12.2 Networks (Graphs)
145
appears, and we can call the array “fully connected”. The remarkable Galam–Mauger
formula gives this critical threshold p Subscript cpc for isotropic lattices:
p Subscript c Baseline equals a left bracket left parenthesis upper D minus 1 right parenthesis left parenthesis upper C minus 1 right parenthesis right bracket Superscript negative bpc = a[(D −1)(C −1)]−b
(12.12)
whereupper DD is the dimension,upper CC the connectivity of the array (i.e., the number of nearest
neighbours of any cell), and aa and bb are constants with values 1.2868 and 0.6160
respectively, allowing one to calculate the critical threshold for many different types
of networks.
12.1.4
Systems Biology
Given that general systems theory has attracted criticism (e.g., Berlinski 1978), one
might also expect systems biology to attract it (e.g., Kirk et al. 2015). System-level
understanding of an organism, which may be a single cell, is prima facie appropri-
ate; Kitano (2002) points out that an understanding of genes and proteins and their
interconnexions is insufficient; both structural and dynamic knowledge is required.
Investigators of the structural features might well feel that they are complicated
enough to be getting on with. 9 One difficulty is knowing where to draw the boundary
that defines the system. Loewe (2016) has provided a thoughtful review of how the
system can sensibly be extended beyond the organism.
The boundaries of systems biology itself are somewhat amorphous and impinge
on many of the bioinformatics topics discussed in this book, such as correlation
among expression levels of genes, investigating how some factor X affects, say, the
proliferation of cell type Y (a causal question), and automated network inference
from expression or other kinds of data.
12.2
Networks (Graphs)
The cellular automata considered above (Sect. 12.1.2) are examples of regular net-
works (of automata): the objects are arranged on a regular lattice and connected in
an identical fashion with neighbours. Consider now a collection of objects (nodes
or vertices) characterized by number, type, and interconnexions (edges or links).
Figure 12.2 represents an archetypical fragment of a network (graph). The connex-
ions between nodes can be given by an adjacency matrixupper AA whose elementsa Subscript i jai j give
the strength of the connexion (in a Boolean network a equals 1a = 1 or 0, respectively con-
nexion present or absent) between nodes ii and j j. In a directed graph upper AA need not be
9 Aloy and Russell (2005).