Basic Examples (2) 

Compute entropy for a nearest neighbor graph of 20 random points:

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Build the nearest neighbor graph of a perfect square lattice with unit lattice constant:

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Compute the graph entropy and verify it is equal to zero as expected by symmetry:

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Scope (3) 

Define a 2D set of point arranged in a square lattice, add distortions to have all edges unequal as in a 2D disordered system and build the corresponding NearestNeighborGraph with all nearest neighbor connections within a given radius:

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Compute the graph entropy:

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Check it is equal to the natural logarithm of the number of distinct edges in the graph:

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Options (2) 

Treat some distance values as equal if they differ by an optional threshold "MinThreshold" set to a given positive value. This introduces some order into the graph layout:

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Check that graph entropy is lower than in the case without rounding factor:

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Applications (2) 

Build a linear graph mapped to the coordinates of a one-dimensional Fibonacci quasiperiodic chain with nit iterations and spacing ratio between nodes tau:

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Computes its graph entropy:

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Check that the numerical value of the graph entropy converges to the universal value 0.665:

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This universal value 0.665 is also independent of the spacing ratio between graph nodes:

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If a point symmetry element of order C_n is present in the graph, then the graph entropy is equal to ln(E/n) where E is the total number of edges in the graph and n is the order of the rotation symmetry.

Start from a slightly disordered square lattice and apply rotations of 0, π/2, π and -π/2 to the starting lattice points to get three additional sub-lattices:

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Build the overall graph connecting all points belonging to the four sub-lattices:

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Compute numerically the graph entropy for this system:

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Check that it is close to the natural logarithm of the number of graph edges divided by the symmetry order:

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Properties and Relations (2) 

Define a grid of points:

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As noise is added to the graph, the entropy increases:

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Neat Examples (2) 

Build a 2D graph mapped to the coordinates of a two-dimensional Fibonacci quasiperiodic array with nit iterations:

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Check that the numerical value of the graph entropy still converges to the universal value 0.665 known from 1D case (computation takes a few seconds):

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A semi-regular set of points obtained by alternatively deleting sites from a square lattice:

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Check that the graph entropy is still zero as expected by symmetry:

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