Michael M. Danziger

Localized attacks on spatially embedded networks with dependencies

Many real world complex systems such as critical infrastructure networks are embedded in space and their components may depend on one another to function. They are also susceptible to geographically localized damage caused by malicious attacks or natural disasters. Here, we study a general model of spatially embedded networks with dependencies under localized attacks. We develop a theoretical and numerical approach to describe and predict the effects of localized attacks on spatially embedded systems with dependencies. Surprisingly, we find that a localized attack can cause substantially more damage than an equivalent random attack. Furthermore, we find that for a broad range of parameters, systems which appear stable are in fact metastable. Though robust to random failures—even of finite fraction—if subjected to a localized attack larger than a critical size which is independent of the system size (i.e., a zero fraction), a cascading failure emerges which leads to complete system collapse. Our results demonstrate the potential high risk of localized attacks on spatially embedded network systems with dependencies and may be useful for designing more resilient systems.Many real world complex systems such as infrastructure, communication and transportation networks are embedded in space, where entities of one system may depend on entities of other systems. These systems are subject to geographically localized failures due to malicious attacks or natural disasters. Here we study the resilience of a system composed of two interdependent spatially embedded networks to localized geographical attacks. We find that if an attack is larger than a finite (zero fraction of the system) critical size, it will spread through the entire system and lead to its complete collapse. If the attack is below the critical size, it will remain localized. In contrast, under random attack a finite fraction of the system needs to be removed to initiate system collapse. We present both numerical simulations and a theoretical approach to analyze and predict the effect of local attacks and the critical attack size. Our results demonstrate the high risk of local attacks on interdependent spatially embedded infrastructures and can be useful for designing more resilient systems.

Robustness of a network formed of spatially embedded networks.

We present analytic and numeric results for percolation in a network formed of interdependent spatially embedded networks. We show results for a treelike and a random regular network of networks each with (i) unconstrained dependency links and (ii) dependency links restricted to a maximum Euclidean length r. Analytic results are given for each network of networks with spatially unconstrained dependency links and compared to simulations. For the case of two fully interdependent spatially embedded networks it was found [Li et al., Phys. Rev. Lett. 108, 228702 (2012)] that the system undergoes a first-order phase transition only for r>r(c) ≈ 8. We find here that for treelike networks of networks (composed of n networks) r(c) significantly decreases as n increases and rapidly (n ≥ 11) reaches its limiting value of 1. For cases where the dependencies form loops, such as in random regular networks, we show analytically and confirm through simulations that there is a certain fraction of dependent nodes, q(max), above which the entire network structure collapses even if a single node is removed. The value of q(max) decreases quickly with m, the degree of the random regular network of networks. Our results show the extreme sensitivity of coupled spatial networks and emphasize the susceptibility of these networks to sudden collapse. The theory proposed here requires only numerical knowledge about the percolation behavior of a single network and therefore can be used to find the robustness of any network of networks where the profile of percolation of a singe network is known numerically.

Percolation and cascade dynamics of spatial networks with partial dependency

Recently, it has been shown that the removal of a random fraction of nodes from a system of interdependent spatial networks can lead to cascading failures which amplify the original damage and destroy the entire system, often via abrupt first-order transitions. For these distinctive phenomena to emerge, the interdependence between networks need not be total. We consider here a system of partially interdependent spatial networks (modelled as lattices) with a fraction q of the nodes interdependent and the remaining 1 − q autonomous. In our model, the dependency links between networks are of geometric length less than r. Under full dependency (q = 1), this system was shown to have a first-order percolation transition if r > rc ≈ 8. Here, we generalize this result and show that for all q > 0, there will be a first-order transition if r > rc(q). We show that rc(q) increases monotonically with decreasing q and limq→0+ rc(q) = ∞. Additionally, we present a detailed description and explanation of the cascading failures in spatially embedded interdependent networks near the percolation threshold pc. These failures follow three mechanisms depending on the value of r. Below rc the system undergoes a continuous transition similar to standard percolation on a lattice. Above rc there are two distinct first-order transitions for finite and infinite r, respectively. The cascading failure for finite r is characterized by the emergence of a critical hole which then spreads through the system while the infinite r transition is more similar to the case of random networks. Surprisingly, we find that this spreading transition can still occur even if p < pc. We present measurements of cascade dynamics which differentiate between these phase transitions and elucidate their mechanisms. These results extend previous research on spatial networks to the more realistic case of partial dependency and shed new light on the specific dynamics of dependencydriven cascading failures.

An Introduction to Interdependent Networks

Many real-world phenomena can be modelled using networks. Often, these networks interact with one another in non-trivial ways. Recently, a theory of interdependent networks has been developed which describes dependency between nodes across networks. Interdependent networks have a number of unique properties which are absent in single networks. In particular, systems of interdependent networks often undergo abrupt first-order percolation transitions induced by cascading failures. Here we present an overview of recent developments and significant findings regarding interdependent networks and networks of networks.

Interdependent Spatially Embedded Networks: Dynamics at Percolation Threshold

Spatially embedded systems of interdependent networks with full dependency (q=1) have been found to have a first-order percolation transition if the dependency link length (the maximum distance in lattice units between a node in one network and the node that it depends on in another network) is longer than a certain critical length rc ≈ 8. We find here that a similar result is valid for any finite value of q with a larger rc as q decreases. We also provide a theoretical approach which correctly predicts the relationship between rc and q. We also examine the dynamics at the percolation threshold pc for varying r and q and find that there are three different mechanisms of failure for every q depending on r. Below rc the system undergoes a continuous transition similar to standard percolation. Above rc there are two distinct first-order transitions for finite or infinite r, respectively. The transition for finite r is characterized by spreading of node failures through the system while the infinite r corresponds to a non-spatial cascading failure similar to the case of random networks. These results extend previous results on spatially embedded interdependent networks to the more realistic cases of partial dependency and shed new light on the specific dynamics of cascading failures in such systems.

The effect of spatiality on multiplex networks

Multilayer infrastructure is often interdependent, with nodes in one layer depending on nearby nodes in another layer to function. The links in each layer are often of limited length, due to the construction cost of longer links. Here, we model such systems as a multiplex network composed of two or more layers, each with links of characteristic geographic length, embedded in 2-dimensional space. This is equivalent to a system of interdependent spatially embedded networks in two dimensions in which the connectivity links are constrained in length but varied while the length of the dependency links is always zero. We find two distinct percolation transition behaviors depending on the characteristic length, ζ, of the links. When ζ is longer than a certain critical value, ζc, abrupt, first-order transitions take place, while for ζ < ζc the transition is continuous. We show that, though in single-layer networks increasing ζ decreases the percolation threshold pc, in multiplex networks it has the opposite effect: increasing pc to a maximum at ζ = ζc. By providing a more realistic topological model for spatially embedded interdependent and multiplex networks and highlighting its similarities to lattice-based models, we provide a new direction for more detailed future studies.Many multiplex networks are embedded in space, with links more likely to exist between nearby nodes than distant nodes. For example, interdependent infrastructure networks can be represented as multiplex networks, where each layer has links among nearby nodes. Here, we model the effect of spatiality on the robustness of a multiplex network embedded in 2-dimensional space, where links in each layer are of variable but constrained length. Based on empirical measurements of real-world networks, we adopt exponentially distributed link lengths with characteristic length ζ. By changing ζ, we modulate the strength of the spatial embedding. When ζ → ∞, all link lengths are equally likely, and the spatiality does not affect the topology. However, when only short links are allowed, and the topology is overwhelmingly determined by the spatial embedding. We find that, though longer links strengthen a single-layer network, they make a multi-layer network more vulnerable. We further find that when ζ is longer than a certain critical value, , abrupt, discontinuous transitions take place, while for the transition is continuous, indicating that the risk of abrupt collapse can be eliminated if the typical link length is shorter than .

Assortativity and leadership emerge from anti-preferential attachment in heterogeneous networks.

Real-world networks have distinct topologies, with marked deviations from purely random networks. Many of them exhibit degree-assortativity, with nodes of similar degree more likely to link to one another. Though microscopic mechanisms have been suggested for the emergence of other topological features, assortativity has proven elusive. Assortativity can be artificially implanted in a network via degree-preserving link permutations, however this destroys the graph’s hierarchical clustering and does not correspond to any microscopic mechanism. Here, we propose the first generative model which creates heterogeneous networks with scale-free-like properties in degree and clustering distributions and tunable realistic assortativity. Two distinct populations of nodes are incrementally added to an initial network by selecting a subgraph to connect to at random. One population (the followers) follows preferential attachment, while the other population (the potential leaders) connects via anti-preferential attachment: they link to lower degree nodes when added to the network. By selecting the lower degree nodes, the potential leader nodes maintain high visibility during the growth process, eventually growing into hubs. The evolution of links in Facebook empirically validates the connection between the initial anti-preferential attachment and long term high degree. In this way, our work sheds new light on the structure and evolution of social networks.

Interdependent resistor networks with process-based dependency

Studies of resilience of interdependent networks have focused on structural dependencies between pairs of nodes across networks but have not included the effects of dynamic processes taking place on the networks. Here we study the effect of dynamic process-based dependencies on a system of interdependent resistor networks. We describe a new class of dependency in which a nodes functionality is determined by whether or not it is actually carrying current and not just by its structural connectivity to a spanning component. This criterion determines its functionality within its own network as well as its ability to provide support-but not electrical current-to nodes in another network. We present the effects of this new type of dependency on the critical properties of σ and , the overall conductivity of the system and the fraction of nodes which carry current, respectively. Because the conductance of current has direct physical effects (e.g. heat, magnetic induction), the development of a theory of process-based dependency can lead to innovative technology. As an example, we describe how the theory presented here could be used to develop a new kind of highly sensitive thermal or gas sensor.

Explosive synchronization coexists with classical synchronization in the Kuramoto model

Explosive synchronization has recently been reported in a system of adaptively coupled Kuramoto oscillators, without any conditions on the frequency or degree of the nodes. Here, we find that, in fact, the explosive phase coexists with the standard phase of the Kuramoto oscillators. We determine this by extending the mean-field theory of adaptively coupled oscillators with full coupling to the case with partial coupling of a fraction f. This analysis shows that a metastable region exists for all finite values of f > 0, and therefore explosive synchronization is expected for any perturbation of adaptively coupling added to the standard Kuramoto model. We verify this theory with GPU-accelerated simulations on very large networks (N ∼ 10(6)) and find that, in fact, an explosive transition with hysteresis is observed for all finite couplings. By demonstrating that explosive transitions coexist with standard transitions in the limit of f → 0, we show that this behavior is far more likely to occur naturally than was previously believed.

Vulnerability of Interdependent Networks and Networks of Networks

Networks interact with one another in a variety of ways. Even though increased connectivity between networks would tend to make the system more robust, if dependencies exist between networks, these systems are highly vulnerable to random failure or attack. Damage in one network causes damage in another. This leads to cascading failures which amplify the original damage and can rapidly lead to complete system collapse.