Examples of New Edge Pruning 46

Since Proposition 2–6 provides us a rule to prune new edges in IE when the network has excess supply and incentive exists over new edges, we can examine some instances where pruning of a new edge may or may not occur.

Observation 2–7: If new edges do not connect to transshipment nodes, then the only possible types of new edges are source to source, source to destination or destination to destination.

New edges that connect source nodes to source nodes create a super supply node. The super supply node may be able to use the combined supply from both nodes to deliver resources to portions of the network that were previously unsatisfied. New edges that connect destination nodes to destination nodes create the equivalent of a super demand node, which would behave like a destination node with a much higher resource requirement. Behavior of super supply and super demand nodes in a network with excess supply can be easily seen by modifying a previous example to have excess supply. Consider Figure 16, with the worst-case attack shown. In Figure 16, the addition of edge (4, 6) creates an equivalent super demand of edges 3,4,5,6 to supply node 2. Since source

node 2 has excess supply, the addition of edge (4, 6) would be considered in the first layer of IE. But, even though node 2 has excess supply, it does not completely satisfy the demand. The second layer of enumeration to add the second defense would consider the addition of edge (1, 2) to the network which creates the equivalent of a super supply node. The new edge (1, 2) is added to the enumeration algorithm because an incentive exists somewhere in the network, specifically, the unsatisfied demand from node 3 that could exist on the path (1, 2, 4, 6, 5, 3). Since incentive exists, the edge (1, 2) is added to the network.

Figure 16. Worst-case Attack with Excess Supply Nodes

The source to destination edge is the last type of new edges that may be added to a network. There are two possibilities for new edges with source to destination nodes.

Possibility 1: The new edge is source to destination with previously unsatisfied

demand. It should be obvious that addition of this new edge could improve the objective function value since the source node has extra resources available to send on the new edge to a destination that was previously penalizing the objective function for having unfulfilled demand. If the cost to transport between source and destination is higher than the penalty for unfulfilled demand, then incentive would not exist.

Possibility 2: Source to destination with previously satisfied demand. There are

Situation 2a: If the cost of moving resources to the destination node can be

improved by adding the new edge to the network, then that new edge would be a candidate for addition.

Situation 2b: If the resource that is already satisfying the demand at the

destination could be better utilized by satisfying some other demand on the network, then addition of the new edge will be an improvement and a candidate for addition.

Situation 2c: If the addition of the new edge does not change the movement of

previous resources, then does the satisfied destination node appear to be the equivalent to a super supply node with the addition of this new edge? The satisfied demand node could look like an equivalent supply node if there is another existing incentive in the network for the edge to be added. While the satisfied demand node itself may not present an incentive for the edge to be added, there may be another efficiency in a different part of the network that will make the satisfied demand node appear to be the equivalent of another supply node.

Situation 2d: If the addition of the new edge does not change the movement of

resources and there is no other existing incentive in the network for the edge to be added, then the candidate edge will not affect a change in the efficiency of the network nor change the worst-case attack. This new edge should be temporarily excluded from consideration in an IE algorithm.

In order to illustrate how new edge pruning works on the test network, consider Figure 17. Figure 17 is a portion of the IE tree on the test network with a defense budget of four units and an attack budget of one unit. The black arrows in Figure 17 show the defenses that could be pruned from the tree. The new edges labeled for pruning have the exact same worst-case attack and objective function value as the parent node. In this case, these defenses do not make any improvement to the function of the system, in terms of system cost or worst-case attack. These potential defenses are ruled to be a waste of defenses since improvement in objective or change in worst-case attack did not occur, and can therefore be pruned from the tree. These edges will be reconsidered as additional defenses in different child nodes later on as the IE strategy continues.

Figure 17. Pruning New Edges in an IE Tree