Random walks with long-range steps generated by functions of Laplacian matrices

We analyze a Markovian random walk strategy on undirected regular networks involving power matrix functions of the type where L indicates a ‘simple’ Laplacian matrix. We refer to such walks as ‘fractional random walks’ with admissible interval . We deduce probability-generating functions (network Green’s functions) for the fractional random walk. From these analytical results we establish a generalization of Polya’s recurrence theorem for fractional random walks on d-dimensional infinite lattices: The fractional random walk is transient for dimensions (recurrent for ) of the lattice. As a consequence, for the fractional random walk is transient for all lattice dimensions and in the range for dimensions . Finally, for , Polya’s classical recurrence theorem is recovered, namely the walk is transient only for lattice dimensions . The generalization of Polya’s recurrence theorem remains valid for the class of random walks with Lévy flight asymptotics for long-range steps. We also analyze the mean first passage probabilities, mean residence times, mean first passage times and global mean first passage times (Kemeny constant) for the fractional random walk. For an infinite 1D lattice (infinite ring) we obtain for the transient regime closed form expressions for the fractional lattice Green’s function matrix containing the escape and ever passage probabilities. The ever passage probabilities (fractional lattice Green’s functions) in the transient regime fulfil Riesz potential power law decay asymptotic behavior for nodes far from the departure node. The non-locality of the fractional random walk is generated by the non-diagonality of the fractional Laplacian matrix with Lévy-type heavy tailed inverse power law decay for the probability of long-range moves. This non-local and asymptotic behavior of the fractional random walk introduces small-world properties with the emergence of Lévy flights on large (infinite) lattices.

Efficient sampling of complex network with modified random walk strategies

Meta-path-based random walk strategy has attracted tremendous attention in heterogeneous network representation, which can capture network semantics with heterogeneous neighborhoods of nodes. Despite the success of meta-path-based random walk strategy in plain heterogeneous networks which contain no attributes, it remains unexplored how meta-path-based random walk strategy could be utilized on attributed heterogeneous networks to simultaneously capture structural heterogeneity and attribute proximity. Moreover, the importance of node attributes and structural relations generally varies across data sets, thus requiring careful considerations when they are incorporated into representations. To tackle these problems, we propose a novel method, Attributed Heterogeneous Network embedding based on Aggregate-path (AHNA), which generates aggregate-path-based random walks on attributed heterogeneous networks and adaptively fuses topological structures and node attributes based on the learned importance. Specifically, AHNA first converts node attributes to additional links in the network to deal with the heterogeneity of structures and attributes, which is followed by an adaptive random walk strategy to strike the importance balance between node attributes and topological structures, thereby generating high-quality representations. Extensive experiments are conducted on three real-world data sets, where AHNA outperforms state-of-the-art approaches by up to 22.7%, 2.6%, and 2.3% on link prediction, community detection, and node classification, respectively. Moreover, our qualitative analysis indicates that AHNA can capture different balances of topological structures and node attributes on various data sets and thus boost the quality of node representations.

We study the diffusive transport of Markovian random walks on arbitrary networks with stochastic resetting to multiple nodes. We deduce analytical expressions for the stationary occupation probability and for the mean and global first passage times. This general approach allows us to characterize the effect of resetting on the capacity of random walk strategies to reach a particular target or to explore the network. Our formalism holds for ergodic random walks and can be implemented from the spectral properties of the random walk without resetting, providing a tool to analyze the efficiency of search strategies with resetting to multiple nodes. We apply the methods developed here to the dynamics with two reset nodes and derive analytical results for normal random walks and Lévy flights on rings. We also explore the effect of resetting to multiple nodes on a comb graph, Lévy flights that visit specific locations in a continuous space, and the Google random walk strategy on regular networks.

Recent researches on multiview learning have received widespread attention due to the increasing generalization of multiview data. As an effective probabilistic model, random walk has also shown encouraging performance in various fields. To further exploit the potential of utilizing random walk schemes to address multiview learning problems, this article proposes a simple yet efficient multigraph random walk scheme for both multiview clustering and semisupervised classification tasks. The proposed model integrates random walk with multiview learning, and recursively learns a globally stable probability distribution matrix from multiple views, on the basis of which the label indicator is obtained in the scene of clustering or semisupervised classification. Furthermore, an adaptive weight vector is learned to incorporate the diversity and complementarity of multiview data. Besides, the relationships between the proposed scheme and spectral clustering, neighborhood embedding and manifold embedding are analyzed theoretically. Finally, comprehensive comparative experiments are conducted with several state-of-the-art multiview clustering and semisupervised classification methods on eight real-world datasets. The experimental results demonstrate the superiority of the proposed method in terms of both clustering and classification performance.

In smart grid, smart meters are deployed to collect power consumption data periodically, and the data are analyzed to improve the efficiency of power transmission and distribution. The collected consumption data may leak the usage patterns of domestic appliances, so that it may damage the behavior privacy of customers. Most related work to protect data privacy in smart grid relies on cryptographic primitives, for example, encryption, which induces a large amount of power consumption overhead. In this paper, we make the first attempt to propose solutions without any cryptographic computation to protect user privacy. The privacy in smart grid is formally defined in the paper. Three schemes are proposed: random perturbation scheme (RPS), random walk scheme (RWS), and distance-bounded random walk with perturbation scheme (DBS). Three algorithms are also proposed in each scheme, respectively. All schemes are ultra-lightweight in terms of computation without relying any cryptographic primitive. The privacy, soundness, and accuracy of proposed schemes are guaranteed and justified by strict analysis.

Grey wolf optimizer (GWO) is a widely applied nature-inspired optimization algorithm with simple search mechanism, which suffers the deficiency of premature convergence. To improve its exploration performance and avoid trapping in to stagnation, we integrate the basic GWO with two random walk strategies, Gaussian random walk and Levy walk, and this new GWO extension is named RWGWO. The performance of RWGWO is compared with those of other comprising GWO variants by using several standard benchmarks. Moreover, RWGWO is applied in solving constrained engineering design problems. With a view to the experimental results, RWGWO has remarkable performance compared with the competitors in this work in terms of convergence speed and accuracy.

A random walk scheme, consisting of alternating phases of regular Brownian motion and Lévy walks, is proposed as a model for run-and-tumble bacterial motion. Within the continuous-time random walk approach we obtain the long-time and short-time behavior of the mean squared displacement of the walker as depending on the properties of the dwelling time distribution in each phase. Depending on these distributions, normal diffusion, superdiffusion, and ballistic spreading may arise.

Many authors defined and extended the beta function in various forms because the beta function has wide uses in different fields of science and applied science. In this article, we define a new more generalized form of the extended beta matrix function via the Wiman matrix function and describe their significant properties and special cases. Furthermore, we define an extension of the Gauss hypergeometric and confluent hypergeometric matrix functions by adopting a novel type of beta matrix function. We also derive their Laplace transform, derivative formula and transformation formulae.

For network graphs, numerous graph features are intimately linked to eigenvalues of the Laplacian matrix, such as connectivity and diameter. Thus, it is very important to solve eigenvalues of the Laplacian matrix for graphs. Similarly, for higher-order networks, eigenvalues of combinatorial Laplacian matrices are also important for invariants of graphs. However, for large-scale networks, it is difficult to calculate eigenvalues of the Laplacian matrix directly because it is either very difficult to obtain the whole network structure or requires a lot of computing resources. Therefore, this article makes the following contributions. Firstly, this paper proposes a random walk approach for estimating the bounds of the greatest eigenvalues of Laplacian matrices for large-scale networks. Considering the relationship between the spectral moments of the adjacency matrix and the closed paths in the network, we utilize the relationship between the adjacency matrix and the Laplacian matrix to establish the relationship between the Laplacian matrix and the closed paths. Then, we employ equiprobable random walks to sample the large graph to obtain the small graph. Through algebraic topology knowledge, we obtain the bounds of the largest eigenvalue of the Laplacian matrix of the large graph by using Laplacian spectral moments of the small graph. Secondly, for high-order networks, this paper proposes a method based on random walks and graph transformations. The graph transformation we propose mainly converts graphs with second-order simplices into ordinary weighted graphs, thereby transforming the problem of solving the spectral moments of the second-order combined Laplacian matrix into solving the spectral moments of the adjacency matrix. Then, we use the aforementioned random walk method to solve bounds of the greatest eigenvalue of the second-order combinatorial Laplacian matrix. Finally, by comparing the proposed method with existing algorithms in synthetic and real networks, its accuracy and superiority are demonstrated.

We consider a generating matrix function H(z)of jumps of a random walk controlled by a finite-state Markov chain. Under the so-called “natural” conditions we prove that the total and partial indices of the matrix function F(z)(which is—H(z)with removed singularity) are zeroes, if the ergodicity condition holds and prove that this condition is necessary and sufficient for ergodicity. Based on the indices being zeroes, we prove that ifF(z)is meromorphic in then the number of roots of det F(z)there is equal to the number of its poles. We show on an example how the partial indices may shape up the complex analytic factorization components of F(z). The roots of on the unit circle are related to a special form of H(z).

Physical insight into superdiffusive dynamics of Sinai billiard through collision statistics

The accurate prediction of uneven rock mass classes is crucial for intelligent operation in tunnel-boring machine (TBM) tunneling. However, the classification of rock masses presents significant challenges due to the variability and complexity of geological conditions. To address these challenges, this study introduces an innovative predictive model combining the improved EWOA (IEWOA) and the light gradient boosting machine (LightGBM). The proposed IEWOA algorithm incorporates a novel parameter l for more effective position updates during the exploration stage and utilizes sine functions during the exploitation stage to optimize the search process. Additionally, the model integrates a minority class technique enhanced with a random walk strategy (MCT-RW) to extend the boundaries of minority classes, such as Classes II, IV, and V. This approach significantly improves the recall and F1-score for these rock mass classes. The proposed methodology was rigorously evaluated against other predictive algorithms, demonstrating superior performance with an accuracy of 94.74%. This innovative model not only enhances the accuracy of rock mass classification but also contributes significantly to the intelligent and efficient construction of TBM tunnels, providing a robust solution to one of the key challenges in underground engineering.

A discrete shuffled frog-leaping algorithm to identify influential nodes for influence maximization in social networks

Trajectory similarity computation is a fundamental problem in various applications (e.g., transportation optimization, behavioral study). Recent researches learn trajectory representations instead of point matching to realize more accurate and efficient trajectory similarity computation. However, these methods can still not be scaled to large datasets due to high computational cost. In this paper, we propose a novel hash learning method to encode the trajectories into binary hash codes and compute trajectory similarities by Hamming distances which is much more efficient. To the best of our knowledge, this is the first work to conduct hash learning for trajectory similarity computation. Furthermore, unlike the Word2Vec model based on random walk strategy, we utilize hypergraph neural networks for the first time to learn the representations for the grids by constructing the hyperedges according to the real-life trajectories, resulting in more representative grid embeddings. In addition, we design a residual network into the multi-layer GRU to learn more discriminative trajectory representations. The proposed <u>H</u>ypergraph <u>H</u>ash <u>L</u>earning for <u>T</u>rajectory similarity commutation is an end-to-end framework and named HHL-Traj. Experimental results on two real-world trajectory datasets (i.e., Porto and Beijing) demonstrate that the proposed framework achieves up to 6.23% and 15.42% accuracy gains compared with state-of-the-art baselines in unhashed and hashed cases, respectively. The efficiency of trajectory similarity computation based on hash codes is also verified. Our code is available at https://github.com/caoyuan57/HHL-Traj.