Antagonistic Phenomena in Network Dynamics

In living cells, the loss of biological function caused by the inactivation of a gene can sometimes be compensated by the inactivation of additional genes. This phenomenon, which has been confirmed experimentally, was first predicted in Reference motter2008predicting in the context of metabolic networks and can be seen as a biological analog of the Braess paradox. For concreteness, consider a single-cell organism for which the biological function of interest is growth (i.e., reproduction) rate, and assume that the cells are fully adapted to their environment, meaning that they maximize growth rate under the given conditions. The knockout of otherwise active metabolic genes leads to the inactivation of the associated metabolic reactions to which the proteins coded by those genes serve as catalyzers. Following such a perturbation, the cells are generally no longer in an optimal growth state. An optimal state may nevertheless be approached when certain additional genes (hence metabolic reactions) are knocked out, which gives rise to the predicted synthetic rescue, as illustrated in Figure 3.

As a minimal description of the phenomenon, we can use flux balance analysis edwards2000escherichia to model the optimal growth state of the cell as a linear programming problem. In this problem, one seeks to maximize the rate of a putative reaction $v_{bio}$ that models the overall biomass production (and hence growth rate) under the constraints imposed by the stoichiometry of the metabolic network, nutrient availability, thermodynamics, and any imposed reaction inactivation:

$$\max_{\{v_{j}\}}~v_{bio}\mbox{ ~subject to~ $\sum_{j}$}S_{ij}v_{j}=0\mbox{ ~$\forall i$}\mbox{ ~and~ }v_{j}^{\min}\leq v_{j}\leq v_{j}^{\max}\mbox{ ~$\forall j$},$$

(1)

where $S_{ij}$ are the entries of the stoichiometric matrix and $v_{j}$ are the reaction fluxes. The suboptimal response to the knockout of a gene can be modeled in its simplest form using the “minimization of metabolic adjustment” hypothesis segre2002analysis , which can be implemented as a quadratic programming problem. The model identifies the available state ${\boldsymbol{v}^{\prime}}=(v^{\prime}_{j})$ in the space of metabolic fluxes that, under the additional constraint imposed by the gene inactivation, is closest to the pre-knockout state ${\boldsymbol{v}}$: $\min_{\{v^{\prime}_{j}\}}||{\boldsymbol{v}^{\prime}}-{\boldsymbol{v}}||^{2}$ subject to constraints in the form of those in Equation 1. Thus, this model effectively predicts that metabolic fluxes are rerouted mostly locally, whereas the new optimal state could require a more global flux rearrangement. The rescue state can then be predicted by applying the same quadratic optimization to the combined perturbation of the primary and rescue gene knockouts. From a biological standpoint, the second knockout effectively precipitates adaptation of the perturbed network that could in principle be eventually achieved by long-term adaptive evolution cornelius2011dispensability .

An interesting potential application of this phenomenon is to the development of pairs of antibiotic drugs that can select against resistant cells. Each gene knockout of a synthetic rescue pair has the potential to inhibit growth when implemented in isolation, but one of them suppresses the impact of the other when applied concurrently. Thus, they serve as targets for antibiotic drugs that would interact antagonistically motter2010improved , and previous research has shown that the combination of two antagonistically interacting antibiotics will select against cells that have developed resistance to the suppressor chait2007antibiotic .

While ordinary materials expand when tensioned, it has been shown that a material can be designed to undergo a negative compressibility transition (i.e., a transition to a contracted phase) in response to increasing tension nicolaou2012mechanical . As in the case of other metamaterials, such a material is engineered to gain its unusual property from its structure rather than composition. In other words, this is a property of the underlying mechanical network, which can in fact be seen as a nontrivial generalization of the spring-string system discussed above. To understand this generalization, we can imagine the constituents of the material as consisting of four particles that interact with each other through attractive forces, which are nevertheless nonlinear, allowing the system to be bistable. When two stable states coexist, one generally corresponds to an expanded configuration while the other corresponds to a contracted one. Ignoring dissipation for simplicity, each such constituent is characterized by a potential of the form

$$V(x,y,h,F)=V_{x}(x)+V_{y}(y)+V_{z}(y-x)+V_{x}(h-y)+V_{y}(h-x)+V_{h}(h)-Fh,$$

(2)

where $h$ is the total length and $F$ is the applied tensional force (see Figure 4a). As $F$ is increased, the stable state corresponding to larger $h$ first becomes marginally stable and then disappears. If the system was initially in this extended state, it will then transition to the other stable state, which corresponds to a smaller $h$.

Mathematically, this transition is determined by a bifurcation analogous to one observed for the potential $U(\xi,{f})=-\xi^{3}/3+{f}\xi$, where ${f}$ is the tunable parameter. This potential has a stable equilibrium point at $\xi^{*}=-\sqrt{f}$ and an unstable one at $\xi^{*}=\sqrt{f}$ for ${f}>0$, has a single (degenerate) equilibrium point at $\xi^{*}=0$ for ${f}=0$, and has no equilibrium point for ${f}<0$. Thus, as ${f}$ is varied from positive to negative, the stable equilibrium point vanishes and the system responds discontinuously. Physically, the disappearance of the occupied stable state is analogous to the cutting of the linking string in Figure 2: it causes the inner particles in Figure 4a to move closer to (and hence attract more strongly) the external particles, which is similar to the transition from a parallel to a series configuration in the spring-string system. A key difference here is that the process can be cycled by varying the tensional force, as shown in the hysteresis diagram of Figure 4b.

The material itself can be formed by aggregating such constituents, as shown in Figure 4c for a square lattice network. In the thermodynamic limit, the bifurcations undergone by the constituents give rise to a transition between the corresponding extended and contracted phases, which can be rigorously characterized at finite temperature using tools from statistical physics nicolaou2013longitudinal . Such materials can find applications in the design of micro-mechanical controls and protective mechanical devices, but they also suggest a more general principle to design metamaterials with inverted responses that can in theory be applied not only to stress-and-strain but also to any pair of thermodynamically conjugated variables.

In the synchronization of coupled oscillators, a parallel with the Braess paradox has been established in which the addition or strengthening of connections between oscillators has the adverse effect of removing a previously existing synchronous state. This possibility has attracted special attention in connection with power-grid networks, where the addition of new line capacity for power transmission could eliminate a phase-locked operating state among power generators and motors in an AC network witthaut2012braess ; coletta2016linear . A minimal model to illustrate this effect is

$$\ddot{\phi}_{i}=P_{i}-\alpha\,\dot{\phi}_{i}-\sum_{j}K_{ij}\sin(\phi_{i}-\phi_{j}),~~i=1,\cdots,n,$$

(3)

where both generator nodes ($P_{i}>0$) and motor nodes ($P_{i}<0$) are represented as damped second-order phase oscillators and $K_{ij}$ represents the network structure as well as the capacity of the transmission lines filatrella2008analysis ; rohden2012self . Of special interest are the frequency-synchronized states satisfying $\dot{\phi}_{1}=\dot{\phi}_{2}=\dots=\dot{\phi}_{n}$ at all times and thus $\Delta_{ij}=\phi_{i}-\phi_{j}=\textsf{const}$ for all $i$ and $j$.

This synchronization condition leads to equations of the form $\sum_{j}K_{ij}\sin\Delta_{ij}=P_{i}$, where $\Delta_{ij}$ is to be determined for given $K_{ij}$ and $P_{j}$. When such a solution for $\Delta_{ij}$ exists, the resulting power flows through the lines are $K_{ij}\sin\Delta_{ij}$ and they automatically respect the line capacities. A new solution for $\Delta_{ij}$ may exist if line capacities are increased (or, in particular, if new lines are added), but therein lies the rub: for the actual state to exist, not only a solution for each $\Delta_{ij}$ has to exist but also each phase angle $\phi_{i}$ has to be uniquely defined, and the latter is not guaranteed when line capacities are increased even if the $\Delta_{ij}$ solution continues to exist. That is, the set of equations $\phi_{i}-\phi_{j}=\Delta_{ij}$ (which must be simultaneously satisfied for every pair of nodes $i$ and $j$ connected by a transmission line with non-zero capacity $K_{ij}$) is no longer guaranteed to have a solution. Indeed, as demonstrated in References witthaut2012braess ; witthaut2013nonlocal , a capacity increase will frequently induce conflicts between the phases in this set of equations, which are reminiscent of the phenomenon of geometric frustration as the conflicts necessarily occur along loops in the network. (In the familiar case of geometric frustration in spin systems, not all pairwise interaction energies can be minimized simultaneously precisely due to geometric constraints similar to the ones considered here.) This effect is illustrated in Figure 5 for a simple network of four identical generators ($P_{i}=1$), four identical motors ($P_{i}=-1$), and lines with identical capacity ($K_{ij}=1.03$ for any line with nonzero capacity).

In networks of diffusively coupled oscillators, which describe the dynamics of power generators in the vicinity of their synchronization manifold motter2013spontaneous , an analogous effect can be observed even in the absence of any loops nishikawa2006synchronization . The simplest such model reads $\dot{\boldsymbol{x}}_{i}={\boldsymbol{F}}({\boldsymbol{x}}_{i})-\sigma\sum_{j}L_{ij}{\boldsymbol{H}}({\boldsymbol{x}}_{j})$, where $L=(L_{ij})$ is the Laplacian matrix representing the (possibly directed) network of interactions, and the stability of a synchronous solution ${\boldsymbol{x}}_{i}(t)={\boldsymbol{s}}(t)$ $\forall i$ is determined by a master stability function that often has a bounded stability region pecora1998master ; nishikawa2006maximum . Accordingly, the system is synchronizable for a wider range of the coupling strength $\sigma$ when the nonidentically-null eigenvalues of the Laplacian matrix $L$ are less scattered; the optimally synchronizable networks are those for which these eigenvalues satisfy $\lambda_{2}=\dots=\lambda_{n}$. As shown in References nishikawa2006synchronization ; nishikawa2006maximum , every network that can be spanned from one of its nodes (a necessary condition for stable synchronization to be possible) can also be converted into an optimally synchronizable network by removing edges or reducing edge strengths. In particular, any unweighted oriented tree spanning the entire network is an optimally synchronizable network. Therefore, starting from an arbitrary network that is not synchronizable, one can always turn it into a synchronizable network by pruning the interactions between oscillators nishikawa2006maximum .

Because such systems can have a bounded stability region, they exhibit a number of other counterintuitive effects as a result of the nontrivial dependence of the eigenvalue spread on the interaction network. For example, it has been shown that otherwise unstable synchronous states can be stabilized by transiently uncoupling the oscillators schroder2015transient ; Schroeder2016 (see also stilwell2006sufficient ; Chen2009onoffcoupling ; jeter2015synchronization ). In different work, it was shown that an otherwise nonsynchronizable network can become synchronizable not only by removing nodes but also by adding nodes despite the resulting increase in the number of eigenvalues that need to fit inside the stability region nishikawa2010network .

A problem of fundamental interest in network dynamics is the one of preventing the loss of resources by means of interventions that are themselves limited to only further removing resources. As a concrete example, consider a food-web network in which a primary extinction triggers a cascade of secondary extinctions. The question of interest is to design a control intervention that if applied after the first extinction (but before the propagation of the cascade) would prevent the other extinctions. An elementary model to conceptualize the problem is the $n$-species Lotka-Volterra predator-prey model cohen1990stochastic :

$$\dot{X_{i}}=X_{i}\left(b_{i}+\sum_{j}a_{ij}X_{j}\right),~~i=1,\cdots,n,$$

(4)

where $X_{i}\geq 0$ is the population of species $i$ and parameter $b_{i}$ is the growth rate for basal species (those that do not feed on others) and the mortality rate for non-basal ones. Upon removal of one species, this system generically has one fixed point at which all other species populations are nonzero when the matrix $A=(a_{ij})$ is invertible (for simplicity let us ignore the possible presence of a nonfixed-point attractor in which all other species survive).

The primary extinction will cause subsequent extinctions if 1) this fixed point is unstable and/or outside the positive ($n-1$)-dimensional orthant or 2) the fixed point is stable and in the positive orthant but the initial extinction laid the network state outside its basin of attraction. Within this simplified picture, control to prevent secondary extinctions should be geared towards manipulating the position and stability of this fixed point and/or directing the state to its basin of attraction. Recognizing that realistic interventions over the relevant time scale cannot directly increase species populations, recent research sahasrabudhe2011rescuing has considered interventions that either temporarily suppress certain species’ populations $\{X_{i}\}$—to bring the state to the desired basin of attraction—or permanently reduce (increase) their growth (mortality) rates $\{b_{i}\}$—to manipulate the fixed point and/or basins of attraction. Figure 6 shows for both types of interventions examples in which they would prevent all secondary extinctions for the model in Equation 4.

The scenario just described is ecologically plausible in view of other antagonistic effects that take place in food-web networks, such as the paradox of enrichment rosenzweig1971paradox , in which increasing food availability to a prey may eventually lead to the extinction of its predator. More important, the concept generalizes to other network processes, including the control of cascading failures in power-grid networks and the reprogramming of intracellular networks, where, owing to scenarios similar to the one above, the most effective beneficial interventions often appear to be deleterious cornelius2013realistic .

The “inefficiency” of the Nash equilibrium dubey1986inefficiency —that follows from the equilibrium not being globally optimal—has been shown to lead to numerous other “more-for-less” paradoxes in networks. To be specific, we focus on variants and implications of the Braess paradox and note that related effects can be recognized across diverse network systems.

For example, in traffic networks exhibiting the Braess paradox, as originally formulated in Reference Braess1969 , the paradox has been shown to actually disappears for sufficiently high traffic demand nagurney2010negation . This means that new routes that inadvertently give rise to the paradox may slow traffic when demand is low and not even be used when demand is high. Other work has shown that, in networks with multiple origin and destination nodes, a decrease in demand can in fact lead to an increase in average travel time fisk1979more ; fujishige2017matroids . An even stronger effect has been established in which an increase in travel time along a route can result in a new equilibrium characterized by the abandonment of a different route connecting the same origin and destination nodes while the original route may continue to be used steinberg1988prevalence . In addition, there are also numerous essentially equivalent restatements of the same underlying phenomena, such as in the conclusion that an increase in the local travel time can lead to a reduction in the global travel time smith1978road , or that the overall transport capacity of a network can be reduced upon addition of capacity to individual links yang1998capacity .

Such paradoxes have also been considered in numerous concrete settings, both in the context of complex networks youn2008price ; skinner2015price ; sole2016congestion and in specific application domains, including computer networks korilis1999avoiding ; kameda2000braess , chemical reaction networks cornelius2011dispensability ; lepore2011computational , and electric networks cohen1991paradoxical ; skinner2015price ; blumsack2007quantitative ; nagurney2016physical . In electric networks, in particular, it has been shown that for graph topologies similar to the one in Figure 1b, the addition of the intermediate (current-carrying) link can create overloads in other links for certain AC circuits blumsack2007quantitative and lead to an increase in voltage drop for a fixed current source in certain two-terminal DC circuits cohen1991paradoxical ; nagurney2016physical .

Among physical systems, a major class of applications concerns the study of fluid networks. Using simplified models of the fluid dynamics, it has been shown that increase in the conductivity of individual pipes in a fluid network can lead to increase in power loss, which can be regarded as a fluid analog of resistance. While generally not observed for two-terminal networks calvert1991braess , this behavior has been predicted for single-source multiple-destination delivery networks of both water calvert1991braess ; keady1995colebrook and natural gas andre2010optimization ; ayala2013braess . This behavior can also be characterized as a need for pressure difference increases to maintain the same outputs following the capacity increase of specific pipes ayala2013braess , thus bearing direct analogy with previous results on simple electric circuits cohen1991paradoxical . Finally, it is interesting to note that similar transport inefficiency has recently been observed also in the quantum regime in mesoscopic material networks pala2012transport ; sousa2013braess , where the addition of a transport path induces a decrease in the overall conductance.

Consider a network of phase oscillators of the form

$$\dot{\theta}_{i}=\omega_{i}+\sigma\sum_{j}A_{ij}\sin(\theta_{j}-\theta_{i}-\alpha),~~i=1,\cdots,n,$$

(5)

where $A_{ij}\geq 0$ and $\sigma\cos(\alpha)>0$. Each individual oscillator is identified by its natural frequency $\omega_{i}$, while the other terms represent interactions between oscillators. What natural frequencies should the oscillators have in order to facilitate complete synchronization of the form ${\theta}_{1}(t)={\theta}_{2}(t)=\dots={\theta}_{n}(t)$? This question can be answered using a small angle approximation in the vicinity of the synchronous state to obtain $\dot{\theta}_{i}=\omega_{i}-\sigma k_{i}\sin(\alpha)+\sigma\cos(\alpha)\sum_{j}A_{ij}(\theta_{j}-\theta_{i})$, where $k_{i}=\sum_{j}A_{ij}$ is the indegree of node $i$. The synchronization condition implies $\dot{\theta}_{i}=\omega_{i}-\sigma k_{i}\sin(\alpha)\equiv\Omega$ $\forall i$ for some constant $\Omega$, which leads to $\dot{\tilde{\theta}}_{i}=-\sigma^{\prime}\sum_{j}L_{ij}\tilde{\theta}_{j}$ for $\tilde{\theta}_{i}=\theta_{i}-\Omega t$ and $\sigma^{\prime}=\sigma\cos(\alpha)$. The synchronous state is stable if and only if all except the identically null eigenvalue of the Laplacian matrix $L$ have positive real parts, which is guaranteed to be the case in any network that can be spanned from a node (as generally assumed in the study of synchronization). On closer examination, the actual condition for synchronization stability is thus that $\omega_{i}=\Omega+\sigma k_{i}\sin(\alpha)$, meaning that the natural frequencies of the individual oscillators have to be nonidentical unless the network has identical indegrees for all nodes. This is intuitive because, in order to achieve an identical state, the oscillators need to be nonidentical to compensate for their nonidentical couplings. A generalization of this argument can be used to optimize synchronization in complex networks in general skardal2014optimal , and analogous results are expected if a different characteristic of the system (e.g., delays, noise level, or coupling strength) is nonuniform across components. For a long time, cases involving such compensatory nonuniformities were the only ones in which differences between the oscillators were found to help minimize differences between their states.

However, it was recently shown that stable identical synchronization can require the oscillators to be nonidentical even when they are identically coupled and indeed equal with respect to all other aspects. A simple scenario in which this was first demonstrated nishikawa2016symmetric was for phase-amplitude oscillators characterized by a phase variable $\theta_{i}$ and an amplitude variable $r_{i}$ such that the system always has one synchronous solution corresponding to $\theta_{1}(t)=\theta_{2}(t)=\dots=\theta_{n}(t)$ and $r_{1}(t)=r_{2}(t)=\dots=r_{n}(t)=1$. The question is whether this solution is stable or not. The uncoupled dynamics of the amplitude variable takes the form $\dot{r}=b_{i}r(1-r)$, where $b_{i}$ is the only parameter allowed to vary from node to node. All other parameters and the network structural properties are identical for all oscillators. As illustrated in Figure 7 for a rotationally invariant network of three such oscillators, there are scenarios for which the synchronous state is not stable for any choice of $b_{1}=b_{2}=b_{3}$, but it becomes stable for specific choices of nonidentical $b_{i}$. This is remarkable because the synchronous state is the state that would reflect the rotational symmetry of the system and, nevertheless, this symmetric state is stable only when the symmetry of the system is broken by making the oscillators nonidentical.

It is instructive to contrast this effect with the well-studied phenomenon of symmetry breaking. The fact that the symmetric system does not exhibit a stable synchronous (hence symmetric) solution can be seen as an ordinary manifestation of spontaneous symmetry breaking. On the other hand, the fact that the symmetry of the stable solution is preserved when (and only when) the symmetry of the system is broken can be seen as the converse of symmetry breaking. More generally, in the same way the former shows that an asymmetric reality may be described by a symmetric theory, the latter shows that a symmetric reality may require an asymmetric theory.

The specific observation that synchronization can be stabilized or enhanced by tuning the oscillators to be nonidentical has potential implications for power-grid networks, whose operation requires frequency synchronization among power generators. As shown in Reference nishikawa2015stability , the stability of the synchronous state of interest can be significantly enhanced when an effective parameter that depends on the damping, inertia and droop coefficients of the power generators is set to be suitably different for different generators.

In networks of coupled oscillators, symmetry breaking itself can lead to rather surprising spatiotemporal patterns formed by two or more domains of qualitatively different dynamics, some in which the oscillators are mutually synchronized and others in which they evolve incoherently. First identified by Kuramoto kuramoto2003reduction and later termed chimeras abrams2004chimera , such states may provide insights into unihemispheric sleep in some animal species rattenborg2000behavioral and fibrillation in the cardiac muscle of ventricles davidenko1992stationary .

The first model in which chimera states were systematically described was a ring network of nonlocally coupled phase oscillators kuramoto2002coexistence ,

$$\frac{\partial\phi(x,t)}{\partial t}=\omega-\int G(x-x^{\prime})\sin\left[\phi(x,t)-\phi(x^{\prime},t)+\alpha\right]dx^{\prime},$$

(6)

where the kernel $G(x-x^{\prime})$ is a decreasing function that determines the distance-dependent strength of the coupling. This equation, which can be derived via phase reduction from a nonlocally coupled complex Ginzburg-Landau equation, describes identically-coupled identical oscillators in the limit of a large number of oscillators. The complete synchronous state is always a solution and can in fact be stable. But its basin of attraction is not global and the system also exhibits persistent chimera states that are approached for other initial conditions. Inspection of the phases of the oscillators along the ring, as in the snapshot in Figure 8, shows a clear separation into a domain of incoherence (center) and a domain of coherence (extremes). Dynamical variants of such patterns, including spiral chimeras in two-dimensional arrays of oscillators, have been known to exist from the very first studies kuramoto2003rotating , and experimental demonstrations of chimeras states have been successful on various systems, including networks of coupled electro-optic hagerstrom2012experimental , chemical tinsley2012chimera , and mechanical martens2013chimera oscillators.

In part because the original theory to describe chimera states was based on a self-consistent mean-field solution, it was initially believed that such states would not emerge in systems with local or global coupling abrams2004chimera and, moreover, that they would be long lived but not permanently stable in finite-size networks wolfrum2011chimera . Subsequent research demonstrated that chimeras can emerge across a surprising range of models and conditions, which include examples of locally and globally coupled systems panaggio2015chimera . Moreover, while the question of stability remains open for many systems, stable chimera states have now been rigorously identified in finite-size networks of chaotic oscillators cho2017stable (see also suda2015persistent ; hart2016experimental ; bick2016chaotic ). Yet, previous work focused exclusively on discrete systems or systems in which the coupling was not strictly local schmidt2014coexistence , leaving open the question of whether locally coupled continuous systems could exhibit chimera states (not to be confused with the continuous representation of discrete systems, such as in Equation (6), whose variables are not spatially continuous, as illustrated in Figure 8). The latter was addressed by recent research, which demonstrated the existence of symmetry-breaking chimera states whose coherent and incoherent phases are analogous to laminar and turbulent phases of a fluid system nicolaou2017continuous , thereby revealing an important connection with the classical field of pattern formation.

Distant oscillators in a network can synchronize stably even when they are connected exclusively through oscillators that are asynchronous. This remote form of cluster synchronization has potential implications for information processing in the brain nicosia2013remote and for secure communication zhang2017incoherence , and has been recognized in many systems. For example, in an initially synchronized undirected star network of diffusively coupled chaotic oscillators, an increase in coupling strength can lead to a short wavelength bifurcation that drives the center node off pace but keeps all the other nodes synchronized; this long-known behavior is observed when the stability region is limited, and has been referred to as a drum-head-mode bifurcation pecora1998master (see also winful1990synchronized ).

More recently it has been noticed that variants of this behavior extend to complex networks in general, largely owing to cluster synchronous states that derive from network symmetry. For example, Reference nicosia2013remote considered the system in Equation 5 for identical $\omega_{i}$ to show that suitable choices of the phase parameter $\alpha$ lead to a frustrated state in which directly connected oscillators do not synchronize whereas certain oscillators that are distant from each other do. The oscillators that synchronize are those symmetrically coupled to the network (i.e., in the same symmetry cluster). This is so because the equation of motion remains invariant under the action of the symmetry group of the network, meaning that the system admits a synchronous solution among those nodes, which in this case is stable even when they are not directly connected.

Several variants of remote synchronization are particularly intriguing. For example, in the study of so-called relay synchronization, it has been shown that two delay-coupled oscillators can synchronize identically (thus without delay) when connected through a third oscillator that lags behind fischer2006zero . In a different study, two chaotic oscillators have been shown to synchronize stably while connected through an intermediate cluster of oscillators that are incoherent both with the outer nodes and with themselves; termed incoherence-mediated remote synchronization zhang2017incoherence , this scenario blends together the properties of remote synchronization with those of chimera states (see Figure 9). It has the advantage of being robust to perturbation of the intermediate oscillators and can in principle be useful for encryption key distribution.

A common theme underlying all forms of remote synchronization just mentioned is indeed symmetry in the network. To appreciate how general symmetry-based remote synchronization can be, it is important to notice that even random networks can exhibit a large number of nontrivial symmetry clusters, and that the nodes in such clusters are often not directly connected macarthur2008symmetry . Recently developed techniques pecora2013symmetries to study the stability of cluster synchronous states while exploring their relation with symmetries promise to be useful in future studies of this phenomenon.

In network systems, the hallmarks of emergent distributed phenomena are found not only in overt manifestations of collective dynamics but also in the associated information transmission and processing. These characteristics are common across numerous systems in biology, physics, and engineering, ranging from neural and biochemical circuits to self-organized communication networks Hopfield1982 ; Cardelli2017 ; Klinglmayr2012applied ; Klinglmayr2012theory . In biological systems, in particular, information handling is often referred to as distributed, but how information may be specifically communicated and dynamically routed in such systems is not yet well understood. Recent work Kirst2012 offers concrete hints on what distributed information routing actually means and what it might condense to, qualitatively and quantitatively.

A theoretical framework for networks of oscillatory units Kirst2016 predicts the patterns of information routing in networks and their dependence on the interaction network and other factors. The framework is established using model systems of the form

$$\dot{\boldsymbol{x}}={\boldsymbol{f}}({\boldsymbol{x}})+{\boldsymbol{\xi}},$$

(7)

where ${\boldsymbol{f}}({\boldsymbol{x}})$ represents the intrinsic time evolution rules and interaction structure of the network and ${\boldsymbol{\xi}}$ represents an external (stochastic) input. The theory determines how routing patterns depend on the dynamical reference state (taken to be a periodic or fixed-point solution of $\dot{\boldsymbol{x}}_{o}={\boldsymbol{f}}({\boldsymbol{x}}_{o})$) in the presence of small external inputs. Because ${\boldsymbol{x}}_{o}$ and ${\boldsymbol{\xi}}$ are system-wide variables, local modifications of individual unit properties, network interactions, and external inputs provide mechanisms to flexibly change information routing throughout the entire network.

As a particularly intriguing property, we mention that local modifications in one part of the network may remotely influence and even reverse the direction of information routing between two other parts (see Figure 10). This points to a potentially general mechanism of remote control that is possible because information routing is ultimately determined by the network dynamics, which are collective and distributed.

Limiting ourselves to synchronization dynamics for concreteness, we note that delays, noise, and correlations between node parameters and network structure have been found to lead to other unanticipated phenomena. For example, scenarios have been described in which time delay in node-to-node communication can facilitate rather than inhibit in-phase (zero-lag) synchronization atay2004delays . This has been observed, for instance, for pulse-coupled oscillators with inhibitory coupling ernst1995synchronization and for relay-coupled oscillators fischer2006zero . For excitatory coupling, on the other hand, networks of pulse-coupled oscillators can be shown to often exhibit attracting periodic orbits with points that are isolated from their basins of attraction, and thus have the peculiar property of being both attractive and unstable timme2002prevalence .

Another notable class of behaviors comes from considering coupled oscillators in the presence of noise, where it has been shown that, due to coupling, modular networks can synchronize in response to noise even when the noise applied to different modules is uncorrelated meng2016independent . In networks of globally coupled oscillators, different work has demonstrated that independent noise at individual nodes can also stabilize otherwise unstable states of partial synchronization clusella2017noise . Moreover, in systems of directionally coupled non-identical oscillators, it has been shown that the phase diffusion in an oscillator can depend non-monotonically on the noise intensity in a coupled oscillator, and thus become more coherent as noise is further strengthened amro2015phase .

Finally, following the discovery of explosive percolation achlioptas2009explosive —characterized by abrupt transitions da2010explosive ; riordan2011explosive ; grassberger2011explosive ; nagler2011impact —analogous transitions (in fact strictly discontinuous ones vlasov2015explosive ) have been identified also in synchronization processes, giving rise to so-called explosive synchronization gomez2011explosive . Explosive synchronization occurs when, as the coupling strength is increased, an otherwise second-order phase transition to synchronization becomes first order. This behavior has been demonstrated for both phase oscillators gomez2011explosive and chaotic oscillators leyva2012explosive in scale-free networks, where the oscillator frequency is positively correlated with the node degree. Such transitions exhibit hysteresis, in which the transformation from coherence to incoherence occurs at a smaller critical coupling than that from incoherence to coherence. The dynamical origin of the hysteretic behavior has been explicitly related to a change in the basin of attraction of the synchronization state zou2014basin .