Elementary trapping sets (ETSs) are the main culprits of the performance of low-density parity-check (LDPC) codes in the error floor region. Due to their large quantities and complex structures, ETSs are difficult to analyze. This paper studies the impact of the distance between cycles on ETSs, focusing on two special graph classes: theta graphs and dumbbell graphs, which correspond to cycles with negative and non-negative distances, respectively. We determine the Turán numbers of these graphs and prove that increasing the distance between cycles can eliminate more ETSs. Additionally, using the linear state-space model and spectral theory, we prove that increasing the length of cycles or distance between cycles decreases the spectral radius of the system matrix, thereby reducing the harmfulness of ETSs. This is consistent with the conclusion obtained using Turán numbers. For specific cases when removing two 6-cycles with distance of -1, 0 and 1, respectively, we calculate the sizes, spectral radii, and error probabilities of ETSs. These results confirm that the performance of LDPC codes improves as the distance between cycles increases. Furthermore, we design the PEG-CYCLE algorithm, which greedily maximizes the distance between cycles in the Tanner graph. Numerical results show that the QC-LDPC codes constructed by our method achieve performance comparable to or even superior to state-of-the-art construction methods.

It is known that windowed decoding (WD) can effectively balance the performance and complexity of spatially coupled low-density parity-check (LDPC) codes. In this study, we show that information can propagate in a wave-like manner at a constant speed under WD. Additionally, we provide an upper bound for the information propagation speed on the binary erasure channel, which can assist in designing the number of iterations required within each window.

Abstract. A [Formula: see text]-graph system [Formula: see text] is a family of not necessarily distinct [Formula: see text]-graphs on the same [Formula: see text]-vertex set [Formula: see text], and a [Formula: see text]-graph [Formula: see text] on [Formula: see text] is said to be [Formula: see text]-transversal provided that there exists an injection [Formula: see text] such that [Formula: see text] for all [Formula: see text]. We show that given [Formula: see text], sufficiently large [Formula: see text], and an [Formula: see text]-vertex [Formula: see text]-graph system [Formula: see text], if [Formula: see text] for each [Formula: see text], then there exists an [Formula: see text]-transversal tight Hamilton cycle. This extends the result of Rödl, Ruciński, and Szemerédi [ Combinatorica, 28 (2008), pp. 229–260] on single [Formula: see text]-graphs.

Abstract. A [Formula: see text]-graph (or [Formula: see text]-uniform hypergraph) [Formula: see text] is uniformly dense if the edge distribution of [Formula: see text] is uniformly dense with respect to every large collection of [Formula: see text]-vertex cliques induced by sets of [Formula: see text]-tuples. Reiher, Rödl, and Schacht [ Int. Math. Res. Not., 2018] proposed the study of the [Formula: see text]-uniform Turán density [Formula: see text] for given [Formula: see text]-graphs [Formula: see text] in uniformly dense [Formula: see text]-graphs. Meanwhile, they [ J. London Math. Soc., 2018] characterized [Formula: see text]-graphs [Formula: see text] satisfying [Formula: see text] and showed that [Formula: see text] “jumps” from 0 to at least [Formula: see text]. In particular, they asked whether there exist 3-graphs [Formula: see text] with [Formula: see text] equal or arbitrarily close to [Formula: see text]. Recently, Garbe, Král’, and Lamaison [ Israel J. Math., 2023] constructed some 3-graphs with [Formula: see text]. In this paper, for any [Formula: see text]-graph [Formula: see text], we give a lower bound of [Formula: see text] based on a probabilistic framework and provide a general theorem that reduces proving an upper bound on [Formula: see text] to embedding [Formula: see text] in reduced [Formula: see text]-graphs of the same density using the regularity method for [Formula: see text]-graphs. By using this result and Ramsey theorem for multicolored hypergraphs, we extend the results of Garbe, Král’, and Lamaison to [Formula: see text]. In other words, we give a sufficient condition for [Formula: see text]-graphs [Formula: see text] satisfying [Formula: see text]. Additionally, we also construct an infinite family of [Formula: see text]-graphs with [Formula: see text].

In this study, we analyze the characteristics of scheduling sequences for layered belief propagation (LBP) that can result in efficient decoding of low-density parity-check (LDPC) codes. Specifically, we claim that scheduling sequences leading to high decoding efficiency should prioritize updating check nodes with lower error probabilities aggregated from neighboring variable nodes. We prove this conclusion separately on both the BEC and the BI-AWGN channels. Some observable characteristics in “good” scheduling sequences regarding row weights, rows connected to punctured columns, and column weights can serve as corollaries to this conclusion. By comprehensively considering these characteristics of good scheduling, we design a multi-sequence nested scheduling scheme of layered decoding for 5G New Radio (NR) LDPC codes. The proposed schemes can obtain scheduling sequences for various rates of rate-compatible LDPC codes using small hardware storage. What’s more, by respectively storing double-sequence, triple-sequence, or more sequences to obtain scheduling sequences at various code rates, a trade-off can be made between decoder storage and decoding performance. Experimental results demonstrate that the proposed scheme achieves performance improvements compared to existing scheduling schemes at nearly all code rates.

The emergence of immune checkpoint inhibitors (ICIs) has significantly advanced cancer treatment. However, only 15-30% of the cancer patients respond to ICI treatment, which stimulates and enhances host immunity to eliminate tumor cells. ICI treatment is very expensive and has potential adverse reactions; therefore, it is crucial to develop a method which enables to accurately and rapidly assess a patient's suitability before ICI treatment. We complied germline whole-genome sequencing (WES) data of 37 melanoma patients who have been treated with ICIs and sequenced in our lab previously, and the WES data of other 700 ICI-treated cancer patients in public domain. Using these data, we proposed a novel double-channel attention neural network (DANN) model to predict cancer ICI-response and validate the predictions. DANN achieved a mean accuracy and AUC of 0.95 and 0.98, respectively, which outperformed traditional machine learning methods. Enrichment analysis of the DANN-identified genes indicated that cancer patients whose in-born genomic variants might mainly affect host immune system in a wide-ranging manner, and then affect ICI response. Finally, we found a set of 12 genes bearing genomic variants were significantly associated with cancer patient survivals after ICI treatment.

Drug combinations are vital for treating complex diseases and advancing drug development, but accurately identifying synergistic combinations remains a significant challenge. Although graph neural networks (GNNs) have recently been used to predict drug combinations, the complex interactions between drugs and multimodal data (e.g., target proteins) and the prevalent high-order relations among drugs have yet to be fully exploited. The hypergraph offers a natural methodology for modeling high-order relations and provides profound insights for multimodal fusion. Here, we introduce the multimodal relational hypergraph neural network (MRHGNN), a novel framework for predicting synergistic drug combinations. Specifically, we design a dual-channel architecture to capture the physicochemical attributes of drugs and their interactive synergies, thereby facilitating the generation of multimodal drug representations. To obtain comprehensive representations of drugs, we use an attention mechanism to explore complementarity among multimodal drug embeddings. In addition, the unified framework jointly learns primary and self-supervised learning tasks, fostering a robust predictive capability. Experimental results demonstrate that MRHGNN accurately predicts synergistic drug combinations, and the effectiveness of the dual-channel setup and motif structures has been validated through ablation studies. Further literature searches illustrate that our model holds significant promise in accelerating the discovery of novel synergistic drug combinations, particularly in cancer therapy. This study not only introduces a novel computational tool but also paves the way for advanced methodologies in drug discovery and development.

Elementary trapping sets (ETSs) are the principal culprits for the performance of LDPC codes in the error floor region. Due to their large quantity, intricate structures, and high computational complexity, determining how to eliminate dominant ETSs in the design of LDPC codes has become a critical issue in improving error floor behavior. In this paper, we address this problem by avoiding particular theta graphs$(\theta(1,2,2)$and$\theta(2,2,2))$in the Tanner graph to eliminate specific ETSs. These can be characterized by a pivotal tool in graph theory - Turán numbers. Theoretically, we derive the exact Turán number for$\theta(1,2,2)$and demonstrate that all$(a, b)$-ETSs in a Tanner graph with variable-reaular degree$d_{L}(v)=\gamma$must satisfy the inequality$b\geq a\gamma-\frac{1}{2}a^{2}$. This result improves the lower bound previously obtained by Amirzade when the girth is 6. For girth 8, by constraining the relationship between any two 8-cycles in the Tanner graph, we establish a similar inequality$b\geq a\gamma-\frac{a(\sqrt{8a-7}-1)}{2}$. Our simulation results indicate that codes designed with these considerations exhibit improved performance and a lower error floor over additive white Gaussian noise channels.

For all integers $k,d$ such that $k \geq 3$ and $k/2\leq d \leq k-1$, let $n$ be a sufficiently large integer ( which may not be divisible by $k$ ) , and let $s\le \lfloor n/k\rfloor-1$. We show that if $H$ is a $k$-uniform hypergraph on $n$ vertices with $\delta_{d}(H)>\binom{n-d}{k-d}-\binom{n-d-s+1}{k-d}$, then $H$ contains a matching of size $s$. This improves a recent result of Lu, Yu, and Yuan and also answers a question of Kühn, Osthus, and Townsend. In many cases, our result can be strengthened to $s\leq \lfloor n/k\rfloor$, which then covers the entire possible range of $s$. On the other hand, there are examples showing that the result does not hold for certain $n, k, d$, and $s= \lfloor n/k\rfloor$.

We introduce a notion of the crux of a graph $G$, measuring the order of a smallest dense subgraph in $G$. This simple-looking notion leads to some generalizations of known results about cycles, offering an interesting paradigm of “replacing average degree by crux.” In particular, we prove that every graph contains a cycle of length linear in its crux. Long proved that every subgraph of a hypercube $Q^m$ (resp., discrete torus $C_3^m$) with average degree $d$ contains a path of length $2^{d/2}$ (resp., $2^{d/4}$) and conjectured that there should be a path of length $2^{d}-1$ (resp., $3^{d/2}-1$). As a corollary of our result, together with isoperimetric inequalities, we close these exponential gaps giving asymptotically optimal bounds on long paths in hypercubes, discrete tori, and more generally Hamming graphs. We also consider random subgraphs of $C_4$-free graphs and hypercubes, proving near optimal lower bounds on the lengths of long cycles.

Research on the expander code shows that for a regular low-density parity-check (LDPC) code, the Tanner graph’s spectrum determines its properties, such as the minimum distance and the size of stopping sets. In this study, we demonstrate theoretically and experimentally that the performance of irregular LDPC codes is related to the graph spectrum. Our observations may provide an efficient metric to search for good irregular LDPC codes.

A k-graph H is called (p, μ)-dense if for all not necessarily disjoint sets A1, …, Ak ⊆ V(H) we have e(A1, …, Ak) ≥ p|A1| ⃛ |Ak| – μ|V(H)|k. This is believed to be the weakest form of quasi-randomness in k-graphs and also known as linear quasi-randomness. In this paper we show that for ℓ < k satisfying (k – ℓ) ∤ k, (p, μ)-denseness plus a minimum (ℓ + 1)-vertex-degree αnk–ℓ–1 guarantees Hamilton ℓ-cycles, but requiring a minimum ℓ-vertex-degree Ω(nk–ℓ) instead is not sufficient. This answers a question of Lenz–Mubayi–Mycroft and characterizes the triples (k, ℓ, d) such that degenerate choices of p and α force ℓ-Hamiltonicity. We actually prove a general result on ℓ-Hamiltonicity in quasi-random k-graphs, assuming a minimum vertex degree and essentially that every two ℓ-sets can be connected by a constant length ℓ-path. This result reduces the ℓ-Hamiltonicity problem to the study of the connection property. Moreover, we note that our proof can be turned into a deterministic polynomial-time algorithm that outputs the Hamilton ℓ-cycle. Our proof uses the lattice-based absorption method in the non-standard way and is the first one that embeds a nonlinear Hamilton cycle in linear quasi-random k-graphs.

BACKGROUND: In the last few decades, cumulative experimental researches have witnessed and verified the important roles of microRNAs (miRNAs) in the development of human complex diseases. Benefitting from the rapid growth both in the availability of miRNA-related data and the development of various analysis methodologies, up until recently, some computational models have been developed to predict human disease related miRNAs, efficiently and quickly. RESULTS: In this work, we proposed a computational model of Random Walk and Binary Regression-based MiRNA-Disease Association prediction (RWBRMDA). RWBRMDA extracted features for each miRNA from random walk with restart on the integrated miRNA similarity network for binary logistic regression to predict potential miRNA-disease associations. RWBRMDA obtained AUC of 0.8076 in the leave-one-out cross validation. Additionally, we carried out three different patterns of case studies on four human complex diseases. Specifically, Esophageal cancer and Prostate cancer were conducted as one kind of case study based on known miRNA-disease associations in HMDD v2.0 database. Out of the top 50 predicted miRNAs, 94 and 90% were respectively confirmed by recent experimental reports. To simulate new disease without known related miRNAs, the information of known Breast cancer related miRNAs was removed. As a result, 98% of the top 50 predicted miRNAs for Breast cancer were confirmed. Lymphoma, the verified ratio of which was 88%, was used to assess the prediction robustness of RWBRMDA based on the association records in HMDD v1.0 database. CONCLUSIONS: We anticipated that RWBRMDA could benefit the future experimental investigations about the relation between human disease and miRNAs by generating promising and testable top-ranked miRNAs, and significantly reducing the effort and cost of identification works.