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