# A Robust Advantaged Node Placement Strategy for Sparse Network Graphs

## TL;DR

This paper introduces a novel hexagonal coordinate system-based node placement strategy to enhance connectivity robustness in sparse heterogeneous networks, addressing NP-hard optimization challenges with efficient approximation algorithms.

## Contribution

The paper proposes a new node placement method using a hexagonal coordinate system to improve robustness in sparse HetNets, along with an approximation solution for an NP-hard problem.

## Key findings

- The proposed algorithm closely approximates exhaustive search solutions.
- It outperforms other methods in cost, runtime, and robustness in sparse networks.
- Classification thresholds for network sparsity levels are derived.

## Abstract

Establishing robust connectivity in heterogeneous networks (HetNets) is an important yet challenging problem. For a HetNet accommodating a large number of nodes, establishing perturbation-invulnerable connectivity is of utmost importance. This paper provides a robust advantaged node placement strategy best suited for sparse network graphs. In order to offer connectivity robustness, this paper models the communication range of an advantaged node with a hexagon embedded within a circle representing the physical range of a node. Consequently, the proposed node placement method of this paper is based on a so-called hexagonal coordinate system (HCS) in which we develop an extended algebra. We formulate a class of geometric distance optimization problems aiming at establishing robust connectivity of a graph of multiple clusters of nodes. After showing that our formulated problem is NP-hard, we utilize HCS to efficiently solve an approximation of the problem. First, we show that our solution closely approximates an exhaustive search solution approach for the originally formulated NP-hard problem. Then, we illustrate its advantages in comparison with other alternatives through experimental results capturing advantaged node cost, runtime, and robustness characteristics. The results show that our algorithm is most effective in sparse networks for which we derive classification thresholds.