On the Robustness of Complex Networks by Using the Algebraic Connectivity

Jamakovic, Almerima; Mieghem, Piet Van

doi:10.1007/978-3-540-79549-0_16

Cited by 85 publications

(61 citation statements)

References 22 publications

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“…As is well known, the algebraic connectivity represents a lower bound for both the edge connectivity and node connectivity of a graph (that is, respectively the minimal number of edges or nodes that should be removed to disconnect the graph) 17 . Indeed, the algebraic connectivity of a graph is often used as a control parameter to make the graph more resilient to random failures of its nodes or edges 29 . Thus, the lower bound of equation (9) represents also a lower bound for the critical percolation threshold measured in ref.…”

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confidence: 99%

Abrupt transition in the structural formation of interconnected networks

2013

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Our world is linked by a complex mesh of networks through which information, people and goods flow. These networks are interdependent on each other, and present structural and dynamical features 1-6 different from those observed in isolated networks 7-9 . Although examples of such dissimilar properties are becoming more abundant-such as in diffusion, robustness and competition-it is not yet clear where these differences are rooted. Here we show that the process of building independent networks into an interconnected network of networks undergoes a structurally sharp transition as the interconnections are formed. Depending on the relative importance of inter-and intra-layer connections, we find that the entire interdependent system can be tuned between two regimes: in one regime, the various layers are structurally decoupled and they act as independent entities; in the other regime, network layers are indistinguishable and the whole system behaves as a singlelevel network. We analytically show that the transition between the two regimes is discontinuous even for finite-size networks. Thus, any real-world interconnected system is potentially at risk of abrupt changes in its structure, which may manifest new dynamical properties.Interacting, interdependent or multiplex networks are different ways of naming the same class of complex systems where networks are not considered as isolated entities but interacting with each other. In multiplex, the nodes at each network are instances of the same entity; thus, the networks are representing simply different categorical relationships between entities, and usually categories are represented by layers. Interdependent networks is a more general framework where nodes can be different at each network.Many, if not all, real networks are coupled with other real networks. Examples can be found in several domains: social networks (for example, Facebook, Twitter and so on) are coupled because they share the same actors 10 ; multimodal transportation networks are composed of different layers (for example, bus, subway and so on) that share the same locations 11 ; the functioning of communication and power grid systems depends one on the other 1 . So far, all phenomena that have been studied on interdependent networks, including percolation 1,3 , epidemics 4 and linear dynamical systems 5 , have provided results that differ much from those valid in the case of isolated complex networks. Sometimes the difference is radical: for example, whereas isolated scale-free networks are robust against failures of their nodes or edges 12 , scale-free interdependent networks are instead very fragile 1,3 .Given such observations, two fundamentally important theoretical questions are in order: Why do dynamical and critical phenomena running on interdependent network models differ

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confidence: 99%

Abrupt transition in the structural formation of interconnected networks

2013

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“…We show these results for G 1 using a conductance argument and for both models using experimental calculation of the resultant spectral gap of the normalized Laplacian, which is a normalized measure of algebraic connectivity [23]. More recently, algebraic connectivity has been noted by network scientists to be an intrinsic measure of the robustness of a complex network to node and link failures [24], thus giving even stronger motivation for our present study.…”

Section: Introductionmentioning

confidence: 68%

More Benefits of Adding Sparse Random Links to Wireless Networks: Yet Another Case for Hybrid Networks

Ercal

2012

International Journal of Distributed Sensor Networks

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We theoretically and experimentally analyze the process of adding sparse random links to random wireless networks modeled as a random geometric graph. While this process has been previously proposed, we are the first to prove theoretical bounds on the improvement to the graph diameter and random walk properties of the resulting graph as a function of the frequency of wires used, where this frequency is diminishingly small. In particular, given a parameter k controlling sparsity, any node has a probability of 1/k 2 nr 2 for being a wired link station. Amongst the wired link stations, we consider creating a random 3-regular graph superimposed upon the random wireless network to create model G 1 , and alternatively we consider a sparser model G 2 as well, which is a random 1-out graph of the wired links superimposed upon the random wireless network. We prove that the diameter for G 1 is O k + log(n) with high probability and the diameter for G 2 is O k log(n) with high probability, both of which exponentially improve the Θ( n/logn) diameter of the random geometric graph around the connectivity threshold, thus also inducing small-world characteristics as the high clustering remains unchanged. Further, we theoretically demonstrate that as long as k is polylogarithmic in the network size, G 1 has rapidly mixing random walks with high probability, which also exponentially improves upon the mixing time of the purely wireless random geometric graph, which yields direct improvement to the performance of distributed gossip algorithms as well as normalized edge connectivity. Finally, we experimentally confirm that the algebraic connectivities of both G 1 and G 2 exhibit significant asymptotic improvement over that of the underlying random geometric graph. These results further motivate future hybrid networks and advances in the use of directional antennas.

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“…Jamakovic and Mieghem proposed to use the second smallest eigenvalue of the Laplacian matrix also known as algebraic connectivity to measure network robustness [18,14]. Malliaros, et al described the relationship between algebraic connectivity and node/edge connectivities [22].…”

Section: Robustnessmentioning

confidence: 99%

R -energy for evaluating robustness of dynamic networks

Gao

Lim

2013

Proceedings of the 5th Annual ACM Web Science Conference

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The robustness of a network is determined by how well its vertices are connected to one another so as to keep the network strong and sustainable. As the network evolves, its robustness changes and may reveal events as well as periodic trend patterns that affect the interactions among users in the network. In this paper, we develop R-energy as a new measure of network robustness based on the spectral analysis of normalized Laplacian matrix. R-energy can cope with disconnected networks, and is efficient to compute with a time complexity of O(|V | + |E|) where V and E are the vertex set and edge set of the network respectively. This makes Renergy more efficient to compute than algebraic connectivity, another well known network robustness measure. Our experiments also show that removal of high degree vertices reduces network robustness (measured by R-energy) more than that of random or small degree vertices. R-energy can scale well for very large networks. It takes as little as 40 seconds to compute for a network with about 5M vertices and 69M edges. We can further detect events occurring in a dynamic Twitter network with about 130K users and discover interesting weekly tweeting trends by tracking changes to R-energy.

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On the Robustness of Complex Networks by Using the Algebraic Connectivity

Cited by 85 publications

References 22 publications

Abrupt transition in the structural formation of interconnected networks

Abrupt transition in the structural formation of interconnected networks

More Benefits of Adding Sparse Random Links to Wireless Networks: Yet Another Case for Hybrid Networks

R -energy for evaluating robustness of dynamic networks

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