 First of all, I'd like to thank the organizer who allowed me to give a talk here. My talk is a little bit different from the previous ones. I'm talking about the some population problem, but it has a long range of connections. The list of my talk is this much, but I don't know how to do, you know, 30 minutes. Okay, so let me first introduce some previous work. In the spin system, as you know, that the soulless propose some wonderful R-square interaction. And when the interaction between the two ising spins, even like this, the very different behavior occurs, and the transition becomes discontinuous even in one dimension, and the correlation lengths diverge with some essential singularity like this. So it exhibits hybrid mixed order phase transition. Recently, in this paper, they are a little bit modified of the original model. The interaction is given within the same domain, and in that, for this case, the correlation lengths diverge in a power-low way, so this case also exhibits the mixed order transition. Let me first introduce the, we are going to consider the similar problem in population. So population transition, population model was introduced to illustrate the metal and insulate transition. At each site who are born randomly pick up, and that this site is occupied by metal, and the fraction of metal is given by P, and P is a control parameter. If we increase the control parameter P, then there is some critical threshold value at which the spanning cluster is generated that is connected from top to the bottom like this. So in the population problem, the order parameter is the fraction of occupied site belonging to the spanning cluster, which exhibits the second order transition with the exponent beta. In one dimension, because if one site in one dimension is missed, then there's no connection to the spanning cluster so that the PC is equal to one, so it's a trivial case. If the dimension is higher than the upper critical dimension, then we can ignore the lattice structure so that it reduced to the so-called random graph model introduced by Erdisch Rayleigh. The system, the N node exist, and we pick up two nodes randomly and occupied with the probability P. So in actually, we select two nodes and occupied P and so on, set that, so the control parameter is number of links L, and the T is defined as L over N, and which is equivalent to the mean degree over two. So this is the control parameter. And then this model exhibits also the population transition and with the second order behavior. In long time ago, about 1983, Schurman introduced some long-range connection population problem. So with the probabilities of the I side and J side connected with a bond with this probability, here the control parameter is a P and S. When S is between one and two, like this, then there exist some finite special value, PC. Above PC, there exist some, the spanning largest cluster, the P. And when the S larger than two, then PC equal one, so there is no giant component in the system. So P infinity equals zero for all P. So in this problem, we have some long-range link between the I side and J side. So this problem already is different from the long-range interaction problem we discussed so far. And later, the Eisenman and Neumann consider this model. So for given finite distance L in one dimension, and if the PI side and J side is occupied with the PIJ with this probability, then there exist the control parameter beta here and P, beta and P. So if beta is larger than one, it can be a population transition, but this is a discontinuous population transition. And so the jump is larger than some beta one minus one half here. So but beta is less than one, then there is no finish transition, and so there is no spanning curse. Later in one dimension, these people already remodified here, instead of P, here is one, and then by this short-range connection, the population transition occurs, population cluster occurs. So in this problem, in this paper, they are interested in the diameter of the population cluster in actually in T dimension. So what they obtain is that S is equal to one. They calculate exactly analytically for one-dimension case, but also they extended and conjecture for general dimension D. That's when S equal D, the diameter proportional to log N over log log N, but in this region, diameter proportional log N to this delta, delta is larger than one, and for this case, it's a power low way, and omega is less than one. Interestingly, when we construct the energy range graph, the mean distance is proportional to log N, also for the scale-free network case, when the degree displacement is even proportional to K to the lambda, lambda is the degree exponent, and when lambda is between two and three, the mean distance is log N over log log N, but lambda is larger than three is log N. So this result, and maybe it has some split, shared a split with this case. So we are interested in some physical characteristics of the problem, one, the energy range model, for them. Let me, so in this case, the long-range connections plays a very important role to determine the types of phase transition. Let me first introduce some epidemic model, so-called SIR model, so susceptible-impacted node, recovered model, so each node on the energy range graph can have three states, susceptible state or impacted state or recovered state. When as a node of labor nodes in susceptible state and impacted state, then this susceptible state becomes impacted by this node with a probability kappa here, and later the impacted node becomes recovered node, so it's no longer impactious. So, for example, this node is impacted here, and this node becomes recovered, then the dynamics stop and no longer proceed next time. And then, we are interested in the SIR model, on the populating cluster, not only at the critical point but above the critical point here, and then for this case, the dynamics proceed in the critical branching tree so that the mean branching ratio becomes one here, so mean branching ratio will be one, so that it usually population transition like this when at the critical point kappa C. At this point, the number of nodes with the state R is scaled as n to the third, so when you normalize it, then it's n to the minus one third, so that in some of the limit, it becomes zero, so ordinary second order transition occurs. So, however, when you introduce some intermediate state W between the susceptible and the infected state, then, so this is the susceptible, weakened and the infected recovered model, so-called SWIR model, then the transition becomes a hybrid transition like this. There is some discontinuous jump and continuously increase. Here, the control parameter kappa, new and lambda, but kappa is most important. So, we can draw the very complicated phase diagram analytically but I would like to say just to say that when the, there is just some critical value of mean degree, kappa k mean degree above which the phase diagram looks like this, so that there is some R A, kappa C. Below kappa C, the order parameter zero and jump like this here and continuously increase. However, when the mean degree is less than some threshold value, then this m star here, m star becomes zero so that the discontinuous transition does not occur. The susceptibility for this case diverge here like this. For this case, here's a, we start with an impact node here, then with some probability kappa with the neighbor, one of the neighbor becomes, a neighbor becomes infected like this, a neighbor can become the weakened state like this and so on. And then after a long time later, so a long time later, impacted the node here and then its neighbor becomes a W node here because there is a long range connection in the Erisirini model when the mean degree is larger than the threshold value. Then there's a long range look here. So this, by this long range connection, this weakened node becomes infected node and so that the process, branching process behave like this. So up to some time here, the critical branching process this way, the impacted node here, critical branching process dynamics proceed up to some here and equal, this process decay like that. However, of long time later here, the W plus i becomes two i process begin to occur here like this process so that the infected node added by this process so that number of recovered node increase above the one, this is a mean branching ratio. So it becomes a super critical behavior so that the order parameter, the order parameter increase suddenly by that super critical behavior like that so that hyper hybrid transition occurs here like this. So that formation of the long range look like here, the major place, the connection of the long range link plays an important role to make the hybrid population transition. Okay, this mechanism can be applied to other hyper hybrid population transition, for example, k-core population. So here is also the W i to the i to the i, two i process increase later here and by this ingredient so that it becomes a super critical behavior so that for k-core population case, there is a sudden drops of the order parameter, these drops is induced by this process like this. So the dynamics is almost the same as I told you for the SWIR model case. Also the cascading failure models in mutually connected multiplex network dynamics case also exhibit some sudden drops of the order parameter. The dynamics can be explained in the same way we as we talk in the SWIR model. So that such a long-range look connection made and so that the order parameter collapse suddenly. Let me move on to the another program. So for the, can we obtain the discontinuous population transition using the Edo Shireini types process. That's the so-called explosive population model. Okay, explosion population model. In 2009, these people introduced some so-called explosive population model. They modify the Edo Shireini model slightly. In the Edo Shireini model, two nodes are pick up, two nodes are pick up randomly and connected by link. But for the data model, we select two pairs of nodes, this one and this one. And we have to determine which node indeed connected to the system. So if we connect this link, then the resulting cluster size is smaller than this one so that we connect to this link and we discard this link and so on. And then they said that there exist some discontinuous transition like this beyond the population threshold of the ordinary population and here, right there. However, later, this continuous behavior valid on only the final system. In infinite system, this jump reduce so that it becomes discontinuous transition. So in the later paper in science, the local modification of the Edo Shireini model cannot guarantee any discontinuous transition. So we need some global types information, modification is needed. Later, our group introduced some so-called avoiding spanning cluster model in each the dynamic rule is defined as follow. So in two dimension, for example, we select two candidates of bond B1 and B2 here. So for example, if we occupy this link, this bond, then there is some spanning cluster between left and the right so that we avoid this B2 link bond instead of B1 link here. So if we occupy the bond B1, then there is no spanning cluster between left and right. And we proceed this dynamics to go on and then at certain time to see that we choose the two bond B1 and B2 and both make the cluster connected so that we have no choice but to choose one of them. So the spanning cluster is generated this way. In this case, in this case, this continuous transition occurs this way. When the M, M, this is the case M equal two. However, we can generalize the four M case. Then if M is large, then then this, there is this, some discontinuous transition. When M is large, that means that we need, we have enough information, global information to choose the one which one is the best way to suppress the spanning cluster. So analytically, we show that when M there exists some MC value, which is proportional to global M, when is M is larger than MC, then global information is enough to guarantee the discontinuous transition. So to get the some discontinuous transition, we need global long range information and we can make some spanning cluster discontinuous. Recently, we introduced another model to show which shows the some hybrid population transition. Again, this model contains some global dynamic rule. We classify the node into two subset. Here, the R subset means that in this subset, the cluster with the smallest size contain. So total number of fraction of the node belonging to this subset is given by G. G is some control parameter, G is less than one. So when G is equal to one, this model reduced to the Erdos-Reynm case. So dynamic rule as followed, at each time step, we pick up the one node from the entire system. So here or here, and the another node is chosen from only the smallest subset, right here. So two nodes pick up from one here and here, then this cluster is merged. If the size is larger than this border, then this cluster will be here like this. And then the hybrid population transition and the cluster size distribution at the transition point follow the power row and the exponent tau depends on the control parameter G. So we can find that the critical exponent of the hybrid population transition depends on the control parameter G. So this hybrid population transition shows the continuous varying, has the continuous varying exponent value. Like for this case or the previous case, for the hybrid population transition, does not guarantee the hyperscaling relation. So okay, this is the summary. I stop here, okay. Thanks a lot for nice talk, so out there.