Rohlf Thimo
Max-Planck-Institute for Mathematics in the Sciences, Inselstrasse 22, D-04103 Leipzig, Germany.
Phys Rev E Stat Nonlin Soft Matter Phys. 2008 Dec;78(6 Pt 2):066118. doi: 10.1103/PhysRevE.78.066118. Epub 2008 Dec 30.
We calculate analytically the critical connectivity K_{c} of random-threshold networks (RTNs) for homogeneous and inhomogeneous thresholds, and confirm the results by numerical simulations. We find a superlinear increase of K_{c} with the (average) absolute threshold mid R:hmid R: , which approaches K_{c}(mid R:hmid R:) approximately h;{2}(2lnmid R:hmid R:) for large mid R:hmid R: , and show that this asymptotic scaling is universal for RTNs with Poissonian distributed connectivity and threshold distributions with a variance that grows slower than h;{2} . Interestingly, we find that inhomogeneous distribution of thresholds leads to increased propagation of perturbations for sparsely connected networks, while for densely connected networks damage is reduced; the crossover point yields a characteristic connectivity K_{d} , that has no counterpart in Boolean networks with transition functions not restricted to threshold-dependent switching. Last, local correlations between node thresholds and in-degree are introduced. Here, numerical simulations show that even weak (anti)correlations can lead to a transition from ordered to chaotic dynamics, and vice versa.
我们通过解析计算得出了具有均匀和非均匀阈值的随机阈值网络（RTN）的临界连通性(K_{c})，并通过数值模拟对结果进行了验证。我们发现(K_{c})随（平均）绝对阈值(R)呈超线性增长，对于较大的(R)，其趋近于(K_{c}(R)\approx h^{2}(2\ln R))，并且表明这种渐近标度对于具有泊松分布连通性和方差增长慢于(h^{2})的阈值分布的RTN是通用的。有趣的是，我们发现阈值的非均匀分布会导致稀疏连接网络中扰动传播增加，而对于密集连接网络，损害会降低；交叉点产生一个特征连通性(K_{d})，这在具有不限于阈值依赖切换的转移函数的布尔网络中没有对应物。最后，引入了节点阈值和入度之间的局部相关性。在此，数值模拟表明，即使是弱（反）相关性也会导致从有序到混沌动力学的转变，反之亦然。
