## 1. Introduction

[2] Even if the use of case studies is common for building theories and the analysis of systems [*Eisenhardt*, 1989], they are nevertheless investigations of only specific pieces of our world, a subset of cases that can differ greatly from one another. Generalizing conclusions from individual investigations of networks is questionable and sometimes inappropriate. The number of available case studies for analysis of water supply networks is restricted due to tedious data collection, difficulties in model setup, and sensitive network data. In order to overcome these obstacles, this paper will present a novel method for the generation of network systems with different characteristics that will assist case study research. The example featured in this work involves the generation of water supply systems (WSSs), but the methodology can likewise be applied to other (natural or man-made) network systems (e.g., rivers, sewers, railroads, electricity).

[3] The field of graph theory defines a network as a directed graph with edges that have capacities. In hydraulic networks the term capacity denotes transport ability and is represented by the pipe diameter and the flow that is delivered from sources to sinks. Simple, complete, and planar graphs; trees; or fractals are examples of possible network shapes. A simple graph has no loops and not more than one edge between any two different vertices. A complete graph is a simple graph in which every pair of distinct vertices is connected by a unique edge. A planar graph is a graph that can be drawn on a plane so that the edges intersect only at the vertices (i.e., two edges do not cross each other at any other point). A tree [*Aldous*, 1993] is a graph in which any two vertices are connected by exactly one path, (i.e., any connected graph without loops is a tree).

[4] A fractal specifies a geometric shape that is composed of smaller components, each of which is a replica of its original shape. Researchers have reported on the fractal nature of rivers [*Tarboton et al.*, 1988; *Rinaldo et al.*, 1992; *Rinaldo et al.*, 1993]. *Tarboton et al.* [1996] presented fractal dimensions by means of Tokunaga cyclicity parameters. As Tokunaga fractals are tree shaped, rivers are assumed to be branched. Building upon this theory, *Cui et al.* [1999] introduced a stochastic Tokunaga model for stream networks.

[5] Urban drainage systems, like rivers, have been found to resemble tree graphs. *Ghosh et al.* [2006] used dendritic and space-filling Tokunaga fractal tree geometry to generate artificial urban drainage systems. *Möderl et al.* [2009] developed a case study generator based on the Galton–Watson branching process to generate virtual urban drainage systems. Similarly, *Urich et al.* [2010] investigated the generation of virtual sewer systems using an agent-based modeling approach. Therein, sewer systems are designed based on a stochastically generated virtual urban fabric detailed by *Sitzenfrei et al.* [2010].

[6] Even though water supply systems consist of loops, their degree of meshing does not allow them to be represented as planar or tree graphs. The aforementioned methods are therefore not appropriate to generating WSSs and other looped systems.

[7] The Modular Design System (MDS) approach was introduced as a consequence of the idea of identifying simple building blocks in complex networks [*Milo et al.*, 2002], *Möderl et al.* [2007]. This method forms the basis for the systematic generation of virtual WSSs. In this paper, the generation of a set of virtual systems is demonstrated first. Second, the characteristics of the systems are analyzed and compared with three real-world networks. Third, as an example for an application of such a set, we analyze one for the effect of an increase in water demand. Other applications that employ sets of case studies are numerous. Their most common use involves the (model-based) test of a range of measures for overcoming various design or management scenarios (e.g., calibration algorithms, sensor placements, rehabilitation strategies). As the sets of case studies encompass a fairly broad range of characteristics, results from their use in research will provide more credibility to the corresponding output. In sensitivity analysis, virtual case studies provide the gradual variation of parameter values required to assess their influence on model output. A final aspect worth mentioning is that sets of virtual case studies are essential in software testing and fixing bugs.