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Given a hereditary property $mathcal H$ of graphs and some $pin[0,1]$, the edit distance function $operatorname{ed}_{mathcal H}(p)$ is (asymptotically) the maximum proportion of edits (edge-additions plus edge-deletions) necessary to transform any graph of density $p$ into a member of $mathcal H$. For any fixed $pin[0,1]$, $operatorname{ed}_{mathcal H}(p)$ can be computed from an object known as a colored regularity graph (CRG). This paper is concerned with those points $pin[0,1]$ for which infinitely many CRGs are required to compute $operatorname{ed}_{mathcal H}$ on any open interval containing $p$; such a $p$ is called an accumulation point. We show that, as expected, $p=0$ and $p=1$ are indeed accumulation points for some hereditary properties; we additionally determine the slope of $operatorname{ed}_{mathcal H}$ at these two extreme points. Unexpectedly, we construct a hereditary property with an accumulation point at $p=1/4$. Finally, we derive a significant structural property about those CRGs which occur at accumulation points.
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The edit distance function of a hereditary property $mathscr{H}$ is the asymptotically largest edit distance between a graph of density $pin[0,1]$ and $mathscr{H}$. Denote by $P_n$ and $C_n$ the path graph of order $n$ and the cycle graph of order $n$, respectively. Let $C_{2n}^*$ be the cycle graph $C_{2n}$ with a diagonal, and $widetilde{C_n}$ be the graph with vertex set ${v_0, v_1, ldots, v_{n-1}}$ and $E(widetilde{C_n})=E(C_n)cup {v_0v_2}$. Marchant and Thomason determined the edit distance function of $C_6^{*}$. Peck studied the edit distance function of $C_n$, while Berikkyzy et al. studied the edit distance of powers of cycles. In this paper, by using the methods of Peck and Martin, we determine the edit distance function of $C_8^{*}$, $widetilde{C_n}$ and $P_n$, respectively.
Given a hereditary property of graphs $mathcal{H}$ and a $pin [0,1]$, the edit distance function ${rm ed}_{mathcal{H}}(p)$ is asymptotically the maximum proportion of edge-additions plus edge-deletions applied to a graph of edge density $p$ sufficient to ensure that the resulting graph satisfies $mathcal{H}$. The edit distance function is directly related to other well-studied quantities such as the speed function for $mathcal{H}$ and the $mathcal{H}$-chromatic number of a random graph. Let $mathcal{H}$ be the property of forbidding an ErdH{o}s-R{e}nyi random graph $Fsim mathbb{G}(n_0,p_0)$, and let $varphi$ represent the golden ratio. In this paper, we show that if $p_0in [1-1/varphi,1/varphi]$, then a.a.s. as $n_0toinfty$, begin{align*} {rm ed}_{mathcal{H}}(p) = (1+o(1)),frac{2log n_0}{n_0} cdotminleft{ frac{p}{-log(1-p_0)}, frac{1-p}{-log p_0} right}. end{align*} Moreover, this holds for $pin [1/3,2/3]$ for any $p_0in (0,1)$.
Scene Graph, as a vital tool to bridge the gap between language domain and image domain, has been widely adopted in the cross-modality task like VQA. In this paper, we propose a new method to edit the scene graph according to the user instructions, which has never been explored. To be specific, in order to learn editing scene graphs as the semantics given by texts, we propose a Graph Edit Distance Reward, which is based on the Policy Gradient and Graph Matching algorithm, to optimize neural symbolic model. In the context of text-editing image retrieval, we validate the effectiveness of our method in CSS and CRIR dataset. Besides, CRIR is a new synthetic dataset generated by us, which we will publish it soon for future use.
This paper proposes a procedure to train a scene text recognition model using a robust learned surrogate of edit distance. The proposed method borrows from self-paced learning and filters out the training examples that are hard for the surrogate. The filtering is performed by judging the quality of the approximation, using a ramp function, enabling end-to-end training. Following the literature, the experiments are conducted in a post-tuning setup, where a trained scene text recognition model is tuned using the learned surrogate of edit distance. The efficacy is demonstrated by improvements on various challenging scene text datasets such as IIIT-5K, SVT, ICDAR, SVTP, and CUTE. The proposed method provides an average improvement of $11.2 %$ on total edit distance and an error reduction of $9.5%$ on accuracy.
Reeb graphs are structural descriptors that capture shape properties of a topological space from the perspective of a chosen function. In this work we define a combinatorial metric for Reeb graphs of orientable surfaces in terms of the cost necessary to transform one graph into another by edit operations. The main contributions of this paper are the stability property and the optimality of this edit distance. More precisely, the stability result states that changes in the functions, measured by the maximum norm, imply not greater changes in the corresponding Reeb graphs, measured by the edit distance. The optimality result states that our edit distance discriminates Reeb graphs better than any other metric for Reeb graphs of surfaces satisfying the stability property.
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