Reticulations

(24) Modeling of trees

Meinhardt [97, Chapter 15] substituted the triangular grid with a square one, and used the resulting cellular space to examine biological hypotheses related to the formation of net-like structures.

In addition to pure branching patterns, his models capture the effect of branch reconnection or anastomosis that may take place between the veins of a lea f.

Greene [54] extended cellular automata to three dimensions, and applied the resulting voxel space automata to simulate growth processes that react to the environment.

For instance, Figure 2.1 presents the growth of a vine over a house. Cohen [15] simulated the development of a branching pattern using expansion rules that operate in a continuous “density field” rather than a discrete cellular or voxel space.

http://algorithmicbotany.org/papers/abop/abop-ch2.pdf

(25) Do biological phenomena obey underlying universal laws of life that can be mathematized so that biology can be formulated as a predictive, quantitative science?

Most would regard it as unlikely that scientists will ever discover “Newton’s laws of biology” that could lead to precise calculations of detailed biological phenomena. Indeed, one could convincingly argue that the extraordinary complexity of most biological systems precludes such a possibility.

The life process covers more than 27 orders of magnitude in mass—from molecules of the genetic code and metabolic machinery to whales and sequoias—and the metabolic power required to support life across that range spans over 21 orders of magnitude.

In marked contrast to the amazing diversity and complexity of living organisms is the remarkable simplicity of the scaling behavior of key biological processes over a broad spectrum of phenomena and an immense range of energy and mass. Scaling as a manifestation of underlying dynamics and geometry is familiar throughout physics.

In biology, the observed scaling is typically a simple power law: Y = Y(0) M(b), where Y is some observable, Y a constant, and M (b)the mass of the organism.

Perhaps of even greater significance, the exponent b almost invariably ap- proximates a simple multiple of 1/4.

Among the many fundamental variables that obey such scaling laws—termed “allometric” by Julian Huxley — are metabolic rate, life span, growth rate, heart rate, DNA nucleotide substitution rate, lengths of aortas and genomes, tree height, mass of cerebral grey matter, density of mitochondria, and concentration of RNA.

An intriguing consequence of these “quarter-power” scaling laws is the emergence of invariant quantities, which physicists recognize as usually reflecting fundamental underlying constraints.

…convincingly showed the predominance of quarter powers across all scales and life forms.

The starting point was to recognize that highly complex, self-sustaining, reproducing, living structures require close integration of enormous numbers of localized microscopic units that need to be serviced in an approximately “democratic” and efficient fashion. To solve that challenge, natural selection evolved hierarchical fractal-like branching networks that distribute energy, metabolites, and information from macroscopic reservoirs to microscopic sites.

We proposed that scaling laws and the generic coarse-grained dynamical behavior of biologi- cal systems reflect the constraints inherent in universal properties of such networks. These constraints were pos- tulated as follows:

1. Networks service all local biologically active regions in both mature and growing biological systems. Such networks are called space-filling.

2. The networks’ terminal units are invariant within a class or taxon.

3. Organisms evolve toward an optimal state in which the energy required for resource distribution is minimized.

Metabolism is organized at a number of levels, and at each level new structures emerge. The result is a hierarchy of networks, each with different physical characteristics and effective degrees of freedom. Yet metabolic rate continues to obey 3/4-power scaling. That invariance is in contrast to the analogous situation in physics.

Scaling, as manifested in structure functions or phase transitions, for example, persists from quarks through hadrons, atoms, and ultimately to matter. Yet no continuous universal behavior emerges: Each level manifests different scaling laws.

The calculations that yield quarter-power scaling depend only on generic network properties. The observation of such scaling at intracellular levels therefore suggests that subcellular structure and dynamics are constrained by optimized space-filling, hierarchical networks. A major challenge, both theoretically and experimentally, is to understand quantitatively the nature and structure of intracellular pathways, about which surprisingly little is known.

Growth and life-history events are, in general, universal phenomena governed primarily by basic cellular properties and quarter-power scaling.

One can derive many scaling relationships within and between plants, including those for conductivity, fluid velocity, and, as first observed by Leonardo da Vinci, area-preserving branching.

Steady-state forest ecosystems, too, can be treated as integrated networks satisfying appropriate constraints. The network elements are not connected physically, but rather by the resources they use. Scaling in the forest as a whole mimics that in individual trees. So, for example, the number of trees as a function of trunk diameter scales just like the number of branches in an individual tree as a function of branch diameter. As figure 6 shows, both scalings are described by predicted inverse-square laws.

Why does the theory work so well? Does some fixed point or deep basin of attraction in the dy- namics of natural selection ensure that all life is organized by a few fundamental principles and that energy is a prime determinant of biological structure and dynamics among all possible variables?

https://jdyeakel.github.io/teaching/ecology/papers/West_Brown_2004.pdf

(26) The Self-Made Tapestry: Pattern Formation in Nature, by Philip Ball.

In the trellis-like shells of microscopic sea creatures we see the same geometry as in the bubble walls of foam, Forks of lightning mirror the branches of a river network or a tree.

This book explains why these are not coincidences. Nature commonly weaves its tapestry without any master plan or blueprint. Instead, these designs build themselves by self-organization.

https://dl.acm.org/doi/abs/10.5555/601160

(27) It is widely assumed that the micro level is causally complete, thus excluding causation at the macro level. However, by measuring effective information—how much a system’s mechanisms constrain its past and future states—we recently showed that causal power can be stronger at macro rather than micro levels.

At a theoretical level, this reductionist “micro” bias is based on the following assumptions (Kim, 1993): (i) once the properties of micro-level physical mechanisms are fixed, macro-level properties are fixed too (supervenience); (ii) causal power resides fully at the microphysical level (micro causal closure); and (iii) if all the causal work is done at the micro level, there is no room for any causal contribution at the macro level (macro causal exclusion). This view of causal power denies the possibility of genuine causal emergence.

https://academic.oup.com/nc/article/2016/1/niw012/2757132

(28) BIFURCATION THEORY

Bifurcation theory is the mathematical study of changes in the qualitative or topological structure of a given family of curves, such as the integral curves of a family of vector fields, and the solutions of a family of differential equations.

a bifurcation occurs when a small smooth change made to the parameter values (the bifurcation parameters) of a system causes a sudden ‘qualitative’ or topological change in its behavior.

Bifurcation types[edit]
It is useful to divide bifurcations into two principal classes:

(1) Local bifurcations, which can be analysed entirely through changes in the local stability properties of equilibria, periodic orbits or other invariant sets as parameters cross through critical thresholds; and

(2) Global bifurcations, which often occur when larger invariant sets of the system ‘collide’ with each other, or with equilibria of the system. They cannot be detected purely by a stability analysis of the equilibria (fixed points).

https://en.wikipedia.org/wiki/Bifurcation_theory

(29) Most phylogenies are typically represented as purely bifurcating. However, as genomic data have become more common in phylogenetic studies, it is not unusual to find reticulation among terminal lineages or among internal nodes (deep time reticulation; DTR).

https://academic.oup.com/sysbio/article/67/5/743/4925328

(30) In multicellular organisms it is of crucial importance to distribute the nutrients and oxygen each cell needs to survive. This can be achieved by simple diffusion when the organism is small and/or thin and the resources are easily accessible (for instance algae). However, for many terrestrial plants this can be achieved more efficiently by using a network of tubes. In leaves of higher plants very diverse shapes of vascular systems are observed ranging from long parallel major veins interconnected by short secondary veins in monocotyledons, through open branching in ferns, to reticulated networks in the dicotyledons.

Reticulated networks may have a robustness advantage against damage due to their redundancies —even when physical damage occurs, fluids can use a different path so the whole leaf is still well supplied (Sack et al., 2003). This gives the leaves more resistance to herbivores and pathogens. The vascular system is formed by a self-assembly process involving complex interactions which are still not well understood.

https://www.sciencedirect.com/science/article/abs/pii/S0022519306002207

(31) Network (Reticulate) Evolution: Biology, Models and Algorithms

Phylogenies are the main tool for representing evolutionary relationships among biological entities at the level of species and above. Because in nearly all cases of biological evolution it is impossible to witness the history of speciation events, biologists, mathematicians, statisticians, and computer scientists have designed a variety of methods for the reconstruction of these events, with the usual model being phylogenetic trees.

There is growing appreciation that phylogeny is an indispensable interpretive framework for studying evolutionary processes, and indeed is central to the organization and interpretation of information on all characteristics of organisms, from structure and physiology to genomics.

For the most part, these methods assume that the phylogeny underlying the data is a tree. However, such is not always the case: for many organisms, a significant level of genetic exchange occurs between lineages, and for some groups, lineages can combine to produce new independent lineages.

These exchanges and combinations transform a tree into a network; the history of life cannot properly be represented as a tree.

Indeed, events such as meiotic and sexual recombination, horizontal gene transfer and hybrid speciation cannot be modeled by bifurcating trees. Meiotic recombination occurs in every generation at the level of individual chromosomes; sexual recombination commonly acts at the population level and recombines the evolutionary histories of genomes. Hybrid speciation is very common in some very large groups of organisms: plants, fish, frogs, and many lineages of invertebrates, and horizontal gene transfer is ubiquitous in bacteria. Although the mixing (reticulation) of evolutionary histories has long been widely appreciated and acknowledged, there has been comparatively little work on computational methods for studying and estimating reticulate evolution, especially at the species level.

https://www.researchgate.net/publication/37441563

(32) Most phylogenies are typically represented as purely bifurcating. However, as genomic data have become more common in phylogenetic studies, it is not unusual to find reticulation among terminal lineages or among internal nodes (deep time reticulation; DTR).

https://academic.oup.com/sysbio/article/67/5/743/4925328

(33) CANALIZATION OF RETICULATED LEAF VEINS

Closed vein loops are commonly observed in plant leaf veins making more resistant to physical damage.

Simple diffusion of nutrients works for small organisms but many plants achieve this more efficiently by using a network of tubes. Reticulated networks have a robust advantage against damage due to their redundancies, so fluids have many alternate paths.

https://academia.edu/resource/work/56656855 –> GOOD MODELING

(34) Reticulate, or non-bifurcating, evolution is now recognized as an important phenomenon shaping the histories of many organisms.

https://givnishlab.botany.wisc.edu/Welcome_files/Sessa%20et%20al.%202012%20MPE.pdf

(35) The traditional view is that species and their genomes evolve only by vertical descent, leading to evolutionary histories that can be represented by bifurcating lineages. However, modern evolutionary thinking recognizes processes of reticulate evolution, such as horizontal gene transfer or hybridization, which involve total or partial merging of genetic material from two diverged species. Today it is widely recognized that such events are rampant in prokaryotes, but a relevant role in eukaryotes has only recently been acknowledged.

https://www.bsc.es/research-and-development/projects/retvolution-reticulate-evolution-patterns-and-impacts-non-vertical

(36) The physics of networks

Statistical analysis of interconnected groups—of computers, animals, or people—yields important clues about how they function and even offers predictions of their future behavior.

In its simplest form, a network is a collection of points, or nodes, joined by lines, or edges.

Figure 1 shows a representation of the internet, the worldwide network of physical data connections between computers. The nodes in the figure represent groups of computers, and the edges represent data connections between those groups.

From looking at the network, it is evident that although the pattern of connections is not a regular one, neither is it completely random.

The network has clear structure, including the prominent starlike formations at the center and the more filamentary connections around the edges.

Figure 2. The distribution of the degrees of nodes on the internet. As indicated, the distribution roughly follows a straight line on a logarithmic plot; that is, it obeys a power law.

When first discovered, the power-law distribution was a surprise to many researchers.

Subsequent studies by various researchers have shown that many other networks, though not always following the power-law pattern precisely, do tend to have skewed degree distributions with a lot of low-degree nodes and a small number of high-degree hubs.

But real-world networks are not random. Many have highly skewed degree distributions, and for them the difference between random and targeted removal of nodes is striking. If you remove nodes purely at random, then most of them have low degree, since most of the nodes in the network have low degree. Thus the removed nodes are connected to few others and have little effect when they are removed.

One of the more remarkable theoretical results to emerge in recent years is that if nodes are removed uniformly at random from a network with a power-law degree distribution, then the network typically remains connected and functional no matter how many nodes are removed.

Of course, the nodes that are removed are themselves no longer connected to the network, but of the nodes left behind, an extensive fraction remain connected. Scale-free networks are thus extremely resilient against random removal or failure of their nodes. The same is not true of purely random networks.

Figure 3. (a) Random removal of nodes from a scale-free network typically has little effect on the overall connectivity. (b) Targeting the highest-degree nodes can have a devastating effect.

For the case of targeted removal of the highest-degree nodes, on the other hand, the reverse is true. The high-degree nodes in a scale-free network are hubs with connections to many other nodes, so the removal of just a few of them can have a substantial effect, as shown in figure 3(b). Analytic calculations, for instance, indicate that no matter what the exponent of the power law, no more than 3% of nodes need to be removed before the entire network becomes disconnected, meaning that the average probability that there is a path connecting any two nodes vanishes.

In the context of a communication network such as the internet, that kind of fragility to a targeted attack could be a bad thing: It’s certainly not desirable for critical infrastructure to be susceptible to failure of, or attacks on, just a few central hubs. In other domains, however, fragility can be good. One reason for the current high level of interest in social networks is their importance in the spread of disease. Diseases travel over networks of contact between individuals just as information travels over the internet, and nodes can be “removed” from those networks by vaccination, assuming a vaccine is available for the disease in question. If vaccination procedures could be targeted toward the highest-degree nodes in a social network, it might in theory be possible to disconnect the network and thus prevent the spread of disease while vaccinating only the tiniest fraction of the population.

It is clear that the degrees of network nodes must play some role in the spread of disease—or other disease-like elements, such as rumors or fads, that pass over networks of contacts between individuals. If no one in a network has any connections, for instance, then diseases of course cannot spread. If everyone has many connections, diseases can spread quickly. One might guess that the speed at which a disease spreads over a network would be determined by the mean degree of a node. Although that is the right basic idea, it turns out to be wrong in detail: The crucial parameter is not the mean degree but the mean squared degree.

This result is again particularly important when applied to scale-free networks. For a power-law degree distribution with an exponent less than 3, the mean squared degree formally diverges, and with it the rate of growth of a disease epidemic. (In a network of finite size, it does not actually diverge, but it does become very large.)

there is little doubt that the existence of network hubs—or “superspreaders,” as they are sometimes called in epidemiology—plays a big role in determining whether and how fast diseases spread.

Perhaps the best known discovery in the study of networks is the so-called small-world effect, the finding that most pairs of people, no matter how distant they may be, are connected by at least one and probably many short chains of acquaintances, (“the six degrees of separation”)

The small-world effect is not confined to social networks and seems to apply to almost all kinds of networks. There are exceptions—networks with special regularities such as low-dimensional networks or lattices, for example—but all the networks mentioned in this article seem to be small worlds.

The fundamental explanation is that the number of people you can reach by taking a given number of steps in your social network increases exponentially with the number of steps you take—a result known as the expander property—so the number of steps needed to reach anyone in the world increases only logarithmically with world population. Since the logarithm is a slowly increasing function, the typical number of steps between any two people in the world is relatively small, even though the population numbers in the billions. The exponential behavior has been confirmed empirically for a wide variety of networks and appears well established.

However, another aspect of the small-world effect really is surprising. Milgram’s experiment reveals not only the existence of short paths between pairs of individuals in social networks but also that people are good at finding those paths.

Traditional Web searches are performed by crawling the Web on a huge scale for information on every conceivable topic, indexing it all, and then searching that index to locate webpages of interest. Spiders take a more customized approach, performing a limited perusal of the network for information on just one topic.

Even after you allow for skewed degree distributions, the connections in typical networks are, unsurprisingly, far from random.

structural differences can have substantial effects on the way a networked system behaves. A disease, for instance, can persist more easily in a positively correlated network by circulating in the dense core where there are many opportunities for it to spread. On the other hand, the below-average density of the periphery makes it harder for the disease to leave the core. In a negatively correlated network, the same disease finds it harder to persist, but if it does persist, then it typically spreads to the whole network.

https://physicstoday.scitation.org/doi/10.1063/1.3027989 –> OUTSTANDING

(37) We investigate tree tensor network states for quantum chemistry. Tree tensor network states represent one of the simplest generalizations of matrix product states and the density matrix renormalization group.

While matrix product states encode a one-dimensional entanglement structure, tree tensor network states encode a tree entanglement structure, allowing for a more flexible description of general molecules.

We describe an optimal tree tensor network state algorithm for quantum chemistry. We introduce the concept of half-renormalization which greatly improves the efficiency of the calculations. Using our efficient formulation we demonstrate the strengths and weaknesses of tree tensor network states versus matrix product states. We carry out benchmark calculations both on tree systems (hydrogen trees and π-conjugated dendrimers) as well as non-tree molecules (hydrogen chains, nitrogen dimer, and chromium dimer).

In general, tree tensor network states require much fewer renormalized states to achieve the same accuracy as matrix product states. In non-tree molecules, whether this translates into a computational savings is system dependent, due to the higher prefactor and computational scaling associated with tree algorithms.

In tree like molecules, tree network states are easily superior to matrix product states. As an illustration, our largest dendrimer calculation with tree tensor network states correlates 110 electrons in 110 active orbitals.

https://www.science.gov/topicpages/t/tree+tensor+networks.html

(38) We present a tree-tensor-network-based method to study strongly correlated systems with nonlocal interactions in higher dimensions.

https://link.aps.org/doi/10.1103/PhysRevB.82.205105

(39) Originally developed in the context of condensed-matter physics and based on renormalization group ideas, tensor networks have been revived thanks to quantum information theory and the progress in understanding the role of entanglement in quantum many-body systems. Moreover, tensor network states have turned out to play a key role in other scientific disciplines.

https://www.nature.com/articles/s42254-019-0086-7

(40) Partition function and space-filling fractal-like networks of branching tubes.

We may think of the probability in quantum mechanics as a sort of fluid that flows from one point to another continuously and without loss or gain. We will utilize this fluid idea and imagine that probability flows through a space-filling fractal-like networks of branching tubes similar to the networks of a general model for the origin of allometric scaling laws in biology.

In the general model, scaling laws arise from the interplay between physical and geometric constraints. This model provides a complete analysis of scaling relations for mammalian circulatory systems that are in agreement with data. We will show that there is a connection between a quantum system in thermal equilibrium and space-filling fractal-like networks. The relationship will be revealed through the calculation of the total fluid (probability) network volume. We will show that this total volume is proportional to the partition function of the related quantum system.

http://ui.adsabs.harvard.edu/abs/2009APS..MARW15012L/abstract

(41) Dirac particles in transparent quantum graphs: Tunable transport of relativistic quasiparticles in branched structures.

https://link.aps.org/doi/10.1103/PhysRevE.101.062208

(42) String-net condensation: A physical mechanism for topological phases.

We show that quantum systems of extended objects naturally give rise to a large class of exotic phases—namely topological phases. These phases occur when extended objects, called “string-nets,” become highly fluctuating and condense. We construct a large class of exactly soluble 2D spin Hamiltonians whose ground states are string-net condensed. Each ground state corresponds to a different parity invariant topological phase. The models reveal the mathematical framework underlying topological phases: tensor category theory. One of the Hamiltonians—a spin- 1∕2 system on the honeycomb lattice—is a simple theoretical realization of a universal fault tolerant quantum computer. The higher dimensional case also yields an interesting result: we find that 3D string-net condensation naturally gives rise to both emergent gauge bosons and emergent fermions. Thus, string-net condensation provides a mechanism for unifying gauge bosons and fermions in 3 and higher dimensions.

https://link.aps.org/doi/10.1103/PhysRevB.71.04511

(43) Tree of life.
https://www.nature.com/articles/540038a

(44) Space filling fractals video.
https://youtu.be/wZVLsJY3MUw