We investigate geometric embeddings among several classes of generalised cone complexes and algorithms, e.g., to compute their homology. Interesting cases arise from moduli spaces of tropical curves. Specifically, via an explicit computation, we show that the tropical honeycomb curves form a contractible sub-locus in the moduli of all tropical K4-curves.

The convex hull of the roots of a classical root lattice is called a root polytope. We determine explicit unimodular triangulations of the boundaries of the root polytopes associated to the root lattices $A_n$, $C_n$, and $D_n$, and we compute their f- and h-vectors. This leads us to recover formulae for the growth series of these root lattices, which were first conjectured by Conway, Mallows, and Sloane and Baake and Grimm and were proved by Conway and Sloane and Bacher, de la Harpe, and Venkov. We also prove the formula for the growth series of the root lattice $B_n$, which requires a modification of our technique.

Finding decompositions of a graph into a family of clusters is crucial to understanding its underlying structure. While most existing approaches focus on partitioning the nodes, real-world datasets suggest the presence of overlapping communities. We present OCA, a novel algorithm to detect overlapped communities in large data graphs. It outperforms previous proposals in terms of execution time, and efficiently handles large graphs containing more than 108nodes and edges.