We perform a high-accuracy moment analysis of the avalanche size, duration and area distribution of the Abelian Manna model on eight two-dimensional and four one-dimensional lattices. The results provide strong support to establish universality of exponents and moment ratios across different lattices and a good survey for the strength of corrections to scaling which are notorious in the Manna universality class. The results are compared against previous work done on Manna model, Oslo model and directed percolation. We also confirm hypothesis of various scaling relations.

The Abelian Manna model of self-organized criticality is studied on various three-dimensional and fractal lattices. The exponents for avalanche size, duration, and area distribution of the model are obtained by using a high-accuracy moment analysis. Together with earlier results on lower-dimensional lattices, the present results reinforce the notion of universality below the upper critical dimension and allow us to determine the coefficients of an ε expansion. By rescaling the critical exponents by the lattice dimension and incorporating the random walker dimension, a remarkable relation is observed, satisfied by both regular and fractal lattices.

We analyze the avalanche size distribution of the Abelian Manna model on two different fractal lattices with the same dimension dg = ln 3/ ln 2, with the aim to probe for scaling behavior and to study the systematic dependence of the critical exponents on the dimension and structure of the lattices. We show that the scaling law D(2 − τ ) = dw generalizes the corresponding scaling law on regular lattices, in particular hypercubes, where dw = 2. Furthermore, we observe that the lattice dimension dg, the fractal dimension of the random walk on the lattice dw, and the critical exponent D, form a plane in 3D parameter space, i.e. they obey the linear relationship D = 0.632(3)dg + 0.98(1)dw − 0.49(3). Although extensive research has been performed on self-organized criticality [1] for models on hypercubic lattices, far less work has been done on fractal lattices [2,3]. It remains somewhat unclear what to conclude from the latter studies. Fractal lattices are important for the understanding of critical phenomena for a number of reasons. Firstly, results for critical exponents in lattices with non-integer dimensions might provide a means to determine the terms of their = 4 − d expansion. Secondly, fractal lattices are particularly suitable for a real space renormalization group procedures, in particular that by . Thirdly, scaling relations that are derived in a straightforward fashion on hypercubic lattices can be put to test in a more general setting. In this Brief Report, we address the first and the third aspect, by examining both numerically and analytically the scaling behaviour of the Abelian version of the Manna model [7][8][9] on two different fractal lattices.The fractal lattices used in this study are generated from the arc-fractal system [10]. The lattice sites are the invariant set of points of the arc-fractal. We consider nearest neighbor interactions among sites. Here, the nearest neighbors of a given site are all sites which have the (same) shortest Euclidean distance to it. Our fractal lattices have no natural boundary; instead, they have only two end points at which two copies can join to form a bigger lattice. The dimension of the lattices is the same as the arc-fractal that generates them.In this study, we shall consider two fractal lattices: the Sierpinski arrowhead and the crab (see Fig. 1). The former is named "Sierpinski arrowhead" because it is the same as the well-known Sierpinski arrowhead [11], whereas the latter is termed "crab" because the overall shape of the generated lattice looks like a crab. These fractal lattices are generated through the arc-fractal system with number of segments n = 3 and opening angle of the arc α = π. For the Sierpinski arrowhead, the rule for orientating the arc at each iteration is "in-out-in", while the rule is "out-in-out" for the crab. Both lattices have the same dimension d g = ln 3/ ln 2 ≈ 1.58. The total number of sites on the lattice at the i-th iteration is N i = 3 i + 1. The coordination number of sites on these lattices varies between two and three. Asy...

Using multiyear satellite rainfall estimates, the distributions of the area, and the total rain rate of rain clusters over the equatorial Indian, Pacific, and Atlantic Oceans was found to exhibit a power law fS()s~s−ζS, in which S represents either the cluster area or the cluster total rain rate and fS(s) denotes the probability density function of finding an event of size s. The scaling exponents ζS were estimated to be 1.66 ± 0.06 and 1.48 ± 0.13 for the cluster area and cluster total rain rate, respectively. The two exponents were further found to be related via the expected total rain rate given a cluster area. These results suggest that convection over the tropical oceans is organized into rain clusters with universal scaling properties. They are also related through a simple scaling relation consistent with classical self‐organized critical phenomena. The results from this study suggest that mesoscale rain clusters tend to grow by increasing in size and intensity, while larger clusters tend to grow by self‐organizing without intensification.

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In this paper, we present an idea of creating fractals by using the geometric arc as the basic element. This approach of generating fractals, through the tuning of just three parameters, gives a universal way to obtain many novel fractals including the classic ones. Although this arc-fractal system shares similar features with the well-known Lindenmayer system, such as the same set of invariant points and the ability to tile the space, they do have different properties. One of which is the generation of pseudo-random number, which is not available in the Lindenmayer system. Furthermore, by assuming that coastline formation is based purely on the processes of erosion and deposition, the arc-fractal system can also serve as a dynamical model of coastal morphology, with each level of its construction corresponds to the time evolution of the shape of the coastal features. Remarkably, our results indicate that the arc-fractal system can provide an explanation on the origin of fractality in real coastline.

This paper examines the development of a road network through time to consider 8 its relationship to processes of urbanisation in Roman and early medieval England. 9Using a popular network measure called PageRank, we classify the importance of 10 nodes in the transport network of roads and navigable waterways to assess the 11 relative location of urban places. Applying this measure we show that there is a 12 strong correlation between the status of towns in both Roman and medieval 13 periods and their proximity to transport nodes with high values of PageRank. 14 Comparison between two temporally distinct networks-Early Roman, and that 15 recorded in the Domesday survey of AD 1086-allows for a discussion about the 16 determinants of urban growth at different times. The applicability of PageRank to 17 other forms of network analysis in archaeology are offered in conclusion. (Gross and Sayama, 2009;Lewis, 2009; Wasserman and 26 Faust, 1994) to other fields have yielded interesting results and proven that promising areas 27 of research can emerge (see for e.g. Barabasi and Oltvai, 2004;Barthelemy, 2011; Borgatti et 28 al., 2009;Dunne et al., 2002;Emirbayer and Goodwin, 1994;Glaisyer, 2004; McPherson et al., 29 2001;Proulx et al., 2005;Singh, 2005;Sorenson et al., 2006;Watts and Dodds, 2007). In 30 archaeology, there is a burgeoning literature utilising such approaches (see, for e.g. 31 Broodbank, 2000;Brughmans, et al., 2016;Collar et al., 2015;Hage and Harary, 1991; 32 Knappett, 2011 32 Knappett, , 2013. Notably, these have included attempts to define network properties 33 from distributions of archaeological materials such as physical monuments (e.g. Johansen, et 34 al. 2004), artefacts of known provenance (e.g. Sindbaek, 2007), historically-attested journeys 35 or physical route networks (e.g. Graham 2006;Isaksen, 2008), and other cultural phenomena 36 such as language and legal traditions (e.g. Collar, 2013;Terrell, 2010). 37 38 Relevant to this paper, are works that have applied network analysis to explore the structure 39 of past transport networks. In an early application of this type, Dicks (1972) evaluated the 40 Roman road system of Britain via path ordering. More recently, Orengo and Livarda (2015) 41 have tested the same evidence to demonstrate the relation between the trading activities of 42 towns and the connections of nearby transport links. 43 44 Extending these debates, this paper seeks to demonstrate the applicability of using a popular 45 network measure called PageRank to archaeological questions. An algorithm used by Google 46 Search, PageRank (Brin and Page, 1998) was originally designed as a way of measuring the 47 relative importance of web pages based on the links among them. The intended purpose of 48 such ranking was to filter the web pages and return the most relevant ones in response to a 49 query given to the search engine. However, the algorithm itself is universal and can usefully 50 be applied to other situations where one seeks to identify the importance of entities in a 5...