable to account for the existence of k > 3 nodes, the prevalence of non-planar bifurcations, and the lack of θ = 2π/3 symmetry. This challenges the validity of the optimal wiring hypothesis for physical networks.
However, we have discovered a direct equivalence between the network manifold minimization problem and higher-dimensional Feynman diagrams in string theory. This novel approach offers fresh insights into the design principles of physical networks and opens up exciting new avenues for future research.