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]]>Clustering is a vital tool when handling data making it a central part of data science. By grouping similar objects together, it helps us find what we are looking for. I don’t go to a bakery to find a book. Clustering is part of a wider idea in science as we are always faced with thousands of potential or actual measurements but we need to focus on the few which are relevant to the process we are trying to understand. I do not need to know the nuclear properties of the constituents of a gas to understand its properties, while measuring temperature, pressure and volume do throw a lot of light on that problem. In whatever branch of science we are working in, we are always trying to reduce the dimensionality of our data, to use the language of statistics and data analysis.

Many of the techniques we use will need a measure of distance and it is most natural to call upon the everyday distance as defined by any ruler – formally the Euclidean distances d where for example d^{2} = x^{2} + y^{2} + z^{2} for the distance between the origin and a point at (x,y,z) in 3-dimensions.

However, what if time is present? Time is very different from space. Mathematically it leads to new types of geometry for space-times, Lorentzian rather than Euclidean. The simplest example is the Minkowski space-time used for studying special relativity. James Clough and I have been using Minkowski space as part of our study of networks which have a sense of time built into them – Directed Acyclic Graphs (see my blog on Time Constrained Networks for instance). Essentially these networks have a time associated with each vertex and then any edges present always point in one direction in time, say from the more recent vertex to an older one. Typically the time is a real physical time but for these types of network one can always construct an effective if artificial time coordinate.

There are many types of data with a directed acyclic graph structure. Citation networks are excellent examples and we will use them to illustrate our ideas in the rest of this article. Each node in a citation network is a document. The edges represent the entries in the bibliography of one document which always reference older documents – our arrow of time. We have worked with several different types of citation network: academic paper networks based on sections of the arXiv paper repository, US Supreme court judgements, and patents. My blog on citation network modelling gives some more background and how I think about citation networks in general.

Combining these two concepts James Clough and I have adapted a well known clustering method, MDS (Multidimensional scaling), so that it works for directed acyclic graphs (Clough and Evans 2016b). Traditional MDS is usually applied to data sets where you have a matrix of distances between each object. For a network, this would usually be the length of the shortest path between each node. MDS then assumes that these objects/nodes are embedded in a Euclidean space and suggests the best set of coordinates for the objects in that space. Clustering can then be performed by looking at which points are close together in this space. We found a way to take account of the fact that two papers on exactly the same topic can be published at the same time in different places. They are clearly ‘close’ together in any common sense definition of close yet there is no direct connection through their citation network. Our method will show that these papers are similar just from the pattern of their citations. Indeed the text could be fairly different (perhaps with two documents on networks, one uses the terms node, link, network while the second uses vertex, edge, graph for the same concepts) but the way these two documents are used by others later, or the way the to documents were based on the same material, indicates they are likely to be working on the same ideas.

Once you have the coordinates of each document in the citation network there are many other standard geometric tools you can use to do other jobs. For instance to recommend similar papers to one you are reading, you just look for other documents close in a geometric sense given the coordinates we have calculated. In the figure we show the top two hundred papers from the first decade of the hep-th part of the arXiv paper repository (this is dominated by string theory). The visualisation uses coordinates found using our Lorentzian MDS technique.

A two dimensional embedding of the 200 most cited papers in the hep-th citation network where coordinates are found using our Lorentzian MDS algorithm. From Clough and Evans 2016b.

Our work with Minkowski space fits into broader programme of looking at networks in terms of the geometry of different types of space, what I call *Netometry* (Networks + Geometry, or perhaps *Neteometry* is better), as exemplified by Krioukov et al 2009. For instance, a good indication that a low dimensional Minkowski space might be a good representation of many citation networks came from our measurements of dimension (Clough and Evans 2016a).

**Bibliography**

Clough, J.R. & Evans, T.S., 2016a. What is the dimension of citation space? Physica A (in press) 2016. [ DOI 10.1016/j.physa.2015.12.053

arXiv:1408.1274 ]

Clough, J.R. & Evans, T.S., 2016b. Embedding graphs in Lorentzian spacetime, arXiv:1602.03103

Krioukov, D., Papadopoulos, F., Kitsak, M., Vahdat, A. and Boguna, M. 2010. Hyperbolic geometry of complex networks. Phys. Rev. E, 82 [ arXiv:1006.5169 ]

Via: Netplexity

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]]>The post Exploring Big Historical Data appeared first on Complexity Science.

]]>I’ve really enjoyed reading my copy of Exploring Big Historical Data: The Historian’s Macroscope (Macroscope for short here) by Shawn Graham, Ian Milligan and Scott Weingart. As the authors suggest the book will be ideal for students or researchers from humanities asking if they can use big data ideas in their work. While history is the underlying context here, most of the questions and tools are relevant whenever you have text based data, large or small. For physical scientists, many of whom are not used to text data, Macroscope prompts you to ask all the right questions. So this is a book which can really cross the disciplines. Even if some readers are like me and they find some aspects of the book very familiar, they will still find some new stimulating ideas. Failing that, will be able to draw on the simplicity of the explanations in Macroscope for their own work. I know enough about text and network analysis to see the details of the methods were skipped over but enough of a broad overview was given for someone to start using the tools. PageRank and tf-idf (term frequency–inverse document frequency) are examples where that practical approach was followed. Humanities has lot of experience of working with texts and a physical scientist like myself can learn a lot from their experience. I have heard this piecemeal in talks and articles over the last ten years or so but I enjoyed having them reinforced in a coherent way in one place. I worry a bit that that the details in Macroscope of how to use one tool or another will rapidly date but on the other hand it means a novice has a real chance to be able to try these ideas out just from this book alone. It is also where the on line resources will come into their own. So I am already planning to recommend this text to my final year physics students tackling projects involving text. My students can handle the technical aspects without the book but even there they will find this book gives them a quick way in.

Staff inthe Physics Department of Imperial College London clustered on the basis of the abstracts of their recent papers.

I can see that this book works as I picked up some of the simpler suggestions and used it on a pet project which is to look at the way that the staff in my department are related through their research interests. I want to see if any bottom-up structure of staff research I can produce from texts written by staff matches up to existing and proposed top-down structures of faculties – departments – research groups. I started using by using python to access to the Scopus api. I’m not sure you can call Elsevier’s pages on this api documentation and even stackoverflow struggled to help me but the blog Getting data from the Scopus API helped a lot. A hand collected list of Scopus author ids enabled me to collect all the abstracts from recent papers coauthored by each staff member. I used python libraries to cluster and display the data, following a couple of useful blogs on this process, and got some very acceptable results. However I then realised that I could use the text modelling discussed in the book on the data I had produced. Sure enough a quick and easy tool was suggested in Macroscope, one I didn’t know, Voyant Tools. I just needed a few more lines to my code in order to produce text files, initially one per staff member containing all their recent abstracts in one document. With the Macroscope book in one hand, I soon had a first set of topics, something easy to look at and consider. This showed me that words like *Physical* and *American*were often keywords, the second of these being quite surprising initially. However, a quick look at the documents with a text editor (a tool that is rightly never far away in Macroscope) revealed that many abstracts start with a copyright statement such as “2015 American Physical Society”, something I might want to remove as this project progresses.

So even for someone like me who has used or knows about sophisticated tools in this area and is (over) confident that they can use such tools, the technical side of Macroscope should provide a very useful shortcut despite my initial uncertainty. Beyond that I found that having the basic issues and ideas behind these approaches reinforced and well laid out was really helpful for me. For someone starting out, like some of my own physical science masters and bachelors students working on some of my social science projects, they will find this book invaluable. A blog or intro document will often show you how to run a tool but they will not always emphasise the wider principles and context for such studies, something you get with Macroscope.

I should make clear that I do have some formal connections with this book. I suggested the general topic of digital humanities and Shawn Graham in particular as a potential author at an annual meeting of the physics and maths advisory committee for ICP (Imperial College Press). For free sandwiches we pass on ideas for topics or book projects to the publisher. I also commented on the formal proposal from all three authors to ICP, for which I often get a free book. My copy of Macroscape was obtained for reviewing a recent book proposal for ICP. Beyond this I get no remuneration from ICP. It is nice to see a topic and an author I highlighted to come together in a real book but the idea is the easy bit and hardly novel in this case. Taking up the idea and making it into a practical publishing project is down to Alice Oven and her ICP colleagues, and to the authors Shawn Graham, Ian Mulligan and Scott Weingart. That’s particularly true here as the book was produced in an unusual open source way and ICP had the guts to go along with the authors to try this different type of approach to publishing.

**References**

Exploring Big Historical Data: The Historian’s Macroscope

Shawn Graham (Carleton University, Canada),

Ian Milligan (University of Waterloo, Canada),

Scott Weingart (Indiana University, USA)

ISBN: 978-1-78326-608-1 (hardback)

ISBN: 978-1-78326-637-1 (paperback)

Via: Netplexity

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]]>The post Modelling the Footprints of Innovation: Citation Networks appeared first on Complexity Science.

]]>When one document refers to another in it’s text, this is called a **citation**. The pattern of these citations is most naturally represented as a network where the nodes are the documents and the links are the citations between the documents. When documents were physical documents, printed or written on paper, then these citations must (almost always) always point back in time to older documents. This arrow of time is imprinted on these citation networks and it leads to interesting mathematical properties.

One of the most interesting features of citations is that they have been carefully curated, sometimes for hundreds of years. The data I use on the US supreme court judgments goes back to the founding of the USA. So citation data is one of the oldest and continuous ‘big data’ sets to study.

The reason why records of citations have been maintained so carefully is that they record the process of **innovation**, be it in patents, law or in academic study. When you try to claim a patent you must by law indicate the prior art, earlier patents with relevant (but presumably less advanced) ideas. When a judge makes a judgement in a case in the USA, they draw on earlier cases which have interpreted, and so created, the law needed to come to a conclusion in the current case. And of course, academics can’t discuss the whole of existing science when explaining their own new ideas so they have to refer back to papers where previous researchers have set out a key idea needed in the current work. Citations are therefore a vital part of knowledge transfer, new ideas build on all the work done in earlier judgments, previous patents or older papers. That is why citations have been so carefully recorded. The network formed by documents and their citations show whose giant shoulders we are standing on when innovations are made, to paraphrase Newton.

From a theoretical point of view there are many interesting features in these networks. If you follow citations from one document to another to another and so on, at each step you will always reach an older paper. So you can never come back to the starting point and such paths. There are no cycles in a citation network (it is an example of a directed acyclic graph). If you look at the number of citations each document gets, how many newer documents refer back to one document, then they follow a fat-tailed distribution – a few documents have most of these citations, while most documents have very few citations each. Derek de Solla Price’s 1965 paper for an early discussion of this feature. Moreover, if you look at documents in the same year, you get roughly the same shape for the number of documents with a given number of citations (see Radicchi et al 2008, Evans et al 2012), at least for well cited documents. Since these networks are of such great interest,many other features have been noted too.

One way for a theorist to understand what is happening is to build a model from a few simple rules which captures as many of the features as possible. One of the first was that of Derek de Solla Price (1965) whose theory of “cumulative advantage” suggested that as new documents were created they would cite existing papers in proportion to the number of citations they already had, that is the richer get richer. This follows a principle used in many other models of fat-tails in other data, and indeed was later rediscovered in the context of the number of links to modern web pages – Barabási and Albert’s preferential attachment (1999). One trouble with this simple model is that the oldest documents are always the ‘rich’ ones with the most links. In reality, each year of publication there are a few documents with many citations (relative to the average number for that year) and most have very few. The Price model does not give this as all documents published in the same year will have roughly the same number of citations. To address this problem we (Sophia Goldberg, Hannah Anthony and myself) searched for a simple model which reproduced this behaviour – fat tails for the citation data of papers published in one field and one year (Goldberg et al, 2015).

The simplest model we found works as follows. At each step we add a new document, representing the evolution in time of our citation network.

A new document (red diamond) is added to an existing set of documents (blue circles) and their citations to earlier documents (arrows).

A new document first looks at recently published documents, as it is well known that citations tend to favour more recent documents. What we mean by recent is set by one of our parameters, a time scale **τ**.

We choose these recent documents partly at random (fraction **(1-p)**) and partly with cumulative attachment (fraction **p**) in which we pick recent papers to cite in proportion to the number of their current citations. This choice of papers is not realistic since it requires the authors to be able to choose from all recent documents while in reality authors only have a limited knowledge. However this stage is meant to capture, statistically at least, the way authors learn about recent developments: a recommendation from a colleague, a talk at a conference, scanning new editions of certain journals and so forth. Sometimes it will be essentially random, sometimes this first choice will reflect the attention papers have or will receive.

Choose to cite a recent document, with probability p use cumulative advantage (preferential attachment) or simply random with probability (1-p).

Once we have chosen these primary documents to cite, our new document then looks at the references within these primary documents.

Each paper cited in the primary paper is then cited, copied, by the new document with probability **q**, the third and last parameter of the model.

The new paper cites the selected secondary papers, so on average q references are copied from the primary paper.

This ‘copying’ process is known to be a way of getting cumulative attachment with only local knowledge of the network (see for instance my paper with Jari Saramäki (Evans & Saramäki 2005) and references therein). That is you only need to read the papers you have already cited, already found, to find these secondary documents to reference. There is no need for the model to know about the whole network at this point, reflecting the limited knowledge of actual authors.

It was only when we added the last copying process that we found our model reproduced the fat-tails seen within the citations to documents published in the same year. Nothing else we tried gave a few well cited papers in every year. Comparing with one data set, taken from the hep-th section of the arXiv repository, we found that the best values for our parameters led to a typical paper in our model of hep-th made up as follows:

- Two primary papers chosen at random from recent papers.
- Two primary papers chosen in proportion to the number of their citations from recent papers.
- Eight secondary papers chosen by copying a reference from one of the first four primary papers.

This may seem like a very high level of papers being copied from the primary ones – on average we found 70% of papers were secondary citations, papers already cited in other papers being cited. One has to ask if the more recent paper, the primary one, contained all the information from the earlier ones as well as innovations being built on in the current paper. Did the new documents really derive useful information from the secondary papers cited? Often you see old ‘classic papers’ gathering citations as they are name checked in the introduction to a new paper. Not clear if the classic paper was even read while performing the current research. This feeling that some papers gain attention and acquire citations that does not reflect any direct influence on the current work is supported by at least two other studies. One was a study by Simkin and Roychowdhury (2005) of the way errors in the bibliographies of papers are copied in later papers. They suggest that this meant 80% of citations came from such copying of references. In another approach, James Clough, Tamar Loach, Jamie Gollings and myself (Clough et al, 2015) exploited the special properties of citation networks and this also suggested that 70%-80% of links were unnecessary for the logical structure of academic citation networks.

Of course constructing simple models will never capture the whole story. Models are, though, a good way to see if we have understood the key principles underlying a system.

- Barabási, A.-L., Albert, R. (1999). Emergence of scaling in random networks. Science, 286, 173.
- Clough, J.R., Gollings, J., Loach, T.V., Evans, T.S. (2015). Transitive reduction of citation networks. J. Complex Networks, 3, 189-203 [doi: 10.1093/comnet/cnu039, arXiv:1310.8224].
- Clough, J.R., Evans, T.S. (2014). What is the dimension of citation space? arXiv:1408.1274.
- Evans, T.S., Saramäki, J.P. (2005). Scale Free Networks from Self-Organisation. Phys.Rev.E, 72, 026138 [doi: 10.1103/PhysRevE.72.026138 , arXiv:1408.2970]
- Simkin, M.V., Roychowdhury, V.P. (2005). Stochastic modeling of citation slips. Scientometrics, 62, 367-384
- Price, D.J.d.S. (1965). The scientific foundations of science policy. Nature, 206, 233-238.
- Radicchi, F., Fortunato, S., Castellano, C. (2008). Universality of citation distributions: Toward an objective measure of scientific impact. PNAS 105, 17268-17272.

Via: Netplexity

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]]>The post Elucidating a Complex Rhythm with a Simple Model appeared first on Complexity Science.

]]>Normally, the heart beats in synchrony so that the upper chambers pump blood into the lower chambers first and then the lower chambers pump the blood out of the heart. Pacemaker cells (found in the upper chambers) rhythmically send out electrical signals to neighbouring muscle cells, telling them to contract. This electrical wave travels across heart muscle cells from the upper chambers to the lower chambers like a smooth Mexican wave (see this video). An abnormal heart rhythm is when these waves travel abnormally, which removes the synchrony of the pumping action.

As we age we all develop fibrosis, which is connective tissue in the heart that disrupts connections between muscle cells and as a result can disrupt the electrical waves. Our risk of AF also increases with age, but the mechanism of this is not fully understood. The increase in fibrosis with age is a potential explanation as to why the risk of AF also increases with age.

We developed a mathematical model that represents how cells are organised and connected within heart muscle tissue and how that changes with age. As the amount of fibrosis increased to a critical point the electrical waves would spontaneously re-organise into circular and spiral patterns, mirroring atrial fibrillation (see video below). We found that “burning” particular regions in the model where the cells were structured in a certain way could stop the fibrillation, however, when there was too much fibrosis the burning was unsuccessful.

The model was able to reproduce features of AF observed in patients, namely, that an increase in fibrosis was related to an increase in the prevalence of AF and that destroying specific regions of tissue could sometimes stop the abnormal rhythm. In addition to this the model also reflects how the disease develops in time: AF initially occurs for short periods and gradually gets longer as the condition progresses. However, medical imaging is unable to identify the structure of tissue at this scale in patients undergoing treatment.

The mathematical model highlights where future research could be targeted. A lot of work and caution is required to translate results from a mathematical model to biological and clinical studies. Furthermore, the study is an interesting example of how simple models can reproduce features of real complex systems.

**References**

K. Christensen, K. A. Manani, N.S. Peters. Simple Model For Identifying Critical Regions in Atrial Fibrillation, 2015. Physical Review Letters 114:028104. DOI: http://dx.doi.org/10.1103/PhysRevLett.114.028104 (open access)

U. Schotten, S. Verheule, P. Kirchhof, A. Goette. Pathophysiological mechanisms of atrial fibrillation: a translational appraisal, 2011. Physiological Reviews, 91(1), 265-325.

**Press Releases**

http://www3.imperial.ac.uk/newsandeventspggrp/imperialcollege/newssummary/news_22-12-2014-15-54-17

http://physics.aps.org/articles/v8/5

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]]>The post Time Constrained Networks appeared first on Complexity Science.

]]>However what if there are constraints on the links made in a network? Surely we should change the way we study networks if space, time or some other constraint is having a significant effect on the formation or use of the network. This has been a major interest of mine over the last few years. Space is one obvious limitation as in some cases long distance are less likely to be made. There has been a lot of work in this area over many decades but I will leave this constraint for another blog.

It is only more recently that the role of time in networks has began to receive more attention. A lot of this recent interest in how to deal with networks where the connections, are made at one time. That is because most communication networks, emails, phone calls and so forth, are of this type. The recent review by Holmes and Saramäki (2012) is such temporal edge networks.

Yet networks are made of two parts: vertices and edges. My recent work has focussed on the case where it is the vertices, not the edges, which are created at a definite time. In such *temporal vertex networks*, causality forces the interactions between nodes to always point in one direction. For example consider a citation network formed by academic papers. The nodes in our network are the academic papers and the links are formed by their bibliographies. So if paper A refers to another paper B then we can be (almost) sure that A was written after B. Information can therefore flow only from B to A. In fact any set of documents can only refer to older ones such networks are common. In law, judges refer to previous judgements to support their arguments. When registering a patent, *prior art* needs to be cited, that is other previously granted work which may have some relevance to the current claim.

The same types of structure occur in several other situations. Any situation where there is a logical flow has the same causal structure. If we have a project where the nodes represent individual tasks then an edge from task S to task T could represent the fact that task T requires task S to have been completed before task T is started. This has been the focus of work on temporal vertex networks in computer science. The logical flow of a mathematical argument or an excel spreadsheet show the same properties. These networks define what is called a *partially ordered set* or *poset* and it is under this title that you find relevant work coming from mathematicians. A final example comes from the Causal Sets approach to quantum gravity (see Dowker 2006 for a review). Here space-time is discrete not continuous, and these discrete points are the nodes of the network. The nodes are connected by edges only if they are causally connected and causality again gives these a common direction.

All of these temporal vertex networks have a key property. That is they contain no loops if you always follow the direction on the edges. You can not go backwards in time. Thus the traditional name for such a network is a *directed acyclic networks *or *DAG* for short.

So the question is how can we adapt traditional network measures to deal with the fact that these networks, DAGs, are constrained by causality? Are there new measures we should employ which give more insights to such networks?

I’ve been looking at these problems with several students (undergraduates in their final year projects and some MSc students), one of whom, James Clough, is now working for his PhD on this topic.

Paths in networks are always important. However one feature of a DAG we have been exploiting is that if we always follow the direction of the arrows, the direction of time, then not all nodes are connected. If we like we could add edges whenever there is such a path connected a late node to an earlier one, a process known as transitive completion. On the other hand we could remove as many edges as we can while leaving the causal relationships intact, a process known as transitive reduction. That is, if there is a path between two nodes in the network before transitive reduction, then there will still be a link afterwards.

What we have done (in *Transitive reduction of citation networks*) is look at how real data from citation networks behaves after transitive reduction. What we find is that different types of citation network behave very differently. The citation network formed from academic papers taken from the arXiv repository and the network of US Supreme Court judgements both show that about 80% of the edges are not needed to retain all the causal relationships. On the other hand the patents network shows the opposite behaviour with all but 15% of edges being essential. The edges removed tend to be the citation to older papers. One interpretation is that academics and and judges may be citing well known early papers and judgements though their current work is only directly related to more recent documents. Perhaps some of these citations do not indicate the early work was needed but reflect other motivations, such as simple copying of popular papers or review in the field which at best only have general relevance. For academic papers this interpretation is supported by the work of Simkins and Roychowdhury In this sense unnecessarily.

The number of citations to a document after transitive reduction certainly gives us a different view of the importance of different documents. For instance paper hep-th/9802109 on the arXiv (*Gauge Theory Correlators** **from Non-Critical String Theor*y by Gubsner et al.) was cited by 1641 papers in the network, but only three citations remained after TR! On the other hand, paper hep-th/9905111 (*Large N Field Theories, String Theory** **and Gravity* by Aharony et al.) has also large number of citations in the raw data, 806, yet after transitive reduction it has 77, so retaining far more of its orifginal citations. Perhaps information in the second paper was used more diversely.

We can find similar examples in the US Supreme Court citation network. The case *Schneider vs. New Jersey* (1939)’ has 144 citations in the original data but this drops to just just one after transitive reduction. *Stromberg vs. California* (1931) also falls from 132 citations to just one. Conversely, the case *Heller vs. New York* (1973) only shows a slight fall after transitive reduction, falling from from 68 to 48 citations and has the most citations in our reduced network. The second most cited case after transitive reduction is *Hamling vs. United States*, which drops from 68 to 38 citations. Wikipedia lists hundreds of Supreme Court cases but the last two are not famous enough to make the Wikipedia list. Our analysis suggests they may have more importance than a simple citation count would suggest. At the very least it might be be worth checking out documents that are highly cited in the essential.

Another way to look at citation networks is to see if we can define a dimension to the network. That is we can try to quantify how much variation there is in the citation process. A low dimension means that there are few directions , few distinct themes relevant for citation in a document. A high dimension indicates that there is a wide range of relevant but distinct directions from which a document will draw on for inspiration. We found (in *What is the dimension of citation space?*) that we were often able to find interesting dimensions. For academic papers, we found that different fields of research have different dimensions. For papers in the hep-th arXiv section (largely string theory) we found a low dimension of around 2 while for theoretical papers closely linked to particle physics experiments (hep-ph section) we found more variation as indicated by a higher dimension of 3. The quant-ph also around 3 while the astro-ph section had a slightly higher dimension of around 3.5. So clearly despite similarities in the main data using standard measures, our time-aware dimension measures show clear differences in the citation behaviour of different areas. String theory in particular seems to be a tightly knit collection of work with each work largely dependent on all the other work, few independent directions can be pursued. The US Supreme Court judgements were more complicated. Small samples (usually from modern judgements) showed a dimension of around 2.5 to 3 but larger samples, typically ranging from modern to the earliest judgements, had lower dimensions, closer to 2. We interpreted this as reflecting the way that there were few early judgements compared to the number produced to day. So that the further back we traced in time to find the influence of judgements on recent ones, the smaller the variation. Perhaps that is not so surprising and we might expect a similar shape if we could follow scientific papers back to the 18th century! patents on the other hand showed a much higher dimension though again these were involved.

It is clear from just the few studies we have made that time makes a crucial difference to the structure of a network. We have tried a few new measures adapted to take account of time and in doing so we have thrown up some intriguing features in real data. There is surely much more to find in when networks are embedded in time.

**References**

Clough, J.R. & Evans, T.S. *What is the dimension of citation space?*, arXiv:1408.1274

Clough, J. R.; Gollings, J.; Loach, T. V. & Evans, T. S., *Transitive reduction of citation networks*, J.Complex Networks to appear 2014, arXiv:1310.8224

Dowker, F. *Causal sets as discrete spacetime*, 2006. Contemporary Physics, **47**, 1-9

Holme, P. & Saramäki, J. 2012. *Temporal Networks* Physics Reports, **519**, 97-125

Simkin M.V. and Roychowdhury V.P., 2003. *Read before you cite!* Complex Systems **14** 269-274

Via: Netplexity

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]]>The post The Many Truths of Community Detection appeared first on Complexity Science.

]]>You do not need to know the detailed properties of every small part making up a gas, it turns out the bulk properties of a gas can be derived from very general principles. In the same way when looking at Facebook data, we might be able to identify groups of people who behave in a similar way. Searching for these groups or clusters in data is central in many areas of physical and social science. It is often easier to understand the behaviour of a large system by looking at these clusters, which are much fewer in number.

In terms of networks, the clustering is based on the structure (topology) of the network and the groups found are called Communities. In this case we might expect a coherent group to be one which has more links between members of the group than it ha to nodes outside the group in other clusters. I have done some work on what is called Community Detection, particularly in methods which assign nodes to the membership of several clusters (e.g. my line graph and clique graph papers referenced below). After all, my social connections are likely to show that I am part of several groups: work colleagues, family relationships, connections made through hobbies or sports.

For some time I have been very wary about the meaning of the clusters found with such methods and particular about claims of one method being able to find “better” communities than another. A recent paper prompted me to think about this again. In *Community detection in networks: structural clusters versus ground truth*, Hric, Darst, and Fortunato from Aalto University in Finland (a big centre for networks research) asked if the network methods were finding different sorts of clusters from those found using other aspects of the data. Typically when testing a community detection method, one sets up artificial networks in which each node is assigned to one community. The edges between nodes are then assigned at random but with a preference for edges to be between nodes from the same community. I can do all the tests I like on artificial data but I am always worried that this approach has introduced some hidden bias. Perhaps we end up choosing the methods that ‘work’ on artificial data but which are perhaps not so good on real messy data? It all comes down to the fact that we have mathematical ways to quantify the difference between community assignments but defining what we mean by “the best” clustering is impossible. Even with artificial networks, the “ground truth” is not generally an absolute truth. Typically the “truth” are input parameters and the actual network generated is partly random. So while the resulting artificial network is correlated with the ground truth it is not designed to be a perfect match. So in this case the “actual truth” will, in almost most cases, be different from the ground truth.

I also worry about what we do when we run network community detection methods on large real data sets where there is no simple ground truth. When I have done this, I can find a variety of possible answers for communities in the data. Many look reasonable but none correlate perfectly with each other or with what I know from other sources. This leaves me wondering if the automatic methods are finding one truth and my other information gives another. Alternatively the automatic methods might be rubbish, good on artificial cases, not so good in reality. There is no simple way of telling.

In any case do real networks have a “ground truth”? Quite often people have data from other sources about real networks and they use this to construct a “ground truth”. The test is then to see if automatic methods can find this ground truth. However what if the other data is wrong? People don’t always tell the truth, they can deliberately mislead or they can misunderstand the problem. Children surveyed about their friendships may tell you who they’d like to be friends with (the most popular person in the class) and not who they actually spend time with.

Zachary Karate Club network clustered using clique graph methods

Take the famous Zachary karate club data set used by many (including myself) as a simple test. This is a network of members of a karate club that split in two during the sociologist’s study. Let us accept that the professionalism of Zachary has produced data that is a true reflection of the situation despite the difficulty of measuring associations in social science. If you look at the published paper it actually gives two truths. One is based on which of two factions the members actually joined, and one based on an automatic community detection method. I suspect most people are using the latter as the ground truth (unwittingly) when testing their work. Perhaps this is a further example supporting the claim that academics only read 20% of their references. Worse the data given in the published karate club paper is not consistent – the unweighted adjacency matrix is not symmetric. So which truth was used for all those papers using the Karate club network?

Another example comes from some work I did on overlapping community methods. Like many other people I downloaded a standard data set from Mark Newman’s web site, an extremely useful resource. The American College Football data was created by Girvan and Newman (in *Community structure in social and biological networks*) and represents the games played between American College Football teams in one season. Also provided are the conference membership of each team. Teams play more games against teams from the their own conference than from any one other conference. In fact this data is so well clustered that surely **no** method should get anything wrong beyond a few independent teams as my visualisations here illustrate (taken from my clique based clustering paper). So I looked at the “mistakes” made by my method. After about two afternoons of wading through interminable web sites of stats on American College football and Wikipedia pages on the College Conference system, I realised that in fact most of the “mistakes” were **not** from the automatic community detection but lay in “the ground truth”, that is in the conferences assigned to teams in the data file. It turns out that the assignments in the original `football.gml` file are for the 2001 season while the file records information about the games played for the 2000 season. For instance the Big West conference existed for football till 2000 while the Sun Belt conference was only started in 2001. There were 11 conferences and 5 independents in 2001 but 10 conferences and 8 independents in 2000. Care is needed as American College athletic conferences cover many sports, with some sports joining or dropped from any one conference time to time. Teams can also switch conferences too. In fact around 10% of the college teams playing American football at the top level changed conferences around 2000-2001.

So often the “ground truth” is just another truth not some absolute truth! The errors in the Zachary Karate club and American College Football data do not matter in one sense as they still provide valid and valuable tests for methods. The conclusions in the hundreds of papers using these data sets and which use these questionable ground truths would not change. Indeed it highlights one role for automatic methods. You can see that where Girvan and Newman’s methods get the “wrong” answer in their original paper (*Community structure in social and biological networks*) they are in fact highlighting problems with their conference data. Validation of data is a very useful if boring job. A final question will always be if there is a single truth. For instance I am in the theoretical physics group of the physics department of the Faculty of Natural Sciences at Imperial College London. That top-down hierarchical truth is important when allocating desks and teaching. However another truth would emerge if you studied my research relationships. Those are with staff and students based in other physics research groups and with colleagues from other departments and even other faculties.

So I was really pleased to see that *Community detection in networks: structural clusters versus ground truth *were questioning the meaning of truth in community detection from a quantitative point of view. Clustering of data, finding communities in data is of tremendous value commercially and for research, but there is still a lot more work to do before we understand these different truths.

**References**

M. Girvan, M.E.J. Newman, *Community structure in social and biological networks,* PNAS (2002) **99**, 7821-782

W. Zachary, *Information-Flow Model For Conflict And Fission In Small-Groups Journal Of Anthropological Research*, (1977) **33** 452–473

D. Hric, R.K. Darst,and S.Fortunato, Community detection in networks: structural clusters versus ground truth, arXiv:1406.0146

T.S. Evans, American College Football Network Files. figshare. (2012). http://dx.doi.org/10.6084/m9.figshare.93179

T.S. Evans, and R. Lambiotte, Line Graphs, Link Partitions and Overlapping Communities Phys.Rev.E, 2009, 80, 016105 [`arXiv:0903.2181`].

T.S. Evans, Clique Graphs and Overlapping Communities J. Stat. Mech. (2010) P12037 `http://dx.doi.org/10.1088/1742-5468/2010/12/P12037`

[`arXiv:1009.0638`]

Via: Netplexity

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]]>The post Why Google scholar is no worse than anything else appeared first on Complexity Science.

]]>*1) Google Scholar Profiles include dirty data*

What is “dirty data”? A site like Impactstory pulling in data from a wide range of non-traditional sources ought to be a little more careful about throwing this term about! One person’s dirty citation is another’s useful lead. It does seem Google scholar is more open to gaming but at least it is easy to spot this using Google scholar if you see some anomalous data. Scopus and Web of Science make their decisions behind closed doors about what to include and what not; how many ‘weak’ journals and obscure conference proceedings are included there, how many book citations are excluded? I’ve heard at least one story about the way bibliometric data was used as a pawn in a dispute between two companies over other commercial interests. I just have no idea how much manipulation of data goes on inside a commercial company. On the altmetrics side of the story, most departments still regard any social media counts as dirty.

*2) Google Scholar Profiles may not last*

Surely a problem with anything, commercial or not. Your institution may switch subscription and cut off your access even if it is still out there. Google certainly has poor reputation on this front. In so many ways, we always gamble when we invest time in a computer product – not sure my PL/1 programming knowledge is much use these days.

*3) Google Scholar Profiles won’t allow itself to be improved upon*

Scopus and Web of Science also carefully control what you can do with their data. In any case you need a subscription before you can start to do anything. So surely this is a criticism of all closed data systems.

*4) Google Scholar Profiles only measure a narrow kind of scholarly impact*

Again, I don’t see Scopus and Web of Science producing much more than bare citation counts and h-indices. The UK 2012 research assessment procedure (REF) only quoted bare citation counts from Scopus. This is a problem of education. Until more people understand how use bibliometric data nothing much will happen and I know h-indices still get thrown about during promotion discussions at my institution (again see an Impactstory blog about why people should stop using the h-index).

**My Approach**

I tend to think of all sources as data. How you interpret should vary as you take each into account. Like all data and measurements, results derived from bibliometric information needs to be checked and validated using several independent sources and alternative methods.

For instance I have access to these three commercial sources, and they tend to give citations counts which differ. Web of Science is generally the most conservative, Scopus is in the middle and Google scholar leads the counts. So I can use all three to give a balanced view and to weed out any problems. They also have different strengths and weaknesses. Google is ahead of the curve and shows where the other two will go a year or two later. My work on archaeology has a large component in books which Google scholar reflects but the other two fail to capture. Scopus is very weak on my early Quantum Field Theory work, while both Web of Science and Google scholar are equally strong in this area and time period.

The tips discussed in “7 ways to make your Google Scholar Profile better” are very useful but many apply to all data sources. For instance Scopus just added two papers by another T.S.Evans working near London to my profile, even though its in a completely different research field, the address is completely different (not even in London) and there is basically zero overlap between these papers and my work. Makes you worry about the quality of the automatic detection software used in commercial bibliometric firms. I can’t fix this myself, I have to email Scopus while I can tidy up Google scholar myself whenever I want. Currently I also feel that the Google scholar recommendations are the most useful source of targeted information on papers I should look at but I am always looking for improvements.

Overall, I feel you need to keep a very balanced approached. Never trust a statistic until you’ve found at least two other ways to back it up independently.

Via: Netplexity

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]]>The post Should I Believe My Conclusion? appeared first on Complexity Science.

]]>Hold on. Huh, 0.001? Am I a genius or am I a fool? Apparently, I am the latter one. Very likely, there is something wrong with my null hypothesis. The p-value of 0.001 is very suspicious.

The thirst of result and publication is terribly unhealthy nowadays. There is certainly pressure on publication in order to show progress and achievement, especially in the current time when job security in academic is weak. Nowadays, there is a tendency to publish a result which has been tested against only a single statistical test. Even worse, many researchers have no knowledge about the limitation of the test they employ for drawing a conclusion.

Confirmation bias is the most toxic psychological substance in modern science. It is not difficult to find a publication which has an eye-catching p-value but questionable conclusion. Researchers tend to publish as soon as possible whenever they get an indicator confirming their hypothesis or theory, forgetting there exists something called false positive.

(The following is a real story appeared in BBC Horizon: How You Really Make Decisions)

In an intelligent training centre in Washington DC, a group of recently recruited analysts were told about an emerging threat about a biological terrorist attack. Given the urgency, the analysts got 15 minutes to identify the terrorist group which is most likely to be responsible for the attack.

Similar to a real situation, the analysts were provided a large pool of real-time information, including tweets from social media and intelligence from government agencies. Hinted by the instructor, a group of terrorist called the Network of Dread was their top suspect of all time. They are a bio-terrorist group which are internationally well-known for their track records. The group has plenty of unknown bioweapon on their disposal and was responsible for a recent attack which caused a number of causalities. There were countless mentions and retweets about the group’s activity. Meanwhile, when the analysts were putting effort on seeking evidence of a potential terrorist attack by the group, two extra clues were fed into the information pool: a bio-medical lab was robbed and a local freight firm was hacked almost at the same time few hours ago.

Time was up. Most of the analysts reported that the Network of Dread was the biggest suspect responsible for the new threat because `evidences’ (based on the amount of mention on twitter?) of their next attack are plenty. Unfortunately, they were all wrong. It turned out that the attacked was organised by a group of cyber hacker called the Master of Chaos.

Up to this point, I hope you have noticed something wried. If not, it is usual. It is the point I want to raise awareness. The point is that the instructor intentionally introduced a bias in the analysts’ heads. They all developed a bias against the all-time suspect called Network of Dread and hence were aiming to look for evidence to support the hypothesis that the group is responsible for the new threat. As the information seemed to confirm their biased mind, the firmly believed their hypothesis with no doubt. The phenomenon is called `Confirmation Bias’. In fact, there are two important clues of which the analysts should have been aware. The theft and hacking reports are obvious. The second one is a bit tricky; if the analysts have read the paper concerning insurgent pattern by Neil Johnson, they should have been aware that the chance of another attack after a fatal terrorist attack is unlikely.

In intelligence analysis, a conclusion can be a life or death matter. But in research, a wrong judgement can also cause a decade-long fallout; the damage to the society is no less than a terrorist attack.

Recently, one of my colleagues was marking project reports of a network course. In the project, the students were asked to verify the power-law tail of the degree distribution of a scale-free network generated by the Barabási–Albert (BA) model.

Deliberately hinted by the instructor (not me), most students picked a statistical measure called Coefficient of Determination, denoted by $latex R^2$, to draw a comparison between the results and the theoretical values. It has the following formula:

$latex R^2=1-\frac{\sum_i (y_i-\bar{y})^2}{f_i-\bar{y}}$

**Figure 1. A typical plot found in the reports where a power-law fit doesn’t hold in the tail.**

So far so good, but then the bizarre thing happen: most students declare a perfect match as they obtained a coefficient of 0.999 despite the dropping tail. Huh?! Something must be wrong. How could an obvious deviation result such a high value?

Any network scientist knows that the BA model does not produce a pure power-law tail; the tail drops in the end due to finite size effect and the dropping region depends on the number of steps. A perfect power law can only be obtained with infinite-many steps. For limited steps size, even $latex 10^6$, the dropping is clear.

**Figure 2. The shift of the dropping region of the BA model with different step sizes.**

So dropping is expected and there is nothing wrong with the plot. The only thing wrong is the measure which suggests a perfect match. As dropping is obvious, how could the students all obtained perfect coefficients? The answer is clear that they forgot they were testing against a fat tail distribution. The difference between the order of magnitude of the head and tail is huge. No matter how much the tail deviate from the expected curve, the contributions from the tail to the coefficient is just insignificant given its tiny values. In other words, the first few data points dominate the value of the coefficient; as long as the first few data points match the theoretical value, the coefficient will certainly close to 1. Using this measure to conclude a power law behaviour will fail for certain.

Every statistical measure has some limitations; certain conditions and some assumptions have to be fulfilled in order to obtain a sturdy measure. In the case of fat-tail distribution, most statistical measures are no longer reliable. Since the central limit theory and in some cases the law of large number do not hold in any fat-tail distribution, many common assumptions in statistical measures are broken. Extreme care is required to obtain a trustworthy measure.

The dropping tail is a conspicuous signal to a failing statistical measure. However, in many cases this matter is obscure. The best way to solve this issue, in my opinion, is to be sceptical to all measures, no matter it is an `industrial standard’ or not. One size cannot fit all; depending on the question, different statistical tools may need to be employed. Instead of seeking confirmation to a theory by getting support from a measure, we should ask why and how a measure works at the first pace. If the hidden assumptions had known, many scientific catastrophes and embarrassment caused by unreproducible experiments could have had been avoided.

In conclusion, scepticism is essential to develop a reliable theory. As truth is robust against any form of test, there is no harm to conduct different statistical tests; no matter the result is positive or not, ask why it happen rather than accepting it as the conclusion.

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]]>The post CitNetExplorer – Citation Network Analyser and Visualisation appeared first on Complexity Science.

]]>One of the interesting things about citation networks is that their vertices have an order given by their publication date. This is a very strong constraint on the system so when you analyse or visualise such a network you should take the time ordering into account. The simplest example is that you should not just look at the vertex degree in these networks, but at in-degree (citation count) and out-degree (length of bibliography). Perhaps the most obvious aspect of the constraint comes when you try to visualise the network. You can just put such networks into a standard network package and treat it as a directed network, which it is. However any standard visualisation will undoubtedly place the vertices all over the two-dimensional surface used for display. Standard visualisations pay no attention to the time-ordering of the vertices yet you almost certainly want to show that information when displaying a citation network as it is such a critical part of the definition. So many of the properties will depend on the age of the publication for instance. I have encountered this myself and played around with a few ad-hoc solutions but came to the conclusion I needed to write something myself, adapting a standard layout method to set one dimension of the vertex coordinates while the second dimensions is set by the vertice’s time. Since the same problem is encountered when making diagrams showing the critical paths in a set of tasks (such as Gantt charts) there are packages which will do this. However you will also want to do different types of analysis on a citation network plus they are likely to be much bigger than a normal Gantt chart.

This is where CitNetExplorer comes in. This comes from Nees Jan van Eck and Ludo Waltman at the CWTS (Centre for Science and Technology Studies) in Leiden, so comes from one of the leading institutes in bibliometric research. Its very early days and I have only had a short play but for me its good points are:

- Free for noncommercial and teaching purposes.
- Cross platform as written in java.
- Stable on my Windows 7 machines.

As it is written in java, it is likely to be stable on other platforms too. - Well presented with a reassuring professional feel.
- Good graphical display.

The publications are laid out using their publication data for the vertical coordinate and a layout algorithm to place the publications horizontally - Good default options.

I got an instantly readable figure every tine I tried it - Good range of graphical output options.

Vector graphics, especially postscript (eps), is essential for me. Note these are all under the Screenshot menu option. - Two basic network format output options.

A pajek .net and a simple text file format (see below) - Various basic analysis tools.

This includes transitive reduction which is something I have been very interested in and can throw up some new insights into the citation counts of papers (see arXiv:1310.8224).

So this looks to be a really nice package. Of course, I am never satisfied so what would I like to see in future versions:

- Open source.

It would be nice to be able to learn from their computational work and to add to this myself. Maybe some type of plug-in could be added to solve the latter problem. I have a few more tricks for citation networks in the pipeline for instance. - More input options.

There are only two and one is tied to Thomson-Reuter’s WoS (Web of Science) database. In the example given by the authors you perform a search on WoS and then save the results in a text file (saverecs.txt). Note you must select the “Web of Science Core Collection” not the “All Databases” option which the example clearly shows but I didn’t read, otherwise the output file will not include the full citation information needed to construct the citation network. This file is a simple text file so you should be able to combine them by hand if like me you are limited to 500 records per file.

The alternative is a pair of relatively simple text files. These are not as yet explained in the documentation. Basically there are two files. First is*name*`pub.txt`file lists the properties of the publications and the order in this file assigns each publication an index (the publication on line 2 is vertex 1, line 3 defines vertex 3 and so on). The second file is called*name*`cite.txt`and is an edge list written in terms of the vertex index. Look at the first few lines of the example data James Clough made from the open source KDD cup arXiv citation network data that we have been using in our recent work. Alternatively if you can produce a file from WoS open it in CitNetExplorer then save it in what is called CitNetExplorer format. These CitNetExplorer files are easy to look at, edit and prepare in a spreadsheet or a basic text editor and appear to be tab separated. - Visualisation editing.

No layout is perfect so it is essential to be able to move the vertices by hand. One of my favourite visualisation packages, visone, shows what you can do in java, and even my own ariadne package built on the jung library gave that functionality automatically.

Rather less seriously, I am not sure about the name. I would pronounce the “Cit” in “CitNetExplorer” as “sit” or perhaps “chit” so I would have kept the “e” in “cite”, CiteNetExplorer, but its not my product. As I’m getting bored typing it it, I’m sure it will become just CNE in any case.

CitNetExplorer http://www.citnetexplorer.nl/

Van Eck, N.J., & Waltman, L. (2014). CitNetExplorer: A new software tool for analyzing and visualizing citation networks. [arXiv:1404.5322]

Van Eck, N.J., & Waltman, L. (2014). Systematic retrieval of scientific literature based on citation relations: Introducing the CitNetExplorer tool. In Proceedings of the First Workshop on Bibliometric-enhanced Information Retrieval (BIR 2014), pages 13-20.

James R. Clough, Jamie Gollings, Tamar V. Loach, Tim S. Evans (2013).

Transitive Reduction of Citation Networks. [arXiv:1310.8224]

Clough, James; Evans, Tim; Loach, Tamar (2013). Transitive Reduction of Citation Networks. (data set) figshare

http://dx.doi.org/10.6084/m9.figshare.834935

Clough, James; Evans, Tim (2014). KDD cup arXiv data for CitNetExplorer. figshare fileset.

http://dx.doi.org/10.6084/m9.figshare.1021647

Via: Netplexity

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]]>The post Sculplexity: sculptures of complexity using 3D printing appeared first on Complexity Science.

]]>One of the ways I learn about a new topic is to set it up as a project for final year undergraduate students or as a masters level project. We have some of the best students in the UK here so all I have to do is come up with an outline and the students usually rise to the challenge.

What I needed was a connection between my work in complexity and 3D printing. This link came from one of those serendipitous moments. Another advantage of being at Imperial is that it is part of a Victorian complex of arts and science institutions so we are surrounded by national museums. One is the V&A (Victoria and Albert museum) which is dedicated to all aspects of design, from all ages and all locations. It also has some amazing tea rooms, covered in tiles designed by William Morris, great when I have a visitor and a little more time. It was on one of those trips that I just happened to walk past something that caught my eye. At one level it is just a table. However I saw this table as a branching process. To the designers, it was inspired by the tree-like structures of nature. The designers had embraced technology to do this, using Computer Aided Design (CAD) and 3D printing. For on further investigation this was the Fractal.MGX table designed by WertelOberfell and Matthias Bär, the first 3D printed object the V&A had acquired.

Branching processes occur in many problems in complex systems and they have a long history of mathematical investigation. So here was the link I was looking for. The question I asked was what other physics models familiar to me could be turned into a 3D printed object? How would you actually do this conversion? Does the tool, the 3D printer, impose its own limitations on the type of mathematical model we can use? Are these new limitations interesting mathematically in their own right? Until now researchers had only seen these mathematical models visualised using two-dimensional representations, often not even using perspective to give the impression of a 3D object. Making a 3D printed object opens up uncharted territory. My project shows one can move from traditional visualisations to new “tactilisations”. So can we gain new insights by using touch rather than vision?

The approach might also be useful for outreach as well as research. The same things that got my students and I interested in the project might intrigue school children, a retired judge or whoever. These objects might be particularly useful when explaining science to those whose sense of touch is better than their sight. However we could also go back to where this project started and see if models of complexity can produce something of aesthetic value alone – hence Sculplexity: sculptures of complexity.

The basic idea is simple. A 3D printer builds up its object in layers. So the height of the object can be thought of as a time. Suppose I have a model which defines a flat (two dimensional) picture. Typically this will be a grid with some squares full and some empty. The model also has to describe how this picture evolves in time. For instance there is a famous example known as Conway’s Game of Life for which are there many 2D visualisations. What I do is use the model at each point in time to define what the printer should print at one height. The next time step in the model will then define what to print on top of the first layer, and so forth.

In fact while the basic idea is straightforward, the implementation turned out to be much much harder than I expected. It is to the real credit of the undergraduate students working with me on this project, Dominic Reiss and Joshua Price, that we pushed this idea through and actually got a final 3D printed object representing our modified forest fire model. OK so our final result is a bit of a lump of black plastic compared to the inspiration for this project, the Fractal.MGX table in the V&A. But this is just the start.

Now that we have shown that this can be done, there is so much more to explore. The possibilities for 3D printing are endless. All we have done is made the first step in terms of 3d printing and mathematical models. We have highlighted the key problems and given at least one way to fix them. I can already see how to extend the existing approach, new solutions to some of the problems, different ways to create an object from a wider variety of theoretical models. Imagination and ingenuity are all that are required.

- Reiss D.S., Price J.J. and Evans T.S., 2013.
*Sculplexity: Sculptures of Complexity using 3D printing*, European Physics Letters**104**(2013) 48001, doi 10.1209/0295-5075/104/48001.

Copy of*Sculplexity: Sculptures of Complexity using 3D printing*on personal web page. - Evans T.S., 2013. Images of 3D printing output for Sculptures of Complexity – Sculpexity. http://dx.doi.org/10.6084/m9.figshare.868866
- Reiss D.S. and Price J.J., 2013. Source-code for Complex Processes and 3D Printing, https://github.com/doreiss/3D_Print, doi 10.6084/m9.figshare.718155 .
- Reiss D.S., 2013. Complex Processes and 3D Printing, project report, http://dx.doi.org/10.6084/m9.figshare.718146.
- Price J.J., 2013. Complex Processes and 3D Printing, project report, http://dx.doi.org/10.6084/m9.figshare.718147.
- 3D printing used as a tool to explain theoretical physics by Laura Gallagher, Imperial College News and Events, 09 December 2013.

Via: Netplexity

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