## Explorations in complex network theory: interferometer networks and the application of generating functions to matrices

dc.contributor.advisor | Diniz Behn, Cecilia | |

dc.contributor.advisor | Carr, Lincoln D. | |

dc.contributor.author | Krawciw, Benjamin A. | |

dc.date.accessioned | 2023-12-04T19:28:22Z | |

dc.date.available | 2023-12-04T19:28:22Z | |

dc.date.issued | 2023 | |

dc.identifier | Krawciw_mines_0052N_12670.pdf | |

dc.identifier | T 9597 | |

dc.identifier.uri | https://hdl.handle.net/11124/178611 | |

dc.description | Includes bibliographical references. | |

dc.description | 2023 Summer. | |

dc.description.abstract | Complex network theory examines the structures that arise from systems like neurons in the brain, members of societies, and pages on the world wide web. I push the field forward in two distinct areas: I address the current inability of complex network theory to handle networks weighted with complex numbers; and I introduce a novel way of expressing networks through the generating function technique. The interest in networks based on complex numbers is motivated by a desire to extend complex network theory to problems in physics which involve traveling signals with phase, such as waves traveling through networks, interfering constructively and destructively depending on their phase at the point of interaction. My new analysis of this class of networks centers around the case of interferometry. I call these complex-valued networks interferometer networks. The work combining generating functions and matrices is motivated by the success generating functions have had in treating large sequences with recursive structure. Large networks can be similarly constructed as a sequence with recursion, making generating functions a possible tool for examining networks. I explore interferometer networks with structures very different to traditional interferometers. Most interferometers compare the phases of only two beams of light. They function by counting the difference in the number of cycles of light waves, which occur at a rate of 1 cycle per 2π radians of phase shift. For more complicated interferometers, I prove that they still only produce 1 cycle per 2π radians of phase shift as long as the phase shift occurs at only one site in the network. However, if this phase shift is split up over multiple places in the network, this limitation no longer holds. I produce an interferometer model, called the N -stage skew-cycle interferometer, which breaks this limitation. The N -stage skew-cycle interferometer serves as a demonstration that large changes in output phase can arise from small changes in inputs. To incorporate the traditional tools of complex network theory, I take two well-known measures in complex network theory, path length and clustering, and generalize them to interferometer networks. These measures are extended to the complex numbers, creating versions that account for the constructive and destructive interference of waves travelling over the network. I call my newly-defined measures apparent path strength and interferometric clustering. There are instances of interferometer networks for which apparent path strength is undefined, especially in the case of feedback loops. This chapter identifies a real-world case where this can arise: the cavity of a Fabry-Perot interferometer. Subsequently, I prove that unique signal solutions exist and are bounded if the `1 norm of the weighted adjacency matrix is less than one. This allows the interferometer network research to continue with a guarantee that my newly-defined measures exist and are bounded. I explore the small-world effect in the context of interferometer networks. The small world effect is a phenomenon that occurs in real-world networks. In the language of clustering and path length, small-world networks tend to have high clustering and short path lengths. This behavior is quantified by a measure called the small-world coefficient. I adapt a model for this behavior to produce interferometer networks. Then, I computationally test those networks using my newly-defined measures. I found that the small-world coefficient, adapted for interferometer networks, depends heavily on the presence of phase in the networks. The small-world coefficient computed with the generalized measures ranges from slightly lower than the small-world coefficient computed with real-valued network measures to several times higher, demonstrating that interferometric network measures are necessary to capture the behavior of the model. This result concludes the investigations into interferometer networks. Subsequently, I explore the application of generating functions to adjacency matrices. I create two representations of networks: the array generating function and the transformation operator. The array generating function stores the entries of an adjacency matrix on a power series of x and y, where powers of x correspond to rows and powers of y correspond to columns. The transformation operator acts on the coefficients of a generating function the same way that matrices act on the entries of a column vector. After defining the array generating function and the transformation operator, I establish some of their properties and demonstrate how to convert between adjacency matrices, array generating functions, and transformation operators. The result of this work is the creation of new ways of expressing networks, which allows convenient ways of generating networks and performing calculations on them. Finally, I discuss the discoveries made throughout the thesis and consider what those results mean for future network research. Interferometer networks, with their associated interferometric measures, have potential applications in quantum random walks, condensed matter models, complex-valued neural networks, and network analysis of alternating-current circuits. Particular problems include analytically modeling the small-world coefficient of the small-world interferometer model and demonstrating the applicability of interferometric measures by applying them to an existing data set. The application of generating functions to matrices likewise creates opportunities for new research: using array generating functions to compute network measures, using generating function techniques to find matrix decompositions, and using repeated applications of the transformation operator to create a generating function that enumerates path lengths on an unlabeled network. | |

dc.format.medium | born digital | |

dc.format.medium | masters theses | |

dc.language | English | |

dc.language.iso | eng | |

dc.publisher | Colorado School of Mines. Arthur Lakes Library | |

dc.relation.ispartof | 2023 - Mines Theses & Dissertations | |

dc.rights | Copyright of the original work is retained by the author. | |

dc.subject | complex networks | |

dc.subject | complex numbers | |

dc.subject | generating functions | |

dc.subject | interferometers | |

dc.subject | small-world networks | |

dc.title | Explorations in complex network theory: interferometer networks and the application of generating functions to matrices | |

dc.type | Text | |

dc.date.updated | 2023-11-30T05:07:11Z | |

dc.contributor.committeemember | Clark, Antwan D. | |

dc.contributor.committeemember | Mehta, Dinesh P. | |

dcterms.embargo.expires | 2024-11-29 | |

thesis.degree.name | Master of Science (M.S.) | |

thesis.degree.level | Masters | |

thesis.degree.discipline | Applied Mathematics and Statistics | |

thesis.degree.grantor | Colorado School of Mines | |

dc.rights.access | Embargo Expires: 11/29/2024 |