Hierarchy Near a Critical Point: From the Ising Model to Gene Networks*

The renormalization group approach involves locally averaging over microscopic variables to obtain coarse-grained variables that describe the macroscopic behavior of the system. We propose that hierarchical clustering corresponds to the same approach. We used the cophenetic correlation coefficient (CCC) to quantify how well the relationships between the components in a system are approximated by a hierarchical construct that captures the macroscopic behavior. Since the behavior of a system near a critical point is dominated by macroscopic variables, we expected the CCC to be higher near such a point. This was verified for the 2D Ising model. We then applied this approach to gene networks with distinct behaviors in different regions of the parameter space, separated by critical surfaces. The CCC was higher near the critical surface for the two gene networks investigated. Further, a higher CCC correlated with higher susceptibility of the networks to perturbations, mirroring the higher susceptibility of physical systems near a critical point. We suggest that the CCC can be a useful quantitative signature of criticality in biological systems.