This document discusses quantifying hierarchy in complex networks. It begins by providing examples of hierarchical structures in the brain, mental lexicon, gene regulation and food webs. It then outlines requirements for measuring hierarchy, such as being based on global topology. It introduces Random Walk Hierarchy as a measure that meets these requirements. Random Walk Hierarchy tracks how information spreads through random walks on the network. The more a network resembles a hierarchy, the more information will converge on root nodes. The measure is applied to different network structures and real-world examples.

1. HOW TO QUANTIFY HIERARCHY?
Dániel Czégel
Dept. of Biological Physics,
Eötvös Loránd University, Budapest
Dept. of Cognitive Science, TU Budapest
IFISC

2. UBIQUITY OF HIERARCHICAL STRUCTURES
2

3. MULTI-LEVEL MODULARITY OF THE BRAIN
3
Meunier, D., Lambiotte, R., Fornito, A., Ersche, K. D., & Bullmore, E. T. (2009).
Hierarchical modularity in human brain functional networks. Frontiers in neuroinformatics, 3.

4. MENTAL LEXICON
4
De Deyne, S., Verheyen, S., Perfors, A., & Navarro, D. J. Evidence for widespread thematic structure in the mental lexicon. (2015)

5. GENE REGULATION, FOOD WEB
5
Ma, H. W., Buer, J., & Zeng, A. P. (2004).
Hierarchical structure and modules in the Escherichia coli
transcriptional regulatory network revealed by a new top-down approach.
BMC bioinformatics, 5(1), 199.

6. LEADER-FOLLOWER & DOMINANCE HIERARCHY
6
https://www.youtube.com/watch?v=FVBiR7pl71U&feature=youtu.be
https://www.youtube.com/watch?v=QMpIvlKPk3k&feature=youtu.be
Nagy, M., Vásárhelyi, G., Pettit, B., Roberts-Mariani, I., Vicsek, T., & Biro, D. (2013).
Context-dependent hierarchies in pigeons. Proceedings of the National Academy of Sciences, 110(32), 13049-13054.

7. 7
1. What is hierarchical
organization exactly?
2. Why is hierarchical
organization so efficient?

8. WHAT IS HIERARCHY?
8
 framework: directed (weighted/unweighted) networks
 hierarchy measure:

9. OUTLINE
 1. Four examples of H
 2. Intuitive requirements for H
 3. Random Walk Hierarchy
9

10. 1. KRACKHARDT’S HIERARCHY
 1. Reachability graph R(G)
 2.
Krackhardt, D. (1994). Graph theoretical dimensions of informal organizations.
Computational organization theory, 89(112), 123-140.
10

11. 2. LINK-FLOW HIERARCHY
Luo, J., & Magee, C. L. (2011). Detecting evolving patterns of self‐organizing networks by
flow hierarchy measurement. Complexity, 16(6), 53-61.
11
~ Reynolds Number

12. GLOBAL REACHING CENTRALITY
 1. Reachability graph R(G)
 2.
 ~Heterogenity in reachability
Mones, E., Vicsek, L., & Vicsek, T. (2012). Hierarchy measure for complex networks.
PloS one, 7(3), e33799.
12

13. TREENESS, FEEDFORWARDNESS, ORDERABILITY
 3 dim morphospace; each dim: different aspect of hierarchy
Corominas-Murtra, B., Goñi, J., Solé, R. V., & Rodríguez-Caso, C. (2013). On the origins of hierarchy in complex
networks. Proceedings of the National Academy of Sciences, 110(33), 13316-13321.
13

14. 14

15. WHAT IS A HIERARCHY MEASURE, THEN?
 1. Based on the global topology; statistics of local
properties (e.g., degree distribution) are not enough!
 2. Intensive:
 3. Minimal for networks where the nodes are
indistinguishable (e.g., directed cycle, full graph)
 4. Maximal for acyclic networks; in particular,
15

16. HOW TO MEASURE HIERARCHY?
 Goal: to define a hierarchy measure , that
 meets all of these requirements
 is easily generalizable to weighted networks
 is computationally efficient
16

17. RANDOM WALK HIERARCHY
 From every node, track the
information backwards through
every possible path
 ~ Random walk on the nodes
 Stationary distribution ~ how
much the given node is the source
of the overall information
 Then:
17
Czégel, D., & Palla, G. (2015). Random walk hierarchy measure:
What is more hierarchical, a chain, a tree or a star? Scientific Reports. (arXiv preprint arXiv:1508.07732.)

18. RANDOM WALK HIERARCHY
 Transition probabilities?
 node j: which node the info have
come from?
 node i: limited resource for
communication
 The two effects combined:
18

19. RANDOM WALK HIERARCHY
 This way: eventually all random
walker would converge to the root
node(s).
 However, we want to measure the
overall structure, not just the
number of root nodes!
 -> Every step: inject the same
amount of random walkers to
every node, then normalize!
19

20. RANDOM WALK HIERARCHY
 The process:
 1. Do infinitely many times:
 a) Inject f/N random walkers to each node:
 b) Transition:
 c) Normalize, i.e., divide by 1+f. (~decay of random walkers)
 2. From the stationary distribution:
20

21. RESULTS: ACYCLIC NETWORKS
 Free parameter: characteristic decay length (or time)
of random walkers:
21

22. RESULTS: REAL NETWORKS
 What kind of structures
are the most (and least)
hierarchical ones?
 -> Compared to their
randomized counterparts:
22

23. APPLICATION: FOOD WEB ANALYSIS
23

24. THANK YOU!
24
The research was partially supported by the European Union and the European Social Fund
through project FuturICT.hu (grant no.:TAMOP-4.2.2.C-11/1/KONV-2012-0013)
and by the Hungarian National Science Fund (OTKA K105447).
czegel_d@yahoo.com