# A natural class of robust networks

(under construction)

Original entry: Ian Burgess, Fall 2009

## References

M. Aldana, P. Cluzel, "A natural class of robust networks," *PNAS* **100**, 8710-8714 (2003).

## Summary

This article does not discuss a specific topic in soft-matter or biophysics, but rather provides a broader mathematical analysis of a specific class of dynamical systems that exists in many biological pathways and processes, and identifies dynamics that are universal to this type of system, thus having implications on the dynamics of many biological processes.

The type of system considered is what is called a scale-free network. A network in general is a collection of elements (genes, proteins etc.) whose interactions obey a certain set of rules in both how one element affects the other and which elements can interact. The toy-model network considered by the authors is that in which each element is a switch with an "on" and "off" state and the type of interaction considered is the induced switching of the state of the target element, conditional on the state of the control element. What distinguishes so-called scale-free networks from other types of networks is the manner in which the elements are allowed to interact, specifically the degree of order in the network. At one end of the spectrum are lattice or Ising type networks where each element can be pictured as being on a lattice with only nearest-neighbor interactions allowed. On the other end there are completely random networks where interactions occur at random and can occur between any two elements. This type of random network has two important parameters: the average number *K* of elements that a given element can interact with and the probability *p* of an element changing its state after interaction. The time-evolution of random networks is found to approach a cyclic behavior. One can draw analogy between such a limit cycle and a cyclic metabolic pathway in a cell. The stability of Scale-free networks lie in between What is identified is a robust stability of specific