On a Class of Stochastic Multilayer Networks

TL;DR

This paper introduces a new stochastic multilayer network model where each layer is a probabilistic subgraph of a base network, analyzes its configuration probabilities, and explores asymptotic independence and Poisson distribution of links.

Contribution

The paper presents a novel stochastic multilayer network model, derives explicit configuration probabilities for specific structures, and demonstrates asymptotic independence and Poisson behavior of links.

Findings

Explicit probability distributions for line and tree structures.

Links become asymptotically independent and Poisson distributed as layers increase.

Numerical results show the impact of multiple layers on network metrics.

Abstract

In this paper, we introduce a new class of stochastic multilayer networks. A stochastic multilayer network is the aggregation of $M$ networks (one per layer) where each is a subgraph of a foundational network $G$. Each layer network is the result of probabilistically removing links and nodes from $G$. The resulting network includes any link that appears in at least $K$ layers. This model is an instance of a non-standard site-bond percolation model. Two sets of results are obtained: first, we derive the probability distribution that the $M$-layer network is in a given configuration for some particular graph structures (explicit results are provided for a line and an algorithm is provided for a tree), where a configuration is the collective state of all links (each either active or inactive). Next, we show that for appropriate scalings of the node and link selection processes in a layer, links are asymptotically independent as the number of layers goes to infinity, and follow Poisson distributions. Numerical results are provided to highlight the impact of having several layers on some metrics of interest (including expected size of the cluster a node belongs to in the case of the line). This model finds applications in wireless communication networks with multichannel radios, multiple social networks with overlapping memberships, transportation networks, and, more generally, in any scenario where a common set of nodes can be linked via co-existing means of connectivity.