Topology of the Erasmus student mobility network

Transcription

1 Topology of the Erasmus student mobility network Aranka Derzsi, Noemi Derzsy, Erna Káptalan and Zoltán Néda

2 The Erasmus student mobility network Aim: study the network s topology (structure) and its characteristics for the directed and non-directed graphs, as well for the weighted and non-weighted graphs reveal the network of professional connections between universities Dataset: Data for year 2003 The number of universities that take part in the program in this period: 2333 The number of exchanged students: The students that take part in an Erasmus mobility form a connection with two universities: in which they study and the one they visit within the framework of the Erasmus program

3 The Erasmus student mobility network o Nodes: universities that are members of the Erasmus framework (2333) o Connections: students that benefit of an Erasmus scholarship and travel from one university to the other Two directed networks derive depending what type of connection is considered: Incoming: university A Outgoing: university A university B university B Two undirected networks derive In or outgoing: university A university B In and outgoing: university A university B For each case a weighted network can be constructed, as well (the strength of the links is proportional with the number of students in the given exchange)

4 The NNESM network topology large number of connections: L= large number of nodes: N=2333 NOT informative visualization

5 The Erasmus Network k-degree component of the NNESM network for k=55. The 180 nodes of this component are presented in hierarchical arrangement. The most highly connected 10 hubs are indicated by their names.

6 Topological properties of the NNESM network degree distribution with exponential tail! Degree-rank plot on semi-log axis for the NNESM network (filled circles). The thick continuous line fits the exponential tail generated by the highly connected universities (more than 50 connections). high global clustering coefficient: CNNESMg=0.183 high local clustering coefficient: CNNESMl=0.292 several connected components: a giant one (99% of nodes) with average distance lnnesm=2.91

7 Topological properties of the NNESM network Cumulative degree distribution function r(k) is the rank-degree distibution function Cumulative distribution exponential => the tail of the degree distribution is also exponential

8 Selective linking (assortative mixing) NO selection based on nation or study language of the participating universities Do highly connected universities tend to connect to universities with large number of connections? Degree of a node neighbors (a) and the local clustering coefficients of the nodes (b) plotted as a function of the degree of a node (filled circles). The continuous line on both panels indicate averages of the points taken with a moving average of length 50 on the horizontal axes. Averages calculated for the network obtained from the configuration model appear with dashed black lines. NO assortative mixing present in the system!

9 Configuration model the model has same network size and degree distribution free-endpoint links Step 1: Preliminary link allocation procedure randomly select two free-end links belonging to different nodes and connect them multiple connections between two nodes are allowed process repeated until all free-end links are connected Step 2: Random local rewiring rewire randomly multiple connections process repeated until all multiple connections are eliminated from the system The generated network s topology few connected components C local C global l NNESM Model The model reproduces well the NNESM network s characteristics!

10 Understanding the exponential degree distribution Our assumption: k cn ~ i i n i ki degree in the NNESM network of a university with rank r=i ni the size of this university (total number of students) c proportionality constant the tail of the size-rank plot of the universities r(n) should follow an exponential trend

11 Random sampling universities from the NNESM network arranged in random order number of students for 150 universities looked up on the internet Degree of the universities in the NNESM network as a function of their sizes. Open circles are results for random sampling on 150 universities, and filled circles present the average for an interval of length 5000 on the horizontal axes. The continuous line is a linear fit of the data points. The tail of the size-rank distribution also follows an exponential trend

12 University size-distribution no data available for European universities University size-distribution by ranking plots: rank-size distribution of the largest UK (a) and US (b) universities. Exponential trend in the system! Exponential size-distribution observed in other group-sizes: Fish-school (J.J. Anderson, Fish. Bull. 79 (1981) ) Cattle-farm (C.T.K. Ching, Am. J. Agric. Econ. 55 (1973) ) (E. Bonabeau, L. Dagorn, Phys. Rev. E 51 (1995) R5220 R5223) (H.-S. Niwa, J. Theoret. Biol. 224 (2003) ) maximum entropy principle

13 The directed and weighted network study Approach: take into account the direction of the mobility between pairs of universities (directedness) include the number of students traveling on each of these directed links (weight) construct two networks based on these ideas: IN (incoming links) network OUT (outgoing links) network construct a randomly directed network by using the undirected configuration network assign weights to the connections

14 The directed and weighted network study Part of the weighted ESM network: the figure shows the largest connected component obtained by using links on which there are at least 15 student exchanges in year This component contains 149 elements, the highlighted nodes are the hubs characterized with the most intense student mobilities.

15 Some characteristics of the directed and weighted network unidirectional links: 2/3 (random direction) bidirectional links: 1/3 partitioning the number of students on the directed links Scaling of the total number of students sent out from a given university as a function of the number of OUT connections the university has. The continuous line shows a power-law fit with exponent 1.17.

16 Adapting the configuration model for the directed and weighted network randomly change the links to symmetric or asymmetric type of connections Partitioning methods: 1. randomly assign the k OUT links 2. preferential linking Rank of universities as a function of their IN and OUT degree for the directed ESM network (filled symbols) and for randomly directed configuration network (open symbols). Comparison of weight distributions for the directed and weighted networks.

17 Conclusions NOT scale-free social network degree distribution function with exponential tail node degree proportional with university size: k ~ N exponential size-distribution explained with maximum entropy principle revealed the most important hubs in network simple random connection models are helpful in understanding the topology A. Derzsi, N. Derzsy, E. Káptalan and Z. Néda; Topology of the Erasmus student mobility network, Physica A, vol. 390, pp (2011)