# Persistent Flows and Non-Reciprocal Interactions in Deterministic Networks

## TL;DR

This paper analyzes deterministic consensus networks with persistent flows and non-reciprocal interactions, establishing conditions for global consensus based on the persistent graph containing a directed spanning tree.

## Contribution

It introduces new balance conditions over time windows for non-reciprocal interactions and proves consensus under weak conditions without requiring positive lower bounds on weights.

## Key findings

- Global consensus occurs if the persistent graph has a directed spanning tree.
- Convergence rates are characterized based on agent interactions.
- Results hold without assuming positive lower bounds on arc weights.

## Abstract

This paper studies deterministic consensus networks with discrete-time dynamics under persistent flows and non-reciprocal agent interactions. An arc describing the interaction strength between two agents is said to be persistent if its weight function has an infinite $l_1$ norm. We discuss two balance conditions on the interactions between agents which generalize the arc-balance and cut-balance conditions in the literature respectively. The proposed conditions require that such a balance should be satisfied over each time window of a fixed length instead of at each time instant. We prove that in both cases global consensus is reached if and only if the persistent graph, which consists of all the persistent arcs, contains a directed spanning tree. The convergence rates of the system to consensus are also provided in terms of the interactions between agents having taken place. The results are obtained under a weak condition without assuming the existence of a positive lower bound of all the nonzero weights of arcs and are compared with the existing results. Illustrative examples are provided to show the critical importance of the nontrivial lower boundedness of the self-confidence of the agents.