Presentation slides for the following paper: Hiroyuki Shimoji, Masato S. Abe, Kazuki Tsuji, Naoki Masuda. Global network structure of dominance hierarchy of ant workers. Journal of the Royal Society Interface, in press (2014).

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