[Paper Review] Vertex Degree of Random Intersection Graph

Bhupendra Gupta|ArXiv.org|Sep 6, 2008

Advanced Graph Theory Research2 references3 citations

TL;DR

This paper analyzes the vertex degree and connectivity thresholds in random intersection graphs, where edges form when vertices share common objects from a set W. Using large deviations and the Borel-Cantelli lemma, it establishes that for $ p = (mn^\alpha)^{-1/2} $, the graph becomes almost surely connected when $ \alpha = 2 $, and derives almost sure bounds on vertex degree scaling as $ n^{1-\alpha} $, with explicit asymptotic constants defined via a logarithmic equation.

ABSTRACT

A random intersection graph is constructed by independently assigning a subset of a given set of objects $W,$ to each vertex of the vertex set $V$ of a simple graph $G.$ There is an edge between two vertices of $V,$ iff their respective subsets(in $W$,) have at least one common element. The strong threshold for the connectivity between any two arbitrary vertices of vertex set $V,$ is derived. Also we determine the almost sure probability bounds for the vertex degree of a typical vertex of graph $G.$

Motivation & Objective

• To determine the strong threshold probability for connectivity in random intersection graphs.
• To derive almost sure upper and lower bounds on the degree of a typical vertex in such graphs.
• To characterize the asymptotic behavior of vertex degrees under varying parameters $ m $, $ n $, and $ p $.
• To establish conditions under which connectivity occurs almost surely or with vanishing probability as $ n \to \infty $.

Proposed method

• Models random intersection graphs via a bipartite graph $ G^*(n,m,p) $, where edges between vertices $ v \in V $ and objects $ w \in W $ are i.i.d. with probability $ p $.
• Defines the random intersection graph $ G(n,m,p) $ with edge $ \{v_i,v_j\} $ iff $ W_{v_i} \cap W_{v_j} \neq \emptyset $.
• Applies the Chernoff-type bound from Lemma 2.1 to control tail probabilities of the binomially distributed vertex degree $ X \sim \text{Bi}(n-1, q_n) $.
• Uses the function $ H(t) = \frac{1}{t}\log t + \frac{1}{t} - 1 $ to express exponential bounds on $ P[X \leq K] $ and $ P[X \geq K] $.
• Applies the Borel-Cantelli lemma to derive almost sure limits of $ X/n^{1-\alpha} $ by analyzing summability of tail probabilities.
• Derives threshold behavior by setting $ p = (mn^\alpha)^{-1/2} $, leading to $ q_n \sim n^{-\alpha/2} $, and studies the sum of edge probabilities over $ n $.

Experimental results

Research questions

• RQ1 What is the threshold probability $ p $ that ensures almost sure connectivity in a random intersection graph?
• RQ2 How does the vertex degree of a typical vertex scale with $ n $, and what are its almost sure upper and lower bounds?
• RQ3 Under what conditions on $ \alpha $ does the edge probability between two vertices become summable or non-summable over $ n $?
• RQ4 How do the asymptotic bounds on vertex degree depend on the parameter $ \alpha $, and what role does the function $ a(c) $ play?

Key findings

• The strong threshold for connectivity is $ p(2) = (m n)^{-1/2} $, such that $ G(n,m,p) $ is almost surely connected when $ \alpha = 2 - \epsilon $, and disconnected with high probability when $ \alpha = 2 + \epsilon $.
• For $ 0 < \alpha < 1 $, the vertex degree $ X $ satisfies $ \limsup_{n \to \infty} X / n^{1-\alpha} \geq a(c) $ almost surely, where $ a(c) $ is the solution in $[1,\infty)$ to $ a \log a - a + 1 = c $.
• For $ 0 < \alpha < 1 $, the vertex degree satisfies $ \liminf_{n \to \infty} X / n^{1-\alpha} \leq a(c) $ almost surely, where $ a(c) $ is the solution in $ (0,1) $ to $ a \log a - a + 1 = c $.
• The edge probability between any two vertices is $ \sim n^{-\alpha/2} $, and this sum diverges for $ \alpha \leq 2 $, leading to infinitely many edges a.s., while it converges for $ \alpha > 2 $.
• The threshold behavior is determined by the summability of edge probabilities: divergence implies connectivity a.s., convergence implies finite edges a.s.
• The asymptotic degree bounds are sharp and depend on the solution to a logarithmic equation involving $ c $, which controls the tail decay rate.