scholarly journals Graph classes with given 3-connected components: Asymptotic enumeration and random graphs

2012 ◽  
Vol 42 (4) ◽  
pp. 438-479 ◽  
Author(s):  
Omer Giménez ◽  
Marc Noy ◽  
Juanjo Rué
Keyword(s):  
Random Graphs ◽  
Connected Components ◽  
Asymptotic Enumeration ◽  
Graph Classes

XIX.—Asymptotic enumeration of connected graphs

1970 ◽  
Vol 68 (4) ◽  
pp. 298-308
Author(s):  
E. M. Wright
Keyword(s):  
Asymptotic Expansion ◽  
Generating Functions ◽  
Binomial Coefficients ◽  
Connected Components ◽  
Asymptotic Enumeration ◽  
Connected Graphs

SynopsisThe number of different connected graphs (with some property P) on n labelled nodes with q edges is fnq. Again Fnq is the number of graphs on n labelled nodes with q edges, each of whose connected components has property P. We consider 8 types of graph for which . We use a known relation between the generating functions of fnq and Fnq to find an asymptotic expansion of fnq in terms of binomial coefficients, valid if (q – ½n log n)/n→∞ as n→∞. This condition is also necessary for the existence of an asymptotic expansion of this kind.

scholarly journals Network dismantling

2016 ◽  
Vol 113 (44) ◽  
pp. 12368-12373 ◽  
Author(s):  
Alfredo Braunstein ◽  
Luca Dall’Asta ◽  
Guilhem Semerjian ◽  
Lenka Zdeborová
Keyword(s):  
Statistical Mechanics ◽  
Large Class ◽  
Random Graph ◽  
Random Graphs ◽  
Degree Distribution ◽  
Connected Components ◽  
Minimal Size ◽  
Epidemic Spreading ◽  
Minimal Set ◽  
Heavy Tailed

We study the network dismantling problem, which consists of determining a minimal set of vertices in which removal leaves the network broken into connected components of subextensive size. For a large class of random graphs, this problem is tightly connected to the decycling problem (the removal of vertices, leaving the graph acyclic). Exploiting this connection and recent works on epidemic spreading, we present precise predictions for the minimal size of a dismantling set in a large random graph with a prescribed (light-tailed) degree distribution. Building on the statistical mechanics perspective, we propose a three-stage Min-Sum algorithm for efficiently dismantling networks, including heavy-tailed ones for which the dismantling and decycling problems are not equivalent. We also provide additional insights into the dismantling problem, concluding that it is an intrinsically collective problem and that optimal dismantling sets cannot be viewed as a collection of individually well-performing nodes.

scholarly journals Limits of multiplicative inhomogeneous random graphs and Lévy trees: limit theorems

2021 ◽  
Author(s):  
Nicolas Broutin ◽  
Thomas Duquesne ◽  
Minmin Wang
Keyword(s):  
Random Graphs ◽  
Metric Spaces ◽  
Companion Paper ◽  
Explicit Construction ◽  
Convergence Condition ◽  
Asymptotic Distributions ◽  
Connected Components ◽  
Inhomogeneous Random Graphs ◽  
The Masses ◽  
The Continuum

AbstractWe consider a natural model of inhomogeneous random graphs that extends the classical Erdős–Rényi graphs and shares a close connection with the multiplicative coalescence, as pointed out by Aldous (Ann Probab 25:812–854, 1997). In this model, the vertices are assigned weights that govern their tendency to form edges. It is by looking at the asymptotic distributions of the masses (sum of the weights) of the connected components of these graphs that Aldous and Limic (Electron J Probab 3:1–59, 1998) have identified the entrance boundary of the multiplicative coalescence, which is intimately related to the excursion lengths of certain Lévy-type processes. We, instead, look at the metric structure of these components and prove their Gromov–Hausdorff–Prokhorov convergence to a class of (random) compact measured metric spaces that have been introduced in a companion paper (Broutin et al. in Limits of multiplicative inhomogeneous random graphs and Lévy trees: the continuum graphs. arXiv:1804.05871, 2020). Our asymptotic regimes relate directly to the general convergence condition appearing in the work of Aldous and Limic. Our techniques provide a unified approach for this general “critical” regime, and relies upon two key ingredients: an encoding of the graph by some Lévy process as well as an embedding of its connected components into Galton–Watson forests. This embedding transfers asymptotically into an embedding of the limit objects into a forest of Lévy trees, which allows us to give an explicit construction of the limit objects from the excursions of the Lévy-type process. The mains results combined with the ones in the other paper allow us to extend and complement several previous results that had been obtained via model- or regime-specific proofs, for instance: the case of Erdős–Rényi random graphs obtained by Addario-Berry et al. (Probab Theory Relat Fields 152:367–406, 2012), the asymptotic homogeneous case as studied by Bhamidi et al. (Probab Theory Relat Fields 169:565–641, 2017), or the power-law case as considered by Bhamidi et al. (Probab Theory Relat Fields 170:387–474, 2018).

scholarly journals Properties of Random Graphs via Boltzmann Samplers

2007 ◽  
Vol DMTCS Proceedings vol. AH,... (Proceedings) ◽  
Author(s):  
Konstantinos Panagiotou ◽  
Andreas Weißl
Keyword(s):  
Random Graphs ◽  
Maximum Degree ◽  
Maximum Size ◽  
Structural Constraints ◽  
Convex Polygons ◽  
Typical Member ◽  
Graph Classes ◽  
International Audience ◽  
Cactus Graphs

International audience This work is devoted to the understanding of properties of random graphs from graph classes with structural constraints. We propose a method that is based on the analysis of the behaviour of Boltzmann sampler algorithms, and may be used to obtain precise estimates for the maximum degree and maximum size of a biconnected block of a "typical'' member of the class in question. We illustrate how our method works on several graph classes, namely dissections and triangulations of convex polygons, embedded trees, and block and cactus graphs.

scholarly journals A Refined View of Causal Graphs and Component Sizes: SP-Closed Graph Classes and Beyond

2013 ◽  
Vol 47 ◽  
pp. 575-611 ◽  
Author(s):  
C. Bäckström ◽  
P. Jonsson
Keyword(s):  
Domain Size ◽  
Additional Restriction ◽  
Connected Components ◽  
Complexity Classes ◽  
Np Hard ◽  
Causal Graph ◽  
Graph Classes ◽  
Causal Graphs ◽  
Np Hardness

The causal graph of a planning instance is an important tool for planning both in practice and in theory. The theoretical studies of causal graphs have largely analysed the computational complexity of planning for instances where the causal graph has a certain structure, often in combination with other parameters like the domain size of the variables. Chen and Giménez ignored even the structure and considered only the size of the weakly connected components. They proved that planning is tractable if the components are bounded by a constant and otherwise intractable. Their intractability result was, however, conditioned by an assumption from parameterised complexity theory that has no known useful relationship with the standard complexity classes. We approach the same problem from the perspective of standard complexity classes, and prove that planning is NP-hard for classes with unbounded components under an additional restriction we refer to as SP-closed. We then argue that most NP-hardness theorems for causal graphs are difficult to apply and, thus, prove a more general result; even if the component sizes grow slowly and the class is not densely populated with graphs, planning still cannot be tractable unless the polynomial hierachy collapses. Both these results still hold when restricted to the class of acyclic causal graphs. We finally give a partial characterization of the borderline between NP-hard and NP-intermediate classes, giving further insight into the problem.

scholarly journals Greed is Good for Deterministic Scale-Free Networks

Algorithmica ◽  
2020 ◽  
Vol 82 (11) ◽  
pp. 3338-3389
Author(s):  
Ankit Chauhan ◽  
Tobias Friedrich ◽  
Ralf Rothenberger
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Power Law ◽  
Real World ◽  
Degree Distribution ◽  
High Probability ◽  
Maximum Independent Set ◽  
Graph Models ◽  
Random Graph Models ◽  
Graph Classes

Abstract Large real-world networks typically follow a power-law degree distribution. To study such networks, numerous random graph models have been proposed. However, real-world networks are not drawn at random. Therefore, Brach et al. (27th symposium on discrete algorithms (SODA), pp 1306–1325, 2016) introduced two natural deterministic conditions: (1) a power-law upper bound on the degree distribution (PLB-U) and (2) power-law neighborhoods, that is, the degree distribution of neighbors of each vertex is also upper bounded by a power law (PLB-N). They showed that many real-world networks satisfy both properties and exploit them to design faster algorithms for a number of classical graph problems. We complement their work by showing that some well-studied random graph models exhibit both of the mentioned PLB properties. PLB-U and PLB-N hold with high probability for Chung–Lu Random Graphs and Geometric Inhomogeneous Random Graphs and almost surely for Hyperbolic Random Graphs. As a consequence, all results of Brach et al. also hold with high probability or almost surely for those random graph classes. In the second part we study three classical $$\textsf {NP}$$ NP -hard optimization problems on PLB networks. It is known that on general graphs with maximum degree $$\Delta$$ Δ , a greedy algorithm, which chooses nodes in the order of their degree, only achieves a $$\Omega (\ln \Delta )$$ Ω ( ln Δ ) -approximation for Minimum Vertex Cover and Minimum Dominating Set, and a $$\Omega (\Delta )$$ Ω ( Δ ) -approximation for Maximum Independent Set. We prove that the PLB-U property with $$\beta >2$$ β > 2 suffices for the greedy approach to achieve a constant-factor approximation for all three problems. We also show that these problems are -hard even if PLB-U, PLB-N, and an additional power-law lower bound on the degree distribution hold. Hence, a PTAS cannot be expected unless = . Furthermore, we prove that all three problems are in if the PLB-U property holds.

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