Entropy and the Complexity of Graphs Revisited

Abstract

This paper presents a taxonomy and overview of approaches to the measurement of graph and network complexity. The taxonomy distinguishes between deterministic (e.g., Kolmogorov complexity) and probabilistic approaches with a view to placing entropy-based probabilistic measurement in context. Entropy-based measurement is the main focus of the paper. Relationships between the different entropy functions used to measure complexity are examined; and intrinsic (e.g., classical measures) and extrinsic (e.g., Körner entropy) variants of entropy-based models are discussed in some detail.

Toward Measuring Network Aesthetics Based on Symmetry

“…A well-known example is the graph entropy measure introduced by Rashevsky [25] and developed extensively by Mowshowitz [21], which is based on the vertex orbits of the automorphism group of a graph. Since automorphisms are permutations of graph elements preserving structure, this measure can be regarded as an index of symmetry [16,26]. Many other information-theoretic and non-information-theoretic graph measures for capturing features of the complexity of a network have been proposed (e.g., [19,20,[27][28][29]).…”

“…This approach is based on comparative graph analysis [15]. In our current work, however, we argue that properties expressing structural features can be measured by single graph complexity measures (e.g., [16]), however difficult it may be to relate the numerical value of a measure to any specific structural features of a graph.…”

“…The measures examined in [13] take account of geometrical properties of networks such as keeping multi-edge paths as straight as possible (i.e., avoiding zig-zag paths). Geometrical features play no role in structural measures of graphs, since such measures only take account of features such as density, symmetry, sparsity, branching, and so forth (e.g., [16][17][18]). This situation is shown by way of example in Figure 1; a purely structural graph measure cannot distinguish those two graphs.…”

Deconvoluting simulated metagenomes: the performance of hard- and soft- clustering algorithms applied to metagenomic chromosome conformation capture (3C)

“…One information-theoretic formulation with acceptable computational complexity is the non-parametric entropy H L (Equation 8) associated with the non-zero eigenvalue spectrum of the normalized Laplacian matrix N = D −1/2 LD −1/2 , where L = D − A is the regular Laplacian matrix, D is the degree matrix and A the adjacency matrix of a graph (Dehmer and Mowshowitz, 2011;Mowshowitz and Dehmer, 2012).…”

New Upper Bound and Lower Bound for Degree-Based Network Entropy

“…After that, many applications of the complex networks based on the entropy associated with structural information were published and various algorithms to analyze the structural complexity were proposed [21][22][23][24][25]. In [26][27][28], the entropy and information theory in graphs and networks were elaborated systematically. The entropy method is one of the most important methods to describe the structural information of the complex networks, especially the degree-based network entropy [29][30][31][32].…”