Characterization of spatial networklike patterns from junction geometry

Alignment with neighbours enables escape from dead ends in flocking models

Coordinated movement in animal groups (flocks, schools, herds, etc.) is a classic and well-studied form of collective behaviour. Most theoretical studies consider agents in unobstructed spaces; however, many animals move in often complicated environments and must navigate around and through obstacles. Here we consider simulated agents behaving according to typical flocking rules, with the addition of repulsion from obstacles, and study their collective behaviour in environments with concave obstacles (dead ends). We find that groups of such agents heading for a goal can spontaneously escape dead ends without wall-following or other specialized behaviours, in what we term 'flocking escapes'. The mechanism arises when agents align with one another while heading away from the goal, forming a self-stable cluster that persists long enough to exit the obstacle and avoids becoming trapped again when turning back towards the goal. Solitary agents under the same conditions are never observed to escape. We show that alignment with neighbours reduces the effective turning speed of the group while letting individuals maintain high manoeuvrability when needed. The relative robustness of flocking escapes in our studies suggests that this emergent behaviour may be relevant for a variety of animal species.

Time Evolution and Spatial Hierarchy of Crack Patterns

Cracks generated due to desiccation of wet colloidal systems are ubiquitous, examples being nanomaterial films, painted walls, cemented floors, mud fields, river beds, and even giant rocks. In all such cases, crack patterns are often appreciably similar but for the length and time scales, which can be widely differing. In this work, we have examined the crack formation more closely to see if there exists some generality with regard to the length scale of parameters and the formation time. Specifically, using a commonly used colloidal dispersion and optimized conditions to form polygonal network patterns rather than isolated cracks (films of subcritical thickness), we have studied the time evolution of the pattern parameters, the area occupied by the cracks, their lengths, and the widths. As is well known, initially, a network of cracks forms, which we term as the primary generation, followed by interconnecting cracks inside the polygonal regions (secondary) and, later, cracks spreading in local regions (tertiary). We find that the area and the width increase nearly linearly with time with the change in the slope corresponding to the change in the generation. When normalized with respect to the final values, the trends obtained for different film thicknesses overlap, the only exception being the pattern containing unconnected cracks. Thus, the time evolution of cracks is shown to be predictable based on width filtering. Including the angle between cracks as further input into the recursive model, the possibility of identifying the hierarchy of crack segments is also shown. The approach may be useful in determining the age, authenticity, and details of old paintings, understanding the stress profile of geological rocks, and analyzing various natural and manmade hierarchical structures.

Mitochondrial Fission and Fusion Dynamics Generate Efficient, Robust, and Evenly Distributed Network Topologies in Budding Yeast Cells

The simplest configuration of mitochondria in a cell is as small separate organellar units. Instead, mitochondria often form a dynamic, intricately connected network. A basic understanding of the topological properties of mitochondrial networks, and their influence on cell function is lacking. We performed an extensive quantitative analysis of mitochondrial network topology, extracting mitochondrial networks in 3D from live-cell microscopic images of budding yeast cells. In the presence of fission and fusion, mitochondrial network structures exhibited certain topological properties similar to other real-world spatial networks. Fission and fusion dynamics were required to efficiently distribute mitochondria throughout the cell and generate highly interconnected networks that can facilitate efficient diffusive search processes. Thus, mitochondrial fission and fusion combine to regulate the underlying topology of mitochondrial networks, which may independently impact cell function.

Generic patterns in the evolution of urban water networks: Evidence from a large Asian city

We examine high-resolution urban infrastructure data using every pipe for the water distribution network (WDN) and sanitary sewer network (SSN) in a large Asian city (≈4 million residents) to explore the structure as well as the spatial and temporal evolution of these infrastructure networks. Network data were spatially disaggregated into multiple subnets to examine intracity topological differences for functional zones of the WDN and SSN, and time-stamped SSN data were examined to understand network evolution over several decades as the city expanded. Graphs were generated using a dual-mapping technique (Hierarchical Intersection Continuity Negotiation), which emphasizes the functional attributes of these networks. Network graphs for WDNs and SSNs are characterized by several network topological metrics, and a double Pareto (power-law) model approximates the node-degree distributions of both water infrastructure networks (WDN and SSN), across spatial and hierarchical scales relevant to urban settings, and throughout their temporal evolution over several decades. These results indicate that generic mechanisms govern the networks' evolution, similar to those of scale-free networks found in nature. Deviations from the general topological patterns are indicative of (1) incomplete establishment of network hierarchies and functional network evolution, (2) capacity for growth (expansion) or densification (e.g., in-fill), and (3) likely network vulnerabilities. We discuss the implications of our findings for the (re-)design of urban infrastructure networks to enhance their resilience to external and internal threats.

Mechanical Stress Induces Remodeling of Vascular Networks in Growing Leaves

Differentiation into well-defined patterns and tissue growth are recognized as key processes in organismal development. However, it is unclear whether patterns are passively, homogeneously dilated by growth or whether they remodel during tissue expansion. Leaf vascular networks are well-fitted to investigate this issue, since leaves are approximately two-dimensional and grow manyfold in size. Here we study experimentally and computationally how vein patterns affect growth. We first model the growing vasculature as a network of viscoelastic rods and consider its response to external mechanical stress. We use the so-called texture tensor to quantify the local network geometry and reveal that growth is heterogeneous, resembling non-affine deformations in composite materials. We then apply mechanical forces to growing leaves after veins have differentiated, which respond by anisotropic growth and reorientation of the network in the direction of external stress. External mechanical stress appears to make growth more homogeneous, in contrast with the model with viscoelastic rods. However, we reconcile the model with experimental data by incorporating randomness in rod thickness and a threshold in the rod growth law, making the rods viscoelastoplastic. Altogether, we show that the higher stiffness of veins leads to their reorientation along external forces, along with a reduction in growth heterogeneity. This process may lead to the reinforcement of leaves against mechanical stress. More generally, our work contributes to a framework whereby growth and patterns are coordinated through the differences in mechanical properties between cell types.

Topological Phenotypes Constitute a New Dimension in the Phenotypic Space of Leaf Venation Networks

The leaves of angiosperms contain highly complex venation networks consisting of recursively nested, hierarchically organized loops. We describe a new phenotypic trait of reticulate vascular networks based on the topology of the nested loops. This phenotypic trait encodes information orthogonal to widely used geometric phenotypic traits, and thus constitutes a new dimension in the leaf venation phenotypic space. We apply our metric to a database of 186 leaves and leaflets representing 137 species, predominantly from the Burseraceae family, revealing diverse topological network traits even within this single family. We show that topological information significantly improves identification of leaves from fragments by calculating a “leaf venation fingerprint” from topology and geometry. Further, we present a phenomenological model suggesting that the topological traits can be explained by noise effects unique to specimen during development of each leaf which leave their imprint on the final network. This work opens the path to new quantitative identification techniques for leaves which go beyond simple geometric traits such as vein density and is directly applicable to other planar or sub-planar networks such as blood vessels in the brain.

Planar reticular networks are ubiquitous in nature and engineering, formed for instance by the arterial vasculature in the mammalian neocortex, urban street grids or the vascular network of plant leaves. We use a topological metric to characterize the way loops are nested in such networks and analyze a large database of 186 leaves and leaflets, revealing for the first time that the nesting of the networks’ cycles constitutes a distinct phenotypic trait orthogonal to previously used geometric features. Furthermore, we demonstrate that the information contained in the leaf topology can significantly improve specimen identification from fragments, and provide an empirical growth model that can explain much of the observed data. Our work can improve understanding of the functional significance of the various leaf vein architectures and their correlation with the environment. It can pave the way for similar analyses in diverse areas of research involving reticulate networks.

Animal transportation networks

Many group-living animals construct transportation networks of trails, galleries and burrows by modifying the environment to facilitate faster, safer or more efficient movement. Animal transportation networks can have direct influences on the fitness of individuals, whereas the shape and structure of transportation networks can influence community dynamics by facilitating contacts between different individuals and species. In this review, we discuss three key areas in the study of animal transportation networks: the topological properties of networks, network morphogenesis and growth, and the behaviour of network users. We present a brief primer on elements of network theory, and then discuss the different ways in which animal groups deal with the fundamental trade-off between the competing network properties of travel efficiency, robustness and infrastructure cost. We consider how the behaviour of network users can impact network efficiency, and call for studies that integrate both network topology and user behaviour. We finish with a prospectus for future research.

Exploring the evolution of London's street network in the information space: a dual approach

We study the growth of London's street network in its dual representation, as the city has evolved over the past 224 years. The dual representation of a planar graph is a content-based network, where each node is a set of edges of the planar graph and represents a transportation unit in the so-called information space, i.e., the space where information is handled in order to navigate through the city. First, we discuss a novel hybrid technique to extract dual graphs from planar graphs, called the hierarchical intersection continuity negotiation principle. Then we show that the growth of the network can be analytically described by logistic laws and that the topological properties of the network are governed by robust log-normal distributions characterizing the network's connectivity and small-world properties that are consistent over time. Moreover, we find that the double-Pareto-like distributions for the connectivity emerge for major roads and can be modeled via a stochastic content-based network model using simple space-filling principles.

The influence of branch order on optimal leaf vein geometries: Murray's law and area preserving branching

Models that predict the form of hierarchical branching networks typically invoke optimization based on biomechanical similitude, the minimization of impedance to fluid flow, or construction costs. Unfortunately, due to the small size and high number of vein segments found in real biological networks, complete descriptions of networks needed to evaluate such models are rare. To help address this we report results from the analysis of the branching geometry of 349 leaf vein networks comprising over 1.5 million individual vein segments. In addition to measuring the diameters of individual veins before and after vein bifurcations, we also assign vein orders using the Horton-Strahler ordering algorithm adopted from the study of river networks. Our results demonstrate that across all leaves, both radius tapering and the ratio of daughter to parent branch areas for leaf veins are in strong agreement with the expectation from Murray’s law. However, as veins become larger, area ratios shift systematically toward values expected under area-preserving branching. Our work supports the idea that leaf vein networks differentiate roles of leaf support and hydraulic supply between hierarchical orders.

The simplicity of planar networks

Shortest paths are not always simple. In planar networks, they can be very different from those with the smallest number of turns - the simplest paths. The statistical comparison of the lengths of the shortest and simplest paths provides a non trivial and non local information about the spatial organization of these graphs. We define the simplicity index as the average ratio of these lengths and the simplicity profile characterizes the simplicity at different scales. We measure these metrics on artificial (roads, highways, railways) and natural networks (leaves, slime mould, insect wings) and show that there are fundamental differences in the organization of urban and biological systems, related to their function, navigation or distribution: straight lines are organized hierarchically in biological cases, and have random lengths and locations in urban systems. In the case of time evolving networks, the simplicity is able to reveal important structural changes during their evolution.

Urban Street Networks, a Comparative Analysis of Ten European Cities

We compare the structural properties of the street networks of ten different European cities using their primal representation. We investigate the properties of the geometry of the networks and a set of centrality measures highlighting differences and similarities between cases. In particular, we found that cities share structural similarities due to their quasiplanarity but that there are also several distinctive geometrical properties. A principal component analysis is performed on the distributions of centralities and their respective moments, which is used to find distinctive characteristics by which we can classify cities into families. We believe that, beyond the improvement of the empirical knowledge on streets' network properties, our findings can open new perspectives into the scientific relationship between city planning and complex networks, stimulating the debate on the effectiveness of the set of knowledge that statistical physics can contribute for city planning and urban-morphology studies.

Quantifying loopy network architectures

Biology presents many examples of planar distribution and structural networks having dense sets of closed loops. An archetype of this form of network organization is the vasculature of dicotyledonous leaves, which showcases a hierarchically-nested architecture containing closed loops at many different levels. Although a number of approaches have been proposed to measure aspects of the structure of such networks, a robust metric to quantify their hierarchical organization is still lacking. We present an algorithmic framework, the hierarchical loop decomposition, that allows mapping loopy networks to binary trees, preserving in the connectivity of the trees the architecture of the original graph. We apply this framework to investigate computer generated graphs, such as artificial models and optimal distribution networks, as well as natural graphs extracted from digitized images of dicotyledonous leaves and vasculature of rat cerebral neocortex. We calculate various metrics based on the asymmetry, the cumulative size distribution and the Strahler bifurcation ratios of the corresponding trees and discuss the relationship of these quantities to the architectural organization of the original graphs. This algorithmic framework decouples the geometric information (exact location of edges and nodes) from the metric topology (connectivity and edge weight) and it ultimately allows us to perform a quantitative statistical comparison between predictions of theoretical models and naturally occurring loopy graphs.