  The structure of the network can be described by motifs, which are subgraphs
that often repeat themselves. In order to understand the structure of network
motifs, it is of great importance to study subgraphs from the perspective of
statistical mechanics. In this paper, we use clustering extensions in
statistical physics to solve the problem of using motifs as network primitives.
By projecting the network motifs to clusters in the gas model, we develop the
partition function of the network, which enables us to calculate global
thermodynamic quantities, such as energy, entropy, and vice versa. Then, we
give the analytic expressions of the number of specific types of motifs and
calculate their correlated entropy. We conduct algebraic experiments on
datasets, both synthetic and in real life, and evaluate the qualitative and
quantitative characterization of motif entropy derived from the partition
function. Our findings show that the motif entropy of networks in real life,
for instance, financial and stock market networks, is of high correlation to
the change of network structure. Hence, our findings are consistent with recent
studies about the similar topic that network motifs can be represented as basic
elements of well-defined information processing functions.
