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Dimension and Euler Characteristics of Graphs

Two representations of a selected graph are shown. On the left, the vertices are labeled with the curvatures. They add up to the Euler characteristic of the graph. On the right, you see the graph with local dimensions. The arithmetic mean is the dimension of the graph.

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Dimension was introduced in [1]. While formally close to the Hausdorff–Uhryson dimension, it is different because the Hausdorff–Uhryson is trivial for graphs.
Curvature goes back to combinatorial curvature introduced by Gromov. It is defined in any dimension in [2]. The statistics for curvature and dimension are covered in [3], where explicit formulas for the average dimension and Euler characteristic of Erdos–Renyi probability spaces can be found. [4] gives an even faster way to compute the Euler characteristic.
The main ingredients are local quantities curvature and local dimension . The sum of all curvatures is the Euler characteristic: this is a Gauss–Bonnet–Chern theorem found in [2], where it is explored in a more geometric setting and where remarkable similarities with differential geometry exist.
The average of all local dimensions is by definition the dimension of the graph. Dimension is a quantity that can be useful to characterize or measure concrete networks. It is fairly well behaved in the sense that also in a probabilistic percolation setting, dimension depends nicely on parameters.

References
[1] O. Knill. "A Discrete Gauss-Bonnet Type Theorem." arXiv. (Sep 13, 2010) arxiv.org/abs/1009.2292.
[2] O. Knill. "A Graph Theoretical Gauss–Bonnet–Chern Theorem." arXiv. (Nov 23, 2011) arxiv.org/abs/1111.5395.
[3] O. Knill. "On the Dimension and Euler Characteristic of Random Graphs." arXiv. (Dec 24, 2011) arxiv.org/abs/1112.5749.
[4] O. Knill. "A Graph Theoretical Poincare–Hopf Theorem." arXiv. (Jan 5, 2012) arxiv.org/abs/1201.1162.
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