Complex network problems in physics, computer science and biology

There is a close relation between physics and mathematics and the exchange of ideas between these two sciences are well established. However until few years ago there was no such a close relation between physics and computer science. Even more, only recently biologists started to use methods and tools from statistical physics in order to study the behavior of complex system. In this thesis we concentrate on applying and analyzing several methods borrowed from computer science to biology and also we use methods from statistical physics in solving hard problems from computer science.; In recent years physicists have been interested in studying the behavior of complex networks. Physics is an experimental science in which theoretical predictions are compared to experiments. In this definition, the term prediction plays a very important role: although the system is complex, it is still possible to get predictions for its behavior, but these predictions are of a probabilistic nature. Spin glasses, lattice gases or the Potts model are a few examples of complex systems in physics.; Spin glasses and many frustrated antiferromagnets map exactly to computer science problems in the NP-hard class defined in Chapter 1. In Chapter 1 we discuss a common result from artificial intelligence (AI) which shows that there are some problems which are NP-complete, with the implication that these problems are difficult to solve. We introduce a few well known hard problems from computer science (Satisfiability, Coloring, Vertex Cover together with Maximum Independent Set and Number Partitioning) and then discuss their mapping to problems from physics.; In Chapter 2 we provide a short review of combinatorial optimization algorithms and their applications to ground state problems in disordered systems. We discuss the cavity method initially developed for studying the Sherrington-Kirkpatrick model of spin glasses. We extend this model to the study of a specific case of spin glass on the Bethe lattice at zero temperature and then we apply this formalism to the K-SAT problem defined in Chapter 1.; The phase transition which physicists study often corresponds to a change in the computational complexity of the corresponding computer science problem. Chapter 3 presents phase transitions which are specific to the problems discussed in Chapter 1 and also known results for the K-SAT problem. We discuss the replica method and experimental evidences of replica symmetry breaking.; The physics approach to hard problems is based on replica methods which are difficult to understand. In Chapter 4 we develop novel methods for studying hard problems using methods similar to the message passing techniques that were discussed in Chapter 2. Although we concentrated on the symmetric case, cavity methods show promise for generalizing our methods to the un-symmetric case.; As has been highlighted by John Hopfield, several key features of biological systems are not shared by physical systems. Although living entities follow the laws of physics and chemistry, the fact that organisms adapt and reproduce introduces an essential ingredient that is missing in the physical sciences. In order to extract information from networks many algorithm have been developed. In Chapter 5 we apply polynomial algorithms like minimum spanning tree in order to study and construct gene regulatory networks from experimental data. As future work we propose the use of algorithms like min-cut/max-flow and Dijkstra for understanding key properties of these networks.