Abstract
Throughout the years, measuring the complexity of networks and graphs has been of great interest to scientists. The Kolmogorov complexity is known as one of the most important tools to measure the complexity of an object. We formalized a method to calculate an upper bound for the Kolmogorov complexity of graphs and networks. Firstly, the most simple graphs possible, those with Kolmogorov complexity, were identified. These graphs were then used to develop a method to estimate the complexity of a given graph. The proposed method utilizes the simple structures within a graph to capture its non-randomness. This method is able to capture features that make a network closer to the more non-random end of the spectrum. The resulting algorithm takes a graph as an input and outputs an upper bound to its Kolmogorov complexity. This could be applicable in, for example evaluating the performances of graph compression methods.
Highlights
Kolmogorov complexity (K-complexity) [1], of a string of characters, is the length of the shortest possible program that, when fed into a universal Turing machine, outputs that string
We built a pool of basic graphs to be used in our code for the Kolmogorov graph covering algorithm
The complexity of graphs was explored by using Kolmogorov complexity
Summary
Kolmogorov complexity (K-complexity) [1], of a string of characters, is the length of the shortest possible program that, when fed into a universal Turing machine, outputs that string This concept, since it was introduced by Kolmogorov, has been used as a measure of complexity and randomness [2]. (n2) bits, is often seen as the upper bound for the K-complexity of all graphs with n nodes [11]. From a more general viewpoint, the code word length of any lossless compression method for graphs can be considered as an upper bound to the K-complexity of graphs [16]. We show that there exists a limited number of graphs on n nodes that can be described to a machine using a constant number of bits These graphs, which will be known as basic graphs, are used for studying the randomness of graphs in general. The applications of this algorithm in the context of evaluating graph compression methods is discussed
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