A new book on circular economy and port ecosystems that includes a chapter written by me and Frank Boons is out. Most of the chapters are in French, but some chapters, including ours, are in English. In our chapter we use principles of congruence analysis and phasic analysis to investigate how well different stage models of industrial symbiosis explain the emergence and development of the Sustainable Connections project in the port of Moerdijk.
During my PhD project I developed a fascination for geometric approaches to exploratory data analysis, such as Multidimensional Scaling (MDS) and Correspondence Analysis. These methods have some very interesting applications. For example, they can help you to strongly reduce the dimensionality of datasets, capturing the most essential patterns in your data. I especially like the ways in which these methods can help you to visually explore concepts that can be captured in terms of ‘distances,’ such as (dis)similarities between people, social groups, events, and etcetera. I thought it would be nice to write a few posts in which I offer basic examples of the methods in use (without going into the technical details), using only open source software. In this post, I focus specifically on MDS, and I offer a basic demonstration of the method, using R and Gephi, two amazing pieces of software.
A very brief introduction into MDS
Let me first say that I am not going to give an in-depth methodological introduction into MDS. I like to keep things at an intuitive level for this post (and I am likely to do so in future posts on this topic). There are plenty of books that offer a great technical introduction into MDS, including Kruskal and Wish (1978) (I love those little green books on Quantitative Applications in the Social Sciences) and Gatrell (1983) (oh, the smell of old books). I should also say that there is more than one type of MDS. For example, there is the distinction between metric and non-metric MDS. I am not going into all these nuances either, but I can at least share with you that, in the example below, I am applying non-metric MDS.
Basically, what MDS can do for you is to take a matrix in which the distances between a set of entities are recorded (i.e., a distance matrix), and try to create a low-dimensional spatial configuration in which the same entities are located in a way that the distances between them are proportional to the distances in the original distance matrix. This may sound trivial, but keep in mind that some types of distances may be very difficult to accurately reproduce in a low-dimensional space. For example, I have experimented with using MDS to visualize the (dis)similarities between social events, based on the actors and issues (e.g, problem and solution definitions) that the events have in common. I usually ended up needing at least 6 dimensions to accurately express these (dis)similarities. Unfortunately, it is impossible to visualize any space with more than three dimensions in one picture, and trying to do so is a serious threat to your mental health.
At this point, some of you might already wonder how we can decide the number of dimensions that we need to accurately represent the contents of a given distance matrix. MDS comes with a measure called ‘stress,’ which you can interpret as a measure of fit between the spatial configuration produced, and the original distance matrix. The higher the stress, the worse the fit. In other words: The higher the stress, the less confidence you should have that the spatial configuration you produced accurately represents the distances in your original distance matrix. Kruskal and Wish (1978) offer an insightful discussion on stress, and on different rules of thumb on how to interpret stress values. A very rough rule of thumb is that stress values below 0.10 usually indicate a good fit, and that values above 0.20 are unacceptable (but don’t cite me on this!).
Our example of today
The example that I will be using is a bit silly, but it serves quite well to show how powerful MDS can be at creating spaces that accurately reflect the distances that you feed into the analysis. For this example, I have taken 12 cities in the Netherlands, and I looked up the distances between all of them. To find the distances, I used this website (thank you, Google). See the distance matrix that I produced below (some of the larger pictures in this post may be easier to read if you click on them).
There is no particular reason why I picked these cities, except for the fact that they are spread out quite nicely throughout the Netherlands. I already know that it will be very easy to represent these distances in a 2-dimensional space, because that is what we do all the time with geographical maps. So the results will not be very surprising, but I still think the results are cool enough.
Let’s get to work!
So, let’s do this! The first thing I did is to save the matrix shown above as a comma-delimited file (.csv extension), because it is easy to import such files into R. If you don’t know how to do this, go to ‘save as’ in Excel, and find comma-delimited in the file options. Comma-delimited files are basically text-files that use commas, or other symbols, to separate columns. I always use semicolons ‘;’ as separators, but you can use whatever you like.
Next, we start up R. My R-session may look at bit weird to you, because I embedded R in Emacs, using Emacs Speaks Statistics, and that is just because I am a nerd. You can of course use the standard GUI for R if you like. R comes with built-in functions that can do MDS, but I like to work with a package called ‘vegan.’ I already had the package installed, but if you don’t, then you can install the package by typing install.packages(“vegan”) into the R-console. After that, you can activate the package by typing library(vegan).
After starting R, I imported the csv-file as a matrix object (see picture further below for a log of all the commands that I used). I called the matrix geoMatrix. After importing the file, I immediately turned the matrix into a so-called dist object (called geoDist), which is something that R uses to store distances. This is also the object that I fed into the MDS analysis.
The ‘vegan’ package comes with a function called metaMDS(). The function takes a lot of arguments, but most of them have default values, and for our example these default values will work just fine. There are two arguments that I would like to draw your attention to: The first argument to the function should be the distance object that you want to use as the input for the MDS analysis. There is another argument k, which indicates the number of dimensions that you want the spatial configuration to have. The default value for k is 2, which is fine for us. However, if you find that with this default value you end up with high stress values, then you might want to consider increasing the value of k. You would then, for example, use the function with the following arguments: metaMDS(geoDist, k = 3).
I use the function with the geoDist object as the only argument, and I tell R to assign the results of the function to an object called geoMDS. The function will start an iterative process in which it attempts to come up with increasingly accurate spatial configurations, and it will spit out the stress value that it ends up with at each run. By default, this function will make up to 20 attempts to come up with a decent configuration (this is also something you can change by changing the arguments given to the function), but in our case the function already reaches a very low stress value at its second run (see log below).
I immediately plotted the resulting configuration, using the standard plot() function. To assign labels to the points in the visualized configuration, I used the function text(). See the screen shot below for a log of all the commands I used (including the arguments given to the different functions) on the right, and the plot of the resulting configuration on the left (if you look below the visualization, you’ll see that I was checking out a discussion on calculating stress in R, just before doing the analysis).
After a visual inspection of the results, I decided that the distances seem pretty close the actual distances you will find on a map. However, the layout of the cities is a bit counterintuitive. Also, the orientation of the space is different from what we are used to with geographical maps, so I decided to import the data into Gephi, and change the layout there, without changing the distances.
Importing the data into Gephi
To do this, I first had to export the coordinates of all the cities in the new space that the MDS analysis created for them. These coordinates are stored somewhere in the geoMDS object. This object has a rather complicated structure that I do not want to get into, and I do not expect you to immediately understand the trick that I use to extract the coordinates from the object. However, let me tell you what I did: I created two new vectors (one for each dimension). The first vector holds all the coordinates in dimension 1, and the second vector holds all the coordinates in dimension 2. In the geoMDS-object, these coordinates are stored in what can be understood as a ‘sub-object’ of geoMDS, which is called ‘points,’ and which is basically a two-column matrix (1 column for each dimension). All coordinates in dimension 1 are stored in the first column, and all coordinates in dimension 2 are stored in the second column. So, if you understand the basics of R, you may also understand that we can access the coordinates in each dimension with the commands geoMDS$points[,1] (for dimension 1) and geoMDS$points[,2] (for dimension 2). After putting the coordinates in two separate vectors, I assembled the vectors into a data.frame. Then I wrote the data.frame to the disk, using the write.table() function. I wrote the results to a file that I named “CoordinatesGeo.csv.” See the screen shot below to see the commands that I used.
Before we can import the resulting file in Gephi, it needs some minor changes. Without changes “CoordinatesGeo.csv” is a table with three colums: One unnamed column with the names of the cities, and two named columns with the coordinates that we created with our MDS analysis. I turned this table into a Gephi nodes list by naming the column with city names “Id”, and by copying this column to a second column that I renamed “Labels” (see screen shot below).
Now we start up Gephi. In Gephi, we start a new project, and we go to the data laboratory. There, we use the ‘import spreadsheet’ function (see below).
This will open a new dialog, where we can indicate how we want to import our data. One thing is very important here, and that is that we want to import the coordinates as numerical data. I usually import numerical data as doubles (see below).
If the settings are like shown in the screen shot above, then you should be okay. We can go back to the overview in Gephi, and the nodes will be randomly distributed in the plot screen. Gephi doesn’t come with a built-in layout algorithm for nodes that have coordinates, but you can use a plugin that I created some time ago, called MDS Layout. You should have this plugin installed before you can proceed with the next steps.
I selected the MDS Layout from the layout menu, and the plugin automatically recognizes the variables ‘dim1’ and ‘dim2’ as appropriate dimensions. The layout that is created if you run the plugin is the same as what we saw when we plotted the results in Gephi (see below).
I wanted the cities to be oriented the same way as in commonly used geographical maps, and for that I used Gephi’s built-in layout algorithm ‘Clockwise rotate’ (interestingly, this algorithm rotates the nodes in the counter-clockwise direction). I set the algorithm to rotate the layout by 90 degrees. After doing that, I immediately saw that the layout of the cities created by the MDS is like a mirror image of the layout that you’ll find on a geographical map (Maastricht is in the West, while it should be in the East, and Middelburg is in the East, while it should be in the West). This is something that we cannot fix in Gephi, but we can easily fix it in Excel.
I reopened the nodes list we created earlier, and after inspecting the coordinates I saw that the coordinates in dimension 2 are the coordinates that are mirrored. To fix this, I created a new column with coordinates for dimension 2. The values in this column are the values of the original column multiplied by -1 (see below).
I imported the adjusted nodes list into Gephi, using the same approach as before. I used the MDS Layout again to create my initial layout. In this case too, I had to use the “Clockwise rotate” layout to rotate the layout by 90 degrees. I also increased the size of the nodes a bit and gave them a color, just for aesthetic reasons. See the result below. If you know the map of the Netherlands, then you’ll see that this is already more like it!
As an ultimate test (okay, this is a bit of an exaggeration) I decided to plot the points on top of a blank map of the Netherlands. I found such a map easily enough through Google Images. I exported the visualization from Gephi as an SVG-file, such that I could adjust the positions of the labels (the city names) in Inkscape (another wonderful piece of open source software). After adjusting the labels, I exported the result as a PNG image. I then opened the blank map of the Netherlands, as well as the PNG file with the cities in of the Netherlands in Gimp (yes, yes, open source) and pasted the cities on the blank map. It took some rescaling and additional rotating to get the layout right, but ladies and gentlemen, the result looks very, very nice.
Just compare the results to an actual map of the Netherlands, and you’ll see that the MDS analysis did a very good job at creating a spatial configuration that accurately represents the distances that we fed into it. The main difference between the layout of the cities in the MDS space, and the layout of the cities in ‘actual’ geographic space is that the layout of the cities on one dimension was mirrored. It reminds me of those crappy Sci-Fi stories in which mirrors are portals to alternative dimensions, but as I mentioned before: I am a nerd.
Like I said, this is a bit of a silly example, but I plan to come up with more interesting ones in the future, where I will explore, for example, the application of MDS to ‘social stuff.’
Hope you enjoyed this!
BTW: Some time ago I created a plugin for Gephi that does MDS, using the path distances between the nodes of a graph as its input. The plugin produces coordinates that can be used to create layouts with the MDS layout algorithm. See here for more details.
Just to warn you: This is not an attempt at serious science. It is just something I made for fun, based on a project that two students are working on.
Let me start by giving some background to what I’ve done. I’m involved in the European GLAMURS project as a postdoctoral researcher. Among other things, we investigate what role bottom-up citizen initiatives can have in sustainability transitions. In the Netherlands, one of the initiatives we work together with is Vogelwijk Energie(k), which is a very interesting local energy initiative in the Hague. One of the things that makes the initiative special is that it is situated in a neighborhood with exceptionally strong investment power, and relatively many connections to important professional networks. Since its early stages (it started in 2009), Vogelwijk Energie(k) has had around 250 members (there are over 2000 households in the neighborhood). One of the ambitions of the board of the initiative is to mobilize a ‘second wave’ of people within and outside the neighborhood, to have them invest in sustainable development. This doesn’t mean that Vogelwijk Energie(k) wants to attract more members; the idea is to raise awareness about sustainability among a broader group of people, and stimulate them to act on that awareness.
We had the opportunity to write an assignment on this problem for a course on Agent Based Modeling (ABM). Two students have been working on that assignment for some time now. They are basically attempting to develop an ABM that models the process through which the mobilization of the ‘second wave’ could take place, allowing for the exploration of different scenarios. Within a few weeks, the two students had developed some very interesting ideas for the model. Their work so far seems to be based on the implicit theory that people can develop a stronger awareness about sustainability by talking about the subject with their neighbors, and that increased awareness may at some point trigger an exploratory process that may, or may not lead to the decision to start investing in sustainability. Their conceptual model is actually more complex, but this is the gist of it. One of the reasons why I like their ideas is that, without knowing, they included some kind of awareness-behavior gap in their model (increased awareness does not immediately lead to changes in behavior).
One of the operational problems that the students are currently facing is how to simulate the interactions between neighbors. Intuitively, we can already say that the likelihood of neighbors interacting with each other is not equal for all pairs of neighbors. Any person is likely to interact frequently with only a small amount of his/her neighbors. Who these neighbors are will also depend (intuitively speaking) on how close they live to each other, and how many opportunities they have to meet (e.g., their children go to the same school, they go to the same supermarket, they are member of the same associations). You’ll get the idea.
One approach to modeling the patterns of interactions between neighbors is the social networks approach. We can visualize neighbors as nodes, and we can visualize frequent interactions between neighbors by drawing edges between those neighbors. In this context, the students had a question for me: “What do you think this network should look like?” This is a rather difficult question to answer without any empirical observations on the actual interactions that take place in the neighborhood. Ideally, we would do a survey on this among the residents of the neighborhood, but that is well beyond the scope of the students’ assignment. I told the students I needed to think about this problem for a while.
Creating a random network based on distances
One of my initial thoughts on the problem described above is that the structure of a social network in a neighborhood will at least depend on the geographical proximity of the neighbors in the network. I realize that there are many other mechanisms that will influence the structure of the network, but it is the geographical dimension that I focused on this evening.
One of the things that I wanted to do is to place neighbors in a space that more or less corresponds with the geographical boundaries of the Vogelwijk neighborhood. I first took a look at a map of the neighborhood on Google Maps. In the visualization below, the neighborhood is marked by the red area.
As you can see, the neighborhood includes two large green areas that have no houses. I used Google Maps to mark the ‘inhabited’ areas, and make a rough calculation of their combined surface area (see below).
Google maps tells us that the area with the black outlines is about 1.12 square kilometers, but to keep things simple, I decided to assume that the area is a rectangular area that is 2.5 kilometers wide, and 0.5 kilometers high, which brings us to a surface area of 1.25 square kilometers.
I also decided to focus on households, rather than neighbors, and I found that there are over 2000 households in the neighborhood. I decided to round the number of households down to 2000.
I created a so-called nodes list in which I listed 2000 households that I simply numbered from 1 to 2000. I randomly assigned X-coordinates and Y-coordinates to each household. The X-coordinates are a random number between -1250 and 1250, and the Y-coordinates are a random number between -250 and 250. See a screen shot of part of the nodes list below.
I wrote a script in R with several functions. One of the functions calculates the Euclidean distances between all the households in the neighborhood, based on the randomly assigned coordinates. The function returns a distance matrix that reports the distances between all pairs of households. Another function normalizes these distances (all distances are converted to proportional values between 0 and 1), and inverts them to create proximities (1-distance). The resulting proximity matrix became the basis for the simulation of the network.
I wrote a simple function in R to simulate the network in the neighborhood, using the proximities of the households as a basis. The logic of the function is very simple: The function considers each pair of households in turn. The proximity of the households (a number between 0 and 1) is multiplied by a random number, for example a number between 0 and 1 (which is used to simulate other influences; I know it is very naive). The resulting number is then compared to a threshold. If the number is below that threshold, then there is no tie between the two households. If the number is equal to, or above the threshold, then a tie between the households in created. I made sure that it is possible for the user of the function to set the threshold, as well as boundaries for the random number that the proximities are multiplied with. Different parameters for this will also lead to different networks. I ran the function numerous times, each time with different parameters. Below, I visualize 2 examples.
Both examples are visualized with Gephi. To visualize a network I opened its adjacency matrix in Gephi, and I imported the nodes list with the household coordinates separately. I used the MDS Layout algorithm to layout the households. Below is a first example (clicking the picture should enlarge it). The size of the nodes is based on their degree. The colors indicate communities, which I identified using Gephi’s built-in modularity algorithm.
In the picture you can see that ties between the households are relatively sparse. This network has a degree distribution of 2.063. In the picture below you can see that most households have one or two connections with other households in the neighborhood. This may not seem like much, but from the few papers on neighborhood networks that I have scanned so far, I understood that ties within a neighborhood tend to be sparse (people are typically more strongly connected with people outside their neighborhood). The visualization also nicely shows the effect of the simple simulation function that I wrote: The connections only exist between households that are relatively proximate.
The pictures below show another example, based on a network that I generated using other parameters. This network has more connections, which can also be seen in the graph of the degree distribution. In this case, most households seem to have connections with 17 other households in their neighborhood, which intuitively sounds unrealistic to me, but it sure creates a pretty picture. I also like how the neighborhood divides up nicely in different communities.
So, that was it. I know this is not a very serious simulation of neighborhood networks, and indeed it is based on intuitions, and not on good science. But it was fun to do!
Earlier, I posted R-scripts that include functions for finding paths in directed, a-cyclic graphs. I (and some colleagues) use this type of graph to model processes (sequences of events) as networks. We refer to this type of graph as event graphs (Boons et al. 2014; Spekkink and Boons, in press), where events are represented by nodes, and the relationships between events are represented by arcs and/or edges. The underlying principles of this approach were based primarily on the work of Abell (1987) on the Syntax of Social Life, where he introduces narrative graphs.
In addition to the path-finding algorithm, I recently created a plugin for Gephi, called Lineage, that identifies all the ancestors and descendants of a node (chosen by the user) in directed graphs. Based on the logic that I used for the Lineage plugin I also did a complete rewrite of the function (for R) that I designed for finding paths in directed, a-cyclic graphs (see this post). I decided to use the same logic to write some additional functions that explore ancestors and descendants of nodes in directed a-cyclic graphs. I wrote four new functions for R:
- GetAncestors(arcs, node): This function returns all the ancestors of a given node, which is submitted by the user by passing it as an argument of the function (node). The other argument (arcs) should be an edge list, structured as a 2-column matrix. The first column should report ‘source’ nodes, and the second column should report the ‘target’ nodes that the ‘source’ nodes link to (see table below for an example of such an edge list). This edge lists represents the network under investigation.
- GetDescendants(arcs, node): This function does the same as the GetAncestors() function, but returns the descendants of the input node, rather than its ancestors.
- CommonAncestors(arcs, nodes): This function finds all the common ancestors of a pair of nodes, or multiple pairs of nodes. In this case too, the arcs argument should be an edge list, as described earlier. The nodes argument can either be a vector of two nodes, or a matrix with multiple pairs of nodes (similar to the edge list displayed in the table below). The function writes a file in which it is reported whether the submitted pairs of nodes have ancestors in common, how many ancestors they have in common, and what these ancestors are. The file is placed in the current working directory of the user.
- CommonDescendants(arcs, nodes): Same as the CommonAncestors() function, but in this case the function returns the common descendants of the pairs of nodes submitted by the user.
All four functions assume that the events are labelled with numbers. Unless the edge lists that are submitted by the user have numerical variables, the functions won’t work as they should.
So what are the functions useful for? I assume that different people may come up with different uses, but I use them specifically as a step in the identification of emergent linkages between events (see Spekkink and Boons, in press). It will take me some paragraphs to explain this.
In my PhD research I introduced a distinction between intentional linkages between events, and emergent linkages between events. An intentional linkage exists between two events A and B if in event B actors intentionally respond to conditions created in event A. For example, event A might concern the development of a plan, and plan B may concern a feasibility study of that plan. I used the idea of intentional linkages to identify different streams of events in a process, that may, for example, represent different projects that unfold independent from each other. In my thesis, I used intentional linkages to identify independent projects that were later assembled into a larger collaborative process, and can thus be understood to serve as building blocks of the collaboratie process. The figure below shows a phase model that I visualize in the conclusions of my thesis, and that also aims to visualize the idea of different building blocks existing independently at first, and being assembled into a collaborative process at a later stage.
Emergent linkages refer to another type of relationship between events. I found that actors involved in independent events often address similar (or the same) issues in their activities. It might, for example, be the case that different actors are working on different projects that are all related to biobased economy, but that actors do not react to each other’s projects. They may not even be aware of each other’s activities. To capture these similarities between events (i.e., the similarities of issues addressed) I introduced the notion of emergent linkages. I defined emergent linkages such that they refer to similarities between events, but that they can only exist between events that are not connected by a path of intentional linkages. The reason for the latter condition is that I was only interested in coincidental similarities. If similarities exist between events that are also connected by a path of intentional linkages, then there is a very strong chance that the similarity is also intentional. To explain what I mean, let me go back to my earlier example, concerning the intentional linkage between event A (a plan is developed), and event B (a feasibility study is performed). Say that the plan concerns the construction of a biofuel factory. In this case it will be logical that both the plan and the feasibility study address issues related to biofuel production. These similarities are not emergent, in my view, and that is why my definition of emergent linkages excludes them.
There are different approaches to identifying similarities between events. The approach that I used is described in Spekkink and Boons (in press). In short, in this approach the similarities between events are calculated as correlations between the column profiles of an incidence matrix, where the rows represent issues, and the columns represent events (a ‘1’ in a given cell indicates that the corresponding issue was addressed in the corresponding cell). The problem with this approach (and other approaches) is that the similarities between all events are calculated this way. This includes the similarities that I do not consider to be emergent. Thus, I need some way of filtering out the non-emergent linkages from the complete list of similarities between events. Based on my definition, I used to filter out the similarities between events that are also on a path of intentional linkages together (this was the reason why I wrote the function for finding paths in directed, a-cyclic graphs). Lately, I have been thinking about expanding this definition, because there are cases where two events are not on a path together, but do have one or more common ancestors. Similarities between these events may also be considered intentional in some cases. For example, consider the development of a large program (event A), the details of which are worked out in multiple, independent working groups (events B, C, and D). Events B, C and D may not be intentionally linked, but they do have a common ancestor (event A), and they all inherit issues from that ancestor. I considered introducing a more restrictive definition of emergent linkages where not only nodes that share a path of intentional linkages are excluded, but also events that have common ancestors. That is the reason why I wrote the CommonAncestors() function. I included the other three functions because it only took small adaptations to make them, and they might be useful for someone.
You can download the script with the functions here.
In an earlier post I described an R-script I wrote for finding paths in directed, a-cyclic graphs. That script had some problems, including efficiency problems. I made a completely new script for finding paths in directed, a-cyclic graphs, and replaced the old script with the new one. The new script also appears to find paths that the old algorithm did not identify, so I must have missed something in the old script.
The logic of the new script is quite simple. It contains one function with a number of parameters (these are also described in the script itself). The function and its possible arguments are as follows:
FindPaths(arcs, origin = 0, originsOnly = F, endsOnly, F, writeToDisk = F)
The first argument (arcs) should be an edge list with two columns, where the first column contains the source nodes, and the second column contains the target nodes of arcs (see table below for an example). All nodes should be represented by numbers for the function to work.
The first step in the function is to identify all origins in the list of arcs (nodes with 0 indegree) and all endpoints (nodes with 0 outdegree). The arguments originsOnly or endsOnly can be set to TRUE (T) to have the function return the list of origins or endpoints respectively. No paths are identified if one of these options is chosen.
The user may also choose a specific origin node (origin), which tells the function to only look for paths that flow from that specific node. This is a wise choice when the graph under investigation is very large and/or complex. If origin is left in its default value of 0, then the function will attempt to identify all paths in the graph.
The function first identifies an initial list of paths with only two nodes. Then this list of paths is fed into a while-loop, in which the paths are completed further (also see comments in script for more detailed descriptions).
The user can choose to write the files to the disk immediately, by setting the function argument writeToDisk to TRUE (T). This creates a file called Paths.csv. If such a file already exists, then new paths will be appended to the existing file. This allows you to easily add paths to the file by using the function in steps (choosing a different origin point at each run), without having to manually merge the files later.
If this argument is set to FALSE (F) (default setting), then the function returns a list of all paths.
The new function can be downloaded here.
See the earlier post here.