Scaling-Laws of Flow Entropy with Topological Metrics of Water Distribution Networks

Abstract

Robustness of water distribution networks is related to their connectivity and topological structure, which also affect their reliability. Flow entropy, based on Shannon's informational entropy, has been proposed as a measure of network redundancy and adopted as a proxy of reliability in optimal network design procedures. In this paper, the scaling properties of flow entropy of water distribution networks with their size and other topological metrics are studied. To such aim, flow entropy, maximum flow entropy, link density and average path length have been evaluated for a set of 22 networks, both real and synthetic, with different size and topology. The obtained results led to identify suitable scaling laws of flow entropy and maximum flow entropy with water distribution network size, in the form of power-laws. The obtained relationships allow comparing the flow entropy of water distribution networks with different size, and provide an easy tool to define the maximum achievable entropy of a specific water distribution network. An example of application of the obtained relationships to the design of a water distribution network is provided, showing how, with a constrained multi-objective optimization procedure, a tradeoff between network cost and robustness is easily identified.

Keywords: flow entropy; power laws; robustness; scaling laws; water distribution networks.

Figure 1

Scatter plots and best fitting power–laws of: (a) entropy vs. number of nodes; (b) entropy vs. number of pipes; (c) entropy vs. link density; (d) entropy vs. network average path length.

Figure 2

Scatter plots and best fitting power–laws of: (a) maximum entropy vs. number of nodes; (b) maximum entropy vs. number of pipes; (c) maximum entropy vs. link density; (d) maximum entropy vs. network average path length.

Figure 3

Scatter plots and best-fitting power law equations: (a) S/m vs. number of nodes; (b) MS/m vs. number of nodes. The dashed lines represent the expected scaling of flow entropy for a network with equiprobable flow paths.

Figure 4

Scaling of maximum flow entropy with number of nodes, for networks with various numbers of loops l. The dashed lines represent maximum flow entropy of networks with fixed average node degree $k=2$ and $k=4$. The dots represent the considered set of 22 WDNs.

Figure 5

Layouts of the water distribution networks for which the multi-objective optimal design procedure based on maximum flow entropy has been applied: (a) Fossolo; (b) Skiathos.

Figure 6

Pareto fronts of the proposed multi-objective optimal network design procedure (flow entropy and total cost of network pipes): (a) Fossolo; (b) Skiathos. The red dots correspond to network layouts before optimization.