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
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
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)
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
nest marked workers
aggressive behaviour (bite and jerk) =
directed linkPhotos by H. Shimoji
• 4 days of observation (5 h/day)
colony # nodes avg deg # bidir links
C1 20 2.9 0/29C2 32 3.4 0/55C3 48 5.6 0/134C4 70 4.5 0/158C5 56 4.8 2/133C6 64 4.3 0/137
“large” network (almost) acyclic?sparse
(almost) directed acyclic graph (DAG)
dominant
subordinate
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
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
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
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
iCR(i)
CR(i) : local reaching centrality of node i
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 =GRC
observed
� µnull
(GRC)
�null
(GRC)
✔ ︎ Similar results for link-reversed dominance networks
colonyReversibility (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
tournamentRandom
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
✔ ︎ Similar results for link-reversed dominance networks
*: p<0.05; **: p<0.01
Let’s look at the degree
attacked by2 workers
(in-degree = 2)
attacks3 workers
(out-degree = 3)
Photo by H. Shimoji
Only the out-degree is heterogeneously distributed (CV = 1.9-3.5)
Out-strength
out-strength = 8
143
5
3
link weight = # observed aggressive behaviour
Photo by H. Shimoji
The top ranker is often not the most frequent attackers.
Out-strength vs worker’s rank
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.
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)
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 0.39C2 0.12 0.23 1 0.23C3 0.13 0.0003 1 0.001C4 0.08 0.0005 1 0.029C5 0.07 0.09 0.96 0.024C6 0.07 0.05 1 0.053
h =12
N3 �N
NX
i=1
✓douti � N � 1
2
◆2
ttri
=4
✓N
transitive
Ntransitive
+Ncycle
� 0.75
◆
(Landau, 1951; Appleby 1983; De Vries, 1995)
(Shizuka & McDonald, 2012)
cycle
transitive