A periodic temporal graph $\mathcal{G}=(G_0, G_1, \dots, G_{p-1})^*$ is an infinite periodic sequence of graphs $G_i=(V,E_i)$ where $G=(V,\cup_i E_i)$ is called the footprint. Recently, the arena where the Cops and Robber game is played has been extended from a graph to a periodic graph; in this case, the copnumber is also the minimum number of cops sufficient for capturing the robber. We study the connections and distinctions between the copnumber $c(\mathcal{G})$ of a periodic graph $\mathcal{G}$ and the copnumber $c(G)$ of its footprint $G$ and establish several facts. For instance, we show that the smallest periodic graph with $c(\mathcal{G}) = 3$ has at most $8$ nodes; in contrast, the smallest graph $G$ with $c(G) = 3$ has $10$ nodes. We push this investigation by generating multiple examples showing how the copnumbers of a periodic graph $\mathcal{G}$, the subgraphs $G_i$ and its footprint $G$ can be loosely tied. Based on these results, we derive upper bounds on the copnumber of a periodic graph from properties of its footprint such as its treewidth.
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For a fixed integer $k\ge 2$, a $k$-community structure in an undirected graph is a partition of its vertex set into $k$ sets called communities, each of size at least two, such that every vertex of the graph has proportionally at least as many neighbours in its own community as in any other community. In this paper, we give a necessary and sufficient condition for a forest on $n$ vertices to admit a $k$-community structure. Furthermore, we provide an $O(n^{2})$-time algorithm that computes such a $k$-community structure in a forest, if it exists. These results extend a result of [Bazgan et al., Structural and algorithmic properties of $2$-community structure, Algorithmica, 80(6):1890-1908, 2018]. We also show that if communities are allowed to have size one, then every forest with $n \geq k\geq 2$ vertices admits a $k$-community structure that can be found in time $O(n^{2})$. We then consider threshold graphs and show that every connected threshold graph admits a $2$-community structure if and only if it is not isomorphic to a star; also if such a $2$-community structure exists, we explain how to obtain it in linear time. We further describe two infinite families of disconnected threshold graphs, containing exactly one isolated vertex, that do not admit any $2$-community structure. Finally, we present a new infinite family of connected graphs that may contain an even or an odd number of vertices without $2$-community structures, even if communities are allowed to have size one.
This paper presents $\textbf{R}$epresentation-$\textbf{C}$onditioned image $\textbf{G}$eneration (RCG), a simple yet effective image generation framework which sets a new benchmark in class-unconditional image generation. RCG does not condition on any human annotations. Instead, it conditions on a self-supervised representation distribution which is mapped from the image distribution using a pre-trained encoder. During generation, RCG samples from such representation distribution using a representation diffusion model (RDM), and employs a pixel generator to craft image pixels conditioned on the sampled representation. Such a design provides substantial guidance during the generative process, resulting in high-quality image generation. Tested on ImageNet 256$\times$256, RCG achieves a Frechet Inception Distance (FID) of 3.31 and an Inception Score (IS) of 253.4. These results not only significantly improve the state-of-the-art of class-unconditional image generation but also rival the current leading methods in class-conditional image generation, bridging the long-standing performance gap between these two tasks. Code is available at //github.com/LTH14/rcg.
We show that the twin-width of every $n$-vertex $d$-regular graph is at most $n^{\frac{d-2}{2d-2}+o(1)}$ and that almost all $d$-regular graphs attain this bound. More generally, we obtain bounds on the twin-width of sparse Erd\H{o}s-Renyi and regular random graphs, complementing the bounds in the denser regime due to Ahn, Chakraborti, Hendrey, Kim and Oum.
We show that a canonical labeling of a random $n$-vertex graph can be obtained by assigning to each vertex $x$ the triple $(w_1(x),w_2(x),w_3(x))$, where $w_k(x)$ is the number of walks of length $k$ starting from $x$. This takes time $O(n^2)$, where $n^2$ is the input size, by using just two matrix-vector multiplications. The linear-time canonization of a random graph is the classical result of Babai, Erd\H{o}s, and Selkow. For this purpose they use the well-known combinatorial color refinement procedure, and we make a comparative analysis of the two algorithmic approaches.
We investigate algorithms for testing whether an image is connected. Given a proximity parameter $\epsilon\in(0,1)$ and query access to a black-and-white image represented by an $n\times n$ matrix of Boolean pixel values, a (1-sided error) connectedness tester accepts if the image is connected and rejects with probability at least 2/3 if the image is $\epsilon$-far from connected. We show that connectedness can be tested nonadaptively with $O(\frac 1{\epsilon^2})$ queries and adaptively with $O(\frac{1}{\epsilon^{3/2}} \sqrt{\log\frac{1}{\epsilon}})$ queries. The best connectedness tester to date, by Berman, Raskhodnikova, and Yaroslavtsev (STOC 2014) had query complexity $O(\frac 1{\epsilon^2}\log \frac 1{\epsilon})$ and was adaptive. We also prove that every nonadaptive, 1-sided error tester for connectedness must make $\Omega(\frac 1\epsilon\log \frac 1\epsilon)$ queries.
Probabilistic graphical models are widely used to model complex systems with uncertainty. Traditionally, Gaussian directed graphical models are applied for analysis of large networks with continuous variables since they can provide conditional and marginal distributions in closed form simplifying the inferential task. The Gaussianity and linearity assumptions are often adequate, yet can lead to poor performance when dealing with some practical applications. In this paper, we model each variable in graph G as a polynomial regression of its parents to capture complex relationships between individual variables and with utility function of polynomial form. Since the marginal posterior distributions of individual variables can become analytically intractable, we develop a message-passing algorithm to propagate information throughout the network solely using moments which enables the expected utility scores to be calculated exactly. We illustrate how the proposed methodology works in a decision problem in energy systems planning.
Optimal transport (OT) theory and the related $p$-Wasserstein distance ($W_p$, $p\geq 1$) are widely-applied in statistics and machine learning. In spite of their popularity, inference based on these tools is sensitive to outliers or it can perform poorly when the underlying model has heavy-tails. To cope with these issues, we introduce a new class of procedures. (i) We consider a robust version of the primal OT problem (ROBOT) and show that it defines the {robust Wasserstein distance}, $W^{(\lambda)}$, which depends on a tuning parameter $\lambda > 0$. (ii) We illustrate the link between $W_1$ and $W^{(\lambda)}$ and study its key measure theoretic aspects. (iii) We derive some concentration inequalities for $W^{(\lambda)}$. (iii) We use $W^{(\lambda)}$ to define minimum distance estimators, we provide their statistical guarantees and we illustrate how to apply concentration inequalities for the selection of $\lambda$. (v) We derive the {dual} form of the ROBOT and illustrate its applicability to machine learning problems (generative adversarial networks and domain adaptation). Numerical exercises provide evidence of the benefits yielded by our methods.
The maximum likelihood threshold (MLT) of a graph $G$ is the minimum number of samples to almost surely guarantee existence of the maximum likelihood estimate in the corresponding Gaussian graphical model. We give a new characterization of the MLT in terms of rigidity-theoretic properties of $G$ and use this characterization to give new combinatorial lower bounds on the MLT of any graph. We use the new lower bounds to give high-probability guarantees on the maximum likelihood thresholds of sparse Erd{\"o}s-R\'enyi random graphs in terms of their average density. These examples show that the new lower bounds are within a polylog factor of tight, where, on the same graph families, all known lower bounds are trivial. Based on computational experiments made possible by our methods, we conjecture that the MLT of an Erd{\"o}s-R\'enyi random graph is equal to its generic completion rank with high probability. Using structural results on rigid graphs in low dimension, we can prove the conjecture for graphs with MLT at most $4$ and describe the threshold probability for the MLT to switch from $3$ to $4$. We also give a geometric characterization of the MLT of a graph in terms of a new "lifting" problem for frameworks that is interesting in its own right. The lifting perspective yields a new connection between the weak MLT (where the maximum likelihood estimate exists only with positive probability) and the classical Hadwiger-Nelson problem.
In 2020, Behr defined the problem of edge coloring of signed graphs and showed that every signed graph $(G, \sigma)$ can be colored using exactly $\Delta(G)$ or $\Delta(G) + 1$ colors, where $\Delta(G)$ is the maximum degree in graph $G$. In this paper, we focus on products of signed graphs. We recall the definitions of the Cartesian, tensor, strong, and corona products of signed graphs and prove results for them. In particular, we show that $(1)$ the Cartesian product of $\Delta$-edge-colorable signed graphs is $\Delta$-edge-colorable, $(2)$ the tensor product of a $\Delta$-edge-colorable signed graph and a signed tree requires only $\Delta$ colors and $(3)$ the corona product of almost any two signed graphs is $\Delta$-edge-colorable. We also prove some results related to the coloring of products of signed paths and cycles.
We provide numerical bounds on the Crouzeix ratiofor KLS matrices $A$ which have a line segment on the boundary of the numerical range. The Crouzeix ratio is the supremum over all polynomials $p$ of the spectral norm of $p(A)$ dividedby the maximum absolute value of $p$ on the numerical range of $A$.Our bounds confirm the conjecture that this ratiois less than or equal to $2$. We also give a precise description of these numerical ranges.
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