10 research outputs found
Diameter of the stochastic mean-field model of distance
We consider the complete graph \cK_n on vertices with exponential mean
edge lengths. Writing for the weight of the smallest-weight path
between vertex , Janson showed that converges in probability to 3. We extend this result by showing
that converges in distribution to a
limiting random variable that can be identified via a maximization procedure on
a limiting infinite random structure. Interestingly, this limiting random
variable has also appeared as the weak limit of the re-centered graph diameter
of the barely supercritical Erd\H{o}s-R\'enyi random graph in work by Riordan
and Wormald.Comment: 27 page
Diameter of the stochastic mean-field model of distance
We consider the complete graph K_n on n vertices with exponential mean n edge lengths. Writing C_{ij} for the weight of the smallest-weight path between vertex i,j \in [n], Janson [17] showed that max_{i,j \in [n]} C_{ij} / log n converges in probability to 3. We extend this results by showing that max_{i,j \in [n]} C_{ij} - 3 log n converges in distribution to some limiting random variable that can be identified via a maximization procedure on a limiting infinite random structure. Interestingly, this limiting random variable has also appeared as the weak limit of the re-centered graph diameter of the barely supercritical Erdös-Rényi random graph in [21]
Long-range first-passage percolation on the complete graph
We study a geometric version of first-passage percolation on the complete
graph, known as long-range first-passage percolation. Here, the vertices of the
complete graph are embedded in the -dimensional torus
, and each edge is assigned an independent transmission time
, where is a rate-one exponential
random variable associated with the edge ,
denotes the torus-norm, and is a parameter. We are interested in
the case , which corresponds to the instantaneous percolation
regime for long-range first-passage percolation on studied by
Chatterjee and Dey, and which extends first-passage percolation on the complete
graph (the case) studied by Janson. We consider the typical
distance, flooding time, and diameter of the model. Our results show a
-type result, akin to first-passage percolation on the complete graph as
shown by Janson. The results also provide a quantitative perspective to the
qualitative results observed by Chatterjee and Dey on .Comment: 16 page
Long paths in first passage percolation on the complete graph II. Global branching dynamics
We study the random geometry of first passage percolation on the complete graph equipped with independent and identically distributed positive edge weights. We consider the case where the lower extreme values of the edge weights are highly separated. This model exhibits strong disorder and a crossover between local and global scales. Local neighborhoods are related to invasion percolation that display self-organised criticality. Globally, the edges with relevant edge weights form a barely supercritical Erdős–Rényi random graph that can be described by branching processes. This near-critical behaviour gives rise to optimal paths that are considerably longer than logarithmic in the number of vertices, interpolating between random graph and minimal spanning tree path lengths. Crucial to our approach is the quantification of the extreme-value behavior of small edge weights in terms of a sequence of parameters (sn)n≥1 that characterises the different universality classes for first passage percolation on the complete graph. We investigate the case where sn→ ∞ with sn= o(n1 / 3) , which corresponds to the barely supercritical setting. We identify the scaling limit of the weight of the optimal path between two vertices, and we prove that the number of edges in this path obeys a central limit theorem with mean approximately snlog(n/sn3) and variance sn2log(n/sn3). Remarkably, our proof also applies to n-dependent edge weights of the form Esn, where E is an exponential random variable with mean 1, thus settling a conjecture of Bhamidi et al. (Weak disorder asymptotics in the stochastic meanfield model of distance. Ann Appl Probab 22(1):29–69, 2012). The proof relies on a decomposition of the smallest-weight tree into an initial part following invasion percolation dynamics, and a main part following branching process dynamics. The initial part has been studied in Eckhoff et al. (Long paths in first passage percolation on the complete graph I. Local PWIT dynamics. Electron. J. Probab. 25:1–45, 2020. https://doi.org/10.1214/20-EJP484); the current paper focuses on the global branching dynamics
Diameter of the stochastic mean-field model of distance
We consider the complete graph K n on n vertices with exponential mean n edge lengths. Writing C ij for the weight of the smallest-weight path between vertices i, j ∈ [n], Janson [18] showed that max i,j∈[n] C ij/logn converges in probability to 3. We extend these results by showing that max i,j∈[n] C ij - 3 logn converges in distribution to some limiting random variable that can be identified via a maximization procedure on a limiting infinite random structure. Interestingly, this limiting random variable has also appeared as the weak limit of the re-centred graph diameter of the barely supercritical Erdos-Rényi random graph in [22]