I am currently looking into the power law, where it comes from and which role it plays in network science (scale-free networks are often characterized by a power law distribution of node degree). I used the web crawler Hyphe to investigate Wikipedia pages on that topic, and Gephi to analyze the links (I know these tools well). You will find here a report of that small experiment, unfolding the method and discussing it a bit.
In a nutshell I obtained a network of Wikipedia pages where we see two main clusters, one about Pareto and statistics, the other one about network science. We see a bridge between both, and as expected the power law is part of it. It validated my implicit hypothesis, and I learned a few additional things. Here is it (you may want to open it full screen and zoom to read the labels).
Let’s start with the elephant in the room: did I learn anything non trivial from this image? Yes. Nothing big, but useful things in a research context. The image above is the entry point I present you to have a quick idea of what I write about. My findings did not come out of just a quick read of that network. They came out of the whole process, and I provide details below. Now that you are warned against this common misunderstanding, here is what I obtained from this work:
- My hypothesis about the power law bridging certain statistical concepts and network science is confirmed. No big surprise, but it is a way to establish it.
- I get oriented in these concepts and I now have a good idea of my next steps. In particular I know which concepts I must prioritize to investigate the relations between the two knowledge areas.
- I have a well-described and argued set of pages (the “Pareto-to-network-science” corpus) that I can repurpose later in a science context, because the process behind is transparent, reproducible and open to criticism.
- For the same reason I have a set of pages defined as bridging my two domains, that I can repurpose later (the “bridge” corpus).
- I also have a better idea of what the two sides are, and in particular the fact that they are asymmetric. I did not expect that (though I should have).
- I had other surprises, and I value them highly because it is a not-so-common occasion to have a clue about my own biases:
- I did not expect the “de Solla Price” bridge
- I did not expect two sub clusters in network science
- I can show an image that summarizes the situation, which might come in handy in a number of situations. Like this post.
In the next sections I will expose the protocol I used to get that network, and my analysis. This is more or less what I would write in a paper. However in addition I will evoke my exploration, that took place before the final protocol, and that is usually not shared in a paper.
1. Starting lists
We start with two manually curated lists of pages related to the two topics we are studying. The two lists have the same amount of pages arbitrary set to 10. Here are the lists:
Pareto and the power law:
At this point there is no crawl or corpus, but since we have seen the final result already, let’s visualize where the starting lists will en up in the final corpus. It will make the analysis easier to understand.
Using the web crawler Hyphe, we define all Wikipedia pages as different web entities and we crawl these 20 pages. We obtain a list of 1874 web entities cited by these, most of which are other Wikipedia pages.
3. Corpus cleaning
We filter out all the web entities cited only by 3 or less of the starting pages, and we remove any web entity that was not a Wikipedia page or was a tool page (categories, help, list of links…) and we crawl them to obtain the hyperlinks between them. At this stage we have 201 Wikipedia pages and the hyperlinks between them.
A quick look into the most linked pages in this corpus shows that many of them are not related to our topics. These “high layer” pages are very generic and they are cited by our two topics just because they are generally cited by many Wikipedia pages. We use a simple criterion to rule them out: we remove any page that does not cite back at least one page of the starting 20. This simple procedure removes half of the pages we had.
Our final corpus consists of 106 Wikipedia pages (and the hyperlinks between them) characterized as:
- Being cited by at least 4 of the starting pages
- Citing back one or more of those starting pages
- Not being a “tool” page (categories, lists of links…)
4. Identifying the two topics
We started with two lists of pages corresponding to two different (but related) topics. We assume that once extended to our final corpus, these two topics are still present and distinguishable. Just looking at the resulting network gives a strong clue that it is indeed the case. However we do not have to rely on a visual interpretation.
We define an extended version of each of the starting set. For each list, the extended version contains all pages that are citing or cited by at least 5 pages of the starting list. In other terms, the extended set contains pages that have a link (citing or being cited) with 50% or the starting pages (of that list). Note that this procedure allows some pages to be on both sides, or on neither, but as the table below shows it is a minority of cases (less than 10%).
Visualizing the “Pareto Power Law” extended set, that we will call “PPL” for brevity, shows that it largely overlaps with the visual cluster on the left, but not totally. A few pages have not been captured by our procedure, while a page on the right has been. That page is “Scale-free network”. Note: I do not attribute the power of being true to visual clusters as opposed to our selection metric, or vice versa. I just observe they generally agree while having a few crucial disagreements.
We will also shorten “Network Science” in “NS”. Visualizing the extended set shows that despite being bigger, it was well captured by our selection procedure. No nodes were missed from the visual cluster, but a node clearly placed on the left side has been caught, it is “Power law”.
If you are familiar with Gephi and its epistemic culture, you might wonder why I did not use modularity clustering to delineate the clusters. I will discuss this point later and remain focused on describing the protocol.
5. Identifying the bridge(s)
First of all, we must note that two pages belong to both sets, which in itself can be seen a strong form of bridging. These two pages are “Scale-free network” and “Power law”.
We then identified bridges by looking at nodes that have connections with at least 10% of each set in a given direction (citing or being cited). This way we make the distinction between 4 types of bridge: cited by one extended set and citing the other (in both directions), cited by both, or citing both. Each page can have multiple bridging roles.
If we just look at the number of different bridging roles played by each pages, we get the following distribution:
5 bridging roles
4 bridging roles
3 bridging roles
2 bridging roles
Watts and Strogatz model
1 bridging role
6. Visualizing results
In Gephi I used a force-driven placement algorithm, Force Atlas 2, to assign node positions as you have seen above. I used the LinLog mode as it emphasizes the clusters, and its drawback (slow convergence) is not really a problem on such a small network. Once the assignment looked to have converged, and only then, I activated the “no overlap” feature to improve readability. As I expected to have to use this “base map” within a text, I chose to rotate it so that it spreads horizontally.
Let’s look at how the links are distributed as a function of our sets. A simple way to do it is to look at density, but this metric is biased by the sizes of clusters and the general density of the network. In order to remove these biases, we will normalize the density the same way we would do with modularity. Like modularity, the normalized densities vary from -0.5 to 1 and the higher, the more links there are compared to the number of links there could be in the context of that network.
The PPL set has an internal normalized density of 0.027, versus an external normalized density of -0.014. Normalized densities are generally low and the important fact here is how internal density dominates external density: there are much more links inside the PPL set than between the PPL set and the rest. The PPL is a cluster in that sense.
Similarly, the NS set has an internal norm. density of 0.117 and external of -0.018. NS is also a cluster, and even better defined.
We see it in the visualization, but it does not rely on the visual representation. We have two well defined topological clusters, with different densities and sizes.
The group of pages defined as “Pareto Power Law” are about statistical laws, probabilities and important figures such as Pareto and Zipf. We suspect that it is part of a much bigger group of pages about statistics, but possibly because the power law is a central concept to that field, our strategy might not have been able to capture that whole group. This set is smaller (30 pages) and less dense (0.027 norm. density) than the “Network Science” set (72 pages and 0.117 norm. density). As a conceptual space it is narrower than network science. We hypothesize that it just the fringe of a larger conceptual space about statistics, and it is possibly not so well defined as a subtopic (a different protocol could test this).
The group of pages defined as “Network Sciences” is larger, better defined, and more interconnected. It is well groomed as a conceptual space, with specific concepts (“Preferential attachment”, “Small-world network”…) intertwined with a body of much more generic concepts (“Internet”, “Social network”…). I am confident that the sub-cluster we identify visually (at the bottom of the cluster) and corresponding to the topic of social networks would be confirmed as such by the same kind of density analysis.
The connexions between the two clusters are multiple. Looking at the direction of links in the different kinds of bridges, we see that there are much more pages where links come from NS and go to PPL than the contrary. This indicates that NS cited the concepts of PPL more than the other way around.
Two pages have a more important bridging role: “Scale-free network” and “Power law”. It is not a big surprise, but I am happy to have established the key role of these two concepts in the circulation of concepts from statistics to network science. The rest of this investigation will rely on a more qualitative approach.
The other bridges that we have identified are a priority for my investigation, and more generally now that the corpus has been scrutinized and that we know that it captures the areas it was intended to, it would be a good idea to read all of these 100 pages. A possible follow-up could be a text analysis of the texts of these pages, and/or their Wikipedia history.
Why not using modularity clustering and betweenness centrality?
Because I did not need to, and it was easier to explain my protocol that way.
This is the alternative protocol: curate a corpus manually so that it captures the two topics. Run modularity clustering in Gephi to find clusters. Run betweenness centrality to find the bridges.
The problem with that protocol is how it depends on abstract concepts for a quite simple thing. Modularity clustering is hard to explain. The visual clustering in the visualization, that is known to be coherent with modularity clustering, is hard to explain. Betweenness centrality is hard to explain. We can explain how it works but not what it does.
Betweenness centrality counts the number of shortest paths, so that a node with a high score is on many shortest paths between other nodes. It means that if you remove such a node, you break many shortcuts, you make distances longer in the network. This is how it works. But not what it does. From my point of view what betweenness centrality does is get both “intuitive bridges” and centers. Centers are nodes that are well connected, connected to other well connected nodes, and that you will find with other metrics such as closeness centrality or just the degree. The “intuitive bridges” are the other ones, left over by other metrics, which are often “in between” clusters. How it works does not tell you what it does, and the justification of the method is obscured.
Sometimes there is no other way. But in this experiment, I designed a strategy that would not rely on these hard-to-explain elements and define our corpus and our bridge in a simpler way. It is just about who cites who, and it still works. But of course I knew it would work beforehand, because I had explored the domain. It was not a bet, as the game was rigged from the start.
Now that I shared a more finalized product, I will open my kitchen and expose my methodological notes. They are just slightly redacted to be readable from you. They were not written to be exposed in extenso, so you will have to pardon the style.
I just started with the following three pages:
First round of exploration: among the Wikipedia pages cited by the 3 entry points, if we except special pages related to scientific content (DOI, ISBN…) and specific to the Wikipedia practice (Main page, Help…) we get 3 other pages: Long Tail, Normal Distribution and Zipf Law. We extend the corpus in that direction.
Second round of exploration, pages cited by 3+ of the above 6. We decide to eliminate list pages (eg. Category: Statistical Law). We find tens of pages about the different statistical distributions. In order to avoid a drifting to statistics in general, we rule them OUT except for the Log-Normal distribution because of the controversy about interpreting real data as power law or log-normal. We just get Benford’s Law, Log-normal distribution, exponential distribution, Generalized Pareto distribution and Zipf-Mandelbrot law.
At this point the corpus is hugely skewed towards statistics. We want to expand it towards networks and management or political science, where Pareto and Juran were influent. We search for specific terms in Hyphe’s Prospect to add sufficiently cited pages:
- “Pareto” gives Vilfredo Pareto, Pareto efficiency and Pareto index
- “Law” gives Power-law and 80-20 law (aliases of pages we had)
- “Network” gives Scale-Free Network and Social Network
- “Watts”and “Barabasi” add Bianconi-Barabasi model, Barabasi-Albert model, Watts and Strogatz model, Albert-Laszlo Barabasi, Duncan J Watts
- “Small-World” brings Small-world network, Small-world experiment and Small-world phenomenon and Small world network (alias).
The following round of exploration allows to find the links to “network science”. We focus on pages cited by 6+ of the above. A lot of graph theory appears and like previously, we try to stay focused on complex / scale free / small world networks and their specificities. By this mean we add 25+ pages mostly related to network science.
The obtained network is pretty clear: the power-law is bridging statistics and in particular Pareto’s law with network science. (see file “Pareto Law exploration.gexf”).
After this exploration we can do a more understandable and more straightforward protocol. We will start with two lists of pages, one on Pareto and one on Network science, crawl both and see what comes out and how it bridges.