Shape of shortest paths in random spatial networks

Vulnerability of transport through evolving spatial networks

Insight into the blockage vulnerability of evolving spatial networks is important for understanding transport resilience, robustness, and failure of a broad class of real-world structures such as porous media and utility, urban traffic, and infrastructure networks. By exhaustive search for central transport hubs on porous lattice structures, we recursively determine and block the emerging main hub until the evolving network reaches the impenetrability limit. We find that the blockage backbone is a self-similar path with a fractal dimension which is distinctly smaller than that of the universality class of optimal path crack models. The number of blocking steps versus the rescaled initial occupation fraction collapses onto a master curve for different network sizes, allowing for the prediction of the onset of impenetrability. The shortest-path length distribution broadens during the blocking process reflecting an increase of spatial correlations. We address the reliability of our predictions upon increasing the disorder or decreasing the fraction of processed structural information.

Indication of long-range correlations governing city size

City systems are characterized by the functional organization of cities on a regional or country scale. While there is a relatively good empirical and theoretical understanding of city size distributions, insights about their spatial organization remain on a conceptual level. Here, we analyze empirically the correlations between the sizes of cities (in terms of area) across long distances. Therefore, we (i) define city clusters, (ii) obtain the neighborhood network from Voronoi cells, and (iii) apply a fluctuation analysis along all shortest paths. We find that most European countries exhibit long-range correlations but in several cases these are anti-correlations. In an analogous way, we study a model inspired by Central Places Theory and find that it leads to positive long-range correlations, unless there is strong additional spatial disorder-contrary to intuition. We conclude that the interactions between cities extend over large distances reaching the country scale. Our findings have policy relevance as urban development or decline can affect cities at a considerable distance.

Boundary effects on topological characteristics of urban road networks

Urban road networks (URNs), as simplified views and important components of cities, have different structures, resulting in varying levels of transport efficiency, accessibility, resilience, and many socio-economic indicators. Thus, topological characteristics of URNs have received great attention in the literature, while existing studies have used various boundaries to extract URNs for analysis. This naturally leads to the question of whether topological patterns concluded using small-size boundaries keep consistent with those uncovered using commonly adopted administrative boundaries or daily travel range-based boundaries. This paper conducts a large-scale empirical analysis to reveal the boundary effects on 22 topological metrics of URNs across 363 cities in mainland China. Statistical results show that boundaries have negligible effects on the average node degree, edge density, orientation entropy of road segments, and the eccentricity for the shortest or fastest routes, while other metrics including the clustering coefficient, proportion of high-level road segments, and average edge length together with route-related metrics such as average angular deviation show significant differences between road networks extracted using different boundaries. In addition, the high-centrality components identified using varied boundaries show significant differences in terms of their locations, with only 21%-28% of high-centrality nodes overlapping between the road networks extracted using administrative and daily travel range-based boundaries. These findings provide useful insights to assist urban planning and better predict the influence of a road network structure on the movement of people and the flow of socio-economic activities, particularly in the context of rapid urbanization and the ever-increasing sprawl of road networks.

Betweenness centrality in dense spatial networks

The betweenness centrality (BC) is an important quantity for understanding the structure of complex large networks. However, its calculation is in general difficult and known in simple cases only. In particular, the BC has been exactly computed for graphs constructed over a set of N points in the infinite density limit, displaying a universal behavior. We reconsider this calculation and propose an expansion for large and finite densities. We compute the lowest nontrivial order and show that it encodes how straight are shortest paths and is therefore nonuniversal and depends on the graph considered. We compare our analytical result to numerical simulations obtained for various graphs such as the minimum spanning tree, the nearest neighbor graph, the relative neighborhood graph, the random geometric graph, the Gabriel graph, or the Delaunay triangulation. We show that in most cases the agreement with our analytical result is excellent even for densities of points that are relatively low. This method and our results provide a framework for understanding and computing this important quantity in large spatial networks.

Random geometric graphs in high dimension

Many machine learning algorithms used for dimensional reduction and manifold learning leverage on the computation of the nearest neighbors to each point of a data set to perform their tasks. These proximity relations define a so-called geometric graph, where two nodes are linked if they are sufficiently close to each other. Random geometric graphs, where the positions of nodes are randomly generated in a subset of R^{d}, offer a null model to study typical properties of data sets and of machine learning algorithms. Up to now, most of the literature focused on the characterization of low-dimensional random geometric graphs whereas typical data sets of interest in machine learning live in high-dimensional spaces (d≫10^{2}). In this work, we consider the infinite dimensions limit of hard and soft random geometric graphs and we show how to compute the average number of subgraphs of given finite size k, e.g., the average number of k cliques. This analysis highlights that local observables display different behaviors depending on the chosen ensemble: soft random geometric graphs with continuous activation functions converge to the naive infinite-dimensional limit provided by Erdös-Rényi graphs, whereas hard random geometric graphs can show systematic deviations from it. We present numerical evidence that our analytical results, exact in infinite dimensions, provide a good approximation also for dimension d≳10.

First-passage percolation under extreme disorder: From bond percolation to Kardar-Parisi-Zhang universality

We consider the statistical properties of arrival times and balls on first-passage percolation (FPP) two-dimensional square lattices with strong disorder in the link times. A previous work showed a crossover in the weak disorder regime, between Gaussian and Kardar-Parisi-Zhang (KPZ) universality, with the crossover length decreasing as the noise amplitude grows. On the other hand, this work presents a very different behavior in the strong-disorder regime. An alternative crossover length appears below which the model is described by bond-percolation universality class. This characteristic length scale grows with the noise amplitude and diverges at the infinite-disorder limit. We provide a thorough characterization of the bond-percolation phase, reproducing its associated critical exponents through a careful scaling analysis of the balls, which is carried out through a continuous mapping of the FPP passage time into the occupation probability of the bond-percolation problem. Moreover, the crossover length can be explained merely in terms of properties of the link-time distribution. The interplay between the characteristic length and the correlation length intrinsic to bond percolation determines the crossover between the initial percolation-like growth and the asymptotic KPZ scaling.

Effect of volume growth on the percolation threshold in random directed acyclic graphs with a given degree distribution

In every network, a distance between a pair of nodes can be defined as the length of the shortest path connecting these nodes, and therefore one may speak of a ball, its volume, and how it grows as a function of the radius. Spatial networks tend to feature peculiar volume scaling functions, as well as other topological features, including clustering, degree-degree correlation, clique complexes, and heterogeneity. Here we investigate a nongeometric random graph with a given degree distribution and an additional constraint on the volume scaling function. We show that such structures fall into the category of m-colored random graphs and study the percolation transition by using this theory. We prove that for a given degree distribution the percolation threshold for weakly connected components is not affected by the volume growth function. Additionally, we show that the size of the giant component and the cyclomatic number are not affected by volume scaling. These findings may explain the surprisingly good performance of network models that neglect volume scaling. Even though this paper focuses on the implications of the volume growth, the model is generic and might lead to insights in the field of random directed acyclic graphs and their applications.