Computing metric hulls in graphs

We prove that, given a closure function the smallest preimage of a closed set can be calculated in polynomial time in the number of closed sets. This confirms a conjecture of Albenque and Knauer and implies that there is a polynomial time algorithm to compute the convex hull-number of a graph, when all its convex subgraphs are given as input. We then show that computing if the smallest preimage of a closed set is logarithmic in the size of the ground set is LOGSNP-complete if only the ground set is given. A special instance of this problem is computing the dimension of a poset given its linear extension graph, that was conjectured to be in P. The intent to show that the latter problem is LOGSNP-complete leads to several interesting questions and to the definition of the isometric hull, i.e., a smallest isometric subgraph containing a given set of vertices $S$. While for $|S|=2$ an isometric hull is just a shortest path, we show that computing the isometric hull of a set of vertices is NP-complete even if $|S|=3$. Finally, we consider the problem of computing the isometric hull-number of a graph and show that computing it is $\Sigma^P_2$ complete.Comment: 13 pages, 3 figure

Equivariant Intersection Cohomology of BXB Orbit Closures in the Wonderful Compactification of a Group

This thesis studies the topology of a particularly nice compactification that exists for semisimple adjoint algebraic groups: the wonderful compactification. The compactifica- tion is equivariant, extending the left and right action of the group on itself, and we focus on the local and global topology of the closures of Borel orbits. It is natural to study the topology of these orbit closures since the study of the topology of Borel orbit closures in the flag variety (that is, Schubert varieties) has proved to be inter- esting, linking geometry and representation theory since the local intersection cohomology Betti numbers turned out to be the coefficients of Kazhdan-Lusztig polynomials. We compute equivariant intersection cohomology with respect to a torus action because such actions often have convenient localization properties enabling us to use data from the moment graph (roughly speaking the collection of 0 and 1-dimensional orbits) to compute the equivariant (intersection) cohomology of the whole space, an approach commonly re- ferred to as GKM theory after Goresky, Kottowitz and MacPherson. Furthermore in the GKM setting we can recover ordinary intersection cohomology from the equivariant inter- section cohomology. Unfortunately the GKM theorems are not practical when computing intersection cohomology since for singular varieties we may not a priori know the local equivariant intersection cohomology at the torus fixed points. Braden and MacPherson address this problem, showing how to algorithmically apply GKM theory to compute the equivariant intersection cohomology for a large class of varieties that includes Schubert varieties. Our setting is more complicated than that of Braden and MacPherson in that we must use some larger torus orbits than just the 0 and 1-dimensional orbits. Nonetheless we are able to extend the moment graph approach of Braden and MacPherson. We define a more general notion of moment graph and identify canonical sheaves on the generalized moment graph whose sections are the equivariant intersection cohomology of the Borel orbit closures of the wonderful compactification

Collective Dynamics of Dark Web Marketplaces

Dark markets are commercial websites that use Bitcoin to sell or broker transactions involving drugs, weapons, and other illicit goods. Being illegal, they do not offer any user protection, and several police raids and scams have caused large losses to both customers and vendors over the past years. However, this uncertainty has not prevented a steady growth of the dark market phenomenon and a proliferation of new markets. The origin of this resilience have remained unclear so far, also due to the difficulty of identifying relevant Bitcoin transaction data. Here, we investigate how the dark market ecosystem re-organises following the disappearance of a market, due to factors including raids and scams. To do so, we analyse 24 episodes of unexpected market closure through a novel datasets of 133 million Bitcoin transactions involving 31 dark markets and their users, totalling 4 billion USD. We show that coordinated user migration from the closed market to coexisting markets guarantees overall systemic resilience beyond the intrinsic fragility of individual markets. The migration is swift, efficient and common to all market closures. We find that migrants are on average more active users in comparison to non-migrants and move preferentially towards the coexisting market with the highest trading volume. Our findings shed light on the resilience of the dark market ecosystem and we anticipate that they may inform future research on the self-organisation of emerging online markets

A linear time algorithm for a variant of the max cut problem in series parallel graphs

Given a graph $G=(V, E)$, a connected sides cut $(U, V\backslash U)$ or $\delta (U)$ is the set of edges of E linking all vertices of U to all vertices of $V\backslash U$ such that the induced subgraphs $G[U]$ and $G[V\backslash U]$ are connected. Given a positive weight function $w$ defined on $E$, the maximum connected sides cut problem (MAX CS CUT) is to find a connected sides cut $\Omega$ such that $w(\Omega)$ is maximum. MAX CS CUT is NP-hard. In this paper, we give a linear time algorithm to solve MAX CS CUT for series parallel graphs. We deduce a linear time algorithm for the minimum cut problem in the same class of graphs without computing the maximum flow.Comment: 6 page