Extended Data Fig. 7: Flow statistics correlate Betweenness Centrality (BC) hierarchy and indicates incompressibility.

(a) Length-weighted distribution of hyphal BC across the entire network demonstrates a long-tailed power-law distribution \(P({BC}) \sim {{BC}}^{-\gamma }\) with \(\gamma \approx 0.3\), indicating hierarchical order. (b) Distribution of hyphal radius extracted from high-resolution videos. (c) Correlation between measured edge observables: radius \(r\), betweenness centrality BC, distance to tip \({d}_{\text{tip}}\), the velocity of particles moving towards the tip \({v}_{+}\), and towards the root \({v}_{-}\). Velocity sign is positive when particles move towards the tip and negative when they move towards the root. Each dot represents the Pearson correlation coefficient computed over all edges belonging to a replicate plate sample. Box represents interquartile range (IQR), with central line indicating median. Whiskers extend from box to minimum and maximum values within 1.5 times IQR. (d-g) Flow speeds are highly correlated across space and time with no detectable lag, consistent with incompressible flow. (d) Close-up of a long hypha imaged at 4x magnification. Colours indicate spatial coordinate along a single hyphal segment across which kymograph analysis was conducted. Scale bar: 1 mm. (e) Kymograph (top) and speed time series (bottom) of flows within the coloured hyphal segment in (d), for a 30 s interval in which rapid changes in flow speed were observed. Arrows indicate spatial position for speed time series of matching colour. (f) Cross-correlation of speed time series at x = 1.2 mm in (e) against all other speed time series in (e). All curves were nearly identical, peaking at a high cross-correlation value centred at a delay close to zero. (g) Cross-correlation delay \(\Delta t\) corresponding to the peak position of each curve in (f), as a function of the distance along the hypha, was very close to zero and demonstrated no dependence on distance, consistent with incompressible flow.