1,720,971 research outputs found
Hamilton cycles in a class of random directed graphs
No full textAbstract: "We prove that almost every 5-in, 5-out digraph is Hamiltonian.
Component structure of the vacant set induced by a random walk on a random graph
No full textWe consider random walks on several classes of graphs and explore the likely structure of the vacant set, i.e. the set of unvisited vertices. Let Γ(t) be the subgraph induced by the vacant set of the walk at step t. We show that for random graphs Gn,p (above the connectivity threshold) and for random regular graphs Gr,r ≥ 3, the graph Γ(t) undergoes a phase transition in the sense of the well-known ErdJW-RSAT1100590x.png -Renyi phase transition. Thus for t ≤ (1 - ε)t*, there is a unique giant component, plus components of size O(log n), and for t ≥ (1 + ε)t* all components are of size O(log n). For Gn,p and Gr we give the value of t*, and the size of Γ(t). For Gr, we also give the degree sequence of Γ(t), the size of the giant component (if any) of Γ(t) and the number of tree components of Γ(t) of a given size k = O(logn). We also show that for random digraphs Dn,p above the strong connectivity threshold, there is a similar directed phase transition. Thus fort ≤ (1 - ε)t*, there is a unique strongly connected giant component, plus strongly connected components of size O(log n), and for t ≥ (1 + ε)t*all strongly connected components are of size O(log n).</p
A Note on the Vacant Set of Random Walks on the Hypercube and Other Regular Graphs of High Degree
No full textWe consider a random walk on a d-regular graph G where d → ∞ and G satisfies certain conditions. Our prime example is the d-dimensional hypercube, which has n = 2d vertices. We explore the likely component structure of the vacant set, i.e. the set of unvisited vertices. Let Λ(t) be the subgraph induced by the vacant set of the walk at step t. We show that if certain conditions are satisfied then the graph Λ(t) undergoes a phase transition at around t* = n loge d. Our results are that if t ≤ (1 − ε)t* then w.h.p. as the number vertices n → ∞, the size L1(t) of the largest component satisfies ≫ n whereas if t ≥ (1 + ε)t* then L1(t) = o(log n).</p
Cover time of a random graph with given degree sequence
No full textIn this paper we establish the cover time of a random graph chosen uniformly at random from the set of graphs with vertex set [n] and degree sequence d. We show that under certain restrictions on d, the cover time of is whp asymptotic to . Here θ is the average degree and d is the effective minimum degree.</p
Vacant sets and vacant nets: Component structures induced by a random walk
No full textGiven a discrete random walk on a finite graph G, the vacant set and vacant net are, respectively, the sets of vertices and edges which remain unvisited by the walk at a given step t.%These sets induce subgraphs of the underlying graph. Let Γ(t) be the subgraph of Ginduced by the vacant set of the walk at step t. Similarly, let Γˆ(t) be the subgraph of G induced by the edges of the vacant net. For random r-regular graphs Gr, it was previously established that for a simple random walk, the graph Γ(t) of the vacant set undergoes a phase transition in the sense of the phase transition on Erd\H{os}-Renyi graphs Gn,p. Thus, for r≥3 there is an explicit value t∗=t∗(r) of the walk, such that for t≤(1−ϵ)t∗, Γ(t) has a unique giant component, plus components of size O(logn), whereas for t≥(1+ϵ)t∗ all the components of Γ(t) are of size O(logn). We establish the threshold value tˆ for a phase transition in the graph Γˆ(t) of the vacant net of a simple random walk on a random r-regular graph. We obtain the corresponding threshold results for the vacant set and vacant net of two modified random walks. These are a non-backtracking random walk, and, for r even, a random walk which chooses unvisited edges whenever available. This allows a direct comparison of thresholds between simple and modified walks on random r-regular graphs. The main findings are the following: As r increases the threshold for the vacant set converges to nlogr in all three walks. For the vacant net, the threshold converges to rn/2logn for both the simple random walk and non-backtracking random walk. When r≥4 is even, the threshold for the vacant net of the unvisited edge process converges to rn/2, which is also the vertex cover time of the process.</p
Random Walks with Look-ahead in Scale-free Random Graphs
No full textIf m ≥ 2 is constant and 0 ≤ r ≤ ε log log n for a small positive constant ε, then whp a random walk with look-ahead r on a scale-free graph G = G(m,n) has cover time CG(r) ∼ (2/(mr−1 (m − 1))) n log n.</p
The cover time of random geometric graphs
No full textWe study the cover time of random geometric graphs. Let I(d) = [0, 1]d denote the unit torus in d dimensions. Let D(x, r) denote the ball (disc) of radius r. Let Υd be the volume of the unit ball D(0, 1) in d dimensions. A random geometric graph G = G(d, r, n) in d dimensions is defined as follows: Sample n points V independently and uniformly at random from I(d). For each point x draw a ball D(x, r) of radius r about x. The vertex set V (G) = V and the edge set E(G) = {{v, w} : w 6= v, w ∈ D(v, r)}. Let G(d, r, n), d ≥ 3 be a random geometric graph. Let CG denote the cover time of a simple random walk on G. Let c > 1 be constant, and let r = (c log n/(Υdn))1/d. Then whp the cover time satisfies CG ∼ c log (c/c − 1)n log n</p
Going Beyond Counting First Authors in Author Co-citation Analysis
Get PDFThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
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