The stoichiometric matrix is a connectivity matrix. Elementary topological properties of the network it represents can be computed directly from the individual elements of S. Direct topological studies are interesting from a variety of standpoints. They focus on relatively easy to understand and intuitive properties of the structure of the network. Elementary topological properties relate to how connected a network is, and how its components participate in forming the connectivity properties of the network. There may be many functional states for a given network structure (see Chapter 16). Topological properties are thus global and less specific than functional states of networks. Some of the differences between functional states and network topology are covered in Part III.

The Binary Form of S

The elementary topological properties are determined based on the non-zero elements in the stoichiometric matrix. Thus, we define the elements of a new matrix ŝ as

ŝij = 0 if sij = 0

ŝij = 1 if sij ≠ 0

which is the binary form of S. This matrix is composed of only zeros and ones. If ŝij is unity, it means that compound i participates in reaction j. Note that in the rare case where a homodimer is formed, i.e., in a reaction of the type 2A → A2, the stoichiometric coefficient of two becomes unity in the binary form of S.

S is a sparse matrix A number of genome-scale stoichiometric matrices have been reconstructed (see [2]). As there are typically only a handful of compounds that participate in a reaction out of hundreds of compounds participating in a network, the stoichiometric matrix is sparse. A sparse matrix is mostly composed of zero elements. For instance, if there are on average 3 compounds that participate in a reaction, but there are m compounds in the network, then the fraction of non-zero elements in the matrix is 3/m. If m is 300, then only 1% of the elements are non-zero and the matrix is sparse.