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Graph stitching

January 2nd, 2016 • Comments Off on Graph stitching

Graph stitching describes a way to merge two graphs by adding relationships/edges between them. To determine which edges to add, a notion of node types is used (based on node naming would be easy :-)). Nodes with a certain type can be “stitched” to a node with a certain other type. As multiple mappings are possible, multiple result/candidate graphs are possible. A good stitch is defined by:

So based on node types two graphs are stitched together, and than a set of candidate result graphs will be validated, to especially satisfy the second bullet.

Let’s use an example to explain this concept a bit further. Assume the electrical “grid” in a house can be described by a graph with nodes like the power outlets and fuses, as well as edges describe the wiring. Some home appliances might be in place and connected to this graph as well. Hence a set of nodes describing for example a microwave (the power supply & magnetron), are in this graph as well. The edge between the power supply and the power outlet describe the power cable. The edge between the power supply and the magnetron is the internal cabling within the microwave. This graph can be seen in the following diagram.

(Click on enlarge)

(Click on enlarge)

The main fuse is connected to the fuses 1 & 2. Fuse 1 has three connected power outlets, of which outlet #2 is used by the microwave. Fuse 2 has two connected power outlets. Let’s call this graph the container from now on.

Now let’s assume a new HiFi installation (consisting of a blu-ray player and an amplifier) needs to be placed within this existing container. The installation itself can again be described using a simple graph, as shown in the following diagram.

(Click to enlarge)

(Click to enlarge)

Placing this request graph into the container graph now only requires that the power supplies of the player and amplifier are connected to the power outlets in the wall using a power cord. Hence edges/relationships are added to the container to stitch it to the request. This is done using the following mapping defition (The power_supply and power_outlet are values for the attribute “type” in the request & container graph):

{
    "power_supply": "power_outlet"
}

As there is more than one possible results for stitching two graphs, candidates (there are 2 power supplies and 5 power outlets in the mix) need to be examined to see if they make sense (e.g. the fuse to which the microwave is connected might blow up if another “consumer” is added). But before getting to the validation, the number of candidates graphs should be limited using conditions.

For example the HiFi installation should be placed in the living room and not the kitchen. Hence a condition as follows (The power outlet nodes in the container graph have an attribute which is either set to ‘kitchen’ or ‘living’) can be defined:

condition = {
    'attributes': [
        ('eq', ('bluray_p', ('room', 'living'))),
        ('eq', ('amp_p', ('room', 'living'))),
    ]
}

Also the amplifier should not be placed in the kitchen while the blu-ray player is placed in the living room. Hence the four nodes describing the request should share the same value for the room attribute. Also it can be defined that the power supplies of the player & amplifier should not be connected to the same power outlet:

condition = {
    'compositions': [
        ('share', ('room', ['amp', 'amp_p', 'bluray', 'bluray_p'])),
        ('diff', ('amp_p', 'bluray_p'))
    ]
}

This already limits the number of candidate resulting graphs which need to validated further. During validation it is determined if a graph resulting out of a possible stitching falls under the definition of a good stitch (see earlier on). Within the container – shown early – the nodes are ranked – red indicating the power outlet or fuse heavy loaded; while green means the power outlet/fuse is doing fine. Now let’s assume no more “consumers” should be added to the second outlet connected to the first fuse as the load (rank) is to high. The high load might be caused by the microwave.

All possible candidate graphs (given the second condition described earlier) are shown in the following diagram. The titles of the graphs describe the outcome of the validation, indicating that adding any more consumers to outlet_2 will cause problems:

(Click to enlarge)

(Click to enlarge)

The container and request are represented as shown earlier, while the stitches for each candidate resulting graph are shown as dotted lines.

Graph stitcher is a simple tool implements a simple a stitching algorithm which generates the possible graphs (while adhering all kinds of conditions). These graphs can than be validated further based on different validators. For example by looking at number of incoming edges, node rank like described before, or any other algorithm. The tool hence can be seen as a simple framework (with basic visualisation support) to validate the concepts & usefulness of graph stitching.