{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:OLPJEH3ZDD3D5DRPIXHREY5HSV","short_pith_number":"pith:OLPJEH3Z","canonical_record":{"source":{"id":"1408.2005","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2014-08-09T01:49:38Z","cross_cats_sorted":["cs.DM","math.CO"],"title_canon_sha256":"77cdad3c3b3bc50e5fbd970337194c525d640713cf73f2f13e276acceaf13e94","abstract_canon_sha256":"89252a8d6ee283039c5450c154146382ef21ee617f9a927b19590580cb96e2e4"},"schema_version":"1.0"},"canonical_sha256":"72de921f7918f63e8e2f45cf1263a79560a7e16aa312c1371ae6d9a8506e70af","source":{"kind":"arxiv","id":"1408.2005","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1408.2005","created_at":"2026-05-18T02:45:30Z"},{"alias_kind":"arxiv_version","alias_value":"1408.2005v1","created_at":"2026-05-18T02:45:30Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1408.2005","created_at":"2026-05-18T02:45:30Z"},{"alias_kind":"pith_short_12","alias_value":"OLPJEH3ZDD3D","created_at":"2026-05-18T12:28:41Z"},{"alias_kind":"pith_short_16","alias_value":"OLPJEH3ZDD3D5DRP","created_at":"2026-05-18T12:28:41Z"},{"alias_kind":"pith_short_8","alias_value":"OLPJEH3Z","created_at":"2026-05-18T12:28:41Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:OLPJEH3ZDD3D5DRPIXHREY5HSV","target":"record","payload":{"canonical_record":{"source":{"id":"1408.2005","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2014-08-09T01:49:38Z","cross_cats_sorted":["cs.DM","math.CO"],"title_canon_sha256":"77cdad3c3b3bc50e5fbd970337194c525d640713cf73f2f13e276acceaf13e94","abstract_canon_sha256":"89252a8d6ee283039c5450c154146382ef21ee617f9a927b19590580cb96e2e4"},"schema_version":"1.0"},"canonical_sha256":"72de921f7918f63e8e2f45cf1263a79560a7e16aa312c1371ae6d9a8506e70af","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:45:30.791001Z","signature_b64":"r0L5eAzN48GuKAnHrWMgtocAwx4wo29TLnwSFoMfDy2X9BgBRgRCsEFqvLVs/pFBc519awiGOi+wIw8HNBAYAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"72de921f7918f63e8e2f45cf1263a79560a7e16aa312c1371ae6d9a8506e70af","last_reissued_at":"2026-05-18T02:45:30.790464Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:45:30.790464Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1408.2005","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":"2026-05-18T02:45:30Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HC+6bbso73FHNnKQsc+sphm1/D7mhpnx2flSK9gGxZj75vf99qiPtIJOmsaro0mlUo9LpYzbniFKBIjhwTNbCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T19:47:25.468176Z"},"content_sha256":"78e20b091359b462eac838c7fcfddab80e3b4d677218166c976e86c6606b0d0c","schema_version":"1.0","event_id":"sha256:78e20b091359b462eac838c7fcfddab80e3b4d677218166c976e86c6606b0d0c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:OLPJEH3ZDD3D5DRPIXHREY5HSV","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the Meeting Time for Two Random Walks on a Regular Graph","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM","math.CO"],"primary_cat":"math.PR","authors_text":"Bhaskar Krishnamachari, Yizhen Zhang, Zihan Tan","submitted_at":"2014-08-09T01:49:38Z","abstract_excerpt":"We provide an analysis of the expected meeting time of two independent random walks on a regular graph. For 1-D circle and 2-D torus graphs, we show that the expected meeting time can be expressed as the sum of the inverse of non-zero eigenvalues of a suitably defined Laplacian matrix. We also conjecture based on empirical evidence that this result holds more generally for simple random walks on arbitrary regular graphs. Further, we show that the expected meeting time for the 1-D circle of size $N$ is $\\Theta(N^2)$, and for a 2-D $N \\times N$ torus it is $\\Theta(N^2 log N)$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1408.2005","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":"2026-05-18T02:45:30Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"lqR2dIuGaytkzQNuKx7cHdcO3dKKe5FvB65196z/j4HDEWRBYgVs511YqCGvQDmxXbj6/h5oXdGASjut11jCBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T19:47:25.468886Z"},"content_sha256":"33ab570a2731705bc22be9d8c973b06f5c2c506fdb80fa1a7fc1219ee1920a3b","schema_version":"1.0","event_id":"sha256:33ab570a2731705bc22be9d8c973b06f5c2c506fdb80fa1a7fc1219ee1920a3b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/OLPJEH3ZDD3D5DRPIXHREY5HSV/bundle.json","state_url":"https://pith.science/pith/OLPJEH3ZDD3D5DRPIXHREY5HSV/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/OLPJEH3ZDD3D5DRPIXHREY5HSV/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":"2026-05-31T19:47:25Z","links":{"resolver":"https://pith.science/pith/OLPJEH3ZDD3D5DRPIXHREY5HSV","bundle":"https://pith.science/pith/OLPJEH3ZDD3D5DRPIXHREY5HSV/bundle.json","state":"https://pith.science/pith/OLPJEH3ZDD3D5DRPIXHREY5HSV/state.json","well_known_bundle":"https://pith.science/.well-known/pith/OLPJEH3ZDD3D5DRPIXHREY5HSV/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:OLPJEH3ZDD3D5DRPIXHREY5HSV","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":"89252a8d6ee283039c5450c154146382ef21ee617f9a927b19590580cb96e2e4","cross_cats_sorted":["cs.DM","math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2014-08-09T01:49:38Z","title_canon_sha256":"77cdad3c3b3bc50e5fbd970337194c525d640713cf73f2f13e276acceaf13e94"},"schema_version":"1.0","source":{"id":"1408.2005","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1408.2005","created_at":"2026-05-18T02:45:30Z"},{"alias_kind":"arxiv_version","alias_value":"1408.2005v1","created_at":"2026-05-18T02:45:30Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1408.2005","created_at":"2026-05-18T02:45:30Z"},{"alias_kind":"pith_short_12","alias_value":"OLPJEH3ZDD3D","created_at":"2026-05-18T12:28:41Z"},{"alias_kind":"pith_short_16","alias_value":"OLPJEH3ZDD3D5DRP","created_at":"2026-05-18T12:28:41Z"},{"alias_kind":"pith_short_8","alias_value":"OLPJEH3Z","created_at":"2026-05-18T12:28:41Z"}],"graph_snapshots":[{"event_id":"sha256:33ab570a2731705bc22be9d8c973b06f5c2c506fdb80fa1a7fc1219ee1920a3b","target":"graph","created_at":"2026-05-18T02:45:30Z","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":"We provide an analysis of the expected meeting time of two independent random walks on a regular graph. For 1-D circle and 2-D torus graphs, we show that the expected meeting time can be expressed as the sum of the inverse of non-zero eigenvalues of a suitably defined Laplacian matrix. We also conjecture based on empirical evidence that this result holds more generally for simple random walks on arbitrary regular graphs. Further, we show that the expected meeting time for the 1-D circle of size $N$ is $\\Theta(N^2)$, and for a 2-D $N \\times N$ torus it is $\\Theta(N^2 log N)$.","authors_text":"Bhaskar Krishnamachari, Yizhen Zhang, Zihan Tan","cross_cats":["cs.DM","math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2014-08-09T01:49:38Z","title":"On the Meeting Time for Two Random Walks on a Regular Graph"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1408.2005","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":"sha256:78e20b091359b462eac838c7fcfddab80e3b4d677218166c976e86c6606b0d0c","target":"record","created_at":"2026-05-18T02:45:30Z","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":"89252a8d6ee283039c5450c154146382ef21ee617f9a927b19590580cb96e2e4","cross_cats_sorted":["cs.DM","math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2014-08-09T01:49:38Z","title_canon_sha256":"77cdad3c3b3bc50e5fbd970337194c525d640713cf73f2f13e276acceaf13e94"},"schema_version":"1.0","source":{"id":"1408.2005","kind":"arxiv","version":1}},"canonical_sha256":"72de921f7918f63e8e2f45cf1263a79560a7e16aa312c1371ae6d9a8506e70af","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"72de921f7918f63e8e2f45cf1263a79560a7e16aa312c1371ae6d9a8506e70af","first_computed_at":"2026-05-18T02:45:30.790464Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:45:30.790464Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"r0L5eAzN48GuKAnHrWMgtocAwx4wo29TLnwSFoMfDy2X9BgBRgRCsEFqvLVs/pFBc519awiGOi+wIw8HNBAYAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:45:30.791001Z","signed_message":"canonical_sha256_bytes"},"source_id":"1408.2005","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:78e20b091359b462eac838c7fcfddab80e3b4d677218166c976e86c6606b0d0c","sha256:33ab570a2731705bc22be9d8c973b06f5c2c506fdb80fa1a7fc1219ee1920a3b"],"state_sha256":"c6d66cdc7bbd6d601ffdd20d628d87771b00d651cb158444a3a813067dc0002d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xVVutzwNhCE760NtY/V74W3NQB/2mcXcyr94lWwOMSapBWXJ1qAwR+n746Jfw0CiRTUy6LwHQn7G7dPCfuuQCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-31T19:47:25.472701Z","bundle_sha256":"ce16e6fed2dc5dd8443db4306e43bd36549a305e7b39a038d591ba3ead1e1efe"}}