The Critical Patch Size Problem in Random Graphs

Nicola Apollonio; Veronica Tora; Davide Vergni

doi:doi:10.11648/j.acm.20261504.14

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The Critical Patch Size Problem in Random Graphs

Nicola Apollonio

, Veronica Tora^*

, Davide Vergni

Published in Applied and Computational Mathematics (Volume 15, Issue 4)

Received: 7 April 2026 Accepted: 22 April 2026 Published: 27 August 2026

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Abstract

The problem of critical patch size --- a threshold condition for population persistence --- is investigated in the context of discrete habitats, modeled as graphs with a distinguished subset of vertices acting as sinks. These sinks impose boundary-like constraints analogous to Dirichlet conditions in continuous domains. The population proliferates locally at the vertices and diffuse across the network through the graph Laplacian. In the sinks the population cannot survive. The Dirichlet eigenvalue of the habitat is defined as the smallest eigenvalue of the principal submatrix of the Laplacian obtained by removing the rows and columns associated with sink vertices. This spectral parameter governs the habitat's viability: survival occurs when the Dirichlet eigenvalue of the habitat lies below a critical reaction-to-diffusion ratio. We study survival conditions for a sequence of random habitats built on binomial random graphs. We establish a law of large numbers for the corresponding sequence of Dirichlet eigenvalues and prove the emergence of a sharp threshold phenomenon: with high probability, a large random habitat is either viable or non-viable, depending on whether the reaction-to-diffusion ratio lies below or above this threshold. Our results provide the first general spectral theory for critical patch size on graphs, with implications for ecology, synthetic biology, and the modeling of processes on brain connectomes.

Published in	Applied and Computational Mathematics (Volume 15, Issue 4)
DOI	10.11648/j.acm.20261504.14
Page(s)	154-161
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2026. Published by Science Publishing Group

Keywords

Critical Patch Size, Graph Laplacian, Dirichlet Eigenvalue, Random Graphs, Chernoff Bounds

References

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[6]	Andrea Marcello Mambuca, Chiara Cammarota, and Izaak Neri. newblock Dynamical systems on large networks with predator-prey interactions are stable and exhibit oscillations. newblock Physical Review E, 105(1): 014305, 2022. https://doi.org/10.1103/PhysRevE.105.014305
[7]	Veronica Tora, Davide Vergni, The critical patch size problem on networks, Mathematical Biosciences, 399, 2026, 109749, https://doi.org/10.1016/j.mbs.2026.109749
[8]	Alan Frieze and Michal Karonski. newblock Introduction to random graphs. newblock Cambridge University Press, 2015.
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[15]	Spencer Greenberg and Mehryar Mohri. newblock Tight lower bound on the probability of a binomial exceeding its expectation. newblock Statistics & Probability Letters, 86: 91-98, 2014. https://doi.org/10.1016/j.spl.2013.12.009

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APA Style

Apollonio, N., Tora, V., Vergni, D. (2026). The Critical Patch Size Problem in Random Graphs. Applied and Computational Mathematics, 15(4), 154-161. https://doi.org/10.11648/j.acm.20261504.14

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ACS Style

Apollonio, N.; Tora, V.; Vergni, D. The Critical Patch Size Problem in Random Graphs. Appl. Comput. Math. 2026, 15(4), 154-161. doi: 10.11648/j.acm.20261504.14

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AMA Style

Apollonio N, Tora V, Vergni D. The Critical Patch Size Problem in Random Graphs. Appl Comput Math. 2026;15(4):154-161. doi: 10.11648/j.acm.20261504.14

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@article{10.11648/j.acm.20261504.14,
  author = {Nicola Apollonio and Veronica Tora and Davide Vergni},
  title = {The Critical Patch Size Problem in Random Graphs},
  journal = {Applied and Computational Mathematics},
  volume = {15},
  number = {4},
  pages = {154-161},
  doi = {10.11648/j.acm.20261504.14},
  url = {https://doi.org/10.11648/j.acm.20261504.14},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20261504.14},
  abstract = {The problem of critical patch size --- a threshold condition for population persistence --- is investigated in the context of discrete habitats, modeled as graphs with a distinguished subset of vertices acting as sinks. These sinks impose boundary-like constraints analogous to Dirichlet conditions in continuous domains. The population proliferates locally at the vertices and diffuse across the network through the graph Laplacian. In the sinks the population cannot survive. The Dirichlet eigenvalue of the habitat is defined as the smallest eigenvalue of the principal submatrix of the Laplacian obtained by removing the rows and columns associated with sink vertices. This spectral parameter governs the habitat's viability: survival occurs when the Dirichlet eigenvalue of the habitat lies below a critical reaction-to-diffusion ratio. We study survival conditions for a sequence of random habitats built on binomial random graphs. We establish a law of large numbers for the corresponding sequence of Dirichlet eigenvalues and prove the emergence of a sharp threshold phenomenon: with high probability, a large random habitat is either viable or non-viable, depending on whether the reaction-to-diffusion ratio lies below or above this threshold. Our results provide the first general spectral theory for critical patch size on graphs, with implications for ecology, synthetic biology, and the modeling of processes on brain connectomes.},
 year = {2026}
}

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TY  - JOUR
T1  - The Critical Patch Size Problem in Random Graphs
AU  - Nicola Apollonio
AU  - Veronica Tora
AU  - Davide Vergni
Y1  - 2026/08/27
PY  - 2026
N1  - https://doi.org/10.11648/j.acm.20261504.14
DO  - 10.11648/j.acm.20261504.14
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 154
EP  - 161
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20261504.14
AB  - The problem of critical patch size --- a threshold condition for population persistence --- is investigated in the context of discrete habitats, modeled as graphs with a distinguished subset of vertices acting as sinks. These sinks impose boundary-like constraints analogous to Dirichlet conditions in continuous domains. The population proliferates locally at the vertices and diffuse across the network through the graph Laplacian. In the sinks the population cannot survive. The Dirichlet eigenvalue of the habitat is defined as the smallest eigenvalue of the principal submatrix of the Laplacian obtained by removing the rows and columns associated with sink vertices. This spectral parameter governs the habitat's viability: survival occurs when the Dirichlet eigenvalue of the habitat lies below a critical reaction-to-diffusion ratio. We study survival conditions for a sequence of random habitats built on binomial random graphs. We establish a law of large numbers for the corresponding sequence of Dirichlet eigenvalues and prove the emergence of a sharp threshold phenomenon: with high probability, a large random habitat is either viable or non-viable, depending on whether the reaction-to-diffusion ratio lies below or above this threshold. Our results provide the first general spectral theory for critical patch size on graphs, with implications for ecology, synthetic biology, and the modeling of processes on brain connectomes.
VL  - 15
IS  - 4
ER  -

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Author Information

Nicola Apollonio

Institute for Applied Mathematics "Mauro Picone", National Research Council of Italy, Rome, Italy

Contact Email

http://orcid.org/0000-0001-6089-1333
Veronica Tora

Institute for Applied Mathematics "Mauro Picone", National Research Council of Italy, Rome, Italy

Contact Email

http://orcid.org/0000-0002-0997-8828
Davide Vergni

Institute for Applied Mathematics "Mauro Picone", National Research Council of Italy, Rome, Italy

Contact Email

http://orcid.org/0000-0003-3227-5765

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APA Style

Apollonio, N., Tora, V., Vergni, D. (2026). The Critical Patch Size Problem in Random Graphs. Applied and Computational Mathematics, 15(4), 154-161. https://doi.org/10.11648/j.acm.20261504.14

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ACS Style

Apollonio, N.; Tora, V.; Vergni, D. The Critical Patch Size Problem in Random Graphs. Appl. Comput. Math. 2026, 15(4), 154-161. doi: 10.11648/j.acm.20261504.14

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AMA Style

Apollonio N, Tora V, Vergni D. The Critical Patch Size Problem in Random Graphs. Appl Comput Math. 2026;15(4):154-161. doi: 10.11648/j.acm.20261504.14

Copy | Download

@article{10.11648/j.acm.20261504.14,
  author = {Nicola Apollonio and Veronica Tora and Davide Vergni},
  title = {The Critical Patch Size Problem in Random Graphs},
  journal = {Applied and Computational Mathematics},
  volume = {15},
  number = {4},
  pages = {154-161},
  doi = {10.11648/j.acm.20261504.14},
  url = {https://doi.org/10.11648/j.acm.20261504.14},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20261504.14},
  abstract = {The problem of critical patch size --- a threshold condition for population persistence --- is investigated in the context of discrete habitats, modeled as graphs with a distinguished subset of vertices acting as sinks. These sinks impose boundary-like constraints analogous to Dirichlet conditions in continuous domains. The population proliferates locally at the vertices and diffuse across the network through the graph Laplacian. In the sinks the population cannot survive. The Dirichlet eigenvalue of the habitat is defined as the smallest eigenvalue of the principal submatrix of the Laplacian obtained by removing the rows and columns associated with sink vertices. This spectral parameter governs the habitat's viability: survival occurs when the Dirichlet eigenvalue of the habitat lies below a critical reaction-to-diffusion ratio. We study survival conditions for a sequence of random habitats built on binomial random graphs. We establish a law of large numbers for the corresponding sequence of Dirichlet eigenvalues and prove the emergence of a sharp threshold phenomenon: with high probability, a large random habitat is either viable or non-viable, depending on whether the reaction-to-diffusion ratio lies below or above this threshold. Our results provide the first general spectral theory for critical patch size on graphs, with implications for ecology, synthetic biology, and the modeling of processes on brain connectomes.},
 year = {2026}
}

Copy | Download

TY  - JOUR
T1  - The Critical Patch Size Problem in Random Graphs
AU  - Nicola Apollonio
AU  - Veronica Tora
AU  - Davide Vergni
Y1  - 2026/08/27
PY  - 2026
N1  - https://doi.org/10.11648/j.acm.20261504.14
DO  - 10.11648/j.acm.20261504.14
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 154
EP  - 161
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20261504.14
AB  - The problem of critical patch size --- a threshold condition for population persistence --- is investigated in the context of discrete habitats, modeled as graphs with a distinguished subset of vertices acting as sinks. These sinks impose boundary-like constraints analogous to Dirichlet conditions in continuous domains. The population proliferates locally at the vertices and diffuse across the network through the graph Laplacian. In the sinks the population cannot survive. The Dirichlet eigenvalue of the habitat is defined as the smallest eigenvalue of the principal submatrix of the Laplacian obtained by removing the rows and columns associated with sink vertices. This spectral parameter governs the habitat's viability: survival occurs when the Dirichlet eigenvalue of the habitat lies below a critical reaction-to-diffusion ratio. We study survival conditions for a sequence of random habitats built on binomial random graphs. We establish a law of large numbers for the corresponding sequence of Dirichlet eigenvalues and prove the emergence of a sharp threshold phenomenon: with high probability, a large random habitat is either viable or non-viable, depending on whether the reaction-to-diffusion ratio lies below or above this threshold. Our results provide the first general spectral theory for critical patch size on graphs, with implications for ecology, synthetic biology, and the modeling of processes on brain connectomes.
VL  - 15
IS  - 4
ER  -

Copy | Download