Random Graphs

Bollobás, Béla

doi:10.1017/cbo9780511814068

Cited by 3,395 publications

(3,142 citation statements)

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“…All the analytical results will concern only very large sparse graphs (N → ∞). Provided the second and higher moments of Q(k) do not diverge, such graphs are locally tree-like in this limit [4,5]. More precisely, call a d-neighborhood of a node i the set of nodes which are at distance at most d from i.…”

Section: B Ensembles Of Random Graphsmentioning

confidence: 99%

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Phase transitions in the coloring of random graphs

2007

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We consider the problem of coloring the vertices of a large sparse random graph with a given number of colors so that no adjacent vertices have the same color. Using the cavity method, we present a detailed and systematic analytical study of the space of proper colorings (solutions).We show that for a fixed number of colors and as the average vertex degree (number of constraints) increases, the set of solutions undergoes several phase transitions similar to those observed in the mean field theory of glasses. First, at the clustering transition, the entropically dominant part of the phase space decomposes into an exponential number of pure states so that beyond this transition a uniform sampling of solutions becomes hard. Afterward, the space of solutions condenses over a finite number of the largest states and consequently the total entropy of solutions becomes smaller than the annealed one. Another transition takes place when in all the entropically dominant states a finite fraction of nodes freezes so that each of these nodes is allowed a single color in all the solutions inside the state. Eventually, above the coloring threshold, no more solutions are available. We compute all the critical connectivities for Erdős-Rényi and regular random graphs and determine their asymptotic values for large number of colors.Finally, we discuss the algorithmic consequences of our findings. We argue that the onset of computational hardness is not associated with the clustering transition and we suggest instead that the freezing transition might be the relevant phenomenon. We also discuss the performance of a simple local Walk-COL algorithm and of the belief propagation algorithm in the light of our results.

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Section: B Ensembles Of Random Graphsmentioning

confidence: 99%

“…In this paper, we study colorings of sparse random graphs [4,5]. Random graphs are one of the most fundamental source of challenging problems in graph theory since the seminal work of Erdős and Rényi [6] in 1959.…”

Section: Introductionmentioning

confidence: 99%

Phase transitions in the coloring of random graphs

2007

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“…The most natural sparse generalization of the complete graph in mathematics is provided by the so-called Erdös-Rényi graphs [21], which in the fifties of the past century defined the concept of a random graph. In modern times, this idea has become redefined as a random network and it has become an important paradigm with many applications.…”

Section: A Cellular Automata Network Model Of the Brainmentioning

confidence: 99%

Firing patterns in a random network cellular automata model of the brain

Acedo¹,

Lamprianidou²,

Moraño³

et al. 2015

Physica A: Statistical Mechanics and its Applications

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One of the main challenges in the simulation of even reduced areas of the brain is the presence of a large number of neurons and a large number of connections among them. Even from a theoretical point of view, the behaviour of dynamical models of complex networks with high connectivity is unknown, precisely because the cost of computation is still unaffordable and it will likely be in the near future. In this paper we discuss the simulation of a cellular automata network model of the brain including up to one million sites with a maximum average of three hundred connections per neuron. This level of connectivity was achieved thanks to a distributed computing environment based on the BOINC (Berkeley Open Infrastructure for Network Computing) platform. Moreover, in this work we consider the interplay among excitatory neurons (which induce the excitation of their neighbours) and inhibitory neurons (which prevent resting neurons from firing and induce firing neurons to pass to the refractory state). Our objective is to classify the normal (noisy but asymptotically constant patterns) and the abnormal (high oscillations with spindle-like behaviour) patterns of activity in the model brain and their stability and parameter * e-mail: luiacrod@imm.upv.es 1 ranges in order to determine the role of excitatory and inhibitory compensatory effects in healthy and diseased individuals.

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“…Unfortunately, vertices having self-loops, as well as multiple edges between vertices, may occur, so that the CM is a multigraph. However, self-loops are scarce when N → ∞ (see e.g., [3] or [8]). …”

Section: The Configuration Modelmentioning

confidence: 99%

“…The above model is a variant of the configuration model [3], which, given a degree sequence, is the random graph with that given degree sequence. The degree sequence of a graph is the vector of which the k th coordinate equals the fraction of vertices with degree k. In our model, by the law of large numbers, the degree sequence is close to the distribution of the nodal degree D of which…”

Section: The Configuration Modelmentioning

confidence: 99%

A Phase Transition for the Diameter of the Configuration Model

Hofstad¹,

Hooghiemstra²,

Znamenski³

2007

Internet Mathematics

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In this paper, we study the configuration model (CM) with i.i.d. degrees. We establish a phase transition for the diameter when the power-law exponent τ of the degrees satisfies τ ∈ (2, 3). Indeed, we show that for τ > 2 and when vertices with degree 1 or 2 are present with positive probability, the diameter of the random graph is, with high probability, bounded from below by a constant times the logarithm of the size of the graph. On the other hand, assuming that all degrees are 3 or more, we show that, for τ ∈ (2, 3), the diameter of the graph is, with high probability, bounded from above by a constant times the log log of the size of the graph.

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Random Graphs

Cited by 3,395 publications

References 0 publications

Phase transitions in the coloring of random graphs

Phase transitions in the coloring of random graphs

Firing patterns in a random network cellular automata model of the brain

A Phase Transition for the Diameter of the Configuration Model

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