 side count is 51, but I only have 15 minutes, but I promise I'll keep this very quick. So essentially my research is made up of different problems in stochastic mechanics and statistical physics that were merged together to get this problem. So here are the different problems that were merged together to get this specific problem versus the problem of the random walk as you all know. It's going by Carl Pearson and here's a simple implementation of a random walk in 2 degrees, showing in Figure 2. So we describe this simple random walk by taking a die, a four-sided die, and the die, rolling the die determines which direction the walker would go in this two-dimensional plane. We can actually extend that further by extending that plane into a graph. So a graph is a structure made out of nodes and edges shown here in Figure 3. So the red dots are the nodes while the black lines connecting these dots are your edges. When you, instead of having a die that has a constant number of sides, you would change this die at each of these red dots. So how would you change them? You would change them by increasing or decreasing the number of sides of this die depending on the out-degree of the node which refers to the number of edges that bleed out of these nodes. So the out-degree is given by the case of i plus while the probability for the walker to move to adjacent nodes is given by its inverse. Now we want to compute for these two quantities that relate to random walks and graphs versus first passage time or also called hitting time. This is the number of steps before a walker is starting node i to reach a target node y while the other one is cover time which is the number of steps it would take for the walker to reach all the nodes in the graph. And then we extend this another even further by considering that the random walk would reset. So resetting would mean that the walker has a probability to move back to its to its initial position at any point during the walk. So here in our figure we have a one-dimensional random walk and at some point in time there is a probability for the walker to move back to its initial position shown by these red dots red dotted lines in the figure. So what are possible applications of resetting random walks? Well we could simulate ecosystems in biological behavior. I had a question for you regarding the resetting. Is there some threshold on the timescale one puts for it to go back to its initial position? Oh for this specific problem there's no specific threshold there's just a probability for it to go back. So there's a set probability. And it can happen over a range of timescales then? Yes. You assume it's a Poissonian process so it's an exponential right hand. So possible applications of resetting random walks would be with simulating ecosystems, optimizing search processes and algorithms, describing diffusive processes, a neural spiking activity, and back to square one mechanics in board games for example. So we extend this another time by considering continuous time random walks. So continuous time this means that the walker is subjected to a waiting time so it waits for a physical amount of time before taking a step. So this waiting time is obtained from an independent random distribution. So now we move on to our formulation of the problem. So we begin with a graph G and from this graph G we obtain a transition matrix capital M here. And this transition matrix is a matrix of probabilities given by this equation one here. It's dependent on the out-degree of the node. We have this simple example here for node graph in figure six. This has the corresponding transition matrix in equation two. Now we want to include resetting on this on this graph here in figure six. To do that we have to modify the transition matrix as well since you're modifying the graph. So to get the transition matrix for a resetting random walk, we take these conditions from equation three. And from our example earlier in figure six, we have now figure seven that demonstrates a resetting random walk where the walker starts at node three. So this has a corresponding transition matrix given by equation four. Now we extend this to continuous time. First we suppose that the walker moves within a physical time t and t plus delta t. This waiting time is obtained from an independent and identically distributed variable phi where we for this problem assume it to be exponentially distributed given by equation five. So from these two distributions here pi of n t and capital phi of n t minus t prime where pi of n t is the number of steps taken by time t from taken from by time t sorry and capital phi n minus t prime is the probability that exactly n steps were made by this difference time difference here. We obtain this simplified distribution since we assume that small phi is exponentially distributed. And from here we obtain the probability mass function the continuous probability mass function for continuous time random walks given by equation nine. So this small p of n here this is the discrete time counterpart of the probability mass function. So what we'll do is we solve for this small p of n for hitting time and for first passage time and copper time. So now we move on to first passage time. To do that sorry so we have two minutes you know for yeah yes yes. So we compute for first passage time by modifying this graph the graph once again and by modifying the transition matrix and we obtain these two cumulative distribution functions and probability mass functions and that essentially again modifies the graph and modifies the transition matrix and we obtain these these plots. We can use the equation earlier for continuous time to simplify this and obtain a continuous first passage time with the corresponding cumulative distribution function and they have these plots as well. Now moving on to cover time we first discussed the concept of first passage time of union of events which means that the walker has to reach to hit all the nodes in the graph individually for it to of course cover. So you take this union of hitting all of the nodes in the graph so that you would get the you would get the eventual eventual expression for the cumulative distribution function of cover time. So we have this these two plots here for the CDF and PMF of cover time since of course these these are just very simple graphs it's not there's not much of more unique behavior so we tried this actually on we tried this actually on randomly generated graphs of course our real life systems consider this formulation here in equation 20 is not actually that efficient if you increase the number of nodes in the number of the number of nodes in the graph so we actually formulated an approximation to this equation and we found alternative computations for obtaining the cover time for graphs that have a specific configuration so we considered the bias path graph, bi-circular graph, and the complete graph as well. So these have different different formulations for cover time that are much more efficient than this more exact equation here in equation 20 but this is nice because we have a baseline comparison using this inclusion exclusion of multiple events. So all in all in my research we have discussed the effects of adding a resetting mechanism to the CDF and PMF of first passage and cover time as I just discussed quickly earlier we've applied an approximation on the expression of cover time and we've found alternative alternative formulations for cover time for the biased path biased circular and complete graph and finally we've devised computational methods that will compute for the CDF for cumulative distribution function and probability mass function of random walks with stochastic resetting on graphs so we've made a method that would accommodate how many increase an increasing amount of nodes but of course again without using the approximation this would become more and more inefficient. So we've concluded that adding a reset mechanism affects the mean first passage time and cover time depending on the initial position of the walker we found out that it generally increases the mean first passage time and cover time as you increase the resetting probability which is expected and the second one is an odd event pattern was almost always seen in the discrete probability mass functions and this is dependent on the topology of the graph and the shortest hit distance possible so that's all for my talk I think so now if anyone is interested to ask questions hello hello hello can I ask a short question we cannot hear you Ali I'm sorry Ali Najee we can't hear you can you hear me now yeah yeah yeah we are yeah we are at 13 minutes we can we can accommodate one quick question a real quick one has someone had one was that I ask if I can ask a short one yeah Maria yes please yeah I'm sorry do you know what the distribution is the distribution of the cover time you showed oh yeah this one yeah yes I'm sorry what do you know what the what sort of distribution it is um well it's dependent heavily on an exponential distribution so you have these very long tails here that would um that would actually approximate sort of and uh and exponential distribution but it doesn't look like gamma right it's not a gamma I don't think okay so yeah we can continue