Network Analysis of Particles and Grains

Papadopoulos, Lia; Porter, Mason A.; Daniels, Karen E.; Bassett, Danielle S.

doi:10.48550/arxiv.1708.08080

Cited by 3 publications

(3 citation statements)

References 343 publications

(756 reference statements)

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“…However, there are some occasions in which it is important to consider higher order networks formed by building blocks that are less densely connected than simplices, i.e cell-complexes formed by gluing convex polytopes. Cell-complexes are of fundamental importance for characterizing self-assembled nanostructures [36] or granular materials [37]. However examples where cell-complexes are relevant also for interdisciplinary applications are not lacking.…”

Section: Introductionmentioning

confidence: 99%

Network Geometry and Complexity

Mulder

Bianconi

2018

J Stat Phys

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Higher order networks are able to characterize data as different as functional brain networks, protein interaction networks and social networks beyond the framework of pairwise interactions. Most notably higher order networks include simplicial complexes formed not only by nodes and links but also by triangles, tetrahedra, etc. More in general, higher-order networks can be cell-complexes formed by gluing convex polytopes along their faces. Interestingly, higher order networks have a natural geometric interpretation and therefore constitute a 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. This model, originally proposed for capturing the evolution of simplicial complexes, is here extended to cell-complexes formed by subsequently gluing different copies of an arbitrary regular polytope. We reveal the interplay between complexity and geometry of the higher order networks generated by the model by studying the emergent community structure and the degree distribution as a function of the regular polytope forming its building blocks. Additionally, we discuss the underlying hyperbolic nature of the emergent geometry and we relate the spectral dimension of the higher-order network to the dimension and nature of its building blocks.

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Section: Introductionmentioning

confidence: 99%

Network Geometry and Complexity

Mulder

Bianconi

2018

J Stat Phys

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“…Network science now branches far into applied fields such as medicine [5], genetics [3], physics [50], sociology [67], ecology [53], and neuroscience [62]. Such breadth is made possible by the discipline's roots in the abstract field of graph theory.…”

Section: Introductionmentioning

confidence: 99%

The importance of the whole: Topological data analysis for the network neuroscientist

Sizemore

Phillips-Cremins

Ghrist

et al. 2019

Network Neuroscience

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177

139

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Data analysis techniques from network science have fundamentally improved our understanding of neural systems and the complex behaviors that they support. Yet the restriction of network techniques to the study of pairwise interactions prevents us from taking into account intrinsic topological features such as cavities that may be crucial for system function. To detect and quantify these topological features, we must turn to algebro-topological methods that encode data as a simplicial complex built from sets of interacting nodes called simplices. We then use the relations between simplices to expose cavities within the complex, thereby summarizing its topological features. Here we provide an introduction to persistent homology, a fundamental method from applied topology that builds a global descriptor of system structure by chronicling the evolution of cavities as we move through a combinatorial object such as a weighted network. We detail the mathematics and perform demonstrative calculations on the mouse structural connectome, synapses in C. elegans, and genomic interaction data. Finally, we suggest avenues for future work and highlight new advances in mathematics ready for use in neural systems.

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“…It is then easy to set up a vibrational model on this graph by considering it as a ball-and-spring system and studying the change of state in it as a result of raising the temperature using the Lindemann criterion [19]. Many granular materials are nowadays studied by using graph-theoretic methods [26].…”

Section: Introductionmentioning

confidence: 99%

Communication Melting in Graphs and Complex Networks

Alalwan,

Arenas,

Estrada

2018

Preprint

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Complex networks are the representative graphs of interactions in many complex systems. Usually, these interactions are abstractions of the communication/diffusion channels between the units of the system. Real complex networks, e.g. traffic networks, reveal different operation phases governed by the dynamical stress of the system. In the case of traffic networks the archetypical transition is from free flow to congestion. A revolutionary approach to ascertain how these transitions emerge is that of using physical models that could account for diffusion process under stress. Here we show how, communicability, a topological descriptor that reveals the efficiency of the network functionality in terms of these diffusive paths, could be used to reveal the transitions mentioned. By considering a vibrational model of nodes and edges in a graph/network at a given temperature (stress), we show that the communicability function plays the role of the thermal Green's function of a network of harmonic oscillators. After, we prove analytically the existence of a universal phase transition in the communicability structure of every simple graph. This transition resembles the melting process occurring in solids. For instance, regular-like graphs resembling crystals, melts at lower temperatures and display a sharper transition between connected to disconnected structures than the random spatial graphs, which resemble amorphous solids. Finally, we study computationally this graph melting process

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Network Analysis of Particles and Grains

Cited by 3 publications

References 343 publications

Network Geometry and Complexity

Network Geometry and Complexity

The importance of the whole: Topological data analysis for the network neuroscientist

Communication Melting in Graphs and Complex Networks

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