Network parameters via equilibrium measures in Schrödinger random walks

Biased Random Search on Directed Networks with Stochastic Resetting

This study investigates biased random walks with stochastic resetting on directed networks. By employing the spectral decomposition of the transition probability matrix, we derive analytical expressions for the stationary distribution and mean first-passage time (MFPT), providing a quantitative framework for characterizing search efficiency in directed systems. Furthermore, we establish a general criterion for optimal resetting and derive sufficient conditions for its existence. The theoretical results are validated through extensive numerical simulations on three representative synthetic networks and three real-world directed networks, demonstrating the applicability of the proposed framework across diverse topologies. These findings provide a systematic understanding of how biased random walks combined with stochastic resetting can improve search efficiency under suitable structural conditions.

2009 Glass transition and random walks on complex energy landscapes

Centrality in Normalized Stochastic Networks

In many applications, network effects are normalized: in opinion dynamics, agents take a weighted average of their friends’ beliefs, or in social media models, users’ adoption decisions depends on the fraction of their peers who also adopt. The outcome of these processes share a common network property: a weighted Katz–Bonacich centrality but one defined over the network’s row-normalized adjacency matrix, which measures relative spillovers. This row-normalized centrality measure is well-understood in deterministic settings in which the full network structure is known, but in many instances, only probabilistic information about the network is available. We show that, under mild regularity conditions, the realized values of row-normalized centralities concentrate around their expectations, and this greatly simplifies the analysis of these stochastic networks. We use this result to further show that optimizing an objective over a stochastic network can be reduced to an optimization problem over an appropriately defined deterministic network. Together, these results yield a general and tractable approach for analyzing network processes and targeting problems in stochastic networks when spillovers are determined by normalized rather than raw connections. We demonstrate the usefulness of these techniques in applications to pricing, network games, and social dynamics.

Component structure and percolation in block models

The stochastic block model is a widely studied model of community structure in networks. Here we study the component structure and percolation properties of networks generated from this model and its variants, using exact methods based on probability generating functions. In particular, we derive expressions for the size of the giant component and the distribution of small components in such networks and for the size of the percolating cluster and position of the percolation threshold for both node and edge percolation, for the original stochastic block model and for its degree-corrected versions. In passing, we also develop a mapping between generating functions for microcanonical and canonical block models that allows us to generalize results for the former to the latter with minimal effort.

Vertex-level diffusion scaling from resistance distance.

The analysis of random walks on networks often relies on global quantities that average over nodes, thereby masking local differences in diffusion speed. This study introduces a vertex-level quantity Hi, defined as the finite-window fitted scaling exponent of the mean squared resistance distance ⟨Ωi2(t)⟩∼Cit2Hi from a given node i. We found nodes with Hi values below 0.5 (echo effect) and above 0.5 (catapult effect). The exponent is computed exactly via matrix powers of the transition matrix. We systematically evaluate Hi on several synthetic network families, generalized Sierpiński graphs, Newman-Watts small-world networks, and a custom grid-path-complete graph, and on two real-world networks (international E-road network and western U.S. power grid). We found nodes with Hi values less than 0.5 (subdiffusive regime) and greater than 0.5 (apparent superdiffusion) in both model networks and real-world networks. Analysis of model networks shows that when a node has an echo effect, its Hi value is less than 0.5, whereas when it has a catapult effect, its Hi value is greater than 0.5. In the two real networks, most nodes are in the subdiffusive regime and the overall heterogeneity of the local diffusion exponents is low, as indicated by Rényi indices of 0.0835 (E-road network) and 0.0555 (power grid). Comparisons with classical centrality measures indicate that Hi provides information not captured by those measures. The local diffusion exponent offers a vertex-level, dynamics-based tool for identifying structural bottlenecks and node roles, complementing global network characterizations.

Graph distance and effective resistance of the random walk trace in four and five dimensions

In this paper, we prove that the fluctuations of the graph distance and the effective resistance on the trace of a random walk in four and five dimensions converge in distribution to a stable law. In previous work, the first and second authors proved that the corresponding fluctuations converge to a Gaussian distribution in dimensions six and higher. Taken together, these results reveal a phase transition between dimensions five and six. Our proof develops a novel coupling with long range percolation, and we expect this technique to find applications in a broad class of related models.