Penalized Maximum Likelihood Inference of Core-Periphery Networks
Abstract
Likelihood-based network models are often fitted under links' independence and low-order constraints, while empirical networks frequently exhibit systematic higher-order structures such as triangles and wedges, characterizing the observed clustering patterns. Real-world core-periphery networks such as the interbank market or the air transportation system represent key examples, with cores displaying complex and nonlinear features. We formalize Penalized Likelihood with Structural Discrepancies (PLSD) as an inference-level correction that trades likelihood fit for agreement with targeted motifs. PLSD augments the negative log-likelihood with a penalty on standardized wedge and triangle discrepancies, yielding a controlled distortion of the likelihood surface that can be interpreted through a linear response analysis. A calibration approach for the unique tuning hyperparameter is introduced to turn off the penalization term when the model is correctly specified, thereby recovering maximum-likelihood estimation in that case. We then introduce a mixed-constraint maximum-entropy core-periphery Exponential Random Graph Model (ERGM), derive unconditional semi-closed motif formulas under Pareto-distributed core fitness, and interpret sparse-regime scaling through a big-jump mechanism for heavy-tailed distributions. We finally corroborate the PLSD inference methodology both with Monte Carlo simulations of a toy stochastic block model and by applying the novel core-periphery ERGM to weekly eMID interbank networks (2009-2015) and monthly US air traffic networks (1991-2000). We show that PLSD describes cores that are not only dense but also higher-order-rich and captures the observed heterogeneous degree distributions and the overexpression of wedges and triangles, at a link-level cost proportional to the model misspecification.
Related
- Consistency of penalized maximum likelihood estimation for multidimensional change-in-velocity detection
- Penalized Likelihood Inference in Highly Multivariate Sparse Spectral Graphical Gaussian Processes
- Exact maximum likelihood inference for drifted multi-sub-fractional Brownian motion at discrete observation
- Global Parameter Identification and Uncertainty Quantification of reduced-order ADM1 using Maximum Likelihood Estimation and Bayesian Inference
- Maximum likelihood, neural networks and posterior mean for inferring fractional Brownian motion parameters
BibTeX
@misc{mosca2026penalized,
title = {{Penalized Maximum Likelihood Inference of Core-Periphery Networks}},
author = {Antonio Mosca and Piero Mazzarisi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.15796},
eprint = {2609.15796},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.15796},
}
RIS
TY - UNPB
TI - Penalized Maximum Likelihood Inference of Core-Periphery Networks
AU - Antonio Mosca
AU - Piero Mazzarisi
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.15796
UR - https://arxiv.org/abs/2609.15796
AB - Likelihood-based network models are often fitted under links' independence and low-order constraints, while empirical networks frequently exhibit systematic higher-order structures such as triangles and wedges, characterizing the observed clustering patterns. Real-world core-periphery networks such as the interbank market or the air transportation system represent key examples, with cores displaying complex and nonlinear features. We formalize Penalized Likelihood with Structural Discrepancies (PLSD) as an inference-level correction that trades likelihood fit for agreement with targeted motifs. PLSD augments the negative log-likelihood with a penalty on standardized wedge and triangle discrepancies, yielding a controlled distortion of the likelihood surface that can be interpreted through a linear response analysis. A calibration approach for the unique tuning hyperparameter is introduced to turn off the penalization term when the model is correctly specified, thereby recovering maximum-likelihood estimation in that case. We then introduce a mixed-constraint maximum-entropy core-periphery Exponential Random Graph Model (ERGM), derive unconditional semi-closed motif formulas under Pareto-distributed core fitness, and interpret sparse-regime scaling through a big-jump mechanism for heavy-tailed distributions. We finally corroborate the PLSD inference methodology both with Monte Carlo simulations of a toy stochastic block model and by applying the novel core-periphery ERGM to weekly eMID interbank networks (2009-2015) and monthly US air traffic networks (1991-2000). We show that PLSD describes cores that are not only dense but also higher-order-rich and captures the observed heterogeneous degree distributions and the overexpression of wedges and triangles, at a link-level cost proportional to the model misspecification.
ER -