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We consider the problem of when one quandle homomorphism will factor through another, restricting our attention to the case where all quandles involved are connected. We provide a complete solution to the problem for surjective quandle…
We construct a pairing, which we call factorization homology, between framed manifolds and higher categories. The essential geometric notion is that of a vari-framing of a stratified manifold, which is a framing on each stratum together…
We define the notion of connectivity set for elements of any finitely generated Coxeter group. Then we define an order related to this new statistic and show that the poset is graded and each interval is a shellable lattice. This implies…
Core-periphery is a key feature of large-scale networks underlying a wide range of social, biological, and transportation phenomena. Despite its prevalence in empirical data, it is unclear whether this property is a consequence of more…
Many of the structural characteristics of a network depend on the connectivity with and within the hubs. These dependencies can be related to the degree of a node and the number of links that a node shares with nodes of higher degree. In…
Homophily, the tendency of individuals to connect with others who share similar attributes, is a defining feature of social networks. Understanding how groups interact, both within and across, is crucial for uncovering the dynamics of…
Detecting and exploiting similarities between seemingly distant objects is without doubt an important human ability. This paper develops \textit{from the ground up} an abstract algebraic and qualitative notion of similarity based on the…
We study the properties of groups that have presentations in which the square of each generator gives the identity and all relations are of even length. We consider the parabolic subgroups of such a group and show that every element has a…
Navigability is a distinctive features of graphs associated with artificial or natural systems whose primary goal is the transportation of information or goods. We say that a graph $\mathcal{G}$ is navigable when an agent is able to…
A necessary condition for uniqueness of factorizations of elements of a finite group $G$ with factors belonging to a union of some conjugacy classes of $G$ is given. This condition is sufficient if the number of factors belonging to each…
The association between tie strength and social structure is a fundamental topic in the social sciences. We study this association by analyzing tie strength in higher-order networks, an increasingly relevant model which can encode group…
Higher-order connectivity patterns such as small induced sub-graphs called graphlets (network motifs) are vital to understand the important components (modules/functional units) governing the configuration and behavior of complex networks.…
A parameterisation of generalised network clustering, in the form of four-motif prevalences, is presented. This involves three real parameters that are conditional on one- two- and three-motif prevalences. Interpretations of these real…
We consider a random geometric hypergraph model based on an underlying bipartite graph. Nodes and hyperedges are sampled uniformly in a domain, and a node is assigned to those hyperedges that lie with a certain radius. From a modelling…
We recall some abstract connectivity concepts, and apply them to special chains in partially ordered sets, called veins, that are defined as order-convex chains that are contained in every maximal chain they meet. Veins enable us to define…
Drawing inspiration from Emmy Noether'set-theoretic foundations for algebra and Charles Ehresmann's topology without points, we adopt a new order-theoretic approach to ideal theory. For this we emphasize the order of divisibility in…
In the context of applying the Lorentz group theory to polarization optics in the frames of Stokes-Mueller formalism, some properties of the Lorentz group are investigated. We start with the factorized form of arbitrary Lorentz matrix as a…
These are the notes of the lectures delivered by the author at CIME in June 2018. The main purpose of the notes is to provide an overview of the techniques used in the construction of the triply graded link homology. The homology is space…
We propose a generalization of small world networks, in which the reconnection of links is governed by a function that depends on the distance between the elements to be linked. An adequate choice of this function lets us control the…
The concept of nestedness, in particular for ecological and economical networks, has been introduced as a structural characteristic of real interacting systems. We suggest that the nestedness is in fact another way to express a mesoscale…