Dynamics of Growing Networks and Evolving Media

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0227670RednerThis award supports theoretical investigations into the structure of growing networks. The networks to be studied grow incrementally through popularity-based rules in which nodes that already have a relatively large number of attached links have a higher propensity for gaining new links. This mechanism appears to underlie a variety of networked systems, such as the world-wide web, the Internet, and various biological networks. The primary goal of this research is to develop a comprehensive quantitative understanding of network structure through the formulation and analysis of the governing rate equations for the evolution of prototypical network models.There are several directions in which research on network structure will be pursued. The role of decay processes, such as the removal of nodes or of links at a finite rate, on the integrity of networks will be studied. Another project will focus on the percolation properties of networks that grow by independent node and link creation events. The rate equation approach will also be extended to treat protein interaction networks in which the growth mechanisms model fundamental processes of small biological organisms, such as yeasts and bacteria. In addition to studying local network properties, such as the degree distribution, effort will be devoted to revealing community structures - tightly linked subnetworks with weaker linkages between communities - within networks. A parallel effort will be devoted to studying porous networks that evolve due to the interaction of a reactive fluid that flows through the medium. In the case of filtration, the pore space is gradually constricted, because of trapping of suspended particles. In dissolution, the pore space is enlarged by a fluid that gradually consumes the solid matrix. The principal goal of this research is to understand the role of the coupling between the pore space geometry and flow properties on long-time dynamical properties.This will be accomplished by constructing simplified models that capture the essence of these processes and upon which analytical calculations can be performed. The basic intellectual goal is to elucidate the role of feedback between the time evolution of the flow and the physical properties of the porous medium. In the problem of filtration, the focus will be to develop a criterion to determine the useful lifetime of a filter and to understand the kinetics of filtration due to the trapping of fine particles. In dissolution, one basic goal is to understand the role of longitudinal gradients on the basic dynamics of single tube growth. Another major goal will be to quantify the transition between homogeneous dissolution and wormhole growth as a function o the relative flow and reaction rates.%%%This award supports theoretical investigations into the structure of growing networks. The networks to be studied grow incrementally through popularity-based rules in which nodes that already have a relatively large number of attached links have a higher propensity for gaining new links. This mechanism appears to underlie a variety of networked systems, such as the world-wide web, the Internet, and various biological networks. The primary goal of this research is to develop a comprehensive quantitative understanding of network structure through the formulation and analysis of the governing rate equations for the evolution of prototypical network models.A parallel effort will be devoted to studying porous networks that evolve due to the interaction of a reactive fluid that flows through the medium. In the case of filtration, the pore space is gradually constricted, because of trapping of suspended particles. In dissolution, the pore space is enlarged by a fluid that gradually consumes the solid matrix. The principal goal of this research is to understand the role of the coupling between the pore space geometry and flow properties on long-time dynamical properties.***