ідея - Mathematics - # Growth Phases in First Passage Percolation

Four Universal Growth Regimes in Degree-Dependent First Passage Percolation on Spatial Random Graphs I

Q: How does one-dependent FPP compare to classical FPP models?

One-dependent first passage percolation (1-FPP) differs from classical independent and identically distributed (iid) first passage percolation (FPP) in several key aspects. In 1-FPP, the transmission time through each edge depends on the direct surroundings of the edge, unlike in classical FPP where edge transmission times are independent and identically distributed random variables. This dependency in 1-FPP is introduced through a penalty function that multiplies the iid transmission time by a polynomial of the expected degrees of the endpoints of the edge. Moreover, while classical FPP typically exhibits Malthusian growth with logarithmic transmission times between uniformly chosen vertices, 1-FPP shows a more diverse range of growth phases. These phases include explosive growth when reaching infinitely many vertices in finite time, polylogarithmic growth where distances increase at most polylogarithmically with respect to graph size without being explosive, polynomial but strictly sublinear growth, and linear growth with exponent equal to 1. In summary, one-dependent FFP introduces spatial dependencies into the spreading process that lead to richer behavior compared to classical FFP models.

Q: What implications do these findings have for real-world phenomena like epidemics?

The findings regarding different growth regimes in degree-dependent first passage percolation (FPP) can have significant implications for understanding real-world phenomena like epidemics. By modeling spreading processes on graphs where transmission times depend on local vertex properties such as expected degrees and spatial distances between vertices, researchers can gain insights into how diseases or information spread within populations. For instance: The sublinear impact of superspreaders observed in various applications can be better captured using penalty functions that introduce sublinear effects based on vertex properties. The identification of distinct growth phases such as explosive, polylogarithmic, polynomial but strictly sublinear, and linear can help explain different patterns seen in epidemic data. The ability to model transitions between these phases based on parameters like tail distributions and long-range connectivity provides a more nuanced understanding of epidemic dynamics. Overall, these results offer a more realistic way to model complex spreading processes observed in epidemics by considering spatial dependencies and varying degrees of influence among individuals or nodes within networks.

Q: How can these results be applied to other graph models beyond first passage percolation?

The results obtained from studying degree-dependent first passage percolation (FPP) can be extended and applied to various other graph models beyond just first passage percolation. Some potential applications include: Robustness Analysis: Applying similar methodologies used in analyzing different growth regimes could help understand robustness properties across various network structures. Epidemic Modeling: Extending the findings to SIR-type epidemic processes or rumor spreading models could provide insights into how information or diseases propagate through networks with varying connectivity patterns. Random Geometric Graphs: Investigating how similar phase transitions manifest in random geometric graphs could shed light on distance-based interactions among nodes. By adapting the concepts learned from studying degree-dependent FFP's rich behavior across different parameter spaces, researchers can explore new avenues for analyzing complex systems represented by diverse graph structures.

Основні поняття

Four universal growth regimes are observed in degree-dependent first passage percolation on spatial random graphs, showcasing unique phases of transmission time growth.

Анотація

The content explores the four distinct growth phases observed in degree-dependent first passage percolation on spatial random graphs. It delves into the underlying model parameters and their impact on the transmission times between vertices. The study reveals a rich behavior with several growth phases and non-smooth phase transitions, providing insights into the dynamics of spreading processes on graph networks.

Introduction

Definition of First Passage Percolation (FPP)
Introduction of One-Dependent FPP

Universality Classes of Transmission Times

Description of one-dependent FPP model parameters and their influence on transmission times

Precise Behavior in the Four Phases

Detailed analysis of each growth phase: explosive, polylogarithmic, polynomial, and linear

New Methodology: Moving to Quenched to Replace FKG-Inequality

Development of a general technique using pseudorandom nets combined with multi-round exposure to replace the FKG inequality

Budget Travel Plan with 3-Edge Bridge-Paths

Explanation of the methodology for constructing connecting paths over multiple iterations

Robustness of Techniques

Discussion on the robustness and applicability of developed techniques across various graph models

Статистика

In this paper we develop new methods to prove the upper bounds in all sub-explosive phases.
We show that as µ increases, different phases occur for the transmission time between vertices.
The cost function C(xy) could also depend on |x − y|.

Цитати

Ключові висновки, отримані з

Four universal growth regimes in degree-dependent first passage percolation on spatial random graphs I

by Júli... о arxiv.org 03-26-2024

https://arxiv.org/pdf/2309.11840.pdf

Four universal growth regimes in degree-dependent first passage percolation on spatial random graphs I

Глибші Запити

How does one-dependent FPP compare to classical FPP models?

One-dependent first passage percolation (1-FPP) differs from classical independent and identically distributed (iid) first passage percolation (FPP) in several key aspects. In 1-FPP, the transmission time through each edge depends on the direct surroundings of the edge, unlike in classical FPP where edge transmission times are independent and identically distributed random variables. This dependency in 1-FPP is introduced through a penalty function that multiplies the iid transmission time by a polynomial of the expected degrees of the endpoints of the edge.
Moreover, while classical FPP typically exhibits Malthusian growth with logarithmic transmission times between uniformly chosen vertices, 1-FPP shows a more diverse range of growth phases. These phases include explosive growth when reaching infinitely many vertices in finite time, polylogarithmic growth where distances increase at most polylogarithmically with respect to graph size without being explosive, polynomial but strictly sublinear growth, and linear growth with exponent equal to 1.
In summary, one-dependent FFP introduces spatial dependencies into the spreading process that lead to richer behavior compared to classical FFP models.

What implications do these findings have for real-world phenomena like epidemics?

The findings regarding different growth regimes in degree-dependent first passage percolation (FPP) can have significant implications for understanding real-world phenomena like epidemics. By modeling spreading processes on graphs where transmission times depend on local vertex properties such as expected degrees and spatial distances between vertices, researchers can gain insights into how diseases or information spread within populations.
For instance:

The sublinear impact of superspreaders observed in various applications can be better captured using penalty functions that introduce sublinear effects based on vertex properties.
The identification of distinct growth phases such as explosive, polylogarithmic, polynomial but strictly sublinear, and linear can help explain different patterns seen in epidemic data.
The ability to model transitions between these phases based on parameters like tail distributions and long-range connectivity provides a more nuanced understanding of epidemic dynamics.
Overall, these results offer a more realistic way to model complex spreading processes observed in epidemics by considering spatial dependencies and varying degrees of influence among individuals or nodes within networks.

How can these results be applied to other graph models beyond first passage percolation?

The results obtained from studying degree-dependent first passage percolation (FPP) can be extended and applied to various other graph models beyond just first passage percolation. Some potential applications include:

Robustness Analysis: Applying similar methodologies used in analyzing different growth regimes could help understand robustness properties across various network structures.

Epidemic Modeling: Extending the findings to SIR-type epidemic processes or rumor spreading models could provide insights into how information or diseases propagate through networks with varying connectivity patterns.

Random Geometric Graphs: Investigating how similar phase transitions manifest in random geometric graphs could shed light on distance-based interactions among nodes.
By adapting the concepts learned from studying degree-dependent FFP's rich behavior across different parameter spaces, researchers can explore new avenues for analyzing complex systems represented by diverse graph structures.

Four Universal Growth Regimes in Degree-Dependent First Passage Percolation on Spatial Random Graphs I