Dynamical phases in a ``multifractal'' Rosenzweig-Porter model

We consider the static and the dynamical phases in a Rosenzweig-Porter (RP) random matrix ensemble with a distribution of off-diagonal matrix elements of the form of the large-deviation ansatz. We present a general theory of survival probability in such a random-matrix model and show that the averaged survival probability may decay with time as a simple exponent, as a stretch-exponent and as a power-law or slower. Correspondingly, we identify the exponential, the stretch-exponential and the frozen-dynamics phases. As an example, we consider the mapping of the Anderson localization model on Random Regular Graph onto the RP model and find exact values of the stretch-exponent \kappaκ in the thermodynamic limit. As another example we consider the logarithmically-normal RP random matrix ensemble and find analytically its phase diagram and the exponent \kappaκ. Our theory allows to describe analytically the finite-size multifractality and to compute the critical length with the exponent \nu_{MF}=1νMF=1 associated with it.

Design and Analysis of Fibonacci based TGO compared with real-time mesh using graph invariant technique

Background: Investigation of graph analytics is one among the foremost established and amazing strategies utilized in taking care of present-day designing issues. during this paper, we applied this procedure to networks. The Connectivity of gadgets is one among the intense issues distinguished in wireless systems. to deal with this issue a completely unique Fibonacci based TGO was proposed for a superior network. Methods: The proposed model takes a shot at the formation of trimet graph dependent on the Fibonacci arrangement implies a bunch is shaped with 3, 5, 8, 13, 21… hubs. Every one of these hubs is again associated recursively by trimet diagram to frame Fibonacci based TGO.practial mesh was invariant with random regular graph. Presently both these meshes are compared with edges, diameter, average degree, average clustering, density and average shortest path Results: Fibonacci TGO has approximately 50 fewer edges at 100 nodes and constant diameter 4.Average degree of Fibonacci TGO is less which is approximately 3 and having 0.7 high average clustering over random Regular. As the number of nodes increases density decreases.TGO is having a better path than random regular. Finally, Fibonacci TGO mesh has better performance and connectivity over realtime meshes in wireless networks. Conclusion: We propose Fibonacci based TGO mesh in the following steps. The major part of this formation was categorized into 2 steps. In the first step based on the input nodes small individual Fibonacci based trimet are generated with 3, 5, 8, 13, and 21… nodes. In the second phase all these trimet will connect to from a Fibonacci based TGO. Now both these meshes are analyzed with network science parameters. It was observed that in all cases Fibonacci based TGO has better connectivity over real time random mesh. Results are generated automatically by using NetworkX package in python language.

The Cover Time of a Biased Random Walk on a Random Regular Graph of Odd Degree

We consider a random walk process on graphs introduced by Orenshtein and Shinkar (2014). At any time, the random walk moves from its current position along a previously unvisited edge chosen uniformly at random, if such an edge exists. Otherwise, it walks along a previously visited edge chosen uniformly at random. For the random $r$-regular graph, with $r$ a constant odd integer, we show that this random walk process has asymptotic vertex and edge cover times $\frac{1}{r-2}n\log n$ and $\frac{r}{2(r-2)}n\log n$, respectively, generalizing a result of Cooper, Frieze and the author (2018) from $r = 3$ to any odd $r\geqslant 3$. The leading term of the asymptotic vertex cover time is now known for all fixed $r\geqslant 3$, with Berenbrink, Cooper and Friedetzky (2015) having shown that $G_r$ has vertex cover time asymptotic to $\frac{rn}{2}$ when $r\geqslant 4$ is even.