1. Small-size, directed networks are relevant for many biological applications, from meta-populations to food webs, from transport flows to evolutionary trees, from epidemics within households to outbreaks of emerging plant pathogens (e.g. Phytophthora ramorum). However, little attention has been paid to dynamic processes in these networks.
2. In the horticultural trade, structural change in hierarchical categories, i.e. the proportion of producers, wholesalers and retailers, can influence the likelihood that plant epidemics will take place in such systems, but it is unclear how.
3. We model disease spread and establishment in directed networks of 100 nodes at four connectance levels in six network structures [local, small-world, random, and scale-free (SF) networks with positive, no, and negative correlation between in- and out-degree (number of incoming and outgoing links)], and study the role of hierarchical categories.
4. For non-SF networks, the correlation coefficient between number of incoming and outgoing links is negatively correlated with the proportion of producers and retailers, and positively correlated with the proportion of wholesalers. Given the previously reported negative correlation between the in–out degree correlation coefficient and the epidemic threshold, adding producers/retailers and removing wholesalers can contribute to making epidemics more difficult in non-SF networks. For SF networks these associations are not generally present, as in these structures epidemic development is driven by the presence of hubs, rather than the features of the majority of the nodes.
5.Synthesis and applications. despite the importance of trade movements of plants for plant epidemics and the emergence of new plant pathogens, little is known about the current contact structure of horticultural networks within and among nations, and about how this is changing. Such information is important for risk assessment and management in plant health regulation. This study thus calls for the long-term collection of data on the number and degree distribution of plant producers, wholesalers and retailers. Our results suggest that plant disease management should focus on the middle-tier of the nursery hierarchy. This result is likely to pertain also to other biological applications of small-size directed networks.
The last decades have seen an unprecedented increase in the number of human beings inhabiting the Earth, now approaching seven billion (Wilson 2002; Cohen 2003; Bongaarts 2009). Part of the food needs generated by this increase has been met by a parallel rise in agricultural crop production, enabled by the growing unsustainable use of fossil energy sources (Ehrlich, Ehrlich & Daily 1993; Conforti & Giampietro 1997; MacKay 2008). More efficient distribution of food has also helped in avoiding recurrent famines, although equity in food distribution is still far from achieved (Daily & Ehrlich 1996; Hazell & Wood 2008; Friel & Baker 2009). At the same time, the growing affluence of many people in both developed and developing countries has generated increasing demand for ornamental plants and new horticultural crops and varieties (Lawson 1996). Increase in crop production has typically not been associated with an increase in the number of growers, as the adoption of free trade policies and the containment of shipping costs have enabled competition from further distances. This enlarged competition has in turn led in many cases to the inevitable adoption of economies of scale with progressive concentration of farms and traders in the hands of a dwindling proportion of the population in most countries. All these structural developments may have affected the likelihood that plant epidemics will occur in the horticultural and ornamental trade (Dehnen-Schmutz et al. 2010). Plant epidemics represent in turn a threat to global and regional food security (Bandyopadhyay & Frederiksen 1999; MacLeod et al. 2010) and to the health of (semi)natural ecosystems (Brasier & Webber 2010; Pautasso et al. in press a).
It is now recognized that, also in the plant trade, the contact structure of networks can profoundly affect the likelihood that an epidemic will take place and the options for controlling it (Jeger et al. 2007; Marder 2007). In today’s globalized world, epidemics are made easier by the presence of long-distance connections (Moore & Newman 2000; Keeling 2005) and super-connected nodes (Pastor-Satorras & Vespignani 2001; May 2006). However, much of the modelling of epidemic development in networks has been conducted in large-size settings. Whether findings thus obtained (e.g. the importance of the presence of super-connected nodes in driving epidemic development) are also relevant for small-size networks (with hundreds rather than tens of thousands or more nodes) is still an open question which is particularly relevant for multiply structured biological populations (Guimarães et al. 2007) and regional trade networks. In addition to the horticultural exchange of plant material, there are many biological applications which can be realistically modelled as small-size networks (Mason & Verwoerd 2007; Thébault & Fontaine 2010). These applications range from social interactions of primates, manakins and scientists (Dunbar 1993; Carillo, Papagni & Capitanio 2008; Ryder et al. 2008), to co-occurrence patterns of epiphytes, mycorrhiza and parasites with host tree species (Southworth et al. 2005; Burns 2007; Vacher et al. 2010).
In the case of network epidemiology, small-size networks are important wherever local epidemics occur within sub-groups of individuals (e.g. households, schools, workplaces and markets; Liu, Wu & Yang 2004; Pellis, Ferguson & Fraser 2009). Clustering of epidemics in small groups of individuals appears to be a frequent phenomenon also for animal and plant diseases (Otterstatter & Thomson 2007; Brooks, Antonovics & Keitt 2008). Although we know that heterogeneities in the number of links among nodes can generally lower epidemic thresholds in complex networks (Colizza & Vespignani 2007; Volz & Meyers 2009), there has been little attention to whether this is the case also for small-size, directed networks. Directed networks are relevant to many asymmetric real-world situations where the presence of a link does not necessarily imply the reverse connection. This is the case for the horticultural trade: plant material is initially propagated by producers; usually sold on to a middle-tier of wholesalers for further growing-on, with possible plant exchange with other wholesalers; and finally sold on to retailers, who will not normally sell back material to producers and the middle-tier. In the general case, for such a hierarchy there are starting nodes with out-links, middle-tier nodes with both in- and out-links (sometimes at the same level), and endpoint nodes with in-links. However, given that in the directed case adjacency matrices are more complicated than for undirected networks, epidemics in directed networks have been relatively rarely studied (Meyers, Newman & Pourbohloul 2006; Kenah & Robins 2007).
With regard to trade in plants and their associated organisms (Jones & Baker 2007; Brenn et al. 2008), little is known about the contact structure of the networks involved. This is of concern, given the many examples of recent emerging plant diseases which have been facilitated by trade movements of infected material among countries (Goss et al. 2009; Jung et al. 2009). For example, Sudden Oak Death in California and Oregon, a regional tree and shrub mortality outbreak, has been the consequence of the inadvertent introduction of Phytophthora ramorum, a newly described oomycete probably originating from East Asia (Holdenrieder et al. 2004; Brasier et al. 2010). This pathogen is also affecting many ornamental species commonly sold for planting in private gardens and public spaces (e.g. Camellia, Magnolia, Pieris, Rhododendron) and has been detected in a number of plant nurseries and garden centres both in the USA and in Europe (Prospero et al. 2009; Xu et al. 2009). Given the presence of P. ramorum-susceptible species and disease-inducing climate in many regions of the world (e.g. the Appalachians, the Mediterranean, New Zealand), movements of infected plant material in the horticultural trade has the potential to make P. ramorum a world-wide plant pathogen emergency (Grünwald, Goss & Press 2008; Hueberli et al. 2008; Moralejo, Garcia-Munoz & Descals 2009). Since the pathogen affects a range of plant species both in the trade and in the semi-natural environment, the ensuing problems for plant health regulation are compounded. There is thus the need to study how different configurations of the network of plant nurseries and retail outlets trading ornamental species susceptible to P. ramorum can affect the likelihood of further pathogen dispersal.
In this study, we investigate how different proportions of plant producers, retailers and the middle-tier (wholesalers, both receiving and shipping plant material) affect the likelihood of epidemic development in small-sized, directed networks. In particular, we focus on the epidemic threshold, i.e. the amount of node-to-node transmission that discriminates between the occurrence of an epidemic or not. Using the same epidemic model, we have shown that heterogeneity in the contact structure still affects the epidemic threshold (the boundary between no epidemic and an epidemic) even in the case of networks of one hundred nodes (Pautasso & Jeger 2008). We have further shown that variations in the epidemic threshold for different network structures and at different levels of connectance can mainly be explained by the correlation coefficient between links to and from nodes and, in some particular cases, by the clustering coefficient of the network (Moslonka-Lefebvre, Pautasso & Jeger 2009). These results are obtained irrespective of the network structure: (1) local, (2) random, (3) small-world (SW), and scale-free (SF) networks with (4) positive (two-way), (5) no (uncorrelated), and (6) negative (one-way) correlation between in- and out-degree (number of incoming and outgoing links).
Local networks are characterized by high levels of clustering (probability that nodes adjacent to any node z are adjacent to each other; analogous to modularity) but large shortest path length (i.e. the minimum number of links that must be crossed to reach a node x from a node y). Epidemics in such networks are considerably limited by local saturation effects (infected nodes only connected to other infected nodes). Local networks are relevant to the environmentally conscious trend towards consumption of local products (Brown, Dury & Holdsworth 2009). Random networks are characterized by low clustering and short average path length, so that local geographical spread is almost absent and the epidemic is dominated by long-range connections. This is the case wherever there is little influence of geographical distance on connectivity (Benczik et al. 2009). SW networks are obtained by adding some long-range connections to local networks. They therefore inherit high clustering properties from the underlying local network, but the presence of just a few long-range connections is sufficient to substantially lower the average path length. Epidemic spread on such networks is therefore characterized by a local wave-like infection front and by jumps to remote parts of the network (Newman, Jensen & Ziff 2002). Finally, SF networks are defined as networks where the degree distribution follows a power-law distribution (the only distribution which is scale-free, i.e. of a form that is invariant under rescaling of the independent variable). Because of the fat tail of the power-law distribution, such networks are characterized by the presence of hubs, i.e. nodes with such a high degree, compared to the others, that epidemic spread is driven by their properties (Dorogvtsev, Goltsev & Mendes 2008). SF networks arise when connections are formed between links based on preferential attachment mechanisms, i.e. the tendency of new nodes to be connected to nodes which already have a higher number of links. Note that a pure power-law distribution is an asymptotic property, and therefore SF networks can be precisely defined only in the limit of an infinite size. By SF networks in the present case of small networks, we mean networks obtained from the preferential-attachment construction algorithm (as defined below).
Using these network structures, we inquire here whether variation in the proportion of nodes which (i) preferentially send rather than receive material (producers), (ii) preferentially receive rather than send material (retailers), and (iii) both receive and send material (wholesalers) explains variation in the correlation coefficient between in- and out-degree and thus in the epidemic threshold. These are broad categories which may in reality be blurred by secondary activities. For example, some producers may also be involved in a small amount of retail, and viceversa (Park & McLaughlin 2000). To account for such fuzziness in the distinction between producers, wholesalers and retailers, we used three different sets of boundaries to define the three categories.
Materials and methods
Materials and methods are simplified from Moslonka-Lefebvre, Pautasso & Jeger (2009). We simulated disease spread and establishment in networks of 100 nodes using six kinds of structure: (1) local (nearest-neighbour transmission), (2) random (nodes connected with probability P), (3) SW (local networks rewired with short-cuts), and SF structure (see Jeger et al. 2007 for a visualization). For SF networks, we considered separately networks with in- and out-degree of nodes (4) positively, (5) not and (6) negatively correlated. The networks were directed, i.e. a link from node a to node b did not imply the reverse connection (Meyers, Newman & Pourbohloul 2006; Foster et al. 2010).
For each network structure, 100 replicates were built in matlab at each level of connectance (100, 200, 400, and 1000 links). Local networks were built starting from a regular ring with 100 links more than the target number of links and by randomly generating 100 gaps. Random directed graphs were generated using the G(N,M) model where M directed links are placed randomly and independently between the N nodes of the graph. SW networks were built with the Watts & Strogatz (1998) algorithm and a rewiring coefficient of 0·25. This rewiring coefficient allowed the construction of SW networks with clustering intermediate between those of random and of local networks. SF networks were built with a preferential attachment algorithm, starting with a seed network and based on five parameters adding nodes and/or links depending on the in-, out-, and total degree of existing nodes as described in Moslonka-Lefebvre, Pautasso & Jeger (2009).
Epidemic development was deterministic, with discrete time-step and fixed contact structure. Networks were not necessarily fully connected, so it is possible that at the lower levels of connectance not all nodes could be reached from all nodes. There were two parameters governing the epidemic development: the probabilities of infection transmission between nodes (Pt) and of infection persistence in a node (Pp). The transmission probability Pt(x,y) from node x to node y was either zero (unconnected nodes) or a constant value Pt, equal for all links. In order to work at the threshold condition (see Pautasso & Jeger 2008), Pt was set differently for each realization of all network types and was equal to the leading eigenvalue of the adjacency matrix (Chakrabarti et al. 2008). We did not build a network structure starting from a certain threshold probability, but we built a network structure and then determined its threshold. Therefore, there is just one replicate for each threshold probability. The persistence probability Pp combined in one single parameter the duration of infectiousness, detection and control measures. We also set Pp to be the same for all nodes. Both Pt and Pp are real variables, ranging between 0 and 1. This can be a realistic assumption for many ecological networks, wherever persistence and transmission are not either switched on or off, but can assume any value between these two extremes.
For each iteration, the infection status Pi(x) of a given node x at time-step i was governed by the following dynamics:
where the sum is over all nodes y.
At the beginning of the epidemic Pi(x) was set to zero for all nodes x except for the starting node s of the epidemic, with P0(s) = 1. For the connection of a node with itself, Pp was used instead of Pt. The biological motivation for self-loops is that nodes which have become infected by a pathogen have a certain probability to remain infected due to the persistence of inoculum through time. The model was thus a Susceptible-Infected-Susceptible (SIS) model. This can be a realistic assumption for many epidemiological systems, wherever nodes are still at risk even after eradication of a disease outbreak if complete immunization is not possible and if there is a continuing trade or contact with susceptible material or inoculum (Jeger et al. 2007; Shishkoff 2007).
The development of the epidemic was assessed on the basis of the sum of Pi(x) across all nodes and on the basis of the number of nodes with Pi(x) higher than an arbitrary value (0·01). The epidemic was started with a single infection of a single node, as the threshold conditions were not affected by whether epidemics are started with a single or with multiple infections (M. Pautasso, unpublished observations). Also, results were consistent using a different starting probability of infection. Although the starting node had a marked influence on the epidemic size at equilibrium (Pautasso, Moslonka-Lefebvre & Jeger in press b), making the epidemic start from different nodes did not affect the threshold conditions (P*p and P*t) which define a boundary between no epidemic and an epidemic. Given that there is a linear threshold in a graph of P*p as a function of P*t (Pautasso & Jeger 2008), we worked at P*p = 0 and assessed the threshold only in terms of P*t. Working at P*p = 0 does not imply that the model changes from SIS to SI. Nodes can be re-infected from connections from other nodes, even if there is no persistence of infection in a given node from time t to time t + 1.
We assumed all nodes to be of equal capacity. Differences between nodes were thus entirely due to their in- and out-degree. We used three boundaries between the hierarchical categories, according to the following equation:
where x is the number of outgoing links and y is the number of incoming links. We defined producers as nodes with Δ of at least 20 (60 or 100)%. Retailers were nodes with Δ lower than −20 (−60 or −100)%. Wholesalers were the remaining nodes. Producers can be visualized as mainly exporters, retailers as mainly importers, and wholesalers as nodes with no preponderance of either export or import, thus linking the other two categories. The model can be applied at the level of individual plant growers and sellers, but also for single countries/regions, which, for a given commodity, will tend to be producers, intermediaries, or consumers. Similar hierarchical categories are present in other ecological applications (e.g. food webs). The proportions of different hierarchical categories are not constant across various structures because of intrinsic differences in the structures and the algorithms used to generate them. For example, local and SW networks tend to have a high proportion of wholesalers because these networks are characterized by local connectivity (with some short-cuts in the case of SW networks), so that the presence of nodes with a high number of incoming and low number of outgoing links (or vice versa) is relatively rare. In contrast, for one-way SF networks, the definition of the network itself (and the algorithm used to create the replicates, which is based on that definition) results in a high proportion of nodes which fulfil the definition of producer/retailer.
We used sas 9·1 (proc GLM) for linear regressions of the threshold P*t and the correlation coefficient between in- and out-degree against the proportion of (1) producers, (2) retailers, and (3) wholesalers in each network replicate for a given (1) network structure, (2) level of connectance, and (3) definition of producers, retailers and wholesalers.
The proportion of producers, wholesalers and retailers varied among network structures and depending on the connectance level and the definition of these node types (Fig. 1). Setting the limit for producers and retailers at 20% resulted for random and two-way SF structures in a higher proportion of producers and retailers and a correspondingly lower proportion of wholesalers than for 60% and 100%, especially at high levels of connectance. For uncorrelated and one-way SF structures, there was a negligible effect of these definitions on the proportion of the three categories. For local and SW structures, this effect was idiosyncratic (Fig. 1).
For a given boundary between producers, wholesalers and retailers (20%, 60% or 100% more outgoing/incoming links), local, random and SW networks had generally a larger proportion of wholesalers than SF structures (Fig. 1). The presence of wholesalers was very low for uncorrelated and one-way SF networks at all connectance levels (Fig. 1). There was a general trend towards a higher proportion of wholesalers with increasing level of connectance, particularly for random and two-way SF structures (Fig. 1).
For most network structures and boundary definitions between node categories, the correlation coefficient between in- and out-degree significantly decreased with an increasing proportion of producers and retailers (Fig. 2; Tables S1 and S3, Supporting Information). This pattern was general for local, random and SW networks, whilst it was present only in exceptional cases for SF networks. This difference between non-SF and SF network structures applied also to the correlation with the proportion of wholesalers. In this case, the correlation coefficient between in- and out-degree significantly increased with increasing proportion of wholesalers (Fig. 2; Supporting Information Table S2), but generally only for local, random and SW networks. These results were robust to changing the connectance level.
Given the previously reported (Moslonka-Lefebvre, Pautasso & Jeger 2009) negative correlation between correlation coefficient between in- and out-degree and the epidemic threshold, the association of the proportion of producers/retailers and wholesalers with the epidemic threshold were reversed compared to the case of the correlation coefficient between in- and out-degree. There was a positive correlation of the epidemic threshold with the proportion of producers and retailers (Tables S4 and S6, Supporting Information) and a negative correlation with the proportion of wholesalers (Table S5, Supporting Information). These associations were generally (with exception of sparsely connected networks) present for local, random and, in part, SW networks and were generally absent for SF networks.
There is much potential in the use of network epidemiology with relevance to applied plant ecology. Such an approach has been particularly fruitful in the case of the international epidemic of P. ramorum. For example, McKelvey, Koch & Smith (2008) built a useful model of disease spread in plant trade networks of the USA, with application to P. ramorum and similar pathogens. They distinguished between nodes that are sources of infection vs. nodes that are at risk of becoming newly infected, rather than hierarchical categories such as producers, wholesalers and retailers. If there is a non-null chance that retailers and wholesalers might re-infect some producers, then the two sub-divisions [(1) sources of infection and nodes at risk of infection; (2) producers/wholesalers/retailers] do not necessarily overlap (also given that the first distinction is dichotomic, the second has three categories). A further model of large-scale P. ramorum spread was developed by Harwood et al. (2009). A realistic model of the UK network trading plants susceptible to P. ramorum was integrated into a spatially explicit simulation software with country-wide data on the distribution of susceptible hosts in the semi-natural vegetation. Results of these simulations confirmed the potential of the trade in making it more difficult to bring this and similar epidemics under control (Harwood et al. 2009). However, there is still the need to test whether structural change in the trade in plants may have an influence on the chances that an epidemic will occur.
Our analysis has four main results to be discussed. These results are likely to pertain also to biological applications of small-size directed networks other than plant trade networks (e.g. food webs, with producers, consumers and predators/parasites). First, directed trade networks have a varying proportion of nodes with preferentially outgoing (producers) or incoming (retailers) links (outgoing links from retailers to consumers are not considered here). This variation is not just objective but also depends on the definition of these node categories. For a given definition, and other things such as number of connections being equal, different network structures are characterized by different average proportions of producers, wholesalers and retailers. Local, random and SW networks have generally a higher proportion of wholesalers (the middle-tier) than SF networks. This is because scale-free networks have super-connected individuals, and, unless these have both a high number of links in and out of them, these hubs will tend to be either producers or retailers. This explains why uncorrelated and one-way SF networks have even more producers and retailers than two-way SF networks (Fig. 1).
Secondly, variation in the proportion of producers, wholesalers and retailers can explain a certain amount of variation in the correlation coefficient between in- and out-degree (Fig. 2). Since this coefficient is in turn negatively correlated with the epidemic threshold (Woolhouse et al. 2005; Moslonka-Lefebvre, Pautasso & Jeger 2009), having fewer producers/retailers (and hence more wholesalers) than another network with similar features (network structure, number of connections) can have a facilitating influence on the epidemic development. This result can be counterintuitive: one could have expected that a limited number of wholesalers would have spread the disease more effectively, by reducing the number of steps from producer to retailer. However, the key variable affected by the reduction in wholesalers (increase in producers/retailers) is the correlation coefficient between links in and out of nodes. This result is broadly independent of the connectance level and appears to be particularly true for non-SF networks, where significant positive associations between proportion of producers/ retailers and epidemic threshold are commonly found (Tables S4–S6). For these network structures, epidemic spread is more difficult if there are more producers and retailers and if the number of wholesalers is reduced. In contrast, for SF networks, only in some cases is there a significant correlation of the proportion of the three categories with the epidemic threshold. This is because in SF networks, epidemic development is driven by the presence of the hubs, rather than by features of the majority of nodes in the network.
A third interesting result is the mismatch between results for the correlation coefficient between in- and out-degree (Fig. 2; Tables S1–S3) and those applying to the epidemic threshold (Tables S4–S6) with regard to the sparsely connected networks (100 links). Whilst for the correlation coefficient between in- and out-degree there are still significant associations with the proportion of producers/retailers (negative) and the proportion of wholesalers (positive), there are no significant correlations for the epidemic threshold. This is likely to be a consequence of the absence of a correlation between in-out degree correlation coefficient and the epidemic threshold we observed at this low level of connectance (Moslonka-Lefebvre, Pautasso & Jeger 2009). This result would imply that the proportion of producers/retailers and of wholesalers can be used to partly predict the epidemic threshold of directed networks, but only if networks are not sparsely connected.
A fourth key outcome of this study relates to the connectance level. Increasing the number of links in networks generally results in an increased proportion of wholesalers for most network structures. Given that increasing the proportion of wholesalers, other things being the same, tends to lower the epidemic threshold, adding links to a trade network has therefore a double negative implication for plant health risk (increasing the connectance level alone lowers the epidemic threshold; Moslonka-Lefebvre, Pautasso & Jeger 2009). This result is not only relevant for plant epidemiology, but is likely to be important also for other small-size systems of connected entities with directed flows and where a continuum of values between two states (without the possibility of node removal due to immunization) can be assumed. Examples of such systems include innovation flows among countries and firms (Paci & Usai 2009), the spread of ideas among disciplines (Bettencourt et al. 2006; Kiss et al. 2010) and disease transmission among livestock farms (Green, Gregory & Munro 2009) or human healthcare settings (Smieszek, Fiebig & Scholz 2009; Grundmann et al. 2010), whenever a cure for the disease is not yet available. It would be interesting to study whether hierarchical categories can play a similar role in SIR models.
Our analysis clearly shows that structural developments in the trade can influence the likelihood of plant epidemics. Worryingly, little is known about the current contact structure of horticultural networks within and among nations, and about how this is changing (Dehnen-Schmutz et al. 2010). The reasons for this are partly due to the fragmentation of the industry and partly to concerns of commercial confidence. This study thus calls for the long-term and standardized collection of data on the number, degree distribution and trade volumes of plant producers, wholesalers and retailers for various regions of the world. Information on the degree distribution of trade networks and the distance, seasonality and strength of connections would allow plant health regulation authorities to determine which network type better describes existing trade networks and thus how existing and new risks from plant pathogens might better be managed. Knowledge on the plant trade hierarchical structure (proportions of producers, wholesalers and retailers) also appears to be important. Our study also calls for an agreement on the definition of these categories. International statistics on the volume and value of traded agricultural crops have long been recorded (e.g. International Statistics Flowers and Plants 2004), but little attention has been paid in such activities to issues specific to networks (e.g. the correlation coefficient between in- and out-degree of countries, traders and farms; whether a certain country, trader and grower is better categorized as a producer, wholesaler or retailer; how to influence network type, clustering and connectance with policy instruments such as agri-environment incentives).
A broad distinction among producers, wholesalers and retailers can be observed for many crops and regions, from cassava (Manihot esculenta) in Africa (Enete 2009) to pot plants in Europe (Ekelund & Axelson 2009). Nonetheless, for a given crop, regional differences in the spatial distribution of hierarchical categories may be present. In the USA, plant wholesalers tend to be located along the coasts, whilst interior regions have often a higher presence of retailers (Behe et al. 2008). In Northern China, horticulture plays a growing role in achieving regional food security, and this is shown by the rising number of producers. There does not seem to be a correspondent increase in wholesaler and retailer activity (Wang et al. 2009). In a survey of the horticultural sector in Germany, Grabnitz & Bokelmann (2004) report that a common prediction of business is that the future will bring an increasing concentration at the retail trade level, a rising importance of large-scale purchasers, and a diminution in the role of wholesalers. According to the results of this study, such a development may result in a diminished risk of spread of invasive species through trade. However, this presupposes that other things remain equal, including long-distance connections, overall volume traded, and network structure. Even if we have shown here that variation in hierarchical categories can affect the epidemic threshold of trade networks, other features such as network structure, capacity, connectance and the presence of hubs remain important (Floerl et al. 2009; Hulme 2009; Kaluza et al. 2010).
Many thanks to K. Dehnen-Schmutz, S. Hehl-Lange, T. Hirsch, O. Holdenrieder, A. Inman, V. Kertesz, E. Lange, A. MacLeod, L. Paul, M. Sharw, D. Slawson and J. Webber for discussions and insights and to C. Donnelly, O. Holdenrieder, T. Matoni and anonymous reviewers for helpful comments on a previous draft. This study was funded by the Department for Environment, Food and Rural Affairs, the Rural Economy and Land Use Programme, UK, and the French Ministry of National Education and Research.