## Analysis Of Network Architecture And Function

Within the mycelial networks of saprotrophic Basidiomycota there is considerable scope for communication, since hyphae maintain continuity with their immediate 'ancestors' and if contact is made with neighbouring regions can become connected via de novo formation of cross-links (anastomoses). This results both radially and tangentially in systems with many connected loops (Figure 4).

The mycelium has evolved differently in different species resulting in a range of network architectures, adapted differently for differing balances of exploration, transport efficiency and resilience to damage. Highly interconnected my-celia are costly to construct but offer alternate transport routes and thus resilience to damage. Sparse networks with fewer interconnections can extend further for a

Figure 4 (a) Mycelium of Phanerochaete velutina after 25 days growth from a 4 cm3 beech wood inoculum (I) across a tray (24 x 24 cm) of compressed non-sterile woodland soil. The fungus has met and colonized a second wood block (B). Digital image courtesy of rory bolton. (b) The digitized network of the same mycelium coloured by log10 of link cross-sectional area. The number of nodes V = 1,738, links E — 2,617 and the number of separate parts G — 1. The number of closed loops (cycles) in the illustrated system is E — V + G — 880, and the fraction of all possible cycles present a — 0.25.

Figure 4 (a) Mycelium of Phanerochaete velutina after 25 days growth from a 4 cm3 beech wood inoculum (I) across a tray (24 x 24 cm) of compressed non-sterile woodland soil. The fungus has met and colonized a second wood block (B). Digital image courtesy of rory bolton. (b) The digitized network of the same mycelium coloured by log10 of link cross-sectional area. The number of nodes V = 1,738, links E — 2,617 and the number of separate parts G — 1. The number of closed loops (cycles) in the illustrated system is E — V + G — 880, and the fraction of all possible cycles present a — 0.25.

given construction cost, but risk the loss of pathways should one part of the network become damaged. Networks can vary not only in their connectedness but also in the strength of their connections. Thick cords confer greater transport capacity and resistance to breakage, but are more costly to produce. While these concepts have been implicit in discussions of fungal foraging strategies and fractal descriptions of mycelia, the architecture of mycelial networks has been little explored until recently (Bebber et al., 2006; Bebber et al., 2006, in press; Fricker and Bebber, in press; Fricker et al., 2007).

### 5.1 Quantifying Network Characteristics

DM is a useful metric for comparing space filling by mycelia (Boddy and Donnelly, in press), but it only expresses a small fraction of the complex architecture of mycelial systems. Tools for analysing networks are, however, emerging from graph theory and statistical mechanics (Albert and Barabasi, 2001; Strogatz, 2001; Dorogovtsev and Mendes, 2002; Newman, 2003; Amaral and Ottino, 2004), that are applicable to mycelial networks (Bebber et al., 2006, in press; Fricker and Bebber, in press; Fricker et al., 2007), and have already proved valuable for understanding the properties of many physical systems that can be described as sets of connected entities, including biological networks such as protein-protein interactions and food webs (Bork et al., 2004; Dunne et al., 2004).

A network is simply a set of nodes, or vertices (V), connected by a set of links (E), or edges. Weights, that define properties such as resistance to breakage or transport capacity, can be associated with either nodes or links, or both. The nodes of a fungal mycelium are the tips, branch points and fusions of hyphae or cords, while the links are the hyphae or cords themselves. Various weights can be assigned to the nodes and links. For example, the mass of a cord can be approximated by its volume, the length multiplied by the cross-sectional area. Similarly, assuming that cords are composed of bundles of hyphae rather than being hollow tubes, the resistance to flow could be a function of the length divided by the cross-sectional area, i.e. long thin tubes have a greater resistance to flow.

The properties of nodes are often defined by the links to which they are connected. In the case of networks without link weights (the majority of examples in the literature lack this information), the number of links per node (termed degree k) is often used to describe something about the connectedness of the network. This measure is unlikely to be of interest in describing mycelial networks, as the majority of nodes will be of degree k — 3 (the branches and fusions) or 1 (the cord tips). Instead, the sum of link weights per node, known as the node strength, is likely to be of greater interest. For example, calculating the node strength for link cross-sectional area could indicate which nodes are likely to be important in transport.

The number of nodes, links and separate parts G, known as disconnected components or subgraphs, G — 1 (for an unbroken network), form the network that can be used to calculate the number of closed loops (cycles) in the network via the simple relation E — V+G (Figure 4). This cyclomatic number is extremely important, for it indicates the number of alternate pathways among points in the network that determine both the resilience to damage and the capability of parallel flow. The cyclomatic number is typically normalized to the maximum number possible for a network of a given size to give the meshedness or alpha coefficient, allowing networks with differing numbers of nodes to be compared.

### 5.2 Modelling Transport

The fungal mycelium is essentially a transport network for nutrients, water and metabolites (Chapter 3). Modelling of transport in the mycelium has been attempted using various approaches, including partial differential equations and autonomous agents (Edelstein and Segel, 1983; Deutsch et al., 1993). Since these methods ignore the network structure of the mycelium, greater insights may be obtained by taking an explicitly network-based approach into the analysis of transport. One way to achieve this is to calculate shortest path distances from each node to every other. If the effective physiological distance, or transport resistance, from one end of a cord to the other is modelled as the cord length divided by its cross-sectional area, the shortest path from one node to another will be the route with the smallest sum of these distances. The shortest path is therefore effectively the path of least resistance. Analysis of shortest paths of P. velutina, growing from wood blocks over soil, shows that the shortest paths from the wood blocks to other nodes of the fungus are smaller than they would be in a network with identical topology (i.e. number and location of links and nodes) but with uniform cord transport capacity (Bebber et al., submitted). The fungus has therefore allocated resources to cords in a way that increases its transport efficiency. The only nodes for which the fungus is less efficient than in the randomized system are those at the periphery of the mycelium, where very fine hyphae are located (Figure 4). Here the fungus has optimized mycelial distribution for searching for new resources rather than optimizing for transport.

The routes taken by shortest paths can reveal other aspects of network transport. For example, the importance of a node can be estimated by its betweenness centrality, which is the proportion of shortest paths that pass through that node (Freeman, 1977). The proportion of paths that pass through the node with the greatest betweenness centrality is the central point dominance. In fungal networks, resources such as wood blocks usually have the greatest betweenness centrality.

The shortest path is usually only one of the several routes that could be taken from one node to another. Transport through a real network will often make use of these alternate routes, in the same way that electricity will flow through each of a set of resistors in parallel. Use of shortest paths to characterize transport necessarily ignores the importance of these parallel pathways through the network. Methods for solving current flow (e.g. Wu, 2004) through networks of electrical resistors can in principle be used to model flow through mycelial networks, for example, by applying a voltage to the inoculum and grounding the hyphal tips. This may provide more realistic models of flow than simply using the shortest path.

### 5.3 Modelling Resilience

In nature, fungal mycelia are threatened by damage from physical disturbance and targeted attack by grazers such as Collembola (Chapter 9). Network architecture plays an important role in resilience to damage, through both route redundancy and the probability of link breakage. Assuming a spatially random mode of attack, the probability of a link being attacked is proportional to its length. If, when attacked, the probability of link breakage is inversely proportional to its cross-sectional area, then the joint probability of a link being attacked and broken is proportional to length divided by cross-sectional area. The effect of attack on transport can be followed by examining the global efficiency, the sum of the reciprocals of all shortest paths (Latora and Marchiori, 2001), as the network disintegrates. Paths that are no longer traversable due to the formation of multiple disconnected components are infinitely long, and thus contribute zero to the global efficiency. Global efficiency therefore declines with increasing proportions of broken links. Another way of characterizing resilience is through the reachability, or availability, of a network (Ross and Harary, 1959). Reachability is the proportion of shortest paths that still exist (i.e. are not infinitely long). Reachability is one for a network that has not been fragmented (i.e. consists of one subgraph), since all nodes are mutually available. Reachability does not depend on path length and is therefore independent of the link breakage probability function, whereas the efficiency will be greater if thick cords are less likely to break than thin ones.

Another way to measure resilience is to measure the proportion of the original network that remains connected to the wood block as increasing numbers of links are broken. In nature, disconnection from a food supply is likely to result in death of the disconnected part. When networks of P. velutina were tested against model networks with uniform link weights, more of the fungal network remained attached to the inoculum when a given proportion of links were broken (Bebber et al., in press). This demonstrated that the allocation of resources to cords not only increases transport efficiency, but also increases the resilience of the network to this kind of random attack. Inspection of network models that have been attacked in this way suggests that the secret to this increased resilience is the maintenance of a connected core structure as peripheral cords are broken. This pattern is intriguingly similar to that obtained in real networks after attack by certain species of Collembola (Chapter 9). Other mycophagous species may attack networks in other ways, depending, for example, on the size of their mouthparts.

### 5.4 Changes in Network Architecture over Time

As already mentioned, in peripheral regions cords are thin and at growing fronts hyphae are not aggregated, and therefore have high resistance to transport and long path lengths to the inoculum. As the network develops, some links become strengthened, such that the effective path lengths become dramatically shortened over time, while other links are removed, leading to an overall decrease in the material cost density over time (Bebber et al., submitted). The expectation would be that such strengthening would be accompanied by an increase in the overall construction cost of the network. However, thinning and removal of extraneous cords actually results in a decrease in the volume of material per unit area covered by the network (Bebber et al., submitted). The mechanism by which certain cords are selected for reinforcement while others are broken down remains unknown. A possible conceptual model is one of Darwinian evolution, in which multiple cords are produced but only the 'fittest', in some sense, survive and produce further growth.

### 5.5 Future Research Direction

One of the most important avenues for further research will be the comparison of network structures and dynamics among the many different cord-forming fungal species. Like any organism, a fungus must partition limited resources among competing requirements. For example, a very dense, highly connected network might have high transport capacity and resilience to damage or attack because of multiple transport pathways. However, it would incur a large material cost of construction per unit area of explored space, and would cover new ground slowly. Conversely, a sparse system could extend further for the same material cost, but would have fewer alternate routes and therefore lower resilience to disconnection. Variation in these tradeoffs among species could reveal important axes of niche differentiation in fungi.

Further, fungi provide one of the few real network systems that can be experimentally manipulated, and that can actually rebuild themselves following damage. Analyses of cord-forming Basidiomycota mycelial systems are therefore likely to reveal a range of evolutionary solutions to network design that may inform the development of other types of transport and infrastructure network, e.g. road, rail and telephone networks.

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