Giant components in three-parameter random directed graphs

1992 ◽  
Vol 24 (4) ◽  
pp. 845-857 ◽  
Author(s):  
Tomasz Łuczak ◽  
Joel E. Cohen
Keyword(s):  
Phase Transition ◽  
Food Webs ◽  
Directed Graph ◽  
Directed Graphs ◽  
Giant Component ◽  
Critical Surface ◽  
Connected Component ◽  
Strongly Connected Component ◽  
Strongly Connected ◽  
Asymptotic Probability

A three-parameter model of a random directed graph (digraph) is specified by the probability of ‘up arrows' from vertexito vertexjwherei < j, the probability of ‘down arrows' fromitojwherei ≥ j,and the probability of bidirectional arrows betweeniandj.In this model, a phase transition—the abrupt appearance of a giant strongly connected component—takes place as the parameters cross a critical surface. The critical surface is determined explicitly. Before the giant component appears, almost surely all non-trivial components are small cycles. The asymptotic probability that the digraph contains no cycles of length 3 or more is computed explicitly. This model and its analysis are motivated by the theory of food webs in ecology.

Giant components in three-parameter random directed graphs

1992 ◽  
Vol 24 (04) ◽  
pp. 845-857 ◽  
Author(s):  
Tomasz Łuczak ◽  
Joel E. Cohen
Keyword(s):  
Phase Transition ◽  
Food Webs ◽  
Directed Graph ◽  
Directed Graphs ◽  
Giant Component ◽  
Critical Surface ◽  
Connected Component ◽  
Strongly Connected Component ◽  
Strongly Connected ◽  
Asymptotic Probability

A three-parameter model of a random directed graph (digraph) is specified by the probability of ‘up arrows' from vertex i to vertex j where i &lt; j, the probability of ‘down arrows' from i to j where i ≥ j, and the probability of bidirectional arrows between i and j. In this model, a phase transition—the abrupt appearance of a giant strongly connected component—takes place as the parameters cross a critical surface. The critical surface is determined explicitly. Before the giant component appears, almost surely all non-trivial components are small cycles. The asymptotic probability that the digraph contains no cycles of length 3 or more is computed explicitly. This model and its analysis are motivated by the theory of food webs in ecology.

scholarly journals Convert a Strongly Connected Directed Graph to a Black-and-White 3-SAT Problem by the Balatonboglár Model

2020 ◽  
Author(s):  
Gábor Kusper ◽  
Csaba Biró
Keyword(s):  
Directed Graph ◽  
Directed Graphs ◽  
Deep Insight ◽  
Communication Graph ◽  
Sat Problem ◽  
Weak Model ◽  
Know How ◽  
Black And White ◽  
Communication Graphs ◽  
Strongly Connected

In a previous paper we defined the Black-and-White SAT problem which has exactly two solutions, where each variable is either true or false. We showed that Black-and-White $2$-SAT problems represent strongly connected directed graphs. We presented also the strong model of communication graphs. In this work we introduce two new models, the weak model, and the Balatonbogl\'{a}r model of communication graphs. A communication graph is a directed graph, where no self loops are allowed. In this work we show that the weak model of a strongly connected communication graph is a Black-and-White SAT problem. We prove a powerful theorem, the so called Transitions Theorem. This theorem states that for any model which is between the strong and the weak model, we have that this model represents strongly connected communication graphs as Blask-and-White SAT problems. We show that the Balatonbogl\'{a}r model is between the strong and the weak model, and it generates $3$-SAT problems, so the Balatonbogl\'{a}r model represents strongly connected communication graphs as Black-and-White $3$-SAT problems. Our motivation to study these models is the following: The strong model generates a $2$-SAT problem from the input directed graph, so it does not give us a deep insight how to convert a general SAT problem into a directed graph. The weak model generates huge models, because it represents all cycles, even non-simple cycles, of the input directed graph. We need something between them to gain more experience. From the Balatonbogl\'{a}r model we learned that it is enough to have a subset of a clause, which represents a cycle in the weak model, to make the Balatonbogl\'{a}r model more compact. We still do not know how to represent a SAT problem as a directed graph, but this work gives a strong link between two prominent fields of formal methods: SAT problem and directed graphs.

Strongly Connected Spanning Subgraph for Almost Symmetric Networks

2017 ◽  
Vol 27 (03) ◽  
pp. 207-219
Author(s):  
A. Karim Abu-Affash ◽  
Paz Carmi ◽  
Anat Parush Tzur
Keyword(s):  
Directed Graph ◽  
Euclidean Distance ◽  
Directed Graphs ◽  
Minimum Weight ◽  
Shortest Paths ◽  
Strong Connectivity ◽  
Spanning Subgraph ◽  
Strongly Connected ◽  
Set Of Points ◽  
Symmetric Networks

In the strongly connected spanning subgraph ([Formula: see text]) problem, the goal is to find a minimum weight spanning subgraph of a strongly connected directed graph that maintains the strong connectivity. In this paper, we consider the [Formula: see text] problem for two families of geometric directed graphs; [Formula: see text]-spanners and symmetric disk graphs. Given a constant [Formula: see text], a directed graph [Formula: see text] is a [Formula: see text]-spanner of a set of points [Formula: see text] if, for every two points [Formula: see text] and [Formula: see text] in [Formula: see text], there exists a directed path from [Formula: see text] to [Formula: see text] in [Formula: see text] of length at most [Formula: see text], where [Formula: see text] is the Euclidean distance between [Formula: see text] and [Formula: see text]. Given a set [Formula: see text] of points in the plane such that each point [Formula: see text] has a radius [Formula: see text], the symmetric disk graph of [Formula: see text] is a directed graph [Formula: see text], such that [Formula: see text]. Thus, if there exists a directed edge [Formula: see text], then [Formula: see text] exists as well. We present [Formula: see text] and [Formula: see text] approximation algorithms for the [Formula: see text] problem for [Formula: see text]-spanners and for symmetric disk graphs, respectively. Actually, our approach achieves a [Formula: see text]-approximation algorithm for all directed graphs satisfying the property that, for every two nodes [Formula: see text] and [Formula: see text], the ratio between the shortest paths, from [Formula: see text] to [Formula: see text] and from [Formula: see text] to [Formula: see text] in the graph, is at most [Formula: see text].

scholarly journals Robust structure measures of metabolic networks that predict prokaryotic optimal growth temperature

2019 ◽  
Vol 20 (1) ◽  
Author(s):  
Adèle Weber Zendrera ◽  
Nataliya Sokolovska ◽  
Hédi A. Soula
Keyword(s):  
Negative Correlation ◽  
Growth Temperature ◽  
Metabolic Networks ◽  
Optimal Growth ◽  
Living Environment ◽  
Connected Component ◽  
Undirected Graphs ◽  
Optimal Growth Temperature ◽  
Strongly Connected Component ◽  
Strongly Connected

Abstract Background Metabolic networks reflect the relationships between metabolites (biomolecules) and the enzymes (proteins), and are of particular interest since they describe all chemical reactions of an organism. The metabolic networks are constructed from the genome sequence of an organism, and the graphs can be used to study fluxes through the reactions, or to relate the graph structure to environmental characteristics and phenotypes. About ten years ago, Takemoto et al. (2007) stated that the structure of prokaryotic metabolic networks represented as undirected graphs, is correlated to their living environment. Although metabolic networks are naturally directed graphs, they are still usually analysed as undirected graphs. Results We implemented a pipeline to reconstruct metabolic networks from genome data and confirmed some of the results of Takemoto et al. (2007) with today data using up-to-date databases. However, Takemoto et al. (2007) used only a fraction of all available enzymes from the genome and taking into account all the enzymes we fail to reproduce the main results. Therefore, we introduce three robust measures on directed representations of graphs, which lead to similar results regardless of the method of network reconstruction. We show that the size of the largest strongly connected component, the flow hierarchy and the Laplacian spectrum are strongly correlated to the environmental conditions. Conclusions We found a significant negative correlation between the size of the largest strongly connected component (a cycle) and the optimal growth temperature of the considered prokaryotes. This relationship holds true for the spectrum, high temperature being associated with lower eigenvalues. The hierarchy flow shows a negative correlation with optimal growth temperature. This suggests that the dynamical properties of the network are dependant on environmental factors.

Network-Based Phase Space Analysis of the El Farol Bar Problem

2021 ◽  
Vol 27 (2) ◽  
pp. 113-130
Author(s):  
Shane St. Luce ◽  
Hiroki Sayama
Keyword(s):  
Decision Making ◽  
Phase Space ◽  
State Transition ◽  
Directed Network ◽  
Connected Component ◽  
Coordination Problem ◽  
New Approach ◽  
Strongly Connected Component ◽  
Strongly Connected ◽  
Transition Network

Abstract The El Farol Bar problem highlights the issue of bounded rationality through a coordination problem where agents must decide individually whether or not to attend a bar without prior communication. Each agent is provided a set of attendance predictors (or decision-making strategies) and uses the previous bar attendances to guess bar attendance for a given week to determine if the bar is worth attending. We previously showed how the distribution of used strategies among the population settles into an attractor by using a spatial phase space. However, this approach was limited as it required N − 1 dimensions to fully visualize the phase space of the problem, where N is the number of strategies available. Here we propose a new approach to phase space visualization and analysis by converting the strategy dynamics into a state transition network centered on strategy distributions. The resulting weighted, directed network gives a clearer representation of the strategy dynamics once we define an attractor of the strategy phase space as a sink-strongly connected component. This enables us to study the resulting network to draw conclusions about the performance of the different strategies. We find that this approach not only is applicable to the El Farol Bar problem, but also addresses the dimensionality issue and is theoretically applicable to a wide variety of discretized complex systems.

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