**The citation network among Wikipedia pages on Dynamical Systems and Mechanics**

“When a documentdicites a documentdj, we can show this by an arrow going from the node representingdito the document representingdj. 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.

**The Dynamical Systems Citation Graph**

So, let us first display 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).

**2. The Mechanics Citation Graph**

Next comes 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 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):

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

- 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.
- The betweenness centrality indices of the common/shared nodes/topics are extremely low, as shown in the following table:

**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*:

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.