The nullity of the net Laplacian matrix of a signed graph

Let $\Gamma = (G, \sigma)$ be a signed graph, where $G = (V(G),E(G))$ is an (unsigned) graph, called the underlying graph. The net Laplacian matrix of $\Gamma$ is defined as $L^{\pm}(\Gamma) = D^{\pm}(\Gamma) - A(\Gamma)$, where $D^{\pm}(\Gamma)$ and $A(\Gamma)$ are the diagonal matrix of net-degrees and the adjacency matrix of $\Gamma$, respectively. The nullity of $L^{\pm}(\Gamma)$, written as $\eta (L^{\pm} (\Gamma))$, is the multiplicity of 0 as an eigenvalue of $L^{\pm}(\Gamma)$. In this paper, we focus our attention on the nullity of the net Laplacian matrix of a connected signed graph $\Gamma$ and prove that $1 \leq \eta (L^{\pm} (\Gamma)) \leq min\{ \beta(\Gamma) + 1, |V(\Gamma)| - 1 \}$, where $\beta(\Gamma) = |E(\Gamma)| - |V(\Gamma)| + 1$ is the cyclomatic number of $\Gamma$. The connected signed graphs with nullity $|V(\Gamma)| - 1$ are completely determined. Moreover, we characterize the signed cactus graphs with nullity $1$ or $\beta(\Gamma) + 1$Comment: 11 pages, 1 figure

Navigation in a small world with local information

It is commonly known that there exist short paths between vertices in a network showing the small-world effect. Yet vertices, for example, the individuals living in society, usually are not able to find the shortest paths, due to the very serious limit of information. To theoretically study this issue, here the navigation process of launching messages toward designated targets is investigated on a variant of the one-dimensional small-world network (SWN). In the network structure considered, the probability of a shortcut falling between a pair of nodes is proportional to $r^{-\alpha}$, where $r$ is the lattice distance between the nodes. When $\alpha =0$, it reduces to the SWN model with random shortcuts. The system shows the dynamic small-world (SW) effect, which is different from the well-studied static SW effect. We study the effective network diameter, the path length as a function of the lattice distance, and the dynamics. They are controlled by multiple parameters, and we use data collapse to show that the parameters are correlated. The central finding is that, in the one-dimensional network studied, the dynamic SW effect exists for $0\leq \alpha \leq 2$. For each given value of $\alpha$ in this region, the point that the dynamic SW effect arises is $ML^{\prime}\sim 1$, where $M$ is the number of useful shortcuts and $L^{\prime}$ is the average reduced (effective) length of them.Comment: 10 pages, 5 figures, accepted for publication in Physical Review

On the eigenvalues and Seidel eigenvalues of chain graphs

In this paper we consider the eigenvalues and the Seidel eigenvalues of a chain graph. An\dbareli\'{c}, da Fonseca, Simi\'{c}, and Du \cite{andelic2020tridiagonal} conjectured that there do not exist non-isomorphic cospectral chain graphs with respect to the adjacency spectrum. Here we disprove this conjecture. Furthermore, by considering the relation between the Seidel matrix and the adjacency matrix of a graph, we solve two problems on the number of distinct Seidel eigenvalues of a chain graph, which was posed by Mandal, Mehatari, and Das \cite{mandal2022spectrum}.Comment: 13 pages, 2 figure

Affine Transformation Edited and Refined Deep Neural Network for Quantitative Susceptibility Mapping

Deep neural networks have demonstrated great potential in solving dipole inversion for Quantitative Susceptibility Mapping (QSM). However, the performances of most existing deep learning methods drastically degrade with mismatched sequence parameters such as acquisition orientation and spatial resolution. We propose an end-to-end AFfine Transformation Edited and Refined (AFTER) deep neural network for QSM, which is robust against arbitrary acquisition orientation and spatial resolution up to 0.6 mm isotropic at the finest. The AFTER-QSM neural network starts with a forward affine transformation layer, followed by an Unet for dipole inversion, then an inverse affine transformation layer, followed by a Residual Dense Network (RDN) for QSM refinement. Simulation and in-vivo experiments demonstrated that the proposed AFTER-QSM network architecture had excellent generalizability. It can successfully reconstruct susceptibility maps from highly oblique and anisotropic scans, leading to the best image quality assessments in simulation tests and suppressed streaking artifacts and noise levels for in-vivo experiments compared with other methods. Furthermore, ablation studies showed that the RDN refinement network significantly reduced image blurring and susceptibility underestimation due to affine transformations. In addition, the AFTER-QSM network substantially shortened the reconstruction time from minutes using conventional methods to only a few seconds

DPR: An Algorithm Mitigate Bias Accumulation in Recommendation feedback loops

Recommendation models trained on the user feedback collected from deployed recommendation systems are commonly biased. User feedback is considerably affected by the exposure mechanism, as users only provide feedback on the items exposed to them and passively ignore the unexposed items, thus producing numerous false negative samples. Inevitably, biases caused by such user feedback are inherited by new models and amplified via feedback loops. Moreover, the presence of false negative samples makes negative sampling difficult and introduces spurious information in the user preference modeling process of the model. Recent work has investigated the negative impact of feedback loops and unknown exposure mechanisms on recommendation quality and user experience, essentially treating them as independent factors and ignoring their cross-effects. To address these issues, we deeply analyze the data exposure mechanism from the perspective of data iteration and feedback loops with the Missing Not At Random (\textbf{MNAR}) assumption, theoretically demonstrating the existence of an available stabilization factor in the transformation of the exposure mechanism under the feedback loops. We further propose Dynamic Personalized Ranking (\textbf{DPR}), an unbiased algorithm that uses dynamic re-weighting to mitigate the cross-effects of exposure mechanisms and feedback loops without additional information. Furthermore, we design a plugin named Universal Anti-False Negative (\textbf{UFN}) to mitigate the negative impact of the false negative problem. We demonstrate theoretically that our approach mitigates the negative effects of feedback loops and unknown exposure mechanisms. Experimental results on real-world datasets demonstrate that models using DPR can better handle bias accumulation and the universality of UFN in mainstream loss methods