Hurwitz equivalence of braid monodromies and extremal elliptic surfaces

We discuss the equivalence between the categories of certain ribbon graphs and subgroups of the modular group $\Gamma$ and use it to construct exponentially large families of not Hurwitz equivalent simple braid monodromy factorizations of the same element. As an application, we also obtain exponentially large families of {\it topologically} distinct algebraic objects such as extremal elliptic surfaces, real trigonal curves, and real elliptic surfaces

Evolutionary constraints on the complexity of genetic regulatory networks allow predictions of the total number of genetic interactions

Genetic regulatory networks (GRNs) have been widely studied, yet there is a lack of understanding with regards to the final size and properties of these networks, mainly due to no network currently being complete. In this study, we analyzed the distribution of GRN structural properties across a large set of distinct prokaryotic organisms and found a set of constrained characteristics such as network density and number of regulators. Our results allowed us to estimate the number of interactions that complete networks would have, a valuable insight that could aid in the daunting task of network curation, prediction, and validation. Using state-of-the-art statistical approaches, we also provided new evidence to settle a previously stated controversy that raised the possibility of complete biological networks being random and therefore attributing the observed scale-free properties to an artifact emerging from the sampling process during network discovery. Furthermore, we identified a set of properties that enabled us to assess the consistency of the connectivity distribution for various GRNs against different alternative statistical distributions. Our results favor the hypothesis that highly connected nodes (hubs) are not a consequence of network incompleteness. Finally, an interaction coverage computed for the GRNs as a proxy for completeness revealed that high-throughput based reconstructions of GRNs could yield biased networks with a low average clustering coefficient, showing that classical targeted discovery of interactions is still needed.Comment: 28 pages, 5 figures, 12 pages supplementary informatio

Mapping tori of free group automorphisms are coherent

The mapping torus of an endomorphism \Phi of a group G is the HNN-extension G*_G with bonding maps the identity and \Phi. We show that a mapping torus of an injective free group endomorphism has the property that its finitely generated subgroups are finitely presented and, moreover, these subgroups are of finite type.Comment: 17 pages, published versio

The Fiedler connection to the parametrized modularity optimization for community detection

This paper presents a comprehensive analysis of the generalized spectral structure of the modularity matrix $B$, which is introduced by Newman as the kernel matrix for the quadratic-form expression of the modularity function $Q$ used for community detection. The analysis is then seamlessly extended to the resolution-parametrized modularity matrix $B(\gamma)$, where $\gamma$ denotes the resolution parameter. The modularity spectral analysis provides fresh and profound insights into the $\gamma$-dynamics within the framework of modularity maximization for community detection. It provides the first algebraic explanation of the resolution limit at any specific $\gamma$ value. Among the significant findings and implications, the analysis reveals that (1) the maxima of the quadratic function with $B(\gamma)$ as the kernel matrix always reside in the Fiedler space of the normalized graph Laplacian $L$ or the null space of $L$, or their combination, and (2) the Fiedler value of the graph Laplacian $L$ marks the critical $\gamma$ value in the transition of candidate community configuration states between graph division and aggregation. Additionally, this paper introduces and identifies the Fiedler pseudo-set (FPS) as the de facto critical region for the state transition. This work is expected to have an immediate and long-term impact on improvements in algorithms for modularity maximization and on model transformations.Comment: 11 pages, 3 figures, 1 tabl

A combinatorial Li-Yau inequality and rational points on curves

We present a method to control gonality of nonarchimedean curves based on graph theory. Let k denote a complete nonarchimedean valued field.We first prove a lower bound for the gonality of a curve over the algebraic closure of k in terms of the minimal degree of a class of graph maps, namely: one should minimize over all so-called finite harmonic graph morphisms to trees, that originate from any refinement of the dual graph of the stable model of the curve. Next comes our main result: we prove a lower bound for the degree of such a graph morphism in terms of the first eigenvalue of the Laplacian and some “volume” of the original graph; this can be seen as a substitute for graphs of the Li–Yau inequality from differential geometry, although we also prove that the strict analogue of the original inequality fails for general graphs. Finally,we apply the results to give a lower bound for the gonality of arbitraryDrinfeld modular curves over finite fields and for general congruence subgroups Γ of Γ (1) that is linear in the index [Γ (1) : Γ ], with a constant that only depends on the residue field degree and the degree of the chosen “infinite” place. This is a function field analogue of a theorem of Abramovich for classical modular curves. We present applications to uniform boundedness of torsion of rank two Drinfeld modules that improve upon existing results, and to lower bounds on the modular degree of certain elliptic curves over function fields that solve a problem of Papikian