Automorphisms of graph products of groups from a geometric perspective

This article studies automorphism groups of graph products of arbitrary groups. We completely characterise automorphisms that preserve the set of conjugacy classes of vertex groups as those automorphisms that can be decomposed as a product of certain elementary automorphisms (inner automorphisms, partial conjugations, automorphisms associated to symmetries of the underlying graph). This allows us to completely compute the automorphism group of certain graph products, for instance in the case where the underlying graph is finite, connected, leafless and of girth at least $5$. If in addition the underlying graph does not contain separating stars, we can understand the geometry of the automorphism groups of such graph products of groups further: we show that such automorphism groups do not satisfy Kazhdan's property (T) and are acylindrically hyperbolic. Applications to automorphism groups of graph products of finite groups are also included. The approach in this article is geometric and relies on the action of graph products of groups on certain complexes with a particularly rich combinatorial geometry. The first such complex is a particular Cayley graph of the graph product that has a quasi-median geometry, a combinatorial geometry reminiscent of (but more general than) CAT(0) cube complexes. The second (strongly related) complex used is the Davis complex of the graph product, a CAT(0) cube complex that also has a structure of right-angled building.Comment: 36 pages. The article subsumes and vastly generalises our preprint arXiv:1803.07536. To appear in Proc. Lond. Math. So

Hypercellular graphs: partial cubes without Q3−Q_3^- as partial cube minor

We investigate the structure of isometric subgraphs of hypercubes (i.e., partial cubes) which do not contain finite convex subgraphs contractible to the 3-cube minus one vertex $Q^-_3$ (here contraction means contracting the edges corresponding to the same coordinate of the hypercube). Extending similar results for median and cellular graphs, we show that the convex hull of an isometric cycle of such a graph is gated and isomorphic to the Cartesian product of edges and even cycles. Furthermore, we show that our graphs are exactly the class of partial cubes in which any finite convex subgraph can be obtained from the Cartesian products of edges and even cycles via successive gated amalgams. This decomposition result enables us to establish a variety of results. In particular, it yields that our class of graphs generalizes median and cellular graphs, which motivates naming our graphs hypercellular. Furthermore, we show that hypercellular graphs are tope graphs of zonotopal complexes of oriented matroids. Finally, we characterize hypercellular graphs as being median-cell -- a property naturally generalizing the notion of median graphs.Comment: 35 pages, 6 figures, added example answering Question 1 from earlier draft (Figure 6.

A combinatorial non-positive curvature I: weak systolicity

We introduce the notion of weakly systolic complexes and groups, and initiate regular studies of them. Those are simplicial complexes with nonpositive-curvature-like properties and groups acting on them geometrically. We characterize weakly systolic complexes as simply connected simplicial complexes satisfying some local combinatorial conditions. We provide several classes of examples --- in particular systolic groups and CAT(-1) cubical groups are weakly systolic. We present applications of the theory, concerning Gromov hyperbolic groups, Coxeter groups and systolic groups.Comment: 35 pages, 1 figur

A Study on Privacy Preserving Data Publishing With Differential Privacy

In the era of digitization it is important to preserve privacy of various sensitive information available around us, e.g., personal information, different social communication and video streaming sites' and services' own users' private information, salary information and structure of an organization, census and statistical data of a country and so on. These data can be represented in different formats such as Numerical and Categorical data, Graph Data, Tree-Structured data and so on. For preventing these data from being illegally exploited and protect it from privacy threats, it is required to apply an efficient privacy model over sensitive data. There have been a great number of studies on privacy-preserving data publishing over the last decades. Differential Privacy (DP) is one of the state of the art methods for preserving privacy to a database. However, applying DP to high dimensional tabular data (Numerical and Categorical) is challenging in terms of required time, memory, and high frequency computational unit. A well-known solution is to reduce the dimension of the given database, keeping its originality and preserving relations among all of its entities. In this thesis, we propose PrivFuzzy, a simple and flexible differentially private method that can publish differentially private data after reducing their original dimension with the help of Fuzzy logic. Exploiting Fuzzy mapping, PrivFuzzy can (1) reduce database columns and create a new low dimensional correlated database, (2) inject noise to each attribute to ensure differential privacy on newly created low dimensional database, and (3) sample each entry in the database and release synthesized database. Existing literatures show the difficulty of applying differential privacy over a high dimensional dataset, which we overcame by proposing a novel fuzzy based approach (PrivFuzzy). By applying our novel fuzzy mapping technique, PrivFuzzy transforms a high dimensional dataset to an equivalent low dimensional one, without losing any relationship within the dataset. Our experiments with real data and comparison with the existing privacy preserving models, PrivBayes and PrivGene, show that our proposed approach PrivFuzzy outperforms existing solutions in terms of the strength of privacy preservation, simplicity and improving utility. Preserving privacy of Graph structured data, at the time of making some of its part available, is still one of the major problems in preserving data privacy. Most of the present models had tried to solve this issue by coming up with complex solution, as well as mixed up with signal and noise, which make these solutions ineffective in real time use and practice. One of the state of the art solution is to apply differential privacy over the queries on graph data and its statistics. But the challenge to meet here is to reduce the error at the time of publishing the data as mechanism of Differential privacy adds a large amount of noise and introduces erroneous results which reduces the utility of data. In this thesis, we proposed an Expectation Maximization (EM) based novel differentially private model for graph dataset. By applying EM method iteratively in conjunction with Laplace mechanism our proposed private model applies differentially private noise over the result of several subgraph queries on a graph dataset. Besides, to ensure expected utility, by selecting a maximal noise level $\theta$, our proposed system can generate noisy result with expected utility. Comparing with existing models for several subgraph counting queries, we claim that our proposed model can generate much less noise than the existing models to achieve expected utility and can still preserve privacy

Graph Theory

This was a workshop on graph theory, with a comprehensive approach. Highlights included the emerging theories of sparse graph limits and of infinite matroids, new techniques for colouring graphs on surfaces, and extensions of graph minor theory to directed graphs and to immersion

Representing Partitions on Trees

In evolutionary biology, biologists often face the problem of constructing a phylogenetic tree on a set X of species from a multiset Π of partitions corresponding to various attributes of these species. One approach that is used to solve this problem is to try instead to associate a tree (or even a network) to the multiset ΣΠ consisting of all those bipartitions {A,X − A} with A a part of some partition in Π. The rational behind this approach is that a phylogenetic tree with leaf set X can be uniquely represented by the set of bipartitions of X induced by its edges. Motivated by these considerations, given a multiset Σ of bipartitions corresponding to a phylogenetic tree on X, in this paper we introduce and study the set P(Σ) consisting of those multisets of partitions Π of X with ΣΠ = Σ. More specifically, we characterize when P(Σ) is non-empty, and also identify some partitions in P(Σ) that are of maximum and minimum size. We also show that it is NP-complete to decide when P(Σ) is non-empty in case Σ is an arbitrary multiset of bipartitions of X. Ultimately, we hope that by gaining a better understanding of the mapping that takes an arbitrary partition system Π to the multiset ΣΠ, we will obtain new insights into the use of median networks and, more generally, split-networks to visualize sets of partitions