{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/101153"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/101153","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Dynamical systems on networks","abstract":"\"We obtain results for two distinct dynamical models: the Kuramoto model, a general model for coupled oscillator systems, and a model for opinion formation in social networks. Our main focus is on understanding the fixed points of these systems and their stability. For many models the stability of such fixed points can be studied with a Laplacian matrix. We give a formula for the inertia of these matrices, characterizing the real parts of the spectrum, by relating them to another matrix depending on the network topology. We then study the Kuramoto model, and in particular, the phenomena of synchronization, when all oscillators rotate at a common frequency, which corresponds to a fixed point. This phenomenon is well-known to depend on the natural frequencies of the oscillators and, more specifically, that the chance of synchronization increases if the natural frequencies are more similar. We then give upper and lower bounds for the volume of the set such frequencies in frequency space. Our bounds can be formulated in terms of sums over spanning trees which we further use to deduce that the volume is intimately related to the number of spanning trees for dense networks. We also characterize the structure of fixed points of the Kuramoto model by showing that every fixed point corresponds to a lattice point in a certain set which records how the phase-angles wrap around cycles in the network. As a consequence, under mild conditions, we derive the rate of growth of the number of fixed points as we consider increasingly large graphs with fixed topology. We also consider a model for opinion formation in social networks. More specifically, we characterize the global minima of an energy functional, intuitively the ``most stable\"\" configurations, when the network is ``balanced\"\" as well as show that the number of stable configurations can increase as we increase the strengths of the relationships in the network. Finally, we describe an algorithm for generating certain random networks. These networks are generalizations of Erd\\H{o}s-R\\'{e}nyi graphs with correlations between pairs of edges depending on the particular pattern they create. We then use this algorithm to study the effect on fixed points of network properties and therefore the dynamics of the Kuramoto model.\"","abstract_html":""We obtain results for two distinct dynamical models: the Kuramoto model, a general model for coupled oscillator systems, and a model for opinion formation in social networks. Our main focus is on understanding the fixed points of these systems and their stability. For many models the stability of such fixed points can be studied with a Laplacian matrix. We give a formula for the inertia of these matrices, characterizing the real parts of the spectrum, by relating them to another matrix depending on the network topology. We then study the Kuramoto model, and in particular, the phenomena of synchronization, when all oscillators rotate at a common frequency, which corresponds to a fixed point. This phenomenon is well-known to depend on the natural frequencies of the oscillators and, more specifically, that the chance of synchronization increases if the natural frequencies are more similar. We then give upper and lower bounds for the volume of the set such frequencies in frequency space. Our bounds can be formulated in terms of sums over spanning trees which we further use to deduce that the volume is intimately related to the number of spanning trees for dense networks. We also characterize the structure of fixed points of the Kuramoto model by showing that every fixed point corresponds to a lattice point in a certain set which records how the phase-angles wrap around cycles in the network. As a consequence, under mild conditions, we derive the rate of growth of the number of fixed points as we consider increasingly large graphs with fixed topology. We also consider a model for opinion formation in social networks. More specifically, we characterize the global minima of an energy functional, intuitively the ``most stable"" configurations, when the network is ``balanced"" as well as show that the number of stable configurations can increase as we increase the strengths of the relationships in the network. Finally, we describe an algorithm for generating certain random networks. These networks are generalizations of Erd\\H{o}s-R\\'{e}nyi graphs with correlations between pairs of edges depending on the particular pattern they create. We then use this algorithm to study the effect on fixed points of network properties and therefore the dynamics of the Kuramoto model."","abstract_has_math":false,"creators":["Ferguson, Timothy"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Bronski, Jared","DeVille, Lee","Zharnitsky, Vadim","Laugesen, Richard"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2018,"date_issued":"2018-09-04T20:34:02Z","date_published":"2018-09-04T20:34:02Z","updated_at":"2026-07-22T22:24:38Z","subjects":["Dynamics","Networks"],"languages":["en"],"rights":["Copyright 2018 Timothy Ferguson"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/101153","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Bronski, Jared","DeVille, Lee","Zharnitsky, Vadim","Laugesen, Richard"]},{"key":"dc:creator","label":"Author","values":["Ferguson, Timothy"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2018-09-04T20:34:02Z","2020-09-05T09:15:29Z","2018-04-13","2018-05"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Dynamics","Networks"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2018 Timothy Ferguson"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/101153"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["\"We obtain results for two distinct dynamical models: the Kuramoto model, a general model for coupled oscillator systems, and a model for opinion formation in social networks. Our main focus is on understanding the fixed points of these systems and their stability. For many models the stability of such fixed points can be studied with a Laplacian matrix. We give a formula for the inertia of these matrices, characterizing the real parts of the spectrum, by relating them to another matrix depending on the network topology. We then study the Kuramoto model, and in particular, the phenomena of synchronization, when all oscillators rotate at a common frequency, which corresponds to a fixed point. This phenomenon is well-known to depend on the natural frequencies of the oscillators and, more specifically, that the chance of synchronization increases if the natural frequencies are more similar. We then give upper and lower bounds for the volume of the set such frequencies in frequency space. Our bounds can be formulated in terms of sums over spanning trees which we further use to deduce that the volume is intimately related to the number of spanning trees for dense networks. We also characterize the structure of fixed points of the Kuramoto model by showing that every fixed point corresponds to a lattice point in a certain set which records how the phase-angles wrap around cycles in the network. As a consequence, under mild conditions, we derive the rate of growth of the number of fixed points as we consider increasingly large graphs with fixed topology. We also consider a model for opinion formation in social networks. More specifically, we characterize the global minima of an energy functional, intuitively the ``most stable\"\" configurations, when the network is ``balanced\"\" as well as show that the number of stable configurations can increase as we increase the strengths of the relationships in the network. Finally, we describe an algorithm for generating certain random networks. These networks are generalizations of Erd\\H{o}s-R\\'{e}nyi graphs with correlations between pairs of edges depending on the particular pattern they create. We then use this algorithm to study the effect on fixed points of network properties and therefore the dynamics of the Kuramoto model.\"","Submission published under a 24 month embargo labeled 'U of I Access', the embargo will last until 2020-05-01","The student, Timothy Ferguson, accepted the attached license on 2018-04-11 at 15:36.","The student, Timothy Ferguson, submitted this Dissertation for approval on 2018-04-11 at 15:43.","This Dissertation was approved for publication on 2018-04-13 at 13:16.","DSpace SAF Submission Ingestion Package generated from Vireo submission #12195 on 2018-08-31 at 17:18:27","Made available in DSpace on 2018-09-04T20:34:02Z (GMT). No. of bitstreams: 3 FERGUSON-DISSERTATION-2018.pdf: 14880827 bytes, checksum: 97feb8f424444798535599fe3d9d82ba (MD5) thesis-ex.tex: 285727 bytes, checksum: b420f1f3e1bbd8690ae9fb8d6c9aa3ea (MD5) LICENSE.txt: 4213 bytes, checksum: 2a37310fbe8cb031e0ea4541a7b45042 (MD5) Previous issue date: 2018-04-13","Embargo set by: Seth Robbins for item 107236 Lift date: 2020-09-04T20:34:13Z Reason: Author requested U of Illinois access only (OA after 2yrs) in Vireo ETD system","Embargo set by: Seth Robbins for item 107236 Lift date: 2020-09-04T20:37:00Z Reason: Author requested U of Illinois access only (OA after 2yrs) in Vireo ETD system","Embargo set by: Seth Robbins for item 107236 Lift date: 2020-09-04T20:42:08Z Reason: Author requested U of Illinois access only (OA after 2yrs) in Vireo ETD system","U of I Only Restriction Lifted for Item 107236 on 2020-09-05T09:15:29Z."]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Dynamical systems on networks"]}]}],"canonical_facts":{"dc:contributor":["Bronski, Jared","DeVille, Lee","Zharnitsky, Vadim","Laugesen, Richard"],"dc:creator":["Ferguson, Timothy"],"dc:date":["2018-09-04T20:34:02Z","2020-09-05T09:15:29Z","2018-04-13","2018-05"],"dc:description":["\"We obtain results for two distinct dynamical models: the Kuramoto model, a general model for coupled oscillator systems, and a model for opinion formation in social networks. Our main focus is on understanding the fixed points of these systems and their stability. For many models the stability of such fixed points can be studied with a Laplacian matrix. We give a formula for the inertia of these matrices, characterizing the real parts of the spectrum, by relating them to another matrix depending on the network topology. We then study the Kuramoto model, and in particular, the phenomena of synchronization, when all oscillators rotate at a common frequency, which corresponds to a fixed point. This phenomenon is well-known to depend on the natural frequencies of the oscillators and, more specifically, that the chance of synchronization increases if the natural frequencies are more similar. We then give upper and lower bounds for the volume of the set such frequencies in frequency space. Our bounds can be formulated in terms of sums over spanning trees which we further use to deduce that the volume is intimately related to the number of spanning trees for dense networks. We also characterize the structure of fixed points of the Kuramoto model by showing that every fixed point corresponds to a lattice point in a certain set which records how the phase-angles wrap around cycles in the network. As a consequence, under mild conditions, we derive the rate of growth of the number of fixed points as we consider increasingly large graphs with fixed topology. We also consider a model for opinion formation in social networks. More specifically, we characterize the global minima of an energy functional, intuitively the ``most stable\"\" configurations, when the network is ``balanced\"\" as well as show that the number of stable configurations can increase as we increase the strengths of the relationships in the network. Finally, we describe an algorithm for generating certain random networks. These networks are generalizations of Erd\\H{o}s-R\\'{e}nyi graphs with correlations between pairs of edges depending on the particular pattern they create. We then use this algorithm to study the effect on fixed points of network properties and therefore the dynamics of the Kuramoto model.\"","Submission published under a 24 month embargo labeled 'U of I Access', the embargo will last until 2020-05-01","The student, Timothy Ferguson, accepted the attached license on 2018-04-11 at 15:36.","The student, Timothy Ferguson, submitted this Dissertation for approval on 2018-04-11 at 15:43.","This Dissertation was approved for publication on 2018-04-13 at 13:16.","DSpace SAF Submission Ingestion Package generated from Vireo submission #12195 on 2018-08-31 at 17:18:27","Made available in DSpace on 2018-09-04T20:34:02Z (GMT). No. of bitstreams: 3 FERGUSON-DISSERTATION-2018.pdf: 14880827 bytes, checksum: 97feb8f424444798535599fe3d9d82ba (MD5) thesis-ex.tex: 285727 bytes, checksum: b420f1f3e1bbd8690ae9fb8d6c9aa3ea (MD5) LICENSE.txt: 4213 bytes, checksum: 2a37310fbe8cb031e0ea4541a7b45042 (MD5) Previous issue date: 2018-04-13","Embargo set by: Seth Robbins for item 107236 Lift date: 2020-09-04T20:34:13Z Reason: Author requested U of Illinois access only (OA after 2yrs) in Vireo ETD system","Embargo set by: Seth Robbins for item 107236 Lift date: 2020-09-04T20:37:00Z Reason: Author requested U of Illinois access only (OA after 2yrs) in Vireo ETD system","Embargo set by: Seth Robbins for item 107236 Lift date: 2020-09-04T20:42:08Z Reason: Author requested U of Illinois access only (OA after 2yrs) in Vireo ETD system","U of I Only Restriction Lifted for Item 107236 on 2020-09-05T09:15:29Z."],"dc:format":["application/pdf"],"dc:identifier":["http://hdl.handle.net/2142/101153"],"dc:language":["en"],"dc:rights":["Copyright 2018 Timothy Ferguson"],"dc:subject":["Dynamics","Networks"],"dc:title":["Dynamical systems on networks"],"dc:type":["text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:24:38Z"}