Function to draw densely interconnected layers in graph (e.g., neural net)

Question

I can use Graph[] to manually draw the shape of a very simple set of interconnected layers (for instance, a small neural network):

Graph[{1 -> 3, 2 -> 3, 1 -> 5, 2 -> 5, 1 -> 4, 2 -> 4, 3 -> 6, 4 -> 6, 5 -> 6}]

produces:

How can I programmatically generate the list of connections from a list of layer sizes? For instance, the above would be: layerCounts = { 2, 3, 1} and I'm hoping for a function Dense that might be invoked like Graph[Dense[layerCounts]]

Answer 1

This one constructs an AdjacencyMatrix that can be input to AdjacencyGraph:

f = Module[{index = {1, 1 + First@#}},
    AdjacencyGraph @ SparseArray[Band[# &[index, index += #]] ->
         ConstantArray[1, #] & /@ Partition[#, 2, 1], {1, 1} Total@#]] &;

where the function # &[index, index += #] is from this post by Mr.Wizard.

Examples:

f@{4, 3, 3, 2, 1} 

SetProperty[f[{5, 3, 1, 3, 5}] ,
 {VertexLabels -> Placed["Name", Center], VertexSize -> Large, VertexLabelStyle -> 20}]

SetProperty[f[#]  , {VertexLabels -> Placed["Name", Center],
    GraphLayout -> {"MultipartiteEmbedding", "VertexPartition" -> #},
    VertexSize -> Large, VertexLabelStyle -> 14}] &@ RandomInteger[{1, 5}, 10]

Answer 2

There might be a built-in that would simplify this, but if not this should get the job done:

fn[lc : {__Integer?Positive}] :=
 Join @@ Tuples /@ 
    Rule @@@ Partition[Internal`PartitionRagged[Range@Tr@lc, lc], 2, 1] // Graph

fn[{2, 3, 1}]

A bit shorter of node names may be arbitrary:

fn[lc : {__Integer?Positive}] :=
 Join @@ Tuples /@ Rule @@@ Partition[MapIndexed[#2 &, Range@lc, {2}], 2, 1] // Graph