{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/97363"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/97363","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Stability thresholds for signed Laplacians on locally-connected networks","abstract":"In this work we are interested in the stability bifurcations of the dynamical systems defined on graphs, and we use signed graph Laplacians as our tool. In chapter 1, we give the formal definition of the Laplacian matrix for a graph, and point out several references on it. In chapter 2, we give the main result from one of the references, along with other preliminaries we need for our results. In chapter 3, we give our first main result -- finding the stable point for the Laplacians of one family of graphs, namely $\\mathcal{C}_n^2$. In chapter 4, we extend the previous definition and question to $\\mathcal{C}_n^{(k)}$, and we give the exact result for $k=3$, along with a procedure to find the answer for general $k>3$. We then give several conjectures about this topic, which are supported by numerical experiments.","abstract_html":"In this work we are interested in the stability bifurcations of the dynamical systems defined on graphs, and we use signed graph Laplacians as our tool. In chapter 1, we give the formal definition of the Laplacian matrix for a graph, and point out several references on it. In chapter 2, we give the main result from one of the references, along with other preliminaries we need for our results. In chapter 3, we give our first main result -- finding the stable point for the Laplacians of one family of graphs, namely <span class=\"etd-inline-math\">\\mathcal{C}<sub>n</sub><sup>2</sup></span>. In chapter 4, we extend the previous definition and question to <span class=\"etd-inline-math\">\\mathcal{C}<sub>n</sub><sup>(k)</sup></span>, and we give the exact result for $k=3$, along with a procedure to find the answer for general $k&gt;3$. We then give several conjectures about this topic, which are supported by numerical experiments.","abstract_has_math":true,"creators":["Cong, Lin"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["DeVille, Lee","Bronksi, Jared","Kirkpatrick, Kay","Rapti, Zoi"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2017,"date_issued":"2017-04-19","date_published":"2017-04-19","updated_at":"2026-07-22T22:24:34Z","subjects":["Graph Laplacian","Dynamics on graphs"],"languages":["en"],"rights":["Copyright 2017 Lin Cong"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/97363","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["DeVille, Lee","Bronksi, Jared","Kirkpatrick, Kay","Rapti, Zoi"]},{"key":"dc:creator","label":"Author","values":["Cong, Lin"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2017-04-19","2017-05","2017-08-10T19:15:07Z"]},{"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":["Graph Laplacian","Dynamics on graphs"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2017 Lin Cong"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/97363"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["In this work we are interested in the stability bifurcations of the dynamical systems defined on graphs, and we use signed graph Laplacians as our tool. In chapter 1, we give the formal definition of the Laplacian matrix for a graph, and point out several references on it. In chapter 2, we give the main result from one of the references, along with other preliminaries we need for our results. In chapter 3, we give our first main result -- finding the stable point for the Laplacians of one family of graphs, namely $\\mathcal{C}_n^2$. In chapter 4, we extend the previous definition and question to $\\mathcal{C}_n^{(k)}$, and we give the exact result for $k=3$, along with a procedure to find the answer for general $k>3$. We then give several conjectures about this topic, which are supported by numerical experiments.","Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2017-08-10 without embargo terms","The student, Lin Cong, accepted the attached license on 2017-04-14 at 22:40.","The student, Lin Cong, submitted this Dissertation for approval on 2017-04-14 at 22:40.","This Dissertation was approved for publication on 2017-04-19 at 12:37.","DSpace SAF Submission Ingestion Package generated from Vireo submission #10774 on 2017-08-10 at 13:40:43","Made available in DSpace on 2017-08-10T19:15:07Z (GMT). No. of bitstreams: 3 CONG-DISSERTATION-2017.pdf: 454832 bytes, checksum: 92ea2fc966a31a20fc9c3772451a6395 (MD5) LICENSE.txt: 4205 bytes, checksum: f5d6f5a8040f918b2bc088ce77a04b46 (MD5) PROQUEST_LICENSE.txt: 4551 bytes, checksum: b41467da8287eba077f38034b803f672 (MD5) Previous issue date: 2017-04-19"]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Stability thresholds for signed Laplacians on locally-connected networks"]}]}],"canonical_facts":{"dc:contributor":["DeVille, Lee","Bronksi, Jared","Kirkpatrick, Kay","Rapti, Zoi"],"dc:creator":["Cong, Lin"],"dc:date":["2017-04-19","2017-05","2017-08-10T19:15:07Z"],"dc:description":["In this work we are interested in the stability bifurcations of the dynamical systems defined on graphs, and we use signed graph Laplacians as our tool. In chapter 1, we give the formal definition of the Laplacian matrix for a graph, and point out several references on it. In chapter 2, we give the main result from one of the references, along with other preliminaries we need for our results. In chapter 3, we give our first main result -- finding the stable point for the Laplacians of one family of graphs, namely $\\mathcal{C}_n^2$. In chapter 4, we extend the previous definition and question to $\\mathcal{C}_n^{(k)}$, and we give the exact result for $k=3$, along with a procedure to find the answer for general $k>3$. We then give several conjectures about this topic, which are supported by numerical experiments.","Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2017-08-10 without embargo terms","The student, Lin Cong, accepted the attached license on 2017-04-14 at 22:40.","The student, Lin Cong, submitted this Dissertation for approval on 2017-04-14 at 22:40.","This Dissertation was approved for publication on 2017-04-19 at 12:37.","DSpace SAF Submission Ingestion Package generated from Vireo submission #10774 on 2017-08-10 at 13:40:43","Made available in DSpace on 2017-08-10T19:15:07Z (GMT). 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