## Abstract

Many biological systems, from fragmented landscapes to host populations, can be represented as networks of connected habitat patches. Links between patches in these connectivity networks can represent equally diverse processes, from individuals moving through the landscape to pathogen transmissions or successive colonization events in metapopulations. Any of these processes can be characterized as stochastic, with functional links among patches that exist with various levels of certainty. This stochasticity then needs to be reflected in the algorithms that aim to predict the dispersal routes in these networks. Here we adapt the concept of reliability to characterize the likelihood that a specific path will be used for dispersal in a probabilistic connectivity network. The most reliable of the paths that connect two patches will then identify the most likely sequence of intermediate steps between these patches. Path reliability will be sensitive to targeted disruptions of individual links that form the path, and this can then be used to plan the interventions aimed at either preserving or disrupting the dispersal along that path. The proposed approach is general, and can be used to identify the most likely dispersal routes in various contexts, such as predicting patterns of migrations, colonizations, invasions and epidemics.

## 1. Introduction

Populations spread in patchy habitats as dispersers move from one habitat patch to another [1–3]. This describes processes as diverse as migrations across landscapes [2,3], pathogens spreading among hosts [4–6], biological invasions [7–9] and colonization in metapopulations [10–13]. Although the definition of a habitat patch may vary greatly depending on the context (table 1), the basic mechanics of dispersal will involve a population from an occupied patch, a source, providing individuals to another patch, a target. If the target also happens to be empty, as in devoid of the kind of dispersers arriving from the source, we can use this information to identify how different patches became occupied.

Dispersal between a source and any target patch will be a stochastic process. The probability that dispersers from a source will occupy a target patch can be represented as a weighted link connecting the two patches [4,10,14,15]. Probabilistic approaches have been previously used to establish the characteristics of connections in a variety of biological contexts, from migrations to epidemics [3,15–21]. Two patches connected by a direct link are said to be adjacent, and the link describing directional connectivity from source to target is called an arc [16]. Combining all patches and their respective arcs provides a connectivity network of a habitat in which a potential for dispersal among patches is represented with arc probabilities. Unlike networks in which links represent stable connections or interactions, such as landscape corridors, networks in which arcs represent a probability of dispersal will not have a fixed structure, as its arcs will essentially be transient [18,22].

The uncertainty about the existence of arcs could reflect the intermittent nature of dispersing forces like ocean currents [23], seasonality of the environment [18], habitat selection decisions [24] or contact rates [25]. All these processes could result in transient links between patches [22], and reflect the expected flow of individuals along a certain route in a way analogous to the current of random walkers moving between two patches [17]. In addition to the total number of dispersers moving between patches, dispersal probability could also include pre- and post-settlement factors such as viability of dispersers, mortality during dispersal and the likelihood of settlement [26]. For example, the probability of catching flu from a sneezing flu patient will include not just the total flow of viral particles but also the susceptibilities of the receiving individuals as well as the virulence of the viral particles. In other words, rather than being a simple measure of flow of dispersers between patches, links in a network could represent complex probabilities that the dispersers from a source will in some way change the state of the target patch. These composite measures of connectivity, such as colonization or transmission probabilities, will depend on the characteristics of source and target patches as well as the characteristics of dispersers (e.g. viability of individuals), and not just on the flow of individuals between patches.

In a complex habitat consisting of many patches, it is unlikely that the dispersers from any one source will be able to reach all other patches directly. To determine how the dispersal process will develop beyond the direct links, we need to consider a sequence of probabilistic dispersal events that will be needed to eventually reach non-adjacent patches. For instance, if the dispersers from patch *A* can affect patch *B* and those from *B* can affect patch *C*, we may want to know the probability that the population on *A* will affect *C* after a certain number of dispersal events have occurred in a system. To obtain this probability, we need to multiply *A*-to-*B* and *B*-to-*C* probabilities. A path between non-adjacent patches can therefore be defined as a series of dispersal events, originating at a source, that are needed to reach a non-adjacent target. As not all individual dispersal events will be equally likely, different paths between the same source–target pair will differ in the probability that they could provide a potential link between them. As such, the paths will also differ in relative importance as routes with which the dispersers could eventually reach a non-adjacent target.

Problems requiring multiplication of successive link weights/probabilities cannot be adequately addressed by simply using classical network search algorithms such as the Dijkstra's algorithm [27]. The core mechanics of these algorithms is the addition of link weights in order to identify the best (usually least costly) path through the network, rather than the multiplication needed to describe successive probabilistic events. Thus, these additive algorithms should not be directly applied to problems where links represent stochastic events, such as the probability of colonization. Moreover, stochastic dispersal means that many different paths could be taken simultaneously with different levels of certainty, for example, due to individual dispersers making different and sometimes sub-optimal choices at each dispersal opportunity. As direct application of additive algorithms does not allow for appropriate characterization of successive probabilistic dispersal events, an approach that takes into account this stochasticity will be needed to characterize the importance of different paths in probabilistic connectivity networks. Once established, the approach characterizing dispersal in probabilistic networks will then be applicable to any network of this kind regardless of its topology, method of construction or the system it represents, in the same manner that classical network search algorithms can be applied to any network in which link addition is biologically informative.

To this end, we propose the analytical approach based on the concept of reliability. Reliability is a term used in systems engineering to describe a probability that the system will function successfully, i.e. that it will not fail, in a given time period [28,29]. In engineering, reliability of a component is determined by its failure rate, while the reliability of a system will depend on the failure rates of individual components. Applying this concept to a network describing probabilistic dispersal, we define the reliability of a link between habitat patches as a probability that a source patch will have some direct effect on the target patch, either due to simple flow of dispersers or including other parameters that could affect this probability. For example, this could be the probability that a population will occupy another patch after dispersal or that the source population will provide genes to the target population, i.e. that the colonization or migration from the source to the target will not fail (see table 1 for other examples). The reliability of a path involving a series of patches will then be the probability that these patches will be colonized in sequence, and calculated as the product of reliabilities of individual events.

While there may be many paths leading from a source to a (non-adjacent or adjacent) target, not all of them will have the same probability of linking the two patches. Ultimately, one path will offer the most likely sequence of dispersal events with which dispersers from a source could reach the target. Comparing reliabilities of possible paths between patches will therefore identify the most reliable/maximum reliability path (MRP) through the network, with the patches comprising the MRP corresponding to the first completed series of dispersal events between source and target.

The MRP will also identify the quickest path to the target patch, as it will represent the lowest expected number of dispersal events needed to reach the target. As not every stochastic dispersal event will be successful, i.e. the population from the source will not necessarily affect the target after each dispersal event, there will be a lag before the next patch can be colonized which will be proportional to the dispersal probability between two patches, with lower dispersal probabilities producing longer lags before the next patch is colonized. For example, if there is a 0.5 probability of dispersal between patches *A* and *B*, and again 0.5 probability of dispersal between patches *B* and *C*, it will take on average two dispersal events to colonize *B* from *A* and four dispersal events in total to colonize *C* from *A*. In terms of traversing a path consisting of several patches, and assuming that the actual time the dispersers travel between individual patches is much lower than the time between successive dispersal events, the overall lag to traverse the path will depend on specific lags on all patches in a path. As MRP is a path expected to require the fewest dispersal events to reach the target, it will also identify the quickest path through which the target is expected to be colonized. MRP will therefore be both the quickest and the most reliable route for network traversal.

Engineering principles and probabilistic computational approaches have also been proposed to characterize other aspects of biological dispersal. For example, pioneering work by Jordán [30] applied some aspects of reliability theory to dispersal and landscape ecology in networks to design more reliable migration corridors (for some other applications of reliability theory to biology, see [31–34]). Link probabilities have also been used analytically in other ecological contexts [3,35–37], for example, to calculate the likelihood that two patches will be mutually reachable [38,39]. Circuit theory is another engineering principle used to characterize dispersal in landscape ecology, with many useful properties such as explicit consideration of the path redundancies and the relative density of random walkers taking specific routes through a landscape [17,40,41]. However, in contrast to density of random walkers as used by the circuit theory applications, the reliability of links can be used to describe any process in which a source can have some effect on the target, such as pathogen transmission [6], colonization in metapopulations [42], or larval dispersal by ocean currents [13], and not just those that record the net flow of dispersers/migrants/random walkers moving across the landscape. In any of these cases (and other examples from table 1), link reliability can represent a composite probability measure which takes into account parameters before, during and after dispersal, which can, but will not necessarily be, correlated with the actual flow of individuals between patches. Our definition of path reliability therefore aims to provide a general analytical approach that will enable the identification of the most likely dispersal routes in a wide variety of biological contexts where the relationship between source and target can be described in probabilistic terms and independently of whether this is directly related to the actual flow of dispersers between the patches.

To this end, we first propose a method to quantify path reliability in habitat networks and identify MRP. We then simulate dispersal process in probabilistic connectivity networks in order to determine the path most often used to occupy a non-adjacent target patch. With these simulations, we show that MRP predicts dispersal routes in probabilistic networks more accurately than the classical least cost/shortest paths. Finally, we demonstrate that path reliability will be sensitive to targeted disruptions of individual links on a path, specifically the weakest link on a path. We discuss these findings in light of potential applications of path reliability to management of fragmented habitats and control of emerging populations.

## 2. Calculating path reliability

### 2.1. Defining path reliability

A non-contiguous habitat consisting of *n* habitat patches can be described as a network defined by a set of nodes *V* = (*v _{1}*,

*v*, … ,

_{2}*v*). Habitat patches between which direct dispersal is possible will have a link/arc connecting them. The probability that the dispersers from

_{n}*v*will occupy

_{i}*v*during a single dispersal event, or the reliability of dispersal from

_{j}*v*to

_{i}*v*, will then be equal to a weighted link

_{j}*w*(

*v*,

_{i}*v*). Many biological exchanges are asymmetric, and adding directional exchange means that

_{j}*v*and

_{i}*v*form an ordered pair (although MRP can just as easily be calculated in undirected networks). As these probabilities represent the likelihood that

_{j}*v*will have an effect on

_{i}*v*(rather than the probability that an individual will choose a specific patch to move to from

_{j}*v*as in random walks), outgoing and incoming probabilities could, but do not need to, add up to 1 for any patch. If a direct exchange between patches is not possible, a target patch may still be occupied by dispersers originating from the source through a path that contains more than one link and one or more intermediate patches. Reliability of such paths, or the probability that a sequence of dispersal events will result in the target being occupied, will be equal to the product of reliabilities of individual dispersal events which together comprise a multi-link path between

_{i}*v*and

_{i}*v*. Therefore, the reliability of path , composed of

_{j}*k*patches between source

*S*=

*v*and target

_{0}*T*=

*v*, will be equal to the product of successive link reliabilities, or2.1with rel(

_{k}*p*) being the reliability of the path

*p*. If

*S*=

*T*then

*p*is a self-loop on

*S*with rel(

*p*) being equal to local retainment or self-seeding at

*S*. If no path between

*S*and

*T*exists then rel(

*p*) = 0; else, the value of rel(

*p*) will range from 0 to 1. The multiplication of individual link reliabilities will mean that paths with fewer links/intermediary steps will generally be more reliable, e.g. if all individual links have equal reliability. As dispersal is stochastic, there will be a lag on each patch

*v*equal to the expected number of dispersal events that will be needed to occupy

_{i}*v*from

_{j}*v*, calculated as 1/rel(

_{i}*w*(

*v*,

_{i}*v*)). For example, if the probability of

_{j}*v*colonizing

_{i}*v*is 0.5, it will on average take two dispersal events to occupy

_{j}*v*. Analogously, the path traversal time can be expressed as 1/rel(

_{j}*p*). For example, if

*v*to

_{i}*v*and

_{j}*v*to

_{j}*v*probabilities are both 0.5, occupying

_{k}*v*from

_{k}*v*will on average require four dispersal events (not counting other sources of lag such as population growth on

_{i}*v*).

_{j}### 2.2. Finding the most reliable path

Most of the existing network search algorithms are designed to find the shortest path in networks in which link weights represent some sort of cost. This is done by finding the path with a minimum sum of weights, as with Dijkstra's algorithm [27], and thus cannot be used to calculate MRP directly. However, a negative log-transformation of the link probabilities provides a network in which the classical shortest path is equivalent to MRP and which can be analysed for MRP by the widely available network search algorithms. In other words, finding the path between *S* and *T* with the maximum product of link weights*,* i.e. , will be equivalent to using the classical network search algorithms to find the path with the minimum sum of the weights in a transformed network, or2.2

In this context, rel(MRP) will be equal to the probability that *v _{i}* will be occupied from

*v*at the first available occasion, i.e. that none of the individual dispersal events will fail due to chance and that

_{i−1}*T*will be occupied after

*k*dispersal events. As such, rel(MRP) will identify the sequence of dispersal events and patches where this will occur most reliably and with the fastest expected traversal time in terms of the number of necessary dispersal events (equal to 1/rel(MRP)).

The proposed method calculates the path through the probabilistic networks while retaining the advantages of classical network search algorithms. As all original link weights/probabilities will have values between 0 and 1, the transformed link weights will never be negative, making their addition straightforward and eliminating potential problems that can arise from having negative link weights [27]. Moreover, the presence of feedback loops (reciprocal weights), self-loops (self-recruitment) or redundancy in network pathways between *S* and *T* will not be an issue for simply finding the path to occupy *T* from *S*, as this will already be handled by the employed network search algorithm. Similarly, additional parameters such as population size, patch quality or dispersers' preferences can be considered as modifiers of individual dispersal probabilities which could alter the link probability values and which path is identified as MRP, but will not have an effect on MRP calculation procedure.

### 2.3. Applications of path reliability to an example network

We illustrate the importance of MRP in a simple five-patch network with a source *S* and a target *T* (figure 1). Here, *T* can be occupied via three paths, and the probability that the next patch will be occupied is expressed as a weight for that individual link. All paths have the same sum of link weights, and therefore can not be differentiated by using the additive algorithms. However, the products of their weights, i.e. their path reliabilities, differ. If we consider *S* to be a source of all dispersers in a system, we can now ask which of those paths will be a more reliable route to occupy *T*. We see that path has the rel(*p _{1}*) = 0.09, path has the rel(

*p*) = 0.16 and path has the rel(

_{2}*p*) = 0.25. This means that after two dispersal events, and for simplicity ignoring any delays that may arise on individual newly occupied patches, there is a 25% chance that

_{3}*T*will be occupied via path

*p*. As such, even though the chance is higher that patches

_{3}*A*and

*B*will be occupied after one step than patch

*C*,

*T*will be more likely to be occupied from

*C*, and therefore

*C*should receive priority if the objective is to prevent

*T*from being occupied, for example, when planning interventions to limit the spread of unwanted species. These considerations would not be possible if the additive algorithms were directly applied to this network. However, once the weights are log-transformed, Dijkstra's algorithm will correctly identify

*p*

_{3}as MRP from

*S*to

*T*.

Using 1/rel(*p*) to assess the expected path traversal time in terms of dispersal events, we can see that it will take on average four dispersal events for *T* to become occupied through *p*_{3}, while path traversal time will be longer for *p*_{1} and *p*_{2} (around six and 11 events, respectively). This is again despite the fact that *A* and *B* have much higher chance of being occupied after the first step than *C*. This will occur because, once *A* is occupied, it will take 10 dispersal events on average before occupying *T*, as the *A*-*T* connection has a relatively low dispersal probability of 0.1. Note that the outgoing connections from *S* do not add up to 1, as these probabilities represent the probability that any of the posterior nodes will be occupied after a single release from *S*, as would be the case after the release of, e.g. pathogens or larvae, which need not be directly proportional to the net flow of dispersers from *S* (e.g. *A* may be more easily colonized than *C* even if they receive the same number of dispersers, etc.).

## 3. Effect of path reliability on simulated dispersal

In order to demonstrate the utility of identifying path reliability and MRP in probabilistic networks, we designed three sets of simulations that use simple dispersal dynamics to show that when dispersal is expressed in probabilistic terms: (i) MRP will be the path most often used to occupy *T*, (ii) MRP will be more often used to occupy *T* than the paths identified by additive algorithms, and (iii) MRP, as the path most often used to occupy *T*, can be sensitive to targeted disruptions.

### 3.1. Maximum reliability path predicts the path most often used to occupy *T*

We designed a series of demonstrative simulations to test whether the predicted MRP indeed corresponds to the path most likely to conduct dispersal through a hypothetical habitat network. The networks for these demonstrations were designed to exhibit the topology of a basic random graph. Random graphs are frequently studied in graph theory and used to show principles that could apply to basic complex networks [43]. As such, they provided an adequate demonstration platform for basic MRP calculations. The networks were constructed with a Poisson degree distribution and a mean number of connections (degree) of 3 (i.e. most patches will have three connections with some having more and some fewer). The networks were finite, fully connected, and each initially consisted of 198 patches. A source patch *S* and a target patch *T* were then added to the network, connecting them to three patches chosen at random while preventing direct *S*–*T* connections, thus bringing the total number of patches in each network to 200. As the definition of a path in graph theory stipulates that nodes could only appear once in a path, the links between patches were left undirected so as to preserve the network properties such as the mean degree. Each link was then assigned a weight randomly selected from a uniform distribution with a minimum of 0.01 and a maximum of 0.5. These weights were then used as probabilities of dispersal between patches, i.e. as a proxy of reliability that the functional link between the patches will exist. In total, 100 networks with distinct link configurations and weights were generated in this way. The predicted MRP between *S* and *T* was then identified for each network by using the negative log-transformation of the weights and applying the Dijkstra's algorithm to identify paths in this transformed network.

The dispersal was modelled as a simple state-based model where patches can have an ‘occupied’ or ‘empty’ state at discrete time steps. This is analogous to the compartmental (‘susceptible–infected’, or SI) models extensively used in epidemiology [44,45] as well as basic metapopulation models [46]. As the outcome of even the most complicated models of dispersal can be ultimately reduced to patches being occupied or empty, this elementary dynamic captures the fundamental dispersal processes in a habitat network.

In simulations, all patches were initially in an ‘empty’ state. The dispersal started at source *S*, and the state of *S* was changed to ‘occupied’ at the start of the simulation. At each subsequent time step, an occupied patch *v _{i}* had the chance to change the state of each empty patch

*v*if there was a direct link between them (

_{j}*w*(

*v*,

_{i}*v*) > 0). The probability of changing the state of

_{j}*v*was stochastic and equal to the link weight

_{j}*w*(

*v*,

_{i}*v*). If

_{j}*v*became occupied at time

_{j}*t*it could affect other empty patches to which it was directly connected at time

*t*+ 1. The order in which occupied patches were processed for dispersal was randomized during each time step so as to prevent the ordering bias. After each time step, we recorded the identities of ordered pairs of patches in which an occupied

*v*changed the state of

_{i}*v*at that step.

_{j}In the first set of simulations, this information was then used to backtrack the path from *S* though which *T* was occupied and determine the exact sequence of patches that were involved in occupying *T*. Each simulation run had 100 time steps, which was enough to occupy *T* in all simulations. A total of 100 simulation runs were performed on each of the 100 networks. By retracing the actual path of dispersal from *S* to *T* in each simulation we determined the number of times each path was used to occupy *T*. We then calculated in how many cases the predicted MRP was used to occupy *T*. The path identified as the one most often used to occupy *T* in simulations but at the same time different from the predicted MRP was then designated as an ‘alternative’ path. As identifying the alternative path was based on simulation outcomes, it was possible for this alternative path to be used more often than MRP.

The simulation results confirmed that the predicted MRP was the path most often used for occupying *T*, and that the number of times this occurred in a network was dependent on the rel(MRP) (figure 2). While the predicted MRP was only used in about a third of the simulations, it was still the path most often used to occupy *T*. MRP was used nearly twice as often as the nearest alternative (figure 2*a*). The number of times MRP was used was strongly associated with the value of rel(MRP) (figure 2*b*). As such, while MRP will be the path most often used to occupy *T*, higher reliability of MRP made this effect more prominent. The reliability of a path will therefore determine its importance for providing a functional link between *S* and *T*, as well as the relative importance of individual patches and links that comprise it.

### 3.2. Maximum reliability path is used more often than additive paths in probabilistic networks

A second set of simulations used networks that had three distinct paths: (i) a MRP, (ii) a path with the maximum sum of weights (MSP) which was distinct from MRP (the path that would be identified if algorithms such as Dijkstra's were used on a probabilistic network to identify the shortest path without first transforming the weights), and (iii) a path distinct from both MRP and MSP with a number of links equal to or less than MRP (a fewest links path that is essentially equivalent to what the additive algorithms would identify as the shortest path in an unweighted network). These networks were generated using the same procedure as the first set, but in this second set only those networks in which these three paths were distinct were used in simulations until a 100 of such networks were available. This provided a set of networks with clear alternatives to MRP that would have been identified by simply applying classical additive network search algorithms to describe dispersal in a probabilistic network. We then performed 100 simulations on each of these networks and recorded how often these paths were used to occupy *T*.

Despite the availability of the distinct additive paths, the predicted MRP was again the path most often used to occupy *T* (figure 3). MRP was used more often than either MSP or the path with the fewest links (figure 3*a*), demonstrating the advantage of identifying MRP as the path most frequently used to occupy *T*. MRP was also used more often than the nearest alternative (again defined as the most often used path other than the predicted MRP) in this set of networks as well. In the simulations, even this alternative path was used more often than either of the additive paths. MRP was the most frequently used path across the entire range of rel(MRP) values (even though the effect becomes more prominent with increasing MRP reliability; figure 3*b*). As such, MRP will predict the path that will be used most often to occupy *T* even if reliability of the path is generally low and in conditions of high redundancy when alternative paths through the network exist.

### 3.3. Maximum reliability path can be sensitive to targeted disruptions

MRP will be important for the persistence of exchanges in a network (see [36]), and will be indicative of the strength of the potential indirect connection between two patches. However, the value of rel(*p*) by itself will not tell us whether a path will be vulnerable to disruptions. While individual dispersal events will all contribute to rel(*p*), not all of these events will be equally important. As path reliability is a product of link weights, the value of the smallest multiplier (the least reliable link) will have the strongest effect on the overall reliability of the path. For example, in figure 1, the reliability of path will be affected the most by a reduction in reliability of link (*S,A*), the least reliable link in that path (reducing (*S,A*) by 0.05 gives rel(*p*_{1}*′*) = 0.07, compared with rel(*p*_{1}′) = 0.0735 for reducing (*A,B*) and rel(*p*_{1}′) = 0.078 for reducing (*B,T*)). Therefore, reducing the reliability of the least reliable link of MRP, or min(*w*(*v _{i}*,

*v*)), for example, by intervening at the patch

_{j}*v*, will have the highest relative potential to disrupt the dispersal between

_{i}*S*and

*T*and interfere with the occupation of

*T*. This link will therefore also be the best place to monitor for disturbances in order to ensure the link between

*S*and

*T*remains possible.

We demonstrate these principles in the third set of simulations, in which we extracted MRPs from the second set of 100 networks. We then simulated putative interventions into these paths in three ways. In the first batch of simulations, we simulated the dispersal in an unmodified path. In the second batch, we halved the dispersal probability of the least reliable link on a path (i.e. min(*w*(*v _{i}*,

*v*)) was now equal to min(

_{j}*w*(

*v*,

_{i}*v*)) − (min(

_{j}*w*(

*v*,

_{i}*v*)))/2). In the third batch, the probability of a random link different from min(

_{j}*w*(

*v*,

_{i}*v*)) was reduced by the same amount (min(

_{j}*w*(

*v*,

_{i}*v*)) − (min(

_{j}*w*(

*v*,

_{i}*v*)))/2) while min(

_{j}*w*(

*v*,

_{i}*v*)) was left untouched. The traversal time in these modified paths was then compared with the traversal time in the corresponding unmodified path in order to assess the effect of intervention in occupying

_{j}*T*. The traversal time was measured as the average number of time steps needed to occupy

*T*from

*S*, where an intervention would disrupt the dispersal and produce a delay before

*T*was occupied. We then compared which intervention had a stronger effect on dispersal between

*S*and

*T*. Dispersal through each path (an unmodified one and two modified ones per network) was simulated 100 times.

Reducing rel(*p*) at min(*w*(*v _{i}*,

*v*)) produced a markedly longer delay before

_{j}*T*was occupied than an intervention performed elsewhere (figure 4). Both of these modifications reduced the reliability of a single link by the same amount, with the only difference being which link was modified. For example, if this intervention is considered to be analogous to preventing the same number of dispersers from successfully occupying the next patch, performing this intervention at min(

*w*(

*v*,

_{i}*v*)) will be more efficient in disrupting the dispersal between

_{j}*S*and

*T*. This effect will increase as min(

*w*(

*v*,

_{i}*v*)) decreases. If a path consists of links with different reliabilities, the dispersal process will therefore be particularly sensitive to anything that disrupts the least reliable link on a path. However, the paths with homogeneous link reliability will be insensitive to the exact location of the intervention and instead will be equally sensitive to the same magnitude of disruption anywhere on the path.

_{j}The advantage of choosing to intervene (or not to intervene) at min(*w*(*v _{i}*,

*v*)) can also be estimated as any other link on the same path would also need to be reduced by the same relative amount to achieve the same effect. For example, as the rel(

_{j}*p*) is a product of link reliabilities, if an intervention halves the probability of min(

*w*(

*v*,

_{i}*v*)), any other link would also need to be halved in order to achieve the same effect. As a single disperser would comprise a greater proportion of dispersal through min(

_{j}*w*(

*v*,

_{i}*v*)), removing one disperser at min(

_{j}*w*(

*v*,

_{i}*v*)) would have a stronger impact on rel(

_{j}*p*). However, this is only true if the effort to remove a single disperser is the same at each link. In other words, the effort to remove a single disperser from min(

*w*(

*v*,

_{i}*v*)) could be higher than achieving the same relative impact elsewhere. As such, the decision on where to invest the intervention effort should be based on where it will have the highest impact on the probability of dispersal between two patches.

_{j}## 4. Discussion

Path reliability is an important metric for characterizing dispersal in stochastic networks. Even though many paths may exist that potentially connect two patches, determining path reliabilities will identify MRP as the most likely sequence of dispersal events between habitat patches that will link a source patch with a target patch. This sequence cannot be identified by directly applying additive network search algorithms to probabilistic networks, and instead requires the proposed multiplicative approach. Identifying the sensitivity of links on a MRP will be important for estimating the robustness of a path to interventions and disturbances in processes that drive metapopulation persistence and conservation efforts [36], as well as control of expansion of emerging metapopulations, such as contagions and epidemics [47]. The proposed concept of path reliability therefore provides a first step towards adopting a new analytical framework for managing probabilistic dispersal in habitat networks.

The simulation outcomes demonstrated that calculating path reliability can predict the path most likely to play a crucial role in the eventual occupation of a specific target of interest. The proposed approach based on link multiplication was clearly superior to applying algorithms that rely on link addition in a probabilistic network. In many biological systems, such as marine metapopulations [13,14], dispersal probabilities are expressed as the probability that sources will provide dispersers to destinations. Despite uncertainty associated with each such dispersal event, the proposed algorithms provide a way to assess reliability of routes through which genes or individuals will spread. Just like additive network search algorithms, path reliability calculations can be applied to any probabilistic network regardless of the method used in its construction, the context it represents or its topology, making them versatile tools for network analysis. Reliability can also be easily translated into path traversal time, which can then be used to assess the effectiveness of interventions in delaying the occupation of targets. The simulation results demonstrate that patch-specific lags in further dispersal could be interpreted in light of the uncertainties inherent in the dispersal process that arise from the stochastic nature of the dispersal process, and independent of the source of this stochasticity.

While intuitive and mathematically uncomplicated, these considerations will nevertheless have significant consequences for developing strategies to manage connectivity in biological networks. Graph theory is a promising tool for conservation planning [13,16,18,48–51], and considerations of path reliability can lead to simple yet potentially powerful and elegant heuristics for designing strategies for control or conservation. In general, dispersal along a path with heterogeneous links will be particularly sensitive to disruptions of its less reliable links, but relatively robust to disruptions of other links. By contrast, dispersal along a path with homogeneous links may be more robust to localized disruptions, but sensitive to disruptions distributed across many links. Identifying locations where link reliability can be reduced by the greatest amount for the invested effort will highlight specific targets, i.e. pairs of patches, where dispersal of invasive species, pests or pathogens could be efficiently disrupted. As the same method can also be used to identify the location most sensitive to unplanned disturbances, this approach can also be used to highlight the links most sensitive to disturbances in connectivity networks representing endangered metapopulations or migrating species.

The current sets of simulations demonstrated the utility of path reliability calculations by modelling dispersal effects with a very simple occupied–empty dynamic on patches. Naturally, any such model can be made more specific by parametrizing the dispersal model for a particular system or species, for example, by implementing density-dependent dispersal that will change the probabilities of reaching the next patch represented by network links. However, our aim was not to provide a solution to a specific biological problem, but rather to demonstrate that any such problem involving probabilistic dispersal should be tackled with multiplicative rather than additive measures that estimate the importance of dispersal routes in the habitat. By using a system-independent framework and algorithms to demonstrate the utility of path reliability calculations in models of stochastic dispersal, we hope that this basic message will spur further interest for ecological implications and management applications of path reliability measures in biological systems.

## Funding statement

This study was supported by a SIEF John Stocker fellowship to K.H. and an ARC Laureate fellowship to P.J.M.

## Acknowledgements

We thank Jason Flower, Maria Eleanor Aurellado, Sandra Hudina and three anonymous reviewers for their helpful comments.

- Received January 4, 2015.
- Accepted January 30, 2015.

- © 2015 The Author(s) Published by the Royal Society. All rights reserved.