 Okay, so well, first of all, I'd like to thanks to thank the organizers for, you know, despite all the odds, managing to put up the conference so thanks for hosting us in all these different fashions. Okay, so, as Emanuel said, I'm presenting work done in collaboration with Andrea Braccialli from the University of Stirling and Ronald Dehan from the University of Amsterdam. Okay, so the focus of the paper is on essentially this sort of consensus model that is used by blockchains like by system like Ripple and Stellar. So these are systems that have been quite influential. I think in terms of market capitalization, if I'm not wrong, Ripple is the fourth and Stellar the 70th largest blockchain company. So they had quite some influence. But on the other hand, they had received relatively little attention from, you know, academic from the academic circles. And also there are some strong criticism about such system because effectively, even though they, well, they present themselves as somewhat somewhat open blockchain system that in fact running as per permission systems at the moment. On the other hand, they are very clear about the ultimate goal in kind of managing to lift their approach to a real permission less setting. So we focused on them, because we thought there were some interesting theoretical questions behind the sort of consensus model they use, and furthermore, sort of considered that type of consensus model is particularly interesting to be investigated by using economic theory tools. So the questions I'm basically the question where we try to address in the paper. So this is consistent and understanding the inherent limitation or if you will, we should extend to which decentralization. So a permission less approach to consensus is possible in that in the consensus model followed by systems like Ripple and Stella. And the underlying model that they use is what is sometimes called a peer to peer trust network. There are several, there are several terminology going around, but the key idea is that the nodes in the system kind of designate, ideally in an autonomous fashion, which other nodes in the system they trust. You can think really of a picture like this, where any every node designates which other nodes is actually listening to and trusting. So this is the kind of main strategy that these systems systems like Ripple and Stella use to achieve some level of civil proofness. Okay, it's the really peer to peer trust network paradigm. So they select every node designates which other nodes to trust, and at the same time, it designates a quota or a threshold, which has to be met among the nodes trust for its own opinion to be determined. So you can think of if a quarter of trusted nodes agree on a value, for instance, so in my talk I will actually use just binary values for simplicity. If a quarter of trusty nodes agree on whether to record the transaction or not then the node that it's listening to those nodes, settles its value on that agreed value. So that's the idea say of validation. Now the question in this, in the setting is what consists of essentially means all honest nodes agreeing in a stable fashion through the trust network they built. So, let's move net then to an abstraction, which we called bits and time trust networks, BTN so this is directly in really following fairly closely, for instance, the sort of mathematical structure that one would find also in the stellar white paper for instance. So we have a set of nodes. We have the set of honest nodes, and then for every member of a H or capital H so for every hopeless node we assign. We have a trust set so it's the subset of nodes that is, it's trusted by I. So for every honest node high. We determine we the bits and time trust network determines a quota. Okay, that's the quote I was mentioned in the previous slides is what would determine the node for to validate precise value. The quota we use here are above 0.75, which is what we, well, what for instance we observe in repo look at repo sets that quota quota to 0.8. So we try to provide a slightly more general analysis. There is a good reason for setting a quota, which is not lower by 0.25 but I will not get to all the details in the talk. So abstractly you can think of nodes making binary decision so should a transaction be included or not in the ledger. And they do that under the influence of the nodes they trust. So if enough trusted nodes have opinion X then that opinion X is validated by the node. Of course, a bit sometime nodes can reveal any opinion to any honest node. So this time. That's how we, we think here of this entire notes so honest notes basically hold a binary opinion, if you wish. This opinion is influenced by the nodes they trust. And bits and time nodes can reveal any opinion to every to any node. Of course listens listens to them. Okay, now I make a shift from that this kind of distributed computing sort of structure to a very similar structure that has been studied in economic theory which goes under the name of command games. So in a command game is very similar structure. Well, apart from the bits and time nodes, but if you put them there in the, in the structure then what we get is something very close to command to BTM to be some time trust that so we have a set of nodes, capital N with a set of honest nodes capital H. We have a set of trusted notes, and then we have this calligraphic C, which essentially is just in the command games literature it would be defined simply as a set of players or set of nodes in this case that if they agree, they can determine my opinion. Okay, so it's terminology from from from that literature from game theory would be the set of winning coalitions for agent or for node I. Of course in the, in this case, what is a winning coalition so what is the coalition that can determine my opinion is just given by the size of that coalition. So the set of matrix C is the set of of nodes, which a set of nodes in among the ones that I trust that have a size that meets the quota of agent I know die. Okay, okay, so in again in game theory terminology that essentially means a command game is simply a set of players or nodes. And to each such player I assign a simple game. So a set of winning coalitions. And this theory was developed by who and sharply in the early 2000s. Okay, so we'll be using this, these structures as a bridge to use ideas from that literature that could. In fact, you can shed some light on on the working of those system like like ripple and stellar. In fact, what I call here command game is pretty much something that you literally would find also in the stellar white paper as referred to refer to as a federated Byzantine agreement systems or FBA yes. It's the same really mathematical structure. Okay, so now let me move to the idea of safety in Byzantine trust networks. So you can think of a fork here simply as being modeled as the situation which to honest notes validate opposite values. So to honest notes, have access to a set of trusted notes, set of notes that trusted by them, that is a size large enough to convince them of opposite values. So we can say that BTN then is safe if such situation doesn't cannot occur. Okay, you can think of then just the picture of a forked BTN would be like this. Okay, so we have two nodes they listen to different sets that there are some which are sufficient to convince them of different values. Okay, of course, interestingly this this notion is really a kind of quite general abstract it's it's protocol independent so I haven't talked yet about how precisely the because this product would be run on this network so it's this notion of safety is proper really of the if you wish the infrastructure of the of the trust network. Okay, then a quest natural question arises all about the necessary structural conditions that can ensure safety. Okay, so that's what I'm moving to now. And this is, of course, directly related to the level of decentralization that can be admitted in such a system. So here I will give you a go very quickly to some theorems that we discuss in the paper. In this theorem, we say, okay, in a uniform BTN so uniform BTN is a bit of an interest network where every quota is the same. Okay, so if we fix for everybody, a quote of that, of in the range given here's a 7508 safety actually can be shown to imply the existence of notes that are trusted by all honest nodes. So in a sense, we can prove that if we under the assumption that everybody uses the same quotas in that given range, then effectively we are requiring the existence of nodes that are trusted by all all honest nodes. And it's an important to notice is that's actually a set of that applies directly to ripple. So in a sense, and in a sense this really tells us gives us a theoretical justification for why ripple is actually working. And as it is doing currently, as far as I know, by means of indeed a set of central set of notes that that then run consensus for the whole for the whole system. So it's a centralized permissioned setup. And well, this is in the bullets you see a kind of a sketch of how the reasoning goes here so we can show that safety implies that any two trust sets should overlap for that fraction one minus q or q is the quota over q of the or their combined size and all pairs of trust set overlap for at least one fourth of their combined size then their intersection should be not empty. So that's essentially how the other reason goes behind these results. So taking a negative kind of perspective you may say that well if this is the setup then fully decentralized consensus is actually not really possible in this setting. And with fully decentralized consensus I really mean full freedom for the nodes to determine who to trust. And as I said, that's not the case for for ripple where everybody now has to link to to select the trust in notes between a specified set provided by ripple itself. Okay, so what happens if we lift known if we lift the uniformity so if we really allow everybody to kind of set their own quotas. So that's the case for instance for stellar. So stellar in principle gives more freedom to know that in fact it stemmed out of the initial ripple proposition. A simple observation we can immediately make is that the next necessary condition for safety in that's in that setting is that I need to self sufficient and I will explain what that means in a moment set of nodes should intersect. So a sufficient set of nodes is what are called the core in stellar, and you can think of them as this so in the picture here in the bottom right of the slide. Well that that can be thought of as a quorum so a quorum is a set of nodes that contains winning coalition for every member of the know of the set. It's kind of set that if it agrees it can agree stably in in validating value for the chain. So what we show in the paper is that. Well of course so if we if we look at this core so this kind of self sufficient set of nodes for the determination of agreement. If we have quora that don't intersect we can easily create forks in the system so a necessary condition for the absence of forks is that quora always intersect. So that always the intersection of to quora always contain is always non empty. Okay, so that's a see that's a condition that was clearly stated in. Starting from the white paper, and it is a condition in a sense, it can be thought of being problematic in several ways but where we tried to put our finger on is the intractability of determining whether, given a Byzantine trust network. We can check whether the quorum intersection property holds. Okay, so for an intersection here and the statement of this theorem means, okay, it's a decision problem in which I give you a Byzantine trust network and I ask you can you check whether whether there are any quora in the in the system that don't intersect. So, why is this an interesting problem because if if we want the system to be safe. Any individual decision by the nodes to select who they want to trust should somehow guarantee quorum intersection. So, in other words, we are putting on the nodes, a requirement to guarantee a kind of global safety property of the network. And our result shows that in fact this decision problem is competition. So it's going to be complete so it's in principle intractable that of course doesn't mean that it's not feasible in practice. It just just says that it is. It is potentially a stumbling block in the pathway to to provide full decentralization to the system. And the last part of my talk I want to talk about the problem of influence. Okay, so we looked at the problem with decentralization from the point of view of, well, for instance, but possible existence of nodes that have to be trusted by all nodes in order to guarantee safety we looked at the problem of guarantee necessary condition for for safety as being intractable computational problems. Now I want to look at what kind of kind of show you how one can understand influence or power in systems like people in Stella. So, this is kind of a understanding influence in this type of blockchain is in a sense trickier than what happens in proof of work or proof of stake block chain where you can say you can think okay the influence of a node in a system in proof of stake roughly is the share of computational power the system, the node invest in the system. And a similar reason you can make for proof of stake. It is aware and clear how to quantify influence on the on the blockchain in systems based on this and time trust networks, Larry Paul and stellar. In the paper we make a proposal with make a proposal on how influence could be understanding such systems, which builds on the idea of command game that I gave you at the beginning. As I showed okay we had in the show you with theorem one, we had that safety in uniform BTN implies the existence of all trusted nodes. So, but what does it mean concretely, okay in terms of the influence that such nodes can have on on consensus values. And this is harder to be determined as in proof of work or proof of stake. So, let's have a look at, again, BTNs and the structure I gave you earlier here phrased as a command game. So this capital L capitals calligraphic C. As I said, can be thought of what in game theory is called a simple game. So it's a set of nodes with a set of winning coalitions. The idea is that those winning collisions are collisions that can determine the value of I so they can determine the value that I validate in the system. So in terms of those as games we can apply what are called in the in theory of single simple games or theory of voting as well power indices. So here in this case I just quickly give you an example of what is called Penrose band of index, which essentially counts the number of times in which each node is pivotal in determining the value of I. So the frequency of in which I on its own can so J on its own can determine the value of I. So now that the details are not so important. But what we can get to is that by using power indices like bands of but there are many others. We can in fact construct what's called an influence matrix, which is a stochastic matrix in this case so the idea of this matrix is if the way in which you can read an entry I for instance I 21 in such a matrix, it tells I 21 gives me the power of agent or node one with respect to a node two. So not one will be trusted by two and we'll have a certain influence in determining the value that to can validate in the system. So this the rows in the in such a matrix will essentially describe how the influence it for the agent or the node in the row is distributed over all the other nodes in the system and it will sum up to one. Now, such such a matrix then describes the kind of direct influence that the nodes that I trust has on on itself. Now what we would be interested in is understanding. So, moving from this simple kind of picture that is given directly by the BTN interpreted as a command game to move from there to an understanding of what is if you wish the long range influence in the system because of course, if I'm if a node J is trusted by I and in turn J trust a node K, this K will have an indirect influence on I. So to capture this idea, we can simply take the limit of such a stochastic matrix, of course when it exists that limits does not always exist. In the situation in which it exists, I can take such a limit to understand the kind of long range influence that every node has on every other node. So, taking this this this idea we can determine that if we take if we consider a safe BTN if we consider a safe Byzantine network so a Byzantine trust network that does not allow for for forking such influence matrix will have a certain property which is called the regularity and which essentially means that it is a matrix in which such a long range limit influence exists. And furthermore that we can we can also say that it is fully regular if there exists at most one Byzantine node where full regularity means that not only that the limit exists but that all rows in the network are identity all rows in the matrix are identical, which intuitively means that every agent so every node in the network is subject to the same pattern of influence by all other nodes. So what does this theorem tells us in practice. So, essentially, it gives us these two messages with respect to this entire network in the case of ripple and stellar. So if we have a situation which there is no Byzantine node. Okay, so H capital H equals and then this we know by theorem one that I should exist still nodes that are trusted by everybody. That's implied by safety. And well, the theorem then also tells us that decent old trusted nodes are the only ones in the system that have positive long range influence or the only ones that can determine the value of consensus. If they, if there are Byzantine nodes, then actually those Byzantine nodes are the only ones that have long range positive influence. And this by a simple artifact because of course Byzantine nodes are not influenced by any other nodes so they are the kind of nodes where the entries in the matrix would for equal the in the diagonal of the matrix would be one. Okay, so they are only influenced by themselves. And as such they are the only ones that can have a long range influence. Okay, so I'll just give a quick So what we tried to provide here was an analysis of inherent to like theoretical limitations of consensus that is based on the trust network paradigm or open quarry system. And we looked in particular the problem decentralization so the extent to which such a model consensus could be decentralized. We looked at the at an attempt to give a put a number on the influence of players or nodes in such systems to understand what effect nodes have what power they have in determining consensus. And of course, this is preaching to the converted but I tried to also show how an economic theory toolbox. The ideas from command games and from power indices can be used for the analysis of systems of this type. Okay, and I conclude here. Thank you.