A new approach to dynamic finite-size scaling

In this work we have considered the Taylor series expansion of the dynamic scaling relation of the magnetization with respect to small initial magnetization values in order to study the dynamic scaling behaviour of 2- and 3-dimensional Ising models. We have used the literature values of the critical exponents and of the new dynamic exponent $x_0$ to observe the dynamic finite-size scaling behaviour of the time evolution of the magnetization during early stages of the Monte Carlo simulation. For 3-dimensional Ising Model we have also presented that this method opens the possibility of calculating $z$ and $x_0$ separately. Our results show good agreement with the literature values. Measurements done on lattices with different sizes seem to give very good scaling.Comment: Latex file with six figures. Accepted for publication in IJM

TPM: Transition probability matrix - Graph structural feature based embedding

summary:In this work, Transition Probability Matrix (TPM) is proposed as a new method for extracting the features of nodes in the graph. The proposed method uses random walks to capture the connectivity structure of a node's close neighborhood. The information obtained from random walks is converted to anonymous walks to extract the topological features of nodes. In the embedding process of nodes, anonymous walks are used since they capture the topological similarities of connectivities better than random walks. Therefore the obtained embedding vectors have richer information about the underlying connectivity structure. The method is applied to node classification and link prediction tasks. The performance of the proposed algorithm is superior to the state-of-the-art algorithms in the recent literature. Moreover, the extracted information about the connectivity structure of similar networks is used to link prediction and node classification tasks for a completely new graph

A Study of Dynamic Finite Size Scaling Behavior of the Scaling Functions-Calculation of Dynamic Critical Index of Wolff Algorithm

In this work we have studied the dynamic scaling behavior of two scaling functions and we have shown that scaling functions obey the dynamic finite size scaling rules. Dynamic finite size scaling of scaling functions opens possibilities for a wide range of applications. As an application we have calculated the dynamic critical exponent ($z$) of Wolff's cluster algorithm for 2-, 3- and 4-dimensional Ising models. Configurations with vanishing initial magnetization are chosen in order to avoid complications due to initial magnetization. The observed dynamic finite size scaling behavior during early stages of the Monte Carlo simulation yields $z$ for Wolff's cluster algorithm for 2-, 3- and 4-dimensional Ising models with vanishing values which are consistent with the values obtained from the autocorrelations. Especially, the vanishing dynamic critical exponent we obtained for $d=3$ implies that the Wolff algorithm is more efficient in eliminating critical slowing down in Monte Carlo simulations than previously reported.Comment: Latex, 24 pages, 13 eps figures. Accepted for publication in Computer Physics Communicatio

Degree-Based Random Walk Approach for Graph Embedding

Graph embedding, representing local and global neighborhood information by numerical vectors, is a crucial part of the mathematical modeling of a wide range of real-world systems. Among the embedding algorithms, random walk-based algorithms have proven to be very successful. These algorithms collect information by creating numerous random walks with a redefined number of steps. Creating random walks is the most demanding part of the embedding process. The computation demand increases with the size of the network. Moreover, for real-world networks, considering all nodes on the same footing, the abundance of low-degree nodes creates an imbalanced data problem. In this work, a computationally less intensive and node connectivity aware uniform sampling method is proposed. In the proposed method, the number of random walks is created proportionally with the degree of the node. The advantages of the proposed algorithm become more enhanced when the algorithm is applied to large graphs. A comparative study by using two networks namely CORA and CiteSeer is presented. Comparing with the fixed number of walks case, the proposed method requires 50% less computational effort to reach the same accuracy for node classification and link prediction calculations