Classical and quantum spreading processes on disordered and complex networks

Thumbnail Image
Malik, Omar, Khawar
Issue Date
Electronic thesis
Research Projects
Organizational Units
Journal Issue
Alternative Title
Spreading processes on networks provide a valuable, abstract framework to study a vast array of problems using a unified formalism, from the spread of wildfires in forests to opinion formation in social groups. This thesis presents three specific problems that relate to different spreading processes on complex networks. The first of these is a study of diffusive persistence on disordered lattices and complex networks to better understand the temporal characteristics and the lifetime of fluctuations in stochastic processes in networks. Diffusive persistence is defined as the probability that the diffusive field at a site (or node) has not changed sign up to a certain time (or in general, that the node remained active/inactive in discrete models). Applications of our research could help one better understand the lifetime and temporal dynamics of activity fluctuations and trends in social networks. We investigated disordered networks (characterized by the fraction of removed edges) and found that the behavior of the persistence depends on the topology of the network. In 2D networks we have found that above the percolation threshold diffusive persistence scales similarly as the original two-dimensional regular lattice, according to a power law with an exponent of 0.186 +- 1.4*10^(-4). At the percolation threshold, the scaling exponent changes to one with 0.141 +- 5.3*10^(-5), as the result of the interplay of diffusive persistence and the underlying structural transition in the disordered lattice at the percolation threshold. In contrast, we found that in random networks without a regular structure, such as Erdős–Rényi networks, no simple power-law scaling behavior exists above the percolation threshold. We also investigate finite-size effects for 2D lattices at the percolation threshold and find that the limiting value obeys a power-law with exponent zθ, where z = 2.56 +- 2.3 * 10^(-2) instead of the value of z=2 normally associated with finite-size effects on 2D lattices. Next, we discuss percolation on quantum networks. Quantum networks describe communication networks that are based on quantum entanglement. A concurrence percolation theory has been recently developed to determine the required entanglement to enable communication between two distant stations in an arbitrary quantum network. Unfortunately, concurrence percolation has been calculated only for very small networks or large networks without loops. Here, we develop a set of mathematical tools for approximating the concurrence percolation threshold for unprecedented large-scale quantum networks by estimating the path-length distribution, under the assumption that all paths between a given pair of nodes have no overlap. We show that our approximate method agrees closely with analytical results from concurrence percolation theory. The numerical results we present include 2D square lattices of 200^2 nodes and complex networks of up to 10^4 nodes. The entanglement percolation threshold of a quantum network is a crucial parameter for constructing a real-world communication network based on entanglement, and our method offers a significant speed-up for the intensive computations involved. Finally, we study how public transportation data can be employed in the modeling of the spread of infectious diseases based on SIR dynamics. We present a model where public transportation data is used as an indicator of broader mobility patterns within a city, including the use of private transportation, walking etc. The mobility parameter derived from this data is used to model the infection rate. As a test case, we study the impact of the usage of the New York City subway on the spread of COVID-19 within the city during 2020. We show that utilizing subway transport data as an indicator of the general mobility trends within the city, and therefore as an indicator of the effective infection rate, improves the quality of forecasting COVID-19 spread in New York City. Our model predicts the two peaks in the spread of COVID-19 cases in NYC in 2020, unlike a standard SIR model that misses the second peak entirely.
School of Science
Full Citation
Rensselaer Polytechnic Institute, Troy, NY
Terms of Use
PubMed ID