We initiate a statistical study of Kalai’s exterior algebraic shifting, focusing on concentration phenomena for random triangulations of a fixed space. First, for a uniform n-vertex refinement of any given graph G, we show that asymptotically almost-surely (a.a.s.) its exterior algebraic shifting is an explicit shifted graph depending only on n and the Betti numbers of G. Next, for any given compact connected Riemannian surface S, sample n points independently at random according to the volume measure, and consider the resulted a.a.s. unique Delaunay triangulation. We prove that a.a.s. its exterior algebraic shifting is an explicit shifted complex depending only on n and the Euler genus of S, and in particular is area-rigid. In both results the expected shifted complex is a homology lex-segment complex, a notion we define combinatorially and characterize numerically à la Björner-Kalai. As a tool to prove the result on surfaces, we prove a universality result on edge contractions: for every fixed surface triangulation K, every dense enough point set in the surface yields a Delaunay triangulation that edge contracts to K.

In the present era, networking has become an essential requirement as individuals seek interconnectivity and efficient means to share data rapidly. When discussing the swift exchange of information, a network characterized by high speed and minimal congestion is favored by all. In networking, Switches play a very important role to transfer packets of information from source to destination, as routers do in any network topology. For switches to exchange data from the source port of one switch to the destination port of another switch, it requires a special arrangement where every switch is connected to another using chassis topology. When packets transfer, in certain cases, it leads to congestion in the output port due to heavy traffic. A method for resolving the congestion is by signaling to the source port(s), sending the traffic causing the congestion, to throttle the traffic towards the congested port. One such signaling may be based on Quantized Congestion Notification (QCN). As the packets might be received from different sources, the notification needs to be replicated to a subset of the source ports sending to the congested target. This paper proposes the analysis of QCN signaling within a chassis topology, based on different cases using an advanced replication architecture i.e., BIER (Bit Index Explicit Replication) to study its effect to achieve high throughput and low latency of the whole system and proposes a novel approach called periodic QCN for handling persistent congestion.

Given a random 3-uniform hypergraph $H=H(n,p)$ on $n$ vertices where each triple independently appears with probability $p$, consider the following graph process. We start with the star $G_0$ on the same vertex set, containing all the edges incident to some vertex $v_0$, and repeatedly add an edge $xy$ if there is a vertex $z$ such that $xz$ and $zy$ are already in the graph and $xzy\in H$. We say that the process propagates if it reaches the complete graph before it terminates. In this paper we prove that the threshold probability for propagation is $p=\frac{1}{2\sqrt{n}}$. We conclude that $p=\frac{1}{2\sqrt{n}}$ is an upper bound for the threshold probability that a random 2-dimensional simplicial complex is simply connected.