## Abstract

Suppose $G$ is a connected graph of $n$ vertices. Let $k > 0$ be some integer, and let $T_i \subseteq V(G)$ be a set of vertices, for $0 \leq i < k$. We say that $T_i$ is the set of "towers" of color $i$.

For every vertex $v \in V(G)$ we denote by $t_i(v)$ the tower $x$ of color $i$ that is closest to $v$ in the graph. (If there is a tie between a few candidates, we break it by picking the tower of color $i$ with lowest id).

Denote the set $T = \cup_{i}{T_i}$ of all the towers. We now define a new directed graph $Q$ with $V(Q) = T$, the set of all towers. For two towers $u, v \in V(Q)$, $(u,v) \in E(Q)$ if and only if $t_i(u) = v$ for some $0 \leq i < k$.

We want to know in which cases $Q$ is a strongly connected or weakly connected (Connected as an undirected graph). In other words, we want to know whether the connections towers form a connected graph.

Our experiments show that for $k \geq \log_{2}{n}$ the resulting overlay graph $Q$ is probably weakly connected, but not necessarily strongly connected.

The experiments are written in Rust and could be found here [github].

## Motivation

Suppose that we are given a large network $G$. Every node is connected directly to a few other nodes, and because of the large size of the network, no node can perceive the full structure of the network. We want to be able to route messages in this network between any pair of nodes.

Our method to do this is by choosing a random set of about $\sqrt{n}$ nodes, called the "towers". Every tower out of the $\sqrt{n}$ towers has some color $c$, where $0 \leq c < k$.

Every node keeps contact with the closest tower of color $c$ for $0 \leq c < k$. This amounts to a total of $c$ towers. Whenever a node $x$ wants to send a message to a remote node $y$ in the network, $x$ sends the message to one of the close towers he keeps contact with. The towers then route the message all the way to one of the towers that keep contact with $y$, and finally that tower passes the message to $y$.

For this scheme to work, we have to make sure that the towers know how to route messages between every two towers. Every node maintains contact with $k$ different towers, and this is also true for tower nodes. We use those connections as an overlay graph between the towers. In this overlay directed graph, every tower is connected to $k-1$ other towers and has a self edge himself. The self edge happens because a tower $x_c$ of color $c$ has $t_c(x_c) = x_c$.

In the picture: The overlay directed graph between the towers (Self edges are omitted). Note that the graph is connected, but not strongly connected. The black small dots represent regular nodes. For example: To route a message from the node $x$ to the node $y$, the message has to pass between various towers along the way.

**If we can assume that the overlay directed graph between towers is somehow
connected**, we should be able to manage routing between the towers. As a simple
solution, as there are only $\sqrt{n}$ towers, this could be done by letting
every tower remember a shortest path to every other tower.

Note that in this setting a tower does more work than a usual node. A tower has to remember shortest path to other $\sqrt{n}$ towers, and in addition maintain contact with $\sqrt{n}$ nodes around him. This is not a fully decentralized setting, but we believe that the work that a tower does should be manageable and some might choose to do it for the right compensation.

In this document we try to evaluate which parameters for this structure of towers (Amount of towers and amount of colors) we should use to make sure that the resulting overlay directed graph between towers is connected.

## Experiment

We experiment as follows: We generate different types of networks of different sizes. For a network of size $n$ nodes we pick $k = 2\cdot\log_{2}{n}$ different colors. For each color we pick $\sqrt{n} / k$ different towers (Towers of different color may overlap). This is the code that picks the towers (written in Rust):

/// Choose nodes to be towers. We pick num_towers towers of every color. There are num_colors /// different tower colors. pub fn choose_towers<Node: Hash + Eq + Clone, R: Rng>(net: &Network<Node>, num_towers: usize, num_colors: usize, rng: &mut R) -> Vec<Vec<usize>> { let mut chosen_towers: Vec<Vec<usize>> = Vec::new(); for _ in 0 .. num_colors { // Pick random towers for a certain color: let mut ctowers = choose_k_nums(num_towers, net.igraph.node_count(), rng) .into_iter() .collect::<Vec<usize>>(); // Sort for determinism: ctowers.sort(); chosen_towers.push(ctowers); } chosen_towers }

Then we calculate iteratively for every node $v$ in the network the closest tower of color $c$ for $0 \leq c < k$. This is how one iteration looks like:

/// Perform one iteration of calculating towers info. /// Return whether any changed happen during this iteration. fn iter_towers_info<Node: Hash + Eq + Clone>(net: &Network<Node>, chosen_towers: &Vec<Vec<usize>>, towers_info: &mut Vec<Vec<Option<LocalTowerInfo>>>) -> bool { let mut changed = false; for node in net.igraph.nodes() { for nei in net.igraph.neighbors(node) { for tower_color in 0 .. chosen_towers.len() { if towers_info[node][tower_color].is_none() { continue } // This is the candidate LocalTowerInfo for nei: let mut candidate_info = towers_info[node][tower_color].clone().unwrap(); candidate_info.distance += 1; candidate_info.gateway = node; // Current nei's LocalTowerInfo: if towers_info[nei][tower_color].is_none() { changed = true; towers_info[nei][tower_color] = Some(candidate_info); continue } let nei_info = towers_info[nei][tower_color].clone().unwrap(); if (candidate_info.distance, candidate_info.gateway, candidate_info.tower_node) < (nei_info.distance, nei_info.gateway, nei_info.tower_node) { changed = true; towers_info[nei][tower_color] = Some(candidate_info); } } } } changed }

Finally, we create a directed overlay graph of the towers and check its connectivity. We check two types of connectivity for the overlay graph: weak connectivity and strong connectivity. Weak connectivity is checked using dfs, and strong connectivity is checked using kosaraju algorithm for finding strongly connected components. Both of those algorithms were taken from the petgraph rust crate.

This is the code that checks connectivity:

/// Check if overlay directed graph of towers is connected. /// Returns (connected, strongly_connected) pub fn is_connected(chosen_towers: &Vec<Vec<usize>>, towers_info: &Vec<Vec<Option<LocalTowerInfo>>>) -> (bool, bool) { // An overlay directed graph of the towers in the network // and the connections between them. // A tower T is connected to a tower T' if T' is the closest tower to T of some color. let mut towers_graph: graphmap::DiGraphMap<usize,()> = graphmap::DiGraphMap::new(); // Add towers as nodes to the graph: for tower_color in 0 .. chosen_towers.len() { for tower_index in 0 .. chosen_towers[tower_color].len() { towers_graph.add_node(chosen_towers[tower_color][tower_index]); } } // For every tower, add all connections to closest local towers as nodes. let graph_nodes = towers_graph.nodes().collect::<Vec<usize>>(); for tower_node in graph_nodes { for tower_color in 0 .. chosen_towers.len() { towers_graph.add_edge(tower_node, towers_info[tower_node][tower_color].clone().unwrap().tower_node,()); } } let sconnected_comps = kosaraju_scc(&towers_graph); (connected_components(&towers_graph) == 1, sconnected_comps.len() == 1) }

The experiment is done for 5 different types of networks:

- rand: An approximated random $G(n,p)$ network with $p = 1.5\log(n)/n$
- 2d: a two dimensional grid.
- rand + 2d: A sum of rand and 2d networks.
- planar: We randomize nodes as points in a large plane. Every point is
connected to the closest $1.5\log(n)$ points [ref] The main bottleneck for
this implementation is the generation of the
`planar`

type network. It is currently done in $O(n^2)$, which takes a long time. [/ref]. - tree: A tree network.

Experiment results ($k = 2\log_{2}{n}$)

$ cargo run --bin towers_scc --release | tee ../results/towers_scc_g_2_weakly_2017_07_26.txt Compiling net_coords v0.1.0 (file:///home/real/projects/d/freedomlayer/freedomlayer_research/network_coords/net_coords) Finished release [optimized] target(s) in 3.30 secs Running `target/release/towers_scc` Checking if local towers overlay graph is strongly connected g= 6; rand ; ni=0 |num_colors = 12 |num_towers = 1 | connected = V| sconnected = V g= 6; rand ; ni=1 |num_colors = 12 |num_towers = 1 | connected = V| sconnected = V g= 6; 2d ; ni=0 |num_colors = 12 |num_towers = 1 | connected = V| sconnected = V g= 6; 2d ; ni=1 |num_colors = 12 |num_towers = 1 | connected = V| sconnected = V g= 6; rand+2d ; ni=0 |num_colors = 12 |num_towers = 1 | connected = V| sconnected = V g= 6; rand+2d ; ni=1 |num_colors = 12 |num_towers = 1 | connected = V| sconnected = V g= 6; planar ; ni=0 |num_colors = 12 |num_towers = 1 | connected = V| sconnected = V g= 6; planar ; ni=1 |num_colors = 12 |num_towers = 1 | connected = V| sconnected = V g= 6; tree ; ni=0 |num_colors = 12 |num_towers = 1 | connected = V| sconnected = V g= 6; tree ; ni=1 |num_colors = 12 |num_towers = 1 | connected = V| sconnected = V g= 7; rand ; ni=0 |num_colors = 14 |num_towers = 1 | connected = V| sconnected = V g= 7; rand ; ni=1 |num_colors = 14 |num_towers = 1 | connected = V| sconnected = V g= 7; 2d ; ni=0 |num_colors = 14 |num_towers = 1 | connected = V| sconnected = V g= 7; 2d ; ni=1 |num_colors = 14 |num_towers = 1 | connected = V| sconnected = V g= 7; rand+2d ; ni=0 |num_colors = 14 |num_towers = 1 | connected = V| sconnected = V g= 7; rand+2d ; ni=1 |num_colors = 14 |num_towers = 1 | connected = V| sconnected = V g= 7; planar ; ni=0 |num_colors = 14 |num_towers = 1 | connected = V| sconnected = V g= 7; planar ; ni=1 |num_colors = 14 |num_towers = 1 | connected = V| sconnected = V g= 7; tree ; ni=0 |num_colors = 14 |num_towers = 1 | connected = V| sconnected = V g= 7; tree ; ni=1 |num_colors = 14 |num_towers = 1 | connected = V| sconnected = V g= 8; rand ; ni=0 |num_colors = 16 |num_towers = 2 | connected = V| sconnected = V g= 8; rand ; ni=1 |num_colors = 16 |num_towers = 2 | connected = V| sconnected = V g= 8; 2d ; ni=0 |num_colors = 16 |num_towers = 2 | connected = V| sconnected = V g= 8; 2d ; ni=1 |num_colors = 16 |num_towers = 2 | connected = V| sconnected = V g= 8; rand+2d ; ni=0 |num_colors = 16 |num_towers = 2 | connected = V| sconnected = V g= 8; rand+2d ; ni=1 |num_colors = 16 |num_towers = 2 | connected = V| sconnected = V g= 8; planar ; ni=0 |num_colors = 16 |num_towers = 2 | connected = V| sconnected = V g= 8; planar ; ni=1 |num_colors = 16 |num_towers = 2 | connected = V| sconnected = V g= 8; tree ; ni=0 |num_colors = 16 |num_towers = 2 | connected = V| sconnected = V g= 8; tree ; ni=1 |num_colors = 16 |num_towers = 2 | connected = V| sconnected = X g= 9; rand ; ni=0 |num_colors = 18 |num_towers = 2 | connected = V| sconnected = V g= 9; rand ; ni=1 |num_colors = 18 |num_towers = 2 | connected = V| sconnected = V g= 9; 2d ; ni=0 |num_colors = 18 |num_towers = 2 | connected = V| sconnected = V g= 9; 2d ; ni=1 |num_colors = 18 |num_towers = 2 | connected = V| sconnected = V g= 9; rand+2d ; ni=0 |num_colors = 18 |num_towers = 2 | connected = V| sconnected = V g= 9; rand+2d ; ni=1 |num_colors = 18 |num_towers = 2 | connected = V| sconnected = V g= 9; planar ; ni=0 |num_colors = 18 |num_towers = 2 | connected = V| sconnected = V g= 9; planar ; ni=1 |num_colors = 18 |num_towers = 2 | connected = V| sconnected = V g= 9; tree ; ni=0 |num_colors = 18 |num_towers = 2 | connected = V| sconnected = V g= 9; tree ; ni=1 |num_colors = 18 |num_towers = 2 | connected = V| sconnected = X g=10; rand ; ni=0 |num_colors = 20 |num_towers = 2 | connected = V| sconnected = V g=10; rand ; ni=1 |num_colors = 20 |num_towers = 2 | connected = V| sconnected = V g=10; 2d ; ni=0 |num_colors = 20 |num_towers = 2 | connected = V| sconnected = V g=10; 2d ; ni=1 |num_colors = 20 |num_towers = 2 | connected = V| sconnected = V g=10; rand+2d ; ni=0 |num_colors = 20 |num_towers = 2 | connected = V| sconnected = V g=10; rand+2d ; ni=1 |num_colors = 20 |num_towers = 2 | connected = V| sconnected = V g=10; planar ; ni=0 |num_colors = 20 |num_towers = 2 | connected = V| sconnected = V g=10; planar ; ni=1 |num_colors = 20 |num_towers = 2 | connected = V| sconnected = V g=10; tree ; ni=0 |num_colors = 20 |num_towers = 2 | connected = V| sconnected = X g=10; tree ; ni=1 |num_colors = 20 |num_towers = 2 | connected = V| sconnected = V g=11; rand ; ni=0 |num_colors = 22 |num_towers = 3 | connected = V| sconnected = V g=11; rand ; ni=1 |num_colors = 22 |num_towers = 3 | connected = V| sconnected = V g=11; 2d ; ni=0 |num_colors = 22 |num_towers = 3 | connected = V| sconnected = V g=11; 2d ; ni=1 |num_colors = 22 |num_towers = 3 | connected = V| sconnected = V g=11; rand+2d ; ni=0 |num_colors = 22 |num_towers = 3 | connected = V| sconnected = V g=11; rand+2d ; ni=1 |num_colors = 22 |num_towers = 3 | connected = V| sconnected = V g=11; planar ; ni=0 |num_colors = 22 |num_towers = 3 | connected = V| sconnected = V g=11; planar ; ni=1 |num_colors = 22 |num_towers = 3 | connected = V| sconnected = V g=11; tree ; ni=0 |num_colors = 22 |num_towers = 3 | connected = V| sconnected = X g=11; tree ; ni=1 |num_colors = 22 |num_towers = 3 | connected = V| sconnected = X g=12; rand ; ni=0 |num_colors = 24 |num_towers = 3 | connected = V| sconnected = V g=12; rand ; ni=1 |num_colors = 24 |num_towers = 3 | connected = V| sconnected = V g=12; 2d ; ni=0 |num_colors = 24 |num_towers = 3 | connected = V| sconnected = V g=12; 2d ; ni=1 |num_colors = 24 |num_towers = 3 | connected = V| sconnected = V g=12; rand+2d ; ni=0 |num_colors = 24 |num_towers = 3 | connected = V| sconnected = V g=12; rand+2d ; ni=1 |num_colors = 24 |num_towers = 3 | connected = V| sconnected = V g=12; planar ; ni=0 |num_colors = 24 |num_towers = 3 | connected = V| sconnected = V g=12; planar ; ni=1 |num_colors = 24 |num_towers = 3 | connected = V| sconnected = V g=12; tree ; ni=0 |num_colors = 24 |num_towers = 3 | connected = V| sconnected = V g=12; tree ; ni=1 |num_colors = 24 |num_towers = 3 | connected = V| sconnected = X g=13; rand ; ni=0 |num_colors = 26 |num_towers = 4 | connected = V| sconnected = V g=13; rand ; ni=1 |num_colors = 26 |num_towers = 4 | connected = V| sconnected = V g=13; 2d ; ni=0 |num_colors = 26 |num_towers = 4 | connected = V| sconnected = V g=13; 2d ; ni=1 |num_colors = 26 |num_towers = 4 | connected = V| sconnected = V g=13; rand+2d ; ni=0 |num_colors = 26 |num_towers = 4 | connected = V| sconnected = V g=13; rand+2d ; ni=1 |num_colors = 26 |num_towers = 4 | connected = V| sconnected = V g=13; planar ; ni=0 |num_colors = 26 |num_towers = 4 | connected = V| sconnected = V g=13; planar ; ni=1 |num_colors = 26 |num_towers = 4 | connected = V| sconnected = V g=13; tree ; ni=0 |num_colors = 26 |num_towers = 4 | connected = V| sconnected = X g=13; tree ; ni=1 |num_colors = 26 |num_towers = 4 | connected = V| sconnected = X g=14; rand ; ni=0 |num_colors = 28 |num_towers = 5 | connected = V| sconnected = V g=14; rand ; ni=1 |num_colors = 28 |num_towers = 5 | connected = V| sconnected = V g=14; 2d ; ni=0 |num_colors = 28 |num_towers = 5 | connected = V| sconnected = V g=14; 2d ; ni=1 |num_colors = 28 |num_towers = 5 | connected = V| sconnected = V g=14; rand+2d ; ni=0 |num_colors = 28 |num_towers = 5 | connected = V| sconnected = V g=14; rand+2d ; ni=1 |num_colors = 28 |num_towers = 5 | connected = V| sconnected = V g=14; planar ; ni=0 |num_colors = 28 |num_towers = 5 | connected = V| sconnected = V g=14; planar ; ni=1 |num_colors = 28 |num_towers = 5 | connected = V| sconnected = V g=14; tree ; ni=0 |num_colors = 28 |num_towers = 5 | connected = V| sconnected = X g=14; tree ; ni=1 |num_colors = 28 |num_towers = 5 | connected = V| sconnected = X g=15; rand ; ni=0 |num_colors = 30 |num_towers = 7 | connected = V| sconnected = V g=15; rand ; ni=1 |num_colors = 30 |num_towers = 7 | connected = V| sconnected = V g=15; 2d ; ni=0 |num_colors = 30 |num_towers = 7 | connected = V| sconnected = V g=15; 2d ; ni=1 |num_colors = 30 |num_towers = 7 | connected = V| sconnected = V g=15; rand+2d ; ni=0 |num_colors = 30 |num_towers = 7 | connected = V| sconnected = V g=15; rand+2d ; ni=1 |num_colors = 30 |num_towers = 7 | connected = V| sconnected = V g=15; planar ; ni=0 |num_colors = 30 |num_towers = 7 | connected = V| sconnected = V g=15; planar ; ni=1 |num_colors = 30 |num_towers = 7 | connected = V| sconnected = V g=15; tree ; ni=0 |num_colors = 30 |num_towers = 7 | connected = V| sconnected = X g=15; tree ; ni=1 |num_colors = 30 |num_towers = 7 | connected = V| sconnected = X g=16; rand ; ni=0 |num_colors = 32 |num_towers = 9 | connected = V| sconnected = V g=16; rand ; ni=1 |num_colors = 32 |num_towers = 9 | connected = V| sconnected = V g=16; 2d ; ni=0 |num_colors = 32 |num_towers = 9 | connected = V| sconnected = V g=16; 2d ; ni=1 |num_colors = 32 |num_towers = 9 | connected = V| sconnected = V g=16; rand+2d ; ni=0 |num_colors = 32 |num_towers = 9 | connected = V| sconnected = V g=16; rand+2d ; ni=1 |num_colors = 32 |num_towers = 9 | connected = V| sconnected = V g=16; planar ; ni=0 |num_colors = 32 |num_towers = 9 | connected = V| sconnected = V g=16; planar ; ni=1 |num_colors = 32 |num_towers = 9 | connected = V| sconnected = X g=16; tree ; ni=0 |num_colors = 32 |num_towers = 9 | connected = V| sconnected = X g=16; tree ; ni=1 |num_colors = 32 |num_towers = 9 | connected = V| sconnected = X g=17; rand ; ni=0 |num_colors = 34 |num_towers = 11 | connected = V| sconnected = V g=17; rand ; ni=1 |num_colors = 34 |num_towers = 11 | connected = V| sconnected = V g=17; 2d ; ni=0 |num_colors = 34 |num_towers = 11 | connected = V| sconnected = X g=17; 2d ; ni=1 |num_colors = 34 |num_towers = 11 | connected = V| sconnected = X g=17; rand+2d ; ni=0 |num_colors = 34 |num_towers = 11 | connected = V| sconnected = V g=17; rand+2d ; ni=1 |num_colors = 34 |num_towers = 11 | connected = V| sconnected = V g=17; planar ; ni=0 |num_colors = 34 |num_towers = 11 | connected = V| sconnected = V g=17; planar ; ni=1 |num_colors = 34 |num_towers = 11 | connected = V| sconnected = X g=17; tree ; ni=0 |num_colors = 34 |num_towers = 11 | connected = V| sconnected = X g=17; tree ; ni=1 |num_colors = 34 |num_towers = 11 | connected = V| sconnected = X g=18; rand ; ni=0 |num_colors = 36 |num_towers = 15 | connected = V| sconnected = V g=18; rand ; ni=1 |num_colors = 36 |num_towers = 15 | connected = V| sconnected = V g=18; 2d ; ni=0 |num_colors = 36 |num_towers = 15 | connected = V| sconnected = X g=18; 2d ; ni=1 |num_colors = 36 |num_towers = 15 | connected = V| sconnected = V g=18; rand+2d ; ni=0 |num_colors = 36 |num_towers = 15 | connected = V| sconnected = V g=18; rand+2d ; ni=1 |num_colors = 36 |num_towers = 15 | connected = V| sconnected = V g=18; planar ; ni=0 |num_colors = 36 |num_towers = 15 | connected = V| sconnected = V g=18; planar ; ni=1 |num_colors = 36 |num_towers = 15 | connected = V| sconnected = V g=18; tree ; ni=0 |num_colors = 36 |num_towers = 15 | connected = V| sconnected = X g=18; tree ; ni=1 |num_colors = 36 |num_towers = 15 | connected = V| sconnected = X g=19; rand ; ni=0 |num_colors = 38 |num_towers = 20 | connected = V| sconnected = V g=19; rand ; ni=1 |num_colors = 38 |num_towers = 20 | connected = V| sconnected = V g=19; 2d ; ni=0 |num_colors = 38 |num_towers = 20 | connected = V| sconnected = V g=19; 2d ; ni=1 |num_colors = 38 |num_towers = 20 | connected = V| sconnected = V g=19; rand+2d ; ni=0 |num_colors = 38 |num_towers = 20 | connected = V| sconnected = V g=19; rand+2d ; ni=1 |num_colors = 38 |num_towers = 20 | connected = V| sconnected = V

About the results table: $g$ means log in base 2 of $n$. In other words, $n
\approx 2^g$. `ni`

means network iteration (We do two iterations for every
specific network type and network size). `num_colors`

means the amount of
colors we have for towers. `num_towers`

is the amount of towers we have of
every color. `connected`

and `sconnected`

indicate whether the resulting
overlay directed graph of towers is weakly connected or strongly connected
respectively. `V`

means yes, `X`

means no. Note that if the overlay directed
graph is strongly connected, it must be also weakly connected. This means that
if we see a `V`

in the last column, we must also see a `V`

in the preceding
column.

We can see in the results that the resulting overlay directed graph of towers is not always strongly connected, but it seems to be weakly connected in all of the examples we have checked. Similar results apply for the choice of $k = \log_{2}{n}$ (The overlay directed graph of towers seems to be weakly connected, but not strongly connected). For $k = 0.5\log_{2}{n}$ the overlay directed graph of towers is sometimes not weakly connected.

The following are the results for $k = \log_{2}{n}$:

$ cargo run --bin towers_scc --release Compiling net_coords v0.1.0 (file:///home/real/projects/d/freedomlayer/freedomlayer_research/network_coords/net_coords) Finished release [optimized] target(s) in 3.34 secs Running `target/release/towers_scc` Checking if local towers overlay graph is strongly connected g= 6; rand ; ni=0 |num_colors = 6 |num_towers = 2 | connected = V | sconnected = V g= 6; rand ; ni=1 |num_colors = 6 |num_towers = 2 | connected = V | sconnected = V g= 6; 2d ; ni=0 |num_colors = 6 |num_towers = 2 | connected = V | sconnected = V g= 6; 2d ; ni=1 |num_colors = 6 |num_towers = 2 | connected = V | sconnected = V g= 6; rand+2d ; ni=0 |num_colors = 6 |num_towers = 2 | connected = V | sconnected = V g= 6; rand+2d ; ni=1 |num_colors = 6 |num_towers = 2 | connected = V | sconnected = V g= 6; tree ; ni=0 |num_colors = 6 |num_towers = 2 | connected = V | sconnected = V g= 6; tree ; ni=1 |num_colors = 6 |num_towers = 2 | connected = V | sconnected = X g= 7; rand ; ni=0 |num_colors = 7 |num_towers = 2 | connected = V | sconnected = V g= 7; rand ; ni=1 |num_colors = 7 |num_towers = 2 | connected = V | sconnected = V g= 7; 2d ; ni=0 |num_colors = 7 |num_towers = 2 | connected = V | sconnected = V g= 7; 2d ; ni=1 |num_colors = 7 |num_towers = 2 | connected = V | sconnected = V g= 7; rand+2d ; ni=0 |num_colors = 7 |num_towers = 2 | connected = V | sconnected = V g= 7; rand+2d ; ni=1 |num_colors = 7 |num_towers = 2 | connected = V | sconnected = V g= 7; tree ; ni=0 |num_colors = 7 |num_towers = 2 | connected = V | sconnected = V g= 7; tree ; ni=1 |num_colors = 7 |num_towers = 2 | connected = V | sconnected = X g= 8; rand ; ni=0 |num_colors = 8 |num_towers = 3 | connected = V | sconnected = V g= 8; rand ; ni=1 |num_colors = 8 |num_towers = 3 | connected = V | sconnected = V g= 8; 2d ; ni=0 |num_colors = 8 |num_towers = 3 | connected = V | sconnected = X g= 8; 2d ; ni=1 |num_colors = 8 |num_towers = 3 | connected = V | sconnected = V g= 8; rand+2d ; ni=0 |num_colors = 8 |num_towers = 3 | connected = V | sconnected = V g= 8; rand+2d ; ni=1 |num_colors = 8 |num_towers = 3 | connected = V | sconnected = V g= 8; tree ; ni=0 |num_colors = 8 |num_towers = 3 | connected = V | sconnected = X g= 8; tree ; ni=1 |num_colors = 8 |num_towers = 3 | connected = V | sconnected = X g= 9; rand ; ni=0 |num_colors = 9 |num_towers = 3 | connected = V | sconnected = V g= 9; rand ; ni=1 |num_colors = 9 |num_towers = 3 | connected = V | sconnected = V g= 9; 2d ; ni=0 |num_colors = 9 |num_towers = 3 | connected = V | sconnected = X g= 9; 2d ; ni=1 |num_colors = 9 |num_towers = 3 | connected = V | sconnected = V g= 9; rand+2d ; ni=0 |num_colors = 9 |num_towers = 3 | connected = V | sconnected = V g= 9; rand+2d ; ni=1 |num_colors = 9 |num_towers = 3 | connected = V | sconnected = V g= 9; tree ; ni=0 |num_colors = 9 |num_towers = 3 | connected = V | sconnected = X g= 9; tree ; ni=1 |num_colors = 9 |num_towers = 3 | connected = V | sconnected = X g=10; rand ; ni=0 |num_colors = 10 |num_towers = 4 | connected = V | sconnected = V g=10; rand ; ni=1 |num_colors = 10 |num_towers = 4 | connected = V | sconnected = V g=10; 2d ; ni=0 |num_colors = 10 |num_towers = 4 | connected = V | sconnected = X g=10; 2d ; ni=1 |num_colors = 10 |num_towers = 4 | connected = V | sconnected = X g=10; rand+2d ; ni=0 |num_colors = 10 |num_towers = 4 | connected = V | sconnected = V g=10; rand+2d ; ni=1 |num_colors = 10 |num_towers = 4 | connected = V | sconnected = V g=10; tree ; ni=0 |num_colors = 10 |num_towers = 4 | connected = V | sconnected = X g=10; tree ; ni=1 |num_colors = 10 |num_towers = 4 | connected = V | sconnected = X g=11; rand ; ni=0 |num_colors = 11 |num_towers = 5 | connected = V | sconnected = V g=11; rand ; ni=1 |num_colors = 11 |num_towers = 5 | connected = V | sconnected = V g=11; 2d ; ni=0 |num_colors = 11 |num_towers = 5 | connected = V | sconnected = X g=11; 2d ; ni=1 |num_colors = 11 |num_towers = 5 | connected = V | sconnected = V g=11; rand+2d ; ni=0 |num_colors = 11 |num_towers = 5 | connected = V | sconnected = V g=11; rand+2d ; ni=1 |num_colors = 11 |num_towers = 5 | connected = V | sconnected = V g=11; tree ; ni=0 |num_colors = 11 |num_towers = 5 | connected = V | sconnected = X g=11; tree ; ni=1 |num_colors = 11 |num_towers = 5 | connected = V | sconnected = X g=12; rand ; ni=0 |num_colors = 12 |num_towers = 6 | connected = V | sconnected = V g=12; rand ; ni=1 |num_colors = 12 |num_towers = 6 | connected = V | sconnected = V g=12; 2d ; ni=0 |num_colors = 12 |num_towers = 6 | connected = V | sconnected = V g=12; 2d ; ni=1 |num_colors = 12 |num_towers = 6 | connected = V | sconnected = X g=12; rand+2d ; ni=0 |num_colors = 12 |num_towers = 6 | connected = V | sconnected = V g=12; rand+2d ; ni=1 |num_colors = 12 |num_towers = 6 | connected = V | sconnected = V g=12; tree ; ni=0 |num_colors = 12 |num_towers = 6 | connected = V | sconnected = X g=12; tree ; ni=1 |num_colors = 12 |num_towers = 6 | connected = V | sconnected = X g=13; rand ; ni=0 |num_colors = 13 |num_towers = 7 | connected = V | sconnected = V g=13; rand ; ni=1 |num_colors = 13 |num_towers = 7 | connected = V | sconnected = V g=13; 2d ; ni=0 |num_colors = 13 |num_towers = 7 | connected = V | sconnected = X g=13; 2d ; ni=1 |num_colors = 13 |num_towers = 7 | connected = V | sconnected = X g=13; rand+2d ; ni=0 |num_colors = 13 |num_towers = 7 | connected = V | sconnected = V g=13; rand+2d ; ni=1 |num_colors = 13 |num_towers = 7 | connected = V | sconnected = V g=13; tree ; ni=0 |num_colors = 13 |num_towers = 7 | connected = V | sconnected = X g=13; tree ; ni=1 |num_colors = 13 |num_towers = 7 | connected = V | sconnected = X g=14; rand ; ni=0 |num_colors = 14 |num_towers = 10 | connected = V | sconnected = V g=14; rand ; ni=1 |num_colors = 14 |num_towers = 10 | connected = V | sconnected = V g=14; 2d ; ni=0 |num_colors = 14 |num_towers = 10 | connected = V | sconnected = V g=14; 2d ; ni=1 |num_colors = 14 |num_towers = 10 | connected = V | sconnected = V g=14; rand+2d ; ni=0 |num_colors = 14 |num_towers = 10 | connected = V | sconnected = V g=14; rand+2d ; ni=1 |num_colors = 14 |num_towers = 10 | connected = V | sconnected = V g=14; tree ; ni=0 |num_colors = 14 |num_towers = 10 | connected = V | sconnected = X g=14; tree ; ni=1 |num_colors = 14 |num_towers = 10 | connected = V | sconnected = X g=15; rand ; ni=0 |num_colors = 15 |num_towers = 13 | connected = V | sconnected = V g=15; rand ; ni=1 |num_colors = 15 |num_towers = 13 | connected = V | sconnected = V g=15; 2d ; ni=0 |num_colors = 15 |num_towers = 13 | connected = V | sconnected = V g=15; 2d ; ni=1 |num_colors = 15 |num_towers = 13 | connected = V | sconnected = X g=15; rand+2d ; ni=0 |num_colors = 15 |num_towers = 13 | connected = V | sconnected = V g=15; rand+2d ; ni=1 |num_colors = 15 |num_towers = 13 | connected = V | sconnected = V g=15; tree ; ni=0 |num_colors = 15 |num_towers = 13 | connected = V | sconnected = X g=15; tree ; ni=1 |num_colors = 15 |num_towers = 13 | connected = V | sconnected = X g=16; rand ; ni=0 |num_colors = 16 |num_towers = 17 | connected = V | sconnected = V g=16; rand ; ni=1 |num_colors = 16 |num_towers = 17 | connected = V | sconnected = V g=16; 2d ; ni=0 |num_colors = 16 |num_towers = 17 | connected = V | sconnected = X g=16; 2d ; ni=1 |num_colors = 16 |num_towers = 17 | connected = V | sconnected = V g=16; rand+2d ; ni=0 |num_colors = 16 |num_towers = 17 | connected = V | sconnected = V g=16; rand+2d ; ni=1 |num_colors = 16 |num_towers = 17 | connected = V | sconnected = V g=16; tree ; ni=0 |num_colors = 16 |num_towers = 17 | connected = V | sconnected = X g=16; tree ; ni=1 |num_colors = 16 |num_towers = 17 | connected = V | sconnected = X g=17; rand ; ni=0 |num_colors = 17 |num_towers = 22 | connected = V | sconnected = V g=17; rand ; ni=1 |num_colors = 17 |num_towers = 22 | connected = V | sconnected = V g=17; 2d ; ni=0 |num_colors = 17 |num_towers = 22 | connected = V | sconnected = X g=17; 2d ; ni=1 |num_colors = 17 |num_towers = 22 | connected = V | sconnected = X g=17; rand+2d ; ni=0 |num_colors = 17 |num_towers = 22 | connected = V | sconnected = V g=17; rand+2d ; ni=1 |num_colors = 17 |num_towers = 22 | connected = V | sconnected = V g=17; tree ; ni=0 |num_colors = 17 |num_towers = 22 | connected = V | sconnected = X g=17; tree ; ni=1 |num_colors = 17 |num_towers = 22 | connected = V | sconnected = X g=18; rand ; ni=0 |num_colors = 18 |num_towers = 29 | connected = V | sconnected = V g=18; rand ; ni=1 |num_colors = 18 |num_towers = 29 | connected = V | sconnected = V g=18; 2d ; ni=0 |num_colors = 18 |num_towers = 29 | connected = V | sconnected = X g=18; 2d ; ni=1 |num_colors = 18 |num_towers = 29 | connected = V | sconnected = V g=18; rand+2d ; ni=0 |num_colors = 18 |num_towers = 29 | connected = V | sconnected = V g=18; rand+2d ; ni=1 |num_colors = 18 |num_towers = 29 | connected = V | sconnected = V g=18; tree ; ni=0 |num_colors = 18 |num_towers = 29 | connected = V | sconnected = X g=18; tree ; ni=1 |num_colors = 18 |num_towers = 29 | connected = V | sconnected = X g=19; rand ; ni=0 |num_colors = 19 |num_towers = 39 | connected = V | sconnected = V g=19; rand ; ni=1 |num_colors = 19 |num_towers = 39 | connected = V | sconnected = V g=19; 2d ; ni=0 |num_colors = 19 |num_towers = 39 | connected = V | sconnected = X g=19; 2d ; ni=1 |num_colors = 19 |num_towers = 39 | connected = V | sconnected = V g=19; rand+2d ; ni=0 |num_colors = 19 |num_towers = 39 | connected = V | sconnected = V g=19; rand+2d ; ni=1 |num_colors = 19 |num_towers = 39 | connected = V | sconnected = V g=19; tree ; ni=0 |num_colors = 19 |num_towers = 39 | connected = V | sconnected = X g=19; tree ; ni=1 |num_colors = 19 |num_towers = 39 | connected = V | sconnected = X g=20; rand ; ni=0 |num_colors = 20 |num_towers = 52 | connected = V | sconnected = V g=20; rand ; ni=1 |num_colors = 20 |num_towers = 52 | connected = V | sconnected = V g=20; 2d ; ni=0 |num_colors = 20 |num_towers = 52 | connected = V | sconnected = X g=20; 2d ; ni=1 |num_colors = 20 |num_towers = 52 | connected = V | sconnected = X g=20; rand+2d ; ni=0 |num_colors = 20 |num_towers = 52 | connected = V | sconnected = V g=20; rand+2d ; ni=1 |num_colors = 20 |num_towers = 52 | connected = V | sconnected = V g=20; tree ; ni=0 |num_colors = 20 |num_towers = 52 | connected = V | sconnected = X g=20; tree ; ni=1 |num_colors = 20 |num_towers = 52 | connected = V | sconnected = X

Note that the results are a bit worse for strong connectivity, because $k$ is smaller. However, all the resulting overlay graphs in this experiment are weakly connected.

## Explaining the weak connectivity

We currently don't have any rigorous proofs for the weak connectivity of the towers overlay directed graph for $k = \log_{2}{n}$. We could apply the following approximate explanation:

Every tower is connected in the overlay graph to a few other towers. If these connections were random, then we would expect that for $k > \log(\sqrt{n})$ the towers overlay graph will start becoming weakly connected.

This is not exact, because the connections between the towers are not really random, as a connection between two towers mean that they are close in the original graph.