## Introduction

Network adaptation reflects the ability of a system to respond to external (e.g., environmental) perturbations in terms of establishing dynamic interactions between the remaining elements. This sort of adaptation based on the coevolution of topologies and states appears in many network systems1,2, including opinion formation3,4, the spread of infectious diseases5, online social networks6, ecological and biological networks7,8,9,10,11, power grids12 and chemistry13. An important class of adaptation dynamics occurs as a consequence of extinction events altering network structure. Examples include the disappearance of companies and technologies due to innovations and market competition14,15, neurons extinction16, gene extinction17, extinction of the least populated molecular species18 local extinction of species in biological networks due to perturbation such as pollution or human disturbance19,20, habitat loss21, climate change22,23 or extinction of interaction partners24,25.

In ecosystems, loss or declining abundance of one species can cause the extinction of other dependent species26,27,28. Understanding how ecosystems respond to species loss has hence been the subject of numerous theoretical and experimental studies29,30. While some communities exhibit density compensation31,32, a common form of adaptation in food webs is represented by the rewiring of feeding links among members of species in response to variation in resources33,34,35,36,37. Functional redundancy can help in the process of rewiring, as species have the ability to replace each other. So, for example, pollination networks have been shown to be fairly robust against species extinctions as a result of bipartite network asymmetry (redundancy in the number of floral visitors per plant25). However, trophic rewiring can lead to overexploitation of resources and aggravate the effects of species loss on food webs38.

From the perspective of the topological structure of the interactions of the species that compose ecosystems, recent studies have revealed key structural features which are believed to influence ecological dynamics39: robustness, defined in terms of the largest connected component40; nestedness, where specialists interact with a subset of the whole set of species that generalists interact with41; modularity, which refers to densely connected non-overlapping subsets of species (modules) with weak interactions between them42,43; and stability, which can be measured as the largest eigenvalue of the appropriate matrix44.

Some studies have analyzed these characteristics as the product of a temporal evolutionary process45,46. Based on previous results, modular structure in species interactions of mutualistic networks hinders the loss of species and promotes the long term persistence of ecological communities, as the compartments buffer the propagation of extinctions across the network47,48,49,50,51. More recent research provided empirical evidence to test this hypothesis experimentally and confirmed that a modular network structure enhances species persistence52. Here we used a dynamical modeling approach to investigate how different extinction-induced rewiring scenarios modify the structure of ecological bipartite networks. In particular, we investigated the effects on topological robustness, measured as the degree of connectivity within the network; network modularity, seen as the compartmentalized nature of the networks, and finally, dynamical stability, measured through the change in the largest eigenvalue of the adjacency matrix.

## Materials and Methods

### Bipartite networks from empirical mutualistic networks

We analyzed 130 empirical mutualistic bipartite networks consisting of 101 plant-pollinator, 25 seed-dispersal, 3 plant-ant, and 1 host-symbiont association (see Table S1, data available in the Web of Life database53 and54,55). We considered here only the presence or absence of the interaction, i.e. unweighted undirected bipartite networks.

Bipartite networks contain two types of nodes: resources and consumers. Resources represent plants (or flowers, seeds, hosts) while consumers represent pollinators (or dispersers, symbionts). They contain, thus, n resources and m consumers, and the adjacency matrix A(n + m, n + m) captures the pairwise interactions between resources and consumers: Aij = 1 if there is a pairwise interaction between resource i and consumer j, and Aij = 0 otherwise. Thus, the adjacency matrix can be written as

$$A=[\begin{array}{cc}{{\bf{0}}}_{n\times n} & {\tilde{A}}_{n\times m}\\ {({\tilde{A}}^{T})}_{m\times n} & {{\bf{0}}}_{m\times m}\end{array}]$$
(1)

where Ã is the incidence matrix, T indicates the matrix transpose and 0m×m is the all-zero matrix and the subindices indicate the number of rows and columns.

At each time step, the dynamics proceeds as follows:

1. 1.

Extinction: a resource node is removed.

2. 2.

Adaptation: each link that was attached to the extinct resource node is rewired to a different resource node with probability r.

The rewiring probability r varies between 0 and 1, where r = 0 represents the no-rewiring case and r = 1 the case where each link is attempted to be rewired.

We explored two scenarios for extinction: (i) random extinction, selecting the resource for extinction at random from the available ones; and (ii) directed extinction, selecting the resource proportionally to its degree, i.e. the number of consumers interacting with it. We considered again two adaptation scenarios, in both cases the consumer is maintained while the resource is rewired: (i) random rewiring, where the extinct resource is replaced by an existing resource at random; and (ii) resource affinity-driven rewiring, where the extinct resource is replaced by another resource selected proportionally to the number of common consumers shared with it. Thus, for instance, in the case of a plant-pollination network, an extinct plant species would be replaced with the largest probability by another species sharing the highest number of pollinators with the extinct one, which is likely to be the phylogenetically most closely related or ecologically redundant species. Hence, if the resource j is selected for extinction, the number of shared consumers with resource k is given by:

$$O(kj)=|{E}_{j}\cap {E}_{k}|$$
(2)

where Ej and Ek are the set of consumers connected with resource j and k. Thus, the probability that resource k is selected as replacement of resource j by resource affinity-driven rewiring is given by

$${p}_{r}(k;j)=\frac{O(kj)}{{\sum }_{l}\,O(lj)}.$$
(3)

### Robustness

Due to the extinctions, the bipartite networks eventually break up in several components. The relative size of the largest connected component in the network after a fraction f of resource nodes are extinct, S(f), reflects the ability of the network to maintain itself connected40,56. The behavior of the relative size of the largest component can be summarized in the robustness ρ, measured as the area under the S(f) curve:

$$\rho =\frac{1}{n}\,\mathop{\sum }\limits_{i=0}^{n}\,S(i/n).$$
(4)

A value of R close to one means that as nodes are being removed the network does not fragment, consisting of a single component all the way; while a value of ρ close to zero signals that the network fragments very easily into small disconnected components.

### Modularity

Modularity measures the presence of densely connected non-overlapping subsets of nodes called communities or modules. For bipartite networks, the modularity can be measured using Barber’s expansion of modularity with specific constraints reflected in the null model57 as

$$Q=\frac{1}{L}\,\mathop{\sum }\limits_{i=1}^{n}\,\mathop{\sum }\limits_{j=1}^{m}\,({\tilde{A}}_{ij}-{B}_{ij})\,\delta ({g}_{i},{h}_{j})=\frac{1}{L}\,\mathop{\sum }\limits_{i=1}^{n}\,\mathop{\sum }\limits_{j=1}^{m}\,\left({\tilde{A}}_{ij}-\frac{{k}_{i}{d}_{j}}{L}\right)\,\delta ({g}_{i},{h}_{j})$$
(5)

where gi and hj are the community indices that resources i and consumers j are assigned to, and L is the number of links in the network. The expected probability of interactions between two types of nodes is given by matrix B. Based on the null model, that considers random pairings between two classes of nodes with given degree sequences, the Bij elements are proportional to the degree of resource ki and the degree of consumer dj in the original network $$\frac{{k}_{i}{d}_{j}}{L}$$. We have used the open-source library BiMat58 to find the community structure that maximizes Eq. (5). To test whether the empirical networks are significantly more modular than random ones, we calculated the modularity of each empirical network and compared them with the modularity of 1000 randomized networks with the same species degree distribution as the empirical network.

The sequence of extinction-rewiring events eventually breaks the bipartite networks in several components. To correct for the network breakout of the bipartite network, we introduced the effective modularity Qe, measured as the product of the modularity and the relative size of the largest connected component (fraction of nodes in the largest connected component): Qe = SQ.

The changes in the community structure of bipartite networks can be quantified with the module persistence Πij defined as the probability that two nodes remain in the same community if they were initially in the same community, Πij(gi = hj, t|gi = hj, t0), where gi (hj) is the community to which node i (j) belongs.

### Stability

To study the relationship between the structure and the stability of an ecosystem, May considered positive and negative interactions to represent the relations among species in an ecological community and analyzed the eigenvalues of the adjacency matrix as a measure of the stability44. The assumption is that the adjacency matrix captures the linear stability of the dynamics of the ecosystem. In our case, the interaction is given by a matrix that reflects the bipartite structure and the mutualistic interactions, that is, all the entries are non-negative. In general, if the dynamics is given by a local dynamics and an interaction term:

$$\begin{array}{rcl}\frac{d{x}_{i}}{dt} & = & {{\mathscr{L}}}_{i}(\overrightarrow{x})\\ & = & f[{x}_{i}]+k\,\sum _{j}\,{A}_{ij}{x}_{j}\end{array}$$
(6)

where xi is the abundance of species i, f(xi) are the local dynamics, k is a coupling strength and Aij the adjacency matrix and therefore we only take into account positive interactions (mutualistic). The stability of this system around a fixed point $${\overrightarrow{x}}^{\ast }$$ can be analyzed by evaluating the largest eigenvalue of the Jacobian matrix $${J}_{ij}=\frac{\partial {{\mathscr{L}}}_{i}(\overrightarrow{x})}{\partial {x}_{j}}=\frac{f({x}_{i})}{\partial {x}_{j}}+k{A}_{ij}$$ at the feasible equilibrium point ($$\overrightarrow{x}={\overrightarrow{x}}^{\ast }$$), for which the off-diagonal elements coincide with kAij. If the local dynamics are the same for all the elements, the diagonal elements are all the same and thus only contribute to the largest eigenvalue as an offset. If all the eigenvalues have a negative real part, then the equilibrium is stable and the perturbed system will go back to the equilibrium, but if any of the eigenvalues have a positive real part, the perturbed system will move away from the equilibrium. Therefore the eigenvalues of the Jacobian can be analyzed in terms of the eigenvalues of the adjacency matrix59.

We measured the change in stability of a mutualistic bipartite network as the relative change of the leading eigenvalue, λ, of the (n + m) × (n + m) adjacency matrix as extinction-adaptation events occur, with respect to the leading eigenvalue of the original matrix, λ0:

$$\Sigma =\frac{{\lambda }_{0}-\lambda }{{\lambda }_{0}}.$$
(7)

Note that the sign reflects an increase (positive) or a decrease (negative) of the stability, i.e., a lower (higher) eigenvalue. In general, we are interested in expected values

$$\langle \Sigma \rangle =\frac{{\lambda }_{0}-\langle \lambda \rangle }{{\lambda }_{0}},$$
(8)

where 〈·〉 denotes an average over realizations of the extinction-adaptation dynamics.

## Results

For each bipartite network analyzed, we reported the results of 100 realizations of the dynamics (unless otherwise indicated) for each scenario. For the sake of clarity, we show results for the five largest networks of different mutualistic interaction types: plant-pollinator (two of them), seed-dispersal, host-symbiont, and plant-ant (Table 1). The results for all the 130 networks are available in the Supporting Information (see Table S1).

### Robustness

For our model, the breakup of the networks depended mostly on the extinction scenario, with a sharp decrease in size of the largest component with a few extinct resources in the directed scenario contrasting with a smoother decay in the random scenario (Fig. 2). Furthermore, for random extinctions, the rewiring mechanism had very little influence on the robustness, while for directed extinctions resource affinity-driven rewiring was more detrimental than random rewiring. The results for the rest of the networks were qualitatively similar (see Fig. S1 in the Supporting Information).

### Modularity

The transformation of the modular structure over time with extinction-adaptation events shows that the removal of nodes and rewiring of links produce changes that affect the global organization of the mutualistic networks, as it can be qualitatively observed in the alluvial maps shown in Fig. 3. To quantify the reorganization of the networks, we measured how the modularity changes with rewiring probability and the fraction of nodes removed.

The effective modularity shows an idiosyncratic effect of the extinction-adaptation, in the sense that it depends mostly on the network analyzed and not on the specific extinction/rewiring scenario. In one extreme, bipartite networks show an increase in Qe with rewiring. This is the case for the Coral Reef and Montane Forest networks (top panel of Fig. 4) and 88% (89/101) of the plant-pollinator and 60% (15/25) plant-seed dispersal networks (see Figs. S2S7 in Supporting Information). On another extreme, 12% (3/25) of plant-seed dispersal networks show a slight decrease in Qe as the rewiring probability is increased, while for the rest it remains rather constant (12%, 12/101, for plant-pollinator networks and 28%, 7/25, for plant-seed dispersal networks). We clustered the empirical mutualistic networks and constructed a dendrogram of the empirical mutualistic networks based on the similarities of the response of the effective modularity to extinctions and rewiring (see Fig. S8 in Supporting Information). In general, most networks show an increment in Qe with rewiring. The directed extinction together with resource affinity rewiring (i.e., the DF scenario) increases Qe more than any of the other scenarios. In the bottom panel of Fig. 4, the difference between RR and DF scenarios is shown.

In order to assess the average persistence of the network, we computed the averages over all node pairs to get the mean persistence PM = 〈Πij(gi = hj, t|gi = hj, t0)〉. The extinction mechanism at play affects the results the most, as was the case for the robustness. The persistence of communities for random extinctions decays almost in a linear fashion as a function of the fraction of extinct resources, with a 50% probability to remain in the same community after 40% of the resources are removed. In contrast, for the directed extinction scenario, persistence decays to a baseline value of less than 30% when less than 20% of resources are removed (values correspond to the coral reef network but results for the rest of networks are qualitatively similar, leading to a higher reconfiguration into new communities with improved compartmentalization. The rewiring scenario increases the persistence of the community structure when the rewiring is driven by resource affinity, which implies that functional replacement helps module persistence. To quantify the gain in persistence due to resource affinity-driven rewiring, we compared the probability that two random nodes (resource or consumer) are in the same module for each extinction scenario but different adaptation, that is, we compared, on the one hand, RR with RF scenarios, and on the other, DR with DF. In both cases, resource affinity-driven rewiring leads to higher mean persistence values, (see Figs. S14 and S15 in Supporting Information).

### Stability

Similarly to robustness and community persistence, stability is affected more heavily by the extinction rule than by the adaptation rule (Fig. 5). Regardless of the adaptation rule, robustness is conserved, as indicated by the relative change of the leading eigenvalue remaining close to zero as long as the network does not break up in components for large values of the fraction of extinct nodes.

The stability of the bipartite networks increases when a high percentage of species are removed, that is, once the original network is reduced to a few species. For directed extinctions, and also irrespective of the adaptation rule, stability increases much more than for random extinctions, with the effect being more noticeable for a lower fraction of extinction-adaptation events. In these cases, the effect also decreases as the rewiring probability r increases. We also note that in targeted extinction scenarios, the average connectivity of the network decreases with fewer extinctions, as we are taking away individuals with large degrees, and thus stability increases. The results for the rest of the networks are qualitatively similar (see Figs. S9S11 in Supporting Information).

### Effective modularity and the normalized maximum degree kmax

In order to know if the transformation of the modular structure of bipartite networks over time can be anticipated by properties of the original network, we computed the average effective modularity 〈Qe〉 of the networks for all rewiring probabilities and all fractions of extinction events (average of the plots in Fig. 4 and correlated it with different initial topological characteristics (see Fig. 6). As Qe behaves very similarly for all scenarios, we only used the DF scenario. The average Qe is positively correlated to the modularity of the original network Q0 (R2 = 0.78, Fig. 6a) but uncorrelated to its robustness when this is measured for random extinctions and no rewiring (R2 = 0.1, Fig. 6b). On the other hand, Qe correlates negatively with nestedness, measured as NODF60 (R2 = 0.68, Fig. 6c) and with the normalized largest degree Kmax, which is the largest degree in the network divided by max(n, m) (R2 = 0.55, Fig. 6d). All these results can thus be reframed in terms of the normalized largest degree, Kmax: the larger Kmax of the original network, the lower the effective modularity and robustness, and the larger the nestedness.

## Discussion

Mutualistic consumer-resource species interactions in ecosystems are well described by bipartite networks, where links inside the two groups of species are not allowed (between consumers and between resources)61,62,63,64. We explored the effects of local species extinction followed by different strategies to adaptation on the robustness, modularity, and stability of 130 empirical mutualistic bipartite ecological networks53,54,55 under the assumptions that i) no new species enter the network and ii) only the binary presence/absence of interactions is considered (not interaction strength). For heterogeneous weight distributions, the results might differ and the dynamical rules have to be formulated to account for the weights. A parsimonious plasticity rule will introduce the rewiring probability as proportional to the weights, thus favoring species with interactions with the largest weights. This will likely introduce rich dynamics and can affect network plasticity depending on whether specialists or generalists have the strongest interactions.

Altogether, we explored four extinction-adaptation scenarios to conclude that the extinction mechanism is crucial to determine the fate of the system concerning its robustness and stability. Directed extinctions lead to new networks that are less robust and more stable, an effect that decreases with growing rewiring probability. This result goes in the same direction as the seminal work by May44: more complex and highly interconnected ecosystems are less likely to be stable. The trend we observe is that resource affinity rewiring leads to an increase of modularity, stability, and robustness in comparison to a random rewiring, which is used as a parsimonious rewiring. All these three properties are desirable for a mutualistic network as it provides resilience to disturbances. Thus resource affinity could be selected as an evolutionary mechanism to increase the survival of ecosystems. Earlier studies suggest that the adaptation mechanism in the topological approach increases the robustness of mutualistic networks to species removal33,35,65,66,67. Our results show that topological plasticity increases robustness, in agreement with those studies. Our simulations also suggest that resource affinity consumer rewiring induced by preferred species extinction is more detrimental to the robustness of the network than random rewiring.

The increase in robustness is connected to changes in other structural metrics: modularity, stability, and robustness are positively correlated among them, but all negatively correlated with nestedness. Our results are consistent with the trends reported in these metrics although the literature shows some conflicting outcomes. Modularity and nestedness are observed to be negatively correlated, as found in several bipartite networks68, at least for large connectance43 and especially for pollination networks69. Nevertheless, note that this relation is not mathematically proven except for the case of some models as in68. Stability is negatively correlated with nestedness as reported in some studies70,71, but positively correlated with modularity, in agreement with previous studies14,72,73. Positive correlations between robustness and modularity have been reported recently in plant-frugivore networks74. However, our results show a non-significant positive correlation between robustness and modularity.

The assumption that resource affinity may affect rewiring is based on the hypothesis that co-occurring species are functionally linked or equivalent. However, resource affinity rewiring patterns are rarely studied in ecosystems. In biological terms, such rewiring might imply that the likelihood that a consumer would use a particular resource used by a different species increases with the number of resources it shares with that species. In real ecosystems, there are several rewiring strategies taking place at the same time75. There is evidence for rewiring based on abundance76,77,78, temporal overlap75,79, trait matching79,80,81,82,83, and trait resemblance84,85. Temporal overlap, based on the phenology of the species, is a necessary condition for a rewiring among species to take place. So is it, to a certain degree, the abundance of species. Given that, species interactions might be constrained by the physiological capacity of one to interact with another, which is described by trait matching. This is the case for example in birds that can only pollinate certain species of flowers if the length of their bills coincides with the corolla length of the flowers. Another different but related case happens when a consumer decides to feed on a resource because another consumer species has a similar trait as his own. This is described by trait resemblance. Resource-affinity rewiring is a proxy for both trait matching and trait resemblance. On the one hand, in the absence of information about traits we expect that trait matching increases with resource overlap, that is, the more common resources are shared between pair of species more likely is that their traits match, On the other hand, we are using a stochastic approach, thus species can rewire to a resource that does not necessarily have the largest overlap, we are taking into account also trait resemblance. Trait matching is stricter than trait resemblance, and thus future innovations will require to consider nonlinear functions of the overlap to capture trait matching more precisely (for example with a threshold value for rewiring). The rewiring based on consumer overlap has been used in some literature, measures as Sorensen similarity index between a pair of plants proportional to the number of sharing pollinators over the sum of the pollinator species of both plants. This approach assumes that topological similarity is associated with phenotypic similarity (trait resemblance)86.

Through analyzing a broad spectrum of bipartite mutualistic networks, we make the argument that evolutionary responses to the loss of species lead networks to become more modular, and the effect is largest when well-connected species were removed from the networks (directed extinctions). These responses could act in the same generation, as a behavioral response, or between generations as an evolutionary mechanism. Our results further demonstrate that adaptation in the form of resource affinity-driven consumer rewiring has the most significant impact on the degree of compartmentalization of the networks. This modular configuration of the network increases the capacity of absorbing disturbance (for example species loss) and hence has a positive effect on the dynamical stability of mutualistic networks. The effective network modularity shows a similar pattern for any of the extinction scenarios but depends on details of the original bipartite network, that is, the effective modularity is idiosyncratic. We also observed a marginal increase in effective modularity with resource affinity driven rewiring. Experiments show that, in agreement with theory, networks with a modular structure of species interactions benefit from localized dispersion of extinction87. Whether this is the case in nature is not resolved, as dynamic rewiring of bipartite mutualistic networks following local extinctions are still poorly documented.

Different scenarios lead a different number of unsuccessful rewiring attempts, thus some scenarios lose more links per extinction event than others. This can happen in two ways, either because the proposed rewired link already exists or because there are no plausible partners for the rewiring. The former happens for both rewirings, while the latter only in the resource-affinity rewiring. Nevertheless, we speculate that this consequence of resource-affinity rewiring can be interpreted as the existence of forbidden links88. Although there are many available resource species in the network, the set of plausible resources to rewire to is empty, as no other consumer that also used the extinct resource uses any of the available resources.

We have shown that for the empirical bipartite networks, the normalized largest degree is correlated with the modularity of the original network. Thus, the success of niche compartmentalization with rewiring probability can be anticipated with the (normalized) largest degree. At the same time, nestedness showed to decrease as modularity increases, nestedness decreases. Finally, stability decreases as the largest degree correlate with stability. How ecosystems face disruptive, extreme events or continuous environmental changes cannot be predicted generically and requires specific knowledge on the species that will be most likely affected and the plasticity of the remaining species to reorganize the whole network. Our experiments with bipartite mutualistic networks consistently show that functional replacement, here modeled as resource affinity, can enhance the resilience of mutualistic networks by increasing the effective modularity as the bipartite networks lose species. Resource affinity is just a proxy for functional redundancy, which has been argued to add resilience to ecosystems under disturbance89,90. The role of functional diversity in ecosystem stability has been previously explored based on simulated random extinctions84, but directed extinctions, as simulated here, were not assessed, nor was resource affinity used as a proxy of functional redundancy in previous simulation studies. Our analysis of the resilience of bipartite mutualistic networks predicts that functional redundancy should help stabilize biological networks against local extinctions. We put forward the hypothesis that resource affinity, as a proxy for functional redundancy, is a likely rewiring mechanism of mutualistic networks in the presence of local extinctions. That is to say, ecosystems have a higher chance of reconfiguration under species extinction. Testing these predictions through longitudinal observations of bipartite mutualistic networks or direct experimental manipulation is required to assess the value of resource affinity for ecosystems resilience, and implementing data-based plasticity mechanisms.