Initialization

The prerequisites in the start configuration for the application of the rules are the following.

• ${\displaystyle \mathrm {N} }$ is the network with nodes and links, whereby each node has a weight ${\displaystyle w}$.
• Each node ${\textstyle n}$ in ${\displaystyle \mathrm {N} }$ node stores the same positive values ${\textstyle MIN}$ and ${\textstyle MAX}$, with ${\textstyle MIN<MAX}$.
• A node ${\textstyle n}$ with weight ${\textstyle w_{n}=MAX}$ is called cluster center.
• ${\textstyle k}$ is ${\textstyle MAX}$ - ${\textstyle MIN}$ and represents the maximum size a cluster can have from the most outer node to the cluster center. The cluster diameter is therefore ${\textstyle k\cdot 2-1}$.
• ${\displaystyle \mathrm {N} (n)}$ returns the direct neighbors of node ${\textstyle n}$.
• ${\textstyle W(\mathrm {N} )}$ is the set of weights of all nodes of ${\displaystyle \mathrm {N} }$.

The following rules describe the state transition for a node ${\textstyle n}$ with weight ${\textstyle w_{n}}$. These rules have to be executed on each node in the order described here.

Rule 1

KHOPCA rule 1

The first rule has the function of constructing an order within the cluster. This happens through a node ${\textstyle n}$ detects the direct neighbor with the highest weight ${\textstyle w}$, which is higher than the node's own weight ${\textstyle w_{n}}$. If such a direct neighbor is detected, the node ${\textstyle n}$ changes its own weight to be the weight of the highest weight within the neighborhood subtracted by 1. Applied iteratively, this process creates a top-to-down hierarchical cluster structure.

if max(W(N(n))) > w_n
    w_n = max(W(N(n))) - 1

Rule 2

KHOPCA rule 2

The second rule deals with the situation where nodes in a neighborhood are on the minimum weight level. This situation can happen if, for instance, the initial configuration assigns the minimum weight to all nodes. If there is a neighborhood with all nodes having the minimum weight level, the node ${\textstyle n}$ declares itself as cluster center. Even if coincidentally all nodes declare themselves as cluster centers, the conflict situation will be resolved by one of the other rules.

if max(W(N(n)) == MIN & w_n == MIN
    w_n = MAX;

Rule 3

KHOPCA rule 3

The third rule describes situations where nodes with leveraged weight values, which are not cluster centers, attract surrounding nodes with lower weights. This behavior can lead to fragmented clusters without a cluster center. In order to avoid fragmented clusters, the node with higher weight value is supposed to successively decrease its own weight with the objective to correct the fragmentation by allowing the other nodes to reconfigure according to the rules.

if max(W(N(n))) <= w_n && w_n != MAX
    w_n = w_n - 1;

Rule 4

KHOPCA rule 4

The fourth rule resolves the situation where two cluster centers connect in 1-hop neighborhood and need to decide which cluster center should continue its role as cluster center. Given any specific criterion (e.g., device ID, battery power), one cluster center remains while the other cluster center is hierarchized in 1-hop neighborhood to that new cluster center. The choice of the specific criterion to resolve the decision-making depends on the used application scenario and on the available information.

if max(W(N(n)) == MAX && w_n == MAX
    w_n = apply criterion to select a node from set (max(W(N(n)),w_n);
    w_n = w_n - 1;

1-D

An exemplary sequence of state transitions applying the described four rules is illustrated below.

2-D

KHOPCA acting in a dynamic 2-D simulation. The geometry is based on a geometric random graph; all existing links are drawn in this network.

3-D

KHOPCA also works in a dynamic 3-D environment. The cluster connections are illustrated with bold lines.