{"id":26507726,"url":"https://github.com/jacksonwalters/orbit-space-homology","last_synced_at":"2025-03-20T23:33:10.388Z","repository":{"id":146379908,"uuid":"81489728","full_name":"jacksonwalters/orbit-space-homology","owner":"jacksonwalters","description":"Computes polyhedral fundamental domains, and the empirical (sample) mean, for the moduli space of unlabeled networks on n vertices.","archived":false,"fork":false,"pushed_at":"2024-03-09T18:08:29.000Z","size":140,"stargazers_count":0,"open_issues_count":0,"forks_count":2,"subscribers_count":3,"default_branch":"master","last_synced_at":"2024-03-10T15:29:44.486Z","etag":null,"topics":["frechet-means","homology","homology-computation","networks","python","sagemath"],"latest_commit_sha":null,"homepage":"","language":"Jupyter Notebook","has_issues":true,"has_wiki":null,"has_pages":null,"mirror_url":null,"source_name":null,"license":"mit","status":null,"scm":"git","pull_requests_enabled":true,"icon_url":"https://github.com/jacksonwalters.png","metadata":{"files":{"readme":"README.md","changelog":null,"contributing":null,"funding":null,"license":"LICENSE","code_of_conduct":null,"threat_model":null,"audit":null,"citation":null,"codeowners":null,"security":null,"support":null,"governance":null,"roadmap":null,"authors":null,"dei":null}},"created_at":"2017-02-09T20:01:15.000Z","updated_at":"2022-12-07T18:45:47.000Z","dependencies_parsed_at":"2024-03-09T15:29:22.508Z","dependency_job_id":"fc325ddf-7c80-48ba-b8a0-1c2aae5c85c3","html_url":"https://github.com/jacksonwalters/orbit-space-homology","commit_stats":null,"previous_names":[],"tags_count":0,"template":false,"template_full_name":null,"repository_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/jacksonwalters%2Forbit-space-homology","tags_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/jacksonwalters%2Forbit-space-homology/tags","releases_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/jacksonwalters%2Forbit-space-homology/releases","manifests_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/jacksonwalters%2Forbit-space-homology/manifests","owner_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/owners/jacksonwalters","download_url":"https://codeload.github.com/jacksonwalters/orbit-space-homology/tar.gz/refs/heads/master","host":{"name":"GitHub","url":"https://github.com","kind":"github","repositories_count":244711713,"owners_count":20497415,"icon_url":"https://github.com/github.png","version":null,"created_at":"2022-05-30T11:31:42.601Z","updated_at":"2022-07-04T15:15:14.044Z","host_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub","repositories_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories","repository_names_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repository_names","owners_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/owners"}},"keywords":["frechet-means","homology","homology-computation","networks","python","sagemath"],"created_at":"2025-03-20T23:33:09.902Z","updated_at":"2025-03-20T23:33:10.382Z","avatar_url":"https://github.com/jacksonwalters.png","language":"Jupyter Notebook","funding_links":[],"categories":[],"sub_categories":[],"readme":"# orbit-space-homology\nGiven a convex polyhedral space X and a finite linear group G, computes the (simplicial) (co)homology of the orbit space X/G. Computes a Dirichlet fundamental domain and refines triangulation by including extra vertices which are fixed by the action of G. \n\nThis code was originally written for the case of unlabled networks with non-negative, real edge weights where $X=\\mathbb{R}_{\\ge 0}^N$, $N = \\binom{n}{2}$ = #edges, $G=\\Sigma_n \\subset \\Sigma_N$, the edge permutations induced by permuting vertices. \n\nREQUIRED SOFTWARE:\n\nSage - http://www.sagemath.org/\n\nlrslib - http://cgm.cs.mcgill.ca/~avis/C/lrs.html\n\nSETUP:\n\nThis code is written in Python for the SageMath software system. \nYou will need to install and configure the latest version of SageMath, available at http://www.sagemath.org/.\n\nThe Polyhedra functions in Sage are able to make use of the following optional Sage package 'lrslib'.\nThis is most easily installed within Sage using the following terminal command: \n\n```sage -i lrslib```\n\nUsing this optional package will increase the speed these computations immensely, very roughly by a factor\nof 10.\n\nRUNTIME:\n\nTo get started, open\n\n```orbit_space_simplicial_homology.ipynb```\n\nin the Sage environment, i.e. a Jupyter notebook with the Sage kernel. You will receive a prompt to set the global variable\n\n```n = \"your desired number of vertices\"```\n\nThe global variable $N = \\binom{n}{2}$. The global variable $G=\\Sigma_n$ will then be computed as a subgroup of the permutation group $\\Sigma_N$. To construct a fundamental domain (for $n=4$, $N=6$) centered at the distinct vector $l=[1,2,3,4,5,6.1]$, simply enter\n\n```F=fund_domain([1,2,3,4,5,6.1])[1]; F```\n\n$F$ is a Sage polyhedron object representing a fundamental Dirichlet domain. Refer to the Sage documentation for the many available options for handling polyhedra:\n\nhttp://doc.sagemath.org/html/en/reference/geometry/sage/geometry/polyhedron/constructor.html\n\nTo compute the cohomology, run the rest of the notebook. It will compute the $k$-faces for each $k$ from $1,\\dots,N$ (very slow for $N \\ge 6$). It will then assemble the boundary maps, and build the chain complex. The homology is $\\text{im}(\\partial_{k+1})/\\text{ker}(\\partial_{k})$. To compute the homology, compute the vertices, the glued faces, the boundary maps, the chain complex, and the homology:\n\n```\n#compute a cross section of the fundamental domain (sum of components is constant)\nF2=cross_section(F[1])\n#compute vertices fixed by G action\nvertices=fixed_verts(F2,QQ); vertices\n#compute glued 0-faces\nzero_faces=glued_faces(0); zero_faces\n#compute glued 1-faces\none_faces=glued_faces(1); one_faces\n...\n#list of all glued face\nglued_face_list=[zero_faces,one_faces,two_faces,…]\n#form the chain complex with truncation at degree 2\n#computes each boundary map\ncc = chain_complex(max_degree=2)\n#compute the homology of the chain complex\ncc.homology()\n```\n\nG-CW complexes: https://math.mit.edu/research/undergraduate/urop-plus/documents/2016/Liu.pdf\n","project_url":"https://awesome.ecosyste.ms/api/v1/projects/github.com%2Fjacksonwalters%2Forbit-space-homology","html_url":"https://awesome.ecosyste.ms/projects/github.com%2Fjacksonwalters%2Forbit-space-homology","lists_url":"https://awesome.ecosyste.ms/api/v1/projects/github.com%2Fjacksonwalters%2Forbit-space-homology/lists"}