A Network Science Summer Course for High School Students
AA Network Science Summer Course for HighSchool Students
Florian Klimm ∗ and Benjamin F. Maier ∗ University of Oxford, Oxford OX2 6GG, United Kingdom Department of Statistics, University of Oxford, 24-29 St Giles’, OxfordOX1 3LB, United Kingdom Department of Mathematics, Imperial College London, London, SW72AZ, United Kingdom MRC Mitochondrial Biology Unit, University of Cambridge, CambridgeBiomedical Campus Hills Road, Cambridge, CB2 0XY, United Kingdom Department of Physics, Humboldt Universit¨at zu Berlin,Newtonstraße 15, D-12489 Berlin, Germany Robert Koch-Institute, Nordufer 20, D-13353 Berlin, GermanyMay 7, 2020
Abstract
We discuss a two-week summer course on Network Science that wetaught for high school pupils. We present the concepts and contents of thecourse, evaluate them, and make the course material available.Keywords: outreach, mathematics, physics, sociology, summer course, teach-ing
Teaching network science [1] has been identified as an important endeavor, becauseit is a concept usually omitted in high school curricula [2, 3, 4]. Hence, scientists ∗ Both authors contributed equally. a r X i v : . [ phy s i c s . e d - ph ] M a y rganized outreach events for teenagers to bring them into contact with the studyof networks, notably in England [5] and Spain [6]. These outreach events consistedof an introductory presentation, followed by work in smaller groups on topics suchas ‘node importance’, ‘disease spread and vaccination strategies’, and ‘why yourfriends have more friends than you do’. Based on the success of these half-dayevents we decided to design a two-week summer course for German high schoolpupils with the topic ‘Networks and Complex Systems’ (German: Netzwerke undkomplexe Systeme ). The course was part of an established German summer schoolcalled
Deutsche Sch¨ulerakademie (German Pupils Academy), whose concepts willbe introduced in Section 2.Due to the longer duration of the course and the higher age of the participantswe were allowed to extend the covered material from the shorter outreach events,go into more mathematical detail with the topics, see Section 3, and include somecomponents of programming with Python. In Section 4 we discuss our approachesand evaluate to what extent they were successful or might need further improve-ments. We make our teaching material available online [7].
Deutsche Sch ¨ulerakademie
The
Deutsche Sch¨ulerakademie (DSA) is an extracurricular summer school for mo-tivated and gifted pupils in the final two years of German high-school [8]. For morethan twenty years every summer multiple of these academies have been organizedwhere each of them consists of six courses for 15–16 pupils each. Our course waspart of the academy ‘Roßleben 2016’ and set in a monastery school in a smalltown in Thuringia. The courses concurrent with ours covered diverse academictopics from analyzing Richard Wagner’s operas to String Theory. Approximatelyone hundred pupils participated in the courses and were encouraged to organizeacademic and non-academic extracurricular activities. Over the two weeks the par-ticipants spent a total of approximately sixty hours in their courses. In this timethey worked under guidance of the lectures. Furthermore, they had to prepare talksfor the participants of other courses explaining the topics of their course, a conceptcalled
Rotation (see Subsection 3.8), and create a written summary of the learnedmaterial, called
Dokumentation (see Subsection 3.9).
In the course we used different didactic elements. Some of them, as the
Rotation and the
Dokumentation , were framed by the DSA structure, others, as pupil presen-tations and programming exercises, were chosen deliberately by us because they2eemed appropriate for conveying the material.
We received a list with contact details of all fifteen participants two months beforethe academy. We expected some heterogeneity in existing knowledge because thepupils were in one of the final two grades in German high school. This discrepancyis enhanced by the federal education system in Germany which gives different focito subjects in different states [9]. To assess the knowledge of the pupils we con-ducted a non-anonymous online survey which was answered by all fifteen partici-pants. We mainly assessed the mathematical knowledge in eleven brief questions,for example, ‘Do you know what a matrix is?’. The results gave us an overviewover the pupils’ abilities: All pupils knew how to operate with vectors and twothirds knew how to work with matrices, almost all of them were able to differenti-ate, but less knew about Riemann integration. The most heterogeneous knowledgewas given for programming with a third of the participants having no programmingexperience and others knowing Python or C-family programming languages rudi-mentarily. Three quarters did not know L A TEX, which we intended to use for the
Dokumentation . We also included an open-ended question about their motivationto choose this particular course. The pupils stated that they were especially inter-ested because the course connects mathematical tools with real-world applicationsand targets questions from different sciences as physics and biology.
Based on the results from the questionnaire we provided the participants with aso-called
Reader as preparation for the course. The main purposes of this doc-ument was to give all participants a common level of mathematical backgroundand to provide them with some programming experience. For this we drew fromtwo sources, (i) the Python online course ‘Spielend programmieren lernen’ (engl.‘Learning Programming Effortlessly’) by the Hasso-Plattner-Institute [10, 11] andone chapter of ‘Physik mit Bleistift’ (engl. ‘Physics with a Pencil’) [12]. The lattergave a brief introduction to vectors and differentiating. In the post-course evalu-ation the pupils were, however, not satisfied with the material as it focused toomuch on mathematical tools that we used only briefly and not enough on learningprogramming. 3 .3 Pupil Presentations
Each of the participants gave a presentation during the course. In preparation ofthe course we suggested a topic to each of them based on their interests as deducedfrom the survey and provided them with reading material. The talks were supposedto be 10–15 min long and we advised the participants to approach the topics froma phenomenological side. For example, Turing patterns were introduced under thetopic ‘how leopards get their patterns’. Our aim was to discuss the mathemat-ics behind the phenomena together after the introduction given by the pupils. Anoverview over all fifteen topics and a brief assessment of their suitability is shownin Table 1. We noted that some pupils considerably overrun the allocated time. Wedecided to not interrupt their talks but let them know afterwards that this might notbe appropriate in other circumstances.
Lectures were a fundamental part of our course. Usually, they covered conceptsthat were earlier introduced by the pupils in their presentations. We subsequentlyformalized the mathematical description and went into more detail. We brieflysummarize the presented topics below.
This lecture was built on the discussion about the representation of graphs as edgelists and adjacency matrices (see Subsubsection 3.5.1). We proceeded to introducesimple network measures as the number of edges m , node degree k , path length L and density ρ . We then discussed some deterministic synthetic networks as theempty graph N n , the path graph P n , the complete graph K n , and the complete bipar-tite graph K n , n and derived formulas of the network measures in dependence of thegraph size n . Later in a programming exercise the student validated these formulasnumerically. Our aim was to teach the pupils network science and classical graph theoreticalconcepts alike.
Graph coloring was an appropriate field for the latter because theunderlying problems are easily accessible, while the proofs can be challenging. Weintroduced node coloring and edge coloring but focussed on the former. The pupilsparticularly enjoyed proving theorems and challenged our proofs if they were toosloppy. In the last part of this lecture we also discussed coloring algorithms, partic-ularly greedy coloring . One task the pupils enjoyed was to compute an upper bound4able 1: Overview of the topics presented by pupils in talks.Topic CommentFour color map problem Good introduction to graph coloring. Allows adiscussion about computer-assisted mathematicalproofsSeven bridges of K¨onigsberg Classic and easily accessible problemTraveling salesman problem Allows introduction to computational complexityKirchhoff’s circuit laws Well-suited to introduce pupils without a physicsbackground to physical phenomenaPercolation theory Easily accessible with e.g. forest tree density andforest firesCybernetics Participant introduced a school project’s workNeuronal networks Interesting topic but too complex for deep discus-sionGraph isomorphism problem Allows discussion of P vs. N P
Disease spreading Excellent for tying together network phenomenaand dynamicsPredator-prey interactions Good introduction to dynamical systems but toocomplex for deep discussionTuring patterns Accessible with, e.g., animal fur patterns but toocomplex for deep discussionDeterministic chaos The logistic map is well-suited for simple analy-ses of chaotic phenomenaFractals Nicely accessible with, e.g. coast lines andKoch’s curveFractal dimension Counter-intuitive but easily accessible with boxcounting method5f the chromatic number χ of a symmetrize food web [13] consisting of n =
63 an-imals in the Everglades by applying greedy coloring many times to different ordersof the nodes. We motivated this question by asking how many different cages arenecessary if a zoo director would like to keep all the animals without one preyingon the other.
The discussion of social networks is well-suited to motivate the real-world appli-cability of network science to the pupils. The participants were rather astonishedby the so-called friendship paradox . We followed Strogatz’s excellent explanationin the New York Times [14].We proceeded by discussing the existence of clusters and other mesoscalestructures in networks. Unfortunately, for privacy reasons, Facebook does not al-low downloading your own friendship network anymore. This used to be an excel-lent way to make pupils familiar with various network characteristics as communi-ties and centrality measures. Therefore, we had to fall back on other available data,e.g. the social graphs from M
OVIE G ALAXIES [15] and B. F. Maier’s Facebookfriends network from 2014 [16]. As another example of a small-world networkwe played the ‘Wikipedia-Game’ to demonstrate short link paths between appar-ently unrelated articles. Its objective is to find the shortest link-path between tworandom Wikipedia articles by only following links on the encountered Wikipediapages [17].
A lot of the pupils showed interest in physics. Therefore we dedicated a day tothe discussion of resistor networks. This topic was first introduced by a pupil’spresentation on Kirchhoff’s circuit laws. Subsequently we aimed at computing theeffective resistance of more complex networks where we followed the discussionin [1]. In particular, we proceeded as follows (see also Exercise Sheet 3).At first we motivated eigenvalues and -vectors of matrices with a rudimentaryexample of a growing population of linearly interacting species. Using this conceptwe were able to motivate the inverse matrix and what it means when a matrix haseigenvalue zero (the concept of singularity), which was necessary to explain thereduction of the graph Laplacian to make it invertible.This topic was very difficult for some of the younger pupils which were not asversatile in working with matrices as the older ones. However, some of the moreengaged participants particularly enjoyed those discussions.6 .4.5 Dynamic Systems and Fractals
In addition to traditional network science we also discussed dynamic systems , e.g.simple exponential growth or more complex Lotka-Volterra equations. The pupilsenjoyed the discussion, although an introduction of mathematical prerequisites,e.g. ordinary differential equations (ODEs), was necessary. Even though introduc-ing those prerequisites took a rather long time, we think it was valuable for thepupils to learn about ODEs using very simple systems. In particular we showedhow to derive the equation of motion for logistic growth given the example of themean-field model of the susceptible-infected infection model. Here, we startedwith discrete time and probabilistic considerations and arrived at the final equationby going to continuous time using ∆ t → In addition to longer lectures we also used a couple of smaller modules, whichhad a stronger component of pupil involvement. We will briefly introduce some ofthem below.
This module was one of the first elements of our course. We separated the pupilsinto two groups. Each of them was given a printout of a small labeled graph withthe aim to communicate the graph in written form to the other group such that theywere able to recreate the same graph.The pupils quickly identified ways to achieve this, for example, as an edge list.After joining both groups together we discussed different approaches and their7imitations, for example, that edge lists do not allow the representation of isolatednodes but an additional node set is required. This naturally led to the introduction ofother representations such as adjacency matrices. One of the graphs was dense andthe other sparse, which lead to the discussion of space and memory considerationswhen working with large graphs in different representations.This simple task was very successful because it clearly motivates the definitionof a graph as node and edge set. Giving the mathematical definitions exclusivelycan intimidate pupils who are not familiar with the notion of sets.
After introducing Erd˝os-R´enyi random graphs G ( n , p ) we gave the participants thetask to create undirected random graphs with the help of six-sided and twenty-sideddice. We discussed how to generate graphs with different connection probabilities p and also identified an appropriate number n of nodes. We settled for n = p ∈ { / , / , / , / } . Each group drew their final graph on theblackboard (see Fig. 1) and we subsequently discussed their properties, e.g. thesize of their giant connected component and their degree distributions P ( k ) , andcompared them with theoretical expectations.This task was praised as ‘very fun’ by the pupils. We evaluated it as successfulfrom a didactical perspective because it playfully connects known material, suchas dice rolls and probabilities, with new concepts, such as random graphs. Further-more, it allows the transition to the creation of random graphs using pseudorandomnumber generators of a computer. However, the introduction of the binomial coef-ficient to explain the random graph’s degree distribution was rather challenging forthe majority of the pupils. In addition to synthetic networks and the networks created from data that we pro-vided, we wanted the participants to experience a more realistic experience of net-work creation. For this we gave them access to a printout of the floor plan of the
Klosterschule Roßleben , which hosted the academy (see Fig. 2). They had the taskto create a network in which nodes represent the n =
161 rooms and edges standfor direct connections between them.We let the pupils organize amongst themselves how to achieve this and howto overcome the problems they faced. They decided to split into groups, each ofthem covering one floor of the building. They labeled each room with a unique8igure 1: Four G ( n = , p ) random graphs as generated by the participants byrolling dice. A twenty-sided dice was used for the probabilities p = /
20 and p =
10 and a six-sided dice for p = / p = / We used the created school building network for the simulation of compartmen-tal models in epidemiology, specifically the susceptible-infected dynamics (SI), asshown in Figure 3. We invented a scenario where another course of the academy,working with bacteria germs, let a hyperinfective strain escape which slowly in-fects the whole building room by room. The pupils then compared the results withthe mean-field approximation introduced in the lectures.We then discussed three vaccination strategies: (i) random vaccination, (ii)vaccination of nodes with high betweenness, and (iii) random next-neighbor vac-9
35 E Y
Figure 2: Left: The monastery school building in Roßleben [18]. Right: Networkillustration of the floor plan of the former monastery school as created by the pupils.Marked are the eastern staircase ‘E’ which has a high betweenness centrality andis close to the epidemic outbreak room ‘A35’, as well as the yard ‘Y’.Figure 3: Left: Simulated SI dynamics on the monastery room network (green)in comparison with a mean-field approximation (blue). Right: Spread of the dis-ease as number of infected rooms over time for random vaccination of one room(green, (1)) and targeted vaccination of staircase ‘E’ (blue, (2)). Both figures wereproduced by one of the participants and thus have German labels. Horizontal axesgive the time steps and vertical axes show the number I ( t ) of infected rooms.cination. The temporal development of the number I ( t ) of infected nodes for theformer two are shown in Figure 3 and the pupils were able to identify that thebetweenness-vaccination is slowing the spread of the disease. The final steadystate, however, shows the same number of infected rooms.This module successfully connected different aspects of the course, between-ness centrality and SI dynamics, with a real-world application. Furthermore, itdemonstrated the effectiveness of vaccination to the pupils and we were able todiscuss the aspect of herd immunity. 10igure 4: Normalized characteristic path length L ( p ) / L ( ) and normalized cluster-ing coefficient C ( p ) / C ( ) for the family of randomly rewired graphs with rewiringprobability p , as discussed by Watts and Strogatz [19]. This figure was producedby a pupil. Besides the introduction of essential concepts for networks we also wanted tofamiliarize the pupils with scientific literature. For this we chose the first halfof Watts’ and Strogatz’ famous paper
Collective Dynamics of ‘small-world’ net-works [19]. The task we gave them consisted of two parts: (i) read the paper andexplain the definition of clustering coefficient and (ii) recreate Figure 2, whichgives the characteristic path length and clustering coefficient of a lattice graph un-der rewiring.This task was challenging but fruitful for the pupils. All of them quickly under-stood the contents of the paper. For the programming exercises they needed morehelp. We were, however, rather impressed by the pupils’ abilities. Reproducing theresults of an important paper, as shown in Figure 4, led to a feeling of achievementas well as motivation for the upcoming lectures. Note that for time limitations weomitted averaging over a large number of realizations, which was used in the orig-inal paper. However, we discussed with the students the impact of and reason forsuch averaging procedures. 11 .6 Problem Sheets
We created six problem sheets that gave the pupils the chance to apply some of thelearned concepts. We initially planned a problem sheet per day but quickly realizedthat our schedule was too ambitious. The problem sheets usually consisted of amixture of pen-and-paper analytical questions, as well as programming exercisesas outlined in the next Subsection. We have made all problem sheets in the originalGerman and a translated English version available online [7].
We used Python as the main and only programming language because it is opensource and easy to learn and implement. In addition to the default Python func-tions we used mainly three libraries,
NETWORK X, MATPLOTLIB , and
NUMPY . Wealso used Gephi [20] for the analysis of social networks and as an easy way tographically access them.Given the results from our initial questionnaire we were aware that the pupilshad very different levels of knowledge of programming. To tackle this problem wecreated two groups, one working on their own on the programming exercises, theother receiving closer guidance.The programming tasks were very diverse in their difficulty. We started withthe creation of the synthetic graphs discussed earlier in the lecture. To make thetask more feasible we gave the code to create a path graph P n of n nodes (see List-ing 1). After a discussion of the code and answering questions like ‘Why is theFOR loop progressing from 0 to n − N n , the cycle graph C n ,and the complete graph K n . We registered that a subset of the pupils had problemswith this task and a careful discussion of the code was necessary. A line-by-linedismantling of the script on the blackboard was especially helpful to increase un-derstanding.Other tasks included the illustration of the networks with the NETWORK X draw function and the computation of network metrics such as degree and betweenness.For each exercise we also had one or two rather complex tasks, such as the imple-mentation of the Dijkstra-algorithm to keep even the most advanced participantsengaged. We clearly communicated that we did not expect the pupils to solve allavailable tasks to ensure they did not feel insufficient. We discussed the findingsof all tasks with the whole class. import numpy as np def path_graph ( n ) : A = np . zeros (( n , n ) ) for u in range (0 , n -1) : A [u , u +1] = 1 A = A + np . transpose ( A ) return A path_10 = path_graph (10) print ( path_10 ) Listing 1: Python code for creating a path graph P with tehn nodes as provided tothe students as starting point for further programming, as the creation of a completegraph C with eight nodes. The
Rotation is an inherent part of each DSA academy. It occurs at half-waythrough the course, thus after approximately seven days of covered material. It isa half-day event in which each course has to organize twenty minute presentationsfor pupils of other courses.We planned the talks in our course and discussed which topics were appropriate13o present to the other pupils. Our participants identified that the fundamentalmathematical definitions should be covered and social networks would be a goodexample to make the topic accessible. They therefore covered the small-worldphenomenon and the friendship paradox.The feedback they received on the presentations was very positive. The partic-ipants as well as the lecturers of the other courses were astonished by how applica-ble and approachable mathematics can be. We think that network science itself canserve as a ‘figurehead’ topic inside mathematics because it easily contradicts theusual stereotypes of mathematics being incomprehensible and overly complicated.
The
Dokumentation (German for documentation) is another inherent part of eachDSA summer school. It is a written report in which each course summarizes thecovered material. Each participant contributed a small one to two page part, whilewe served as editors.Despite most participants having no knowledge of L A TEX we decided to writethe whole report with S
HARE L A T E X, an online L A TEX editor that allows real-timecollaboration and online compiling [21]. This came with the known advantagesof L A TEX such as easy incorporation of formulas, figures, and citations. It further-more allowed the collaboration of all pupils and removed compatibility issues. Weintroduced the use of L A TEX in a special session and accompanied the whole writingprocess to efficiently dismantle encountered problems.The first step was the discussion of an overall structure of the Dokumentation.We did not impose any approach but discussed different potential approaches withthe pupils. The final structure was similar but not identical to the one we chose forthe course overall. The second step was the individual discussion with each pupilabout the material they wanted to cover and what illustrations were necessary. Allfigures in this paper were created by pupils as part of the documentation.The creation of the Dokumentation was a very novel challenge for the partici-pants. One encountered problem was the space limitation, such that the presentedinformation had to be condensed, while staying comprehensible. Furthermore, theappropriate referencing was unfamiliar to the pupils. Most of them did not en-counter these aspects before and thus found them challenging. Overall however,we were impressed with the result and needed only two to three iterations of editingper pupil before the texts met the appropriate quality. There are plenty of other L A TEX online editors available, such as Overleaf, Papeeria, and Au-thorea. Conclusions
We were overall content with the course, the pupils’ involvement, and their progress.We think that network science is a appropriate topic to give pupils an introductionto university-level science and mathematics, as we noted from the feedback duringthe Rotation, even for those that are not naturally interested in such topics.We encountered the disparity in background, especially concerning program-ming, as the most challenging part, because it made it more difficult to keep every-body engaged. As two instructors, we were able to split the class into a beginnerand advanced half, which helped in some cases. A better, more detailed, discus-sion of programming for those that had never programmed before might have beenfruitful.A crucial part of our teaching was the flexibility in the covered topics. Some-times the pupils had problems with concepts that we thought were taught at school,e.g. the binomial coefficient, while other new concepts took much less time to dis-cuss than we anticipated.One weakness of our preparation was the reader. We should have tailored itmore to the specific course content and given ‘homework’ problems instead ofreading material only. This would have allowed us to challenge the pupils withmore advanced topics, such as more complicated differential equations, and thusgo into more detail in the course itself.Although the discussion of a broad range of topics was praised by the pupils wefelt that a focus on networks under omission of a majority of the dynamic systemspart would have been beneficial. The latter is an interesting and complex branchof science and would provide enough material for a course by itself. In retrospect,we would definitely drop the discussion of Lotka-Volterra systems, Turing patterns,chaos, and fractals while a discussion of stability and infection dynamics might stillbe interesting as an example for the application of network science to real-worldproblems.We also want to note that the organization of such a course was also very bene-ficial for ourselves. It allowed us to recapture many different topics in graph theoryand network science and think about ways to motivate and connect them with eachother.Finally, as well as [5, Harrington at al. ] and [6, Sanchez et al. ], we encouragefellow network scientists to use the presented material to tailor their own courseto spread the fascination about networks and dynamic systems amongst teenagers,and to contact us in case there are any questions.15
Acknowledgements
We would like to thank all the pupils who participated in the course for their en-gagement and criticism. It was a great pleasure working with them for the twoweeks of the program. Further thanks goes to the central organization of the DSAin Bonn, as well as the academy administration and fellow course organizers at theDSA Roßleben 2016. We thank our doctoral advisers Mason A. Porter and DirkBrockmann for supporting us in this endeavor and for their fruitful discussion. Weare grateful to Sarah M. Griffin for proofreading the manuscript.
This work is supported by the EPSRC and MRC (FK, grant numbers EP/L016044/1,EP/R513295/1, EP/N014529/1) and by the Joachim Herz Stiftung (BFM).
References [1] Mark Newman.
Networks: An Introduction . Oxford University Press, 2010.[2] Catherine Cramer, Mason A. Porter, Hiroki Sayama, Lori Sheetz,and Stephen Uzzo. Network literacy – essential concepts andcore ideas. https://sites.google.com/a/binghamton.edu/netscied/teaching-learning/network-concepts, 2015.[3] Catherine Cramer, Lori Sheetz, Hiroki Sayama, Paul Trunfio, H Eugene Stan-ley, and Stephen Uzzo. Netsci high: bringing network science research tohigh schools. In
Complex Networks VI , pages 209–218. Springer, 2015.[4] Hiroki Sayama, Catherine Cramer, Mason A. Porter, Lori Sheetz, andStephen Uzzo. What are essential concepts about networks?
Journal ofComplex Networks , page cnv028, 2015.[5] Heather A Harrington, Mariano Beguerisse-D´ıaz, M Puck Rombach, Laura MKeating, and Mason A Porter. Commentary: Teach network science toteenagers.
Network Science , 1(02):226–247, 2013.[6] Angel S´anchez and Cristina Br¨andle. More network science for teenagers. arXiv preprint arXiv:1403.3618 , 2014.[7] Florian Klimm and Benjamin F. Maier. Teaching material forsummer school on network science for high school students.https://github.com/floklimm/DSAMaterialNetworks, 2017.168] Elke V¨olmicke. Deutsche Sch¨ulerakademie, 10 2017.[9] Bridgette Lohmar and Thomas Eckhardt. The education system in the federalrepublic of germany 2012/2013: A description of the responsibilities, struc-tures and developments in education policy for the exchange of informationin europe. In
Bonn: Secretariat of the Standing Conference of the Ministersof Education and Cultural Affairs of the L¨ander in the Federal Republic ofGermany , 2014.[10] Christoph Meinel and Stefanie Schweiger. openHPI–Das MOOC-Angebotdes Hasso-Plattner-Instituts. In
Veranstaltungen 4.0 , pages 195–226.Springer, 2017.[11] Martin von L¨owis. Spielend Programmieren lernen!, 10 2017.[12] Hermann Schulz. Physik mit Bleistift.
Verlag Harri Deutsch , 2004.[13] Robert E. Ulanowicz and Donald L. DeAngelis. Network analysis of trophicdynamics in south florida ecosystems.
US Geological Survey Program on theSouth Florida Ecosystem , 114, 2005.[14] Steven H. Strogatz. Friends you can count on.
New York Times , 2012.[15] Jermain Kaminski, Michael Schober, Raymond Albaladejo, Oleksandre Zas-tupailo, and Cesar Hidalgo. Moviegalaxies-social networks in movies.
Con-sult´e sur http://moviegalaxies. com , 2012.[16] Benjamin F. Maier and Dirk Brockmann. Cover time for random walks onarbitrary complex networks.
Physical Review E , 96(4):042307, October 2017.[17] Robert West, Joelle Pineau, and Doina Precup. Wikispeedia: An online gamefor inferring semantic distances between concepts. In
Proceedings of theTwenty-First International Joint Conference on Artificial Intelligence , pages1598–1603, 2009.[18] Michael Sander. Photography of the monastery school roßleben.” https://de.wikipedia.org/wiki/Datei:Klosterschule_Ro%C3%9Fleben.JPG ”, 8 2010.[19] Duncan J Watts and Steven H Strogatz. Collective dynamics of ’small-world’networks.