On connectivity, conductance and bootstrap percolation for a random k‐out, age‐biased graph

Abstract

A uniform attachment graph (with parameter k), denoted Gn,k in the paper, is a random graph on the vertex set [n], where each vertex v makes k selections from [v − 1] uniformly and independently, and these selections determine the edge set. We study several aspects of this graph. Our motivation comes from two similarly constructed, well‐studied random graphs: k‐out graphs and preferential attachment graphs. In this paper, we find the asymptotic distribution of its minimum degree and connectivity, and study the expansion properties of Gn,k to show that the conductance of Gn,k is of order false(normallognfalse)−1. We also study the bootstrap percolation on Gn,k, where r infected neighbors infect a vertex, and show that if the probability of initial infection of a vertex is negligible compared to false(normallognfalse)−rfalse/false(r−1false) then with high probability (whp) the disease will not spread to the whole vertex set, and if this probability exceeds false(normallognfalse)−rfalse/false(r−1false) by a sub‐logarithmical factor then the disease whp will spread to the whole vertex set.

On Bollobás‐Riordan random pairing model of preferential attachment graph

“…Using the union bound, we obtain: with probability 1 − O( −(m−1) ), P(∃a ∶ X(a) ≤ (9m) −1 (log (2) n) 1) .…”

“…Liberally citing [8], consider m = 1 first. The graphs are nested: G 1 1 ⊂ G 2 1 ⊂ • • •, with G t 1 having vertex set [t] and t edges/loops, and with precisely 1 edge/loop incident to vertex t. Therefore for s < t the graph G s 1 is a subgraph of G t 1 induced by [s]. In particular, G 1 1 has a single vertex 1 and a single loop.…”

“…A loop contributes 2 to a vertex degree. It is postulated that, given G t− 1 1 , vertex t attaches by an edge to a vertex s ∈ [t − 1] (t develops a loop resp.) with probability proportional to the degree of s in G t− 1 1 (with probability proportional to 1 resp.).…”

“…It is postulated that, given G t− 1 1 , vertex t attaches by an edge to a vertex s ∈ [t − 1] (t develops a loop resp.) with probability proportional to the degree of s in G t− 1 1 (with probability proportional to 1 resp.). For m > 1, m edges emanating from t are added, one at a time, to G t−1 m , at each of m steps "counting the previous edges as well as the 'outward half' of the edge being added as already contributing to the degrees."…”

“…The bound (1.4), for any ≥ E[X], holds also for the negatively associated Bernoulli random variables. 1) .…”