 No, okay, so good morning everybody. Welcome to the second seminar of January of our statistical physics joint STP system meeting. And it's my pleasure today to introduce to you Ginestra Bianconi from Queens Mary University of London and other to the Institute of London. Ginestra is an expert in several aspects of statistical mechanics in particular she has pioneered ideas in network theory that span several fields. And she has also had a steam tier before setting in London for several years so it's also double pleasure for us to introduce a year, and today she will be telling about their recent research on the DRAC operator and dynamics of topological signals. Thank you so much for the kind. Oh, sorry. This on. Okay, so thank you so much for the kind invitation is a real pleasure to be back in Trieste and meet so many friends. Today I want to tell you about work that we have been doing during COVID. So, you will see we we took a route that is quite original probably due to the isolation that we experienced. And in particular, I want to tell you about research work on topological signals and the DRAC operator. This works parts from the interest on higher order networks and higher order networks are characterizing interaction between two or more nodes in a complex systems typically, and they come in different flavors so they can be hyper graph in which you have interaction that are not only per wise but includes many body, I get order interaction. You have simple shell complex that are a special type of hyper graph that are amenable to topological treatment, and then network with triadic So it's going to be interesting and research has been is been done here in ICTP regarding those and, but I will not speak about those today. For instance, higher order interaction are attracting a lot of interesting brain research when you can have three region of the brain that are correlated per wise, or activated at the same time and this leads to a higher order interaction, which is depicted in a triangle formed by three nodes and trillings, but a field triangle in which you have this higher order interaction. Another very intuitive example of higher order network is for instance collaboration network where you have team that collaborated together so each paper is co author by more than true author typically. And so this has sparked the interest on higher order network. And of course one of the central team in network science has been the interplane between structure and dynamics, and there has been in network science a lot of research in this direction, in particular focusing on the combinatorial and statistical property of networks for instance how the scale free distribution change the critical behavior of critical system critical phenomena such as the easy model percolation and so on. But when you look at higher order networks not only combinatorial and statistical property are important, but also network topology and network geometry. And this is the central team of my talk today. And this is kind of the message of a little book that I published by Cambridge University Press. And this is a very personal account of higher order networks, but actually if you want to see some wider, wider point of view from a more comprehensive set of people. You can look at our nature physics perspective article on higher order network, in which we highlight the importance, the importance of topological signals. The dynamical variable, not only associated to the node of the network, as it's usual, but also associated to the links to the triangles to the theater Idra. And this dynamical variable here depicted as a clock on the links can be projected up and down on triangle or on nodes using topological tools and I will speak about that. So here we will look at higher order network in the term of simplicial complex. Simplicies are set of D plus one node, it's the dimensional synthesis are set of D plus one nodes. And they encode these higher order interaction. So it's zero simplex is one node, a one simplex is a link, a two simplex is a triangle, and so on. And they encode these higher order interaction which I mentioned before, but also they can, they are amenable to to be treated in geometry and geometry. And a face of a simple shell complex is a simplex, which is built by a subset of the node. So if you have a tetrahedra, the faces, or a three simplex the faces are four nodes, six links, and for triangle. And these are important because a simple shell complex is a set of simplices, but close under the inclusion of the phases of is simplex. So if you have a triangle, then you should also include all the links and all the nodes of this triangle in your, in your set in your simple shell complex and this is a little, a little condition in order to apply all the beautiful tool of topology to this object. Here I want to mention that so although now this talk is mostly about dynamics structure and dynamics. We have also worked a lot on structure and on modeling and this is a simple shell complex since they describe this network geometry, they, they are very interesting object to characterize network topology and geometry from data. For instance, you know, you know, fiber in the brain are play are forming manifold, or you can have fungi and things like that. One question is whether the network geometry of complex system is a priori prerequisite of the network evolution so where nodes know that are embedded into space or where there is the special aspect of this network is an emerging phenomenon of network dynamics. And this leads to the concept of emergent geometry. So a mathematical tool which allows to describe a network that are geometrical, but starting only for poorly combinatorial and statistical property of the of the model. So let's look on this approach that of course is a wide history. And, and here we have looked at a very simple model for instance a model in which you stuck a triangle to a link randomly with this one one to partner move. When you do that deterministic deterministically choosing this link randomly, and with the probability P you do this other move that is closing this pie which is also close to one partner move in a way that each link is in is incident to at most two triangle. And the network that result is this kind of very interesting manifold that is generated because the constraint that each link is incident at most to triangle is the, in addition to the connectivity of the of this, of this simple shell complex is a condition to build manifold is a combinatorial condition to build manifold so we can have this emergent geometry here, and then we, this is two dimensional. And so we went with model with higher or higher dimensional. So not only stuck in triangle to links, but also having to try to and stuck in to try to triangles. And we do all only this move of stacking a triangle to a link and we can generate a kind of the selection of hyperbolic plane. So these emergent hyperbolic play deep geometry, it's emergent in this way or you can have three, three D manifold. And when you add feature to to the faces or to the notes, then you can have transition also of of the dynamics from something that grows in every direction more or less uniformly to something that chooses one direction is also called a kind of kind of spine direction, or you can have also very cramped configuration. So, yeah, so for this model we have theoretical understanding of the homogeneous face. And like this, this, this, this space transition this fine thing is, is related to my previous work on both Einstein condensation on complex network so is a similar transition, but is also well also in network from the mathematical perspective that the transition to the both Einstein condensation is not fully understood. So this is still still research ongoing but okay so for instance you look at the diameter them out the diameter scale with the network science. This is a small world, so it scales like the logarithm of the network science here, like a beta lattice. And in this case is polynomial. This is finite dimensional but you know in one direction. Another approach to modeling is is is maximum entropy and so a configuration model and so we have also this kind of approach for simple shell complexes. Okay, so let me just now go back so this model you can take it as model of reference where we can run dynamical process if we want. And let's now look back at this diagram out this different aspect to shape the interaction between structure and dynamics. And let me focus on the in the role of topology in shaping higher order dynamics. And so topology is of course, the study of shapes and their invariant. So they connected component of a network or the cavity for instance the one dimensional cavity of a circle or the two dimensional cavity of a sphere. And these are accounted for instance from Betty numbers. And when you look at the network for instance this fungal network is clear that the cycle structure is important there. So I want to use topology to do that, and there are ways to do that for instance persistent homology you can, you can study the significance of the cyclic pattern in the structure. But actually what I want to tell you here is something different how to use topology to study dynamics topological signals. The topological signals are dynamical variable sitting not only on the notes, but also on the links or eventually on triangle or the simple as you, you like, also in cell complex you can have a square or a cube. So, these topological signals are attracting increasing attention in the context of a complex systems. There are data about it. So for instance, you can think of a citation that a team of older gods, you know, not the team of three older or pair of two older got on the paper could live together, or you can have fluxes in biological representation networks that are defined on the links synaptic signals between the nodes. There is all of sports that are pushing on edge signals at the level of brain region. And you can have also vector field so vector define on on the node for instance of a simple shell complex, but then as for instance speed of wind at given location or current at given location in the ocean, and you have a tessellation of the surface of the earth. So you project this vector on the tessellation so you define. You treat these vector field, especially as a co change so as a function defined on the links of the triangulation. In order to treat topological signals we need algebraic topology. So, a simple as I said is a set of the simple as a set of D plus one nodes. So, simplices have an orientation. So, because in algebraic topology because they are considered as the element of a vector space. So, for instance, you have a link i j and this has a different orientation of the link j i opposite orientation of the link. This is here is indicated with our role but it's not the direction is all in orientation and a triangle you can have a anti clockwise direction 123 and then, you know, you have the opposite orientation that is. With in the clockwise direction. So, with the synthesis with oriented system places you can construct and chain. So linear combination of these oriented synthesis with coefficient either in the integer or in the real. And from this chain, you can have a kind of interpretation of what is an unchanged so for instance you have 13 minus 23 plus 24, you go from 123 from three to two because there is a minus in front and then from 224 so you can construct can have algebraic expression for instance also cycle or things like that. So, on this and chain you can define a boundary operator and this is an important operator that maps each and say and chain to n minus one chain. And so it can be defined by his action on each simplex of the simple shell complex. And if you apply to a link. The boundary operator gives the two and points with opposite with two minus one. If you apply the boundary operator to a triangle 123 you get the three link at its boundary and for this is the name. You have 123 and minus 13 so you go from three to one along along the orientation of the triangle. And there's nothing to be very scared about it because these boundary matrices are only represented by matrices rectangular matrices that have no the links. So here, the boundary of the link one two is one is two minus one, and the boundary of the triangle is 123 minus 123. You construct this rectangular matrices, and then this boundary operator actually have also geometrical interpretation so be one can be interpreted interpreted as a discrete divergence be one transpose as the discrete gradient be to transpose as the discrete discrete curve. And you have this property that is a very important topological property that the boundary of the boundaries new. So because if you have a triangle, the boundaries the links at the boundary of the triangle but this is a close cycle. So the boundary of the boundaries. So practically only one dimension up or one dimension down with with the boundary operator. So, by concatenating the boundary operator you can construct a large Laplacian, which describe diffusion from and synthesis to and synthesis either going one dimension down and minus one, or one dimension up. So the for the node you can only go one dimension up and this is the graph Laplacian for the links you can go down and up and if you are if we have only know the links and triangles for for for the, for the triangle you can only go down to the link. And so this odd Laplacian can be written in this way so you go either one dimension up or one dimension down so you can go from no from link to link to note or from link to link through triangle. And so, certainly this odd Laplacian encode topology because the dimension of the kernel is then Betty number, and actually also the again vector of the kernel can be. There is a basis of the kernel, they can vector of the kernel in which they can vector are localized on the cavity on the n dimensional cavity. And this Ellen down and Ellen up they have very interesting property because the image of one is in the kernel of the other. And this leads to a very nice property that is called odd the composition, which tells you that if you have a seniors on the links that is a unique way to write it as a gradient flow, going away from a source to a sink and harmonic component along the cavity and a curl flow or solid on your component. Okay, so this is a very nice tool that is taking a lot of interest in in higher order networks to treat topological signals. And what we did is we use it to define a kura motto model on for topological signals. So the kura motto model is this model proposed by kura motto in 75. So you have faces associated to the nodes of the network. So the faces also laid at their own frequency, and then they are coupled in such a way that each node can try to align to the face of the nearby node. And in a fully connected network in the infinite limit you have a phase transition between an unsynchronized phase to a synchronized phase with disorder parameter. And despite each phase as his own intrinsic frequency that is drawn randomly from a Gaussian distribution for instance so they, there is no coherence if there is no interaction. Then if the strength the coupling constant is larger than a bit there is this finite fraction of oscillator that starts to oscillate together. And the question is, can we define this kura motto model for fancy places associated to the links. Okay, so, so what we did is it's quite simple so we started by an expression of the above the above equation for the kura motto model in the boundary operator this is exactly equal to the equation that kura motto road. And so, in terms of the boundary operator you see that the coupling couple face of the node if they share a link in this non linear type of coupling. And so what we did is we def. So this is a vector of the phases of the node with this is the vector of the frequencies internal frequency of the links and this is the coupling term. So what we did is we consider a phase associated to the links now intrinsic frequency associated to the links and then the coupling such that link to link try to align to each other. So if they share a triangle or if they share a node. And the linearized dynamics is. Can they be that there are sites which are not coupled by anybody. Well, if they are easily isolated. It means, not to any triangle like. Yeah, but it is, there is the other term. So they can be con links can be. Are coupled. If they are connected by triangle or they are connected by a link and if they are connected by. So the linearized dynamics is the one defined by the large Laplacian. And from this is apparent that the harmonic moment. So when the fee is aligned into the kernel of the Laplacian, the harmonic model are the one that are free to oscillate and the other modes are dumped. The synchronization that remains if the other mode freeze is actually only a synchronization along the cavity. So the dynamics select the cavity and oscillate only along the cavities. So in order to measure what happens to the other mode, we, we filter so we project the face on the links to the triangles or to the nodes, and the space. And from this projected dynamics that is now totally the couple, we can see that essentially this other mode that are not harmonic freeze with with when we raise the coupling constant sigma. And we have two other parameter, one for the faces projected on the node for instance and one for the faces projected on the links. So what we do then is what we can kind of modulate the coupling constant of one term coupling links that share triangles with the older parameter of the projected dynamics on the node and vice versa. And the two projected dynamics are not anymore the couple and we get an explosive transition discontinues. So this is a configuration model of simple shell complexes, which I mentioned before, is a kind of random graph, but we did it also on NGF, and we did it also. On the on real data set. So we can have real connect on we build a simple shell complex from in a way that is called a click complex, and we find this phenomenon so it seems to be quite independent on on the topology, although I mean we want to to go back to this so I can check it carefully. So in particular, I think the cavity distribution would maybe in on a random graph. Yes. So this is on a fully connected network on a random graph. Yes, you have a transition. The analytic tool to characterize this transition are not exact, but there are approximations so there are the annealed approximation which is called that we also use to to you for our model gives good result on random graph for instance, doesn't need to be fully connected. Yeah, yeah. Although with the network with spectral dimension, you might have some some instability of the fully synchronized state. So there is a critical spectral dimension that should be, you can have some kind of frustrated localized synchronization. So the question is, can we now have a dynamics, the couple node and links. And so for this we need, we need the data cooperator, we need to leave the ocean laplacian and treat the odd, the data cooperator. So here, I just wanted to put this slide to be sure so the data cooperator is not something we invented or I invented. I saw it fast in a paper by satellite garnet on and Zanardi in 2016, which is a very nice paper proposing quantum algorithm for topological data analysis. They don't cite anybody for the data cooperator. So I worked in my group for a long time without having other reference for that. But recently I find a very interesting reference by Olaf post in 2009, which is a mathematician, which identifies a lot of property of the data cooperator. And also he doesn't cite anybody. And but I could go back and see that Davis as in non commutative geometry as a paper in which he mentioned the data con graph. And this is as far as I could go back. Because there are also other paper that sites this paper but and and we are working quite intensively on that, and starting from my 2021 paper. And I will, I will speak about this work. So the data cooperator us on a topological spinner. So you have, for instance, if you have no links and triangles, you have a node signal, a link signals and a triangle signal. And the data cooperator is defined in terms of the boundary. And so this is a block structure between node links and triangle. This is when you have a simple shell complex of two dimension but you can go higher. And so when you applied the data cooperator on the topological spinners. The links to be projected on the nodes, the nodes to be projected on the links and so on so you can allow cross talking about these signals in different dimension using the data cooperator. What is it called the data cooperator, because it's most characteristic property as did I can use paper was looking for is that the square of the data cooperator is the laplacian so is a matrix was diagonal. And so what are the large laplacian. And. Okay, so of course they can value of the data cooperator will be the square of the again value of the laplacian, the laplacian so you have positive and negative again values not anymore positive definite. And so you can decompose in the one and the two the one only couple northern links, the two only couple links and triangle. And for this, direct the composition that we call you, you can play the same game of odd the composition so you can have that that anything, any topological spinners and not links and triangles can be decomposing a unique way into northern links links and triangles or an harmonic component. And so the DRAC operator as the spectrum that are these square plus or minus the square root of the laplacian, and then plus zero again value which has this generously the sum of the Betty number. And they can vector are they can vector either of the one or the two or harmonic and they can vector of the one are related by chirality. The, the again vector of the one of this type either you won we won or we won minus we won where you won and we won are the matrices that indicate the left and the right singular value of the boundary operator. Let's start to consider on a network on a network. This definition of the data cooperator is can be written as D plus the star where D is the serial derivative and the star is is dual. So this is a self adjoint operator. And so this is the most basic way to define the DRAC on a network. For the moment not. Okay, but. Okay, so let me, let me go. I think, I think DRAC is correct in this case so you can put a complex number in front. As long as this modulus is one, the square will be the laplacian as well so this complex number as we will see plays the role of a gamma metrics and as if you work for instance in one dimension or two dimension. This is enough. And if you want to have like this instead you need to kind of upgrade to a gamma function. You have a topological spinner that now is defined on node and links. So, the vector defined on all the nodes and all the links so this is a spinner but as a geometrical interpretation. So let me start writing the equation in DRAC equation in an Hamiltonian form. So with this DRAC operator that I defined before this D plus the star and beta is this kind of matrix familiar matrix where D and beta anticommute. And these of course is related to the chirality of of the again back store of the. Just to show you that you can do this, you can study this, this again, again value problem. And when you do that you see that key is an again back to the graph laplacian with a and the, and see is a game back to the L one down laplacian. And the dispersion relation is the relativistic one. So, the energy state are of course positive and negative. And so you have this kind of matter anti matter symmetry but the state of energy equal to M, they correspond to the harmonic again vector. So, the pattern might not be symmetric on a chain you have the the harmonic again back to the nodes and the links is the same, because for the quality a chain and no then the links are the same. But in general for a network you can have, you might have more states at value energy M or than energy minus M or vice versa. So, they get back to it can be represented on any network, you know for a certain node and links you can plot it play with this. And of course, it's also interesting you can play with weights weights of the node and weights of the need because well, of course I mentioned to you the laplacian but that is also the way that laplacian or the normalize laplacian so you can introduce some weight. And, and have a data cooperator in which now here you have the odd to all of the, this is, this is the co boundary and this is the odd to all of the co boundary. We applied this to many classical problem. So, for instance synchronization again, we find this discontinuous transition forward transition and continuous backward transition, but as a function of Z of this Z so with this non linear coupling, we can have to this continuous transition, and here the node and the links are coupled locally so the node feels only the nearby links and the link feel only the nearby nodes. It's quite different from what I show you before with this global coupling constant that that was depending on the order parameter here is the dynamics is local and you have this phenomena. The linearized dynamics is this one. The harmonic component are the one that remain unchanged, while the other component here. We treat the, the, the component along the vector lambda and minus lambda together. So they, they have a, they don't freeze with a dumping with a stable fixed point but they have a stable focus. So there is a kind of rhythm at merging and also in the non linear model. And this emergence of low frequency that we are quite fascinated with. So this is treatment that emerged from the dynamics. For the moment we don't have a theoretical understanding. In addition to to the fact that the linearized dynamics is a stable focus but for the fully non linear model. This emergence of low frequency it may just a numerical observation for the moment. Yeah. So, so if you put that equal to zero. These dynamics the couple and will be the dynamics of note, the standard model, and the standard link link connected to note. So this z, then you have this local coupling of northern links so this will start speaking together. So, yeah, in some sense you can go back to. We also consider touring patterns. Not all topology satisfy the condition to have touring pattern on their structure, but square lattice do so we can have stripes we can have more complex pattern. And we are also applying to signal processing. We have signals on the nodes and on the links and you want to do signal processing jointly. But now I want to tell you something more about three plus one dimensional like this. So, if you have a. We see that we have D plus the star, this is the data cooperate on a graph, but we can play with this complex number. And so this complex number can be used to play and distinguish between link index direction or in the wide direction so when I do the boundary on the x direction I put to be equal to one and when I do the boundary on the y direction I want to be equal to I so that I can distinguish between the two direction, but in three dimension. I need more. And also in four dimension three plus one dimension I need more. So this is what I tried to do. So I want to distinguish between four different direction x, y, z and T. I want to have weight on the links, eventually. So I now change a bit to this scenario. And I have the function on the links is now not to define on each link. So in such a way that is known localized. But each link is directed so you have I to J. And I have a function defined on the link I to J, which in some sense describe a flux going from I to J. And a function defined on J to I such that they define the opposite direction flux so then this way they, the faith function is now localized so I don't have any more this problem that the theory is not localized. And since I want to distinguish between these different direction, I have a part on the node a part on the directed link, but the part on the node will be double. So I will have to function define on the node and to function define on the links. And the, the matrix the weighted matrices here are e to the a one and e to the zero. So in algebraic topology these metrics are taken real and diagonal here I took the liberty to take it more general. Although there might be geometric constraint that that I'm not taking into account yet that that needs to apply. I found a re operator that is for each link is one and minus one one link at one and minus the link on the other land, but these are only on link in the x direction in the y direction in the z direction. And I make it weighted with this weighted matrices so G zero is a node by node matrix G one is the link by link matters. So we can have the laplacian which is be one be one dagger now and in direction x and the laplacian in the old network is the sum of the Laplacian x y z and eventually also LT. If the network is not weighted this Laplacian commute, but if it's weighted in an arbitrary metric this Laplacian might not commute. Okay. Instead of having the complex number I have now the gamma matrices that take this form and they're expressed in terms of the Pauli matrices. And the, the data cooperator the serial derivative and links to all are indicated in this way so where I practically double because I have two. And two zero crocheting so this is a matrix that is. Two by two in in this block form. And so the data cooperator now as the gamma D mu plus D mu star. So we have to contract. And the way we contract the indices is by assuming that the metrics is Euclidean so we absorb the complex number into the, the, the signature of the metric into the definition of the gamma matrices. The square of the directional data cooperator is the Laplacian for space spatial one. But for the time, time, like directional operator, the square is the Laplacian with the minus in front, as we want. So that I noticed in 2001 is that this data cooperator do not anti commute and do not commute either. So it's not like the derivative that commute is in many cases is like a derivative but they don't commute. So, because if you go from, if you go from a node to a link index direction. And then you have an operator that go from a link in the y direction to a node you find zero because you are on the X links on the X direction, but if you are from the link index you go on the node, and then you go back in the link on the why, then you have a non zero term. And so you have this non zero common anti commutator and commutator. And one can interpret the commutator as a curvature. This is non zero only in the link link sector. There are already two other indices hidden here that are the indices of the link one non zero in the X direction here and one non zero in the y direction. So, one might want to define a riches color by summing over these. These matrices the element in the link mu and the element in the link new. And so here it becomes more and more speculating my argument but I think it's a kind of program direction so you can have a anti commutator that is also non negative and from from the spatial direction I call it. magnetic magnetic field. And so you can define the data cooperator where the, the, the nodes, the node part follows practically the claim Gordon equation. And depending on the laplacian in time and in space commuting or anti commuting you can have different different discussion relation because this p square might not be the sum of px plus p y plus p z because if they don't commute is the Agen vector of their sum. And when they all commute you get the, the relativistic dispersion relation, and, and then I want to mention here, one calculation which is quite particular when you have the time and the space laplacian with not which, which commute but the space laplacian commute between each other. And then we have key that follows the claim Gordon equation psi follows the shreddinger equation with the general magnetic constant to and the game, the magnetic field is played by the by the anti commutator of the data cooperator this is why we call it magnetic fields and interesting enough. The key is non negligible is of order e to e minus m. So, we can write some action for the metrics are contractor at new new. And then we can try to minimize or contract it four times along, for instance a placate. And of course one can play the game and try to think, propose some equation, which would be an equation of motion of both the matter fields so the topological spinner and the weight. So this matrix a one a zero because these action is only determine in term of the way. So let's go back to complexity. Of course they're not only three dimensional lattices. Of course, in general in a graph you would like to have a gamma metrics the right algebra for any generic graph and this is a very challenging thing to do. If you applied the side one can have also multi layer networks in which you can have two nodes connected by two link only one link or only the other link in the other layer. So you can consider these three different type of links and treat them with the directional data cooperator. And in this case, the, the Laplacian do not commute with each other typically. So I reached my conclusion. Thank you so much for your attention. So, here we have covered a series of work on topological signals which are. I believe I hope you believe you see is an amazing objects. So they are dynamical variable not only associated to the note but also to the links. And they can undergo synchronization and this synchronization as a relation with topology and in particular with the homology of the simple shell complex. And when we want to treat topological signal of different dimension we can couple with the data cooperator. And this data cooperator as a lot of interesting use they can use in classical dynamics in a very wide set of scenario. But I think also exploring the fact that the data cooperator my propose a field theory in which the spinner is as a geometrical interpretation is quite interesting and of course, possibly this kind of idea could be maybe also simulated in in some scenario so just constructed on, I don't know, quantum quantum quantum material right to construct an artificial field theory of this type. Okay, so I need to thank all my collaborators. My group is rather small, I should admit, but these are people that have worked with me over the year. So, at each time, the team has been very small. And this is the selected recent reference. Thank you so much for your attention. Let me start. When you define the possible actions that you get for this spirit, I would have said okay one just take the trace of the black cat. And the which is kind of the square term. And the trace of that we know if you know locally on a single day which is the square term. Why don't you keep the simple choices that are also at this first time. So, yeah, so the history is that I took this, this fast to action fast. Yes, so these are along the placets. So this, this can can be a kind of curvature going along the placets. And this kind of electromagnetic like inspired expression. Because the other one. Yeah, so it's very strange because this f this commutator and anti commutator are quite interesting. And of course the anti commutator as, as the, the term in which the demo demo right that is known zero. But then, when you do a new new a new new. And when you go and do this calculation together is a bit like the electromagnetic interaction also is like you have also the magnetic part because because of this particular structure so practically what changes only the sign in between. And when you multiply this anti commutator with itself, essentially, then you have, you go from Y link to X links and then you need to contract with X links with Y links. And so you have only a sign of difference, maybe between the two, when you do the square of this one. It's very strange. Yeah, it is like they are strictly related to this to quantity so writing in term of the anti commutator is, it's not so different from writing in term of the commutator. Yes. Yeah, so. Okay, I let me repeat the question the question is that, if you have no synchronization. So you have this synchronization could amount of like, and then for a random graph, similar to kind of like, but then if the network have a geometry and a spectral dimension. Then this synchronization might be unstable and then you have this frustrated synchronization. And so we have work on this and we have seen also these uppers and spectral dimension and year for direct synchronization essentially we did not check carefully on simply shell complex with the spectral dimension it might be that when you have the complex synchronization. It might be that it is important not only the graph spectral dimension but also the simple shell complex spectral dimension. There might be an interplay because one can have spectral dimension both of graph and I wrote that I wrote the dynamics so there might be a way we are exploring that. Yeah, let me say that she will be around the week so in case you want to know. Thank you next again. Thank you.