Two Models of Random Intersection Graphs for Classification

Component Evolution in General Random Intersection Graphs

“…Indeed, this inequality implies (17), because ρ −τ → ρ(Q) as τ ↓ 0, see (12). We fix τ ∈ (0, 1) and prove (43).…”

“…Moreover, we assume that the distribution of S(i) is a mixture of uniform distributions. That is, for every k, conditionally on the event |S(i)| = k, the random set S(i) is uniformly distributed in the class of all subsets of W of size k. In particular, with P * denoting the distribution of |S(i)|, we have, for every A ⊂ W , P(S(i) = A) = studied in Godehard and Jaworski [12]. They are applied in the analysis of secure wireless networks [9], [11], [14], social networks [8], and statistical classification [12].…”

“…That is, for every k, conditionally on the event |S(i)| = k, the random set S(i) is uniformly distributed in the class of all subsets of W of size k. In particular, with P * denoting the distribution of |S(i)|, we have, for every A ⊂ W , P(S(i) = A) = studied in Godehard and Jaworski [12]. They are applied in the analysis of secure wireless networks [9], [11], [14], social networks [8], and statistical classification [12]. We study the component evolution of random intersection graphs G(n, m, P * ) in the case where m is much larger than n. Note that if we choose P * such that P * ( cm/n) = 1, where c ≥ 0 is a constant, then the expected value of the degree of a typical vertex is approximately c, see [12], [17].…”

“…They are applied in the analysis of secure wireless networks [9], [11], [14], social networks [8], and statistical classification [12]. We study the component evolution of random intersection graphs G(n, m, P * ) in the case where m is much larger than n. Note that if we choose P * such that P * ( cm/n) = 1, where c ≥ 0 is a constant, then the expected value of the degree of a typical vertex is approximately c, see [12], [17]. Therefore, one may expect to observe a fast growth of the largest connected component in G(n, m, P * ) when |S(i)| (the size of a typical random set) is of stochastic order Θ P ( m/n).…”

Epidemics on Random Graphs with Tunable Clustering

“…This construction can be generalized in various ways-see, e.g. Godehardt and Jaworski (2002) and Deijfen and Kets (2007)-but in this paper we will stick to the above formulation. We will also specialize to the case where m = βn α for some constants α, β > 0 (where · denotes the integer part); see Karoński et al (1999) for a motivation of this choice of m. In fact, to get a graph with tunable clustering, we will soon take α = 1.…”