Structural transition in interdependent networks with regular interconnections.

In smart power grid, power grid and communication network are connected and mutually dependent. The failure in power grid might cause failures in communication network, and vice versa. A tiny failure in either of them could trigger cascade of failures within the entire system. In this paper, we focus on understanding the structure of smart power grid and studying the underlying network model, their interactions, relationships and how cascading failures occur in the system. We propose a practical model for smart power grid as interdependent complex network. The interdependency between two networks is `oneto-multiple': each node in the communication network has only one support link from the power grid, while each node in power grid is connected to multiple communication nodes. We study the effect of cascading failures using percolation theory, and present detailed mathematical analysis of failure propagation in the system. We analyze the robustness of our model caused by random attacks or failures by calculating the size of functioning parts in both networks. Using simulations, we prove that there exists a threshold for the proportion of faulty nodes, beyond which the system collapses. Also we determine the critical values for different system parameters. To the best of our knowledge, this is the first work that models smart grid as interdependent complex networks and studies its fault tolerance.

Network science has attracted much attention in recent years due to its interdisciplinary applications. We witnessed the revolution of network science in 1998 and 1999 started with small-world and scale-free networks having now thousands of high-profile publications, and it seems that since 2010 studies of ‘network of networks’ (NON), sometimes called multilayer networks or multiplex, have attracted more and more attention. The analytic framework for NON yields a novel percolation law for n interdependent networks that shows that percolation theory of single networks studied extensively in physics and mathematics in the last 50 years is a specific limit of the rich and very different general case of n coupled networks. Since then, properties and dynamics of interdependent and interconnected networks have been studied extensively, and scientists are finding many interesting results and discovering many surprising phenomena. Because most natural and engineered systems are composed of multiple subsystems and layers of connectivity, it is important to consider these features in order to improve our understanding of such complex systems. Now the study of NON has become one of the important directions in network science. In this paper, we review recent studies on the new emerging area—NON. Due to the fast growth of this field, there are many definitions of different types of NON, such as interdependent networks, interconnected networks, multilayered networks, multiplex networks and many others. There exist many datasets that can be represented as NON, such as network of different transportation networks including flight networks, railway networks and road networks, network of ecological networks including species interacting networks and food webs, network of biological networks including gene regulation network, metabolic network and protein–protein interacting network, network of social networks and so on. Among them, many interdependent networks including critical infrastructures are embedded in space, introducing spatial constraints. Thus, we also review the progress on study of spatially embedded networks. As a result of spatial constraints, such interdependent networks exhibit extreme vulnerabilities compared with their non-embedded counterparts. Such studies help us to understand, realize and hopefully mitigate the increasing risk in NON.

General matrices can be split uniquely into Frobenius-orthogonal components: a constant row and column sum (type S) part, a vertex cross sum (type V) part and a weight part. We show that for square matrices, the type S part can be expressed as a sum of squares of type V matrices. We investigate the properties of such decomposition under matrix multiplication, in particular how the pseudoinverses of a matrix relates to the pseudoinverses of its component parts. For invertible matrices, this yields an expression for the inverse where only the type S part needs to be (pseudo)inverted; in the example of the Wilson matrix, this component is considerably better conditioned than the whole matrix. We also show a relation between matrix determinants and the weight of their matrix inverses and give a simple proof for Frobenius-optimal approximations with the constant row and column sum and the vertex cross sum properties, respectively, to a given matrix.

Universal spectral properties of multiplex networks allow us to assess the nature of the transition between disease-free and endemic phases in the SIS epidemic spreading model. In a multiplex network, depending on a coupling parameter, p, the inverse participation ratio (IPR) of the leading eigenvector of the adjacency matrix can be in two different structural regimes: (i) layer-localized and (ii) delocalized. Here we formalize the structural transition point, p*, between these two regimes, showing that there are universal properties regarding both the layer size n and the layer configurations. Namely, we show that IPR ∼ n −δ , with δ ≈ 1, and revealed an approximately linear relationship between p* and the difference between the layers’ average degrees. Furthermore, we showed that this multiplex structural transition is intrinsically connected with the nature of the SIS phase transition, allowing us to both understand and quantify the phenomenon. As these results are related to the universal properties of the leading eigenvector, we expect that our findings might be relevant to other dynamical processes in complex networks.

Network of networks: A bibliometric analysis

Considering the real-world scenarios that there are interactions between edges in different networks and each network has different topological structure and size, we introduce a model of interdependent networks with arbitrary edge-coupling strength, in which q A and q B are used to represent the edge-coupling strength of network A and network B respectively. A mathematical framework using generating functions is developed based on self-consistent probabilities approach, which is verified by computer simulations. In particular, we carry out this mathematical framework on the Erdös–Rényi edge-coupled interdependent networks to calculate the values of phase transition thresholds and the critical coupling strengths which distinguish different types of transitions. Moreover, as contrast to the corresponding node-coupled interdependent networks, we find that for edge-coupled interdependent networks the critical coupling strengths are smaller, and the critical thresholds as well, which means the robustness of partially edge-coupled interdependent networks is better than that of partially node-coupled interdependent networks. Furthermore, we find that network A will have hybrid percolation behaviors as long as the coupling strength q A belongs to a certain range, and the range does not affected by average degree of network A. Our findings may fill the gap of understanding the robustness of edge-coupled interdependent networks with arbitrary coupling strength, and have significant meaning for network security design and optimization.

The maximum absolute value of the determinant of n x n nonsinguiar (0,1) matrices that have constant line sums (i.e., row sums and column sums) k is investigated. For n≠4k=2, this maximum determinant is determined to be 2t if n=3t or 3t+2, and 2t−1 if n=3t+1. Restriction to a subset of these matrices, namely those that are symmetric and have zero trace (their graphs are regular of degree k), leaves the maximum unchanged for k=2. For this restricted class, when n≥7k=n−3, the maximum absolute value of the determinant is (n−3)3[n/4]−-1, This maximum gives a lower bound for the maximum absolute value of the determinant for the larger class, but in general this bound is not tight. Other determinantal values and bounds for specific n and k are given.

Understanding the correlation between the underlie network structure and overlay cascade effects in the interdependent networks is one of major challenges in complex network studies. There are some existing metrics that can be used to measure the cascades. However, different metrics such as average node degree interpret different characteristic of network topological structure, especially less metrics have been identified to effectively measure the cascading performance in the interdependent networks. In this paper, we propose to use a combined Laplacian matrix to model the interdependent networks and their interconnectivity, and then use its effective resistance metric as an indicator to its cascading behavior. Moreover, we have conducted extensive comparative studies among different metrics such as average node degree, and the proposed effective resistance. We have found that the effective resistance metric can describe more accurate and finer characteristics of topological structure of the interdependent networks than average node degree which is widely adapted by the existing research studies for measuring the cascading performance in the interdependent networks.

It is shown that for real,m x n matricesA andB the system of matrix equationsAX=B, BY=A is solvable forX andY doubly stochastic if and only ifA=BP for some permutation matrixP. This result is then used to derive other equations and to characterize the Green’s relations on the semigroup Ω n of alln x n doubly stochastic matrices. The regular matrices in Ω n are characterized in several ways by use of the Moore-Penrose generalized inverse. It is shown that a regular matrix in Ω n is orthostochastic and that it is unitarily similar to a diagnonal matrix if and only if it belongs to a subgroup of Ω n . The paper is concluded with extensions of some of these results to the convex setS n of alln x n nonnegative matrices having row and column sums at most one.

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 .

In this note, we show that the norm of an $n\times n$ random jointly exchangeable matrix with zero diagonal can be estimated in terms of the norm of its $n/2\times n/2$ submatrix located in the top right corner. As a consequence, we prove a relation between the second largest singular values of a random matrix with constant row and column sums and its top right $n/2\times n/2$ submatrix. The result has an application to estimating the spectral gap of random undirected $d$-regular graphs in terms of the second singular value of {\it directed} random graphs with predefined degree sequences.

Who influences whom and how in river-basin governance?A participatory stakeholder and social network analysis in Zayandeh-Rud basin, Iran

The benchmark consensus protocol is used by the most notable distributed algorithms for controlling multiagent systems through local interaction rules, where it yields to the well-known Laplacian matrix whose null space is fixed and is spanned by the vector of ones provided that the underlying graph is connected. Since this protocol is the key building block for a wide range of other existing distributed algorithms, its extensions are likewise based on this Laplacian matrix. However, the fixed nature of the null space of this Laplacian matrix limits the composition of complex (i.e. reconfigurable) cooperative behaviours for multiagent systems. Motivated by this standpoint, the contribution of this paper is threefold: (i) For undirected and connected graphs, we introduce a new Laplacian matrix, whose null space is spanned by a user-assigned vector composed of nonzero elements. We give its mathematical definition and show that it inherits key properties of the aforementioned well-known Laplacian matrix. (ii) We next present distributed control architectures for convergence to the desired null space as well as for convergence to a specific vector within that null space. (iii) We finally multiplex information networks to distributively control the null space of this new Laplacian matrix, where this is the most important contribution of this paper. We specifically show that the resulting null space can vary through local interaction rules as desired between the spans of two or more user-assigned vectors to pave the way for composing complex cooperative behaviours for multiagent systems.

Traffic demand prediction using a social multiplex networks representation on a multimodal and multisource dataset

A detailed understanding of cellular mechanisms requires knowledge of structure and dynamics of the involved biomacromolecules at atomic resolution. NMR spectroscopy uniquely allows determination of static and dynamic processes at atomic level, including structured states often represented by a single state as well as by unstructured conformational ensembles. While a high-resolution description of structured states may also be obtained by other techniques, the characterization of structural transitions occurring during biomolecular folding is only feasible exploiting NMR spectroscopic methods. The NMR methodical strategy includes the fast initiation of a folding reaction in situ and the possibility to detect the induced process with sufficient time resolution on the respective NMR time scale. In the case of ligand-induced structural transitions of RNA, the initiation of the folding reaction can be achieved by laser-triggered deprotection of a photolabile caged ligand whose release induces folding of a riboswitch RNA. The strategy discussed here is general and can also be transferred to other biological processes, where at least one key reagent or substrate, e.g., ions, ligands, pH, or one specific conformational state, can be photochemically caged. The rates of reversible and irreversible reactions or structural transitions that can be covered by real-time NMR methods range from milliseconds up to hours.In this chapter, we discuss the application of a time-resolved NMR strategy to resolve the ligand-induced folding of the guanine-sensing riboswitch aptamer domain of the B. subtilis xpt-pbuX operon.