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We extend a result of Goldreich and Ron about estimating the collision probability of a hash function. Their estimate has a polynomial tail. We prove that when the load factor is greater than a certain constant, the estimator has a gaussian…
Hash functions are a basic cryptographic primitive. Certain hash functions try to prove security against collision and preimage attacks by reductions to known hard problems. These hash functions usually have some additional properties that…
Convolutional layers within graph neural networks operate by aggregating information about local neighbourhood structures; one common way to encode such substructures is through random walks. The distribution of these random walks evolves…
Graph convolutional networks are a popular class of deep neural network algorithms which have shown success in a number of relational learning tasks. Despite their success, graph convolutional networks exhibit a number of peculiar features,…
Biological processes underlying the basic functions of a cell involve complex interactions between genes. From a technical point of view, these interactions can be represented through a graph where genes and their connections are,…
Graph-structured data arise in wide applications, such as computer vision, bioinformatics, and social networks. Quantifying similarities among graphs is a fundamental problem. In this paper, we develop a framework for computing graph…
Random geometric graphs consist of randomly distributed nodes (points), with pairs of nodes within a given mutual distance linked. In the usual model the distribution of nodes is uniform on a square, and in the limit of infinitely many…
Charles, Goren, and Lauter [J. Cryptology 22(1), 2009] explained how one can construct hash functions using expander graphs in which it is hard to find paths between specified vertices. The set of solutions to the classical Markoff equation…
Given a network, the critical node detection problem finds a subset of nodes whose removal disrupts the network connectivity. Since many real-world systems are naturally modeled as graphs, assessing the vulnerability of the network is…
Probabilistic dependency graphs (PDGs) are a flexible class of probabilistic graphical models, subsuming Bayesian Networks and Factor Graphs. They can also capture inconsistent beliefs, and provide a way of measuring the degree of this…
The Deligne-Ogus-Shioda theorem guarantees the existence of isomorphisms between products of supersingular elliptic curves over finite fields. In this paper, we present methods for explicitly computing these isomorphisms in polynomial time,…
We investigate the computational complexity of the exponential random graph model (ERGM) commonly used in social network analysis. This model represents a probability distribution on graphs by setting the log-likelihood of generating a…
We propose and study a hierarchical algorithm to generate graphs having a predetermined distribution of cliques, the fully connected subgraphs. The construction mechanism may be either random or incorporate preferential attachment. We…
We study the modular curves defined by Weber functions, and associated modular polynomials, action of $\mathrm{SL}_2(\mathbb{Z})$, and parametrizations of elliptic curves with a view to the study of the isogeny graphs that they determine,…
We study stochastic graph optimization problems in a novel distributed setting. As in the standard centralized setting, a random subgraph $G^*$ of a known base graph $G$ is realized by including each edge $e$ independently with a known…
Typical R-convolution graph kernels invoke the kernel functions that decompose graphs into non-isomorphic substructures and compare them. However, overlooking implicit similarities and topological position information between those…
We consider soft random geometric graphs, constructed by distributing points (nodes) randomly according to a Poisson Point Process, and forming links between pairs of nodes with a probability that depends on their mutual distance, the…
Isogeny volcanoes are graphs whose vertices are elliptic curves and whose edges are $\ell$-isogenies. Algorithms allowing to travel on these graphs were developed by Kohel in his thesis (1996) and later on, by Fouquet and Morain (2001).…
Wireless sensor networks are often modeled in terms of a dense deployment of smart sensor nodes in a two-dimensional region. Give a node deployment, the \emph{critical geometric graph (CGG)} over these locations (i.e., the connected…
We study computing geometric problems on uncertain points. An uncertain point is a point that does not have a fixed location, but rather is described by a probability distribution. When these probability distributions are restricted to a…