flow op• Data kinds: matrix × signal → table
• Call: import fullseye as fs; fs.ledger.graph_physarum_transport(lengths, supply, dt=0.1, max_iters=5000, tol=1e-06, gap_tol=1e-06) (to call the implementation directly, import physarum_search; physarum_search.graph_physarum_transport(lengths, supply, dt=0.1, max_iters=5000, tol=1e-06, gap_tol=1e-06); from the registry, opsgraph.get("graph_physarum_transport"))
L1 optimal transport (the Beckmann problem) on a weighted graph by the slime mould's tube dynamics.
`lengths is a symmetric (n, n) matrix of tube lengths (0` = no tube, diagonal 0) and
`supply an (n,) vector that sums to 0: mass enters at the nodes with supply > 0` and
leaves where it is `< 0`. The pressures solve the weighted Laplacian with this right-hand
side (Kirchhoff), each tube carries `Q_e = D_e (p_u - p_v) / L_e` and adapts
`dD_e/dt = |Q_e| - D_e (explicit Euler, step dt`). Bonifaci (2017) and Facca,
Karrenbauer, Kolev and Mehlhorn (2020) prove that the flow converges to a minimiser of
`sum_e L_e |q_e|` under flow conservation, i.e. to the 1-Wasserstein distance between
`supply+ and supply-` in the graph metric (Facca, Cardin and Putti 2018 for the
continuum). With one unit source and one sink this is :func:graph_physarum_path.
Returns `cost = sum_e L_e |Q_e|` (the transport cost of the current flow: an upper bound
on the distance), `dual_bound = supply . p' with p'` the pressure scaled to be
1-Lipschitz along every tube (Kantorovich–Rubinstein: a lower bound on the distance at any
iteration, no external solver needed), `gap = cost - dual_bound` (0 at the optimum;
it closes like `tol / dt` divided by the smallest non-zero flow, because the scaling is
set by the tube whose conductance still lags its flow the most),
`conductance and flow ((n, n)` matrices, flow antisymmetric; the net outflow of
every node equals its supply), `potential ((n,)` pressures, 0 at one node per connected
component), `iters, converged (max |dD| fell below tol), gap_converged` (the
run stopped because `gap <= gap_tol * cost: the cost is certified within gap_tol` of the
optimum; when several flows are optimal the conductances keep wandering while the pressures
settle, so this is the stop that fires; the gap is checked every 10 iterations), `history`,
`cost_history and dual_history ((k, 2)`: iteration and lower bound at each check).
Raises `ValueError`: matrix not square / symmetric / finite / non-negative, self-loops,
no edge; `supply not an (n,)` finite numeric vector, all zero, not summing to 0
(relative `1e-9`) or unbalanced inside a connected component (mass with nowhere to go);
`dt outside (0, 1]; max_iters < 1; tol < 0; gap_tol < 0`.
• Sample-data catalog (download URLs / licences) — 2-D uses skimage.data (BSD/public domain) plus synthetic images; 3-D lists download URLs for real data sources (Stanford, PDS, …).
• Operator provenance and references — the sources of the research/methods this op family came from.
• The canonical algorithm (author, year) and its uses are named in the family usage guide above.
• poc_physarum_transport — py -3.11 examples/poc_physarum_transport.py
table as input)graph_edge_consensus · graph_core_persistence · tree_morphometry · tree_sholl · tree_run_length
flow)graph_physarum_path · physarum_route · physarum_transport_image
*Provenance: physarum_search.py — GRAPH operator registry. This per-op note is generated by tools/opdocs.py md (do not hand-edit).*
© 2026 Kazufumi Furuse — Fullseye operator documentation. Licensed under Apache-2.0.