However, a topology with a fast consensus rate, e.g., the exponential graph, generally has a large maximum degree, which incurs significant communication

The scale of these problems calls for efficient distributed algorithms across multiple computingnodes.

A.2 Proof of Theorem 1 Before showing properties of W defined by (3), we provide two lemmas as follows. Referring to Theorem 1.6 of [32], we have the following result for a sequence of random matrices. One-peer undirected graphs generated in Alg. 2 with s = 3 and u = 1, . . . From the node's perspective, an equivalent version of Alg. 2 is presented in Alg. 4. In the remainder In Alg. 4, we compute This yields the equivalence between Alg.2 and Alg. 4 for the case of v We first provide the following three lemmas. Averaging the above equations yields the result.

Decentralized optimization is an emerging paradigm in distributed learning in which agents achieve network-wide solutions by peer-to-peer communication without the central server. Since communication tends to be slower than computation, when each agent communicates with only a few neighboring agents per iteration, they can complete iterations faster than with more agents or a central server. However, the total number of iterations to reach a network-wide solution is affected by the speed at which the information of the agents is ``mixed'' by communication. We found that popular communication topologies either have large degrees (such as stars and complete graphs) or are ineffective at mixing information (such as rings and grids). To address this problem, we propose a new family of topologies, EquiTopo, which has an (almost) constant degree and network-size-independent consensus rate which is used to measure the mixing efficiency.In the proposed family, EquiStatic has a degree of $\Theta(\ln(n))$, where $n$ is the network size, and a series of time-varying one-peer topologies, EquiDyn, has a constant degree of 1. We generate EquiDyn through a certain random sampling procedure. Both of them achieve $n$-independent consensus rate. We apply them to decentralized SGD and decentralized gradient tracking and obtain faster communication and better convergence, both theoretically and empirically.

However, a topology with a fast consensus rate, e.g., the exponential graph, generally has a large maximum degree, which incurs significant communication

We generate EquiDyn through a certain random sampling procedure.

This paper explores the use of large language models (LLMs), here represented by GPT 3.5-Turbo (henceforth "GPT"), to perform coding for a thematic analysis. Coding is highly labor intensive, making it infeasible for most researchers to conduct exhaustive thematic analyses of large corpora. Recent advances in large language models (LLMs) have opened the door to novel approaches for automating aspects of qualitative research, including thematic analysis (TA). Prior work has shown that LLMs can generate plausible thematic codes for text data (Dai, Xiong, and Ku, 2023; Morgan, 2023; De Paoli, 2024). This paper focuses on the development and evaluation of an AI-assisted coding methodology designed to enhance the thematic coding of text passages using large language models.

We present a collaborative framework where multiple large language models, namely GPT-4-0125-preview, Meta-LLaMA-3-70B-Instruct, Claude-3-Opus, and Gemini-1.5-Flash, work together to generate and respond to complex PhD-level probability questions in the absence of definitive ground truth. This study explores how inter-model consensus enhances response reliability and serves as a proxy for assessing the quality of generated questions. To quantify agreement and consistency, we employ statistical methods including chi-square tests, Fleiss' Kappa, and confidence interval analysis, measuring both response precision and question clarity. Our findings highlight that Claude and Gemini generate well-structured and less ambiguous questions, leading to higher inter-model agreement. This is reflected in their narrower confidence intervals and stronger alignment with answering models. Conversely, LLaMA demonstrates increased variability and lower reliability in question formulation, as indicated by broader confidence intervals and reduced consensus rates. These results suggest that multi-model collaboration not only enhances the reliability of responses but also provides a valuable framework for assessing and improving question quality in the absence of explicit ground truth. This research offers meaningful insights into optimizing AI-driven reasoning through collaborative large-language model interactions.

We explore the collaborative dynamics of an innovative language model interaction system involving advanced models such as GPT-4-0125-preview, Meta-LLaMA-3-70B-Instruct, Claude-3-Opus, and Gemini-1.5-Flash. These models generate and answer complex, PhD-level statistical questions without exact ground-truth answers. Our study investigates how inter-model consensus enhances the reliability and precision of responses. By employing statistical methods such as chi-square tests, Fleiss' Kappa, and confidence interval analysis, we evaluate consensus rates and inter-rater agreement to quantify the reliability of collaborative outputs. Key results reveal that Claude and GPT-4 exhibit the highest reliability and consistency, as evidenced by their narrower confidence intervals and higher alignment with question-generating models. Conversely, Gemini and LLaMA show more significant variability in their consensus rates, as reflected in wider confidence intervals and lower reliability percentages. These findings demonstrate that collaborative interactions among large language models (LLMs) significantly improve response reliability, offering novel insights into autonomous, cooperative reasoning and validation in AI systems.