Generalization of Tensors
Generalization of Tensors
Put a scalar field on a graph: just weight the nodes (e.g. put there)
V
r
Put a tensor field by weighting the edges of a directed graph (e.g. the aggregate from ends of an edge)
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r
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GridGraph[{10,10}]
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Vector field: assign a value to every edge
Rank 2 tensor: assign values to pairs of edges
[[ Exactly like the velocity field in a CA fluid ]] [ discrete particle on a directed edge ]
Example of an intrinsic computation: difference of between nodes at the ends of an edge
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r
Don’t get to pick up individual indices; only sum or project in the direction of a geodesic
Sum over a graph neighborhood of T[i,j]
δ
ij
Integral over a ball of an isotropic tensor would then be
What is the analog of spherical harmonics for a graph?
Imagine a PDE or a CA operating on the system
Trivial spreading CA gives volume.....
Look at evolution of some distribution on edges etc.
Parallel transport around cycles in graph??
Sierpinski
Sierpinski
Sierpinski graph:
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GraphData[{"SierpinskiTetrahedron",5}]
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GraphPlot3D[%]
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GraphPlot3D[GraphData[{"SierpinskiTetrahedron",2}],GraphLayout"SpringElectricalEmbedding"]
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SierpinskiMesh[4]
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MeshConnectivityGraph[%]
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