Dedicated to Professor Emeritus Paul Luzio.
RETICULATIONS
This essay explores the underpinnings of my sculpture called Reticulations. While this piece has a superficial similarity to the Quantum Fluctuations sculpture, the differences are important to note. Quantum Fluctuations are gyrating vortexes which have no directionality and spread in spacetime in all directions. By contrast, Reticulations are unidirectional tubular forms, with anastomoses and interconnections. They function as hydraulic systems in nature, and serve as channels to convey nutrients, energy and information.
This essay is rather technical but it attempts to summarize the science that forms the background to this sculpture. Yes, it is technical. That is how I work. From concepts and science to the visual image.
For me, Reticulations are a fascinating aspect in nature and, the more you look, the more you see. Even when occurring in very diverse phenomena — plant leaves, lightning, river watersheds, blood vessels — their structural similarities suggest that that these are all governed by similar laws and geometric constraints. This is deeply intriguing.
THE BIG PICTURE
“Branching is probably the most common mode of growth in Nature. From plants to river networks, from lung and kidney to snow-flakes or lightning sparks, branches grow and blossom everywhere, in every realm of Nature. When Galileo Galilei stated that the geometry of Nature was written in terms of planes, cones and spheres, he missed one essential pattern of Nature: the tree.” (49)
Consider the following string of ideas:
This is simply a way to reason that the universe is constructed as a whole, not as a disconnected assembly of parts. These nested networks, seen from the smallest quantum level to the structure of galaxies are not tree-like and bifurcating, but reticulate and complex.
In 1635, Juan Eusebio Nieremberg lucidly articulated a similar viewpoint:
“Nature rises up by connections, little by little and without leaps, as though it proceeds by an unbroken web, it proceeds in a leisurely and placid uninterrupted course. There is no gap, no break, no dispersion of forms: they have, in turn, been connected, ring within ring. That very golden chain is universal in its embrace.” (14)
The theme of a deeply interconnected cosmos, where the sum is greater than the parts, and where cumulative complexity leads inevitably to emergent properties unrelated to the component parts is a theme that deeply moves me. It also makes sense at a scientific level.
The emblem, to me, of this interconnectedness of diverse entities is the reticulated form. This is what this sculpture is about.
UBIQUITY OF RETICULATIONS
Reticulations are everywhere. Reticulated networks are employed by nature for a myriad of purposes. These include the transport of rainwater in geologic drainage basins or deltas entering the sea; the distribution of nutrients and wastes in plants; numerous body organs including the liver and blood vessels; and sub-cellular organelles like the Golgi apparatus. These networks appear to have similar rules governing them:
“The ramiform patterns of blood vessels are similar to the branchlike patterns of rivers.” (8).
SELF-ORGANIZING SYSTEMS
Self-organization is a common property of all matter that is prevalent and obvious in living matter. Philip Ball’s book, The Self-Made Tapestry: Pattern Formation in Nature is a treatise on self-organizing patterns, many of which are reticulate networks:
“ 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.” (26)
PHYLOGENIES
We all know the natural forms of tree networks. This is especially true of biological classification systems.
Ernst Haekel’s famous tree of life is an old image dating to Darwin and his overturning of the christian hierarchical Great Chain of Being (47).
But, as biology became more sophisticated through genetics and molecular biology, the tree became replaced by the shrub, or a reticulated network where inheritance was not just vertical but also horizontal through gene transfer from viruses and bacteria:
“The evolutionary relationships among organisms have traditionally been represented using rooted phylogenetic trees. However, due to reticulate processes such as hybridization or lateral gene transfer, evolution cannot always be adequately represented by a phylogenetic tree.” (22)
ROLE OF RETICULATIONS IN MORPHOGENESIS
Living things can be most simply characterized as tubes through which food passes.
Tubes are the foundation of compartments and the separation of reagents which allows the energetic processes of life to exist. In a homogeneous soup with no compartments, life would simply not be possible.
Gradients created by differential charges or concentrations of molecules drive reactions that generate molecular energy. And of course, nature has extensively improvised on the theme of tubes, most dramatically with reticulate structures. These structures are key to development, morphogenesis, growth, and function, across a wide range of phyla and scales (16, 17).
“The functional anatomy of many organ systems relies on networks of tubes. Although their final morphologies are different, the genetic programs activated during branching morphogenesis are surprisingly similar across organs and phyla. At its core, branching morphogenesis is directed by the interplay between an inductive stimulus and its counterbalancing inhibitor. These opposing molecular forces instruct large-scale cellular rearrangements that drive branches forward.” (17)
And like so many cases in the natural world, enormous diversity is generated by relatively simple rules and a small number of building blocks:
“Circulatory, respiratory, and secretory organs are all built from networks of tubes. Whereas tubular organs differ vastly in shape and function, only a handful of strategies to make tubes have been revealed.” (18)
MECHANISM OF BRANCHING
How does branching occur across such diverse systems?
Branching and anastomosis are the key features of reticulate systems.
“An anastomosis is a connection or opening between two things (especially cavities or passages) that are normally diverging or branching, such as between blood vessels, leaf veins, or streams.” (48)
There is quite a body of knowledge regarding the formation of these biological systems; branched ducts can be generated by various different mechanisms including growth, cell rearrangements, contractility, adhesion changes, programmed cell death, and other mechanisms (15).
One mechanism is growth inhomogeneity (i.e. differential growth rates). This is the most important factor for generating complexity in plant morphogenesis. Although we should not forget that programmed cell death also contributes to the formation of tubular structures, such as aerenchyma (soft plant tissue containing air spaces, found especially in many aquatic plants), and tracheary elements, such as xylem tubes for transporting liquids up plant stems (23).
The structural uniformity of many reticulate systems is a combination of genetics and stochastic environmental or local cues. Early branching is pre-programmed genetically, setting the pattern for predictable forms of successive rounds of branching (19). These mature branched structures in 3D can contain 30-40 generations of branching in incredibly complex forms (11). A number of these, such as the liver and many plant organs, can self-heal upon damage.
FRACTAL DISTRIBUTION NETWORKS
Fractals appear in nature with frequency. Dendritic or fractal networks are abundant in nature plants and trees, river drainage basins, human and animals cardiovascular systems, crystals, lightnings and many more (7). They can appear similar at different scales, showing self-similarity with unfolding symmetry. These are effectively ducted distribution networks (13).
A vast survey of both mammalian cardiovascular systems and plant architecture, suggest that the architecture associated with length scaling is guided by the principles of space-filling fractals (4).
Space filling fractals (see lovely video in reference 44) are spontaneous geometric forms which self-assemble into remarkable patterns. Nature uses simple template motifs to generate these beautiful patterns.
The fractal patterns in 3 dimensions are also characterized by a volume-filling Tokunaga fractal tree structure composed of nodes and branches (6). These take the 2D template system into three dimensions.
But how 3D living reticulations like the smooth endoplasmic reticulum of cell Golgi apparatus (2) are constructed — where anastomoses are regulated geometrically without the ability to sense the location of a candidate fusible arm — is a complete mystery. It imputes a keen sense of location in space by the parts of an organism.
The complexity of the branching structure also depends on the frequency of branch generation (i.e., branching times); therefore, to prevent the problem of tangled branches or tubes, branch generation or elongation needs to be restricted. This restriction has effect to maintain the simplicity of branch shape. (23)
The relatively simple template system is also very space-efficient:
“Furthermore, formation of a fractal structure by the scale-down of repeating units enables efficient use of the limited space, thus avoiding branch overlap without interrupting the generation of complexity.” (23)
In many cases these networks are self-similar and exhibit fractal scaling (5, 7, 8).
HIERARCHY OF NETWORKS
These systems of fractal networks can be nested into hierarchies. For instance:
“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.” (25)
Scaling, it seems, is a way that the cosmos is interconnected in non-intuitive ways, in a series of levels:
“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.” (25)
At a macroscopic level, there are fractal network effects:
“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.” (25)
And scale is always a factor:
“Sometimes the branching network is not just a part of the organism, it is the organism, as in many fungal systems.” (12).
TYPES OF NETWORKS
There are at least two distinct types of biological networks (21):
RESILIENCY AND REDUNDANCY
What functions do reticulate networks serve? It appears that conserving space is one function, and a second is ensuring that the reticulate system is resistant to damage and functional collapse.
A sophisticated analysis of the vulnerability of networks of many types (36) led to some very interesting conclusions:
“The internet, social media, biology, geological processes, all employ reticulated net systems of nodes and connections and multiple branch points. These networks are widely employed due to their efficiency and redundancy. The functional efficiency — be it fluid transport or social media — is greater in these complex systems. The redundancy makes these networks considerably more resistant to collapse if just a few nodes are damaged. Plant leaves, for instance, contain looped reticulate systems that are quite resistant to herbivore damage. Similarly, the internet structure is not vulnerable to collapse if a few key nodes are removed.” (36)
Structural resilience can be achieved through simple iterative redundancy.
“Adaptive rules are believed to guide development of these networks and lead to a reticulate, hierarchically nested topology that is both efficient and resilient against perturbations.” (13)
Simple diffusion of nutrients and wastes work in small single cell organisms but larger more complex creatures and plants have evolved greater efficiencies by the network of tubes, whose redundancies provide alternative paths upon damage. (33)
SCALING LAWS
To ensure democratic distribution of resources (energy, metabolites and information) within a complex organism, nature is governed by simple scaling laws (1) that govern fractal branching networks. The predominance of quarter powers scaling laws occur across all scales and all life forms. (25)
These systems follow simple allometric scaling laws to achieve complex outcomes, by self-organization following simple rules. They form hierarchical nested systems.
“Adaptive rules are believed to guide development of these networks and lead to a reticulate, hierarchically nested topology that is both efficient and resilient against perturbations. However, as of yet, no mechanism is known that can generate such networks on all scales.” (8)
In its simplest form, a network is a collection of points, or nodes, joined by lines, or edges. In the case of the internet “the pattern of connections is not a regular one, but neither is it completely random.” It surprisingly obeys a power law . ( 36)
This drives to the very heart of resiliency of many reticulated networks:
“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.” (36)
“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.” (36)
“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.” (36)
These network effects are even important in epidemiology:
“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.” (36)
BRANCHING THEORY
Branching is the basis of a reticulated structure. It is intimately related functionally to efficient space conservation:
“Branching usually arises where there is a reason to maximize the total area of contact between a structure and the environment that surrounds it, particularly where there is also a reason to pack this area into a small volume.” (16)
In one study of the properties of hierarchies of blood vessels (8), it was concluded that recurrence or recursion underpinned self-similarity; that this recursive structure has symmetry; and lastly, the organism tends to imitate or mirror similar structures in the environment.
“It tells you that complex branched epithelial structures develop as a self-organised process, reliant upon a strikingly simple, but generic, rule, without recourse to a rigid, pre-determined sequence of genetically programmed events.” (11)
QUANTUM RETICULATIONS
Since reticulations are so ubiquitous in nature both macroscopically and microscopically, does this hint that they also are pervasive at the quantum level, since the micro informs the macro? Are quantum fluctuations actually a reticulated network of fields as suggested in the famous image?
Quantum fluctuations (45).
In general, quantum physics has not greatly explored the idea of reticulate underpinnings, although the following passage is very interesting:
“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.” (40)
In more exotic phases of matter, there are suggestions that there are reticulate systems:
“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.” (42)
“Dirac particles in transparent quantum graphs: Tunable transport of relativistic quasiparticles in branched structures.” (41)
In astrophysics, the existence of a cosmic web (46) looks remarkably like a reticulated structure:
SUMMARY
Nature is configured on a theme of increasing complexity. Our creation story of the big bang began with absolute density and uniformity, and in the course of nearly 14 billion years developed a dazzling array of astrophysical features and extraordinarily strange phases of matter. Nature also developed a vast diversity of living matter of which we have only catalogued a few percent
At the heart of natural complexity is the development of a diverse toolbox since nature is driven to complexify by hybridization both in the animate and the inanimate spheres. Complex systems require complex structures to effect their functionality, and branching reticulated networks have evolved to meet these needs. In the process, and by their very nature, these networks interconnect the cosmos into one unified structure whose sum is greater than the sum of the parts.
#1. BIOLOGY
(1) Life’s Universal Scaling Laws . Biological systems have evolved branching networks that transport a variety of resources. We argue that common properties of those networks allow for a quantitative theory of the structure, organization, and dynamics of living systems.
The search for universal quantitative laws of biology that supplement or complement the Mendelian laws of inheritance and the principle of natural selection might seem to be a daunting task.
After all, life is the most complex and diverse physical system in the universe, and a systematic science of complexity has yet to be developed. 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.
Throughout those immense ranges, life uses basically the same chemical constituents and reactions to create an amazing variety of forms, processes, and dynamical behaviors. All life functions by transforming energy from physical or chemical sources into organic molecules that are metabolized to build, maintain, and reproduce complex, highly organized systems.
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.
Phase transitions, chaos, the unification of the fundamental forces of nature, and the discovery of quarks are a few of the more significant examples in which scaling has illuminated important universal principles or structure.
In biology, the observed scaling is typically a simple power law: Y = Y 0 M b, where Y is some observable, Y 0 a constant, and M the mass of the organism. 1–3 Perhaps of even greater significance, the exponent b almost invariably approximates a simple multiple of 1/4.
Among the many fundamental variables that obey such scaling laws—termed “allometric” by Julian Huxley 4 —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
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.
Examples include animal circulatory systems, plant vascular systems, and ecosystem and intracellular networks. We proposed that scaling laws and the generic coarse-grained dynamical behavior of biological systems reflect the constraints inherent in universal properties of such networks. These constraints were postulated as follows :
▸Networks service all local biologically active regions in both mature and growing biological systems. Such networks are called space-filling.
▸The networks’ terminal units are invariant within a class or taxon.
▸Organisms evolve toward an optimal state in which the energy required for resource distribution is minimized.
These properties, which characterize an idealized biological organism, are presumed to be consequences of natural selection.
Guided by the three postulates, we and our colleagues built on earlier work to derive analytic models of the mammalian circulatory and respiratory systems and of plant vascular systems. The theory enables one to address the types of questions we raised at the beginning of this section and predicts quarter-power scaling of diverse biological phenomena even though the networks and associated pumps are very different.
https://physicstoday.scitation.org/doi/10.1063/1.1809090
(2) Smooth endoplasmic reticulum: the archetypical reticulate network.
https://www.genome.gov/genetics-glossary/Endoplasmic-Reticulum-Smooth
(3) Branching in Nature. Dynamics and Morphogenesis of Branching Structures, from Cell to River Networks
Branching is probably the most common mode of growth in Nature. From plants to river networks, from lung and kidney to snow-flakes or lightning sparks, branches grow and blossom everywhere, in every realm of Nature. When Galileo Galilei stated that the geometry of Nature was written in terms of planes, cones and spheres, he missed one essential pattern of Nature: the tree.
https://link.springer.com/book/10.1007/978-3-662-06162-6
(4) Branching in vascular networks and in overall organismic form is one of the most common and ancient features of multicellular plants, fungi and animals. By combining machine-learning techniques with new theory that relates vascular form to metabolic function, we enable novel classification of diverse branching networks—mouse lung, human head and torso, angiosperm and gymnosperm plants.
We analyse the largest-ever compilation of branching network data, with over 58 distinct networks and approximately 8000 vessels or tree limbs. We collected these data over the last decade for both mammalian cardiovascular systems and plant architecture in both angiosperms and gymnosperms.
Specifically, our empirical finding of the primacy of information based on scaling ratios of radii strongly suggests that hydrodynamic principles are the primary drivers of vascular branching patterns and overall network form.
The majority of plant networks adhere to area-preservation while exhibiting a greater tendency than mammals to branch asymmetrically. Within the plants we find that differentiation is driven at the species level, unrelated to plant categorization as angiosperm or gymnosperm.
The inability of the length scale factors to inform classification between networks suggests several scenarios. Two contrasting and extreme scenarios are that either a universal architecture or a completely random architecture is being followed by both the mammals and plants [9]. This result is unlike the radial scaling that is strongly coupled to hydraulics. Current theory suggests that the architecture associated with length scaling is guided by the principles of space-filling fractals. However, large deviations are observed between the joint distributions of the length-scale factors and the theoretical curves determined by the space-filling conservation equation (figure 4a). A third scenario is that there exists a disconnect between how length-scale factors are conventionally defined in simplified models versus how they are measured in complicated natural systems.
In contrast to previous theory and importantly for understanding how diverse branching architectures could lead to universal scaling exponents, we find near constancy of the metabolic scaling exponent despite large fluctuations in length scaling.
These predictions are both in agreement with respiration-based studies of mammals [16], and demonstrate the need for theories of metabolic scaling that incorporate the finite size of the network.
Incorporating topological features—connectivity and loops—and branching angles could enhance categorization methods.
https://royalsocietypublishing.org/doi/10.1098/rsif.2020.(5) There are many examples of branching networks in biology. Examples include the structure of the plants and trees as well as cardiovascular and bronchial systems. In many cases these networks are self-similar and exhibit fractal scaling. In this paper we introduce the Tokunaga taxonomy for the side branching of networks and his parameterization of self-similar side-branching. We introduce several examples of deterministic branching networks and show that constructions with the same fractal dimension can have different side-branching parameters.
https://www.sciencedirect.com/science/article/pii/S0022519398907238
(6) An illustration of a volume-filling Tokunaga fractal tree. Nodes and branch construction.
https://www.researchgate.net/publication/11742693_Symmetries_in_geology_and_geophysics/figures?lo=1
(7) Dendritic networks (or fractal networks) are widely occurring in nature. Multiple examples can be found in geomorphology and terrestrial biosphere: plants and trees, river drainage basins, human and animals cardiovascular systems, crystals, lightnings and many more.
https://www.sciencedirect.com/science/article/abs/pii/S0019103518301143
(8) An analogy between the fractal nature of networks of arteries and that of systems of rivers has been drawn in the previous works. However, the deep structure of the hierarchy of blood vessels has not yet been revealed. This paper is devoted to researching the fractals, allometric scaling, and hierarchy of blood vessels.
There is an analogy between the structure of hierarchies of blood vessels of mammals and that of network of rivers and streams. Many studies showed that river and stream networks bear fractal properties.
The ramiform patterns of blood vessels are similar to the branchlike patterns of rivers.
The basic properties of hierarchies of blood vessels can be revealed as follows.
https://arxiv.org/pdf/1511.02276.pdf –> INTERESTING
(9) We find that branch alignment follows a generic scaling law determined by the strength of global guidance, while local interactions influence the tissue density but not its overall territory.
https://www.nature.com/articles/s41467-021-27135-5
(10) Phylogenetic network analysis as a parsimony optimization problem
Many problems in comparative biology are, or are thought to be, best expressed as phylogenetic “networks” as opposed to trees. In trees, vertices may have only a single parent (ancestor), while networks allow for multiple parent vertices.
There are two main interpretive types of networks, “softwired” and “hardwired.” The parsimony cost of hardwired networks is based on all changes over all edges, hence must be greater than or equal to the best tree cost contained (“displayed”) by the network.
This is in contrast to softwired, where each character follows the lowest parsimony cost tree displayed by the network, resulting in costs which are less than or equal to the best display tree. Neither situation is ideal since hard-wired networks are not generally biologically attractive (since individual heritable characters can have more than one parent) and softwired networks can be trivially optimized (containing the best tree for each character).
https://bmcbioinformatics.biomedcentral.com/articles/10.1186/s12859-015-0675-0 –> EXCELENT ANALYSIS
(11) In the centenary year of the publication of a seminal treatise on the physical and mathematical principles underpinning nature – On Growth and Form by D’Arcy Wentworth Thompson – a Cambridge physicist has led a study describing an elegantly simple solution to a puzzle that has taxed biologists for centuries: how complex branching patterns of tissues arise.
“On the surface, the question of how these structures grow – structures that may contain as many as 30 or 40 generations of branching – seems incredibly complex,” says Professor Ben Simons, who led the study, published today in the journal Cell.
Professor Simons holds positions in the University of Cambridge’s Cavendish Laboratory and Wellcome Trust/Cancer Research UK Gurdon Institute.
It tells you that complex branched epithelial structures develop as a self-organised process, reliant upon a strikingly simple, but generic, rule, without recourse to a rigid, pre-determined sequence of genetically programmed events.”
(12) Transport networks are vital components of multicellular organisms, distributing nutrients and removing waste products. Animal cardiovascular and respiratory systems, and plant vasculature are branching trees whose architecture is linked to universal scaling laws in these organisms. ( And also see Archimedes law )
By contrast, it is not clear whether the transport systems of multicellular fungi will fit into this conceptual framework, as they have evolved to explore and exploit a patchy environment rather than ramify through a three-dimensional organism (West et al. 1999a). Unlike all the other biological transport systems studied, the fungal network is not part of the organism, it is the organism.
https://royalsocietypublishing.org/doi/10.1098/rspb.2007.0459
(13) Complex distribution networks are pervasive in biology.
Examples include nutrient transport in the slime mold Physarum polycephalum as well as mammalian and plant venation.
Adaptive rules are believed to guide development of these networks and lead to a reticulate, hierarchically nested topology that is both efficient and resilient against perturbations. However, as of yet, no mechanism is known that can generate such networks on all scales.
https://link.aps.org/doi/10.1103/PhysRevLett.123.248101
(14) Nature rises up by connections, little by little and without leaps, as though it proceeds by an unbroken web, it proceeds in a leisurely and placid uninterrupted course. There is no gap, no break, no dispersion of forms: they have, in turn, been connected, ring within ring. That very golden chain is universal in its embrace. – Juan Eusebio Nieremberg, 1635.
From very early in the Middle Eastern and European religious and intellectual traditions, chains, cords, ladders and stairways served as metaphors for order in the world, or between earth and heaven [2–6]. The image of a tree sometimes served in the same metaphorical sense [[5], pp.319-329; [6], p.22]. A linear order in nature was compatible, for example, with the hierarchical arrangement of creation implied by emanationist cosmology, correspondences between spiritual and earthly bodies, and the literal or figurative ascent of the soul or mind toward God.
“A reticulated tree, or net, which might more appropriately represent life’s history”, by W. Ford Doolittle. Figure reproduced from [171] by permission of the American Association for the Advancement of Science. (Also see https://www.researchgate.net/publication/38092957_Trees_and_networks_before_and_after_Darwin/figures?lo=1)
Network representation of vertical inheritance and lateral exchange among prokaryotes, by Tal Dagan and William Martin. Reproduced from Figure 2(e) of [167] by permission of the Royal Society.
https://link.springer.com/article/10.1186/1745-6150-4-43
(15) Branching morphogenesis is ubiquitous and important in creating bulk transport systems. Branched ducts can be generated by several different mechanisms including growth, cell rearrangements, contractility, adhesion changes, and other mechanisms.
https://www.sciencedirect.com/science/article/abs/pii/S0070215307810088
(16) Branching Morphogenesis edited by Jamie Davies
The development of repeatedly branched structures is an important mechanism of morphogenesis across a wide range of phyla and scales.
Branching usually arises where there is a reason to maximize the total area of contact between a structure and the environment that surrounds it, particularly where there is also a reason to pack this area into a small volume.
https://books.google.com/books/about/Branching_Morphogenesis.html?id=TORHlaSLUAcC
(17) The ramified architectures of organs such as the mammary gland and lung are generated through branching morphogenesis, a developmental process through which individual cells bud and pinch off of pre-existing epithelial sheets.
The formation, branching, and overall architecture of tubular structures are essential for the development, growth, and function of most multicellular organisms. Tubes give rise to several organs and organ systems, including the gut and gastrointestinal tract, the heart and vessels of the circulatory system, and the reproductive organs including the uterus and fallopian tubes. Ducts and vessels serve as the primordial architecture for organs that undergo further morphogenesis and as the fundamental building blocks for other organs including the lung, kidney, and salivary gland. For these ramified organs, proper function is defined by their branched architecture.
The functional anatomy of many organ systems relies on networks of tubes. Although their final morphologies are different, the genetic programs activated during branching morphogenesis are surprisingly similar across organs and phyla. At its core, branching morphogenesis is directed by the interplay between an inductive stimulus and its counterbalancing inhibitor. These opposing molecular forces instruct large-scale cellular rearrangements that drive branches forward.
https://pubmed.ncbi.nlm.nih.gov/22524386/
(18) TISSUE ORIGAMI: SCULPTING A TUBULAR ARCHITECTURE
Circulatory, respiratory, and secretory organs are all built from networks of tubes.
Despite the diversity and complexity of the epithelial tubular structures throughout an organism, the lumen of the mature duct is always defined by polarized cells. Apicobasal polarity is generated and maintained by polarity complexes including PAR, Crumbs, and Scribble (130). These complexes organize and modulate the microtubule cytoskeleton and membrane trafficking, in part, by regulating Rho GTPase through GAPs and GEFs.
Whereas tubular organs differ vastly in shape and function, only a handful of strategies to make tubes have been revealed.
In stereotyped organs such as the vertebrate lung, the iterative use of a few basic branching patterns produces complex self-similar geometries that form patterns with a fractal dimension of 1.6 (y ∝ x1.6).
Cellular contractility regulates tube formation and branching.
http://cmngroup.princeton.edu/56_Gleghorn_ARBME_2012.pdf
(19) Branching morphogenesis involves the restructuring of epithelial tissues into complex and organized ramified tubular networks. Early rounds of branching are controlled genetically in a hardwired fashion in many organs, whereas later, branching is stochastic, responding to environmental cues.
Somewhat as a surprise, it turns out that a significant portion of the branching pattern in many of these organs is controlled genetically in a hardwired fashion, giving rise to successive rounds of branching in a predictable manner.
https://www.sciencedirect.com/science/article/pii/S1534580702004100
(20) Classification, Evolution, and the Nature of Biology
By Alec L. Panchen
It is taken for granted today, at least by zoologists, that systematic classifications of organisms can be represented by branching diagrams (dendrograms) that represent hierarchical arrangements — Darwin’s (1859) “groups within groups”…nested groups of taxa.
Modern biological classification is a process of “ordinally stratified hierarchical clustering ”, and the result is an aggregated hierarchy in which the units, usually species, which constitutes its lowest rank, are aggregated in successively higher ranks. The hierarch is also an inclusive one, as opposed to an exclusive one:
“ Military ranks from private, corporal, sergeant, lieutenant, captain, up to general are a typical example of an exclusive hierarchy. A lower rank is not a subdivision of a higher rank. The scala naturae is another good example of an exclusive hierarchy. Each level of perfection was considered an advance (or degradation) from the next lower (or higher) level in the hierarchy, but did not include it.”
Taxonomic hierarchy is divergent , so that a taxon of a specific rank belongs only to one taxon of higher rank. Also, the hierarchy is usually irregular; it is not expected that the whole will necessarily have a fixed symmetrical pattern. Thus when represented by a dendrogram, the treelike pattern will not have branches that rejoin after separating, and the tree will also be irregular.
https://books.google.com/books/about/Classification_Evolution_and_the_Nature.html?id=r-6SFNkjqKAC
Excellent book
(21) Network analysis is becoming the core methodology to treat complex biological systems. Complex networks are found in all parts of biology, but there are at least two distinct types of biological networks.
(A) The most common type the nodes and edges are empirically observed, and the network analysis involves summarizing the characteristics of the network.
(B) In the second type, only the leaf nodes are observed, And the the internal nodes and all of the edges must be inferred from information available about the leaf nodes.
Perhaps the most widespread of this inferred type of network is the phylogenetic network, which illustrates the genealogical history connecting all of life. Evolution involves a series of unobservable historical events, each of which is unique, and we can neither make direct observations of them, nor perform experiments to investigate them…there is no mathematical algorithm for discovering unique historical accidents.
https://www.researchgate.net/profile/Wenjun
(22) The evolutionary relationships among organisms have traditionally been represented using rooted phylogenetic trees.
However, due to reticulate processes such as hybridization or lateral gene transfer, evolution cannot always be adequately represented by a phylogenetic tree, and rooted phylogenetic networks that describe such complex processes have been introduced as a generalization of rooted phylogenetic trees.
In fact, estimating rooted phylogenetic networks from genomic sequence data and analyzing their structural properties is one of the most important tasks in contemporary phylogenetics.
https://link.springer.com/article/10.1007/s00285-022-01746-y
(23) Theoretical models for branch formation in plants
Growth inhomogeneity (i.e., differential growth rates) is the most important factor for generating complexity in plant morphogenesis, although programmed cell death also contributes to the formation of tubular structures, such as aerenchyma and tracheary elements.
Branches are initiated through growth inhomogeneity, and subsequently some branches elongate after their generation.
Mechanical stresses and their feedback via the cytoskeleton are considered to be one factor that controls the growth inhomogeneity in plants: inhomogeneities occur against regular boundaries, although the inhomogeneities are disorganized.
Molecular-based positional information, especially existing as periodic patterns, is important for plant branch formation. As suggested by Turing (1952), the mechanisms underlying the formation of such patterns can be explained by reaction–diffusion (RD) systems. On the cellular level, ROP proteins are considered to be crucial factors that provide the positional information observed in root hairs.
The complexity of branching depends on the frequency of branch generation (i.e., branching times); therefore, to prevent tangled branches, branch generation or subsequent elongation needs to be restricted. This restriction has effect to maintain the simplicity of branch shape.
Furthermore, formation of a fractal structure by the scale-down of repeating units enables efficient use of the limited space, thus avoiding branch overlap without interrupting the generation of complexity.
Diversity of branches. a–d Development of diversity in divarication: from left to right, disk-like architectures grown with equally spaced periodic patterns, as described by Harrison and Kolář (1988), Holloway and Harrison (1999), and Nakamasu et al. (2014). Branches gradually develop during growth processes. a No side branches, b, c bifurcation, and d monopodial branching. As branch development proceeds, the branches tend to overlap. c, d Scale-downs of iteratively added units to avoid collision is included. e–j Representations of three-dimensional branching with particular branching rules were generated based on Honda’s I-model (Borchert and Honda 1984); e, h bifurcation; divergent angle is 90° and branch angle of two daughter branches is 45°; f, i alternate phyllotaxis; divergent angle is 137.5° and branch angle is 45°; and g, j opposite phyllotaxis; branch angles of two lateral branches are 45° and divergent angle is 90°. e–g Branch lengths are the same for the whole tree. h–j Branch lengths decrease dependent on the branch hierarchies with ratio 0.8.
The two-dimensional branch architectures termed divarications are often observed in leaves. Such divaricated leaves are categorized as serrations, lobes, and leaflets mainly according to their degree of protrusion.
https://link.springer.com/article/10.1007/s10265-019-01107-9 –> Excellent review
(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.
(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
(45) Quantum fluctuations video, by “Derek Leinweber”. www.physics.adelaide.edu.au.
https://en.m.wikipedia.org/wiki/Quantum_fluctuation
(46) Cosmic web.
https://www.scientificamerican.com/article/canadian-telescope-delivers-deepest-ever-radio-view-of-cosmic-web/#
(47) The Great Chain of Being.
https://en.m.wikipedia.org/wiki/Great_chain_of_being
(48) Anastamosis.
https://en.m.wikipedia.org/wiki/Anastomosis
(49) https://link.springer.com/