Privacy has nowadays become a major concern in large-scale data processing. Releasing seemingly harmless statistics of a dataset could unexpectedly leak sensitive information of individuals (see e.g.

Regarding the minor clarity issues, we will adjust Figure 1 according to these suggestions and fix the typos stated. If we understand correctly, the reviewer's main concerns are that the numerical results are not comprehensive. We compared COMP/DD/SSS/LP experimentally because these all use the same test matrix (i.i.d. Reviewer 3's suggestions, we believe it would belong in the supplementary material and not the main body.

Hyperparameters V alue Number of encoder (decoder) layers 6 Number of layers in the feed forward network 2 Number of hidden units in the feed forward network 128 Mask filter size 3 Mask number of filters 16 Ratio of residual connection 1.5 Dropout rate 0.1 Optimizer Adam Warm-up steps 4000 Learning rate p d min ( p t, t 4000 Unless otherwise specified, the task performed in this section is selection sort (Section 4). Figure 6 shows the sorting performance of the transformers w/o mask supervision. Figure 7 shows sorting performances with different encoding schemes. In Figure 9, we show the strong generalization performance of the different architectures. While some changes are able to improve performance in this regime, the performance ultimately drops steeply as the length of the test sequence increases. The symbol e represents the end token.

Hyperparameters V alue Number of encoder (decoder) layers 6 Number of layers in the feed forward network 2 Number of hidden units in the feed forward network 128 Mask filter size 3 Mask number of filters 16 Ratio of residual connection 1.5 Dropout rate 0.1 Optimizer Adam Warm-up steps 4000 Learning rate p d min ( p t, t 4000 Unless otherwise specified, the task performed in this section is selection sort (Section 4). Figure 6 shows the sorting performance of the transformers w/o mask supervision. Figure 7 shows sorting performances with different encoding schemes. In Figure 9, we show the strong generalization performance of the different architectures. While some changes are able to improve performance in this regime, the performance ultimately drops steeply as the length of the test sequence increases. The symbol e represents the end token.

We study the problem of robustly estimating the edge density of Erd\H{o}s-R\'enyi random graphs $G(n, d^\circ/n)$ when an adversary can arbitrarily add or remove edges incident to an $\eta$-fraction of the nodes. We develop the first polynomial-time algorithm for this problem that estimates $d^\circ$ up to an additive error $O([\sqrt{\log(n) / n} + \eta\sqrt{\log(1/\eta)} ] \cdot \sqrt{d^\circ} + \eta \log(1/\eta))$. Our error guarantee matches information-theoretic lower bounds up to factors of $\log(1/\eta)$. Moreover, our estimator works for all $d^\circ \geq \Omega(1)$ and achieves optimal breakdown point $\eta = 1/2$. Previous algorithms [AJK+22, CDHS24], including inefficient ones, incur significantly suboptimal errors. Furthermore, even admitting suboptimal error guarantees, only inefficient algorithms achieve optimal breakdown point. Our algorithm is based on the sum-of-squares (SoS) hierarchy. A key ingredient is to construct constant-degree SoS certificates for concentration of the number of edges incident to small sets in $G(n, d^\circ/n)$. Crucially, we show that these certificates also exist in the sparse regime, when $d^\circ = o(\log n)$, a regime in which the performance of previous algorithms was significantly suboptimal.

We give the first polynomial-time, differentially node-private, and robust algorithm for estimating the edge density of Erd\H{o}s-R\'enyi random graphs and their generalization, inhomogeneous random graphs. We further prove information-theoretical lower bounds, showing that the error rate of our algorithm is optimal up to logarithmic factors. Previous algorithms incur either exponential running time or suboptimal error rates. Two key ingredients of our algorithm are (1) a new sum-of-squares algorithm for robust edge density estimation, and (2) the reduction from privacy to robustness based on sum-of-squares exponential mechanisms due to Hopkins et al. (STOC 2023).

With the ubiquitous presence of networks in many areas of science and technology, a multitude of new challenges have gained importance in the statistical analysis of networks. One such area, termed network archaeology (Navlakha and Kingsford [26]) studies problems about unveiling the past of dynamically growing networks, based on present-day observations. In order to develop a sound statistical theory for such problems, one usually models the growing network by simple stochastic growth dynamics. Perhaps the most prominent such growth model is the preferential attachment model, advocated by Albert and Barabási [2]. In these models, vertices of the network arrive one by one and a new vertex attaches to one or more existing vertices by an edge according to some simple probabilistic rule. Arguably the simplest problem of network archaeology is that of root finding, when one aims at estimating the first vertex of a random network, based on observing the (unlabeled) network at a much later point of time.

This paper studies the problem of recovering the hidden vertex correspondence between two edge-correlated random graphs. We focus on the Gaussian model where the two graphs are complete graphs with correlated Gaussian weights and the Erd\H{o}s-R\'enyi model where the two graphs are subsampled from a common parent Erd\H{o}s-R\'enyi graph $\mathcal{G}(n,p)$. For dense graphs with $p=n^{-o(1)}$, we prove that there exists a sharp threshold, above which one can correctly match all but a vanishing fraction of vertices and below which correctly matching any positive fraction is impossible, a phenomenon known as the "all-or-nothing" phase transition. Even more strikingly, in the Gaussian setting, above the threshold all vertices can be exactly matched with high probability. In contrast, for sparse Erd\H{o}s-R\'enyi graphs with $p=n^{-\Theta(1)}$, we show that the all-or-nothing phenomenon no longer holds and we determine the thresholds up to a constant factor. Along the way, we also derive the sharp threshold for exact recovery, sharpening the existing results in Erd\H{o}s-R\'enyi graphs. The proof of the negative results builds upon a tight characterization of the mutual information based on the truncated second-moment computation and an "area theorem" that relates the mutual information to the integral of the reconstruction error. The positive results follows from a tight analysis of the maximum likelihood estimator that takes into account the cycle structure of the induced permutation on the edges.