Boolean dynamics of networks with scale-free topology

Compensatory interactions to stabilize multiple steady states or mitigate the effects of multiple deregulations in biological networks

“…1, the damage probability after double knockout is rather high regardless of the degree distribution and is higher than the damage probability after single knockout. As one can see, the double-knockout damage probability is higher in a network with higher average degree, which is consistent with the established conclusion that the complexity of the dynamics increases with larger average node in-degree [26][27][28]. The NKK model with K=1 is an exception; here the damage probability is 1 whether one or two nodes are damaged.…”

“…Two states that initially differ in a single node's state will diverge on average in the chaotic phase. The critical boundary is average degree <K> = 2 when considering unbiased Boolean logic (all Boolean functions) and using an annealed approximation (at every time step the input nodes and Boolean functions are randomized for each node) [25][26][27][28][29][30]. We note that our setting of a steady state damaged by a single node knockout is different from what was considered in previous work on random Boolean network ensembles.…”

“…To make sure that the generated ensemble has the desired topology and degree distribution, we only accept at least weakly connected networks and use effective rules when assigning a Boolean function to each node. We study ensembles with average degree <K> = 1, 2 and 3, which would be in frozen phase for <K> = 1 and chaotic phase for <K> = 2 or 3 when considering the annealed approximation, the infinite network size limit and unbiased effective Boolean rules [26][27][28][29][30]. Note that knowing the phase is not enough to predict the damage probability.…”

Dynamics-driven evolution to structural heterogeneity in complex networks

“…Networks with pronounced power-law degree distributions also emerged in other model systems with comparable dynamicl and evolutionary time scales [16,17]. In real-world systems, it was noticed by Aldana [26] that a major fraction of scale-free complex networks has their decay exponent γ in the tiny range of γ ∈ [2.0, 2.5].…”

Graphs as Models of Large-Scale Biochemical Organization

“…Figure 18 shows a small graph corresponding to the regulatory network of E. coli (left) and the degree distribution for a the bigger network of S. cerevisiae (right). This new finding has stimulated research in Boolean networks which have a power-law degree distribution [Aldana, 2003, Aldana andCluzel, 2003], further extending the results on random Boolean networks.…”