Create continuous_graph_embedding.md

continuous_graph_embedding.md

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+# Continous Graph Embedding
+
+labels: ricci_tensor, riemannian_geometry, experimental, gnn, topology, differential_geometry
+
+Model the graph as a continuous manifold, model entities on the graph as points on the manifold, and a learnable shell parameter. 
+The shell functionally defines the entity as a region of the manifold.
+Define edge(from: Node, to: Node) as an indicator function that returns true if the veridical graph has an edge from x to y. 
+
+```python
+def edge(from: Node, to: Node):
+  """an indicator function that returns true if the veridical graph has an edge from x to y."""
+  d = distance(from, to)
+  return d < from.shell.projected_onto(d).norm()
+```
+
+for a simple spherical representation, we can let the shell be a radius term. Any nodes inside the shell's radius, we draw an edge to.
+
+## Expected Properties
+
+### Semantic arithmetic over graphs
+
+semantic arithmetic where we are conditioning on a graph over inputs. this is actually completely identical to a normal conditioning vector.
+
+BUT: we can take advantage of the shell term to operate as a weighting function, and invert the spatial warps to normalize the weightings in the topology of the space.
+
+Apply this representation to a dataset to learn the local warp tensor. Ricci tensor.
+
+Can we use this to fit just the ricci tensors on a frozen representation? use that to measure the local curvature of the space?