Exact analysis of dynamics on networks using permutation group theory

Exact analysis of dynamics on networks using permutation group theory<br /> <br /> Many social, economic and natural phenomena, including the spread of epidemics and voting intentions, can be modelled as dynamical processes on networks. Mathematical analyses of such models usually resort to low-dimensional &lsquo;mean-field&rsquo; approximations, typically based on intuitive probabilistic reasoning rather than rigorous mathematics. In contrast, dynamics on small networks can be formulated exactly as Markov chains, enabling more detailed analyses. Exact analysis of larger networks can be achieved by &lsquo;lumping&rsquo; states together to reduce the state-space size using network symmetries, or graph automorphisms, i.e. permutations of the vertices that leave the network unchanged.<br /> <br /> In this PhD, we will investigate graph and group structures that give rise to significant lumping, i.e. a significant reduction in the size of state-space for Markov chain dynamics on networks. Different types of graph structure can be produced using 'graph products', where multiple graphs are combined according to specific rules about which edges are included. In this way, families of graphs can be identified for which the automorphism group is known and the amount of lumping can be derived analytically. In addition, we would also like to develop a general theoretical approach to identify lumped states (and hence the size of the lumped state-space) based on permutation group theory, whereby the automorphism group is sub-divided into intransitive, imprimitive and primitive group structures. The challenge will be to pick apart how these group structures can be used to identify a set of representative lumped states in a computationally efficient way.<br /> <br /> This PhD will build expertise at the interface between pure and applied mathematics, particularly in the areas of network science, Markov chains, graph theory and permutation group theory. The project will involve a stimulating mix of computational and theoretical work, supported by supervisors in both the Leeds Applied Nonlinear Dynamics and Logic/Algebra groups. The successful applicant will benefit from the large and vibrant groups in these areas at Leeds, as well as regular seminar series and specialist lecture courses via the MAGIC consortium, which spans 20 UK universities.

<p>Formal applications for research degree study should be made online through the&nbsp;<a href="https://www.leeds.ac.uk/research-applying/doc/applying-research-degrees">University&#39;s website</a>. Please state clearly in the Planned Course of Study section that you are applying for <em><strong>PHD Applied Mathematics FT</strong></em> and in the research information section&nbsp;that the research degree you wish to be considered for is <strong><em>Exact analysis of dynamics on networks using permutation group theory</em></strong>&nbsp;as well as&nbsp;<a href="https://eps.leeds.ac.uk/maths/staff/4092/dr-jon-ward">Dr Jonathan Ward</a> as your proposed supervisor.</p> <p>If English is not your first language, you must provide evidence that you meet the University&#39;s minimum English language requirements (below).</p> <p><em>As an international research-intensive university, we welcome students from all walks of life and from across the world. We foster an inclusive environment where all can flourish and prosper, and we are proud of our strong commitment to student education. Across all Faculties we are dedicated to diversifying our community and we welcome the unique contributions that individuals can bring, and particularly encourage applications from, but not limited to Black, Asian, people who belong to a minority ethnic community, people who identify as LGBT+ and people with disabilities. Applicants will always be selected based on merit and ability.</em></p> <p class="MsoNoSpacing">Applications will be considered on an ongoing basis. &nbsp;Potential applicants are strongly encouraged to contact the supervisors for an informal discussion before making a formal application. &nbsp;We also advise that you apply at the earliest opportunity as the application and selection process may close early, should we receive a sufficient number of applications or that a suitable candidate is appointed.</p> <p>Please note that you must provide the following documents at the point you submit your application:</p> <ul> <li>Full Transcripts of all degree study or if in final year of study, full transcripts to date</li> <li>Personal Statement outlining your interest in the project</li> <li>CV</li> <li>Funding information including any alternative sources of funding that you are applying for or if you are able to pay your own fees and maintenance</li> </ul>

Applicants to research degree programmes should normally have at least a first class or an upper second class British Bachelors Honours degree (or equivalent) in an appropriate discipline. The criteria for entry for some research degrees may be higher, for example, several faculties, also require a Masters degree. Applicants are advised to check with the relevant School prior to making an application. Applicants who are uncertain about the requirements for a particular research degree are advised to contact the School or Graduate School prior to making an application.

The minimum English language entry requirement for research postgraduate research study is an IELTS of 6.0 overall with at least 5.5 in each component (reading, writing, listening and speaking) or equivalent. The test must be dated within two years of the start date of the course in order to be valid. Some schools and faculties have a higher requirement.

<p style="margin-bottom:12px"><strong>Self-Funded or externally sponsored students are welcome to apply.</strong></p> <p><strong>UK</strong>&nbsp;&ndash;&nbsp;The&nbsp;<a href="https://phd.leeds.ac.uk/funding/209-leeds-doctoral-scholarships-2022">Leeds Doctoral Scholarships</a>, <a href="https://phd.leeds.ac.uk/funding/234-leeds-opportunity-research-scholarship-2022">Leeds Opportunity Research Scholarship</a> and <a href="https://phd.leeds.ac.uk/funding/55-school-of-mathematics-scholarship">School of Mathematics Scholarships</a> are available to UK applicants. <a href="https://phd.leeds.ac.uk/funding/60-alumni-bursary">Alumni Bursary</a> is available to graduates of the University of Leeds.</p> <p><strong>Non-UK</strong> &ndash;<a href="https://phd.leeds.ac.uk/funding/55-school-of-mathematics-scholarship">The School of Mathematics Scholarships</a> are available to all International applicants.&nbsp; The&nbsp;<a href="https://phd.leeds.ac.uk/funding/48-china-scholarship-council-university-of-leeds-scholarships-2021">China Scholarship Council - University of Leeds Scholarship</a>&nbsp;is available to nationals of China. The&nbsp;<a href="https://phd.leeds.ac.uk/funding/73-leeds-marshall-scholarship">Leeds Marshall Scholarship</a>&nbsp;is available to support US citizens. <a href="https://phd.leeds.ac.uk/funding/60-alumni-bursary">Alumni Bursary</a> is available to graduates of the University of Leeds.</p> <p><strong>Important:&nbsp;</strong> Any costs associated with your arrival at the University of Leeds to start your PhD including flights, immigration health surcharge/medical insurance and Visa costs are not covered under this studentship.</p> <p>Please refer to the <a href="https://www.ukcisa.org.uk/">UKCISA</a> website for information regarding Fee Status for Non-UK Nationals.</p>

<p>For further information about this project, please contact Dr Jonathan Ward<br /> e:&nbsp;<a href="mailto:j.a.ward@leeds.ac.uk">j.a.ward@leeds.ac.uk</a>, t: +44 (0)113 343 5157.</p> <p>For further information about your application, please contact Doctoral College Admissions<br /> e:&nbsp;<a href="mailto:maps.pgr.admissions@leeds.ac.uk">maps.pgr.admissions@leeds.ac.uk</a></p>