Optimal interdependence enhances robustness of complex systems

TL;DR

This paper demonstrates that increasing interdependence between coupled networks can enhance their robustness by creating stable activity attractors, contrary to the usual association of interdependence with fragility.

Contribution

It introduces a novel perspective showing that optimal interdependence can promote robustness in interconnected dynamical systems, supported by analysis of coupled network dynamics.

Findings

System-wide persistence of activity at optimal interdependence levels

Formation of stable activity attractors ('islands') in the global dynamics

Counterintuitive robustness enhancement through interdependence

Abstract

While interdependent systems have usually been associated with increased fragility, we show that strengthening the interdependence between dynamical processes on different networks can make them more robust. By coupling the dynamics of networks that in isolation exhibit catastrophic collapse with extinction of nodal activity, we demonstrate system-wide persistence of activity for an optimal range of interdependence between the networks. This is related to the appearance of attractors of the global dynamics comprising disjoint sets ("islands") of stable activity.

Equations

$$\begin{gathered} z_i^{\mu }(n+1)=(1-D)F(z_i^{\mu }(n))[1+\sum _{j=1}^N{J}_{ij}^{\mu }{z}_j^{\mu }(n)] \\ +\frac{D}{(G-1)}\sum _{\nu \ne \mu }^{G}F(z_i^{\nu }(n))[1+\sum _{j=1}^N{J}_{ij}^{\nu }{z}_j^{\nu }(n)]. \end{gathered}$$

Full text

Many complex systems that occur in biological Pocock2012 , technological Rinaldi2001 and socio-economic Schweitzer2009 contexts are strongly influenced by the behavior of other systems Helbing2013 . Such interdependence can result in perturbations in one system propagating to others, potentially resulting in a cascading avalanche through the network of networks vespignani ; Gao2011 . Recent studies of percolation of failure processes in a system of two Buldyrev2010 ; Parshani2010 ; Wei2012 or more Gao2012 connected networks have suggested that interdependence makes the entire system fragile. However, a proper appraisal of the role of interdependence on the stability of complex systems necessarily needs to take into account the dynamical processes occurring on them Barzel2013 ; Danziger2015 . Compared to a purely structural approach (such as percolation, that considers the effect of removing nodes or links), a dynamical systems perspective provides a richer framework for assessing the robustness of systems. Indeed, investigating how fluctuations from the equilibrium state in a local region of a complex system can propagate to other regions forms the basis for addressing the dynamical stability of the system May1972 . Extending this framework to the context of interdependent networks can potentially offer us insights on why such systems are ubiquitous in the real world in spite of their structural fragility (as indicated by percolation).

In this paper we show that strong interdependence between networks can increase the robustness of the system in terms of its dynamical stability. In particular, we show for a pair of networks, whose nature of inter-network coupling differs qualitatively from the intra-network links, that there exists an optimal range of interdependence which substantially enhances the persistence probability of active nodes. By contrast, decreasing the inter-network coupling strength so that the networks are effectively independent results in a catastrophic collapse with extinction of activity in the system almost in its entirety. The increased persistence at optimal coupling is seen to be related to the appearance of attractors of the global dynamics comprising disjoint sets of stable activity. Our results also suggest that the nature of inter-network interactions is a crucial determinant of the role of interdependence on the robustness of complex systems. For example, increasing the intensity of nonlinear interactions between nodes leads to loss of stability and subsequent transition of the nodes to a quiescent state, while stronger diffusive coupling between the networks can make a global state corresponding to persistent activity extremely robust.

Let us consider a model system comprising $G$ interdependent networks. Each network has $N$ dynamical elements connected to each other through a sparse random topology of nonlinear interactions. Interdependence is introduced by diffusively coupling an element $i$ in a network to the corresponding $i$-th element of other network(s). This framework can be used to represent, for instance, dispersal across $G$ neighboring habitat patches of $N$ interacting species in an ecological system. A continuous dynamical variable $z_{i}^{\mu}$ ($i=1,\ldots,N;\mu=1,\ldots,G$) is associated with each node of the multiple coupled networks. In the above-mentioned example, it can be interpreted as the relative mass density of the $i$-th species in the $\mu$-th patch. We consider generalized Lotka-Volterra interactions between the nodes within a network as this is one of the simplest and ubiquitous types of nonlinear coupling Goel1971 ; sinha . The dynamical evolution of the system can then be described in terms of the $GN$ coupled equations:

[TABLE]

Here Jμ is the interaction matrix for the $\mu$-th network, while $D$ is a measure of the strength of interdependence via diffusive coupling between networks. The range of the variable $z_{i}^{\mu}$ is decided by the function $F()$ governing the dynamics of individual elements in the system. Here we consider $F$ to be a smooth unimodal nonlinear map defined over a finite support and having an absorbing state. This class of dynamical systems is quite general and are capable of exhibiting a wide range of behavior including equilibria, periodic oscillations and chaos Feigenbaum1978 . For the results shown here we have used the logistic form logisticmaps : $F(z)=rz(1-z)$ if $0<z<1$, and $=0$ otherwise, such that $z=0$ is the absorbing state, and $r$ is a nonlinearity parameter that determines the nature of the dynamics. Unlike most studies involving logistic map that consider $r\in[0,4]$ so that the dynamics is confined to the unit interval, here we specifically use $r>4$. This implies that $F()$ maps a finite subinterval within $[0,1]$ directly to the absorbing state. Iterative application of $F()$ will result in only a set of measure zero remaining in (0,1) Brin2002 . Thus, in isolation, the nodes will almost always converge to the absorbing state, resulting in their extinction.

However, interaction with other nodes can maintain a node in an active state ($z>0$) indefinitely, and we define a measure for the global stability of the system as the asymptotic fraction of nodes in each network that have not reached the absorbing state, viz., $f_{active}={\rm Lt}_{n\rightarrow\infty}f_{active}(n)$, where $f_{active}(n)=\sum_{i=1}^{N}\Theta[F(z^{\mu}_{i}(n))]/N$ (with $\Theta[x]=1~{}\text{for}~{}x>0,~{}\text{and}~{}0~{}\text{otherwise}$). Thus, we explicitly investigate conditions under which interdependence between networks can result in persistent activity in at least a subset of the nodes comprising the system. Using an ecological analogy, our focus is on the long-term survival of a finite fraction of the ecosystem as a function of the degree of dispersal between neighboring patches rather than the intrinsic stability of individual species populations.

The degree of interdependence between the networks can be varied by changing the number of pairs of corresponding nodes $M$ ($0\leq M\leq N$) that are linked via dispersion. The interaction matrix Jμ in each network is considered to be sparse, such that only $C$ fraction of the matrix elements are non-zero with their interaction strengths chosen randomly from a $Normal(0,\sigma^{2})$ distribution. For simplicity, we shall focus on a pair of interdependent networks (i.e., $G=2$) schematically shown in Fig. 1 (a), both networks being chosen from the same ensemble so as to have identical parameters $r$, $C$ and $\sigma$. We distinguish between the population densities $z$ of the $N$ species in the two networks by denoting them as $x_{i}$ and $y_{i}$ ($i=1,\ldots,N$) respectively, their initial values being chosen at random from the uniform distribution [0,1].

Fig. 1 (b-c) show the time-evolution of the state of the dynamical variables $x_{i}$ and the global stability measure $f_{active}(n)$ for one of the networks ($N=256$, $C=0.1$, $\sigma=0.01$) where the nonlinearity parameters $r_{i}$ are distributed uniformly in [4.0,4.1].

As mentioned above, this distribution of $r_{i}$ implies that the individual node dynamics would almost certainly converge to the absorbing state, and this is indeed what is observed when the networks are isolated, i.e., $D=0$. However, when the interdependence is increased, e.g., to $D=0.15$, we observe that a finite fraction of nodes persist in the active state, although for much lower (e.g., $D=0.1$) and higher (e.g., $D=0.2$) interdependence the system exhibits complete extinction of activity [Fig. 1(b-c)]. Thus an optimal diffusive coupling between corresponding nodes in the two networks provides global stability to the system. This suggests, for instance, that ecological niches which in isolation are vulnerable to systemic collapse resulting in mass extinction, can retain species diversity if connected to neighboring habitats through species dispersal. Indeed, for this to happen, it is not even required that all species in the network be capable of moving between the different habitats. As seen from Fig. 1 (d), if only a subset of $M$ nodes (out of $N$) are coupled between the two networks through diffusion, the system exhibits enhanced persistence with $f_{active}$ increasing with $M$. However, enhancing the intensity of nonlinear interactions within each network by increasing either their connectivity $C$ or range of interaction strengths (measured by the dispersion $\sigma$) decreases the survival probability of active nodes, as has been observed in earlier studies of global stability of independent networks Sinha05 ; sinha .

Fig. 2 (a) shows in detail the contrasting contribution of intra- and inter-network interactions to the robustness of the network in terms of maintaining persistent activity. The probability that a node persists in the active state asymptotically is seen to vary non-monotonically with increasing interdependence $D$ between the networks at different values of the parameters $C$, $\sigma$ and $r$ that determine intra-network dynamics. For reference let us focus on the curve for $C=0.1$, $\sigma=0.01$ and $r\in[4.0,4.1]$ [shown using circles in (a)]. We observe that when diffusion is either too low ($D<0.09$) or high ($D>0.2$) all activity in the network ceases within the duration of simulation. However, for the intermediate range of values of $D$, activity continues in at least a part of the network with the persistence probability reaching a peak around $D\simeq 0.16$. Varying the other parameters, such as network connectivity $C$, intra-network interaction strength $\sigma$ or the nonlinearity parameter $r$, has a simpler outcome, viz., a decrease in the probability that activity will persist in the network at long times. This is shown by the other curves where we increase in turn $C$ (triangles), $\sigma$ (squares) and $r$ (diamonds). Thus, our results indicate that there exists an attractor corresponding to persistent activity in the network for an optimal range of interdependence (in the neighborhood of $D=0.15$) which coexists with the attractor corresponding to the extinction of network activity, relatively independent of other parameters.

To understand this in detail, we first note that even when $N=1$, this much simpler system of two diffusively coupled elements exhibits qualitatively similar features when subjected to noise [Fig. 2 (b)]. The multiplicative noise of strength $\epsilon$ in the nonlinearity parameter, viz., $r=r_{0}(1+\epsilon\eta)$ where $\eta$ is a Gaussian random process with zero mean and unit variance, is introduced in lieu of the perturbations that each map will feel when connected to a much larger network through nonlinear interactions (Eq. 1) note1 . As in the case of the network, we choose $r_{0}>4$ so that an isolated node will almost always converge to the absorbing state, resulting in its extinction. Upon coupling two nodes, however, we observe that the probability of long-term survival of activity in the system becomes finite at an intermediate range of diffusive coupling strengths (around $D=0.15$), similar to that observed for a $N=256$ network in Fig. 2 (a). Thus, understanding the genesis of diffusion-induced persistence for a pair of coupled logistic maps subject to noise linz , may provide an explanation for the same phenomenon observed in the system of interdependent networks described earlier.

Fig. 3 shows the regions in the phase space $I^{2}:[0,1]\times[0,1]$ of the system of two coupled maps that correspond to initial states which move out of $I^{2}$ and into the absorbing state as a result of the dynamics. When the maps are isolated ($D=0$), successive iterations result in these regions increasing in size and eventually taking over the entire domain so that extinction will always happen. Similar behavior is seen for weak coupling ($D=0.02$) although the shape of the regions are now modulated as a result of the interaction between the two maps. For high values of coupling also ($D=0.3$), we observe the total extinction of activity in the asymptotic limit. However, for an intermediate value of coupling ($D=0.16$), the complement region defined by trajectories starting from anywhere inside it remaining within $I^{2}$, retains a finite measure even at long times, thereby ensuring persistence of activity. Introducing multiplicative noise in the dynamics does not significantly change the structure of the basins shown in Fig. 3 for low values of the noise strength $\epsilon$.

To understand the above results we represent the dynamics of the system as $x_{n+1}=(1-D)F(x_{n})+DF(y_{n})$, $y_{n+1}=(1-D)F(y_{n})+DF(x_{n}),$ where $x,y$ are dynamical variables and $F(x)=rx(1-x)$. The first term of each evolution equation can be interpreted as a logistic map with growth rate $(1-D)r$ while the second term represents a contribution from the other map that is diffusively coupled to it. If the sum of the two terms exceed 1 for any of $x,y$, the corresponding variable goes to the absorbing state. In the weak coupling limit of low $D$ where the first term dominates, it follows that if the effective growth rate $(1-D)r$ exceeds 4, the system will exit the unit interval almost surely. Thus a lower bound for the range of $D$ in which persistence can be observed is obtained by ensuring that $D>D_{c1}=1-(4/r)$. For example, for $r=4.1$, $D_{c1}\simeq 0.024$. The upper bound of $D$ for persistence is obtained by observing that when the two coupled maps synchronize their activity upon strong coupling, the dynamics reduces to that of an effective 1-dimensional map with $r>4$ whose trajectories will eventually exit the unit interval with probability 1. Whether synchronization occurs can be investigated by looking at the dynamics of the difference of the two variables, $\delta=y-x$, viz., $\delta_{n+1}=r(1-2D)\delta_{n}[1-(x_{n}+y_{n})]$. If $D>D_{c2}=(1/2)(1-(1/r))$, e.g., $D_{c2}\simeq 0.37$ for $r=4.1$, the difference goes to zero asymptotically resulting in synchronization of the two maps and convergence to the absorbing state. Thus, the system has a possibility of persistence only in the intermediate range $D_{c1}<D<D_{c2}$. In this region, where the individual maps exhibit periodic attractors, persistence can arise through out-of-phase oscillations in the two maps, each alternately visiting two disjoint intervals in [0,1] such that the sum of terms never exceed 1. Thus, regions of $[0,1]\times[0,1]$ domain which yield the in-phase solution lead to the absorbing state (extinction), while those giving rise to the out-of-phase solution lead to persistence, resulting in a complex basin of attraction for the persistent activity state [shown in Fig. 3 (i-l)]. To show that such stable out-of-phase period-2 solutions exist for an optimal range of $D$, we can solve the coupled equations $x^{*}_{1,2}=(1-D)F(x^{*}_{2,1})+DF(x^{*}_{1,2}),$ and check for stability, thereby obtaining an implicit equation involving the parameters $r$ and $D$ for which $0<x^{*}_{1},x^{*}_{2}<1$. For specific choices of $r$ and $D$, we can numerically verify that these solutions are stable, thereby providing confirmatory evidence of the proposed mechanism by which an optimal coupling induces persistence.

The bifurcation diagrams shown in Fig. 4 (a-c) indicate how the range of diffusive coupling strengths over which persistent activity is observed changes as we move from the simple case of two coupled maps ($N=1$) to interdependent networks ($N>>1$). As already discussed, diffusively coupling two logistic maps having $r>4$ allow their states to remain in the unit interval (corresponding to the nodes being active) provided the strength of coupling $D$ remains within an optimal range [Fig. 4(a)]. Note that within this range there exists a region, approximately between (0.11,0.18), in which the attractor of the dynamical state of the node occupies a much smaller region of the available phase space $I$. It is intuitively clear that for such values of $D$, introducing noise is much less likely to result in the system dynamics going outside the unit interval (thereby making the node inactive). If we now introduce multiplicative noise of low intensity (i.e., small $\epsilon$), the range of $D$ over which persistent activity occurs shrinks [Fig. 4(b)]. However, noise does not completely alter the nature of the system dynamics even though the the bifurcation structure is now less crisp. In particular, the system appears to be particularly robust in the region referred to earlier where the attractor covers only a small volume of the unit interval. We can compare this case with that of two interdependent networks, each comprising a large number of nodes [Fig. 4(c)] where the intra-network interactions are considered effectively to be ‘noise’. We observe a reasonable similarity between their bifurcation structures, with the region of persistent activity spanning approximately the same range of $D$. As in the case of coupled maps with noise, in the case of networks also the system is most robust in the region where the attractor for the unperturbed system of two diffusively coupled maps is confined within a small subinterval inside $I\times I$. The validity of considering the dynamics of a coupled nodes embedded within a network as equivalent to the pair being perturbed by an effective external noise is further established by the strong resemblance between the return maps and time-series for the two cases [Fig. 4(d-e)] for a diffusive coupling strength in the optimal range that leads to persistent activity. As mentioned earlier, to survive indefinitely the dynamical state of each map switches alternately between two disjoint intervals of the unit interval in an out-of-phase arrangement [see the time-series in the lower panels of Fig. 4(d-e)], corresponding to a trajectory that jumps between two “islands” of the basin for the attractor corresponding to persistent activity attractor of the coupled system shown in Fig. 3 (l).

The above analysis, apart from explaining why populations that go extinct rapidly in isolation will survive for long times upon being coupled optimally, also helps us understand how the persistence behavior in the system will be affected by increasing the number of interacting components. As can be observed from Eq. (1), increasing $N$ keeping $C$, $\sigma$ unchanged corresponds to the summation in the interaction term being performed over more components. This suggests that there will be stronger fluctuations, that can be interpreted as a larger effective noise applied to the individual elements resulting in a higher probability of reaching the absorbing state and thereby lowering the survival fraction $f_{active}$. We have confirmed this through explicit numerical calculations in which $N$ is systematically increased. To ensure that the results reported here are not sensitively dependent on the specific details of the model that we have considered here, we have also carried out simulations with (a) different forms of unimodal nonlinear maps, e.g., $F(x)=(x-l)e^{r(1-x)}~{}\text{for}~{}x>l;~{}0~{}\text{otherwise}$ Ricker1954 , and (b) different types of connection topologies for the initial network, e.g., those with small-world properties Watts1998 or having scale-free degree distribution Barabasi99 . We find in all such cases that that the qualitative features reported here are unchanged, with the network connecting the surviving nodes becoming homogeneous asymptotically irrespective of the nature of the initial topology, suggesting that the enhanced persistence of activity in optimally interdependent networks is a generic property.

To conclude, we have investigated the role of interdependence between constituent networks on the stability of the entire system in a dynamical framework. Unlike percolation-based approaches where failure is often identified exclusively with breakdown of connectivity so that increasing interdependence necessarily enhances fragility vespignani , our dynamical perspective that considers processes on nodes with a diverse spectrum spanning equilibrium, periodic and chaotic behavior, reaches a strikingly different conclusion. In particular, we show that if the physical interdependence between two networks has the form of diffusive coupling between corresponding nodes, the system has a much higher likelihood of survival when the coupling strength is in an optimal range, even if, in isolation both networks face almost certain catastrophic collapse. Thus, interdependence between complex networks need not always have negative repercussions. Instead its impact depends strongly on the context (e.g., the nature of the interdependence and the type of dynamics being considered). It could well be the case that there are several real systems, e.g., in ecology, where interdependence is essential for maintaining diversity in the presence of persistent fluctuations that are potentially destabilizing. Instead of a simple equation between interdependence and fragility, future investigations into systems exhibiting diverse dynamics and variety of possible couplings may reveal a more nuanced picture of the benefits and drawbacks of interdependence and the trade-offs involved in specific settings.

We would like to thank Trilochan Bagarti, Abhijit Chakraborty, Deepak Dhar, V. Sasidevan and Amit Sharma for helpful discussions.