An example of graph limits of growing sequences of random graphs

Janson, Svante; Severini, Simone

doi:10.4310/joc.2013.v4.n1.a3

Cited by 4 publications

(6 citation statements)

References 18 publications

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“…Our main object of study will be the construction given in the following definition. This is a special case of a construction already presented in [2].…”

Section: Graph Likelihoodmentioning

confidence: 93%

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A notion of graph likelihood and an infinite monkey theorem

Banerji¹,

Mansour²,

Severini³

2013

J. Phys. A: Math. Theor.

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We play with a graph-theoretic analogue of the folklore infinite monkey theorem. We define a notion of graph likelihood as the probability that a given graph is constructed by a monkey in a number of time steps equal to the number of vertices. We present an algorithm to compute this graph invariant and closed formulas for some infinite classes. We have to leave the computational complexity of the likelihood as an open problem.

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“…Our main object of study will be the construction given in the following definition. This is a special case of a construction already presented in [2].…”

Section: Graph Likelihoodmentioning

confidence: 93%

“…In this note, we consider an infinite monkey theorem, but for graphs rather than strings. Our setting involves a device which performs "non-preferential" attachment [2]. At time step t + 1, a new vertex is added to a graph G tthe process starts from the single vertex graph, G 1 .…”

Section: Introductionmentioning

confidence: 99%

A notion of graph likelihood and an infinite monkey theorem

Banerji¹,

Mansour²,

Severini³

2013

J. Phys. A: Math. Theor.

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“…The double random process is when we choose even the sequence at random, choosing d i with some distribution from the interval [0, i]. Note that Janson and Severini [11] study the double random process from a different point of view. The following corollary states that, in some sense, almost all double random processes will result in the Rado graph.…”

Section: Proof Of Corollary 24mentioning

confidence: 99%

“…The Janson-Severini process. Janson and Severini [11] introduced a process that also includes ours. Their construction is the following.…”

Section: Introductionmentioning

confidence: 99%

Generating Infinite Random Graphs

Bíró

Darji

2018

Proceedings of the Edinburgh Mathematical Society

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We define a growing model of random graphs. Given a sequence of nonnegative integers {dn} ∞ n=0 with the property that d i ≤ i, we construct a random graph on countably infinitely many vertices v 0 , v 1 . . . by the following process: vertex v i is connected to a subset of {v 0 , . . . , v i−1 } of cardinality d i chosen uniformly at random. We study the resulting probability space. In particular, we give a new characterization of random graph and we also give probabilistic methods for constructing infinite random trees.

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“…Another extension of the model could be in the same spirit of Ref. [47]. The probability of attachment is determined by the outcome of a process external to the network.…”

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confidence: 99%

Co-evolution of networks and quantum dynamics: a generalization of preferential attachment

et al. 2013

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We propose a model of network growth in which the network is co-evolving together with the dynamics of a quantum mechanical system, namely a quantum walk taking place over the network. The model naturally generalizes the Barabási-Albert model of preferential attachment and it has a rich set of tunable parameters, such as the initial conditions of the dynamics or the interaction of the system with its environment. We show that the model produces networks with two-modal power-law degree distributions, super-hubs, finite clustering coefficient, small-world behaviour and non-trivial degree-degree correlations. PACS numbers: 89.75.Hc, 03.67.-a, 89.75.-kModels of graph growth are important for studying and simulating the behaviour of a large variety of real-world phenomena and for the in silico construction of networks with given properties. Growth processes occur ubiquitously in social systems, technology, and nature. See, for example, Ref. [1][2][3] for overviews on complex networks growth. Among the most extensively studied models of network growth are those based on preferential attachment -or "the rich get richer"scheme -together with its many variations [4]. Predating networks science, the basic idea behind preferential attachment goes back to the 1920s and the work of the statistician Yule [5]. The set-up usually consists of two ingredients: an iterative process in which new nodes are sequentially added to an existing graph; and a mechanism for choosing the neighbours of newly arrived nodes. Only when the preference on the neighbours is a linear function of the degrees of the nodes, then the degree distribution of the growing graph turns out to be a power-law, as those observed in the majority of real-world complex networks. The literature contains many variants of preferential attachment, respectively defined by local

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An example of graph limits of growing sequences of random graphs

Cited by 4 publications

References 18 publications

A notion of graph likelihood and an infinite monkey theorem

A notion of graph likelihood and an infinite monkey theorem

Generating Infinite Random Graphs

Co-evolution of networks and quantum dynamics: a generalization of preferential attachment

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