Abstract
We study the joint components in a random 'double graph' that is obtained by superposing red and blue binomial random graphs on $n$~vertices. A joint component is a maximal set of vertices that supports both a red and a blue spanning tree. We show that there are critical pairs of red and blue edge densities at which a giant joint component appears. In contrast to the standard binomial graph model, the phase transition is first order: the size of the largest joint component jumps from $O(1)$ vertices to $\Theta(n)$ at the critical point. We connect this phenomenon to the properties of a certain bicoloured branching process.
Highlights
In recent years there has been a growing interest in ‘multilayer networks’ as a model for large real-world structures [1]
Attention is focused on properties of a multilayer network that arise from interactions between the layers
We refer to E1 as the set of red edges and E2 as the set of blue edges
Summary
In recent years there has been a growing interest in ‘multilayer networks’ as a model for large real-world structures [1]. Molloy [12] provided a rigorous proof for the size of the giant joint component in the special case when λ1 = λ2, and stated what should be the generalisation to unequal edge densities and even to graphs formed from three or more distinguished edge sets Both the heuristic and rigorous results approach the giant joint component from above, by repeatedly stripping vertices that cannot be contained in it. Our method differs as we relate the size of the joint-giant in the double graph G(n, λ1, λ2) to a bicoloured branching process, where every particle in the process has Po(λ1) red offspring and independently Po(λ2) blue offspring. We show that whp any non-trivial joint component of sublinear size contains exactly two vertices, i.e. it is a pair of vertices connected by a red and a blue edge.
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