# 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)

V

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

V

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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