Coprime networks of the composite numbers: pseudo-randomness and synchronizability

TL;DR

The paper constructs a deterministic network with vertices as composite numbers up to $n$, connecting pairs that are coprime. It derives analytic expressions for edge density, average degree, maximum degree, diameter, and clustering, and proves the graph sequence is weakly pseudo-random with $p=6/\pi^2$, indicating random-like coprimality structure. It also analyzes the Laplacian spectrum to reveal lower synchronizability compared to standard random networks, linking these findings to ecological and predator-prey dynamics. The work combines number-theoretic insights with graph-theoretic measures, offering potential applications in modeling non-synchronizing dynamics and providing publicly available code for replication.

Abstract

In this paper, we propose a network whose nodes are labeled by the composite numbers and two nodes are connected by an undirected link if they are relatively prime to each other. As the size of the network increases, the network will be connected whenever the largest possible node index $n\geq 49$. To investigate how the nodes are connected, we analytically describe that the link density saturates to $6/π^2$, whereas the average degree increases linearly with slope $6/π^2$ with the size of the network. To investigate how the neighbors of the nodes are connected to each other, we find the shortest path length will be at most 3 for $49\leq n\leq 288$ and it is at most 2 for $n\geq 289$. We also derive an analytic expression for the local clustering coefficients of the nodes, which quantifies how close the neighbors of a node to form a triangle. We also provide an expression for the number of $r$-length labeled cycles, which indicates the existence of a cycle of length at most $O(\log n)$. Finally, we show that this graph sequence is actually a sequence of weakly pseudo-random graphs. We numerically verify our observed analytical results. As a possible application, we have observed less synchronizability (the ratio of the largest and smallest positive eigenvalue of the Laplacian matrix is high) as compared to Erdős-Rényi random network and Barabási-Albert network. This unusual observation is consistent with the prolonged transient behaviors of ecological and predator-prey networks which can easily avoid the global synchronization.

Figures (7)

• Figure 1: Coprime networks of composite numbers: (a) $n=25$, (b) $n=30$. Here, the size of each node is proportional to its degree and the color of each node is given according to its clustering coefficient with lighter blue nodes as nodes with higher value of local clustering. For $n=30$, the network is disconnected. Later we will show that the constructed network will be connected if $n\geq 49$.
• Figure 2: The average degree $\overline{d(n)}$ is plotted by changing the highest possible node index $n$. Here the average degree increases monotonically by adding the nodes. The data has been plotted using log-log scale. The scaling exponent is 1.
• Figure 3: The ratio of the computed and theoretical max degree($\Delta/\Delta'$) is plotted by changing the highest possible node index $n$. The ratio getting closer to 1 indicates the accuracy of the theoretical maximum of the degree.
• Figure 4: The average of the local clustering coefficients of the nodes is plotted by changing the highest possible node index $n$. It shows the average of the local clustering coefficients approaches near $\frac{6}{{\pi}^2}(\approx 0.61)$ with the increase in $n$. We have used log-scale for $x$-axis.
• Figure 5: The ratio of the maximum eigenvalue $\lambda_1$ of the adjacency matrix and N(n)p is plotted by changing the highest possible node index $n$. The ratio approaching 1 provides evidence to $\lambda_1=(1+o(1))N(n)p$.

Theorems & Definitions (16)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Theorem 7
• Theorem 8
• Lemma 1
• Lemma 2