Impacts of Node Connectivity on Reliability of Water Distribution Networks - Pattern Recognition for Reliability Assessment of Water Distribution Networks - page 140

randomized (range of) multipliers for major hydraulic parameters such as nodal discharge,
pipe diameters, or PDD threshold. All results are saved in the text file readable by MS Excel,
which enables easy graphical presentation.

6.4 ANALYSIS OF NETWORK CONNECTIVITY BASED ON GRAPH THEORY

As mentioned in Chapter 4, in the terminology of graph theory, water distribution networks
consisting of nodes and links are actually graphs composed of vertices and edges ,
respectively. Equally, the degree of each vertex, showing the number of corresponding edges,
will mean the number of pipe connections in each node. Graph will be directed if the edges
have fixed direction (for instance, pipes of fixed flow/velocity direction). In this case the
degree of vertex can be in- or out-degree, showing the number of edges directed in or out of
the vertex, respectively (i.e. the pipes with inflow or outflow to the node). If the edges have
arbitrary direction, the graph is undirected .

Besides the vertex degree, several other parameters exist in the analysis of graph structures
i.e. network configurations, which can be used to investigate particular structural
characteristics. As most of these can be measured in different ways, the final decision on
appropriate parameters may depend on the specific domain of the networks and can even be a
result of trial and error process. The list of the most common connectivity terminology and
related aspects of network configuration is given in Table 6.7, followed by short description
of the parameters specifically analysed in this research. The terminology typical for water
distribution field is further extensively used.

Table 6.7 Connectivity terminology and network configuration (adapted from http://www.itee.uq.edu.au/)

Parameter Aspects of network configuration

Degree (in- and out-) Shows importance of a node, based on how connected it is.

Degree distribution Set of related properties, such as average shortest path length; indicates probability

of creating disconnected components through node/link removal.

Shortest path length Distance between two nodes; shows degree of mutual influence.

Clique Identifies highly interconnected sub-networks.

Cluster Identifies highly connected nodes.

Pivots, Cut-points Identifies nodes crucial to keep the network connected.

Bridges Identifies links crucial to keep the network connected.

Node-connectivity,
Link-connectivity

Indicates buffer the network layout has before becoming disconnected.

Centralization Indicates how much the network centres on a single node or group of nodes.

Betweenness centrality,
Closeness centrality

Shows importance of a node based on its relationship to other nodes in the network.

6.4.1 Node Degree

In network of n nodes, the node degree, Deg i , indicates the number of connections to node i .
If the network links are directed, in- and out-degree will show the number of connections to-
and from the node, respectively. The average nodal degree has the same meaning as the