{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:GQX2E3BQWKPH4MWMEO6BOMXVGM","short_pith_number":"pith:GQX2E3BQ","canonical_record":{"source":{"id":"1809.07698","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2018-09-15T17:06:02Z","cross_cats_sorted":["math.DG"],"title_canon_sha256":"f7f1b16c18ec0e2efe3368d2e6c3eb2d5373360037e79460606d082cdfe62d68","abstract_canon_sha256":"e6946364e9b937b43595c88479ac3a6cd1720d3fd86dac8ec9733fd5b4f0e79b"},"schema_version":"1.0"},"canonical_sha256":"342fa26c30b29e7e32cc23bc1732f5332db1454c80c4500a835b80fb9a521389","source":{"kind":"arxiv","id":"1809.07698","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1809.07698","created_at":"2026-05-17T23:57:39Z"},{"alias_kind":"arxiv_version","alias_value":"1809.07698v1","created_at":"2026-05-17T23:57:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1809.07698","created_at":"2026-05-17T23:57:39Z"},{"alias_kind":"pith_short_12","alias_value":"GQX2E3BQWKPH","created_at":"2026-05-18T12:32:25Z"},{"alias_kind":"pith_short_16","alias_value":"GQX2E3BQWKPH4MWM","created_at":"2026-05-18T12:32:25Z"},{"alias_kind":"pith_short_8","alias_value":"GQX2E3BQ","created_at":"2026-05-18T12:32:25Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:GQX2E3BQWKPH4MWMEO6BOMXVGM","target":"record","payload":{"canonical_record":{"source":{"id":"1809.07698","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2018-09-15T17:06:02Z","cross_cats_sorted":["math.DG"],"title_canon_sha256":"f7f1b16c18ec0e2efe3368d2e6c3eb2d5373360037e79460606d082cdfe62d68","abstract_canon_sha256":"e6946364e9b937b43595c88479ac3a6cd1720d3fd86dac8ec9733fd5b4f0e79b"},"schema_version":"1.0"},"canonical_sha256":"342fa26c30b29e7e32cc23bc1732f5332db1454c80c4500a835b80fb9a521389","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:57:39.897726Z","signature_b64":"9xXgmH25HiqY9XtMn0jxxUaFss7GuEdg09QeWciwY/K2LRYtXjCYc4aSzhqClbohWNDVko18znrR7nl89L6MAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"342fa26c30b29e7e32cc23bc1732f5332db1454c80c4500a835b80fb9a521389","last_reissued_at":"2026-05-17T23:57:39.897217Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:57:39.897217Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1809.07698","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:57:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Fo0GeoR1hmUunhay6WCg6rF5wefgRTjpamP57GRS6c8uIU+MTpFxHccacW0sXUU12Fzp61gfk7dRD+QnB4hCCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-06T14:50:02.418231Z"},"content_sha256":"effc7254f7c7300e01748420bb69d9ddbbaa671e747c717f28ddae2bdcd1e5e4","schema_version":"1.0","event_id":"sha256:effc7254f7c7300e01748420bb69d9ddbbaa671e747c717f28ddae2bdcd1e5e4"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:GQX2E3BQWKPH4MWMEO6BOMXVGM","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Discrete Ricci curvatures for directed networks","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG"],"primary_cat":"math.MG","authors_text":"Areejit Samal, Emil Saucan, J\\\"urgen Jost, R.P. Sreejith, R.P. Vivek-Ananth","submitted_at":"2018-09-15T17:06:02Z","abstract_excerpt":"A goal in network science is the geometrical characterization of complex networks. In this direction, we have recently introduced Forman's discretization of Ricci curvature to the realm of undirected networks. Investigation of this edge-centric network measure, Forman-Ricci curvature, in diverse model and real-world undirected networks revealed that the curvature measure captures several aspects of the organization of undirected complex networks. However, many important real-world networks are inherently directed in nature, and the definition of the Forman-Ricci curvature for undirected networ"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.07698","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:57:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gU+c7h/KQHj9pNTJN7v2OrbEwjiJM+cB5yaJZzYjiZTIC/wwq2NJNOhT+qlgy/WZMHsDgxzdvw5Mf5CBzRUnDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-06T14:50:02.418993Z"},"content_sha256":"8f56ea7f49e8d128ef9eb4d240f06e150ed5af61bd1857458091f9834576433c","schema_version":"1.0","event_id":"sha256:8f56ea7f49e8d128ef9eb4d240f06e150ed5af61bd1857458091f9834576433c"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/GQX2E3BQWKPH4MWMEO6BOMXVGM/bundle.json","state_url":"https://pith.science/pith/GQX2E3BQWKPH4MWMEO6BOMXVGM/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/GQX2E3BQWKPH4MWMEO6BOMXVGM/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-06T14:50:02Z","links":{"resolver":"https://pith.science/pith/GQX2E3BQWKPH4MWMEO6BOMXVGM","bundle":"https://pith.science/pith/GQX2E3BQWKPH4MWMEO6BOMXVGM/bundle.json","state":"https://pith.science/pith/GQX2E3BQWKPH4MWMEO6BOMXVGM/state.json","well_known_bundle":"https://pith.science/.well-known/pith/GQX2E3BQWKPH4MWMEO6BOMXVGM/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:GQX2E3BQWKPH4MWMEO6BOMXVGM","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"e6946364e9b937b43595c88479ac3a6cd1720d3fd86dac8ec9733fd5b4f0e79b","cross_cats_sorted":["math.DG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2018-09-15T17:06:02Z","title_canon_sha256":"f7f1b16c18ec0e2efe3368d2e6c3eb2d5373360037e79460606d082cdfe62d68"},"schema_version":"1.0","source":{"id":"1809.07698","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1809.07698","created_at":"2026-05-17T23:57:39Z"},{"alias_kind":"arxiv_version","alias_value":"1809.07698v1","created_at":"2026-05-17T23:57:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1809.07698","created_at":"2026-05-17T23:57:39Z"},{"alias_kind":"pith_short_12","alias_value":"GQX2E3BQWKPH","created_at":"2026-05-18T12:32:25Z"},{"alias_kind":"pith_short_16","alias_value":"GQX2E3BQWKPH4MWM","created_at":"2026-05-18T12:32:25Z"},{"alias_kind":"pith_short_8","alias_value":"GQX2E3BQ","created_at":"2026-05-18T12:32:25Z"}],"graph_snapshots":[{"event_id":"sha256:8f56ea7f49e8d128ef9eb4d240f06e150ed5af61bd1857458091f9834576433c","target":"graph","created_at":"2026-05-17T23:57:39Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"A goal in network science is the geometrical characterization of complex networks. In this direction, we have recently introduced Forman's discretization of Ricci curvature to the realm of undirected networks. Investigation of this edge-centric network measure, Forman-Ricci curvature, in diverse model and real-world undirected networks revealed that the curvature measure captures several aspects of the organization of undirected complex networks. However, many important real-world networks are inherently directed in nature, and the definition of the Forman-Ricci curvature for undirected networ","authors_text":"Areejit Samal, Emil Saucan, J\\\"urgen Jost, R.P. Sreejith, R.P. Vivek-Ananth","cross_cats":["math.DG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2018-09-15T17:06:02Z","title":"Discrete Ricci curvatures for directed networks"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.07698","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:effc7254f7c7300e01748420bb69d9ddbbaa671e747c717f28ddae2bdcd1e5e4","target":"record","created_at":"2026-05-17T23:57:39Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"e6946364e9b937b43595c88479ac3a6cd1720d3fd86dac8ec9733fd5b4f0e79b","cross_cats_sorted":["math.DG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2018-09-15T17:06:02Z","title_canon_sha256":"f7f1b16c18ec0e2efe3368d2e6c3eb2d5373360037e79460606d082cdfe62d68"},"schema_version":"1.0","source":{"id":"1809.07698","kind":"arxiv","version":1}},"canonical_sha256":"342fa26c30b29e7e32cc23bc1732f5332db1454c80c4500a835b80fb9a521389","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"342fa26c30b29e7e32cc23bc1732f5332db1454c80c4500a835b80fb9a521389","first_computed_at":"2026-05-17T23:57:39.897217Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:57:39.897217Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"9xXgmH25HiqY9XtMn0jxxUaFss7GuEdg09QeWciwY/K2LRYtXjCYc4aSzhqClbohWNDVko18znrR7nl89L6MAw==","signature_status":"signed_v1","signed_at":"2026-05-17T23:57:39.897726Z","signed_message":"canonical_sha256_bytes"},"source_id":"1809.07698","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:effc7254f7c7300e01748420bb69d9ddbbaa671e747c717f28ddae2bdcd1e5e4","sha256:8f56ea7f49e8d128ef9eb4d240f06e150ed5af61bd1857458091f9834576433c"],"state_sha256":"f4896fb3ee41a1e10e2c27b5106037e64808f74f8d0d5359c55d5b6ef159206b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Ltx59/APajMexa2dT5Azr6BmQgcW6b1fOr598M3iReBb5L/ScN7XowEAFpIN6VjLDLjnhicG2+jFoSzzLFM0BQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-06T14:50:02.422737Z","bundle_sha256":"44299245a4657885b8bd4936597a8be94d693d8c044635e13d5167f0b65bc314"}}