Percolation on branching simplicial and cell complexes and its relation to interdependent percolation

Bianconi, Ginestra; Kryven, Ivan; Ziff, Robert M.

doi:10.1103/physreve.100.062311

Cited by 30 publications

(20 citation statements)

References 51 publications

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“…Many of the presented results are higher-dimensional analogues of the well-known phenomena in random graphs related to connectivity, giant component, cycles, etc., therefore we preamble, where possible, the higher-dimensional statements with brief recollections of their one-dimensional counterparts. The focus on mathematics taken in this review precludes us unfortunately from covering many exciting subjects, such as models of growing complexes [2][3][4][5][6][7], their applications, and phenomena in them including percolation [8][9][10][11][12][13]. Yet in the concluding Section 6 that outlines our view on interesting future directions, we also comment briefly on some applications and their implications for different models of random simplicial complexes.…”

Section: Introductionmentioning

confidence: 99%

Random Simplicial Complexes: Models and Phenomena

Bobrowski,

Krioukov

2021

Preprint

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We review a collection of models of random simplicial complexes together with some of the most exciting phenomena related to them. We do not attempt to cover all existing models, but try to focus on those for which many important results have been recently established rigorously in mathematics, especially in the context of algebraic topology. In application to real-world systems, the reviewed models are typically used as null models, so that we take a statistical stance, emphasizing, where applicable, the entropic properties of the reviewed models. We also review a collection of phenomena and features observed in these models, and split the presented results into two classes: phase transitions and distributional limits. We conclude with an outline of interesting future research directions.

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Section: Introductionmentioning

confidence: 99%

Random Simplicial Complexes: Models and Phenomena

Bobrowski,

Krioukov

2021

Preprint

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“…We note that these results do not exclude a priori that spatial effects might also be important elements determining the power-law increase in the number of cases. In particular, hierarchical and hyperbolic networks describing nested communities of people during lockdown can be responsible for a broadening of the critical region in which one can observe the power-law critical behaviour [27] similarly to what happens for percolation on the same type of networks [28,29,30,31,32].…”

Section: Introductionmentioning

confidence: 90%

“…Here, λ c indicates the so-called epidemic threshold. However, it has to be noted that in hyperbolic and hierarchical structures the critical region may stretch out for a finite range of values of the infectivity [27], which corresponds to the fact that in these networks one can observe two percolation thresholds [28,29,30,31,32]. When the onset of the epidemic is started from a single infected individual, the latter three dynamical regions are characterised by different dynamical properties: the supercritical region is characterised by an exponential increase of the number of infected individuals, the critical regimeby a power-law increase with exponent 2, while the subcritical regime -by finite size stochastic fluctuations.…”

Section: The Major Properties Of the Sir Modelmentioning

confidence: 99%

Critical time-dependent branching process modelling epidemic spreading with containment measures

Sun,

Kryven,

Bianconi

2022

Preprint

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During the COVID pandemic, periods of exponential growth of the disease have been mitigated by containment measures that in different occasions have resulted in a power-law growth of the number of cases. The first observation of such behaviour has been obtained from 2020 late spring data coming from China by Ziff and Ziff in Ref. [1]. After this important observation the power-law scaling (albeit with different exponents) has also been observed in other countries during periods of containment of the spread. Early interpretations of these results suggest that this phenomenon might be due to spatial effects of the spread. Here we show that temporal modulations of infectivity of individuals due to containment measures can also cause power-law growth of the number of cases over time. To this end we propose a stochastic wellmixed Susceptible-Infected-Removed (SIR) model of epidemic spreading in presence of containment measures resulting in a time dependent infectivity and we explore the statistical properties of the resulting branching process at criticality. We show that at criticality it is possible to observe power-law growth of the number of cases with exponents ranging between one and two. Our asymptotic analytical results are confirmed by extensive Monte Carlo simulations. Although these results do not exclude that spatial effects might be important in modulating the power-law growth of the number of cases at criticality, this work shows that even well-mixed populations may already feature non trivial power-law exponents at criticality.

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“…In other words, that the structure of a dynamic network may have memory of its past. Some examples of evolving network models include Price’s (1965) preferential attachment model 1 , the vertex copying model 2 , network optimisation models 3 , and branching simlicial complexes 4 , 5 . Preferential attachment, vertex copying models, and branching simplicial complexes all result in heavy-tailed degree distributions.…”

Section: Introductionmentioning

confidence: 99%

Coloured random graphs explain the structure and dynamics of cross-linked polymer networks

Schamboeck

Iedema

Kryven

2020

Sci Rep

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Step-growth and chain-growth are two major families of chemical reactions that result in polymer networks with drastically different physical properties, often referred to as hyper-branched and crosslinked networks. in contrast to step-growth polymerisation, chain-growth forms networks that are history-dependent. Such networks are defined not just by the degree distribution, but also by their entire formation history, which entails a modelling and conceptual challenges. We show that the structure of chain-growth polymer networks corresponds to an edge-coloured random graph with a defined multivariate degree distribution, where the colour labels represent the formation times of chemical bonds. The theory quantifies and explains the gelation in free-radical polymerisation of cross-linked polymers and predicts conditions when history dependance has the most significant effect on the global properties of a polymer network. As such, the edge colouring is identified as the key driver behind the difference in the physical properties of step-growth and chain-growth networks. We expect that this findings will stimulate usage of network science tools for discovery and design of cross-linked polymers. Already in the early days of network science it has been realised that in dynamic networks the entire time trajectory of network formation may reflect on the topological features of the structure that is formed at the end. In other words, that the structure of a dynamic network may have memory of its past. Some examples of evolving network models include Price's (1965) preferential attachment model 1 , the vertex copying model 2 , network optimisation models 3 , and branching simlicial complexes 4,5. Preferential attachment, vertex copying models, and branching simplicial complexes all result in heavy-tailed degree distributions. In this work, we introduce an evolving network model with an arbitrary degree distribution to show that the different effects induced by the two most common polymerisation processes on the resulting materials are provoked by the presence or absence of memory in the underlaying network structures. Step-growth and chain-growth polymerisation are two major families of chemical reactions that result in polymer networks. For step-growth, such polymers include polyethylene (PE), polyvinyl chloride (PVC), polypropylene (PP) and polyacrylates 6 , which have a variety of applications: including paints and adhesives 7,8 , food packaging 9 , biomaterials and medical devices 10,11. Well-known examples of chain-growth polymers are gels that find applications as absorbents for medical, chemical and agricultural purposes 12 , and coatings made by photopolymerisation 13,14. The reason for the difference in the physical and mechanical properties of the step-and chain-growth derived polymers lies in the distinct network structures. Moreover, the structure of chain-growth networks can be manipulated by adjusting polymerisation conditions and species concentrators in order to optimise physical and mechanical properties. ...

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Percolation on branching simplicial and cell complexes and its relation to interdependent percolation

Cited by 30 publications

References 51 publications

Random Simplicial Complexes: Models and Phenomena

Random Simplicial Complexes: Models and Phenomena

Critical time-dependent branching process modelling epidemic spreading with containment measures

Coloured random graphs explain the structure and dynamics of cross-linked polymer networks

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