A universal transition in the robustness of evolving open systems

Temporal inactivation enhances robustness in an evolving system

We study the robustness of an evolving system that is driven by successive inclusions of new elements or constituents with m random interactions to older ones. Each constitutive element in the model stays either active or is temporarily inactivated depending upon the influence of the other active elements. If the time spent by an element in the inactivated state reaches T _W , it gets extinct. The phase diagram of this dynamic model as a function of m and T _W is investigated by numerical and analytical methods and as a result both growing (robust) as well as non-growing (volatile) phases are identified. It is also found that larger time limit T _W enhances the system's robustness against the inclusion of new elements, mainly due to the system's increased ability to reject 'falling-together' type attacks. Our results suggest that the ability of an element to survive in an unfavourable situation for a while, either as a minority or in a dormant state, could improve the robustness of the entire system.

Enhanced robustness of evolving open systems by the bidirectionality of interactions between elements

Living organisms, ecosystems, and social systems are examples of complex systems in which robustness against inclusion of new elements is an essential feature. A recently proposed simple model has revealed a general mechanism by which such systems can become robust against inclusion of elements with totally random interactions when the elements have a moderate number of links. The interaction is, however, in many systems often intrinsically bidirectional like for mutual symbiosis and competition in ecology. This study reports the strong reinforcement effect of the bidirectionality of the interactions on the robustness of evolving systems. We show that the system with purely bidirectional interactions can grow with twofold average degree, in comparison with the purely unidirectional system. This drastic shift of the transition point comes from the reinforcement of each node, not from a change in structure of the emergent system. For systems with partially bidirectional interactions we find that the regime of the growing phase gets expanded. In the dense interaction regime, there exists an optimum proportion of bidirectional interactions for the growth rate at around 1/3. In the sparsely connected systems, small but finite fraction of bidirectional links can change the system's behaviour from non-growing to growing.

Structural instability of large-scale functional networks

We study how large functional networks can grow stably under possible cascading overload failures and evaluated the maximum stable network size above which even a small-scale failure would cause a fatal breakdown of the network. Employing a model of cascading failures induced by temporally fluctuating loads, the maximum stable size n_max has been calculated as a function of the load reduction parameter r that characterizes how quickly the total load is reduced during the cascade. If we reduce the total load sufficiently fast (r ≥ r_c), the network can grow infinitely. Otherwise, n_max is finite and increases with r. For a fixed r(< r_c), n_max for a scale-free network is larger than that for an exponential network with the same average degree. We also discuss how one detects and avoids the crisis of a fatal breakdown of the network from the relation between the sizes of the initial network and the largest component after an ordinarily occurring cascading failure.

Punctuated equilibrium as an emergent process and its modified thermodynamic characterization

We address evolutionary dynamics and consider under which conditions the ecosystem interaction network allows punctuated equilibrium (i.e., alternation between hectic and quasi-stable phases). We focus on the links connecting various species and on the strength and sign of those links. For this study we consider the Tangled Nature model, which allows considerable flexibility and plasticity in the analysis of interspecies interactions. We find that it is necessary to have a proper balance of connectivity and interaction intensities so as to establish the kind of mutual cooperation and competition found in nature. It suggests evolutionary punctuated equilibrium as an emergent process, thus displaying features of complex systems. To explicitly demonstrate this fact we consider an extended form of thermodynamics, defining (for the present context) relevant out-of-equilibrium "collective" functions. We then show how to characterize the punctuated equilibrium through entropy-like and free energy-like quantities. Finally, from a close analogy to thermodynamic systems, we propose a protocol similar to simulated annealing. It is based on controlling the species' rate of mutation during the hectic periods, in this way enhancing the exploration of the genome space (similar to the known behavior of bacteria in stressful environments). This allows the system to more rapidly converge to long-duration quasi-stable phases.

Structural Robustness of Unidirectional Dependent Networks Based on Attack Strategies

Current works have been focused on the robustness of single network and interdependent networks. However, to be more correct, the dependence of many real systems should be described as unidirectional. To study the structural robustness of networks with unidirectional dependence, the dependent networks named UDN are proposed, the description of the propagation of failures in them is given, as well as the introduction of the attack strategies that the probability of a node being attacked depends on the degree (DP attack) or on the betweenness (BP attack) of this node. The simulated results show that UDN is more vulnerable to BP attack when is first attacked a node with high betweenness. Compared with the Interacting Networks (IN), the UDN is more fragile under the two attack’s strategies.