Professor Ernesto Estrada - Università Ca' Foscari Venezia · Let G be a connected graph. Then,...

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Professor Ernesto EstradaDepartment of Mathematics & Statistics

University of Strathclyde

Glasgow, SCOTLAND

www.estradalab.org

Professor Ernesto EstradaDepartment of Mathematics & Statistics

University of Strathclyde

Glasgow, UK

www.estradalab.org

7

11 km

11 k

m

3.7 km21

.6 k

m

8

10 cm

25 cm

9Estrada, Sheerin, Random Rectangular Graphs,

Phys. Rev. E 91, 042805, 2015. arXiv preprint arXiv:1502.02577.

10

00

,

uu

uLu

tG

Eji

Eji

jik

ji

i

,

,

0

1,L

(8)

(9)

n 210

Let the eigenvalues and orthonormalised eigenvectors of L(G) be:

nU

21

Mesbahi, Egerstedt, Graph Theoretic Methods in Multiagent Networks, Pricenton Univ. Press, 2010, pp. 42-48.

11

Definition. The agreement set is the subspace span{1}, that is

nA

.,,| jiuuuA ji

n (10)

Theorem 1. Let G be a connected graph. Then, the (undirected) agreement model (8) converges to the agreement set (10) witha rate of convergence that is dictated by . 2

Mesbahi, Egerstedt, Graph Theoretic Methods in Multiagent Networks, Pricenton Univ. Press, 2010, pp. 42-48.

12Estrada, Sheerin, Consensus Dynamics on Random Rectangular Graphs,

Physica D, Nonlinearity 323-324, 2016, 20-26.

13

Theorem 2. Let GR(n,a,r) be an RRG with n nodes embedded in a rectangle of sides lengths a and b=a-1, and connection radius r. Then, the algebraic connectivity is bounded by

.log1

18 2

24

2

2 na

arnGR

(11)

Estrada, Sheerin, Consensus Dynamics on Random Rectangular Graphs,


14

bound (11)



15

Proposition 3. Let GR(n,a,r) be an RRG with n nodes embedded in a rectangle of sides lengths a and b=a-1, and connection radius r. Then, the diameter of an RRG is bounded by

.14

ar

aD

(6)

Sketch of the proof



16

bound (6)



17

Sketch of the proof

c

r

cnd

r

ar

aGD

r

ba

r

cnGD d

14

22



18

Sketch of the proof

Lemma 4 (Alon-Milman). Let G be a connected graph with n nodes. Then, the algebraic connectivity is bounded by

.log8 2

22

max2 n

D

kGR

.1

18log

1

84

2

2

24

2

max2

a

arnn

a

arkGR

N. Alon and V.D. Milman. J. Comb. Theory, B, 383, 88, 1985

Then, it is straightforward that

(12)

19

Theorem 5. Let G be connected graph with n nodes. Then, the time at which the consensus state is reached is bounded by

where d is the stopping criterion.

,ln

1

1

022

2

n

p

c

up

nt

d

(13)



20

Let GR be an RRG with a fixed number of nodes and radius.Then, as a:

ct 02 RG

bound (13)

21Estrada, Sheerin, Consensus Dynamics on Random Rectangular Graphs,


22

Fast consensus

23

Theorem 6. Let GR(n,a,b,r) be an RRG with n nodes embedded in a rectangle of sides lengths a and b, and connection radius r. Then, the expected average degree is

where

,1

2ab

fnk i

.

,

,0

22

3

2

1

bara

arb

br

f

f

f

fi

(1)

(2)

Estrada, Sheerin, Random Rectangular Graphs, Phys. Rev. E 91, 042805, 2015. arXiv preprint arXiv:1502.02577.

24

. 2

1

3

4 432

1 rrbaabπrf

.sin23

2

3

4

3

4

6

1 122222

2234

2

r

babrbr

brabrarbf

.sincos23

2

3

4

3

2

3

4

2

1

6

1

1122222

2222

444222

3

r

a

r

babrbr

bra

arar

brbabarf


25

observed

estimated


26

observedestimated


27

Proposition 7. Let GR(n,a,b,r) be an RRG with n nodes embedded in a rectangle of sides lengths a and b, and connection radius r. If

then

, ;log1

2

n

ab

fn i

.exp ,,,lim

econnectedisrbanGP R

n

(3)


28

,explog1

lim2

en

ab

fnP i

n

.explog1

expexp2

en

ab

fn i

That is

which means that we have the following lower bound:

(4)

(5)


29

bound (5)


32

Proposition 8. Let GR(n,a,r) be an RRG with n nodes embedded in a rectangle of sides lengths a and b=a-1, and connection radius r. For n and sufficiently small r, the distribution of the node degrees tends to a Poisson distribution

.!k

ekkp

kk


34

Proposition 9. Let GR(n,a,r) be an RRG with n nodes embedded in a rectangle of sides lengths a and b=a-1, and connection radius r. Then, the average path length is bounded by

.2

14

ar

aral

(7)


35

bound (7)


37

bound (7)


38

Sketch of the proof

.,

1

1

,

ji

jidnn

l

j

k jkdn

l ,1

1

The average path length of a graph is defined by:

Let

Then , where .kk

i ll maxill

39

Sketch of the proof

c1

c2

vi

Let vi be the node at one of the corners of the RRG.

r

40

Sketch of the proof

k

ki

dd

ddl

1

1 2121

Then the APL of vi can be expressed as:

where .r

cd i

i

1d

41

Sketch of the proof

.1111 2112211 kkk dddddddddd

sddd 21Let . Then,

1111

1

21

2211

d

ddd

dddddd

k

kk

Thus:

2

1

2

111 1

21

2211

d

ddd

ddddddl

k

kki

42Estrada, Unpublished.

n

j

jn 1

1d

43

Theorem 10. Let GR(n,a,b,r) be an RRG with n nodes embedded in a rectangle of sides with lengths a and b, and connection radius r. Then, the spectral radius of GR is bounded by

.1

1 iR fab

nG

Estrada, Unpublished.

44

Sketch of the proof

Lemma 11. Let G be a connected graph with n nodes. Then,

where C is the clique number of the graph.

C

mCGR

121

Corollary 12.

n

mnGR

121

Wilf, J. London Math. Soc. 42 330 (1967).

45

Sketch of the proof

.1

11 iR fab

nnkG

It is easy to see that

46

Theorem 13. Let GR(n,a,r) be an RRG with n nodes embedded in a rectangle of sides lengths a and b=a-1, and connection radius r. Then, as a

01 RG

Estrada, Unpublished.

47

1

1

48

observed

lower bound

49

-1

50

D’Arcy W. Thompson2 May 1860 – 21 June 1948

Professor Ernesto Estrada - Università Ca' Foscari Venezia · Let G be a connected graph. Then,...

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