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To improve transaction rates, many cryptocurrencies have implemented so-called ''Layer-2'' transaction protocols, where payments are routed across networks of private payment channels. However, for a given transaction, not every network…
Finding community structures in networks is important in network science, technology, and applications. To date, most algorithms that aim to find community structures only focus either on unipartite or bipartite networks. A unipartite…
Models such as Word2Vec and GloVe construct word embeddings based on the co-occurrence probability $P(i,j)$ of words $i$ and $j$ in text corpora. The resulting vectors $W_i$ not only group semantically similar words but also exhibit a…
Diagrammatic techniques to compute perturbatively the spectral properties of Euclidean Random Matrices in the high-density regime are introduced and discussed in detail. Such techniques are developed in two alternative and very different…
Wei's celebrated Duality Theorem is generalized in several ways, expressed as duality theorems for linear codes over division rings and, more generally, duality theorems for matroids. These results are further generalized, resulting in two…
The study of complex networks is a significant development in modern science, and has enriched the social sciences, biology, physics, and computer science. Models and algorithms for such networks are pervasive in our society, and impact…
This paper introduces combinatorial representations, which generalise the notion of linear representations of matroids. We show that any family of subsets of the same cardinality has a combinatorial representation via matrices. We then…
This thesis deals with the enumerative study of combinatorial maps, and its application to the enumeration of other combinatorial objects. Combinatorial maps, or simply maps, form a rich combinatorial model. They have an intuitive and…
The use of complex networks as a modern approach to understanding the world and its dynamics is well-established in literature. The adjacency matrix, which provides a one-to-one representation of a complex network, can also yield several…
We study the connections between three seemingly different combinatorial structures - "uniform" brackets in statistics and probability theory, "containers" in online and distributed learning theory, and "combinatorial Macbeath regions", or…
We introduce a technique for establishing and amplifying gaps between parameters of network coding and index coding. The technique uses linear programs to establish separations between combinatorial and coding-theoretic parameters and…
Various statistics on wreath products are defined via canonical words, "colored" right to left minima and "colored" descents. It is shown that refined counts with respect to these statistics have nice recurrence formulas of…
Co-occurrence matrices, such as co-citation, co-word, and co-link matrices, have been used widely in the information sciences. However, confusion and controversy have hindered the proper statistical analysis of this data. The underlying…
The convolution of indicators of two conjugacy classes on the symmetric group S_q is usually a complicated linear combination of indicators of many conjugacy classes. Similarly, a product of the moments of the Jucys--Murphy element involves…
We introduce combinatorial objects named matricubes that provide a generalization of the theory of matroids. As matroids provide a combinatorial axiomatization of hyperplane arrangements, matricubes provide a combinatorial axiomatization of…
We introduce the notion of a network's conduciveness, a probabilistically interpretable measure of how the network's structure allows it to be conducive to roaming agents, in certain conditions, from one portion of the network to another.…
Let $M$ be a matroid. We study the expansions of $M$ mainly to see how the combinatorial properties of $M$ and its expansions are related to each other. It is shown that $M$ is a graphic, binary or a transversal matroid if and only if an…
Product search is one of the most popular methods for customers to discover products online. Most existing studies on product search focus on developing effective retrieval models that rank items by their likelihood to be purchased. They,…
In this paper we investigate the problem of learning evolving concepts over a combinatorial structure. Previous work by Emamjomeh-Zadeh et al. [2020] introduced dynamics into interactive learning as a way to model non-static user…
Recurrence plots provide a graphical representation of the recurrent patterns in a timeseries, the quantification of which is a relatively new field. Here we derive analytical expressions which relate the values of key statistics, notably…