1 The language of graphs and networks
The first thing that needs to be clarified is that the terms graphs and networks are used
indistingtly in the literature. In this Chapter we will reserve the term graph for the abstract
mathematical concept, in general referred to small, artificial formations of nodes and edges.
The term network is then reserved for the graphs representing real-world objects in which the
nodes represent entities of the system and the edges represent the relationships among them.
Therefore, it is clear that we will refer to the system of individuals and their interactions as a
‘social network’ and not as a ‘social graph’. However, they should mean exactly the same.
For the basic concepts of graph theory the reader is recommended to consult the
introductory book by Harary (1967). We start by defining a graph formally. Let us consider a
finite set V v1 , v2 ,, vn of unspecified elements and let V V be the set of all ordered
pairs vi , v j of the elements of V . A relation on the set V is any subset E V V . The
relation E is symmetric if vi , v j E implies v j , vi E and it is reflexive if
v V , v, v E . The relation E is antireflexive if vi , v j E implies vi v j . Now we can
define a simple graph as the pair G V , E , where V is a finite set of nodes, vertices or
points and E is a symmetric and antireflexive relation on V , whose elements are known as
the edges or links of the graph. In a directed graph the relation E is non-symmetric. In
many physical applications the edges of the graphs are required to support weights, i.e., real
numbers indicating a specific property of the edge. In this case the following more general
definition is convenient. A weighted graph is the quadruple G V , E,W , f where V is a
finite set of nodes, E V V e1, e2 ,, em is a set of edges, W w1 , w2 ,
, wr is a set of
weights such that wi and f : E W is a surjective mapping that assigns a weight to
each edge. If the weights are natural numbers then the resulting graph is a multigraph in
which there could be multiple edges between pairs of vertices. That is, if the weight between
nodes p and q is k N it means that there are k links between the two nodes.
In an undirected graph we say that wo nodes p and q are adjacent if they are joined
by an edge e p, q . In this case we say that the nodes p and q are incident to the link e ,
and the link e is incident to the nodes p and q . The two nodes are called the end nodes of
the edge. Two edges e1 p, q and e2 r , s are adjacent if they are both incident to at
least one node. A simple but important characteristic of a node is its degree, which is defined
as the number of edges which are incident to it or similarly the number of nodes adjacent to
it. Slightly different definitions apply to directed graphs. The node p is adjacent to node q if
there is a directed link from p to q , e p, q . We also say that a link from p to q is
incident from p and incident to q ; p is incident to e and q is incident from e .
Consequently, we have two different kinds of degrees in directed graphs. The in-degree of a
node is the number of links incident to it and its out-degree is the number of links incident
1.1 Graph operators
The incidence and adjacency relations in graphs allow us to define the following graph
operators. We consider an undirected graph for which we construct its incidence matrix with