Fig 1.
Distribution networks from mycelial fungi and rodent vasculature.
(A) An example of a network from the fungus Phallus impudicus (P.I.). The cords making up the mycelial network are represented as edges (gray lines) and connect to form branching, fusion, or end points (red nodes). The top left node is the inoculum. (B) An example of a vasculature network from the surface of a rat brain in the region of the middle cerebral artery. Vessel segments are represented as edges (gray lines) and connect to form branching points on the surface backbone (red nodes) or connect to penetrating arterioles (blue nodes). In both (A) and (B), the figures on the right are magnified sections of the full networks.
Table 1.
The number of nodes and edges for all studied networks.
The first grouping corresponds to the mycelial networks and the second grouping corresponds to the vasculature networks.
Fig 2.
Construction of spatial null model networks.
(A,B) The minimum spanning tree MST and (C,D) the greedy triangulation GT for the mycelial network and vasculature network shown in Fig 1.
Table 2.
Comparison of various network properties between the mycelial and vasculature networks.
The first column indicates the network property. The second and third columns give the mean ± the standard error of the mean for each network property, averaged over the population of fungal networks and the population of vasculature networks, respectively. The last column indicates the level of significance from a two-sample t-test used to assess statistical differences in the mean values of each network property between the two types of distribution networks; **p-value <0.001, ***p-value <0.0001.
Fig 3.
Comparison of topological network metrics between the mycelial and vasculature systems.
In each panel, the x-axis labels the kind of network, with the curly braces grouping the mycelia (first 4 networks) and vasculature (last 2 networks). The y-axis measures the value of a given property for each network. A p-value is displayed if we found a statistically significant difference (as determined by a two-sample t-test) in the mean value of the given property between the population of mycelial networks and the population of vasculature networks. (A) The mean degree 〈k〉 was significantly larger in the fungi compared to the vasculature. (B) The normalized clustering coefficient was significantly larger in the fungi compared to the vasculature. (C) The alpha index α was significantly larger in the fungi compared to the vasculature. (D) The normalized topological efficiency
was significantly larger in the vasculature compared to the fungi. See the main text for more details on each measure.
Fig 4.
Topological vs. physical edge-betweenness centrality.
(A) For each kind of network (displayed on the x-axis), the data shows the Spearman rank correlation coefficient ρ for the relationship between the topological edge betweenness centrality and the physical edge betweenness centrality
. The curly braces group the fungi (first 4 networks) and vasculature (last 2 networks). All networks exhibited significant positive rank correlations, but the displayed p-value from a two-sample t-test indicates that the mean rank correlation coefficient of the mycelial networks was significantly smaller than the mean rank correlation coefficient of the vasculature networks. (B) Topological edge betweenness centrality
vs. physical edge betweenness centrality
for the network with the highest Spearman correlation between these quantities (Mouse 5). (C) Topological edge betweenness centrality
vs. physical edge betweenness centrality
for the network with the lowest Spearman correlation between these quantities (R.B.). See the text for more details on these measures.
Fig 5.
Rentian scaling analysis of the distribution networks.
(A,B) Examples of log m vs. log n computed from a topological partitioning of a network from the mycelium P.V. 1 and from a topological partitioning of an arterial network from a rat brain, respectively. (C,D) Examples of log m vs. log n computed from a physical partitioning of a network from the mycelium P.V. 1 and from a physical partitioning of an arterial network from a rat brain, respectively. In panels (A–D) the black lines correspond to a simple linear regression, from which we estimate the displayed scaling exponents t or p, describing either the topological or physical Rentian scaling relationships, respectively. The r-value is the Pearson correlation coefficient and the r2-value is the coefficient of determination. Additional examples of these relationships for other networks are shown in Fig. F and Fig. H in the S1 Text.
Fig 6.
Comparison of Rentian scaling analysis in the mycelial networks and vasculature networks.
In each panel, the x-axis labels the kind of network, with the curly braces grouping the mycelia (first 4 networks) and the vasculature (last 2 networks). The y-axis measures the value of a given property for each network. A p-value is displayed if there was a statistically significant difference (as determined by a two-sample t-test) in the mean value of the given property between the population of mycelial networks and the population of vasculature networks. (A) The normalized topological Rent exponent was significantly smaller in the population of mycelial networks compared to the population of vasculature networks. (B) The normalized physical Rent exponent
was significantly larger in the population of mycelial networks compared to the population of vasculature networks. (C) The ratio pmin/p of the theoretical minimum physical Rent exponent to the true physical Rent exponent was signficantly smaller in the population of mycelial networks compared to the population of vasculature networks. See the main text for more details on each measure.
Fig 7.
Mycelial and vasculature networks exhibit distinct tradeoffs between wiring, physical efficiency, and structural robustness.
In panels (A-C) the x-axis labels the kind of network, with the curly braces grouping the mycelia (first 4 networks) and the vasculature (last 2 networks). The y-axis measures the value of a given property for each network. A p-value is displayed if there was a statistically significant difference (as determined by a two-sample t-test) in the mean value of the given property between the population of mycelial networks and the population of vasculature networks. (A) The mean relative wiring for all networks was significantly larger for the population of mycelial networks compared to the population of vasculature networks. (B) The difference in the mean relative physical efficiency of the mycelial networks and the vasculature networks was not statistically significant. (C) The mean relative structural robustness was significantly larger for the population of mycelial networks compared to the population of vasculature networks. (D) A scatterplot of the relative physical efficiency vs. the relative wiring for each network. (E) A scatterplot of the relative structural robustness vs. the relative wiring for each network. For all plots, the relative quantities Wrel, , and Rrel were determined by normalizing each network with respect to its own spatial null models, which then allowed for a meaningful comparison of the different types of distribution networks to one another.
Fig 8.
Decline of physical efficiency with random edge removal.
(A) Ensemble average of the ratio of the physical efficiency of damaged networks to the physical efficiency of the original network, as a function of the edge fraction removed. The efficiency of the vasculature networks declined more rapidly than that of the fungal networks. (B) The drop-off in physical efficiency was approximated for each network as the slope of a linear fit to Ep(f)/Ep(0) vs. f, computed between f = 0 and f = 0.1. There was a positive correlation between the slope and the relative wiring of the networks. (C) An example of a network from P.V. 2, where the red lines correspond to a random selection of f = 0.15 of the total number of edges, and the gray lines correspond to the remaining edges in the network. (D) An example of a network of a mouse vasculature system, where the red lines correspond to a random selection of f = 0.15 of the total number of edges, and the gray lines correspond to the remaining edges in the network.