Extreme degrees in random graphs

Pałka, Zbigniew

doi:10.1002/jgt.3190110202

Cited by 6 publications

(3 citation statements)

References 13 publications

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“…In particular he analysed the span between the minimum and the maximum degree of sparse G n,p . Similar results were obtained independently by Bollobás [130] (see also Palka [640]). Bollobás [132] answered the question for what values of p(n), G n,p w.h.p.…”

Section: Notessupporting

confidence: 89%

Introduction to Random Graphs

2015

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From social networks such as Facebook, the World Wide Web and the Internet, to the complex interactions between proteins in the cells of our bodies, we constantly face the challenge of understanding the structure and development of networks. The theory of random graphs provides a framework for this understanding, and in this book the authors give a gentle introduction to the basic tools for understanding and applying the theory. Part I includes sufficient material, including exercises, for a one semester course at the advanced undergraduate or beginning graduate level. The reader is then well prepared for the more advanced topics in Parts II and III. A final part provides a quick introduction to the background material needed. All those interested in discrete mathematics, computer science or applied probability and their applications will find this an ideal introduction to the subject.

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Section: Notessupporting

confidence: 89%

Introduction to Random Graphs

2015

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“…In the other two probability models on bipartite graphs, G p and G k , two types of results are known: those on the minimum and maximum degrees [5,7,36] and those on the number of vertices with a given degree [23,33,34]. For results in the digraph counterpart G p see [37] (and below).…”

Section: Historical Notesmentioning

confidence: 99%

Degree sequences of random digraphs and bipartite graphs

McKay

Skerman²

2016

Journal of Combinatorics

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We investigate the joint distribution of the vertex degrees in three models of random bipartite graphs. Namely, we can choose each edge with a specified probability, choose a specified number of edges, or specify the vertex degrees in one of the two colour classes.This problem can alternatively be described in terms of the row and sum columns of random binary matrix or the in-degrees and out-degrees of a random digraph, in which case we can optionally forbid loops. It can also be cast as a problem in random hypergraphs, or as a classical occupancy, allocation, or coupon collection problem.In each case, provided the two colour classes are not too different in size nor the number of edges too low, we define a probability space based on independent binomial variables and show that its probability masses asymptotically equal those of the degrees in the graph model almost everywhere. The accuracy is sufficient to asymptotically determine the expectation of any joint function of the degrees whose maximum is at most polynomially greater than its expectation.

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“…2 we exhibit the relative size n (k,L) as well as the normalized number of edges l (k,L) for the G(k, L)-core for varied degree parameter k and multi-hop parameter L in ER networks. In the calculation of rate equation, we set q max = 40 as P (q max ) is less than n −1 in all our networks (In fact, the maximum degree of such an ER network is only around (ln ln n) −1 ln n with high probability [31].) Several interesting observations are as follows.…”

Section: Erdős-rényi Networkmentioning

confidence: 99%

Multi-Hop Generalized Core Percolation on Complex Networks

Shang

2020

Advs. Complex Syst.

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Recent theoretical studies on network robustness have focused primarily on attacks by random selection and global vision, but numerous real-life networks suffer from proximity-based breakdown. Here we introduce the multi-hop generalized core percolation on complex networks, where nodes with degree less than [Formula: see text] and their neighbors within [Formula: see text]-hop distance are removed progressively from the network. The resulting subgraph is referred to as [Formula: see text]-core, extending the recently proposed [Formula: see text]-core and classical core of a network. We develop analytical frameworks based upon generating function formalism and rate equation method, showing for instance continuous phase transition for [Formula: see text]-core and discontinuous phase transition for [Formula: see text]-core with any other combination of [Formula: see text] and [Formula: see text]. We test our theoretical results on synthetic homogeneous and heterogeneous networks, as well as on a selection of large-scale real-world networks. This unravels, e.g., a unique crossover phenomenon rooted in heterogeneous networks, which raises a caution that endeavor to promote network-level robustness could backfire when multi-hop tracing is involved.

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Extreme degrees in random graphs

Cited by 6 publications

References 13 publications

Introduction to Random Graphs

Introduction to Random Graphs

Degree sequences of random digraphs and bipartite graphs

Multi-Hop Generalized Core Percolation on Complex Networks

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