Introduction

In this appendix we provide basic concepts aimed at introducing the formalism of networks. We first introduce graphs and make simple examples, and then discuss the topological properties of networks. Other general presentations of network structure can be found in review articles and books.

A network (or graph in a more mathematical language) is defined as a set of Nvertices (or nodes) connected by links (or edges). Links, and consequently the whole graph, can be either directed (oriented), if a direction is specified as in Fig. A.1a, or undirected (not oriented), if no direction is specified as in Fig. A.1b. More precisely, undirected links are rather bidirectional ones, since they can be traversed in both directions. For this reason an undirected graph can always be thought of as a directed one where each undirected link is replaced by two directed links pointing in opposite directions (see Fig. A.1c). A link in a directed network is said to be reciprocated if another link between the same pair of vertices, but with opposite direction, is there. Therefore, an undirected network can be regarded as a special case of a directed network where all links are reciprocated. The links of a network may also carry a number, referred to as the weight of the edge, representing the strength of the corresponding interaction. In such a case one speaks of a weighted network. In the present appendix we do not consider weighted networks explicitly.