School of Mathematical Sciences, Queen Mary University of London, London E1 4NS, United Kingdom and The Alan Turing Institute, The British Library, London NW1 2DB, United Kingdom.
Mathematical Institute, Utrecht University, PO Box 80010, 3508 TA Utrecht, The Netherlands.
Phys Rev E. 2019 Dec;100(6-1):062311. doi: 10.1103/PhysRevE.100.062311.
Network geometry has strong effects on network dynamics. In particular, the underlying hyperbolic geometry of discrete manifolds has recently been shown to affect their critical percolation properties. Here we investigate the properties of link percolation in nonamenable two-dimensional branching simplicial and cell complexes, i.e., simplicial and cell complexes in which the boundary scales like the volume. We establish the relation between the equations determining the percolation probability in random branching cell complexes and the equation for interdependent percolation in multiplex networks with interlayer degree correlation equal to one. By using this relation we show that branching cell complexes can display more than two percolation phase transitions: the upper percolation transition, the lower percolation transition, and one or more intermediate phase transitions. At these additional transitions the percolation probability and the fractal exponent both feature a discontinuity. Furthermore, by using the renormalization group theory we show that the upper percolation transition can belong to various universality classes including the Berezinskii-Kosterlitz-Thouless (BKT) transition, the discontinuous percolation transition, and continuous transitions with anomalous singular behavior that generalize the BKT transition.
网络几何形状对网络动态具有很强的影响。特别是，离散流形的基础双曲几何形状最近已被证明会影响它们的临界渗流特性。在这里，我们研究了不可约二维分支单纯形和胞腔复形中链路渗流的性质，即边界按体积缩放的单纯形和胞腔复形。我们建立了确定随机分支胞腔复形中渗流概率的方程与具有等于 1 的层间度相关的多重网络中相依渗流方程之间的关系。通过使用这种关系，我们表明分支胞腔复形可以显示出超过两个渗流相变：上渗流相变、下渗流相变以及一个或多个中间相变。在这些额外的转变中，渗流概率和分形指数都具有不连续性。此外，我们利用重整化群理论表明，上渗流相变可以属于各种具有普遍性的类别，包括 Berezinskii-Kosterlitz-Thouless (BKT) 相变、不连续渗流相变以及具有广义 BKT 相变的异常奇异行为的连续相变。
