01.12.2019  Regular article  Ausgabe 1/2019 Open Access
Tracing patterns and shapes in remittance and migration networks via persistent homology
 Zeitschrift:
 EPJ Data Science > Ausgabe 1/2019
Publisher’s Note
1 Introduction
Order  Country  Abbrev.  Order  Country  Abbrev. 

1  Afghanistan  AF  26  Lebanon  LB 
2  Armenia  AM  27  Malaysia  MY 
3  Azerbaijan  AZ  28  Maldives  MV 
4  Bahrain  BA  29  Mongolia  MN 
5  Bangladesh  BD  30  Myanmar  MM 
6  Bhutan  BT  31  Nepal  NP 
7  Brunei Darussalam  BN  32  Oman  OM 
8  Cambodia  KH  33  Pakistan  PK 
9  China  CN  34  Philippines  PH 
10  Hong Kong  HK  35  Qatar  QA 
11  Macau  MO  36  Republic of Korea  KR 
12  Cyprus  CY  37  Saudi Arabia  SA 
13  Dem. People’s Rep. of Korea  KP  38  Singapore  SG 
14  Georgia  GE  39  Sri Lanka  LK 
15  India  IN  40  State of Palestine  PS 
16  Indonesia  ID  41  Syria  SY 
17  Iran  IR  42  Tajikistan  TJ 
18  Iraq  IQ  43  Thailand  TH 
19  Israel  IL  44  TimorLeste  TI 
20  Japan  JP  45  Turkey  TR 
21  Jordan  JO  46  Turkmenistan  TM 
22  Kazakhstan  KZ  47  United Arab Emirates  AE 
23  Kuwait  KW  48  Uzbekistan  UZ 
24  Kyrgyzstan  KG  49  Vietnam  VN 
25  Laos  LA  50  Yemen  YE 
2 Dataset description
3 Methods
3.1 Directed clique complexes
3.2 Directed clique homology
3.3 Filtrations, persistence, and perpetuity
3.4 Maxtomin weight filtration
3.5 Variation on persistence barcodes
3.6 Another toy example
4 Patterns and shapes in Asian net migration and remittance networks
5 Summary and conclusion

Classifying 1cycles as circuits (no sink nor source), Type 1 (with unique sink and unique source), or Type 2 (with multiple sinks and sources). These different types of 1cycles encode different flow patterns. The definition of the directed clique also allows for the detection of the circuit with three vertices—a feature not commonly detected in other TDA methods.

Characterizing topological features of directed network by perpetuity. While persistent homology on undirected networks tracks longlived topological features along a filtration, persistent directed clique homology in addition detects topological features of the entire directed network by tracking cycle generators that never get killed off.

Distinguishing topological features of the same dimension by introducing coloring schemes for the barcodes. For the 0dimensional barcode, coloring the bars according to eventual connected component membership encodes the clustering of the vertices at the end of the filtration similar to the final clustering output of single linkage hierarchical clustering. For dimensions of at least 1, coloring the bars according to variation in the edge weights within the cycles captures information on similarity or disparity among internal flows in the cycle. It is worth noting that persistence is agnostic of such characteristic in detected cycles.