# Directed Acyclic Graphs

## Background

The principle of Directed Acyclic Graphs (DAGs) in blockchain is to present a way to include traditional off-chain blocks into the ledger, which is governed by mathematical rules. The main problems to be solved by the DAG derivative protocols are:

1. inclusion of orphaned blocks (decrease the negative effect of slow propagation); and
2. mitigation against selfish mining attacks.

## Braiding the Blockchain

Dr. Bob McElrath
Ph.D. Theoretical Physics

### Summary

"Braiding the Blockchain" by Dr. Bob McElrath, Scaling Bitcoin, Hong Kong, December 2015.

This talk discusses the motivation for using Directed Acyclic Graphs (DAGs), which are orphaned blocks, throughput and more inclusive mining. New terms are defined to make DAGs applicable to blockchain, as it needs to be more specific: Braid vs. DAG, Bead vs. block, Sibling and Incest.

The Braid approach:

• incentivizes miners to quickly transmit beads;
• prohibits parents from containing conflicting transactions (unlike GHOST or SPECTRE);
• constructs beads to be valid Bitcoin blocks if they meet the difficulty target.

### Video

Note: Transcripts are available here.

## GHOST-DAG

Aviv Zohar
Prof. at The Hebrew University and Chief Scientist @ QED-it

### Summary

"The GHOST-DAG Protocol" by Yonatan Sompolinsky, Scaling Bitcoin, Tokyo, October 2018.

This talk discusses the goal going from chain to DAG being to get an ordering of the blocks that does not change when time between blocks approaches block propagation time; and security that does not break at higher throughputs. Terms introduced here are Anticone (the view of the DAG a block sees in the past and the future) and $k​$-cluster (a set of blocks with an Anticone at most $k​$). The protocol also makes use of a greedy algorithm in order to find the optimal $k​$-cluster at each step as it attempts to find the overall optimal $k​$-cluster.

### Video

Notes:

• Start watching at 30min or open the video in a separate tab here.
• Transcripts are available here.

## SPECTRE

Aviv Zohar
Prof. at The Hebrew University and Chief Scientist @ QED-it

### Summary

"Scalability II - GHOST/SPECTRE" by Dr. Aviv Zohar, Technion Summer School on Decentralized Cryptocurrencies and Blockchains, 2017.

This talk discusses the application of DAGs in the SPECTRE protocol. Three insights into the Bitcoin protocol are shared: DAGs are more powerful; Bitcoin is related to voting; and amplification. These are discussed in relation to SPECTRE, while properties sought are consistency, safety and weak liveness. Voting outcomes are strongly rejected, strongly accepted or pending.

## PHANTOM

Yonatan Sompolinsky
Founding Scientist, Daglabs

### Summary

""BlockDAG Protocols - SPECTRE, PHANTOM"* by Yonatan Sompolinsky, Blockchain Protocol Analysis and Security Engineering, Stanford, 2018.

This talk introduces the BlockDAG protocol PHANTOM, and motivates it by virtue of blockchains not being able to scale and BlockDAGs being a generalization of blockchains. The mining protocol references all tips in the DAG (as opposed to the tip of the longest chain) and also publishes all mined blocks as soon as possible (similar to Bitcoin). Blocks honestly created (i.e. honest blocks) will only be unconnected if they were created at approximately the same time. PHANTOM's goal is to recognize honest $k$-clusters, order the blocks within and disregard the rest.