scholarly journals A random hierarchical lattice: the series-parallel graph and its properties

2004 ◽  
Vol 36 (03) ◽  
pp. 824-838 ◽  
Author(s):  
B. M. Hambly ◽  
Jonathan Jordan
Keyword(s):  
Phase Transition ◽  
Random Graphs ◽  
Effective Resistance ◽  
First Passage Percolation ◽  
Hierarchical Lattice ◽  
First Passage ◽  
In Series ◽  
Parallel Graph

We consider a sequence of random graphs constructed by a hierarchical procedure. The construction replaces existing edges by pairs of edges in series or parallel with probability p. We investigate the effective resistance across the graphs, first-passage percolation on the graphs and the Cheeger constants of the graphs as the number of edges tends to infinity. In each case we find a phase transition at

scholarly journals A random hierarchical lattice: the series-parallel graph and its properties

2004 ◽  
Vol 36 (3) ◽  
pp. 824-838 ◽  
Author(s):  
B. M. Hambly ◽  
Jonathan Jordan
Keyword(s):  
Phase Transition ◽  
Random Graphs ◽  
Effective Resistance ◽  
First Passage Percolation ◽  
Hierarchical Lattice ◽  
First Passage ◽  
In Series ◽  
Parallel Graph

We consider a sequence of random graphs constructed by a hierarchical procedure. The construction replaces existing edges by pairs of edges in series or parallel with probability p. We investigate the effective resistance across the graphs, first-passage percolation on the graphs and the Cheeger constants of the graphs as the number of edges tends to infinity. In each case we find a phase transition at

scholarly journals First passage percolation on random graphs with finite mean degrees

2010 ◽  
Vol 20 (5) ◽  
pp. 1907-1965 ◽  
Author(s):  
Shankar Bhamidi ◽  
Remco van der Hofstad ◽  
Gerard Hooghiemstra
Keyword(s):  
Random Graphs ◽  
First Passage Percolation ◽  
First Passage

scholarly journals Scaling Limits of Random Graphs from Subcritical Classes: Extended abstract

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Konstantinos Panagiotou ◽  
Benedikt Stufler ◽  
Kerstin Weller
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Scaling Factor ◽  
Scaling Limits ◽  
First Passage Percolation ◽  
First Passage ◽  
Outerplanar Graphs ◽  
Nous Montrons ◽  
Analytic Expressions ◽  
International Audience

International audience We study the uniform random graph $\mathsf{C}_n$ with $n$ vertices drawn from a subcritical class of connected graphs. Our main result is that the rescaled graph $\mathsf{C}_n / \sqrt{n}$ converges to the Brownian Continuum Random Tree $\mathcal{T}_{\mathsf{e}}$ multiplied by a constant scaling factor that depends on the class under consideration. In addition, we provide subgaussian tail bounds for the diameter $\text{D}(\mathsf{C}_n)$ and height $\text{H}(\mathsf{C}_n^\bullet)$ of the rooted random graph $\mathsf{C}_n^\bullet$. We give analytic expressions for the scaling factor of several classes, including for example the prominent class of outerplanar graphs. Our methods also enable us to study first passage percolation on $\mathsf{C}_n$, where we show the convergence to $\mathcal{T}_{\mathsf{e}}$ under an appropriate rescaling. On s’int´eresse au comportement asymptotique du graphe aleatoire $\mathsf{C}_n$ sur $n$ sommets pris uniformément d’une classe sous-critique des graphes sur n sommets. Dans cette contribution nous montrons que le graphe normalisée$\mathsf{C}_n / \sqrt{n}$ converges vers un arbre aléatoire brownien continue Te multiplie par une constante qui dépends de la classede graphes considérée. Nous calculons l’expression analytique pour cette constante dans plusieurs cas parmi la classefameuse des graphes planaire extérieure. En plus, on montre que le diamètre $\text{D}(\mathsf{C}_n)$ et la hauteur $\text{H}(\mathsf{C}_n^\bullet)$ de l’équivalent racine de $\mathsf{C}_n$ sont bornes par des bornes sous gaussiens. Notre méthode nous permettons aussi de l’étudier la percolation du premier passage sur $\mathsf{C}_n$. Nous montrons que $\mathcal{T}_{\mathsf{e}}$ sujet a une changement d’échelle appropriée

scholarly journals Competing first passage percolation on random graphs with finite variance degrees

2019 ◽  
Vol 55 (3) ◽  
pp. 545-559
Author(s):  
Daniel Ahlberg ◽  
Maria Deijfen ◽  
Svante Janson
Keyword(s):  
Random Graphs ◽  
Finite Variance ◽  
First Passage Percolation ◽  
First Passage

scholarly journals Speeding up non-Markovian first-passage percolation with a few extra edges

2018 ◽  
Vol 50 (3) ◽  
pp. 858-886 ◽  
Author(s):  
Alexey Medvedev ◽  
Gábor Pete
Keyword(s):  
Random Graphs ◽  
Spanning Trees ◽  
Real Life ◽  
First Passage Percolation ◽  
Power Law Distribution ◽  
First Passage ◽  
Critical Window ◽  
Expected Time ◽  
Si Model ◽  
Passage Times

Abstract One model of real-life spreading processes is that of first-passage percolation (also called the SI model) on random graphs. Social interactions often follow bursty patterns, which are usually modelled with independent and identically distributed heavy-tailed passage times on edges. On the other hand, random graphs are often locally tree-like, and spreading on trees with leaves might be very slow due to bottleneck edges with huge passage times. Here we consider the SI model with passage times following a power-law distribution ℙ(ξ>t)∼t-α with infinite mean. For any finite connected graph G with a root s, we find the largest number of vertices κ(G,s) that are infected in finite expected time, and prove that for every k≤κ(G,s), the expected time to infect k vertices is at most O(k1/α). Then we show that adding a single edge from s to a random vertex in a random tree 𝒯 typically increases κ(𝒯,s) from a bounded variable to a fraction of the size of 𝒯, thus severely accelerating the process. We examine this acceleration effect on some natural models of random graphs: critical Galton--Watson trees conditioned to be large, uniform spanning trees of the complete graph, and on the largest cluster of near-critical Erdős‒Rényi graphs. In particular, at the upper end of the critical window, the process is already much faster than exactly at criticality.

scholarly journals THE NEAREST UNVISITED VERTEX WALK ON RANDOM GRAPHS

2021 ◽  
pp. 1-17
Author(s):  
David J. Aldous
Keyword(s):  
Random Graphs ◽  
Finite Graph ◽  
Metric Entropy ◽  
Challenging Problem ◽  
First Passage Percolation ◽  
Cover Time ◽  
First Passage ◽  
Entropy Measures ◽  
Order Of Magnitude

We revisit an old topic in algorithms, the deterministic walk on a finite graph which always moves toward the nearest unvisited vertex until every vertex is visited. There is an elementary connection between this cover time and ball-covering (metric entropy) measures. For some familiar models of random graphs, this connection allows the order of magnitude of the cover time to be deduced from first passage percolation estimates. Establishing sharper results seems a challenging problem.

scholarly journals First passage percolation on sparse random graphs with boundary weights

2019 ◽  
Vol 56 (2) ◽  
pp. 458-471
Author(s):  
Lasse Leskelä ◽  
Hoa Ngo
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Time Scale ◽  
Network Size ◽  
First Passage Times ◽  
First Passage Percolation ◽  
First Passage ◽  
Connectivity Structure ◽  
Random Weights ◽  
Passage Times

AbstractA large and sparse random graph with independent exponentially distributed link weights can be used to model the propagation of messages or diseases in a network with an unknown connectivity structure. In this article we study an extended setting where, in addition, the nodes of the graph are equipped with nonnegative random weights which are used to model the effect of boundary delays across paths in the network. Our main results provide approximative formulas for typical first passage times, typical flooding times, and maximum flooding times in the extended setting, over a time scale logarithmic with respect to the network size.

scholarly journals The front of the epidemic spread and first passage percolation

2014 ◽  
Vol 51 (A) ◽  
pp. 101-121 ◽  
Author(s):  
Shankar Bhamidi ◽  
Remco van der Hofstad ◽  
Júlia Komjáthy
Keyword(s):  
Random Graphs ◽  
Epidemic Models ◽  
Random Networks ◽  
Epidemic Spread ◽  
First Passage Percolation ◽  
First Passage ◽  
Bounded Degree ◽  
Epidemic Curve ◽  
Second Moments ◽  
Bounded Degree Graphs

We establish a connection between epidemic models on random networks with general infection times considered in Barbour and Reinert (2013) and first passage percolation. Using techniques developed in Bhamidi, van der Hofstad and Hooghiemstra (2012), when each vertex has infinite contagious periods, we extend results on the epidemic curve in Barbour and Reinert (2013) from bounded degree graphs to general sparse random graphs with degrees having finite second moments asn→ ∞, with an appropriateX2log+Xcondition. We also study the epidemic trail between the source and typical vertices in the graph.

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