A Large Scale Analysis of Information-Theoretic Network Complexity Measures Using Chemical Structures

Abstract

This paper aims to investigate information-theoretic network complexity measures which have already been intensely used in mathematical- and medicinal chemistry including drug design. Numerous such measures have been developed so far but many of them lack a meaningful interpretation, e.g., we want to examine which kind of structural information they detect. Therefore, our main contribution is to shed light on the relatedness between some selected information measures for graphs by performing a large scale analysis using chemical networks. Starting from several sets containing real and synthetic chemical structures represented by graphs, we study the relatedness between a classical (partition-based) complexity measure called the topological information content of a graph and some others inferred by a different paradigm leading to partition-independent measures. Moreover, we evaluate the uniqueness of network complexity measures numerically. Generally, a high uniqueness is an important and desirable property when designing novel topological descriptors having the potential to be applied to large chemical databases.

Mapping and discrimination of networks in the complexity-entropy plane

“…However, while the large variety of present metrics allows for a quantification of a network's particular macroscopic and microscopic structure, it still remains a subject of current research to (i) assess the actual complexity of a network based on these sets of characteristics [18], and (ii) to determine distinct sets of properties for certain classes of networks, such as infrastructure or social networks, in order to objectively and comprehensively distinguish between them. While there exists a variety of such complexity measures [19], most of them are tailored to specific applications and have so far not been successfully applied to intercompare different types or classes of networks as in this respect they often lack a meaningful interpretation [20].…”

Entropy of Weighted Graphs with Randi´c Weights

“…To the best of our knowledge, this problem has received very little attention so far with only a few exceptions, e.g., [61]. We examined extremal properties of our entropy definition for special graph classes.…”

Information Theory of Networks

“…Also, it attains maximum entropy for asymmetric graphs. However, it has been shown [41] that these symmetry-based measures possess little discrimination power. The reason for this is that many non-isomorphic graphs have the same orbit structure and, hence, they can not be distinguished by this index.…”

“…As the measure depends on correlations between degrees of pairs of vertices [65], it is not surprising that its discrimination power is low, see [41].…”

“…An important problem is to evaluate the degree of degeneracy of a measure by several quantities such as the sensitivity measure due to Konstantinova [40]. Note that the discrimination power of a measure clearly depends on the graph class in question, see [38,41].…”