Minimum spanning trees across dense cities

Consider~$n$ nodes distributed independently across~$N$ cities contained with the unit square~$S$ according to a distribution~$f.$ Each city is modelled as an~$r_n \times r_n$ square contained within~$S$ and~$MSTC_n$ denotes the length of the minimum spanning tree containing all the~$n$ nodes. We use approximation methods to obtain variance estimates for~$MSTC_n$ and prove that if the cities are well-connected and densely populated in a certain sense, then~$MSTC_n$ appropriately centred and scaled converges to zero in probability. Using the proof techniques, we alternately derive corresponding results for the length~$MST_n$ of the minimum spanning tree for the usual case when the nodes are independently distributed throughout the unit square~$S.$ In particular, we obtain that the variance of~$MST_n$ grows at most as a power of the logarithm of~$n$ and use a subsequence argument to get almost sure convergence of~$MST_n$ appropriately centred and scaled.Comment: arXiv admin note: text overlap with arXiv:1801.0269

Rates of linear codes with low decoding error probability

Consider binary linear codes obtained from bipartite graphs as follows. There are~$k \geq 1$ left nodes each representing a message bit and there are~$m = m(k)$ right nodes each representing a parity bit, generated from the corresponding set of message node neighbours. Both the message and the parity bits are sent through a memoryless binary input channel that either retains, flips or erases each transmitted bit, independently. Based on the received set of symbols, the decoder at the receiver obtains an estimate of the original message sent. If the decoding error probability~$P_k \longrightarrow 0$ and the average degree per parity node remains bounded as~$k \rightarrow \infty,$ then the rate of the code~$\frac{k}{k+m} \longrightarrow 0$ as~\(k \rightarrow \infty.\

Long Paths and Hamiltonian paths in Inhomogenous Random Graphs

In this paper, we study long paths and Hamiltonian paths in inhomogenous random graphs. In the first part of the paper, we consider an inhomogenous Erd\H{o}s-R\'enyi random graph $G_E$ with average edge density $p_n.$ We prove that if $np_n^2 \longrightarrow \infty$ as $n \rightarrow \infty,$ then the longest path contains at least $n-ne^{-\delta_1 np_n^2}$ nodes with high probability (i.e., with probability converging to one as $n \rightarrow \infty$), for some constant $\delta_1> 0 .$ In particular, if $np_n^2 = M\log{n}$ for some constant $M > 0$ large, then $G_E$ is Hamiltonian with high probability; i.e., the longest path contains all the nodes of $G_E.$ In the second part of the paper, we consider a random geometric graph $G_R$ consisting of $n$ nodes, each independently distributed according to a (not necessarily uniform) density $f.$ If $r_n$ is the connectivity radius and $nr_n^2 \longrightarrow \infty,$ then with high probability, the longest cycle contains at least $n-ne^{-\delta_2 nr_n^2}$ nodes for some constant $\delta_2 > 0.$ As a consequence of our proof, we obtain that if $nr_n^2 = \log{n} + 7\log{\log{n}} + \omega_n$ and $\omega_n \longrightarrow \infty$ as $n \rightarrow \infty,$ then with high probability $G_R$ contains a Hamiltonian cycle

Duality in percolation via outermost boundaries III: Plus connected components

Tile $\mathbb{R}^2$ into disjoint unit squares $\{S_k\}_{k \geq 0}$ with the origin being the centre of $S_0$ and say that $S_i$ and $S_j$ are star adjacent if they share a corner and plus adjacent if they share an edge. Every square is either vacant or occupied. In this paper, we use the structure of the outermost boundaries derived in Ganesan (2017) to alternately obtain duality between star and plus connected components in the following sense: There is a star connected cycle of vacant squares attached to and surrounding the finite plus connected component containing the origin

Randomized detection and detection capacity of multidetector networks

In this paper, we study the following detection problem. There are $n$ detectors randomly placed in the unit square $S = \left[-\frac{1}{2},\frac{1}{2}\right]^2$ assigned to detect the presence of a source located at the origin. Time is divided into slots of unit length and $D_i(t) \in \{0,1\}$ represents the (random) decision of the $i^{\rm th}$ detector in time slot $t$. The location of the source is unknown to the detectors and the goal is to design schemes that use the decisions $\{D_i(t)\}_{i,t}$ and detect the presence of the source in as short time as possible. We first determine the minimum achievable detection time $T_{cap}$ and show the existence of \emph{randomized} detection schemes that have detection times arbitrarily close to $T_{cap}$ for almost all configuration of detectors, provided the number of detectors $n$ is sufficiently large. We call such schemes as \emph{capacity achieving} and completely characterize all capacity achieving detection schemes

Duality in percolation via outermost boundaries I: Bond Percolation

Tile (\mathbb{R}^2\) into disjoint unit squares (\{S_k\}_{k \geq 0}\) with the origin being the centre of $S_0$ and say that (S_i\) and (S_j\) are star adjacent if they share a corner and plus adjacent if they share an edge. Every square is either vacant or occupied. Outermost boundaries of finite star and plus connected components frequently arise in the context of contour analysis in percolation and random graphs. In this paper, we derive the outermost boundaries for finite star and plus connected components using a piecewise cycle merging algorithm. For plus connected components, the outermost boundary is a single cycle and for star connected components, we obtain that the outermost boundary is a connected union of cycles with mutually disjoint interiors. As an application, we use the outermost boundaries to give an alternate proof of mutual exclusivity of left right and top bottom crossings in oriented and unoriented bond percolation

Graph extensions, edit number and regular graphs

A graph G on n vertices is said to be extendable if G can be modified to form a new graph H on more than n vertices, while preserving the degrees of the vertices common to G and H. The added vertices all have the same degree and we define edit numbers to quantify the amount of modification needed to obtain the extended graph. Characterizing graphs with least possible edit numbers, we obtain that graphs with zero edit number can be extended using regular graphs. We also describe an iterative algorithm to construct connected regular graphs on arbitrarily large vertex sets, starting from the complete graph on a fixed set of vertices

Existence of connected regular and nearly regular graphs

For integers $k \geq 2$ and $n \geq k+1$, we prove the following: If $n\cdot k$ is even, there is a connected $k$-regular graph on $n$ vertices. If $n\cdot k$ is odd, there is a connected nearly $k$-regular graph on $n$ vertices

Outermost boundaries for star-connected components in percolation

Tile $\mathbb{R}^2$ into disjoint unit squares $\{S_k\}_{k \geq 0}$ with the origin being the centre of $S_0$ and say that $S_i$ and $S_j$ are star-adjacent if they share a corner and plus-adjacent if they share an edge. Every square is either vacant or occupied. If the occupied plus-connected component $C^+(0)$ containing the origin is finite, it is known that the outermost boundary $\partial^+_0$ of $C^+(0)$ is a unique cycle surrounding the origin. For the finite occupied star-connected component $C(0)$ containing the origin, we prove in this paper that the outermost boundary $\partial_0$ is a unique connected graph consisting of a union of cycles $\cup_{1 \leq i \leq n} C_i$ with mutually disjoint interiors. Moreover, we have that each pair of cycles in $\partial_0$ share at most one vertex in common and we provide an inductive procedure to obtain a circuit containing all the edges of $\cup_{1 \leq i \leq n} C_i.$ This has applications for contour analysis of star-connected components in percolation

Recurrence region of multiuser Aloha

In this paper, we provide upper and lower bounds for the region of positive recurrence for a general finite user Aloha network