Complexity

Volume 2019, Article ID 2096749, 16 pages

https://doi.org/10.1155/2019/2096749

## A Century of Topological Coevolution of Complex Infrastructure Networks in an Alpine City

^{1}Unit of Environmental Engineering, Department of Infrastructure, University of Innsbruck, Technikerstrasse 13, 6020 Innsbruck, Austria^{2}Lyles School of Civil Engineering, Purdue University, 550 Stadium Mall Drive, West Lafayette, IN 47907, USA^{3}Urban Computing Business Unit, JD Finance No. 18 Kechuang 11 Street, Beijing, China

Correspondence should be addressed to Jonatan Zischg; ta.ca.kbiu@ghcsiz.natanoj

Received 24 May 2018; Revised 29 November 2018; Accepted 3 December 2018; Published 6 January 2019

Academic Editor: Albert Diaz-Guilera

Copyright © 2019 Jonatan Zischg et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In this paper, we used complex network analysis approaches to investigate topological coevolution over a century for three different urban infrastructure networks. We applied network analyses to a unique time-stamped network data set of an Alpine case study, representing the historical development of the town and its infrastructure over the past 108 years. The analyzed infrastructure includes the water distribution network (WDN), the urban drainage network (UDN), and the road network (RN). We use the dual representation of the network by using the Hierarchical Intersection Continuity Negotiation (HICN) approach, with pipes or roads as nodes and their intersections as edges. The functional topologies of the networks are analyzed based on the dual graphs, providing insights beyond a conventional graph (primal mapping) analysis. We observe that the RN, WDN, and UDN all exhibit heavy tailed node degree distributions with high dispersion around the mean. In 50 percent of the investigated networks, can be approximated with truncated [Pareto] power-law functions, as they are known for scale-free networks. Structural differences between the three evolving network types resulting from different functionalities and system states are reflected in the and other complex network metrics. Small-world tendencies are identified by comparing the networks with their random and regular lattice network equivalents. Furthermore, we show the remapping of the dual network characteristics to the spatial map and the identification of criticalities among different network types through co-location analysis and discuss possibilities for further applications.

#### 1. Introduction

Many complex systems can be described as networks [1], and with recent increases in computing power it is now feasible to investigate the topologies of entire networks consisting of high-resolution data [2]. Examples of these types of investigations range from molecular interaction networks (e.g., protein interactions of cells) and social networks (e.g., communication between humans) to global transportation systems and individual human mobility [3–6].

Despite the differences in various types and representations of these networks, important commonalities exist. The analysis of complex networks gives insight to structural morphologies, similarities, recurring patterns, and scaling laws [7, 8]. The applications are multifaceted: identification of central nodes; prediction of future developments and network growth; information transfer; identification of vulnerabilities to enhance security [9]; and improvement of network resilience [10, 11]. Complex network analyses of critical infrastructure, such as water distribution networks (WDNs) and urban drainage networks (UDNs), provide valuable insights beyond the traditional engineering approaches, to design and operate systems in a more reliable way and to help build-up structural resiliency [12, 13].

In the past, most structural features in complex networks were investigated based on a conventional graph representation (so-called “primal space”), where pipes or conduits are the edges and their intersections the vertices of a mathematical graph [14, 15]. Conversely, different approaches, based for example on common attribute classification (i.e., road name or pipe size) or intersection continuity (i.e., maximum angle of deflection), consider the network structure in its “dual space”, i.e., functional components (e.g., pipes with same diameter) which belong together, represent the vertices and their intersection the edges of the graph [16, 17]. Further explanations are provided in the next section. Unlike the conventional primal representation, dual mapping approaches may also consider the continuity of links (pipes or conduits) over a variety of edges and hierarchy (e.g., pipe diameter; isolation valves; maximum designed flow; speed limits; road class) for further graph analysis.

There exist different ways of creating the dual graph of a network, taking into account physical (e.g., geometric) and/or behavioral (e.g., symbolic) considerations. The street name (SN) approach, for example, uses the historical naming conventions to create the dual graph, but neglects the geometrical properties of the network. Hybrid approaches, like the Hierarchical Intersection Continuity Negotiation (HICN) [16], combine geometric (e.g., maximum angle of deflection of connected roads) and hierarchical (e.g., road class) attributes, to better capture the structural network topology resulting from top-down (centralized designs) and bottom-up local-planning actions (self-organization).

Previous studies using the dual mapping approach were mainly performed on road networks (RNs) [16–18], but some also on water distribution and urban drainage networks [19–21]. In principle, an extension to each network type is possible. Masucci et al. [16] investigated the road network growth for the city of London and found stable statistical properties to describe the topological network dynamics. Krueger et al. [20] applied the HICN principle for the first time to the evolving sewer networks in a large Asian city with 4 million people. The authors found that sewer network types quickly evolve to become scale-free in space and time. In Jun and Loganathan [19] a dual mapping approach was used to describe the connectivity of isolation zones in water distribution networks.

Klinkhamer et al. [21] examined the co-location of existing road and sewer networks in a large Midwestern US city and homoscedasticity of subnets across the city but did not examine temporal evolution of these networks. In Mair et al. [22] the geospatial co-location of roads, pipes, and sewers was investigated using data set for three Alpine case studies, finding strong similarities between these networks. Studies on the coevolution of water infrastructure networks (water distribution and urban drainage) and road network are crucial when investigating functional interdependencies and cascading vulnerabilities across multiplex network layers. Examples are the flood-induced change in road traffic or the collapse of entire road segments causing flow disruptions in all networks to different extents.

In this paper, we present for the first time a topological analysis of three infrastructure networks coevolving over a century. The results of the dual mapping for a unique dataset of 11 time-stamped water distribution and urban drainage network states and 8 time-stamped road networks of the medium-size Alpine case study city, as the town and its infrastructure, evolved during the past 108 years, and the population tripled from about 40,000 to about 130,000. First results of this case study are presented in Zischg [23]. We investigated network topological metrics using the HICN dual mapping approach [16]. We observe that some infrastructure networks show node degree distributions that behave like truncated power-laws under the dual representation. However, this “scale-free” network characteristics depend on the network type and change over time. With the presented methodology, differences and similarities of patterns (e.g., vertex connectivity) and trends for the infrastructure development are obtained. This study includes an investigation of the sensitivity of the dual mapping approach, using different criteria to build the new graph. The reflected structural features, such as the backbone of the networks, were uncovered for each network type and remapped to the spatial map. A further analysis shows the pairwise co-location of high node degree components (“network hubs”) across different infrastructure network types, which builds the basis for analyzing disturbances and structural resilience.

#### 2. Data Analyses

##### 2.1. Network Connectivity

Node degree distribution is a significant topological property of complex networks. The degree of a node in an undirected network describes the number intersecting links and is calculated through the network’s adjacency matrix , where the degree of node is defined by the sum of the -th row of . For example, the node degree in social networks represents the number of contacts. Scale-free networks show node degree distributions that follow a Pareto power-law distribution [20, 21], with for , whereas random networks have Poisson distributed node degrees. We use the method proposed by Clauset et al. [24] to test the power-law hypothesis and determine scaling parameters of the node degree distributions for the various network states. By calculating the* p* value, an indicator for the goodness-of-fit is determined. In case the* p* value is greater than 0.1, the power-law is a plausible hypothesis for the data within given ranges. However, the definitive recipe to fit power-law distributions does not yet exist [25]. The mean node degree for undirected networks is defined as , where is the total number of edges and is the total number of vertices. In the limits a mean node degree of 2 indicates a tree-like network structure, and grid patterns or cyclic structures have mean node degrees around 4 [26]. Higher statistical moments of are also important, including the variance that reflects the dispersion around the mean [27].

Along with the node degree distribution, the characteristic path length (or average path length) is an important and robust measure of network topology. It quantifies the level of integration/segregation throughout the network. In water infrastructure and power grid networks energy losses are dependent on the characteristic path length. It is calculated by the average shortest path distance between all couples of nodes as follows:where is the number of vertices and denotes the shortest path between vertex and [3]. The probability density function of the shortest path lengths (between all couples of nodes for RNs; between all terminal nodes and the source or sink node, for WDNs and UDNs, respectively), can, for example, be considered as the approximation of the travel-time distribution with a nearly consistent distribution of flow velocities.

The local clustering coefficient of node describes the connectivity (number of edges* m*) among its* k* neighbors. A perfect cluster/clique indicates a full connection of all nodes/individuals. If an isolation of one node in the cluster occurs, the other nodes remain connected. Conversely, a indicates that node holds together all its neighboring nodes. Regarding infrastructures, a higher clustering coefficient indicates the existence of local and alternative flow paths in the network. In undirected graphs, for an individual node is defined as follows: The overall (average) clustering coefficient can be determined by averaging the local clustering across all nodes. When the network reveals a high , which is typical for regular lattice networks (high local efficiency), and has a small , as found in random networks (high global efficiency), then the network can be characterized as small-world network [30]. Telesford et al. [31] developed methodology to test the “small worldness” by comparing the network with its equivalent random and regular lattice networks, which have the same node degree distribution, as follows:where is the characteristic path length of the randomized network equivalent and is the clustering coefficient of the regularized network equivalent. A -value of 0 indicates a network that is in perfect balance between normalized values of high clustering and low characteristic path length. Negative values indicate a graph with more regular characteristics, whereas positive values indicate more random graph characteristics [31]. Small-world networks are significantly more clustered (segregated) than random networks and have the same characteristic path length as random (integrated) networks, making them locally and globally efficient, for example, for optimal information transfer [30].

##### 2.2. HICN Principle for Dual Mapping

The HICN approach emphasizes the functional topology of the network by aggregating components (e.g., pipes, conduits, and roads) with identical attributes (e.g., pipe diameter, pipe segments, and road type), while also maintaining a certain level of straightness (e.g., road sections) [16]. After reducing the network complexity with this “generalization model,” the aggregated edges are converted into vertices and the intersections are converted into edges. The resulting graph is the so-called “dual” (mapped) representation of the “primal” graph (see Figure 1). In addition to the edge class, the angular threshold is a second criterion used for the generalization model. It defines the maximum exterior convex angle of connected edges being merged [17].