Measures of Complexity of a Graph or Network
Quantitative Measures of Network Complexity
Algorithmic complexity of a graph
Correlation of automorphism group size and topological properties with program-size complexity evaluations of graphs and complex networks They show that: Kolmogorov complexity can capture group-theoretic and topological properties of abstract and empirical networks, ranging from metabolic to social networks, to small synthetic networks.
We derive these results via two different Kolmogorov complexity approximation methods applied to the adjacency matrices of the graphs and networks. The methods used are the traditional lossless compression approach to Kolmogorov complexity, and a normalised version of a Block decomposition method (BDM) based on algorithmic probability theory.
Complexity and edge density
Complexity is minimal for empty or complete graphs
Kolmogorov Random Graphs and the Incompressibility Method
Complexity vs symmetry of the graph
The symmetry is measured by the cardinality of the Graph automorphism group. The following plot from empirical complex networks shows that they are indeed negatively correlated. The graph automorphism is normalized, and NBDM refers to the normalized BDM.
Information content in a graph
Entropy and the Complexity of Graphs Revisited
Information Content of Colored Motifs in Complex Networks