arXiv subjects

Huaning Liu

Publications and source records attributed to Huaning Liu.

10 recordsLinked to original sources

arXiv2026

Regulation of Rumor Propagation via (Multi-Leader) Stackelberg Graphon Games

We study the control of rumor propagation in large networked populations by using Stackelberg graphon games. We first introduce a principal who wants to incentivize the spread of her preferred news and discourage the spread of non-preferred news. We define the Stackelberg graphon game equilibrium (SGGE), characterize the graphon game Nash equilibrium (GGNE) with a forward-backward differential equation system, and establish existence results. We further formulate a multi-leader model with two competing principals, each incentivizing her own preferred news. Finally, we propose a bi-level algorithm for computing (multi-leader) Stackelberg graphon game equilibria and conclude with numerical experiments where we show that existence of competing principals will result in strong opinion divisions in the population.

math.OC↗

arXiv2026

Modeling Epidemic Spread with Strategic Vaccination and Socialization: a Mean Field Game Analysis

We study a game-theoretic model of epidemic control in a large population with finitely many groups and non-cooperative individuals. In the model, individuals jointly choose their socialization levels and vaccination rates, and vaccination is subject to a linear individual cost structure. We derive a forward-backward ordinary differential equations (FBODE) system that characterizes the mean field Nash equilibrium, show that the equilibrium vaccination rate exhibits an at-most one-jump bang-bang structure, and establish the existence of a Carath\'eodory solution to the FBODE. This establishes a realistic interpretation of the vaccination decisions, meaning individuals decide to vaccinate until a time point which is determined by model parameters and then stop after. We further consider a population-awareness extension in which individuals incorporate population infection information into their objective functions, and we prove a similar at-most one-jump bang-bang property under suitable conditions. Finally, we propose a numerical algorithm for solving the FBODE and conduct simulations to validate the theoretical findings. The experiments highlight two main insights: the trade-off between socialization and vaccination, and the greater importance of quarantining infected individuals instead of restricting susceptible individuals.

math.OC↗

arXiv2026

On the Scalability of Quasi-Complementary Sequence Sets: Quadratic and Cubic Laws

This work is concerned with the fundamental scaling laws of quasi-complementary sequence sets (QCSSs) by understanding how large the set size (denoted by $M$) can grow with the flock size ($K$) and the sequence length ($N$). We first establish a geometric framework that transforms a QCSS into a complex unit-norm codebook, through which and by exploiting the density thresholds of the codebooks, certain polynomial upper bounds of the QCSS set size are obtained. Sharp quadratic and cubic scaling laws are then introduced. Specifically, we show that asymptotically optimal QCSSs with tightness factor $\rho=1$ satisfy $M \le (1+o(1))K^2N$, while asymptotically near-optimal QCSSs satisfy $M \le (1+o(1))K^3N^2$ for $\rho < {(1+\sqrt{5})}/{2}$. To validate these upper bounds, we further propose explicit additive-character and mixed-character based constructions for QCSSs that achieve $M = K^2N + K$ and $M = K^3N^2 + 2K^2N + K$, respectively, thereby showing that the quadratic and cubic scaling laws are asymptotically tight. Our proposed constructions admit flexible parameter choices, and their maximum correlation estimates are shown to be tight through explicit extremal examples. Additionally, it is conjectured that the cubic scaling law is universal for all $1<\rho\le 2$, i.e., any asymptotically near-optimal QCSSs should satisfy $M \le (1+o(1))K^3N^2$. This identifies a fundamental cubic barrier for QCSS scalability.

math.CO↗

arXiv2026

An Explicit Upper Bound of Generalized Quadratic Gauss Sums and Its Applications for Asymptotically Optimal Aperiodic Polyphase Sequence Design

This work is motivated by the long-standing open problem of designing asymptotically order-optimal aperiodic polyphase sequence sets with respect to the celebrated Welch bound. Attempts were made by Mow over 30 years ago, but a comprehensive understanding to this problem is lacking. Our first key contribution is an explicit upper bound of generalized quadratic Gauss sums which is obtained by recursively applying Paris' asymptotic expansion and then bounding it by leveraging the fast convergence property of the Fibonacci zeta function. Building upon this major finding, our second key contribution includes four systematic constructions of order-optimal sequence sets with low aperiodic correlation and/or ambiguity properties via carefully selected Chu sequences and Alltop sequences. For the first time in the literature, we reveal that the full Alltop sequence set is asymptotically optimal for its low aperiodic correlation sidelobes. Besides, we introduce a novel subset of Alltop sequences possessing both order-optimal aperiodic correlation and ambiguity properties for the entire time-shift window.

cs.IT↗

arXiv2025

Incorporating Authority Perception, Economic Status, and Behavioral Response in Infectious Disease Control

We introduce a multi-population mean field game framework to examine how economic status and authority perception shape vaccination and social distancing decisions under different epidemic control policies. We carried out a survey to inform our model and stratify the population into six groups based on income and perception of authority, capturing behavioral heterogeneity. Individuals adjust their socialization and vaccination levels to optimize objectives such as minimizing treatment costs, complying with social-distancing guidelines if they are authority-followers, or reducing losses from decreased social interactions if they are authority-indifferents, alongside economic costs. Public health authorities influence behavior through social-distancing guidelines and vaccination costs. We characterize the Nash equilibrium via a forward-backward differential equation system, provide its mathematical analysis, and develop a numerical algorithm to solve it. Our findings reveal a trade-off between social-distancing and vaccination decisions. Under stricter guidelines that target both susceptible and infected individuals, followers reduce both socialization and vaccination levels, while indifferents increase socialization due to followers' preventative measures. Adaptive guidelines targeting infected individuals effectively reduce infections and narrow the gap between low- and high-income groups, even when susceptible individuals socialize more and vaccinate less. Lower vaccination costs incentivize vaccination among low-income groups, but their impact on disease spread is smaller than when they are coupled with social-distancing guidelines. Trust-building emerges as a critical factor in epidemic mitigation, underscoring the importance of data-informed, game-theoretical models that aim to understand complex human responses to mitigation policies.

math.OC↗

arXiv2025

New Constructions of Locally Perfect Nonlinear Functions and Their Application to Sequence Sets With Low Ambiguity Zone

Low Ambiguity Zone (LAZ) sequences play a pivotal role in modern integrated sensing and communication (ISAC) systems. Recently, Wang \textit{et al.} [arXiv:2501.11313] proposed a definition of locally perfect nonlinear functions (LPNFs) and constructed three classes of both periodic and aperiodic LAZ sequence sets with flexible parameters by applying such functions and interleaving techniques. Some of these LAZ sequence sets are asymptotically optimal with respect to the Ye-Zhou-Fan-Liu-Lei-Tang bounds under certain conditions. In this paper, we present constructions of three new classes of LPNFs with new parameters. Based on these LPNFs, we further propose a series of LAZ sequence sets that offer more flexible parameters. Furthermore, our results show that some of these classes are asymptotically optimal in both the periodic and aperiodic cases, respectively.

cs.IT↗

arXiv2025

Modeling of Rumor Propagation in Large Populations with Network via Graphon Games

In this paper, we propose a graphon game model to understand how rumor (such as fake news) propagates in large populations that are interacting on a network and how different policies affect the spread. We extend the SKIR model that is used to model rumor propagation and implement individual controls and weighted interactions with other agents to have controlled dynamics. The agents aim to minimize their own expected costs non-cooperatively. We give the finite player game model and the limiting graphon game model to approximate the Nash equilibrium in the population. We give the graphon game Nash equilibrium as a solution to a continuum of ordinary differential equations (ODEs) and give existence results. Finally, we give a numerical approach and analyze examples where we use piecewise constant graphon.

math.OC↗

arXiv2024

On the cross-correlation of Golomb Costas permutations

In the most interesting case of safe prime powers $q$, Gómez and Winterhof showed that a subfamily of the family of Golomb Costas permutations of $\{1,2,\ldots,q-2\}$ of size $φ(q-1)$ has maximal cross-correlation of order of magnitude at most $q^{1/2}$. In this paper we study a larger family of Golomb Costas permutations and prove a weaker bound on its maximal cross-correlation. Considering the whole family of Golomb Costas permutations we show that large cross-correlations are very rare. Finally, we collect several conditions for a small cross-correlation of two Costas permutations. Our main tools are the Weil bound and the Szemerédi-Trotter theorem for finite fields.

math.CO↗

arXiv2021

Balance and pattern distribution of sequences derived from pseudorandom subsets of $\mathbb{Z}_q$

Let $q$ be a positive integer and $\mathcal{S}=\left\{x_0,x_1,\ldots,x_{T-1}\right\}\subseteq\mathbb{Z}_q=\{0,1,\ldots,q-1\}$ with $$0\leq x_0<x_1<\ldots< x_{T-1}\leq q-1.$$ We derive from $\mathcal{S}$ three (finite) sequences. 1. For an integer $M\geq 2$ let $(s_n)$ be the $M$-ary sequence defined by \begin{eqnarray*} s_n\equiv x_{n+1}-x_n \bmod M, \qquad n=0,1,\ldots, T-2. \end{eqnarray*} 2. For an integer $m\geq 2$ let $(t_n)$ be the binary sequence defined by \begin{eqnarray*} t_n=\left\{\begin{array}{ll} 1, & \hbox{if } 1\leq x_{n+1}-x_n\leq m-1, \\ 0, & \hbox{otherwise}, \end{array}\right. \qquad n=0,1,\ldots, T-2. \end{eqnarray*} 3. Let $(u_n)$ be the characteristic sequence of $\mathcal{S}$, \begin{eqnarray*} u_n=\left\{\begin{array}{ll} 1, & \hbox{if } n\in \mathcal{S}, \\ 0, & \hbox{otherwise}, \end{array}\right. \qquad n=0,1,\ldots, q-1. \end{eqnarray*} We study the balance and pattern distribution of the sequences $(s_n)$, $(t_n)$ and $(u_n)$. For sets $\mathcal{S}$ with desirable pseudorandom properties, more precisely, sets with low correlation measures, we show the following: 1. The sequence $(s_n)$ is (asymptotically) balanced and has uniform pattern distribution if $T$ is of smaller order of magnitude than $q$. 2. The sequence $(t_n)$ is balanced and has uniform pattern distribution if $T$ is approximately $\left(1-\frac{1}{2^{1/(m-1)}}\right)q$. 3. The sequence $(u_n)$ is balanced and has uniform pattern distribution if $T$ is approximately $\frac{q}{2}$. These results are motivated by earlier results for the sets of quadratic residues and primitive roots modulo a prime. We unify these results and derive many further (asymptotically) balanced sequences with uniform pattern distribution from pseudorandom subsets.

math.NT↗

arXiv2021

Correlation measures of binary sequences derived from Euler quotients

Fermat-Euler quotients arose from the study of the first case of Fermat's Last Theorem, and have numerous applications in number theory. Recently they were studied from the cryptographic aspects by constructing many pseudorandom binary sequences, whose linear complexities and trace representations were calculated. In this work, we further study their correlation measures by using the approach based on Dirichlet characters, Ramanujan sums and Gauss sums. Our results show that the $4$-order correlation measures of these sequences are very large. Therefore they may not be suggested for cryptography.

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