Random Self-similar Networks Research Articles

Statistical Tests of Random Self-similar Networks Using Digital Elevation Models

Recently, a new class of construction network models, called the random self-similar network model, has been developed to provide a better approximation of the self-similar structure of real river networks. Random self-similar networks are generated by two-type Galton–Watson branching processes. The model has been employed to study the scaling behavior of river networks and to develop a geophysical scaling theory of floods. In this manuscript, the ability of the random self-similar network model to approximate real river networks was statistically tested. The random self-similar networks were parameterized with river networks extracted from digital elevation model data. Firstly, statistical tests were performed to test the invariance of generator distributions. Secondly, values of scaling ratios of extracted river networks were also compared with those that were predicted by the random self-similar model. Finally, whether the appearances of generators follow a geometric distribution was tested using goodness-of-fit test. The results show that not all of the tested basins have invariant generator distributions. The random self-similar network model estimated values of Horton ratios are consistently greater than those obtained by the linear regression method, while the model estimated values of the Hack ratio fall in the reasonable range which was reported by previous researchers. The hypothesis that generator distributions are geometric is rejected for all basins. More empirical research is needed on the comparison between the random self-similar network model and real river networks.

Extending generalized Horton laws to test embedding algorithms for topologic river networks

River networks in the landscape can be described as topologic rooted trees embedded in a three-dimensional surface. We examine the problem of embedding topologic binary rooted trees (BRTs) by investigating two space-filling embedding procedures: Top-Down, previously developed in the context of random self-similar networks (RSNs), and Bottom-Up, a new procedure developed here. We extend the concept of generalized Horton laws to interior sub catchments and create a new set of scaling laws that are used to test the embedding algorithms. We compare the two embedding strategies with respect to the scaling properties of the distribution of accumulated areas Aω and network magnitude Mω for complete order streams ω. The Bottom-Up procedure preserves the equality of distributions Aω/EAω=dMω/EMω; a feature observed in real basins. The Top-Down embedded networks fail to preserve this equality because of strong correlations of tile areas in the final tessellation. We conclude that the presence or absence of this equality is a useful test to diagnose river network models that describe the topology/geometry of natural drainage systems. We present some examples of applying the embedding algorithms to self similar trees (SSTs) and to RSNs. Finally, a technique is presented to map the resulting tiled region into a three-dimensional surface that corresponds to a landscape drained by the chosen network. Our results are a significant first step toward the goal of creating realistic embedded topologic trees, which are also required for the study of peak flow scaling in river networks in the presence of spatially variable rainfall and flood-generating processes.

Scaling of peak flows with constant flow velocity in random self-similar networks

Abstract. A methodology is presented to understand the role of the statistical self-similar topology of real river networks on scaling, or power law, in peak flows for rainfall-runoff events. We created Monte Carlo generated sets of ensembles of 1000 random self-similar networks (RSNs) with geometrically distributed interior and exterior generators having parameters pi and pe, respectively. The parameter values were chosen to replicate the observed topology of real river networks. We calculated flow hydrographs in each of these networks by numerically solving the link-based mass and momentum conservation equation under the assumption of constant flow velocity. From these simulated RSNs and hydrographs, the scaling exponents β and φ characterizing power laws with respect to drainage area, and corresponding to the width functions and flow hydrographs respectively, were estimated. We found that, in general, φ > β, which supports a similar finding first reported for simulations in the river network of the Walnut Gulch basin, Arizona. Theoretical estimation of β and φ in RSNs is a complex open problem. Therefore, using results for a simpler problem associated with the expected width function and expected hydrograph for an ensemble of RSNs, we give heuristic arguments for theoretical derivations of the scaling exponents β(E) and φ(E) that depend on the Horton ratios for stream lengths and areas. These ratios in turn have a known dependence on the parameters of the geometric distributions of RSN generators. Good agreement was found between the analytically conjectured values of β(E) and φ(E) and the values estimated by the simulated ensembles of RSNs and hydrographs. The independence of the scaling exponents φ(E) and φ with respect to the value of flow velocity and runoff intensity implies an interesting connection between unit hydrograph theory and flow dynamics. Our results provide a reference framework to study scaling exponents under more complex scenarios of flow dynamics and runoff generation processes using ensembles of RSNs.

Testing statistical self‐similarity in the topology of river networks

Recent work has demonstrated that the topological properties of real river networks deviate significantly from predictions of Shreve's random model. At the same time the property of mean self‐similarity postulated by Tokunaga's model is well supported by data. Recently, a new class of network model called random self‐similar networks (RSN) that combines self‐similarity and randomness has been introduced to replicate important topological features observed in real river networks. We investigate if the hypothesis of statistical self‐similarity in the RSN model is supported by data on a set of 30 basins located across the continental United States that encompass a wide range of hydroclimatic variability. We demonstrate that the generators of the RSN model obey a geometric distribution, and self‐similarity holds in a statistical sense in 26 of these 30 basins. The parameters describing the distribution of interior and exterior generators are tested to be statistically different and the difference is shown to produce the well‐known Hack's law. The inter‐basin variability of RSN parameters is found to be statistically significant. We also test generator dependence on two climatic indices, mean annual precipitation and radiative index of dryness. Some indication of climatic influence on the generators is detected, but this influence is not statistically significant with the sample size available. Finally, two key applications of the RSN model to hydrology and geomorphology are briefly discussed.