Mycelial networks.

Progress in image processing and analysis has allowed for automated extraction of digitized networks from images of mycelium [17]. In this study, we examine such networks from three different species. For a clean comparison against the rodent vasculature, we study a subset of networks that were ungrazed and grown without any additional resources aside from source innocula. In addition, since the vasculature was sampled at only one time point, for each type of mycelium, we use the most mature (i.e., the oldest) networks. Finally, we only consider networks with at least 500 nodes, so as to obtain good estimates of Rentian scaling (see Network measures). In all, we examine 4 different networks formed by Phallus impudicus (P.I.) grown from a single inoculum and sampled at 46 days, 3 networks from Phanerochaete velutina (P.V. 1) grown from 5 inocula and sampled between 30 days and 35 days, 5 networks from Phanerochaete velutina (P.V. 2) grown from 1 inocula and sampled between 39 days and 46 days, and 1 network from Resinicium bicolor (R.B.) grown from a single inoculum and sampled at 31 days. All of the data for the fungal networks was collected from several studies and made available online [19]. More information on details of these networks can be found in [19], the references therein, and in the documentation within the database.

Rodent vasculature networks.

The rodent vasculature dataset consists of the pial networks in the region of the middle cerebral artery from four rats and five mice. To map out the networks, the vasculature was imaged and then traced by hand (further details can be found in the original study [27]). The rodent vasculature networks were kindly provided by Pablo Blinder and the group of David Kleinfeld at the University of California, San Diego.

Standard topological graph metrics.

In our analysis, we begin by first characterizing the different distribution networks using a set of widely-utilized graph metrics that quantify the connectivity—or topology—of a network, without utilizing information about the physical locations of nodes and edges. In particular, we will examine the mean degree 〈k〉, the average clustering coefficient C, the alpha index α, the topological efficiency E^t, and the topological edge betweeness centrality . In short, the mean degree measures the average number of edges incident to nodes in a network, the clustering coefficient quantifies the occurrence of fully connected triplets of nodes in a network, the alpha index measures the density of elementary cycles of all lengths in a planar network, the topological efficiency is defined as the average of the inverse of the shortest topological path lengths between all pairs of network nodes, and the topological edge betweenness centrality measures the number of shortest topological paths that go through a given edge. As these are widely used measures in the network science literature, we leave more detailed descriptions and definitions to the S1 Text.

Rentian scaling.

We also investigate Rentian scaling [32–37] in the distribution systems, which is an empirical scaling relationship between the number of nodes n in a partition of a network and the number of edges m crossing the boundary of that partition, such that m ∝ n^r where 0 ≤ r ≤ 1 is the Rent exponent. Since its initial discovery in very large scale integrated (VLSI) circuits [38], this phenomena has been observed in biological neural systems [39, 40] and in urban transportation networks [3]. As described further below, Rentian scaling can be examined in both the topological and physical space of a network, and the presence of Rentian scaling indicates a type of hierarchical organization and can be used to assess network complexity.

Topological Rentian scaling is indicated by a relationship of the form, m ∝ n^t, where n is the number of nodes inside a topological partition of the network, m is the number of edges crossing the partition boundary, and 0 ≤ t ≤ 1 is the topological exponent. More specifically, one asks whether this relationship holds across partitions of varying sizes. Here, we assess networks for topological Rentian scaling using the hyper-graph partitioning package hMETIS [41, 42], in which a recursive min-cut bi-partitioning algorithm recursively splits the network into halves, quarters, etc. in a way that attempts to minimize the number of edges passing from one partition to another (see Fig. D(i) in the S1 Text). After each round of partitioning, we track the number of nodes n in each partition, and the number of edges m crossing the partition boundaries; if the relationship between these quantities obeys the form m ∝ n^t, then the network is said to show topological Rentian scaling, and we can estimate the scaling exponent t from the slope of a best-fit line through log m vs. log n (Fig. D(ii) in the S1 Text). See the S1 Text for further details on how this analysis is carried out. Rather than considering t in isolation, it is also useful to recall the relationship between topological scaling and network complexity, as developed in VLSI theory. In particular, the topological dimension of a network d_T is related to t via ; thus, higher topological exponents indicate higher dimensional network topologies [33, 36].

The presence of physical Rentian scaling can be examined by testing for a relationship of the form m ∝ n^p, where m is the number of network edges crossing the boundary of a contiguous physical partition of the network, n is the number of nodes inside the partition, and p is the physical Rent exponent. More specifically, one examines whether this relationship holds over many partitions across different length scales in real space. To empirically test for this relationship in a network, we consider 5000 square partitions of the network, where the side length and center position of each partition is chosen at random (see Fig. D(iii) in the S1 Text). In order to avoid boundary effects due to the finite size of the network, we only sample boxes contained within the network’s convex hull. We then compute the number of nodes n inside each partition and the number of edges m crossing the partition boundary (Fig. D(iv) in the S1 Text), and test whether these quantities are related by a power law of the form m ∝ n^p, where p is the physical Rent exponent (see Fig. D(v) in the S1 Text). If such a relationship holds, then the exponent p can be estimated from the slope of a best-fit line through log m vs. log n (see S1 Text for more details on how this analysis is carried out). The existence of physical Rentian scaling signifies hierarchical structure in the spatial layout of the network, and furthermore, because the scaling is determined from a random sampling of the network, its existence signifies a degree of spatial homogeneity and space-filling capacity. Importantly, physical Rentian scaling extends ideas of self-similarity (such as the fractal dimension [43]), which are agnostic to the spatial positioning of network elements. Moreover, the theoretical minimum value for p is related to the Euclidean dimension d_E into which a network is embedded and the topological Rent exponent t via [34, 39]. The theory of VLSI circuit design predicts that low values of p (i.e., values close to the theoretical minimum) are associated with networks that have been cost-efficiently embedded into physical space, in terms of wiring length [32, 34, 39].

Biophysically-motivated network measures.

In addition to standard measures of network topology, we can also compute measures of network organization that are more directly related to the functional requirements of and constraints on biological distribution networks. We describe these biophysically-motivated network metrics in the remainder of this section.

Wiring length.

For spatially embedded networks such as the transport systems considered in this study, one can approximate the total wiring length of the networks, which in turn provides an estimate of material cost. Given a distance matrix D_ij that encodes the Euclidean distance between all pairs of nodes, and an undirected, unweighted adjacency matrix A_ij that represents network connectivity, we define the total wiring length of the network as (2) Note that this definition of wiring length assumes that vessels or cords are straight, and neglects additional wiring length that may result from tortuosity.

Physical efficiency.

For spatial networks, we can also examine paths along network edges using physical distances, rather than topological distances. In particular, the shortest physical path between a pair of nodes i and j is taken to be the path that minimizes the sum of the lengths of the edges traversed in a path connecting nodes i and j. This is in contrast to the shortest topological path between the same nodes, which neglects geometric information. Using physical path lengths rather than topological path lengths, we can compute the average physical efficiency . This quantity is defined as (3) where is the length of the shortest physical path along network edges between nodes i and j. Dividing by the corresponding value for a fully connected network with the same number of nodes yields the global physical efficiency E^p, where 0 ≤ E^p ≤ 1. Shortest path and efficiency measures that explicitly incorporate the geometric layout of a network have previously been used to study spatial networks such as ant galleries [44], street patterns in cities [45, 46], brain networks [47], slime mould [48], and fungal networks [17, 20–23], but to the best of our knowledge, have not yet been used to quantify animal vasculature networks.

Physical edge betweenness centrality.

From the shortest physical paths between all node pairs, we can also calculate a physical edge betweenness centrality . Letting be the fraction of shortest paths (measured in physical units of distance) between q and r that pass through edge (i, j), we compute (4) For an edge (i, j), this quantity is thus the fraction of shortest physical paths between all node pairs that traverse the edge (i, j).

Network robustness.

Finally, we note that a desirable feature for both natural and artificial networks is robustness. Here, we consider structural robustenss in the sense of how well-connected a network remains when subjected to damage. We probe this by removing a fraction of the total number of edges at random from a network, and then track how the size of the largest connected component evolves as the fraction of removed edges increases. In particular, we define the robustness R of a network to be the percentage of edges removed in order for the size of the largest connected component to drop to half of its original value [44, 45]. Since edges are removed at random, there can be some variation in the results when the same procedure is run again on the same network. We thus compute R a total of 20 times for each network in order to generate a representative ensemble average. We again note that in our analysis, we consider the robustness of the entire vasculature network, rather than just the backbone, as considered in [27]. This allows for a cleaner comparison between the vasculature and fungi, and takes into consideration the many vessels that branch directly off of the backbone to penetrating arterioles.

Randomly-rewired benchmarks.

One null model we consider is a type of random graph—sometimes called the configuration model—constructed by randomly rewiring a given empirical network while preserving its degree distribution. To generate these randomized benchmarks, we use the Brain Connectivity Toolbox [50], which contains a function based on the algorithm introduced in [51], and that also maintains the connectedness of the rewired network. Each edge is rewired approximately 15 times. This null model is employed in the first part of our analysis, in order to better compare certain topological metrics across the set of empirical distribution networks. In particular, for each empirical network, we generate an ensemble of 10 randomly rewired networks. We then compute the mean value 〈X_rewire〉 of a given metric across the ensemble of randomly rewired networks, and normalize the empirical value X by the reference value 〈X_rewire〉. We can then compare the normalized values . This allows us to ask if different kinds of distribution networks exhibit differences in specific network properties, while controlling for variability in network size and degree distribution. Note that while this null model preserves certain topological features of an empirical network, it is agnostic to spatial features.

Spatial null models for wiring, efficiency, and robustness.

In spatial networks, the nodes and edges exist in real space, and this physical embedding often has significant consequences for the network topology [5]. An important constraint for biological systems in particular is the material and energetic costs associated with building and maintaining the physical structure of the network. But a competing constraint stems from the fact that distribution networks must be able to move resources efficiently and be robust to damage. In order to gain an understanding of how the spatial embedding of these networks might affect their architecture, and how distinct types of biological transport systems may differentially balance certain pressures, in the second part of our analysis we compare the empirical networks to two idealized planar null model networks: the minimum spanning tree (MST) and the greedy triangulation (GT). The same or similar null models have previously been used in the analysis of fungal systems, [17, 20–23], slime mould [48], networks of ant galleries [44], and urban street networks [45, 46].

The MST is a graph that connects all of the nodes in a network such that the sum of the total edge weights is minimal. By construction, the MST also contains the minimum number of edges M needed to connect a network (M = N − 1, where N is the number of nodes). In the biological distribution networks studied here, the relevant edge weights are their physical lengths. Therefore, for the MST null model, we preserve the true spatial locations of all nodes in the empirical network, and compute the MST on the matrix of Euclidean distances, D_ij, between all node pairs. The resulting network is planar and minimizes the total wiring length W of the network (Eq 2), given the spatial distribution of nodes. The GT represents the opposite extreme, and is a maximally connected—in terms of the number of edges—planar network. In particular, following [44–46], we compute GTs on the true node locations of all real networks by iteratively connecting pairs of nodes in ascending order of their distance while ensuring that no edges cross. As with the MST, this null model is also constructed under a geometric constraint, but since it contains many more edges, it represents an effective upper bound on the wiring length of a planar network with given node locations. In Fig 2, we show the MST and GT for the mycelial and vasculature networks depicted in Fig 1.

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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.

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Relative measures of wiring length, spatial efficiency, and robustness.

In the second part of our analysis, we examine a set of network measures that explicitly take into account spatial constraints and that are more directly pertinent to the function of transport networks. These are the wiring length, physical efficiency, and robustness (see Biophysically-motivated network measures). It is important to note that these kinds of metrics have previously been used to quantify and compare the structure of mycelial networks [17, 20–23]. Here, we will build on and extend prior analyses by examining tradeoffs and relationships between these quantities, and most importantly, we focus on comparing the wiring length, network efficiency, and structural robustness across the vasculature and mycelial networks.

While useful measures to consider in the context of distribution systems, when considered in isolation, it is difficult to understand how these metrics are related to one another and how they compare across similar networks of different size or of completely different type. In order to better understand how the network architecture of the vasculature and mycelial systems might be differentially organized for each of these measures, we thus use a set of normalized quantities that characterize how the wiring length, efficiency, and structural robustness of a given empirical network compare to approximate limiting values for a network of the same size and with the same node locations. In particular, following [44–46], we define a relative cost, physical efficiency, and structural robustness using the MST and GT null models.

The relative cost is (5) where W, W_MST, and W_GT denote the total wiring length of the real, MST, and GT networks, respectively. In a similar manner, the relative global physical efficiency, , is given by (6) and the relative robustness R_rel is (7) By definition, the MST is minimally wired, but since it contains no redundancy or shortcuts, we also expect it to have low efficiency. It is also clear that the removal of any edge will break the MST into disconnected components. In contrast, the GT is a highly expensive network to build, but this increase in network density for the same set of nodes should improve the robustness as well as the efficiency of the network, since it allows for shortcuts between pairs of otherwise more distant nodes. Thus, the MST is an effective lower bound for wiring length, efficiency and robustness, and the GT is a good approximation for a planar network that achieves upper bounds with respect to the same measures. The relative measures are normalized between 0 and 1 for the MST and GT, respectively. This normalization allows for a direct quantification of where the real distribution networks lie in comparison to the limiting values achievable for a fixed spatial distribution of nodes. Furthermore, the relative quantities can be used to understand and properly contrast the tradeoffs between wiring, efficiency, and resistance to random damage between the vasculature and fungi.

Standard network metrics.

To begin, we characterize the vasculature and mycelia using a series of standard topological network metrics.

Focusing first on measures that assess local connectivity in the immediate neighborhood of nodes in a network, we found that the mean degree 〈k〉 was highly constrained, falling between 2 and 3 for all networks. This is expected for biological flow networks, in which junctions typically develop from the branching, fusion, or anastomosis of vessels or cords [17, 52]. However, even within this narrow range, we observed statistically significant differences in 〈k〉 for the vasculature vs. fungi (p-value = 7.8 × 10⁻⁸, Fig 3A), with the mycelial networks having larger mean degrees () than the vasculature systems (). We also examined the clustering coefficient, which measures transitivity in a network (i.e., the density of the shortest possible cycles in a network—those of length three). Considering, in particular, the normalized clustering coefficient (Fig 3B), we found that most of the empirical networks tended to have larger clustering relative to their randomly-rewired null models, although two of the vasculature networks had C = 0. We note that, especially for the vasculature, many instantiations of the randomly rewired networks also yielded clustering coefficients of zero, pointing out that triangles are generally rare in randomized networks of the same size and degree distribution. Nevertheless, in total across the two kinds of distribution systems, we found that the mycelial networks had higher normalized clustering coefficients () than the vasculature networks (), and this difference was statistically significant (p-value = 6.8 × 10⁻⁵). This result indicates that compared to an ensemble of size and degree-distribution preserving null models, the mycelial networks tend to have a higher density of 3-edge cycles than the vasculature networks.

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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.

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While the clustering coefficient assesses a network for the existence of cycles of length three, we can also consider loops (or cycles) on larger topological length scales. This is particularly relevant for distribution systems, as past work on empirical vasculature networks [27] and models of flow networks [53] have found that loops can serve to protect the system from occlusions or damage to veins. To quantify the presence of cycles of all lengths in the mycelial and vasculature networks, we computed the alpha index α [5, 45], which is equal to 0 for a network that is a tree (no cycles) and is equal to 1 for a maximal planar network. The alpha index is plotted for all networks in Fig 3C. One observes that the vasculature networks tend to have lower alpha indices () compared to the fungal networks (), and we found that this difference was statistically significant (p-value = 7.4 × 10⁻⁸). Thus, compared to maximal planar networks of the same size, the mycelial networks have relatively more loops than the vasculature networks. As expected [5], the alpha index gives similar information to the mean degree.

We next examined the topological efficiency E^t, which—in contrast to the previous metrics—evaluates larger-scale organization in a network by considering the shortest (topological) pathways between all pairs of nodes. To compare the topological efficiency across the vasculature and fungi, for each network we computed the normalized quantity that measures how the global topological efficiency of each distribution network compares to a collection of rewired networks of the same size and degree distribution (Fig 3D). We found a statistically significant difference in the population averages of the normalized topological efficiencies of the vasculature vs. mycelial networks (, ; p-value = 9.8 × 10⁻⁶). This result suggests that the two kinds of distribution networks can also be distinguished in terms of this measure of their global organizational structure.

As previously noted, for spatially embedded networks, we can also measure path lengths in terms of Euclidean distances. An interesting question one can then ask is: how much overlap is there between counterpart physical and topological measures of network organization? Do they contain the same (or similar) information, or measure something distinct about the system? Here, we examine the relationship between the topological edge betweenness centrality and the physical edge betweenness centrality , which measure the fraction of shortest topological and physical paths, respectively, that pass through a given edge in a network. For each distribution network, we computed the Spearman rank correlation coefficient ρ between and to assess the strength and direction of a monotonic relationship between the two quantities (see Fig 4A). Interestingly, while there was a significant, positive rank correlation between and for all networks, we also observed a clear and statistically significant difference (p-value = 7.5 × 10⁻⁸) in the average rank correlation coefficient of the population of mycelial networks compared to that of the vasculature networks. In particular, we found that the mycelial networks had smaller ρ values than the vasculature networks (, ). Thus, in the vasculature networks, the topological and physical edge betweenness centralities contain similar information in that edges that participate in many of the shortest topological paths also participate in many of the shortest physical paths through the network. But this overlap is not nearly as strong in the mycelial networks, suggesting that topological and physical correlates of network architecture need not necessarily be redundant. Fig 4B and 4C show vs. for the networks with the highest (Mouse 5) and lowest (R.B.) Spearman correlations, respectively. These findings motivate the second major part of our analysis (Comparative analysis of network organization using biologically-motivated measures), in which we characterize and compare and contrast the vasculature and mycelial networks using spatial and more biologically-motivated measures of network structure that take into account the physical nature of these systems.

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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.

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Rentian scaling.

Important questions regarding biological transport systems are whether distribution networks of different types or from different organisms exhibit universal characteristics that allow for high functionality, and in particular if and how they achieve embeddings into physical space that are energetically favorable while maintaining topologies that support efficiency and robustness. For example, it is thought that across technological [32], neural [39, 40], and distribution [54, 55] networks alike, one such common feature is hierarchical structure. To begin investigating these questions here, we next assessed each network for the presence of topological and physical Rentian scaling, following the methods described in Rentian scaling.

We first computed topological Rentian scaling. Fig 5A and 5B show examples of log m vs. log n obtained from a min-cut bi-partitioning method applied to a rat vasculature network and a mycelial network, respectively. (Fig. F in the S1 Text shows examples from the other distribution networks as well). Visual inspection suggested that m and n scale with one another in log-log space, permitting the estimation of a topological Rent exponent t. We also calculated the linear correlation coefficient and a linear regression between log m and log n (depicted as the black lines through the data in the examples of Fig 5A and 5B) for each distribution network. When averaged over several runs of the topological partitioning process, we found r ≥ 0.991 and r² > 0.981 for all networks, providing further evidence of Rentian scaling in topological space. (See S1 Text for further details regarding this analysis).

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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.

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We turn next to physical Rentian scaling. In the context of the distribution networks, this analysis evaluates whether the spatial layout of a network is such that the number m of vessels or cords passing into or out of a given contiguous region of a network scales with the number of nodes n inside that region, and whether such a relation is conserved over different length scales and spatial domains of a network. Fig 5C and 5D show examples of log m vs. log n in a mycelial system and a vasculature system, respectively, and Fig. H in the S1 Text shows examples from each of the other kinds of distribution networks as well. Visual observation provided initial evidence for the existence of physical Rentian scaling in the distribution networks, and to further assess each network for this relationship, we also computed a linear correlation and a linear-regression between log m vs. log n (displayed as the black lines over the data in Fig 5C and 5D). For all networks, we found Pearson’s correlation coefficients of r > 0.9 and coefficients of determination r² > 0.82, which further indicated that the functional form m ∝ n^p was a relatively good fit to the data and that we could estimate physical Rentian scaling exponents p for each network from the linear-regression. (See S1 Text for further details on this analysis). The presence of physical Rentian scaling suggests that the connectivity of the distribution networks exhibits a type of hierarchical organization and space-filling capacity in terms of their physical arrangement and embedding into real space.

Though we can estimate physical and topological Rent exponents for the empirical networks from best-fit lines to log m vs. log n, it is important to note that because the networks considered here are relatively small, the values of the exponents should be considered carefully, as should a direct comparison across different networks. In order to give context and meaning to the exponent values, we thus compared the scaling relationships from the empirical networks to those of their randomly rewired benchmarks (see Null models). Keeping the node locations for the randomly-rewired null models the same as the corresponding empirical network, we considered the ratios and . Comparing between the vasculature and mycelial systems (Fig 6A) revealed a statistically significant distinction between the two kinds of networks (p-value = 1.2 × 10⁻⁴). In particular, the population average of the topological Rent exponent ratio was smaller for the set of mycelial networks compared to the set of vasculature networks (, ), suggesting that in terms of topological complexity, the vasculature networks are on average more similar to their randomly-rewired benchmarks. We also found a clear relative difference in the physical Rent exponent ratios of the vasculature vs. fungal networks (Fig 6B), with the population of vasculature networks having smaller mean physical Rent exponent ratio (, ). The difference between the two groups was statistically significant (p-value = 2.0 × 10⁻⁸), and indicates that, compared to the mycelial networks, the physical Rent exponents of the vasculature networks are even smaller than the mean exponent from their ensemble of random null models.

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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 p^min/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.

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Because we could estimate physical and topological Rent exponents for each empirical network and their randomly rewired null models, we also examined how the physical Rent exponent of each network compared to its theoretical minimum value (see Rentian scaling). We first considered the quantity for each network, where p is the true physical Rent exponent of an empirical network and p^min is its theoretical minimum value, and where p_rewire is the true physical Rent exponent of a corresponding randomly rewired benchmark network and is its theoretical minimum value; the angled brackets indicate an average over the ensemble of null-model networks. As perhaps expected, we found for all networks. Thus, the empirical distribution networks have physical Rent exponents closer to their minimum possible values, indicating that they achieve more efficient spatial layouts relative to their set of randomly-rewired null-models. We next compared the quantity p^min/p across the vasculature and mycelial networks (Fig 6C), finding a statistically significant difference between the two groups (, , p-value = 1.8 × 10⁻¹¹). The vasculature networks yielded values of p^min/p closer to one (i.e., had physical Rent exponents closer to their theoretical minimum) compared to the fungal networks, suggesting that they may be especially efficiently embedded. We explore this idea further in the next section. Also see the S1 Text for an analysis of topological and physical Rentian scaling in simulated networks developed from biologically inspired optimization principles [53], in which we also find evidence of these relationships as well as exponent values similar to those obtained from the empirical data.

Tradeoffs between physical network efficiency and wiring.

We begin by examining the physical efficiency E^p and the wiring length W of each network. The physical efficiency (Eq 3) is a measure that reflects the routing capabilities of a network [56], and importantly, because E^p is defined from the Euclidean lengths of network edges—rather than topological distances—it takes into account the spatial layout of a system. Intuitively, higher efficiencies may allow for improved transport of fluid and nutrients throughout a network, which are desirable capabilities for both vasculature and mycelial systems. In general, it is expected that increases in the number of connections between the same set of nodes will improve network efficiency by creating shortcuts between pairs of otherwise more distant regions. But, an addition of edges will in turn increase the amount of material used to construct the network. Note that here we estimate the material cost—or more precisely, the wiring length—to be the sum of the physical lengths of all edges in the system (Eq 2). As noted previously and in the discussion section below, an improved quantification of network cost would also include radial information as well, but this data was unavailable for the vasculature systems.

In order to quantify the extent to which each network was organized for low wiring length or high physical efficiency—and to compare and contrast the balance between these competing network features across the two types of distribution networks—we considered the relative quantities W_rel (Eq 5) and (Eq 6). These measures reflect how close a given network is to two spatially-informed null models [44–46], the minimum spanning tree MST and the greedy triangulation GT. Given a set of node locations, the MST represents a low cost but also low efficiency planar graph, and the GT represents a high cost but also high efficiency planar graph [44]. Recall that and W_rel are bounded between 0 and 1 to reflect the similarity of the empirical networks to their MST (where the relative measures are equal to zero) and the GT (where the relative measures are equal to one), respectively (see Null models for more details on the null models and relative measures).

The relative wiring length W_rel (Fig 7A) satisfied W_rel < 0.4 for all networks, suggesting the presence of general constraints on the structure and wiring usage in these systems. Grouping together all mycelial networks into one set and all vasculature networks into another set, we used a two-sample t-test to further determine whether there were statistical differences in the mean of W_rel between the two populations. We found that the mean relative wiring was indeed significantly different between the two groups (p-value = 1.5 × 10⁻¹¹), with the population of vasculature networks having a lower mean relative wiring compared to that of the population of mycelial networks (, ). We next examined the relative physical efficiency , which was intermediately valued for all networks (Fig 7B), ranging between 0.25 and 0.61. Interestingly, we found that the difference in the mean relative efficiency measure was not statistically significant across the two groups (, , p-value >0.05). Thus, while both sets of networks appear more similarly optimized for physical efficiency, our results indicate that the wiring constraints are stronger for the vasculature networks than for the fungi. This suggests that there may be different kinds of pressures driving the network organization of the two types of distribution systems.

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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 W_rel, , and R_rel 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.

https://doi.org/10.1371/journal.pcbi.1006428.g007

To further investigate the tradeoffs between the relative efficiency and relative wiring, we plotted vs. W_rel (Fig 7D). We observed that the vasculature consistently satisfied , with a significant margin between the two quantities. A similar finding held for several of the fungi, but we also found that some of the mycelial networks sat just above, on, or sometimes below the line of equality (). This result implies that for a given wiring, the vasculature networks can achieve a relatively greater physical efficiency—suggesting an economical, well-organized architecture—compared to several of the fungal networks, which appeared to have a less-optimal use of extra wiring, as measured by the ability of that wiring to increase the relative efficiency of the network. Moreover, for significantly lower W_rel, the rat vasculature achieved similar or higher values than many of the fungi, indicating an especially beneficial use of material in the vasculature systems. Finally, we note that the rodent data was much more tightly clustered in this phase space compared to the more varied fungal data; this points to consistent, regularized network structure in the vasculature in contrast to more diversity in the mycelial networks.

Differences in network robustness.

To better understand the implications of the wiring-efficiency tradeoff, we also considered the structural robustness of each network, which—like efficiency—should be facilitated by increased wiring. Robustness is an important consideration in both mycelial and vasculature networks. In mycelia, damage can occur from factors such as rough environmental conditions or from predation [17, 29–31], and in vasculature, a common source of damage is from the formation of blood clots that block vessels and prevent flow, leading to stroke. Previous work on fungal networks has shown that these organisms can achieve high levels of robustness to attack [17, 21]. In a study on rodent vasculature [27], interconnected loops were found to provide robustness to the backbone of surface vessels under edge deletion, and after occlusion to a surface arteriole, the backbone was able to re-route blood flow and preserve the surrounding neuronal tissue.

Here, we let the robustness R of a network be the percentage of edges removed in order for the size of the largest connected component to drop to half of its original value [44, 45], and we considered the relative robustness R_rel (see Biophysically-motivated network measures for details) for comparison of the fungi and vasculature. Fig 7C shows the relative robustness for all networks, which ranged between 0.08 and 0.50. Using a two-sample t-test to compare R_rel between the population of fungal networks and the population of vasculature networks, we found that the means of the two groups were significantly different (, ; p-value = 9.4 × 10⁻⁹). This clear separation between the mycelial networks and the vasculature networks suggests that the mycelia may be better optimized for resistance to damage, and further points to the fact that although comparable in certain ways, the two types of transport networks indeed exhibit different structural organization and tradeoffs. We also observed a positive trend between R_rel and W_rel across all networks (Fig 7E), indicating that increasing the amount of wiring used to connect a given set of nodes strongly correlates with improved resistance to network damage. (As might be expected, we additionally found a positive trend between R_rel and a relative measure of network density, where instead of using the total wiring length in Eq 5, we used the total number of edges in each network). There was not such a strong relationship between and W_rel, suggesting that while increased connectivity may partially improve network efficiency, the specific placement of additional wiring in the network can play a role as well.

Given that both the vasculature and fungal systems are subject to certain forms of impairment, another important question is how the physical efficiency of the underlying networks changes as a result of random damage. In other words, how functional does the topology of these networks remain when edges are removed? In order to examine this question, we tracked the change in the global physical efficiency of each network as a function of edge fraction removed. More specifically, if we let f represent the fraction of edges removed, we considered a range f = 0 to f = 0.6, in steps of Δf = 0.004. At each step, we removed the corresponding fraction of edges at random from each network, and computed the global physical efficiency (Eq 3) E^p(f) of the perturbed system and also the ratio E^p(f)/E^p(0). This process was repeated 20 times for each fraction f, and we considered the ensemble average 〈E^p(f)/E^p(0) 〉 over the 20 trials. The structure of the resulting curves of 〈E^p(f)/E^p(0)〉 vs. f (see Fig 8A) provides insight into how the transport capabilities of each network are altered under random damage.

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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 E^p(f)/E^p(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.

https://doi.org/10.1371/journal.pcbi.1006428.g008

We observed that the efficiency of the vasculature networks falls off more rapidly than the efficiency of any of the mycelial networks (Fig 8A). In order to help visualize the effect of the edge removal, we show an example of a robust fungal network (Fig 8C) and more delicate vasculature network (Fig 8D); in each case, the red lines correspond to a fraction f = 0.15 of edges selected at random. To quantify the steepness of the decline in physical efficiency for realistic removal fractions, we considered the slope of linear fits to 〈E^p(f)/E^p(0)〉 vs. f between f = 0 and f = 0.1. Then, we asked whether the steepness of the fall-off (estimated by the slope of the linear fit to each curve) was related to the relative wiring of the network. We found that W_rel was indeed a good predictor of the decay in physical efficiency (Pearson correlation r = 0.95, p-value = 2.85 × 10⁻¹¹; Fig 8B). This analysis highlights the fragility of the vasculature systems, whose ability to maintain their transport efficiency is impaired quickly with damage compared to the fungal networks, which appear better able to maintain this function in the face of edge loss.