An introduction to Network Science and Graph Theory

Network Science emerged as an essential mathematical framework for the description of complex systems. This paper is a concise introduction to the main concepts of Graph Theory, the foundation of the Network Science. Starting with an historical introduction, we present the basic definition of graph and its properties such as average degree, adjacency matrix and components. Then we present some famous graph models showing its definitions and properties. We finish this paper with a description of percolation on networks showing the synergy between Network Science and Physics, while dealing with concepts like phase transition and criticality.

Keywords:
Network; graph; degree; components; random network; scale-free network