 Okay guys, for the final presentation of today, so it's going to be in charge of Tuana and Rihanna as I mentioned, the supervisors are Fernando Metz and Jacopo Grilli from ICTP. The last one, the first one is from Brazil, and Tuana and Rihanna are going to talk about stability analysis of ecosystems with random model interactions. Please Tuana, go ahead and start. Thank you. So hello everyone, I am Rihanna. Together with Tuana, we will be presenting our work on the stability analysis of ecosystems with random modular interactions. And our supervisors here are Professor Fernando Metz and Professor Jacopo Grilli. Next to one please. So for the other presentation, one please next. Okay. We will provide first a motivation or a background for our study. Then from these ideas and premises, we will show you our model for this project. Then we will show you our theoretical calculations and accompanying these calculations. We will be also showing you our results using numerical simulations and finally wrap up everything in the conclusions. Next please. So the starting point of our project is on the population dynamics in ecosystem. So if we let XSI be the populations of species I, we can create a first order nonlinear differential equation on the population of each species shown here. So the XI over DT will be a function that is dependent also on the population of the other species since species are interacting. Like for example, in here, in this figure, you can see that the third source population is dependent to the population of the prey. Next please. So in this first order nonlinear differential equation, we can get the equilibrium points in which in this point, the function XSI by is the extremum. So these equilibrium points can either be stable or unstable by stable. We mean that the equilibrium point when you give a perturbation to the system, it will return to the original equilibrium point. But if it's unstable, it will go far away. Like for example, in here in this pendulum, there are two equilibrium points. But if you perturb the pendulum here, you find that the pendulum is swinging back and forth. And eventually return to this equilibrium point. Well for this configuration, you will never find it returning to this unstable equilibrium point. Next please, the one. So from that, from that equation before, if we compile all the nonlinear differential equations for all species, we have this equation. And we perform a multivariate Taylor and Taylor series. We can find this leading term, which is a partial differentiation of all the species of the population species. And from this partial differential equation, we can get what is the Jacobian matrix, which is essential to linear stability analysis. So this Jacobian matrix includes all the change in the population of species I with regards to its interaction to population A to species J. And if we plug in these equilibrium points, we find this what we call community matrix, which includes all the interaction strengths between species. And it contains only constants. Next one, please. So from that, by using that community matrix and we let x tilde be x minus the equilibrium point, we can, we can find this first order linear differentiation at the differential equation. So in the end, we will find that we have this solution for x tilde and x tilde will denote how far through time will we be from the equilibrium point when the system is perturbed. So as you can see, it is just a superposition of exponentials with exponents lambda t. Lambda is the eigenvalue of this community matrix and by convention, we will set the this convention where we arrange it according to the greatest real real component because lambda is a complex number next to one piece. So you see, if we have a lambda that is, if we have a lambda that has a real part that is less than zero, we find the system to approach a stable stable state. Because if we break down the lambda into real imaginary parts where they imagine imaginary parts will give the oscillation to the system because e to the i something is just a sign and cosine. And if e is negative a is raised to the negative power you can find it that it will be decaying so you can see it will approach to a certain value. Well, on the other hand, if the real part of lambda one is greater than zero, you will find it exploding because the e raised to a certain exponent that is positive will just keep on growing over time. Note that here we used lambda one because as you can see if lambda one is assured to be negative, since it has greatest real part, then the other eigenvalue follows that they are negative. So we can be assured that the system is generally stable. Otherwise, we cannot say really if it's stable. Lambda one, we will coin a term later that we will call this the right most eigenvalue because if we plot it in the real and imaginary axis, it will be on the right most side of the plane. Next please. Next please. So again, I'm returning to the community matrix. We note that it contains the interaction of the species. So this, in fact, this matrix can be used as an adjacency matrix for a certain network that we wish to construct. Next please. Okay, so notice that if we use this community matrix, we note that if n is very large like for example 10,000, it will be difficult to note the interaction of each species to another species. So the genius Robert may propose to use random matrices as community matrix. And he found that in his work, the condition for stability for a generally a general and fully connected random matrix has this equation. So it means here that the stability of the system is related to the complexity of the network. So for example, size of the matrix if the different if there are different more numbers species in the network, you can find that it will be more unstable. And also the connectance of the matrix the interaction the number of interactions in the system and finally the variance of the elements variance of the elements of the matrix. Next please. I think what is powerful for random matrices is that it's property called this on self averaging, and that if we have a random matrix with a certain structure. And if we get the lambda, the eigenvalues, the spectrum of eigenvalues, we can see that for 100 runs, we will find a the same or not really the same spectrum but somehow follows a certain behavior as long as they have the same structure, the same structure. And here we note that there is a bulk of eigenvalues here, which will be called the spectrum of the absolute continuous spectrum which will be useful later on into one's discussion. Next please. The problem of maze approach is that it is not quite realistic, because in reality, ecological communities has different network structures. And furthermore, the connection between nodes are not really fully connected, because not really not all animals eat all the other animals in the system. So, for large food webs, there is actually a sparse connection between species. And that's one of the questions that we one of the questions that we would like to answer. What happens if the matrix becomes sparser and more structured. Next please. So, here in reality, these are not all the structures, but here are some interesting structures that are present in an ecosystem. It could be modular, for example, if we have a network of a river ecosystem and our infrastructure system. We know that the species inside these ecosystems interact with each other, but we also know that some animals interact with another ecosystem. So, basically, these two communities interact with each other. But for modular case, we emphasize that the connection within the communities is greater than the interaction between communities. And the opposite part of the opposite version of the modular network is called the bipartite, where in which the interaction between communities is greater than the interaction within communities and to the very extreme case, there is no interaction within the system. We also find a corporate structure in which the other community is denser than the other and that they interact with each other. Next please. So, from these ideas, we build our model and the main idea of our model is that we have two communities interacting within and interacting with each other. We emphasize that we will be using directed random graphs and the connection inside a community is controlled by some connectivity parameter P in which P is equals to C sub use of B over N where you and bear the communities and we divide it by N to make sure that the connection is sparse. So, here in community one, it will be controlled by C11 or we will call C out one for C for community two, it will be C to two or C out two. Then the interaction between these two communities will be controlled by C21 or C12 or CI or we will just make it CI, we will just make it equal. So, in general, with this network structure we find a black matrix of this form. So, on the diagonal components of this black matrix is the connectivity within the system and the off diagonal terms will note the connectivity between the system. And in general, we have this formula of A in which C is the connectivity matrix, meaning this will take care of the off diagonal terms of this A and D is the self regulation terms. Which will be the whole diagonal part of this matrix A. Note that if C is entirely zero, we will also note that C is just zero and one and we won't put a strength on the interactions for now. Note that if C is entirely zero, you will have only a diagonal of this matrix. And in this diagonal, we will have a, we will have their eigenvalues as its diagonal itself. So we can say that for species that are not interacting with each other. It is stable because the eigenval, the x tilde here will be e to the minus d for all eigenvalues and it will just, and e to the minus something will just decay and we note that that system is stable. Perhaps it, it may be the reason why it is called a self regulation term. And next. Okay, so can we, and for the introduction part, let me now introduce you to the result and discussion part. We built our work up on theory, analytical approach recently introduced by Philando and his collaborator in 2019. So the main idea of his method is to consider an ensemble of directed random graph, which are locally Chi like. And using that for any note into graph. His local neighborhood look like a cheese, and then using this Chi like locally Chi like structure. One important observation is that if you remove any of the note in the graph, for instance, if I remove the note J here, I will decouple the dynamic of the species. Of this note chase. So for example, if I remove chase, so dynamic of LK one and K2 become uncoupled. And this means that I can cheat the eigenvector centrality for all the note LK one and K2 as a statistically independent variable independent and random variable. And Philando has the rise is general formula, in which I chase is a eigenvector elements chase. Now is the full eigenvector and J is the J components of this full eigenvector vectors, and then the eigenvector centrality of a note chase is a sum over the eigenvector centrality of all the note K belonging to the our component of chase. And our component of the note chase, it determine as a setup on the note K, which receive a link or directed link from J's. So, and then we wait to the contributions by the factor of one over Z plus T, where these belong to the complex plane, but not belong to the continuous part of the spectrum. So the main point when we introduce this formula is we will try to find non trivial solution of the set of recursive formula. So this is the basic analytical tool we will apply in in our study, let now specify what we need to know in our model. So as we had so to do, actually for a note belonging to one of the two company committee for instance the note one, the note that I highlighted now here. So it can have one link to a note belong to the same committee but also link to the note belong to different committee. So this means that we, since we have two communities, we can split up the eigenvector into two parts, our one and our two, and for the eigenvalue, eigenvector value of any note in the first communities, we need to take care of the two contributions. What was the first one from the note belong to the same committee as this note and the second one from the note belonging to different committees. So with that in mind, we can write down these set of two equation, which are coupled together because you can see that as I show in the previous slide you have the term which coming from the contribution of the note belonging to the same committee as the note j here and also some contribution coming from the different committees. So we are interested in finding the non trivial solution of the set of equation. And seen as a note before that own in our locally Chi-Li graph, we can uncouple the dynamic of the species at different note, thanks to the locally Chi-Li structure so we can derive an equation for the distribution of the eigenvector and then from that equation we can write out the couple equation for the first moment. And here is a equation for the average value of the eigenvector centrality for the note belonging to the first committee and likewise for the note belonging to the second committees. And now you can easily solve this equation because it's the first order linear equation. And you can use the same method to derive equation for the second and the third so on and so forth moment. So it just become quite important complicated so I don't solve them here. And so the first of the non trivial solution to the first moment equation will give you the position of the outlier. So since we have a quadratic equation we actually have two outlier corresponding to one to plus and the second one to the minus side here from this equation so you remember that T is a diagonal term C1 and C2 is the link density as a mean our degree from the second and the first committee and here C i is the inter-communities links. So this is for the outlier and if you aim to look for the non trivial solution for the second moment equation you will find the boundary of the continuous part of the spectrum and it look almost the same as this equation if you may notice but actually the difference is here when actually the model of this D plus D square is equal to the right hand side. And before going to the general result let me show you some particular cases in which we verify our theoretical fighting. So on the left figure it says no zero diagonal term and on the right is the D equal 3 and you see that in the absence of D the system is unstable and standard if you look at the scatterplot of the eigenvalue empirical distribution of the eigenvalue you see that there are two outlier the same as what we predicted and the right mode eigenvalue is well positive so that the system is unstable and when we plug in the regularization D we can see the spectrum towards the left and make it stable and here we go fighting you also agree with the simulation so the tick line here is what we calculated from our theory and the second cases we look at is the corporate free network actually in the corporate free network was a result by similar to the two model network the only difference is for the corporate free one of the two committees are densely connected so this is why you observe more like the same figure as a last one, but more interestingly is when you look at the bipartite structure so for the bipartite structure to set on C1 and C2 equals zero and you only introduce a link between communities and now you can see that there is one outlier on the right mode eigenvalue and another outlier which now like behind the moon of the spectrum is quite interesting compared to the corporate free and the modular and so now since you have made some impression of what the spectrum and how the system stable and unstable under some condition some particular condition of the parameter of the model I will show you the general results that we obtained during this project so first we get stability instability diagram so in this diagram we plot on the icon Y axis the value of C1 and C2 and we made the contour plot for different value of CI so going from CI equal zero to CI equal three and the region above any of these curve is indicating this is unstable and below the curve is unstable phase so this means that as long as you increase CI you swing the stability regions so for CI equal three the stability region is just this one and on the outer part of the parameter space just unstable giving you an unstable system and another interesting picture that we see is from our analytical calculation we predict that dependent on C1 and C2 so if we take C1 as a function of C2 let's say we need to we necessarily observe a non-linear dependent between C1 and C2 on the boundary of the stable and unstable system transition and so the theory agree with the experiment numerical simulation here and another important feature that we also many to look at is another type of transition system is a transition of the spectrum of the topological structure of the spectrum itself so I think that in all the previous slides clear to do that were typically we observed a kind of continuous spectrum of all the way right with the densely located the eigenvalue and now liar but that's continuous spectrum actually can only exist above some critical value of the C1 and C2 even some constant value of CI so for example here if I set CI code one on this part have continuous spectrum but below this like below this rectangular shape here you will have the discrete spectrum and to give you an impression of how the spectrum discrete spectrum can look like this is what I plotted here for a very simple system in which on C1 C2 and CI set equal to 0.5 and the system actually consists of a set of discrete point located at random but you don't see any kind of dense continuous part of spectrum anymore. And actually when we look at CI equals zero so this mean that the transition happen with C1 equals to equal one either C1 equal one and C2 equals zero and by versa this mean that this boundary of the transition between two types spectrum actually identical to the percolation threshold so this is quite interesting to observe and now we will combine what we obtained from the stability in stability analysis and the discrete and continuous spectrum transition and we combine them into one single phase diagram and we plot it for two different value of the D and first of all we should remark that discrete and continuous transition spectrum transition does not depend on the value of the diagonal term so it's own appear here and you see here in this phase diagram but stability region does depend on the value of D so in the in select case when these small the stable phases this small reason and also rest unstable but when you increase the D of course you can make the system more stable and and this is why the unstable reasons become smaller. So, let me close with some conclusion and and future directions that we would like to go further. So, in our work we have found the analytical result regarding the conditions for the existence of the gap between the alliance of both. And secondly we find the condition under which a system is stable and the most important fighting is with able to show that model of structure in most tables on the corporate 31 and for future work we would prefer to go to study generalization the Eigen vector or to study a generalization of our setting now so we have only two community but it's quite natural to consider many more communities and we also can try to extend our approach to more general processes of ensemble of weighted and degree correlated graph. So, thank you for your attention and thank you. Thank you for supervision and the school organizer for giving us an opportunity to study this interesting topic. Thank you. Thank you to one and thank you again for this actually very nice presentation again very interesting project in in it. So, we have time five minutes or so for for questions so colleagues and students you know the drill. Oh, raise your hand. I mean, but if you raise it now I might not say it. Questions. Can you go one slide back. There was something that, since I'm, you know, jiggling with many laptops and dextors and right. So it was not clear for me this diagram what the different regions were present. Can you look it to me again please. So, let me just clarify that below this dash slide here is a stable region and above it is unstable and the zero here is the discrete spectrum case and above this slide is the continuous spectrum case. Okay, but the transition to stability to non stability is the is the dash line. Yes, it's the last slide is the last slide. So simply is that the mechanism. No, let's, let's, sorry. I think it's. I have email saying over done. Sorry, there is, I'm not sure whether Alex is there's some noise or somebody wants to, to contribute to the discussion. I'm not entirely sure either I thought it was just noise so I've muted it, but if it wants to come back on because they're making a contribution to the school then they can. Okay, okay. So, so, so sorry again so so then to add the transition from stability to non stability one stable is the dash line. Yes, it's a last slide. Then the continuous line is just simply that the mechanics is different. I mean the spectrum changes from continuous to discrete and then this is the discrete part. Yeah. Excellent. More questions. If not, last time to Anna and Rian and the supervisors for this fantastic job. Thank you very much guys. Excellent.