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Hypernetworks for reconstructing the dynamics of multilevel systems
Networks are fundamental for reconstructing the dynamics of many systems, but have the drawback that they are restricted to binary relations. Hypergraphs extend relational structure to multi-vertex edges, but are essentially set-theoretic and unable to represent essential structural properties. Hypernetworks are a natural multidimensional generalisation of networks, representing n-ary relations by simplices with n vertices. The assembly of vertices to make simplices is key for moving between levels in multilevel systems, and integrating dynamics between levels. It is argued that hypernetworks are necessary, if not sufficient, for reconstructing the dynamics of multilevel complex systems
Embracing <i>n</i>-ary Relations in Network Science
Most network scientists restrict their attention to relations between pairs of things, even though most complex systems have structures and dynamics determined by n-ary relation where n is greater than two. Various examples are given to illustrate this. The basic mathematical structures allowing more than two vertices have existed for more than half a century, including hypergraphs and simplicial complexes. To these can be added hypernetworks which, like multiplex networks, allow many relations to be defined on the vertices. Furthermore, hypersimplices provide an essential formalism for representing multilevel part-whole and taxonomic structures for integrating the dynamics of systems between levels. Graphs, hypergraphs, networks, simplicial complex, multiplex network and hypernetworks form a coherent whole from which, for any particular application, the scientist can select the most suitable
Forman's Ricci curvature - From networks to hypernetworks
Networks and their higher order generalizations, such as hypernetworks or
multiplex networks are ever more popular models in the applied sciences.
However, methods developed for the study of their structural properties go
little beyond the common name and the heavy reliance of combinatorial tools. We
show that, in fact, a geometric unifying approach is possible, by viewing them
as polyhedral complexes endowed with a simple, yet, the powerful notion of
curvature - the Forman Ricci curvature. We systematically explore some aspects
related to the modeling of weighted and directed hypernetworks and present
expressive and natural choices involved in their definitions. A benefit of this
approach is a simple method of structure-preserving embedding of hypernetworks
in Euclidean N-space. Furthermore, we introduce a simple and efficient manner
of computing the well established Ollivier-Ricci curvature of a hypernetwork.Comment: to appear: Complex Networks '18 (oral presentation
Complexity and robustness in hypernetwork models of metabolism
Metabolic reaction data is commonly modelled using a complex network approach, whereby nodes represent the chemical species present within the organism of interest, and connections are formed between those nodes participating in the same chemical reaction. Unfortunately, such an approach provides an inadequate description of the metabolic process in general, as a typical chemical reaction will involve more than two nodes, thus risking over-simplification of the the system of interest in a potentially significant way. In this paper, we employ a complex hypernetwork formalism to investigate the robustness of bacterial metabolic hypernetworks by extending the concept of a percolation process to hypernetworks. Importantly, this provides a novel method for determining the robustness of these systems and thus for quantifying their resilience to random attacks/errors. Moreover, we performed a site percolation analysis on a large cohort of bacterial metabolic networks and found that hypernetworks that evolved in more variable enviro nments displayed increased levels of robustness and topological complexity
Synchronization of dynamical hypernetworks: dimensionality reduction through simultaneous block-diagonalization of matrices
We present a general framework to study stability of the synchronous solution
for a hypernetwork of coupled dynamical systems. We are able to reduce the
dimensionality of the problem by using simultaneous block-diagonalization of
matrices. We obtain necessary and sufficient conditions for stability of the
synchronous solution in terms of a set of lower-dimensional problems and test
the predictions of our low-dimensional analysis through numerical simulations.
Under certain conditions, this technique may yield a substantial reduction of
the dimensionality of the problem. For example, for a class of dynamical
hypernetworks analyzed in the paper, we discover that arbitrarily large
networks can be reduced to a collection of subsystems of dimensionality no more
than 2. We apply our reduction techique to a number of different examples,
including a class of undirected unweighted hypermotifs of three nodes.Comment: 9 pages, 6 figures, accepted for publication in Phys. Rev.
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