Métodos espectrais de partição de grafos

Abstract

The Theory of Graphs has great potential for the description and mathematical modeling of phenomena associated to networks. In problems of this nature a recurring difficulty is the size of the network, i.e. the number of vertices and edges. To overcome these difficulties, an alternative is to decompose the graph associated with the problem by grouping the vertices with similar properties, thus generating a new graph with fewer vertices but still representing the same phenomenon. These groupings of vertices will be called communities. Another aspect, that cannot be overlooked, is the good presentation of data that graphs offer. In this context, the detection of communities has the role of synthesizing this information even further. However, community detection can rarely be done empirically, especially for large graphs. Therefore, an analytical treatment of the graph is made necessary , with mathematical rigor. This mathematically rigorous treatment is a positive point because it will require the use of more developed theories, such as linear algebra. Thus, we will have a greater amount of tools to approach a theme that might not be familiar to us. This work aims to present the maximization of modularity and the minimization of the cutting function, using spectral analysis of the graph as an alternative to partitioning or decomposition of graphs.