Daily Papers

The widespread application of graph data inevitably brings significant privacy risks, as its unprotected use can lead to the leakage of sensitive information. These risks are particularly acute in the setting with an untrusted curator, where the data remains decentralized and each user only holds the connections to their neighbors. To mitigate such privacy risks, we adopt local differential privacy (LDP) to collect users' private information and generate a synthetic graph. However, existing methods suffer from either excessive noise injection by perturbing the local adjacency lists or significant structural information loss due to the simplistic graph encoding process. To address these issues, we propose LDPGraph, an effective graph synthesis algorithm that takes one step further by exploiting neighborhood structures under LDP. To obtain neighborhood statistics beyond degrees, LDPGraph aggregates a noisy global view from perturbed adjacency lists and combines it with projected local connections to estimate node-level triangle counts. To correct the structural inconsistency caused by separately perturbing degree and triangle count, LDPGraph jointly refines them into feasible degree-triangle targets. To reconstruct a global graph from these estimated targets, LDPGraph adopts a triangle-first strategy that first preserves local clustering structures and then fulfills remaining degree requirements. Extensive experiments on four real-world datasets and multiple commonly used graph metrics validate the superiority of LDPGraph. Source code is available at https://github.com/ZJU-TrustAID/LDPGraph.

For a vertex v of a graph G, the triangle-degree td_G(v) is the number of triangles containing v. A graph is triangle-distinct, or K_3-irregular, if its vertex triangle-degrees are pairwise distinct. Chartrand, Erdős, and Oellermann asked whether a regular K_3-irregular graph exists. We prove that such graphs exist for every regularity rge 9. More precisely, for every integer kge 15 we construct a 2k-regular triangle-distinct graph on 4k+2 vertices; complementation gives a (2k+1)-regular example of the same order. The construction uses two antiregular threshold blocks joined by a zero-one matrix with prescribed margins, followed by matrix 2-switches that preserve those margins. After a uniform reference perturbation, exactly three triangle-degree collisions remain. Switches chosen according to parity remove two of them, and the last possible collision is controlled by a short quadratic discriminant argument. Odd kge 17 and even kge 62 are handled symbolically, while the remaining 24 values are settled by exact finite verification. Together with the known examples for 9le rle 29, this closes the positive existence problem for every rge 9.

Triangle centrality is introduced for finding important vertices in a graph based on the concentration of triangles surrounding each vertex. It has the distinct feature of allowing a vertex to be central if it is in many triangles or none at all. We show experimentally that triangle centrality is broadly applicable to many different types of networks. Our empirical results demonstrate that 30% of the time triangle centrality identified central vertices that differed with those found by five well-known centrality measures, which suggests novelty without being overly specialized. It is also asymptotically faster to compute on sparse graphs than all but the most trivial of these other measures. We introduce optimal algorithms that compute triangle centrality in O(mbarδ) time and O(m+n) space, where barδle O(m) is the average degeneracy introduced by Burkhardt, Faber, and Harris (2020). In practical applications, barδ is much smaller than m so triangle centrality can be computed in nearly linear time. On a Concurrent Read Exclusive Write (CREW) Parallel Random Access Machine (PRAM), we give a near work-optimal parallel algorithm that takes O(log n) time using O(mm) CREW PRAM processors. In MapReduce, we show it takes four rounds using O(mm) communication bits and is therefore optimal. We also derive a linear algebraic formulation of triangle centrality which can be computed in O(mbarδ) time on sparse graphs.

We determine the 4-point correlation function and amplitude in planar, maximally supersymmetric Yang-Mills theory to 12 loops. We find that the recently-introduced 'double-triangle' rule in fact implies the previously described square and pentagon rules; and when applied to 12 loops, it fully determines the 11-loop correlator and fixes all but 3 of the (22,024,902) 12-loop coefficients; these remaining coefficients can be subsequently fixed using the '(single-)triangle' rule. Not only do we confirm the Catalan conjecture for anti-prism graphs, but we discover evidence for a greatly generalized Catalan conjecture for the coefficients of all polygon-framed fishnet graphs. We provide all contributions through 12 loops as ancillary files to this work.

Generalised object counting aims to estimate the number of instances of an arbitrary object category from a single image, but many recent methods can struggle on structurally complex objects due to limited spatial modelling. We present UpCount, a class-agnostic counter designed to better preserve spatial structure. UpCount strengthens the visual representation by extracting multi-layer features from a ViT-B/16 encoder and reassembling them into a refined multi-scale pyramid that is spatially refined using Dense Prediction Transformers and FeatUp, yielding features with improved structural and spatial sensitivity; a proposal--verification counting head then identifies repeated patterns and produces a density map for the final count. On FSC-147, UpCount achieves 12.39 MAE and 100.89 RMSE on the test set, and it transfers effectively to vehicle counting on CARPK (6.27 MAE, 8.79 RMSE). Code: https://github.com/r28112072-rgb/upcount

A classical problem in convex and discrete geometry asks for the convex polyhedron of greatest volume whose vertices are chosen from the unit sphere S^2. For a prescribed number N of vertices, the problem is known only in a small number of cases. In this paper we resolve the next outstanding case, N=9. We prove that every convex polyhedron with at most nine vertices on S^2 has volume at most 32sqrt{3-3}, with equality, up to rotation, precisely for a triaugmented triangular prism of an explicitly determined shape. The proof combines combinatorial and geometric reductions with sharp volume estimates. By a theorem of Berman and Hanes (Mathematische Annalen, 1970), a volume maximizer must be simplicial, reducing the 2,606 combinatorial types of 9-vertex polyhedra to 50. We prove that a maximizer cannot have a trivalent vertex, leaving only five combinatorial types, which are treated using geometric and combinatorial arguments. In particular, we determine the exact maximizer within the triaugmented triangular prism class, and characterize the equality case.

The flexibility and accuracy of methods for automatically counting objects in images and videos are limited by the way the object can be specified. While existing methods allow users to describe the target object with text and visual examples, the visual examples must be manually annotated inside the image, and there is no way to specify what not to count. To address these gaps, we introduce novel capabilities that expand how the target object can be specified. Specifically, we extend the prompt to enable what not to count to be described with text and/or visual examples, introduce the concept of `pseudo-exemplars' that automate the annotation of visual examples at inference, and extend counting models to accept visual examples from both natural and synthetic external images. We also use our new counting model, CountGD++, as a vision expert agent for an LLM. Together, these contributions expand the prompt flexibility of multi-modal open-world counting and lead to significant improvements in accuracy, efficiency, and generalization across multiple datasets. Code is available at https://github.com/niki-amini-naieni/CountGDPlusPlus.

A rational spherical triangle is a triangle on the unit sphere such that the lengths of its three sides and its area are rational multiples of π. Little and Coxeter have given examples of rational spherical triangles in 1980s. In this work, we are interested in determining all the rational spherical triangles. We introduce a conjecture on the solutions to a trigonometric Diophantine equation. An implication of the conjecture is that the only rational spherical triangles are the ones given by Little and Coxeter. We prove some partial results towards the conjecture.

Consider the following process: Take any four-digit number which has at least two distinct digits. Then, rearrange the digits of the original number in ascending and descending order, take these two numbers, and find the difference between the two. Finally, repeat this routine using the difference as the new four-digit number. In 1949, D. R. Kaprekar became the first to discover that this process, known as the Kaprekar Routine, would always yield 6174 within 7 iterations. Since this number remains unchanged after an application of the Kaprekar Routine, it became known as Kaprekar's Constant. Previous works have shown that the only base 10 Kaprekar's Constants are 495 and 6174, the 3-digit and 4-digit case. However, little attention has been given to other bases or determining which digit cases and which bases have a Kaprekar's Constant. This paper analyzes the behavior of the Kaprekar Routine in the 3-digit case, deriving an expression for all 3-digit Kaprekar Constants. In addition, the author developed a series of C++ programs to analyze the paths integers followed to their respective Kaprekar's Constant. Surprisingly, it was determined from this program that the most commonly required number of iterations required to reach Kaprekar's Constant for 3-digit integers was consistently 3, regardless of base. When loaded as a matrix, the iteration requirement data demonstrates a precise recurring relationship reminiscent of Pascal's Triangle.

The recursive-line and signed Zarankiewicz numbers maximize the number of squares in augmentations of a maximum C_4-free base, subject to two sufficient irreducibility criteria. The count includes one square per base cell and one per selected pair of unused cells. We compare these parameters with the second-order number, which uses irreducibility itself. Every maximum mtimes n base admits a recursive-line augmentation with at least mn/2-Cmax(m,n) squares, for an absolute constant C. Combining this bound with a two-column extension of known fixed-width families, we show that all three parameters are asymptotically equivalent, uniformly as the larger dimension tends to infinity. For individual displays, a transfer graph shows that once the signed closure identifies every selected pair, the signed criterion is equivalent to irreducibility. We determine the second-order number for every six-column rectangle and give eventual exact formulas for all three numbers at widths seven, nine and eleven. We also prove signed and recursive-line equality at 8times7 and, together with earlier values, whenever the shorter side is at most six, except possibly at 14times4. The exact-value results are computer-assisted, using exhaustive enumeration, checked propositional refutations and symbolic certificates with a proved lifting argument for arbitrary lengths.

Seminal research in the field of graph neural networks (GNNs) has revealed a direct correspondence between the expressive capabilities of GNNs and the k-dimensional Weisfeiler-Leman (kWL) test, a widely-recognized method for verifying graph isomorphism. This connection has reignited interest in comprehending the specific graph properties effectively distinguishable by the kWL test. A central focus of research in this field revolves around determining the least dimensionality k, for which kWL can discern graphs with different number of occurrences of a pattern graph P. We refer to such a least k as the WL-dimension of this pattern counting problem. This inquiry traditionally delves into two distinct counting problems related to patterns: subgraph counting and induced subgraph counting. Intriguingly, despite their initial appearance as separate challenges with seemingly divergent approaches, both of these problems are interconnected components of a more comprehensive problem: "graph motif parameters". In this paper, we provide a precise characterization of the WL-dimension of labeled graph motif parameters. As specific instances of this result, we obtain characterizations of the WL-dimension of the subgraph counting and induced subgraph counting problem for every labeled pattern P. We additionally demonstrate that in cases where the kWL test distinguishes between graphs with varying occurrences of a pattern P, the exact number of occurrences of P can be computed uniformly using only local information of the last layer of a corresponding GNN. We finally delve into the challenge of recognizing the WL-dimension of various graph parameters. We give a polynomial time algorithm for determining the WL-dimension of the subgraph counting problem for given pattern P, answering an open question from previous work.