We show how a network of interconnections between nodes can be constructed to have a specified distribution of nodal degrees. This is achieved by treating the network as a thermodynamic system subject to constraints and then rewiring the system to maintain the constraints while increasing the entropy. The general construction is given and illustrated by the simple example of an exponential network. By considering the constraints as a cost function analogous to an internal energy, we obtain a characterisation of the correspondence between the intensive and extensive variables of the network. Applied to networks in living organisms, this approach may lead to macroscopic variables useful in characterising living systems.

The explosion of interest in non-random graphs as representations of real-world networks of connections between nodes has generated a large and growing literature, both in the characterisation of observed networks and on methods of construction of specific models [1]. In all of these cases, the specific contexts, in which networks may represent metabolic reactions [2], genetic regulation [3], linguistic relations [4], social graphs [5], internet connections [6] and so on, are abstracted into a general relation between the connectivity of the nodes. To this end, we define the degree of a node as the number of (undirected) connections to or from that node. The models can then be characterised by the distribution of nodal degrees. For example, in the now well-known, scale-free networks, the number of nodes nr with r connections is given by a power law nr ∝ r −β . Although many ways are known to obtain specific examples of non-random networks by evolving the connections according to various schemes [1], there is as yet in general no systematic way of constructing the connections in a network to obtain a given nodal distribution, the current models being found largely by trial and error.