Abstract
We study the resilience of a multiplex socio-ecological system (SES) which we structure from the spheres composing the sustainability Venn diagram. The SES network is subject to dynamics of spread of a global reform through the knock-on effect. The model outcomes reveal that high probability of reform completion on an SES layer through nodes previously reformed on other SES layers is necessary and sufficient to obtain positive density of reformed nodes on that layer. Full density can only be reached in the absence of risk of reform abrogation. The opposite case prevents the equilibrium density from reaching a steady state. The numerical simulation results show that the combination of likely probability of reform completion and of proportional influence of all layers yields the maximum magnitude of efficiency of the knock-on effect. We thus provide a formalized argument in favor of giving equal weight to all aspects of sustainable development.

To answer that question, we consider sustainability to be achievable, by means of a reform 45 initiated by public authorities or the civil society, whenever the states of SES components 46 2 A system is considered to be resilient when its structure adapts to perturbations while continuing to function, be it at the expense of changes (Liu et al., 2007) . 3 The concept of reform assumes the presence of a crisis which could be solved through corrective actions. density from reaching a steady state. The numerical simulation results show that the combination of likely probability of reform completion and of proportional influence of 79 all layers yields the maximum magnitude of efficiency of the knock-on effect. We thus 80 provide a formalized argument in favor of giving equal weight to all aspects of sustainable 81 development. Our clarification also opens an interesting debate on sustainability issues. 82 After this starting section, the dynamic behavior of the multiplex network, studied at 83 the levels of a layer and of multiple layers, is modeled in Section 2. Section 3 is devoted is distributed among L n layers, where n = 1, ..., 3. Each layer contains N nodes, with 90 i = 1, ..., N , with different intra-layer connectivity. Let A n , for n = 1, ..., 3, be the 91 adjacency matrix of L n with nonnegative elements (a n ij ) N ×N , for i = 1, ..., N . Two nodes are connected when a n ij = 1; and a n ij = 0 otherwise. Each node in L n is connected to 93 its counterparts in L −n , such that there exists a systematic link between the nodes of 94 different layers.
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Agents can either be target nodes or reformed nodes, which, in the latter case, have 96 previously been targets of the reform. In order to get reformed, a target node has to be 97 sufficiently open for reform. In case a target node from L n is connected to at least one 98 intra-layer reformed node, let β n ∈ [0, 1], n = 1, ..., 3, be the probability of openness for 99 reform on layer n. Should the reform be called into question, let µ n ∈ [0, 1], n = 1, ..., 3, 100 be the risk that a reformed node from L n gets abrogated, such that it returns back to its 101 original state.
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Given the inter-layer connectivity of the multiplex network, let k n , such that 3 n=1 k n = 103 1, be the parameter of influence emanating from either layer. For example, whenever a 104 6 For illustrative purposes, the population of agents can be interpreted as a set of countries, the actors of which evolve in different SES layers, that are pursuing common reforms, or as a set of stakeholders, representing various spheres of the society, committed to the same purpose.
Figure 2:
Example of a multiplex network composed of economic (blue), social (red) and environmental (green) layers. Each of them is composed of six connected nodes, that is L n = {1, 2, 3, 4, 5, 6}, for n = 1, ..., 3.
reform comes from the counterparts in other layers, the weighted probability that a target 105 node from L n is open for reform issued by other layers amounts to (1 − k n )β n .
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Finally, consider p n,i (t) ∈ [0, 1] to be the probability that node i from L n gets reformed 107 at time t, such that its complement corresponds to the probability that i remains a target Before moving forward to multiplex networks, let us start with a single layer in order to 112 study the intra-layer connectivity. The evolution of p for node i from L n is formalized in 113 the form of a dynamical equation
The network model puts emphasis on the fact that agents do not necessarily engage in a binding cooperative game, but instead follow (allow themselves to be influenced by) their neighborhood or (by) their counterparts evolving on other layers. In point of fact, in dynamic game theory, the spread of a strategy, or that of a practice, takes place after the individual comparison between alternative payoffs, where high-payoff strategies propagate in the population of players.
for n = 1\ {2, 3} and i = 1, ..., N , where q n,i (t) = N j=1 1 − β n a n ij p n,j (t) represents is the associated homogeneous recurrence relation with a solution of c 1 (q n,i (t) − µ n ) t−1 .
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The nonhomogeneous part yields c 2 =
from which we obtain the 126 stationarity expression of
(2) for n = 1\ {2, 3} and i = 1, ..., N . After considering the above, three cases may be 128 observed.
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The first case corresponds to p n,i (t) = 0 ⇔ q n,i (t) = µ n . The probability that node 130 i ∈ L n gets reformed at time t by any neighbor from the layer is null if the probability 131 that a target node does not get reformed is equal to the probability that a reformed node 132 becomes abrogated.
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The second case corresponds to p n,i (t) > 0 ⇔ q n,i (t) < µ n . The probability that node 134 i ∈ L n gets reformed at time t by any neighbor from the layer is positive if the probability 135 that a target node does not get reformed is less than the probability that a reformed node 136 becomes abrogated.
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The third case corresponds to p n,i (t) = 1 ⇔ µ n = 0. The certainty that node i ∈ L n 138 gets reformed at time t by any neighbor from the layer occurs if the probability that a 139 reformed node becomes abrogated is equal to zero. The following proposition ensues.
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Proposition 1 In a network exclusively dependent on intra-layer connectivity, although 141 the risk of abrogation might annul a reform project conducted on that layer, a higher level 142 than that of reform failure is necessary and sufficient to achieve the possibility of reform;
the certainty of reform implies the absence of risk of abrogation.
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The necessity is straightforward from the expression of p n,i (t). The sufficiency comes 145 from the construction of probability q n,i (t), which is itself dependent on p n,i (t). The probability dynamics previously obtained enable us to study the density of reformed 148 agents in a network exclusively dependent on intra-connectivity.
ability that node i, despite being open for reform, does not get reformed by neighbor 151 j.
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We observe that ρ i (t) = 0 ⇔ q n,i (t) = {1, µ n }. This implies that the density of 153 reformed nodes is equal to zero in case of certainty that node i has not been reformed by 154 any neighbor from L n at time t, or when its probability of not being reformed equals that 155 of being abrogated. In greater depth, these properties give the following.
for n = 1\ {2, 3} and i = 1, ..., N . While a n ij p n,j (t) = 1/β n represents the eigenvalue
for n = 1\ {2, 3} and i = 1, ..., N , where the spillover threshold amounts to 1−µ 1/N n a n ij p n,j (t) .
We have lim p n,j (t)→0
1−µ 1/N n a n ij p n,j (t) = ∞ and lim p n,j (t)→1
In the first case, 162 as the probability of reforming node i by neighbor j at time t goes to zero, the spillover 163 threshold goes to the unattainable level of infinity, which yields a zero density of reformed 164 nodes. In the second case, as the probability of reforming node i by neighbor j at time t 165 tends to one, the spillover threshold amounts to 1 − µ 1/N n , that is zero for large values of 166 N . We thus fall on the same property in both cases.
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Likewise, we observe that ρ i (t) > 0 ⇔ q n,i (t) < µ n . The result implies that the 168 density of reformed nodes is strictly positive when the probability that node i has not 169 been reformed by any neighbor at time t is less than that of being abrogated. In detail, 170 we have the following
for n = 1\ {2, 3} and i = 1, ..., N . We have lim p n,j (t)→0
1−µ 1/N n a n ij p n,j (t) , 1 a n ij p n,j (t) = (∞, ∞) 172 and lim p n,j (t)→1
1−µ 1/N n a n ij p n,j (t) , 1 a n ij The consideration of an interacting multiplex network makes the density both dependent 186 on intra-and inter-layer connectivities. We have
for n = 1, ..., 3 and i = 1, ..., N , where q n,i (t) = N j=1 1 − k −n β n a n ij p n,j (t) is the 188 probability that node i, despite being open for reform, does not get reformed by neighbor 189 j either through intra-or inter-layer connectivity. This time, q n,i (t) is also dependent on 190 k −n , be it the influence coming from the reformed counterparts in other layers.
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Once again, we observe that ρ i (t) = 0 ⇔ q n,i (t) = {1, µ n }. More specifically, k −n β n =
192
(1 − k n )β n = 1 a n ij p n,j (t) = 0. Despite the influence of nodes from both the neighborhood of 193 node i and from layers L −n through k −n , the policy knock-on effect will be vain. When 194 q n,i (t) = µ n , the spillover threshold amounts to β n = 1−µ 1/N n k −n a n ij p n,j (t) = 1−µ 1/N n (1−kn)a n ij p n,j (t) . We 195 have lim kn→0 1−µ 1/N n (1−kn)a n ij p n,j (t) = 1−µ 1/N n a n ij p n,j (t) . By that, when the combined influence from layers 196 L −n is high enough, their policy knock-on effect will depend on the probability that node 197 i gets reformed by node j via k −n at time t. As p n,j (t) → 1, the spillover threshold 198 is zero for large values of N . In parallel, we have lim kn→1 1−µ 1/N n (1−kn)a n ij p n,j (t) = ∞, be it 199 another unattainable threshold level. In both cases, zero density of reformed nodes will 200 be achieved.
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Again, we observe that ρ i (t) > 0 ⇔ q n,i (t) < µ n . This comes down to 1−µ 1/N n (1−kn)a n ij p n,j (t) < β n < 1 (1−kn)a n ij p n,j (t) or β n ∈ (0, 1), when p n,j (t) → 1, for large values of N . When the likelihood of reforming node i by node j through k −n at time t is close to certainty, the 204 spillover threshold lies within zero and one for large values of N . Thereby, ρ i (t) > 0 can 205 be obtained through high probability of achieving reform in other layers. 206 As for ρ i (t) = 1 ⇔ µ n = 0, reaching full density of reformed nodes implies a risk of 207 reform abrogation equal to zero. 
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Let us now analyze the stability of equilibrium density by considering ρ i (t) as a Lya-214 punov function candidate. The latter is then assumed to be a rate function (Mesquita
We know, by definition of q n,i (t), that its derivative resumes to that of −p n,j (t) < 0.
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As a consequence, whenever q n,i (t) < µ n ≤ 1, which corresponds to the criterion for 218 obtaining positive density of reformed nodes, ρ i (t) > 0, such that the equilibrium density 219 is unstable in the sense of Lyapunov. It implies that the reform spread on layers can be 220 withdrawn in time. The result is in accordance with our previous results, for positive 221 density also depends on the tradeoff between the risks of failing to reform and that of 222 abrogating the reform. In fact, according to the model outcomes, high probability of 223 abrogation signifies that the reform has been previously adopted by a number of nodes.
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Albeit, what triggers the reform diffusion also prevents it from attaining stationarity. Fig. 3 illustrates the spillover threshold values above which the policy knock-on effect is 234 operational. We observe a sequence of decreasing convex curves with a corner equilibrium, 235 at p n,j (t), µ n = (0, 1), from which arise the belt-shaped areas, that delimit the levels of 236 β n , colored in shades of blue. It verifies the property of µ n = q n,i (t). The same can be 237 noticed for p n,i (t) = 1, where µ n = 0. 238 Figure 3 : Levels of spillover thresholds β n in a single layer. The x-axis corresponds to the probability (p n,j (t)) that node i from L n is reformed by neighbor j at time t. The y-axis denotes the probability (µ n ) that a reformed node from L n gets abrogated. While the light blue area corresponds to higher values of spillover threshold, that is lim p n,j (t)→0,µn→0 β n = 0.23, dark blue areas match with levels of spillover threshold of lim p n,j (t)→1,µn→1 β n = 0.00 + .
The substitutability between the probability of reform and that of abrogation is less 239 pronounced for low values of p n,i (t) and µ n . This can be explained by the fact that when 240 the probability of reforming a node is low, the possibility to abrogate that reform is low 241 as well, such that the two parameters evolve in a complementary way. As the eventuality 242 of abrogation increases, the probability of reform decreases, so that both parameters turn 243 substitutable.
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For high values of p n,j (t), β n is invariably around zero. A spillover threshold close to 245 but different from zero implies that the reform can easily spread through the intra-layer 246 connectivity. As both p n,i (t) and µ n tend to zero, the spillover threshold increases, and 247 the spread by means of intra-layer connectivity becomes less reachable as well.
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Result 1 In a network exclusively dependent on intra-layer connectivity, likely probability 249 of reform completion irrespective of the probability of reform abrogation is necessary and 250 sufficient to initiate the knock-on effect.
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The first result implies that a decentralized spread of reform on a layer can be con- Let us now take a closer look at the combined influence from k −n on the spillover threshold. 
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One interesting result is that β n only exists for k n ≤ 1/3 when p n,j (t) → 0. Thereby, 263 whenever the influence from L −n is less than 2/3, at the levels of probabilities of reform 264 -be it through the inter-layer connectivity -close to zero, the knock-on effect fails to 265 function. For k n → 1, β n only exists for p n,j (t) ≥ 2/3. In this case, the knock-on effect 266 will not take place either.
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For all other configurations, the spread of reform should be operational, with a maxi-268 mum magnitude of efficiency for p n,j (t) > 1/2 and k n ≤ 1/3. 269 Figure 4 : Levels of spillover thresholds β n in a multiplex network. The x-axis corresponds to the probability (p n,j (t)) that node i from L n is reformed by neighbor j through layers L −n . The y-axis denotes the magnitude of influence (k n ) from L n . While the light blue area corresponds to high levels of spillover threshold, that is lim p n,j (t)→0,kn→1 β n = 0.99, dark blue areas correspond to low but positive levels of spillover threshold, that is lim p n,j (t)→1,kn→0 β n = 0.00 + .
Result 2 In an interacting multiplex network both dependent on intra-and inter-layer inactions as paramount, has often been relegated to a secondary role (Seghezzo, 2009). By 292 modeling sustainability through multiplex networks, we implicitly address sustainability 293 in a holistic manner, in that we attempt to take into account its different aspects, without 294 omitting to subject them to time dynamics. From a broader perspective, our results 295 should be viewed as a proof that multiplex networks can be put to good use to apprehend 296 the topics relative to the sustainability of SES. To a lesser degree, our framework also 297 succeeded in measuring the magnitude of spillover effects, which have previously been importance does not seem to be of clear evidence yet. 313 Indeed, good reforms offer critical insights on conflict between the various spheres 314 of economy, society and ecology (Brennan, 2008) . For example, Estapé-Dubreuil et al.
of three dimensions of sustainability. It implies that advancing two objectives requires for sustainable development (Pearce, 1988) , in which prices observed on markets fully in-

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