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IV. Ecosmomics: An Independent Source Script of Generative, Self-Similar, Complex Network Systems

1. Network Physics: A Vital Anatomy and Physiology

Mokhlissi, Raihana, et al. The Structural Properties and Spanning Trees Entropy of the Generalized Fractal Scale-Free Lattice. Journal of Complex Networks. Online August, 2019. RM, Dounia Lotfi, and Mohamed El Marraki, Mohammed V University, Rabat, Morocco and Joyati Debnath, Winona State University, USA mathematicians post a sophisticated description of nature’s innate geometries. While invisible, their linkages are truly present as they unite and vivify all the overt objects and entities.

Enumerating all the spanning trees of a complex network is theoretical defiance for mathematicians, electrical engineers and computer scientists. In this article, we propose a generalization of the Fractal Scale-Free Lattice and study its structural properties. As its degree distribution follows a power law, we prove that the proposed generalization does not affect the scale-free property. In addition, we use equivalent transformations to count the number of spanning trees in the generalized Fractal Scale-Free Lattice. Finally, in order to evaluate the robustness of the generalized lattice, we compute and compare its entropy with other complex networks. (Abstract)

Dr. Joyati Debnath is a Full Professor of Mathematics and Statistics at Winona State University. She received an M. S. in Pure Mathematics and Ph. D. in Applied Mathematics from Iowa State University. She received numerous Honors and Awards including the Best Teaching Award from Iowa State University, and the Outstanding Woman of Education Award. Dr. Debnath has research interest in the areas of Topological Graph Theory, Integral Transform Theory, Partial Differential Equations and Boundary Value Problem, Associations of Variables, Discrete Mathematics, and Software Engineering Metrics. (WSU page)

Molkenthin, Nora and Marc Timme. Scaling Laws in Spatial Network Formation. Physical Review Letters. 117/168301, 2016. While many papers nowadays report a specific instance of complex systems (galaxies, brains, language), herein MPI Network Dynamics and Self-Organization physicists distill an independent, universally recurrent constancy which can be seen to take on self-similar forms and flows everywhere.

Geometric constraints impact the formation of a broad range of spatial networks, from amino acid chains folding to proteins structures to rearranging particle aggregates. How the network of interactions dynamically self-organizes in such systems is far from fully understood. Here, we analyze a class of spatial network formation processes by introducing a mapping from geometric to graph-theoretic constraints. Combining stochastic and mean field analyses yields an algebraic scaling law for the extent (graph diameter) of the resulting networks with system size, in contrast to logarithmic scaling known for networks without constraints. (Abstract)

Taken together, we uncovered an algebraic scaling law for network formation processes under geometric constraints. We have analyzed a spatial network formation model by mapping geometric constraints in space to purely graph-theoretical constraints on the topological changes of a network. Direct numerical simulations as well as analytic mean field calculations strongly indicate a scaling law with the graph diameter growing algebraically with system size, representing spatially self-similar (`fractal') networks. This algebraic law scaling is largely independent of the details of the model setup and clearly induced by geometric constraints. More generally, our results may suggest that geometric constraints generically induce algebraic (rather than logarithmic) scaling laws of networks in space. (4-5)

Moreno, Yamir and Matjaz Perc, eds. Focus on Multileyer Networks. New Journal of Physics. Circa 2018,, 2019. University of Zaragoza, Spain and University of Maribor, Slovenia physicists open a special collection with this title, as the quote notes. We note, for example, Inter-Layer Competition in Adaptive Multiplex Network by Elena Pitsik (20/075004) and Communicability Geometry of Multiplexes by Ernesto Estrada (21/015004, 2019).

In the later past century and early 2000's, the availability of data about real-world systems made it possible to study the topology of large networks. This work has revealed the structure, dynamics and functions of complex networks, as well as new models for synthetic networks. During the last 5 years, also backed up by new results, scientists have realized that many systems and processes cannot be described with single-layer nets since they have a multilayer geometry made up of many layers. The study of these multiplex networks has pointed out that their structure, dynamics, and evolution exhibit non-trivial relationships and interdependencies that give rise to new phenomena. (Scope)

Motter, Adilson and Yang Yang. The Unfolding and Control Network Cascades. Physics Today. January, 2017. A Northwestern University astrophysicist and a biochemical engineer initially record how these active interconnective topologies have now been found to occur throughout nature and society. As a result, generic, universally applicable structures and dynamics, e.g. neural net behavior, can be distilled. Because they are so pervasive and important the paper proposes novel ways for their salutary management. See concurrently Understanding the Controllability of Complex networks from the Microcosmic to the Macrocosmic by Peng Sun and Xiaoke Ma in the New Journal of Physics (19/013022, 2017), and Networks in Motion by Motter and Reka Albert in Physics Today (April 2012).

A characteristic property of networks is their ability to propagate influences, such as infectious diseases, behavioral changes, and failures. An especially important class of such contagious dynamics is that of cascading processes. These processes include, for example, cascading failures in infrastructure systems, extinctions cascades in ecological networks, and information cascades in social systems. In this review, we discuss recent progress and challenges associated with the modeling, prediction, detection, and control of cascades in networks.

Mulder, Daan and Ginestra Bianconi. Network Geometry and Complexity. arXiv:1711.06290. Queen Mary University of London mathematicians continue their project (search GB) to express a common, independent network form and function in a way that can be given a nonlinear, nonequilibrium physical basis. See also Non-Euclidean Geometry in Nature by Sergei Nechaev at 1705.08013.

Recently higher order networks describing the interactions between two or more nodes are attracting large attention. Most notably higher order networks include simplicial complexes formed not only by nodes and links but also by triangles, tetrahedra, etc. glued along their faces. Simplicial complexes and in general higher order networks are able to characterize data as different as functional brain networks or collaboration networks beyond the framework of pairwise interactions. Interestingly higher order networks have a natural geometric interpretation and therefore constitute the natural way to explore the discrete network geometry of complex networks. Here we investigate the rich interplay between emergent network geometry of higher order networks and their complexity in the framework of a non-equilibrium model called Network Geometry with Flavor. (Abstract excerpt)

Newman, Mark. Networks: An Introduction. New York: Oxford University Press, 2018. In this second edition, the University of Michigan systems theorist provides a further essential guide to nature’s ubiquitous propensity for relational nodes and links in dynamic communities from life’s origin, genomics and proteomics, to local and global environments.

The study of networks, including computer, social , and biological networks, has attracted enormous interest in the last few years. The study of networks is broadly interdisciplinary and central developments have occurred in many fields, including mathematics, physics, computer and information sciences, biology, and the social sciences. Topics covered include the measurement of networks; methods for analyzing network data, including methods developed in physics, statistics, and sociology; fundamentals of graph theory; computer algorithms; mathematical models of networks, including random graph models and generative models; and theories of dynamical processes taking place on networks.

Newman, Mark. The Physics of Networks. Physics Today. November, 2008. The Paul Dirac Collegiate Professor of Physics at the University of Michigan provides a tutorial on the ubiquitous mathematics of interconnected groups, characterized by arrays of nodes and edges or links. See also the extensive volume: The Structure and Dynamics of Networks edited by Newman, et al (Princeton UP, 2006).

The observation of a power-law distribution thus indicates that the placement of edges in the network is, in a sense, far from being random. (34)

Nicolaides, Christos, et al. Self-Organization of Network Dynamics into Local Quantized States. arXiv:1509.05243. We cite this posting by MIT and Technical University of Madrid scientists as a 2015 representation of a generic nonlinear complex system via a reciprocity of unitary agents and informed relations. As the paper alludes, its independent, universal presence can be identified across natural and social realms, which can then provide a guide for future designs.

Self-organization and pattern formation in network-organized systems emerges from the collective activation and interaction of many interconnected units. A striking feature of these non-equilibrium structures is that they are often localized and robust: only a small subset of the nodes, or cell assembly, is activated. Understanding the role of cell assemblies as basic functional units in neural networks and socio-technical systems emerges as a fundamental challenge in network theory. A key open question is how these elementary building blocks emerge, and how they operate, linking structure and function in complex networks. Here we show that a network analogue of the Swift-Hohenberg continuum model---a minimal-ingredients model of nodal activation and interaction within a complex network---is able to produce a complex suite of localized patterns. Hence, the spontaneous formation of robust operational cell assemblies in complex networks can be explained as the result of self-organization, even in the absence of synaptic reinforcements. Our results show that these self-organized, local structures can provide robust functional units to understand natural and socio-technical network-organized processes. (Abstract)

Pajevic, Sinisa and Dietmar Plenz. The Organization of Strong Links in Complex Networks. Nature Physics. Online March, 2012. As a good example of the nascent advance to detect such deep similarities, National Institute of Health systems theorists find the same dynamics and topologies to hold for genomic, neuronal, social webs, linguistic, vehicle transport, and scientific collaborations. In so doing, a notable common quality is suggested. The nodal components of each domain - neurons, truck drivers, word usage – engage in a “local learning” from which arises an “integrative weight organization.” Once again, this grand natural reciprocity of entity and whole, self and group, accrues everywhere, to the benefit of both phases.

Many complex systems reveal a small-world topology, which allows simultaneously local and global efficiency in the interaction between system constituents. Here, we report the results of a comprehensive study that investigates the relation between the clustering properties in such small-world systems and the strength of interactions between its constituents, quantified by the link weight. For brain, gene, social and language networks, we find a local integrative weight organization in which strong links preferentially occur between nodes with overlapping neighbourhoods.. Our findings identify a general organization for complex systems that strikes a balance between efficient local and global communication in their strong interactions, while allowing for robust, exploratory development of weak interactions. (Abstract, 429) The predominance of integrative weight organization in natural, complex networks seems to reflect a general local weighting principle that results in networks which maintain robust functionality and efficient communication while adapting their weights to changing environments. (435)

Papadopoulos, Lia, et al. Network Analysis of Particles and Grains. arXiv:1708.08080. We cite this entry as another example of nature’s innate propensity to form an anatomy and physiology of multi-connective webs everywhere. It is also notable because coauthors Karen Daniels, Mason Porter and Danielle Bassett achieve this through creative studies and applications of neural network architectures and performance.

The arrangements of particles and forces in granular materials and particulate matter have a complex organization on multiple spatial scales that range from local structures to mesoscale and system-wide ones. The theoretical study of particle-level, force-chain, domain, and bulk properties requires the development and application of appropriate mathematical, statistical, physical, and computational frameworks. Recently, tools from network science have emerged as powerful approaches for probing and characterizing heterogeneous architectures in complex systems, and a diverse set of methods have yielded fascinating insights into granular materials. In this paper, we review work on network-based approaches to studying granular materials and explore the potential of such frameworks to provide a useful description of these materials and to enhance understanding of the underlying physics. (Abstract)

Perc, Matjaz. Diffusion Dynamics and Information Spreading in Multilayer Networks: An Overview. European Physical Journal Special Topics. 228/2351, 2019. The University of Maribor, Slovenia theorist (search) emphasizes how nature’s multiplex networks not only engender neural, physiological, and social structures but also, by their title features, serve life’s vital communicative conveyance. It is then said that a better working knowledge of network phenomena can help avoid problems with power grids, traffic flow, and so on.

Perotti, Juan, et al. Emergent Self-Organized Complex Network Topology out of Stability Constraints. Physics Review Letters. 103/108701, 2009. In an endeavor to understand the robust effectiveness of these ubiquitous webworks, scientists from Cordoba and Chicago, including Dante Chialvo, say they grow and flourish because new agents or nodes are admitted based on how they contribute to their overall viability. Which could be a good example of a natural principle of much advantage for social guidance. Rather than liberal or socialist vs. conservative libertarian, life’s vitality at every stage professes a mutual reciprocity of entity and group.

Although most networks in nature exhibit complex topologies, the origins of such complexity remain unclear. We propose a general evolutionary mechanism based on global stability. This mechanism is incorporated into a model of a growing network of interacting agents in which each new agent’s membership in the network is determined by the agent’s effect on the network’s global stability. It is shown that out of this stability constraint complex topological properties emerge in a self-organized manner, offering an explanation for their observed ubiquity in biological networks. (108701-1)

Complex networks of interacting agents are ubiquitous, in a wide range of scales, from the microscopic level of genetic, metabolic, and proteins networks to the macroscopic human level of the Internet. (108701-1)

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