The travelling salesman problem is to find a shortest path from the travelling salesman’s hometown, make the round of all the towns in the set, and finally go back home. This paper investigates the travelling salesman problem with fuzzy random travelling time. Three concepts are proposed: expected shortest path, (α, β)-path and chance shortest path according to different optimal desire. Corresp...

Extinction appears ubiquitously in many fields, including chemical reactions, population biology, evolution and epidemiology. Even though extinction as a random process is a rare event, its occurrence is observed in large finite populations. Extinction occurs when fluctuations owing to random transitions act as an effective force that drives one or more components or species to vanish. Although...

In 1962, Pósa conjectured that a graph G = (V,E) contains a square of a Hamiltonian cycle if δ(G) > 2n/3. Only more than thirty years later Komlós, Sárközy, and Szemerédi proved this conjecture using the so-called Blow-Up Lemma. Here we extend their result to a random graph setting. We show that for every > 0 and p = n−1/2+ a.a.s. every subgraph of Gn,p with minimum degree at least (2/3+ )np co...

The famous Pósa conjecture states that every graph of minimum degree at least 2n/3 contains the square of a Hamilton cycle. This has been proved for large n by Komlós, Sarközy and Szemerédi. Here we prove that if p ≥ n−1/2+ε, then asymptotically almost surely, the binomial random graph Gn,p contains the square of a Hamilton cycle. This provides an ‘approximate threshold’ for the property in the...

We use Fibonacci heaps to improve a parametric shortest path algorithm of Karp and Orlin, and we combine our algorithm and the method of Schneider and Schneider’s minimum-balance algorithm to obtain a faster minimum-balance algorithm. For a graph with n vertices and m edges, our parametric shortest path algorithm and our minimum-balance algorithm both run in O(nm + n logn) time, improved from O...

Multivariate random fields whose distributions are invariant under operatorscalings in both time-domain and state space are studied. Such random fields are called operator-self-similar random fields and their scaling operators are characterized. Two classes of operator-self-similar stable random fields X = {X(t), t ∈ R} with values in R are constructed by utilizing homogeneous functions and sto...

We study the distribution of optimal path lengths in random graphs with random weights associated with each link ("disorder"). With each link i we associate a weight tau(i) = exp (a r(i)), where r(i) is a random number taken from a uniform distribution between 0 and 1, and the parameter a controls the strength of the disorder. We suggest, in an analogy with the average length of the optimal pat...

We examine how three different communication processes operating through social networks are affected by homophily – the tendency of individuals to associate with others similar to themselves. Homophily has no effect if messages are broadcast or sent via shortest paths; only connection density matters. In contrast, homophily substantially slows learning based on repeated averaging of neighbors’...

A geodesic in a graph G is a shortest path between two vertices of G. For a specific function e(n) of n, we define an almost geodesic cycle C in G to be a cycle in which for every two vertices u and v in C, the distance dG(u, v) is at least dC(u, v) − e(n). Let ω(n) be any function tending to infinity with n. We consider a random d-regular graph on n vertices. We show that almost all pairs of v...

We show that p = √ e n is a sharp threshold for the random graph Gn,p to contain the square of a Hamilton cycle. This improves the previous results of Kühn and Osthus and also Nenadov and Škorić.