http://bactra.org/notebooks Cosma's Notebooks en http://bactra.org/notebooks/2009/04/28#nk-networks <P>See also: <a href="adaptation.html">Adaptation</a>; <a href="alife.html">Artificial Life</a>; <a href="biological-order.html">Biological Order and Levels of Organization</a>; <a href="cellular-automata.html">Cellular Automata</a>; <a href="chaos.html">Chaos and Non-linear Dynamics</a>; <a href="dissipative-structures.html">Dissipative Structures</a>; <a href="edge-of-chaos.html">Edge of Chaos</a>; <a href="physical-biology.html">Physical Principles in Biology</a>; <a href="self-organization.html">Self-organization</a>; <a href="stat-mech.html">Statistical Mechanics</a> <ul>Recommended (essential): <li><a href="http://www.santafe.edu/sfi/People/kauffman/">Stuart Kauffman</a>, <cite>The Origins of Order</cite> [Highly interesting, but to be read with some caution. The writing is often confused, and, frankly, after more than a decade many of the ideas have not panned out. Still extremely important.] </ul> <ul>Recommended, more specialized: <li>Carlos Gershenson, Stuart A. Kauffman, and Ilya Shmulevich, "The Role of Redundancy in the Robustness of Random Boolean Networks", <a href="http://arxiv.org/abs/nlin.AO/0511018">nlin.AO/0511018</a> </ul> <ul>To read: <li>Andrew Berdahl, Amer Shreim, Vishal Sood, Maya Paczuski, Joern Davidsen, "Random sampling vs. exact enumeration of attractors in random Boolean networks", <a href="http://arxiv.org/abs/0904.3948">arxiv:0904.3948</a> <li>Madalena Chaves, Reka Albert and Eduardo D. Sontag, "Robustness and fragility of Boolean models for genetic regulatory networks", <a href="http://arxiv.org/abs/q-bio.MN/0401037">q-bio.MN/0401037</a> <li>S. N. Coppersmith, "The computational complexity of Kauffman nets and the P versus NP problem", <a href="http://arxiv.org/abs/cond-mat/0510840">cond-mat/0510840</a> <li>L. Correale, M. Leone, A. Pagnani, M. Weigt, R. Zecchina, "Computational core and fixed-point organisation in Boolean networks", <a href="http://arxiv.org/abs/cond-mat/0512089">cond-mat/0512089</a> <li>Barbara Drossel, "On the number of attractors in random Boolean networks", <a href="http://arxiv.org/abs/cond-mat/0503526">cond-mat/0503526</a> = <a href="http://dx.doi.org/10.1103/PhysRevE.72.016110"><cite>Physical Review E</cite> <strong>72</strong> (2005): 016110</a> <li>Richard Durrett and Vlada Limic, "Rigorous Results for the NK Model", <cite>The Annals of Probability</cite> <strong>31</strong> (2003): 1713--1753 <li>Carlos Gershenson <ul> <li>"Introduction to Random Boolean Networks", <a href="http://arxiv.org/abs/nlin.AO/0408006">nlin.AO/0408006</a> <li>"Phase Transitions in Random Boolean Networks with Different Updating Schemes", <a href="http://arxiv.org/abs/nlin/0311008">nlin/0311008</a> </ul> <li>Florian Greil and Barbara Drossel, "The dynamics of critical Kauffman networks under asynchronous stochastic update", <a href="http://arxiv.org/abs/cond-mat/0501081">cond-mat/0501081</a> = <a href="http://dx.doi.org/10.1103/PhysRevLett.95.048701"><citE>Physical Review Letters</cite> <strong>95</strong> (2005): 048701</a> <li>Leo Kadanoff, Susan Coppersmith and Maximino Aldana, "Boolean Dynamics with Random Couplings," <a href="http://arxiv.org/abs/nlin.AO/0204062">nlin.AO/0204062</a> <li>Viktor Kaufman and Barbara Drossel, "Relevant components in critical random Boolean networks", <a href="http://arxiv.org/abs/cond-mat/0606507">cond-mat/0606507</a> <li>Viktor Kaufman, Tamara Mihaljev and Barbara Drossel, "Scaling in critical random Boolean networks", <a href="http://arxiv.org/abs/cond-mat/0506807">cond-mat/0506807</a> <li>Juha Kesseli, Pauli R&auml;m&ouml;, and Olli Yli-Harja, "Tracking perturbations in Boolean networks with spectral methods", <a href="http://dx.doi.org/10.1103/PhysRevE.72.026137"><cite>Physical Review E</cite> <strong>72</strong> (2005):026137</a> <li>Constantin Klemm and Stefan Bornholdt, "Stable and unstable attractors in Boolean networks", <a href="http://arxiv.org/abs/cond-mat/0411102">cond-mat/0411102</a> <li>Michele Leone, Andrea Pagnani, Giorgio Parisi and Osvaldo Zagordi, "Finite size corrections to random Boolean networks", <a href="http://dx.doi.org/10.1088/1742-5468/2006/12/P12012"><citE>Journal of Statistical Mechanics: Theory and Experiment</cite> <strong>2006</strong>: P12012</a> <li>Vlada Limic and Robin Pemantle, "More rigorous results on the Kauffman-Levin model of evolution", <a href="http://dx.doi.org/10%2E1214/009117904000000081"><cite>Annals of Probability</cite> <strong>32</strong> (2004): 2149--2178</a> = <a href="http://arxiv.org/abs/math.PR/0308282">math.PR/0308282</a> <li>Bartolo Luque, Fernando J. Ballesteros and Manuel Fern&aacute;ndez, "Variances as order paramaeter and complexity measure for random Boolean networks", <a href="http://dx.doi.org/10.1088/0305-4470/38/5/005"><citE>Journal of Physics A: Mathematical and General</cite> <strong>38</strong> (2005): 1031--1038</a> <li>James F. Lynch, "Critical Points for Random Boolean Networks," <a href="http://arxiv.org/abs/nlin.AO/0110036">nlin.AO/0110036</a> <li>H. Mahmoudi, A. Pagnani, M. Weigt and R. Zecchina, "Propagation of external regulation and asynchronous dynamics in random Boolean networks", <a href="http://dx.doi.org/10.1063/1.2742931"><cite>Chaos</cite> <strong>17</strong> (2007): 026109</a> <li>Tamara Mihaljev, Barbara Drossel, "Scaling in a general class of critical random Boolean networks", <a href="http://arxiv.org/abs/cond-mat/0606612">cond-mat/0606612</a> = <a href="http://dx.doi.org/10.1103/PhysRevE.74.046101"><cite>Physical Review E</cite> <strong>74</strong> (2006): 046101</a> <li>Andre A. Moreira and Luis A. N. Amaral, "Canalizing Kauffman networks: non-ergodicity and its effect on their critical behavior", <a href="http://arxiv.org/abs/cond-mat/0504722">cond-mat/0504722</a> = <a href="http://dx.doi.org/10.1103/PhysRevLett.94.218702"><cite>PRL</cite> <strong>94</strong> (2005): 218702</a> <li> U. Paul, V. Kaufman, B. Drossel, "The properties of attractors of canalyzing random Boolean networks", <a href="http://arxiv.org/abs/cond-mat/0511049">cond-mat/0511049</a> <li>Pauli R&auml;m&ouml;, Juha Kesseli, and Olli Yli-Harja, "Stability of functions in Boolean models of gene regulatory networks", <a href="http://dx.doi.org/10.1063/1.1996927"><cite>Chaos</cite> <strong>15</strong> (2005): 034101</a> <li>Cynthia J. Olson Reichhardt and Kevin E. Bassler, "Canalization and Symmetry in Boolean Models for Genetic Regulatory Networks", <a href="http://arxiv.org/abs/q-bio.QM/0610011">q-bio.QM/0610011</a> <li>Chikoo Oosawa, Kazuhiro Takemoto, Shogo Matsumoto, Michael A. Savageau, "Local cause of coherence in Boolean networks", <a href="http://arxiv.org/abs/nlin.CG/0611049">nlin.CG/0611049</a> <li>Bjoorn Samuelsson and Carl Troein, "Random maps and attractors in random Boolean networks", <a href="http://arxiv.org/abs/cond-mat/0505481">cond-mat/0505481</a> <li>Steffen Schober and Martin Bossert, "Analysis of random Boolean networks using the average sensitivity", <a href="http://arxiv.org/abs/0704.0197">arxiv:0704.0197</a> <li>Agnes Szejka and Barbara Drossel, "Evolution of Canalizing Boolean Networks", <a href="http://arxiv.org/abs/q-bio.PE/0701025">q-bio.PE/0701025</a> </ul>