Graph entropy

In information theory, the graph entropy is a measure of the information rate achievable by communicating symbols over a channel in which certain pairs of values may be confused.^[1] This measure, first introduced by Körner in the 1970s,,^[2]^[3] has since also proven itself useful in other settings, including combinatorics.^[4]

Definition

Let $G=(V,E)$ be an undirected graph. The graph entropy of $G$ , denoted $H(G)$ is defined as

H(G)=\min _{X,Y}I(X;Y)

where $X$ is chosen uniformly from $V$ , $Y$ is an independent set in G, and $I(X;Y)$ is the mutual information of $X$ and $Y$ .

Properties

Monotonicity. If $G_{1}$ is a subgraph of $G_{2}$ on the same vertex set, then $H(G_{1})\leq H(G_{2})$ .
Subadditivity. Given two graphs $G_{1}=(V,E_{1})$ and $G_{2}=(V,E_{2})$ on the same set of vertices, the graph union $G_{1}\cup G_{2}=(V,E_{1}\cup E_{2})$ satisfies $H(G_{1}\cup G_{2})\leq H(G_{1})+H(G_{2})$ .
Arithmetic mean of disjoint unions. Let $G_{1},G_{2},\cdots ,G_{k}$ be a sequence of graphs on disjoint sets of vertices, with $n_{1},n_{2},\cdots ,n_{k}$ vertices, respectively. Then $H(G_{1}\cup G_{2}\cup \cdots G_{k})={\tfrac {1}{\sum _{i=1}^{k}n_{i}}}\sum _{i=1}^{k}{n_{i}H(G_{i})}$ .

Additionally, simple formulas exist for certain families classes of graphs.

Edge-less graphs have entropy $0$ .
Complete graphs on $n$ vertices have entropy $\lg n$ , where $\lg$ is the binary logarithm.
Complete balanced k-partite graphs have entropy $\lg r$ where $r$ is the binary logarithm. In particular, complete balanced bipartite graphs have entropy $1$ .
Complete bipartite graphs with $n$ vertices in one partition and $m$ in the other have entropy $H\left({\frac {n}{m+n}}\right)$ , where $H$ is the binary entropy function.

Example

Here, we use properties of graph entropy to provide a simple proof that a complete graph $G$ on $n$ vertices cannot be expressed as the union of fewer than $\lg n$ bipartite graphs.

Proof By monotonicity, no bipartite graph can have graph entropy greater than that of a complete bipartite graph, which is bounded by $1$ . Thus, by sub-additivity, the union of $k$ bipartite graphs cannot have entropy greater than $k$ . Now let $G=(V,E)$ be a complete graph on $n$ vertices. By the properties listed above, $H(G)=\lg n$ . Therefore, the union of fewer than $\lg n$ bipartite graphs cannot have the same entropy as $G$ , so $G$ cannot be expressed as such a union. $\blacksquare$

Notes

↑ Matthias Dehmer; Abbe Mowshowitz; Frank Emmert-Streib (21 June 2013). Advances in Network Complexity. John Wiley & Sons. pp. 186–. ISBN 978-3-527-67048-2.
↑ Körner, János (1973). "Coding of an information source having ambiguous alphabet and the entropy of graphs.". 6th Prague conference on information theory: 411–425.
↑ Niels da Vitoria Lobo; Takis Kasparis; Michael Georgiopoulos (24 November 2008). Structural, Syntactic, and Statistical Pattern Recognition: Joint IAPR International Workshop, SSPR & SPR 2008, Orlando, USA, December 4-6, 2008. Proceedings. Springer Science & Business Media. pp. 237–. ISBN 978-3-540-89688-3.
↑ Bernadette Bouchon; Lorenza Saitta; Ronald R. Yager (8 June 1988). Uncertainty and Intelligent Systems: 2nd International Conference on Information Processing and Management of Uncertainty in Knowledge Based Systems IPMU '88. Urbino, Italy, July 4-7, 1988. Proceedings. Springer Science & Business Media. pp. 112–. ISBN 978-3-540-19402-6.

General References

Matthias Dehmer; Frank Emmert-Streib; Zengqiang Chen; Xueliang Li; Yongtang Shi (25 July 2016). Mathematical Foundations and Applications of Graph Entropy. Wiley. ISBN 978-3-527-69325-2.

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