Random Graphs

Gilbert, E. N.

doi:10.1214/aoms/1177706098

Cited by 1,145 publications

(789 citation statements)

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“…The static version of the model is well studied, as most of its features (e.g., degree distribution, expected number of edges) are already known [32,33]. The temporal version of this model …”

Section: A Erdős-rényi Temporal Modelmentioning

confidence: 99%

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Error and attack vulnerability of temporal networks

2012

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The study of real-world communication systems via complex network models has greatly expanded our understanding on how information flows, even in completely decentralized architectures such as mobile wireless networks. Nonetheless, static network models cannot capture the time-varying aspects and, therefore, various temporal metrics have been introduced. In this paper, we investigate the robustness of time-varying networks under various failures and intelligent attacks. We adopt a methodology to evaluate the impact of such events on the network connectivity by employing temporal metrics in order to select and remove nodes based on how critical they are considered for the network. We also define the temporal robustness range, a new metric that quantifies the disruption caused by an attack strategy to a given temporal network. Our results show that in real-world networks, where some nodes are more dominant than others, temporal connectivity is significantly more affected by intelligent attacks than by random failures. Moreover, different intelligent attack strategies have a similar effect on the robustness: even small subsets of highly connected nodes act as a bottleneck in the temporal information flow, becoming critical weak points of the entire system. Additionally, the same nodes are the most important across a range of different importance metrics, expressing the correlation between highly connected nodes and those that trigger most of the changes in the optimal information spreading. Contrarily, we show that in randomly generated networks, where all the nodes have similar properties, random errors and intelligent attacks exhibit similar behavior. These conclusions may help us in design of more robust systems and fault-tolerant network architectures.

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Section: A Erdős-rényi Temporal Modelmentioning

confidence: 99%

“…For the binomial distribution B(N,p), it is known [32][33][34] that the average degree is (N − 1)p. Hence the average node degree in the Markov temporal model is (N − 1)Pr [ON]. Proof of Lemma 3.…”

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confidence: 99%

Error and attack vulnerability of temporal networks

2012

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“…They are very ubiquitous as null models for studying real-world networks. The first model is the Erdős-Rï¿oenyi G (n, p) [36] also known as the Gilbert model [37], in which a graph with n nodes is constructed by connecting nodes randomly in such a way that each edge is included in G (n, p) with probability p independent from every other edge. The second model was introduced by Barabï¿oesi and Albert [38] on the basis of a preferential attachment process.…”

Section: H Index Of Random Networkmentioning

confidence: 99%

Exploring the “Middle Earth” of network spectra via a Gaussian matrix function

Estrada

Alhomaidhi

Al-Thukair

2017

Chaos: An Interdisciplinary Journal of Nonlinear Science

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We study a Gaussian matrix function of the adjacency matrix of artificial and real-world networks. In particular, we study the Gaussian Estrada index-an index characterizing the importance of eigenvalues close to zero. This index accounts for the information contained in the eigenvalues close to zero in the spectra of networks. Here we obtain bounds for this index in simple graphs, proving that it reaches its maximum for star graphs followed by complete bipartite graphs. We also obtain formulas for the Estrada Gaussian index of Erdős-Rï¿oenyi random graphs as well as for the Barabï¿oesi-Albert graphs. We also show that in real-world networks this index is related to the existence of important structural patterns, such as complete bipartite subgraphs (bicliques). Such bicliques appear naturally in many real-world networks as a consequence of the evolutionary processes giving rise to them. In general, the Gaussian matrix function of the adjacency matrix of networks characterizes important structural information not described in previously used matrix functions of graphs. 2The spectrum of a network-the set of its eigenvalues-provides important information about the structural and dynamical properties of the corresponding system. Most of the functions used to study network spectra give more weight to the largest modular eigenvalues. Then, the information contained in the eigenvalues close to the centre of the spectra, i.e, those close to zero, has remained totally unexplored in the study of graph spectra. Here we study a Gaussian matrix function that gives more weights to the eigenvalues closest to the centre of the spectrum of a network. Using this function we extract important structural information hidden in the spectra of networks, such as emergence of complete bipartite subgraphs (bicliques) which appear naturally in many real-world networks as a consequence of the evolutionary processes giving rise to them. These bicliques are also ubiquitous in random networks generated by preferential attachment mechanisms, such as the Barabï¿oesi-Albert model. In this work we provide a series of analytical results that pave the way for further analysis and uses of this Gaussian matrix function to understand network structure and dynamics.3

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“…In the so-called G(n, p) model, which is also known as the Gilbert model [24], there are n vertices in the graph, and each undirected edge between two vertices i and j is present in the edge set E with probability p, independently of other edges [10]. Because each undirected edge is added with probability p independently, a node degree has a binomial(n−1, p) distribution.…”

Section: Erdös-rényi Random Graphsmentioning

confidence: 99%

A New Random Graph Model with Self-Optimizing Nodes: Connectivity and Diameter

Kabkab

2015

Internet Mathematics

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We introduce a new random graph model. In our model, n, n ≥ 2, vertices choose a subset of potential edges by considering the (estimated) benefits or utilities of the edges. More precisely, each vertex selects k, k ≥ 1, incident edges it wishes to set up, and an undirected edge between two vertices is present in the graph if and only if both of the end vertices choose the edge. First, we examine the scaling law of the smallest k needed for graph connectivity with increasing n and prove that it is (log(n)). Second, we study the diameter of the random graph and demonstrate that, under certain conditions on k, the diameter is close to log(n)/ log(log(n)) with high probability. In addition, as a byproduct of our findings, we show that, for all sufficiently large n, if k > β log(n), where β ≈ 2.4626, there exists a connected Erdös-Rényi random graph that is embedded in our random graph, with high probability.

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Random Graphs

Cited by 1,145 publications

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Error and attack vulnerability of temporal networks

Error and attack vulnerability of temporal networks

Exploring the “Middle Earth” of network spectra via a Gaussian matrix function

A New Random Graph Model with Self-Optimizing Nodes: Connectivity and Diameter

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