You cannot select more than 25 topics
Topics must start with a letter or number, can include dashes ('-') and can be up to 35 characters long.
30 lines
3.8 KiB
TeX
30 lines
3.8 KiB
TeX
\section{Detecting Small-Worldness}
|
|
|
|
As we have seen, many real technological, biological, social, and information networks fall into the broad class of \emph{small-world} networks, a middle ground between regular and random networks: they have high local clustering of elements, like regular networks, but also short path lengths between elements, like random networks. Membership of the small-world network class also implies that the corresponding systems have dynamic properties different from those of equivalent random or regular networks. \s
|
|
|
|
\nd However, the existing \emph{small-world} definition is a categorical one, and breaks the continuum of network topologies into the three classes of regular, random, and small-world networks, with the latter being the broadest. It is unclear to what extent the real-world systems in the small-world class have common network properties and to what specific point in the \emph{middle-ground} (between random and regular) a network generating model must be tuned to genuinely capture the topology of such systems. \s
|
|
|
|
\nd The current \emph{state of the art} algorithm in the field of small-world network analysis is based on the idea that small-world networks should have some topological structure, reflected by properties such as an high clustering coefficient. On the other hand, random networks (as the Erd\H{o}s-R\'enyi model) have no such structure and, usually, a low clustering coefficient. The current \emph{state of the art} algorithms can be empirically described in the following steps:
|
|
|
|
\begin{enumerate}
|
|
\item Compute the average shortest path length $L$ and the average clustering coefficient $C$ of the target system.
|
|
\item Create an ensemble of random networks with the same number of nodes and edges as the target system. Usually, the random networks are generated using the Erd\H{o}s-R\'enyi model.
|
|
\item Compute the average shortest path length $L_r$ and the average clustering coefficient $C_r$ of each random network in the ensemble.
|
|
\item Compute the normalized average shortest path length $\lambda := L/L_n$ and the normalized average clustering coefficient $\gamma := C/C_n$
|
|
\item If $\lambda$ and $\gamma$ are close to 1, then the target system is a small-world network.
|
|
\end{enumerate}
|
|
|
|
\nd One of the problems with this interpretations is that we have no information on how the average shortest path scales with the network size. Specifically, a small-world network is defined to be a network where the typical distance $L$ between two randomly chosen nodes (the number of steps required) grows proportionally to the logarithm of the number of nodes $N$ in the network.
|
|
$$ L \propto N $$
|
|
But since we are working with a real-world network, there is no such thing as "same network with different number of nodes". So this definition, can't be applied in this case. \s
|
|
|
|
\nd Furthermore, let's try to take another approach. We can consider a definition of small-world network that it's not directly depend of $\gamma$ and $\lambda$, e.g:
|
|
|
|
\begin{center}
|
|
\emph{A small-world network is a spatial network with added long-range connections}.
|
|
\end{center}
|
|
|
|
\nd Then we still cannot make robust implications as to whether such a definition is fulfilled just using $\gamma$ and $\lambda$ (or in fact other network measures). The interpretation of many studies assumes that all networks are a realization of the Watts-Strogatz model for some rewiring probability, which is not justified at all! We know many other network models, whose realizations are entirely different from the Watts-Strogatz model. \s
|
|
|
|
\nd The above method is not robust to measurement errors. Small errors when establishing a network from measurements suffice to make, e.g., a lattice look like a small-world network. See \cite{https://doi.org/10.48550/arxiv.1111.4570} and \cite{10.3389/fnhum.2016.00096}. \s
|