# Reciprocity-driven Sparse Network Formation

## TL;DR

This paper introduces decentralized algorithms for resource exchange networks that optimize sparsity and reciprocity, using convex programming and reweighted l1 minimization, addressing NP-hardness in network formation.

## Contribution

It proposes novel decentralized algorithms based on convex programming and reweighted l1 minimization to approximate sparsest reciprocal exchanges in resource networks.

## Key findings

- Algorithms effectively balance sparsity and reciprocity.
- The induced graphs exhibit desirable structural properties.
- Trade-offs between sparsity and reciprocity are characterized.

## Abstract

A resource exchange network is considered, where exchanges among nodes are based on reciprocity. Peers receive from the network an amount of resources commensurate with their contribution. We assume the network is fully connected, and impose sparsity constraints on peer interactions. Finding the sparsest exchanges that achieve a desired level of reciprocity is in general NP-hard. To capture near-optimal allocations, we introduce variants of the Eisenberg-Gale convex program with sparsity penalties. We derive decentralized algorithms, whereby peers approximately compute the sparsest allocations, by reweighted l1 minimization. The algorithms implement new proportional-response dynamics, with nonlinear pricing. The trade-off between sparsity and reciprocity and the properties of graphs induced by sparse exchanges are examined.