I have code that generates a Matrix `Mat` (`Mat =` [data](https://pastebin.com/bTtguLL2)) that, when the graph is plotted, `AdjacencyGraph[Mat]` gives a graph.      

Now, there are many inbuilt functions for `GraphLayout`. Once we fix a particular `GraphLayout`. For a particular `GraphLayout`, we get the vertex coordinates when we use `GraphEmbedding`.     


Let us take a simple example. If I have a matrix for a square geometry (square lattice). Then its graph will have a square face. Following the previous paragraph strategy, I will get the coordinates for the vertices like `{{0,0}, {0,1},...{0,Lx},...{Ly,Lx}}`, where `Lx` and `Ly` is the number of vertices along `x` and `y`. That is related to the size of the matrix which is `(Lx X Ly) X (Lx X Ly)`. One can do this for honeycomb or triangle lattices.


However, my graph has higher connectivity that doesn't fall in the above case (meaning it's not Euclidean). If I use the existing `GraphLayout`, then the coordinates are not as per the hyperbolic plane. For instance, if I use the `GraphLayout -> "SpringElectricalEmbedding"` then it uses the particular mechanism (minimizing the spring energy, not important here) to have a graph layout. But it starts to be a problem if I plot different (hyperbolic) graphs.      

As an input, I will give the matrix that has a particular connectivity between the vertices. For example below, I give as an input a lattice `{8,3}` in the matrix form, which is a polygon with 8 sides, and at each vertex, three such polygon (octagon) meets. As an output, I want the coordinates for all the vertices so that the coordinates satisfy a hyperbolic distance formula `distance[z1,z2]`. So is there a way to create a custom `GraphLayout` that does it?    
       
On idea from my side. Maybe we can create an `EdgeWeight` that encodes the distance function that in turn dictates how two edges should be connected? (no idea how good it is, maybe not at all)     

**MWE**:      

    Mat = data;
    graphMat = GraphEmbedding[AdjacencyGraph[ConnectMat, VertexCoordinates -> Coords, VertexLabels -> None, ImageSize -> Large, GraphLayout -> "SpringElectricalEmbedding"]
     distance[z1_, z2_] :=  1./2 ArcCosh[1 + (2 Abs[z1 - z2]^2)/((1 - Abs[z1]^2) (1 - Abs[z2]^2))];
[![Graph for the above data][1]][1]

Update: Can we use `EdgeWeight` function perhaps? Where this `EdgeWeight` accounts for the hyperbolic distance?


  [1]: https://i.sstatic.net/NSwiX.png