I will start my question by describing a particular graph.

1. There are a finite, fixed, number of nodes. labeled {A,B....Z} (so for this example there are 24 nodes).

2. Each node consists of another graph (tree) with exactly 3 nodes in 3 depths.

That means we have a graph where the nodes themselves are graphs.

3. We will take the labels for each of these new lower graphs from the set (a,b,....z),

where we can use any small letter from the alphabet any number of times.

4. Since there are 72 lower letter nodes in total derived from the set of 24 letters of the alphabet, the nodes (A,B....Z) share nodes.

E.g. A={a,d,u}

B={b,u,p}

C={c,b,a};

the nodes "A" and "B" share the lower level node labelled "u".

In fact, a crucial point is that this is the exact same node.

Additionally the rule for populating the nodes {A,B,....Z} with smaller nodes is,

take the exact same letter as the Original node and label the root of the smaller graph with it in small letters,

so for "A" it's "a",

then choose any two other letters at random from the set {a,b....z}.

E.g. B={b,g,t}

Once we have done this for all {A,B...Z}, we save this configuration and use it from now on.

So the topology of this graph takes on a new form.

We could not simply have taken the smaller nodes and connected them and had a normal graph, because in some sense, the nodes in the small node normal graph are grouped in three's, and this grouping has an order .

And the real interest in all of this is that we want to connect the larger nodes,( A,B...Z) with rules derived from considering the underlying graph with the smaller nodes.

note: We have introduced a random element in that we arranged the paths between "a,b...z" randomly (while populating the graph), but we are going to start with a problem framed in terms of nodes (A,B...Z), and generate a solution by iteratively adjusting the (a...z). path arrangement till it is no longer random.

It is clear that the nodes (A,B..Z) are connected. Also the paths between any two Nodes "A" to "N" becomes highly abstract. If we take the path in a line from say "A" to "N", the path that the underlying small letter nodes (a,....x) , will have will not itself be a straight line. It will consist of a 3-dimensional lattice that is unique to the exact path taken from "A" to "N".

Another point;

If we select a path with the nodes (A,B...Z) , and the representation of the underlying graph (a,b...z) was initialized somehow,

then each path A"" to "N" will have associated with it a graph with nodes (a to z).

Also, if we take any sub graph from the graph (A,B...Z) the underlying graph will be highly complex and highly fractal, even if its initialization was random.

Now for the problem I've been leading to:

if we select a node "A" and form the root of the overlying graph, {A,B,...Z} and then take another instance of this overlying graph and raise it to a select B as its root,

what is the most optimal (simplest) function we can have that can transform the graph with A as its root into the one with B as its root?

Because they contain exactly the same material, there will be many solutions, so we stick with the simplest.

Another question would be

how could we adjust the arrangement we chose at random for the underlying layer of small letter nodes to make this function simpler.

1. There are a finite, fixed, number of nodes. labeled {A,B....Z} (so for this example there are 24 nodes).

2. Each node consists of another graph (tree) with exactly 3 nodes in 3 depths.

That means we have a graph where the nodes themselves are graphs.

3. We will take the labels for each of these new lower graphs from the set (a,b,....z),

where we can use any small letter from the alphabet any number of times.

4. Since there are 72 lower letter nodes in total derived from the set of 24 letters of the alphabet, the nodes (A,B....Z) share nodes.

E.g. A={a,d,u}

B={b,u,p}

C={c,b,a};

the nodes "A" and "B" share the lower level node labelled "u".

In fact, a crucial point is that this is the exact same node.

Additionally the rule for populating the nodes {A,B,....Z} with smaller nodes is,

take the exact same letter as the Original node and label the root of the smaller graph with it in small letters,

so for "A" it's "a",

then choose any two other letters at random from the set {a,b....z}.

E.g. B={b,g,t}

Once we have done this for all {A,B...Z}, we save this configuration and use it from now on.

So the topology of this graph takes on a new form.

We could not simply have taken the smaller nodes and connected them and had a normal graph, because in some sense, the nodes in the small node normal graph are grouped in three's, and this grouping has an order .

And the real interest in all of this is that we want to connect the larger nodes,( A,B...Z) with rules derived from considering the underlying graph with the smaller nodes.

note: We have introduced a random element in that we arranged the paths between "a,b...z" randomly (while populating the graph), but we are going to start with a problem framed in terms of nodes (A,B...Z), and generate a solution by iteratively adjusting the (a...z). path arrangement till it is no longer random.

It is clear that the nodes (A,B..Z) are connected. Also the paths between any two Nodes "A" to "N" becomes highly abstract. If we take the path in a line from say "A" to "N", the path that the underlying small letter nodes (a,....x) , will have will not itself be a straight line. It will consist of a 3-dimensional lattice that is unique to the exact path taken from "A" to "N".

Another point;

If we select a path with the nodes (A,B...Z) , and the representation of the underlying graph (a,b...z) was initialized somehow,

then each path A"" to "N" will have associated with it a graph with nodes (a to z).

Also, if we take any sub graph from the graph (A,B...Z) the underlying graph will be highly complex and highly fractal, even if its initialization was random.

Now for the problem I've been leading to:

if we select a node "A" and form the root of the overlying graph, {A,B,...Z} and then take another instance of this overlying graph and raise it to a select B as its root,

what is the most optimal (simplest) function we can have that can transform the graph with A as its root into the one with B as its root?

Because they contain exactly the same material, there will be many solutions, so we stick with the simplest.

Another question would be

how could we adjust the arrangement we chose at random for the underlying layer of small letter nodes to make this function simpler.

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