Abstract
This work exhibits a novel phase transition for the classical stochastic block model (SBM). In addition we study the SBM in the corresponding near-critical regime, and find the scaling limit for the component sizes. The two-parameter stochastic process arising in the scaling limit, an analogue of the standard Aldous’ multiplicative coalescent, is interesting in its own right. We name it the (standard) Interacting Multiplicative Coalescent. To the best of our knowledge, this object has not yet appeared in the literature.
Highlights
The multiplicative coalescent is a process constructed in [1]
The multiplicative coalescent takes values in the space of collections of blocks with mass (a number in (0, ∞)) and evolves according to the following dynamics: each pair of blocks of mass x and y merges at rate xy into a single block of mass x + y
If A = (Ai,j )i,j is a family of i.i.d. exponential random variables, and the relation R∗ is maximal RMMt(x; A, R∗) has the law of the multiplicative coalescent started from ord(x) and evaluated at time t
Summary
The Aldous’ standard multiplicative coalescent is the scaling limit of near-critical Erdos-Rényi graphs. The scaling limit of Theorem 3.1, is to the best of our knowledge, a completely new stochastic object, of an independent interest to probability theory and applications We name it the (standard) interacting multiplicative coalescent. The multiplicative coalescent is a process taking values in l2 If it starts from an initial value x in l1, it will reach the constant state There are uncountably many different non-standard extreme eternal multiplicative coalescent entrance laws and they have been classified in [2], and linked further in [15, 14] to an analogous family of Lévy-type processes.
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
