Abstract-Communication networks are vulnerable to natural disasters, such as earthquakes or floods, as well as to physical attacks, such as an Electromagnetic Pulse (EMP) attack. Such realworld events happen in specific geographical locations and disrupt specific parts of the network. Therefore, the geographical layout of the network determines the impact of such events on the network's connectivity. In this paper, we focus on assessing the vulnerability of (geographical) networks to such disasters. In particular, we aim to identify the most vulnerable parts of the network. That is, the locations of disasters that would have the maximum disruptive effect on the network in terms of capacity and connectivity. We consider graph models in which nodes and links are geographically located on a plane, and model the disaster event as a line segment or a circular cut. We develop algorithms that find a worstcase line segment cut and a worst-case circular cut. Then, we obtain numerical results for a specific backbone network, thereby demonstrating the applicability of our algorithms to real-world networks. Our novel approach provides a promising new direction for network design to avert geographical disasters or attacks.

Abstract-Communication networks are vulnerable to natural disasters, such as earthquakes or floods, as well as to physical attacks, such as an Electromagnetic Pulse (EMP) attack. Such realworld events happen in specific geographical locations and disrupt specific parts of the network. Therefore, the geographical layout of the network determines the impact of such events on the network's connectivity. In this paper, we focus on assessing the vulnerability of (geographical) networks to such disasters. In particular, we aim to identify the most vulnerable parts of the network. That is, the locations of disasters that would have the maximum disruptive effect on the network in terms of capacity and connectivity. We consider graph models in which nodes and links are geographically located on a plane, and model the disaster event as a line segment or a circular cut. We develop algorithms that find a worstcase line segment cut and a worst-case circular cut. Then, we obtain numerical results for a specific backbone network, thereby demonstrating the applicability of our algorithms to real-world networks. Our novel approach provides a promising new direction for network design to avert geographical disasters or attacks.

Abstract-Fiber-optic networks are vulnerable to natural disasters, such as tornadoes or earthquakes, as well as to physical failures, such as an anchor cutting underwater fiber cables. Such real-world events occur in specific geographical locations and disrupt specific parts of the network. Therefore, the geography of the network determines the effect of physical events on the network's connectivity and capacity.In this paper, we develop tools to analyze network failures after a 'random' geographic disaster. The random location of the disaster allows us to model situations where the physical failures are not targeted attacks. In particular, we consider disasters that take the form of a 'random' line in a plane. Using results from geometric probability, we are able to calculate some network performance metrics to such a disaster in polynomial time. In particular, we can evaluate average two-terminal reliability in polynomial time under 'random' line-cuts. This is in contrast to the case of independent link failures for which there exists no known polynomial time algorithm to calculate this reliability metric. We also present some numerical results to show the significance of geometry on the survivability of the network and discuss network design in the context of random line-cuts. Our novel approach provides a promising new direction for modeling and designing networks to lessen the effects of geographical disasters or attacks.

Abstract-Communication networks are vulnerable to natural disasters, such as earthquakes or floods, as well as to human attacks, such as an electromagnetic pulse (EMP) attack. Such real-world events have geographical locations, and therefore, the geographical structure of the network graph affects the impact of these events. In this paper we focus on assessing the vulnerability of (geographical) networks to such disasters. In particular, we aim to identify the location of a disaster that would have the maximum effect on network capacity. We consider a geometric graph model in which nodes and links are geographically located on a plane. Specifically, we model the physical network as a bipartite graph (in the topological and geographical sense) and consider the set of all vertical line segment cuts. For that model, we develop a polynomial time algorithm for finding a worst possible cut. Our approach has the potential to be extended to general graphs and provides a promising new direction for network design to avert geographical disasters or attacks.

Abstract-Failures in fiber-optic networks may be caused by natural disasters, such as floods or earthquakes, as well as other events, such as an Electromagnetic Pulse (EMP) attack. These events occur in specific geographical locations, therefore the geography of the network determines the effect of failure events on the network's connectivity and capacity.In this paper we consider a generalization of the mincut and max-flow problems under a geographic failure model. Specifically, we consider the problem of finding the minimum number of failures, modeled as circular disks, to disconnect a pair of nodes and the maximum number of failure disjoint paths between pairs of nodes. This model applies to the scenario where an adversary is attacking the network multiple times with intention to reduce its connectivity. We present a polynomial time algorithm to solve the geographic min-cut problem and develop an ILP formulation, an exact algorithm, and a heuristic algorithm for the geographic max-flow problem.

The aim of this paper is twofold. First, we show that a certain concatenation of a proximity operator with an affine operator is again a proximity operator on a suitable Hilbert space. Second, we use our findings to establish so-called proximal neural networks (PNNs) and stable tight frame proximal neural networks. Let $$\mathcal {H}$$ H and $$\mathcal {K}$$ K be real Hilbert spaces, $$b \in \mathcal {K}$$ b ∈ K and $$T \in \mathcal {B} (\mathcal {H},\mathcal {K})$$ T ∈ B ( H , K ) a linear operator with closed range and Moore–Penrose inverse $$T^\dagger $$ T † . Based on the well-known characterization of proximity operators by Moreau, we prove that for any proximity operator $$\mathrm {Prox}:\mathcal {K}\rightarrow \mathcal {K}$$ Prox : K → K the operator $$T^\dagger \, \mathrm {Prox}( T \cdot + b)$$ T † Prox ( T · + b ) is a proximity operator on $$\mathcal {H}$$ H equipped with a suitable norm. In particular, it follows for the frequently applied soft shrinkage operator $$\mathrm {Prox}= S_{\lambda }:\ell _2 \rightarrow \ell _2$$ Prox = S λ : ℓ 2 → ℓ 2 and any frame analysis operator $$T:\mathcal {H}\rightarrow \ell _2$$ T : H → ℓ 2 that the frame shrinkage operator $$T^\dagger \, S_\lambda \, T$$ T † S λ T is a proximity operator on a suitable Hilbert space. The concatenation of proximity operators on $$\mathbb R^d$$ R d equipped with different norms establishes a PNN. If the network arises from tight frame analysis or synthesis operators, then it forms an averaged operator. In particular, it has Lipschitz constant 1 and belongs to the class of so-called Lipschitz networks, which were recently applied to defend against adversarial attacks. Moreover, due to its averaging property, PNNs can be used within so-called Plug-and-Play algorithms with convergence guarantee. In case of Parseval frames, we call the networks Parseval proximal neural networks (PPNNs). Then, the involved linear operators are in a Stiefel manifold and corresponding minimization methods can be applied for training of such networks. Finally, some proof-of-the concept examples demonstrate the performance of PPNNs.

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Continuous image morphing is a classical task in image processing. The metamorphosis model proposed by Trouvé, Younes and coworkers [39,53] casts this problem in the frame of Riemannian geometry and geodesic paths between images. The associated metric in the space of images incorporates dissipation caused by a viscous flow transporting image intensities and its variations along motion paths. In many applications, images are maps from the image domain into a manifold (e.g. in DTI imaging the manifold of symmetric positive definite matrices with a suitable Riemannian metric). In this paper, we propose a generalized metamorphosis model for manifold-valued images, where the range space is a finite-dimensional Hadamard manifold. A corresponding time discrete version was presented in [42] based on the general variational time discretization proposed in [13]. Here, we prove the Mosco-convergence of the time discrete metamorphosis functional to the proposed manifold-valued metamorphosis model, which implies the convergence of time discrete geodesic paths to a geodesic path in the (time continuous) metamorphosis model. In particular, the existence of geodesic paths is established. In fact, images as maps into Hadamard manifold are not only relevant in applications, but it is also shown that the joint convexity of the distance function -which characterizes Hadamard manifolds -is a crucial ingredient to establish existence of the metamorphosis model.