{"title":"具有宏观延迟的网络演化:渐进与凝聚","authors":"Sayan Banerjee, Shankar Bhamidi, Partha Dey, Akshay Sakanaveeti","doi":"arxiv-2409.06048","DOIUrl":null,"url":null,"abstract":"Driven by the explosion of data and the impact of real-world networks, a wide\narray of mathematical models have been proposed to understand the structure and\nevolution of such systems, especially in the temporal context. Recent advances\nin areas such as distributed cyber-security and social networks have motivated\nthe development of probabilistic models of evolution where individuals have\nonly partial information on the state of the network when deciding on their\nactions. This paper aims to understand models incorporating \\emph{network\ndelay}, where new individuals have information on a time-delayed snapshot of\nthe system. We consider the setting where one has macroscopic delays, that is,\nthe ``information'' available to new individuals is the structure of the\nnetwork at a past time, which scales proportionally with the current time and\nvertices connect using standard preferential attachment type dynamics. We\nobtain the local weak limit for the network as its size grows and connect it to\na novel continuous-time branching process where the associated point process of\nreproductions \\emph{has memory} of the entire past. A more tractable `dual\ndescription' of this branching process using an `edge copying mechanism' is\nused to obtain degree distribution asymptotics as well as necessary and\nsufficient conditions for condensation, where the mass of the degree\ndistribution ``escapes to infinity''. We conclude by studying the impact of the\ndelay distribution on macroscopic functionals such as the root degree.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"1 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Network evolution with Macroscopic Delays: asymptotics and condensation\",\"authors\":\"Sayan Banerjee, Shankar Bhamidi, Partha Dey, Akshay Sakanaveeti\",\"doi\":\"arxiv-2409.06048\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Driven by the explosion of data and the impact of real-world networks, a wide\\narray of mathematical models have been proposed to understand the structure and\\nevolution of such systems, especially in the temporal context. Recent advances\\nin areas such as distributed cyber-security and social networks have motivated\\nthe development of probabilistic models of evolution where individuals have\\nonly partial information on the state of the network when deciding on their\\nactions. This paper aims to understand models incorporating \\\\emph{network\\ndelay}, where new individuals have information on a time-delayed snapshot of\\nthe system. We consider the setting where one has macroscopic delays, that is,\\nthe ``information'' available to new individuals is the structure of the\\nnetwork at a past time, which scales proportionally with the current time and\\nvertices connect using standard preferential attachment type dynamics. We\\nobtain the local weak limit for the network as its size grows and connect it to\\na novel continuous-time branching process where the associated point process of\\nreproductions \\\\emph{has memory} of the entire past. A more tractable `dual\\ndescription' of this branching process using an `edge copying mechanism' is\\nused to obtain degree distribution asymptotics as well as necessary and\\nsufficient conditions for condensation, where the mass of the degree\\ndistribution ``escapes to infinity''. We conclude by studying the impact of the\\ndelay distribution on macroscopic functionals such as the root degree.\",\"PeriodicalId\":501245,\"journal\":{\"name\":\"arXiv - MATH - Probability\",\"volume\":\"1 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-09-09\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - Probability\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2409.06048\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Probability","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.06048","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Network evolution with Macroscopic Delays: asymptotics and condensation
Driven by the explosion of data and the impact of real-world networks, a wide
array of mathematical models have been proposed to understand the structure and
evolution of such systems, especially in the temporal context. Recent advances
in areas such as distributed cyber-security and social networks have motivated
the development of probabilistic models of evolution where individuals have
only partial information on the state of the network when deciding on their
actions. This paper aims to understand models incorporating \emph{network
delay}, where new individuals have information on a time-delayed snapshot of
the system. We consider the setting where one has macroscopic delays, that is,
the ``information'' available to new individuals is the structure of the
network at a past time, which scales proportionally with the current time and
vertices connect using standard preferential attachment type dynamics. We
obtain the local weak limit for the network as its size grows and connect it to
a novel continuous-time branching process where the associated point process of
reproductions \emph{has memory} of the entire past. A more tractable `dual
description' of this branching process using an `edge copying mechanism' is
used to obtain degree distribution asymptotics as well as necessary and
sufficient conditions for condensation, where the mass of the degree
distribution ``escapes to infinity''. We conclude by studying the impact of the
delay distribution on macroscopic functionals such as the root degree.