arXiv subjects

Muchen Ju

Publications and source records attributed to Muchen Ju.

4 recordsLinked to original sources

arXiv2026

Majority Dynamics on Resampled Sparse Erd\H{o}s--R\'enyi Graphs: Gaussian Winner Selection and Pace to Unanimity

We study the two-opinion majority dynamics process: at each time step, every vertex adopts the majority opinion among its neighbors, retaining its current opinion if there is a tie. Independently at each step, the interaction graph is resampled from the sparse Erd\H{o}s--R\'enyi model $\mathbb G(N,p)$ with $p=b\log N/N$ and fixed $b>1$. Our results identify three regimes governed by the initial advantage $\Delta_0=|B_0|-|R_0|$, where $|B_0|$ and $|R_0|$ denote the initial blue and red camps, respectively. First, an initial blue advantage above an explicit constant multiple of $N/\sqrt{\log N}$ leads to blue unanimity within two updates with high probability. Second, throughout the intermediate regime $\sqrt{N/\log N}\ll\Delta_0\lesssim N/\sqrt{\log N}$, we obtain explicit high-probability upper and lower bounds on the blue-unanimity time. Finally, uniformly in the critical window $\Delta_0\sqrt p=O(1)$, the blue- and red-unanimity probabilities equal $\Phi(\sqrt{2/\pi}\,\Delta_0\sqrt p)+o(1)$ and $\Phi(-\sqrt{2/\pi}\,\Delta_0\sqrt p)+o(1)$, respectively, and unanimity is reached within $(1+o(1))\log N/\log\log N$ many updates with high probability. This resolves the resampled version of the \emph{optimal power-of-few} conjecture raised by Tran and Vu (2025).

math.PR↗

arXiv2026

Majority Dynamics on Assortative Sparse Stochastic Block Models

Majority dynamics is a two-opinion process in which each vertex repeatedly updates to the majority opinion among its neighbors. We study this process on a resampled sparse binary stochastic block model in the assortative regime. At each time step, a graph is sampled from the current opinion partition: vertices with the same opinion are joined with probability $\alpha=a\log N/N$, while vertices with differing opinions are joined with probability $\beta=b\log N/N$, where $a>b>1$. Let $B_t$ and $R_t$ denote the blue and red camps at time $t$. We show that the weighted advantage $\widetilde{\Delta}_t =b|B_t|-a|R_t|$, rather than the unweighted advantage $\Delta_t=|B_t|-|R_t|$ alone, governs the pace to unanimity. Our results, which hold with high probability as $N\to\infty$, identify three regimes for blue unanimity under the initial blue advantage, i.e., $\Delta_0>0$: constant time, subpolynomial time, and polynomial time. First, when $\widetilde{\Delta}_0 \gtrsim -N/\sqrt{\log N}$, blue unanimity occurs within three updates. Second, when $\widetilde{\Delta}_0 < 0$ and $|\widetilde{\Delta}_0| = o(N)$, blue unanimity occurs within $N^{o(1)}$ updates. Furthermore, when $\widetilde{\Delta}_0 < 0$, $|\widetilde{\Delta}_0| = O(N)$, and $\Delta_0\gg\sqrt{N/\log N}$, blue unanimity still occurs within $N^{I_0+o(1)}$ updates, where \[ I_0= \left(\mathbf{ReLU}\Big(\sqrt{a\frac{|R_0|}{N}}-\sqrt{b\frac{|B_0|}{N}}\Big)\right)^2, \] and $\mathbf{ReLU}(x)=\max\{x,0\}$. Conversely, away from the weighted threshold, when $|B_0|/|R_0|\le a/b-\kappa$ and $\Delta_0>0$, $N^{I_0 - o(1)}$ updates are necessary for blue unanimity. Our analysis relies on detailed estimates for one-vertex flip probabilities in sparse binomial differences, which could be of independent interest.

math.PR↗

arXiv2025

Sharp Inner Product Correlations for Hypercube Bijections

We resolve a conjecture of Rob Morris concerning bijections on the hypercube. Specifically, we show that for any bijection $f : \{-1,1\}^n \to \{-1,1\}^n$, \[ \Pr_{x,y \in \{-1,1\}^n}\big[ \langle x,y \rangle \ge 0 \;\text{and}\; \langle f(x),f(y) \rangle \ge 0 \big] \;\;\ge\; \tfrac{1}{4} - O(1/\sqrt{n}), \] implying the same lower bound for the joint event under any two bijections. Our proof proceeds by applying the spectral decomposition of the Hamming association scheme, which allows us to reformulate the problem as a linear program over the Birkhoff polytope. This makes it possible to isolate the contribution of the nontrivial spectrum, which we show is asymptotically negligible, leaving the dominant contribution arising from the principal eigenvalue.

math.CO↗

arXiv2024

Arborescences of Random Covering Graphs

A rooted arborescence of a directed graph is a spanning tree directed towards a particular vertex. A recent work of Chepuri et al. showed that the arborescences of a covering graph of a directed graph G are closely related to the arborescences of G. In this paper, we study the weighted sum of arborescences of a random covering graph and give a formula for the expected value, resolving a conjecture of Chepuri et al.

math.CO↗