1. Dominance hierarchy of
worker ants as directed
networks
Hiroyuki Shimoji (Univ. Ryukyus, Japan & Univ. Tokyo, Japan)
Masato S. Abe (Univ. Tokyo)
Kazuki Tsuji (Univ. Ryukyus)
Naoki Masuda (University of Bristol, UK)
Ref: Shimoji, Abe, Tsuji & Masuda, J. R. Soc. Interface, in press (2014);
arXiv:1407.4277; data available online

2. Dominance hierarchy
• Pecking order of hens (Schjelderup-Ebbe, 1922)
• Automise the access to food/mates/space/shelter
• Reduce aggregation
• Keep workers to work for the colony’s benefit
Thorleif Schjelderup-Ebbe
(1894-1976)
- ✔︎ ✔︎ ✔︎ ✔︎ ✔︎
- ✔︎ ✔︎ ✔︎ ✔︎
- ✔︎ ✔︎ ✔︎
- ✔︎ ✔︎
- ✔︎
-
self
peer
Icon and picture from Freepik.com and Wikipedia

3. Dominance hierarchy as
network
• Most studies have focused on
• How close data are to “linear” hierarchy
• How to rank individuals in a group
• Small groups
• Network analysis of dominance hierarchy
has been surprisingly rare.
• Some recent work as undirected networks
• Triad census (Shizuka & McDonald, 2012)

4. Diacamma sp.
• Monogynous
• A colony contains at most one
(functional, not morphological) queen.
• 20-300 workers, i.e., “large” groups
• Suitable for observing behaviour:
• Large body size
• Many previous studies

5. nest marked workers
aggressive behaviour
(bite and jerk) =
directed link
Photos by H. Shimoji

6. • 4 days of observation (5 h/day)
colony # nodes avg deg # bidir links
C1 20 2.9 0/29
C2 32 3.4 0/55
C3 48 5.6 0/134
C4 70 4.5 0/158
C5 56 4.8 2/133
C6 64 4.3 0/137
“large” network (almost) acyclic?sparse

7. (almost) directed acyclic graph (DAG)
dominant
subordinate

8. A B C D E F G
A
B
C
D
E
F
G
6 1 4 6 8 5
5 5 2 1
2 2 1
1 15 1 11 1
4 2
DAG hierarchy is not trivial
1. In large groups, linear hierarchy is often
violated.
data from Appleby, Animal Behaviour, 1983
winner dominant dominant
red deer stags
A B C D E F G
A
B
C
D
E
F
G
✓ ✓ ✓ ✓ ✓ ✓
✓
✓
✓ ✓ ✓ ✓
✓
A F G E B D C
A
F
G
E
B
D
C
✓ ✓ ✓ ✓ ✓ ✓
✓ ✓ ✓ ✓
✓
✓
✓
subordinatesubordinateloser

9. DAG hierarchy is not trivial
2. There are various DAGs.
• Variation in link density
• Even for a fixed link density, various
DAGs
linear tournament arborescence

10. Quantifications of DAGs
(link weight ignored)
1. Reversibility (Corominas-Murtra,
Rodríguez-Caso, Goñi, Solé, 2010)
• Information necessary to reversely
travel to the most dominant nodes
2. Hierarchy (their 2011) ν ∈ [-1, 1]
• ν = 0 ⟺ lack of hierarchy in either
direction

11. Quantifications of DAGs (cnt’d)
3. Global reaching centrality (Mones, Vicsek,
Vicsek, 2012):
• Large GRC ⟺ directed paths starting from
a small fraction of nodes reach a majority of
nodes
• Directed star: GRC = 1
• 0 ≤ GRC ≤ 1
4. Network motif (Milo et al. 2002)
GRC =
1
N 1
NX
i=1
[Cmax
R CR(i)] , where Cmax
R = max
i
CR(i)
CR(i) : local reaching centrality of node i

12. Null model networks
• Randomised DAGs (Goñi, Corominas-Murtra,
Solé, Rodríguez-Caso, 2010)
• In-degree and out-degree of each node are
fixed.
• Thinned linear tournament (= cascade model by J.
E. Cohen & C. M. Newman, 1985)
• Number of links matched
• Does not conserve in/out- degree of each
node
• Then, calculate the Z score: e.g.,
p=0.6
Z =
GRCobserved µnull(GRC)
null(GRC)

13. ✔︎ Similar results for link-reversed dominance networks
colony
Reversibility (H ≥ 0)Reversibility (H ≥ 0)Reversibility (H ≥ 0) Hierarchy (0 ≤ ν ≤ 1)Hierarchy (0 ≤ ν ≤ 1)Hierarchy (0 ≤ ν ≤ 1) GRC (0 ≤ GRC ≤ 1)GRC (0 ≤ GRC ≤ 1)GRC (0 ≤ GRC ≤ 1)
colony
Value
Thinned
tournament
Random
DAG Value
Thinned
tournament
Random
DAG Value
Thinned
tournament
Random
DAG
C1 0.28 -2.36* - 0.59 3.68** -0.33 0.94 4.45** 1.01
C2 1.41 1.86 1.76 0.14 1.05 -1.70 0.71 2.72** -2.11*
C3 1.73 0.24 2.33* 0.31 3.32** 0.05 0.88 4.93** -1.40
C4 1.33 -0.36 -1.33 0.32 3.90** -1.08 0.96 6.60** 1.66
C5 2.37 4.98** 0.20 0.28 3.15** 0.74 0.86 4.82** -0.89
C6 2.02 4.09** 1.69 0.14 1.72 0.66 0.82 4.54** -0.64
*: p<0.05; **: p<0.01

14. ✔︎ Similar results for link-reversed dominance networks
*: p<0.05; **: p<0.01

15. Let’s look at the degree
attacked by
2 workers
(in-degree = 2)
attacks
3 workers
(out-degree = 3)
Photo by H. Shimoji

16. Only the out-degree is heterogeneously distributed (CV = 1.9-3.5)

19. Out-strength
out-strength = 8
1
4
3
5
3
link weight = # observed aggressive behaviour
Photo by H. Shimoji

20. The top ranker is often not the most frequent attackers.
Out-strength vs worker’s rank

21. Summary of the observations
• Empirical dominance networks are close to
random DAGs.
• Similar to citation networks (Karrer &
Newman, PRL, PRE 2009)
• Not close to the thinned linear tournament
• Sparse
• Out-degree: heterogeneous, in-degree: not so
much
• Most aggressive workers are near the top (but
not necessarily the very top) of the hierarchy.

22. Discussion
• How is the link density regulated?
• Cost of attacking
• Benefit of keeping hierarchy: workers work for the
colony (so-called indirect fitness)
• Why (evolutionarily) does the DAG-like dominance
hierarchy form?
• For high rankers, more chances to reproduce (direct
fitness)
• For low rankers in the bottom of hierarchy, why?
• Why does the top ranker limit the number of direct
subordinates?
• Generative models?
Ref: Shimoji, Abe, Tsuji & Masuda, J. R. Soc. Interface, in press (2014)

23. Discussion (cnt’d)
• Linearity is not detected by previous methods
due to sparseness.
colony h’ P(h’)
’)
ttri P(ttri)
C1 0.21 0.18 1 0.39
C2 0.12 0.23 1 0.23
C3 0.13 0.0003 1 0.001
C4 0.08 0.0005 1 0.029
C5 0.07 0.09 0.96 0.024
C6 0.07 0.05 1 0.053
h =
12
N3 N
NX
i=1
✓
dout
i
N 1
2
◆2
ttri =4
✓
Ntransitive
Ntransitive + Ncycle
0.75
◆
(Landau, 1951; Appleby
1983; De Vries, 1995)
(Shizuka & McDonald, 2012)
cycle
transitive