Regularity of squarefree monomial ideals

We survey a number of recent studies of the Castelnuovo-Mumford regularity of squarefree monomial ideals. Our focus is on bounds and exact values for the regularity in terms of combinatorial data from associated simplicial complexes and/or hypergraphs.Comment: 23 pages; survey paper; minor changes in V.

Matchings, coverings, and Castelnuovo-Mumford regularity

We show that the co-chordal cover number of a graph G gives an upper bound for the Castelnuovo-Mumford regularity of the associated edge ideal. Several known combinatorial upper bounds of regularity for edge ideals are then easy consequences of covering results from graph theory, and we derive new upper bounds by looking at additional covering results.Comment: 12 pages; v4 has minor changes for publicatio

Regularity of Edge Ideals and Their Powers

We survey recent studies on the Castelnuovo-Mumford regularity of edge ideals of graphs and their powers. Our focus is on bounds and exact values of $\text{ reg } I(G)$ and the asymptotic linear function $\text{ reg } I(G)^q$, for $q \geq 1,$ in terms of combinatorial data of the given graph $G.$Comment: 31 pages, 15 figure

Some problems in combinatorial topology of flag complexes

In this work we study simplicial complexes associated to graphs and their homotopical and combinatorial properties. The main focus is on the family of flag complexes, which can be viewed as independence complexes and clique complexes of graphs. In the first part we study independence complexes of graphs using two cofibre sequences corresponding to vertex and edge removals. We give applications to the connectivity of independence complexes of chordal graphs and to extremal problems in topology and we answer open questions about the homotopy types of those spaces for particular families of graphs. We also study the independence complex as a space of configurations of particles in the so-called hard-core models on various lattices. We define, and investigate from an algorithmic perspective, a special family of combinatorially defined homology classes in independence complexes. This enables us to give algorithms as well as NP-hardness results for topological properties of some spaces. As a corollary we prove hardness of computing homology of simplicial complexes in general. We also view flag complexes as clique complexes of graphs. That leads to the study of various properties of Vietoris-Rips complexes of graphs. The last result is inspired by a problem in face enumeration. Using methods of extremal graph theory we classify flag triangulations of 3-manifolds with many edges. As a corollary we complete the classification of face vectors of flag simplicial homology 3-spheres

Building Information Filtering Networks with Topological Constraints: Algorithms and Applications

We propose a new methodology for learning the structure of sparse networks from data; in doing so we adopt a dual perspective where we consider networks both as weighted graphs and as simplicial complexes. The proposed learning methodology belongs to the family of preferential attachment algorithms, where a network is extended by iteratively adding new vertices. In the conventional preferential attachment algorithm a new vertex is added to the network by adding a single edge to another existing vertex; in our approach a new vertex is added to a set of vertices by adding one or more new simplices to the simplicial complex. We propose the use of a score function to quantify the strength of the association between the new vertex and the attachment points. The methodology performs a greedy optimisation of the total score by selecting, at each step, the new vertex and the attachment points that maximise the gain in the score. Sparsity is enforced by restricting the space of the feasible configurations through the imposition of topological constraints on the candidate networks; the constraint is fulfilled by allowing only topological operations that are invariant with respect to the required property. For instance, if the topological constraint requires the constructed network to be be planar, then only planarity-invariant operations are allowed; if the constraint is that the network must be a clique forest, then only simplicial vertices can be added. At each step of the algorithm, the vertex to be added and the attachment points are those that provide the maximum increase in score while maintaining the topological constraints. As a concrete but general realisation we propose the clique forest as a possible topological structure for the representation of sparse networks, and we allow to specify further constraints such as the allowed range of clique sizes and the saturation of the attachment points. In this thesis we originally introduce the Maximally Filtered Clique Forest (MFCF) algorithm: the MFCF builds a clique forest by repeated application of a suitably invariant operation that we call Clique Expansion operator and adds vertices according to a strategy that greedily maximises the gain in a local score function. The gains produced by the Clique Expansion operator can be validated in a number of ways, including statistical testing, cross-validation or value thresholding. The algorithm does not prescribe a specific form for the gain function, but allows the use of any number of gain functions as long as they are consistent with the Clique Expansion operator. We describe several examples of gain functions suited to different problems. As a specific practical realisation we study the extraction of planar networks with the Triangulated Maximally Filtered Graph (TMFG). The TMFG, in its simplest form, is a specialised version of the MFCF, but it can be made more powerful by allowing the use of specialised planarity invariant operators that are not based on the Clique Expansion operator. We provide applications to two well known applied problems: the Maximum Weight Planar Subgraph Problem (MWPSP) and the Covariance Selection problem. With regards to the Covariance Selection problem we compare our results to the state of the art solution (the Graphical Lasso) and we highlight the benefits of our methodology. Finally, we study the geometry of clique trees as simplicial complexes and note how the statistics based on cliques and separators provides information equivalent to the one that can be achieved by means of homological methods, such as the analysis of Betti numbers, however with our approach being computationally more efficient and intuitively simpler. Finally, we use the geometric tools developed to provide a possible methodology for inferring the size of a dataset generated by a factor model. As an example we show that our tools provide a solution for inferring the size of a dataset generated by a factor model