Weighted Graph Compression using Genetic Algorithms

Networks are a great way to present information. It is easy to see how different objects interact with one another, and the nature of their interaction. However, living in the technological era has led to a massive surge in data. Consequently, it is very common for networks/graphs to be large. When graphs get too large, the computational power and time to process these networks gets expensive and inefficient. This is common in areas such as bioinformatics, epidemic contact tracing, social networks, and many others. Graph compression is the process of merging nodes that are highly connected into one super-node, thus shrinking the graph. The goal of graph compression is to merge nodes while mitigating the amount of information lost during the compression process. Unweighted graphs are largely studied in this area. However, in this thesis, we extend the approaches to compress weighted graphs via genetic algorithms and analyse the compression from an epidemic point of view. It is seen that edge weights provide vital information for graph compression. Not only this, but having meaningful edge weights is important as different weights can lead to different results. Moreover, both the original edge weights and adjusted edge weights produce different results when compared to a widely used community detection algorithm, the Louvain Algorithm. However, the different results may be helpful to public health officials. Lastly, the NSGA-II algorithm was implemented. It was found that NSGA-II is more suitable as a pre-processing tool, in order to find a target compression that introduces a comfortable level of distortion, and then using the single-objective genetic algorithm to achieve an improved solution for the target

3-Factor-criticality of vertex-transitive graphs

A graph of order $n$ is $p$-factor-critical, where $p$ is an integer of the same parity as $n$, if the removal of any set of $p$ vertices results in a graph with a perfect matching. 1-Factor-critical graphs and 2-factor-critical graphs are factor-critical graphs and bicritical graphs, respectively. It is well known that every connected vertex-transitive graph of odd order is factor-critical and every connected non-bipartite vertex-transitive graph of even order is bicritical. In this paper, we show that a simple connected vertex-transitive graph of odd order at least 5 is 3-factor-critical if and only if it is not a cycle.Comment: 15 pages, 3 figure