This work extends Preferential attachment (PA) to spatial co-movement through a directed PA model whose out- and in-node weights follow temporally persistent Gaussian-process lognormal fields, and derives a strictly concave inverse that recovers the in-weights from terminal degree proportions.
Abstract
Spatially embedded directed networks, such as airline networks, often exhibit simultaneous high activity at nearby nodes. Preferential attachment (PA) explains hub dominance. We extend it to spatial co-movement through a directed PA model whose out- and in-node weights follow temporally persistent Gaussian-process lognormal fields. Under sublinear PA, out-degree proportions converge to explicit normalized powered weights, whereas self-loop exclusion yields a coupled in-degree limit. We derive a strictly concave inverse that recovers the in-weights from terminal degree proportions. For ordered network histories, we develop a minorization-maximization (MM) weight estimator and profile likelihood for the PA exponent; temporal pre-whitening and a spatial quasi-likelihood estimate the latent covariance. Simulations verify transmission of distance-decaying dependence and show how random segment volume creates a distance-independent common mode in raw degrees. An analysis of U.S. domestic flights (2015-2019) separates network-wide volume variation from a short-range spatial component. An observed-volume reconstruction reproduces the raw-degree common mode, and the fitted field yields an exploratory co-exceedance transition scale of roughly 150 km. A per-carrier analysis of European air traffic also reveals the same decomposition.
Networks with nearly identical degree distributions can place their hubs in sharply different neighborhoods. We develop a model diagnostic based on the mean degree of the neighbors of a degree-$k$ vertex. Under rank-one inhomogeneous random graphs, this statistic has degree-invariant centering and $k^{-1/2}$ fluctuations. Under non-rank-one kernels, posterior uncertainty about the root type can instead determine both centering and scale. Under linear preferential attachment, the statistic grows as $(m+\delta)\log k$. We turn these model-specific limits into goodness-of-fit tests for specified sparse-graph nulls and a weighted log-degree slope test for residual hub-neighborhood trends. Simulations evaluate null calibration, degree-distribution misspecification, and power against degree-matched preferential-attachment alternatives. Applications to high-school contact and arXiv coauthorship networks show that the method separates level misspecification from disassortative and positive residual trends. Reddit interaction networks provide a further appendix example.
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.
We introduce the Zealot–Preacher model, a generalized diffusion framework on networks with simultaneous heterogeneity at the node and edge levels. Standard diffusion models typically assume homogeneous susceptibility across nodes and homogeneous transmission strength across edges. We relax both assumptions by assigning to each node a latent receptiveness parameter and to each edge a latent conductiveness parameter governing influence propagation. The framework accommodates both binary contagion and continuous consensus dynamics and nests standard homogeneous diffusion as a special case. We characterize the resulting weighted consensus process and show that low-receptiveness nodes disproportionately influence long-run network outcomes. To estimate the model in large networks, we develop a hierarchical latent-variable approach in which receptiveness and conductiveness follow Beta distributions, with hyperparameters estimated by maximum likelihood and individual effects recovered through Bayesian updating. An application to U.S. interlocking director networks shows that diffusion of board gender diversity is constrained primarily by widespread node-level resistance rather than by the absence of conductive transmission channels. Female shared directors exhibit higher average conductiveness than male shared directors, suggesting gendered heterogeneity in diffusion channels.
Ricardo Gimeno, Ruth Mateos de Cabo· Mathematics· 0 citations
Experiments on networks observed over time face network spillovers, temporal carryover, and dependence deliberately introduced by the design. We propose COSTA---Covariance-Optimized Spatiotemporal Treatment Allocation---a joint Bernoulli design for unit--time assignments. Under common treatment marginals and a nonnegative linear network--temporal exposure model, Horvitz--Thompson bias for the sustained all-treated versus all-control contrast is exactly the negative expected weight of an assignment cut. A covariance-level variance envelope yields an MSE bound that can be optimized directly over assignment covariance. To scale this design, we introduce a thresholded-Gaussian Kronecker parameterization that mirrors the network and temporal exposure operators while preserving valid Bernoulli marginals. We next develop inference theory for the joint effects of designed treatment dependence and interference-induced outcome dependence. Canonical correlations between latent blocks generating separated HT contributions supply the coefficients required by graph-$\psi$ central limit and network-HAC theory; a spectral-floor and far-row-mass condition gives a primitive sufficient check. The framework covers sparse, block, Kronecker, locally factored, and other structured covariance sequences satisfying these conditions. Semi-synthetic RetailRocket and MovieLens experiments show substantial default-setting RMSE reductions and well-calibrated model-assisted design-centered intervals across linear, nonlinear, and demand-substitution outcome surfaces.
This paper studies the problem of steering collective beliefs in social networks when only a small fraction of nodes can be directly influenced. We propose a sparse optimal control framework built on the Network Drift-Diffusion Model (NDDM). Two intervention mechanisms are considered: direct control and latent (indirect) control. To select which nodes to actuate, we compare six centrality measures---Degree, Betweenness, Eigenvector, Closeness, PageRank, and K-shell---and keep only the top 5\%--30\% as control inputs. The optimal feedback law follows from the Hamilton-Jacobi-Bellman (HJB) equation, which reduces to solving Riccati-type differential equations. We test our approach on Erdős-Rényi (ER), Barabási-Albert (BA), and Watts-Strogatz (WS) networks. The results show that the best centrality choice depends strongly on the network topology, and the system undergoes phase transitions as control parameters vary.
Bo Wang· Frontiers in Computing and I...· 0 citations
This paper presents a computational study of language competition dynamics on Erd\H{o}s--R\'enyi random networks, extending the foundational Abrams--Strogatz model through two novel contributions: (i) a dynamic edge-weighting mechanism that reinforces social ties between co-minority speakers by an additive increment $\Delta$, and (ii) a probabilistic agent-based framework governing language switching via a weighted majority rule. Phase boundaries separating the dominance and coexistence regimes are identified across a two-dimensional parameter space $(p, \Delta)$, where $p$ denotes the network connectivity probability. We further characterise anomalous persistence zones within predicted dominance regions, attributing them to the formation of isolated minority speaker clusters. Scaling study across network sizes $N \in \{50, 100, 250, 500, 1000\}$ reveal that average cluster size decreases with $N$ and that phase boundaries diffuse with increasing stochastic noise. Finally, we discuss extensions to a tripartite bilingual model and heterogeneous prestige/volatility to more faithfully capture real sociolinguistic contact scenarios.
