The citation network among Wikipedia pages on Dynamical Systems and Mechanics

“When a document di cites a document dj, we can show this by an arrow going from the node representing di to the document representing dj. In this way the documents from a collection D form a directed graph, which is called a ‘citation graph’ or ‘citation network’”

— L. Egghe, and R. Rousseau (Introduction to informetrics: Quantitative methods in library, documentation and information science. Amsterdam, 1990, Elsevier).

In the work (with Sergios T. Lenis), the collection of documents that we are considering is a corpus of the Wikipedia (WikiProject) Lists of topics (https://en.wikipedia.org/wiki/Wikipedia:WikiProject_Lists_of_topics). In other words, our corpora consists of a number of (web) pages, each one referring to a (thematic) topic from the certain Wikipedia list of topics. For example, if we are dealing with the (hypothetical) Wikipedia list of topics LTi (having a URL of the form https://en.wikipedia.org/wiki/LTi), then LTi is a list of (web) pages, each one of which being identified to a topic LTiTj (typically, having a URL of the form https://en.wikipedia.org/wiki/LTiTj). Therefore, when considering a number of different Wikipedia lists of topics, the aggregated (web) pages (are called) and will form a set of topics. (Notice that there might be common topics in different Wikipedia lists of topics.) In this sense, when there is a hyper-link from topic LTiTj to topic LTmTn, we will say that “topic LTiTj cites topic LTmTn” or that “topic LTmTn is cited by topic LTiTj.” Thus, what we get from a corpora of Wikipedia lists of topics {LTi: i = 1, …, I} is a directed graph G = (V, E), where the set of nodes V = V(G) is the set of topics {LTiTj: i = 1, …, I, j = 1, …, J} and the set of (directed) edges E = E(G) is a set of citations from topic LTαTκ to topic LTβTλ, for some α, β in {1, …, I} and some κ, λ in {1, …, J}. For obvious reason, graph G will be called citation graph (or citation network). Furthermore, the reciprocity coefficient of the citation graph is defined as the ratio of reciprocated edges/citations over all edges/citations of the graph. Moreover, notice that graph G might contain self-loops or internal citations inside topic LTαTκ (which is the case for anchor tags inside the web page of topic LTαTκ).

In what follows, we are going to consider two specific Wikipedia lists of topics:

(1) the list of dynamical systems and differential equations topics (https://en.wikipedia.org/wiki/List_of_dynamical_systems_and_differential_equations_topics) and

(2) the list of mathematical topics in classical mechanics (https://en.wikipedia.org/wiki/List_of_mathematical_topics_in_classical_mechanics).

The citation graph produced from the former list of topics will be simply referred as Dynamical Systems (citation) graph, the latter as Mechanics (citation) graph and the citation graph produced from the.combined (joined) list of dynamical systems and differential equations topics and list of mathematical topics in classical mechanics will be called joined Dynamical Systems and Mechanics (citation) graph.

Before tackling the joined graph, let us first show the properties of the individual citation graphs.

  1. The Dynamical Systems Citation Graph

So, let us first display the Dynamical Systems citation graph:

The Dynamical Systems citation graph.

This is a directed graph of 163 nodes/topics and 1140 edges/citations with density 0.0432. It is weakly connected but not strongly connected (in fact, it includes 35 strongly connected components). Moreover, the Dynamical Systems graph contains 2 self-loops (internal citations) and the reciprocity coefficient is 0.5316 (i.e., 53.1% of citations are reciprocated). Notice that 154 nodes/topics (represented as blue triangles) are citing/cited only by Dynamical Systems topics, while there exist 9 nodes/topics (represented as blue stars) which are also citing/cited by Mechanics topics too.

Below is a table showing the top 14 Dynamical Systems nodes/topics with regards to their out-degree (i.e., most citing topics) together with the corresponding in-degree (i.e., counts of other topics in the Dynamical Systems graph which cite the former topics).

The top 14 nodes/topics receiving the highest number of citations (in degree).

2. The Mechanics Citation Graph

Next comes the Mechanics citation graph:

The Mechanics citation graph.

Again, this is a directed graph of 61 nodes/topics and 448 edges/citations with density 0.1224. It is also weakly connected but not strongly connected (in fact, it includes 11 strongly connected components). Moreover, the Mechanics graph contains 2 self-loops (internal citations) and the reciprocity coefficient is 0.6071 (i.e., 60.7% of citations are reciprocated). Notice that 52 nodes/topics (represented as red squares) are citing/cited only by Mechanics topics, while there exist 9 nodes/topics (represented as red stars) which are also citing/cited by Dynamical Systems topics too.

Below is a table showing the top 12 Mechanics nodes/topics with regards to their out-degree (i.e., most citing topics) together with the corresponding in-degree (i.e., counts of other topics in the Mechanics graph which cite the former topics).

3. The Joined Dynamical Systems and Mechanics Citation Graph

Finally, let us focus on the joined Dynamical Systems and Mechanics citation graph:

The joined Dynamical Systems and Mechanics citation graph.

The joined Dynamical Systems and Mechanics citation graph is a directed graph of 215 nodes/topics and 2039 edges/citations with density 0.0439. It is weakly connected but not strongly connected (in fact, it includes 44 strongly connected components). Moreover, the joined Dynamical Systems and Mechanics graph contains 4 self-loops (internal citations) and the reciprocity coefficient is 0.5365 (i.e., 53.6% of citations are reciprocated).

We call common or shared nodes/topics the 9 topics which occur both in the list of Dynamical Systems topics and the Mechanics topics. In the graph visualization above, the common nodes/topics are denoted as stars, while the Dynamical Systems nodes/topics as triangles and the Mechanics nodes/topics as squares. Concerning the nodal colors that we are using, these colors are allocated according to the community (detected through the label propagation algorithm) into which the nodes/topics belong (in fact, 24 such communities have been detected).

Therefore, one can assign an attribute on the nodes/topics of the joined Dynamical Systems and Mechanics graph - let us call it “type”- such that the attribute of type takes the values:

  • DynS on 154 nodes/topics of the graph (i.e., those nodes/topics coming from the list of Dynamical Systems topics but not belonging to the list of Mechanics topics),
  • Mech on 52 nodes/topics of the graph (i.e., those nodes/topics coming from the list of Mechanics topics but not belonging to the list of Dynamical Systems topics),
  • DynS&Mech on 9 nodes/topics of the graph (i.e., those nodes/topics which co-occur in both the list of Mechanics topics and the list of Dynamical Systems topics).

4. Bridges and Brokers of Citations

Corresponding to the partition of nodes/topics according to the values that the attribute of type takes on them, one can consider a partition of the edges/citations of the joined Dynamical Systems and Mechanics graph. In fact, edges/citations among topics of the DynS type and topics of the Mech type are called bridges (or traversals) of citations among nodes/topics of the joined Dynamical Systems and Mechanics graph. In particular, one may partition the set of edges/citations of this graph as follows:

  • 858 citations among pairs of topics both of the DynS type,
  • 237 citations among pairs of topics both of the Mech type,
  • 34 citations among pairs of topics both of the DynS&Mech type,
  • 235 citations from topics of the DynS type to topics of the Mech type,
  • 248 citations from topics of the Mech type to topics of the DynS type,
  • 142 citations from topics of the DynS type to topics of the DynS&Mech type,
  • 106 citations from topics of the DynS&Mech type to topics of the DynS type,
  • 101 citations from topics of the Mech type to topics of the DynS&Mech type,
  • 76 citations from topics of the DynS&Mech type to topics of the Mech type.

Thus, those nodes/topics partaking in the set of bridges of citations are divided into the following types of topical brokerage:

  • 88 out-brokers of topics of the DynS type citing topics of the Mech type, which are also in-brokers of topics of the Mech type being cited by topics of the DynS type,
  • 108 in-brokers of topics of the DynS type being cited by topics of the Mech type,which are also out-brokers of topics of the Mech type citing topics of the DynS type,
  • 121 double-brokers (i.e., both out- and in-brokers) of topics citing/cited among topics of the DynS&Mech type and topics of the Mech type.

Similarly, one may decompose the out- and in-degrees of nodes/topics of the joined Dynamical Systems and Mechanics graph in two parts:

  • the out-in-degree of a node/topic i is the number of edges/citations from i to other topics in the same list of topics with the one where i belongs,
  • the out-out-degree of a node/topic i is the number of edges/citations from i to other topics of the different list of topics than the one where i belongs,
  • the in-in-degree of a node/topic i is the number of edges/citations citing i from other topics in the same list of topics with the one where i belongs,
  • the in-out-degree of a node/topic i is the number of edges/citations citing i from other topics of the different list of topics than the one where i belongs.

Clearly, for any node/topic i the sum of its out-in- and out-out-degree is simply its out-degree and the sum of its in-in- and in-out-degree is simply its out-degree. The following table shows these degrees for 14 topics of the joined Dynamical Systems and Mechanics graph (sorted according to their in-degree, i.e., the number of citations that these topics receive):

Degrees of 14 topics sorted in the in-degree.

The following table shows the centrality indices of 10 nodes/topics of the joined Dynamical Systems and Mechanics graph (sorted according to their PageRank):

Centrality indices of 10 topics sorted in the PageRank.

5. Attribute Assortativity

Moreover, the attribute assortativity coefficient of the joined Dynamical Systems and Mechanics graph with regards to the (discrete) attribute of type was computed to be equal to 0.1848. This means that this graph tends to be rather disassortotative (mixed) due to the following two reasons:

  1. There exist 448 bridges of citations among topics of the Dynamical Systems list and the Mechanics list, i.e., a considerable percentage (23.7%) of citations is “mixing” the two Wikipedia lists.
  2. The betweenness centrality indices of the common/shared nodes/topics are extremely low, as shown in the following table:
The betweenness centrality indices of the common/shared nodes/topics.

6. Topical Access, BFS-trees and Arborescences

Given two nodes/topics i, j in the joined Dynamical Systems and Mechanics citation graph G, we say that topic i accesses (or reaches) topic j if there exists a directed path (sometimes called dipath) from i to j. Such a path is a sequence of edges/citations which connect a sequence of nodes/topics from i to j, but with the added restriction that all the intermediate edges/citations are all directed in the same direction.

For example, the topic Attractor may reach the topic Symbolic_dynamics through a directed path of length 2 ([Attractor, Horseshoe_map, Symbolic_dynamics]). Notice that not any topic is accessed from any other topic (since the graph is not strongly connected). For instance, the topics Floquet_theory and Swarm_intelligence are not mutually accessible.

In fact, for any node/topic of the joined Dynamical Systems and Mechanics graph, one may construct the Breadth First Search tree (or BFS-tree) with root this topic, which reaches to other topics (but not all) following directed paths of minimum length (geodesic paths). The following is the BFS-tree from the topic Many-body_problem:

The BFS-tree with root the topic Many-body_problem.

However, focusing on the giant strongly connected component Gg of the joined Dynamical Systems and Mechanics citation graph G (Gg consists of 164 nodes/topics and 1805 edges/citations), one may detect the existing arborescences in Gg using Edmonds’ algorithm which is an algorithm for finding a minimum spanning tree in a directed graph. In graph theory, an arborescence is a directed graph in which, for a vertex i called the root and any other vertex j, there is exactly one directed path from i to j. In particular, for the giant connected component of the joined Dynamical Systems and Mechanics citation graph Gg, every topic of Gg, except the topic Dissipative_system, may be an arborescent root.

THOSE WHO WANT TO ACCESS DIRECTLY THE ZEPPELIN NOTEBOOK WHERE ALL THE ABOVE COMPUTATIONS/VISUALIZATIONS HAVE BEEN IMPLEMENTED, PLEASE GO TO:

URL: http://150.140.171.51:9980/#/

User Name: wikipedia

Password: wikipedia

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