Antonio Mosca, Piero Mazzarisi
arXiv 14 Sep 2026 · physics.soc-ph
arXiv:2609.15796 · PDF · Extracted main text
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.
appendix boundary found by appendix_command · 88% of the source is main text. Read the extracted text to check this.
The works this paper leans on most, across its whole bibliography — not restricted to papers in our corpus. Ranked by composite intensity, which combines how often a work is mentioned, how many sections mention it, and how much of that falls in the main text rather than the appendix.
| Reference | Intensity | Mentions | Sections | Main text | |
|---|---|---|---|---|---|
| 1 | Barucca, Paolo and Lillo, Fabrizio (2016) Disentangling bipartite and core-periphery structure in financial networks | 0.644 | 4 | 1 | 100% |
| 2 | Garlaschelli, Diego and Loffredo, Maria I (2008) Maximum likelihood: Extracting unbiased information from complex networks | 0.585 | 3 | 1 | 100% |
| 3 | Milo, Ron and Shen-Orr, Shai and Itzkovitz, Shalev and Kashtan, Nada… (2002) Network motifs: simple building blocks of complex networks | 0.585 | 3 | 1 | 100% |
| 4 | Newman, Mark (2010) Networks: An Introduction | 0.585 | 3 | 1 | 100% |
| 5 | Barucca, Paolo and Lillo, Fabrizio (2018) The organization of the interbank network and how the ECB unconventional measures affected the e-MID overnight market | 0.511 | 2 | 1 | 100% |
| 6 | Boyd, Stephen and Vandenberghe, Lieven (2004) Convex Optimization | 0.511 | 2 | 1 | 100% |
| 7 | Caldarelli, Guido and Capocci, Andrea and De Los Rios, Paolo and Muñ… (2002) Scale-Free Networks from Varying Vertex Intrinsic Fitness | 0.511 | 2 | 1 | 100% |
| 8 | Chatterjee, Sourav and Diaconis, Persi and Sly, Allan (2011) Random graphs with a given degree sequence | 0.511 | 2 | 1 | 100% |
| 9 | Fricke, Daniel and Lux, Thomas (2015) Core-periphery structure in the overnight money market: Evidence from the e-MID trading platform | 0.511 | 2 | 1 | 100% |
| 10 | Jaynes, Edwin T (1957) Information theory and statistical mechanics | 0.511 | 2 | 1 | 100% |
Showing the top 10 of 46 scored citations.