Dissertation Transport and Percolation in Complex Networks Thank the Staff at Gene's Polymer Studies Group and Bu Physics Department, Especially

Acknowledgments In the path for pursuing my Ph.D, there are many people supporting me, guiding me and helping me. Ph.D, all the years offering the financial support to me along the path to achieving it and teaching me to prepare excellent talks. I feel lucky in his research group that I have the opportunities to exchange and cooperate with many other research groups from all over the world. Prof. Shlomo Havlin is the main collaborator in my research career for these years. He is always guiding me in academic field and he has been an exemplary mentor to me. I express my deep thankfulness to him. They are very helpful friends sharing knowledge and experience with me generously. Bob Tomposki and Mirtha Cabello, for helping me all these years. ABSTRACT To design complex networks with optimal transport properties such as flow efficiency, we consider three approaches to understanding transport and percolation in complex networks. We analyze the effects of randomizing the strengths of connections, randomly adding long-range connections to regular lattices, and percolation of spatially constrained networks. Various real-world networks often have links that are differentiated in terms of their strength, intensity, or capacity. We study the distribution P (σ) of the equivalent conduc-tance for Erd˝ os-Rényi (ER) and scale-free (SF) weighted resistor networks with N nodes, for which links are assigned with conductance σ i ≡ e −ax i , where x i is a random variable with 0 < x i < 1. We find, both analytically and numerically, that P (σ) for ER networks exhibits two regimes: (i) For σ < e −apc , P (σ) is independent of N and scales as a power law P (σ) ∼ σ k/a−1. Here p c = 1/ k is the critical percolation threshold of the network and k is the average degree of the network. (ii) For σ > e −apc , P (σ) has strong N dependence and scales as P (σ) ∼ f (σ, ap c /N 1/3). Transport properties are greatly affected by the topology of networks. We investigate the transport problem in lattices with long-range connections and subject to a cost constraint, seeking design principles for optimal transport networks. Our network is built from a regular d-dimensional lattice to be improved by adding long-range connections with probability P ij ∼ r −α ij , where r ij is the lattice distance between site i …