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 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.