Complex Networks: Structure and Dynamics

In this dissertation, we present a study of complex systems with underlying network topology. Complex systems are formed due to multiple interactions among many different elements. The interactions among the elements not only changes the properties of individual element but also guides them to self-organize. Implications of understanding both the functional and the structural aspects would be felt in a wide range of problem involving social interactions, neural process, disease spread, genome assembly and many more. Three research problems are addressed here: (A) structure of evolving networks; (B) emergence of fairness on complex networks which is understanding dynamics on a given network; and (C) modular synchronization which investigates evolution of network structure based on dynamics. A: It is now well established that complex networks share common properties in terms of their non-trivial network structure known as the network topology. We presented a statistical explanation for widely occurring complex network topologies with the assumption that the effect of various attributes, which determine the ability of each node to attract other nodes, is multiplicative. This composite attribute or fitness is shown to be lognormally distributed and is used in forming the complex network. By varying the parameters of the lognormal distribution, this construction generates many types of real-world networks.B: In the cortex of neural network, for instance, the neurons exhibit collective synchronization within each module rather than global synchronization. To explain this important phenomenon, we consider a well-connected network of neurons, each of which is described by the Hindmarsh-Rose model. The neurons in the system were coupled using adaptive coupling. Numerical simulations on the network demonstrates that modular synchronization emerges in a self-organized fashion.C: Understanding the emergence of fairness is crucial in many biological and social systems. Previous theoretical studies using the evolutionary ultimatum game do not explain the observed experimental behavior. We model this phenomenon using an ultimatum game with a nonlinear utility function. Our study of the game on complex networks show good agreement with the experiments. We also show that the clustering in the network and the nonlinear utility function play important roles in predicting the outcome