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Medium worlds theories I

M.R. Pinheiro

PO Box 12396, A’beckett St., Melbourne, Victoria 8006, Australia

Abstract

Reﬁning the results from the work of Comellas et al. regarding deterministic small-world networks, and intending to

apply results deduced from theirs into traﬃc networks, we introduce new constraints, extend their work to networks of

circulants, criticize the choice of the name ‘small worlds’ for large circulants, with a number greater than 64 for their ver-

tices and, as a side result, we introduce a new form of graph: starants as a replacement of the circulants for the case of

disease spread and social networks. In order to reach our goals we make use of standard combinatorial tools, graph anal-

ysis, and general algebraic procedures.

2006 Elsevier Inc. All rights reserved.

Keywords: Small-world; Communication networks; Networks; Combinatorial problems

1. Introduction

Deterministic small-world communication networks were introduced by Comellas et al. [1]. They are sup-

posed to have strong local clustering (nodes have many neighbors in common), small diameter (largest of the

shortest distances between nodes must be small), and would be located between regular lattices, which are

highly clustered, large worlds, where the diameter, or characteristic path length, grows linearly with the num-

ber of nodes, and random networks, which are poorly clustered, small worlds, where the diameter grows log-

arithmically with the number of nodes. We shall name them ‘medium worlds’. Circulant graphs are considered

part of the deterministic small-world communication networks, once they have strong local clustering, but

large average distance between pairs of nodes. They are included in the class of structured networks. In this

paper, we want to contribute to Comellas et al. ﬁndings, which focused on replacing probabilistic models with

deterministic small-world networks and non-random interconnection patterns, by adding some constraints to

their variables plus proposing a new form of graph: starants. Small diameters not only make calculations eas-

ier but are consistent with the concept of small-world networks besides being more relevant than average dis-

tance for the study of small ‘communication networks’, such as traﬃc networks, most of the time. Following

Comellas et al., our model will be more adequate for situations where the number of neighbors of a node must

be ﬁxed because of technical considerations. Contrary to Comellas et al. work, though, we would like to con-

sider any possible number of nodes for our graphs. From Comellas et al. [1], we use:

0096-3003/$ - see front matter 2006 Elsevier Inc. All rights reserved.

doi:10.1016/j.amc.2006.05.213

E-mail address: mrpprofessional@yahoo.com

Applied Mathematics and Computation 188 (2007) 1061–1070

www.elsevier.com/locate/amc

Lemma 1. Let S be a segment of C

n,d

;deven, with length (k 1)d+1<l

s

6kd+1, kP1. The maximum

distance between any node of S and one of the end nodes of S is k.

•The maximum distance between any node of Sand one of the end nodes of Sis k.

Theorem 1. Given C

n,d

;deven, and D<DCn;d, the number of hubs required to construct a new graph G with

diameter at most D

G

6D from C

n,d

by using a graph H of diameter D

H

to interconnect the hubs is

he¼2n

dðDDHÞþ2

lm

or

ho¼2ðndÞ

dðDDH1Þþ2

lm

8

>

<

>

:

depending on (D D

H

) being even (h

e

) or odd (h

o

).

Theorem 2. Given C

n,d

;deven, let H denote a graph with h nodes and diameter D

H

. There is a graph G with n

nodes and h hubs (using graph H to interconnect the hubs) which has diameter D

G

62k + D

H

, where k¼

n

h

de

1

d

.

If the condition n [(k 1)d+ 1](h 1) 6kd+ 1 is also satisﬁed, then the diameter is D

G

62k 1+D

H

.

The above results rely on Lemma 1 and Comellas et al. constraints are enough for them. We also use the

result below [1] regarding clustering:

Proposition 1. The clustering parameter of C

n,d

is

CCn;d3

4

ðd2Þ

ðd1Þ:

This result relies on the fact that nodes i and (i + j) have d(j + 1) common neighbors, 16j6d

2.

2. Notation and some deﬁnitions

1. C

n,d

– circulant graph of nnodes and d(even degree) links per node such that each node iis adjacent to

nodes (i± 1), ði2Þ;...;id

2

, where the labeling obeys the rules for the group mod n. This graph has

got diameter Dn

d

whenever d52 and Dn

2

otherwise.

2. Star graph – rooted tree containing n nodes with a central node (root) of degree (n1).

3. Complete graph on nnodes – graph where every node has got degree (n1).

4. SC

n;d– string of ncirculant graphs connected by means of K

2

exactly dtimes for each circulant graph

added (taking away the ﬁrst and the last graph on the string which will use K

2

exactly d

2times to make

the connection): each vertex ‘i’ is connected to d

2ki

;16k6d

2.

5. CC

n;d– circle of circulant graphs connected by means of K

2

exactly dtimes for each graph added: each

vertex ‘i’ is connected to d

2ki

;16k6d

2.

6. SC

n– string of ncirculant graphs connected by means of K

2

as many times as we like for each circulant

graph added.

7. CC

n– circle of ncirculant graphs connected by means of K

2

as many times as we like for each circulant

graph added.

8. SC

n,d

– ‘Starant’ graph, that is, a circulant graph with degree dcontaining a star inside of it whose ver-

tices coincide with the vertices of the circulant graph, a total of nvertices.

9. SS

n;d– string of nstar-graphs connected by means of K

2

exactly dtimes for each star-graph added (taking

away the ﬁrst and the last graph on the string which will use K

2

exactly d

2times to make the connection):

each vertex ‘i’ is connected to d

2ki

;16k6d

2.

10. SC

n;d– string of ncirculant graphs connected by means of K

2

exactly dtimes for each circulant graph

added (taking away the ﬁrst and the last graph on the string, which will use K

2

exactly d

2times to make

the connection): each vertex ‘i’ is connected to d

2ki

;16k6d

2.

1062 M.R. Pinheiro / Applied Mathematics and Computation 188 (2007) 1061–1070

11. CSC

n;d– circle of nstarant graphs connected by means of K

2

exactly dtimes for each starant graph added:

each vertex ‘i’ is connected to d

2ki

;16k6d

2.

12. CS

n;d– circle of stars connected by means of K

2

exactly dtimes for each star-graph added: each vertex ‘i’

will be connected to d

2ki

;16k6d

2.

13. SS

n– string of nstar-graphs connected by means of K

2

as many times as we like for each star-graph

added.

14. CS

n– string of nstar-graphs connected by means of K

2

as many times as we like for each star-graph

added.

Section 2introduces constraints on Comellas et al. work. Section 3 is about networks formed by circulant

graphs. Section 3discusses the choice of hubs in order to decrease the diameter of a network to a size DG. Sec-

tion 4deals with clustering and regularity. Section 5brings networks of star graphs. Section 6is about reduction

of diameter whilst, in Section 7, we introduce the Starant graphs. Section 9 deals with clustering in respect to the

newly created types of networks. The work in the paper ends with a summary of its contents and some predic-

tion of future work. In the next sections we make use of the new result below, introduced by us:

Theorem 3 (New). The longest path that there may exist between two nodes of a circulant graph of degree 2

measures n

2

.

Proof. We use Lemma 2 from [1] and the fact that when d= 2 the distance between nodes is maximum. See:

l

s

= min(jijj,njijj) according to Lemma 2 from [1]. This way, considering nodes iand j, and the ver-

tices of our circulant graph to be labeled with the elements of the group mod(n), plus assuming that j>i,we

have

nðjiÞ6ðjiÞ() n

26ðjiÞ

ðjiÞ<nðjiÞ() n

2>ðjiÞ

:

Therefore

ðjiÞPn

2() lsnðjiÞ

ðjiÞ<n

2() lsðjiÞ

:

But

ðjiÞPn

2)nðjiÞ6n

2)ls6n

2

and

ðjiÞ<n

2)ls<n

2:

Therefore, ls¼n

2is an upper bound for l

s

.Ifnis even, take k¼n

2¼b

n

2cin Lemma 2 from [1]; otherwise, take

k¼ðn1Þ=2¼b

n

2c. In any case, one can say that k¼b

n

2c.h

3. Limiting d(new result)

In [1], Comellas et al. assume that nodes iand (i+j) have d(j+ 1); 1 6j6d

2common neighbors. Hence,

when iand jare d

2apart, they will have d

21

neighbors.

One of the counter-examples to the above theory is C

9,6

: any vertices nand (n+ 3) should have two com-

mon neighbors but, in fact, they have three common neighbors.

Lemma 2. If dis the maximum degree that a vertex may have in the circulant graph C

n

(C being allowed to be

either regular or random), and n is the number of vertices in C

n,d

,d<2n

3.

Proof. If iand jhave common neighbors to the right of i, for instance, there should not be common neighbors

to its left. Analogously, when iand jhave common neighbors to the left of i, there should not be common

M.R. Pinheiro / Applied Mathematics and Computation 188 (2007) 1061–1070 1063

neighbors to its right, and the proof of one case is analogous to the other. We shall, therefore, consider just the

ﬁrst case in our proof. According to Comellas et al., 0 <ðjiÞ6d

2. The situation to be excluded, taking into

account our ﬁrst supposition regarding neighborhood is, therefore, (iu)(mod n)=(j+v)(modn) where

0<u;v6ðjiÞ;fu;vgN

what implies that

0<ðjiþuþvÞ6d

2þ2ðjiÞ

what implies that

0<ðjiþuþvÞ63d

2

ﬁnally implying that we should exclude the cases where

0<kn 63d

2

what brings us to our constraint once the lowest value for kis 1. h

4. Networks formed by one set of circulant graphs (extension of results)

4.1. String of circulant graphs

We connect them in a sequence: ‘m’with‘(m+ 1)’, but we stop connecting them when nis reached. We

make use of K

2

just once for each graph added, departing from the most extreme vertex in each graph in rela-

tion to the last chosen hub.

Theorem 4. In this case, the new diameter is

DSC

n;2¼X

n

m1

nm

dm

þðn1Þ:

Proof. The result is trivial, one just has to consider adding an external edge to the graph and recall that the

longest of the shortest paths of a circulant graph is dn

de.h

4.2. Circle of circulant graphs

We connect the circulant graphs in a sequence, as before, but we add an extra connection, made with k

2

,to

join graph ‘n’ to graph ‘1’.

Theorem 5. The diameter of a circle of circulant graphs is

DCC

n;2¼Pn

m¼1

nm

dm

lm

þn

2

6

6

6

47

7

7

5:

Proof. Trivial. h

5. Diameter reduction (the lemma is a new result, the theorem is an extension of an improved version of an

existing result)

All proofs on hubs and reduction of diameter of [1] are based on the use of the results concerning segments:

1064 M.R. Pinheiro / Applied Mathematics and Computation 188 (2007) 1061–1070

Deﬁnition 1. Segment is a graph induced by two consecutive hubs, considering all vertices and edges between

them. Hubs for us, so far, were the chosen vertices in each star/circulant to be connected by means of k

2

. From

now on, hubs will be vertices chosen amongst the hubs used to connect graphs in a circular way to be inter-

linked in a direct way using a graph of known diameter. Therefore, a segment will still be deﬁned as a graph

induced by two consecutive hubs.

Lemma 3. If S is a segment of CC

n;d;deven; 2(k 1) <l

s

62k + 1; k P1 will mean that the maximum distance

between any node of S and one of its ﬁnal nodes (hub) is k.

Proof. We know that the diameter of CC

n;2is DCC

n;2¼Pn

m¼1

nm

dm

de

þn

2

and this is the maximum diameter that one

can have for CC

n;d, since the connection between any two circulants is minimum in this case. Therefore, since

n0

2

¼Pn

m¼1

nm

dm

lm

þn

2

6

6

6

47

7

7

5¼pif n0¼2pþR¼ls;0<R<2;

pif n0¼2p¼ls;

n062k+ 1 implies k=p, that is, kis the maximum distance between a node and an endpoint, as required. h

Theorem 6. Take D<DCC

n;2. The number of hubs required to build a new graph G with diameter D

G

6D from CC

n;d

using a graph of diameter D

H

to interconnect the hubs is

he¼Pn

m¼1

nm

dm

de

n

DDH

or

ho¼Pn

m¼1

nm

dm

de

n

DDH

8

>

>

>

<

>

>

>

:

depending on whether (D D

H

) is even (h

e

) or odd (h

o

).

Proof. We have to split the proof into cases:

•Case 1: If (DD

H

) is even, choose ‘h’ circulant graphs of CC

n;dto be hubs such that each segment Shas got

length 2(k1) + 1 < l

s

6(2k+ 1) for some k61. Thus, the maximum distance from any node of Cto a

hub is k(Lemma 1), provided that kis minimum, and the maximum distance between any two hubs of Gis

D

H

. Let us worry about hnow. Let h¼Pn

m¼1

nm

dm

de

n

k

. All segments have length (2k+ 1) at most. This way,

the diameter DG62kþDH¼D)k¼DDH

h)h¼Pn

m¼1

nm

dm

de

n

DDH

.

•Case 2: If (DD

H

) is odd, do h¼Pn

m¼1

nm

dm

de

þn2

2kþ1

(this way, we will cover the whole graph minus 2

2kþ1).

Thus, (h1) segments will have length (2k+ 1) and the remaining segment will have length

Pn

m¼1

nm

dm

lm

þnðh1Þð2kþ1Þ. But if h¼Pn

m¼1

nm

dm

de

n

k

or

hPPn

m¼1

nm

dm

lm

þn2

2kþ1

or

hð2kþ1ÞPX

n

m¼1

nm

dm

þn2

thus

hð2kþ1Þð2kþ1ÞPX

n

m¼1

nm

dm

þn2ð2kþ1Þ

M.R. Pinheiro / Applied Mathematics and Computation 188 (2007) 1061–1070 1065

or ðh1Þð2kþ1ÞPPn

m¼1

nm

dm

lm

þn32kwhat implies that Pn

m¼1

nm

dm

lm

þnðh1Þð2kþ1Þ6

Pn

m¼1

nm

dm

lm

þnPn

m¼1

nm

dm

lm

nþ3þ2kthat is, the length of the remaining segment is less than or equal

to 3 + 2k=2(k+ 1) + 1. Well, DPD

G

)D=D

H

+2l

s

PD

G

according to our choice for DD

H

even,

but this will not work here. Now we need to have an odd number instead of 2l

s

. Since subtracting one may

return 0, we add one unit to the result. This is why our last segment had to be less than (k+ 1). Therefore,

(2k+ 1) is the maximum distance from point to hub, and DG6ð2kþ1ÞþDH¼D)k¼DDH1

2)

Pn

m¼1

nm

dm

de

þn2

DDH

.h

Theorem 7. Consider CC

n;2, H being a graph with ‘n’ nodes (h 6n) and diameter D

H

. Consequently, there is a

graph G with ‘n’ nodes and ‘h’ hubs (using H to connect hubs) which has diameter D

G

62k + D

H

where

k¼Pn

m¼1

nm

dm

de

þn

h

1

2

2

6

6

63

7

7

7

:

If the condition Pn

m¼1dnm

dmeþnð2k1Þðh1Þ6ð2kþ1Þis also satisﬁed, the diameter is DG62k1þDH.

Proof. Since there are hhubs, one can divide CC

n;dinto segments of length Pn

m¼1

nm

dm

de

n

h

at most. Let k2Zsuch

that

2k1<Pn

m¼1

nm

dm

lm

þn

h

2

6

6

63

7

7

7

62kþ1

and k¼Pn

m¼1dnm

dmþn

h

1

h

2

6

6

63

7

7

7

. Build a graph Gusing Hto interconnect hubs of CC

n;d.ByLemma 1, the distance

between any node of Gand a hub is kat most and the distance between any two hubs of Gis D

H

at most.

This way, D

G

62k+D

H

.If

X

n

m¼1

nm

dm

þnð2k1Þðh1Þ62kþ1;

we can choose the hubs such that (h1) of the segments have maximum length (2k1) and the remaining

segment has maximum length 2k+1.ByLemma 1, the distance to a hub is (k1) at most for every node in

(h1) of the segments and kat most for all nodes of the remaining segment. This gives

D

G

6D

H

+2k1. h

6. Clustering and regularity (extension of results)

Theorem 8. The clustering parameter of CC

n;dis

CCC

n;d¼Pn

m¼1dm

dm

2

Pn

m¼1dmþ

d

2

d

nþ1;

where d

m

stands for degree of graph m from the circle and dstands for the degree of the circle itself.

Proof. Since the number of triangles containing node ‘i’ is still 3

4dd

21

for each circulant internally, but also

for each hub, we just apply the concept from [1] to calculate our C

v

,C

v

being the quotient of the number of

edges between a vertex neighbors and the possible number of edges. This way, for each circulant,

1066 M.R. Pinheiro / Applied Mathematics and Computation 188 (2007) 1061–1070

CC

v¼

3

4dðd

21Þ

dðd1Þ

2

¼3

2

d

21

d1

and for the circle of circulants, we consider the average of C

v

over all vertices, that is

CCC

n;d¼Pn

m¼1

3

4dmðdm

21Þ

Pn

m¼1

dm

2ðdm1Þþ3

2

d

21

d1

nþ1

that is

CCC

n;d¼Pn

m¼1dm

Pn

m¼1dmþ

d

2

d

nþ1:

7. Networks formed by one set of star graphs (new results)

7.1. String of stars

We connect them in a sequence, making use of K

2

once for each graph added (but the last one) in a regular

way, joining the furthest vertices from one graph to the furthest ones in the other.

Theorem 9. In this case, our new diameter is DS

Sn;2¼2nþðn1Þ¼3n1.

Proof. Trivial. h

7.2. Circle of stars

We build a string of stars and connect the last star-graph to the ﬁrst one.

Theorem 10. In this case, our new diameter is DS

Cn;2¼3n

2

.

Proof. Trivial. h

7.3. Reduction of diameter

It is very important to recover the concept of segment for sets of stars and circulants.

Deﬁnition 2. We called hubs the chosen vertices in each star/circulant in terms of connecting it to K

2

. We now

re-deﬁne hubs to be vertices chosen to be inter-linked in a direct way using a graph of known diameter. This

way, a segment is still a graph induced by two consecutive hubs.

Lemma 4. If S is a segment of CS

n;d;deven; 2(k 1) <l

s

62k + 1; k P1 means that the maximum distance

between any node of S and one of its ﬁnal nodes (hub) is k.

Proof. The proof is the same as for the circle of circulants. h

Theorem 11. Take D<DCS

n;2. The number of hubs required to build a new graph G with diameter D

G

6D from

CS

n;2using a graph of diameter D

H

to interconnect the hubs is

he¼3n

DDHþ1

lm

if ðDDHÞis even;

ho¼3n2

DDH

lm

if ðDDHÞis odd:

8

>

<

>

:

M.R. Pinheiro / Applied Mathematics and Computation 188 (2007) 1061–1070 1067

Proof. We again have to split the proof in cases:

•Case 1: If (DD

H

) is even, choose ‘h’ nodes of CSn;2, each node being a circulant (and we can always choose

them such that the length of Sis as required) to become hubs such that each segment Shas got length

2(k+1)+1<l

s

<(2k+ 1) for some k61. Thus, the maximum distance from any node Gto a hub is k

(Lemma 1), provided that kis minimum, and the maximum distance between any two hubs of Gis D

H

.

Let us determine the value of hnow. Let h¼3n

2kþ1

lm

. All segments have length (2k+ 1) at most. This

way, the diameter DG62kþDH¼D)k¼DDH

2)h¼3n

DDHþ1

lm

.

•Case 2: If (DD

H

) is odd, do h¼3n2

2kþ1

lm

(this way, we will cover the whole graph minus 2

2kþ1). Thus, (h1)

segments will have length (2k+ 1) and the remaining segment will have length 3n(h1)(2k+ 1). That

is,

DG6ð2kþ1ÞþDH¼D;

k¼DDH1

2)h¼3n2

DDH

:

Theorem 12. Consider CS n; 2, H being a graph with ‘n’ nodes and ‘h’ hubs (using H to connect hubs) which has

diameter D

G

62k + D

H

where k¼

3n

h

de

1

2

. If the condition

3nð2k1Þðh1Þ62kþ1

is also satisﬁed, the diameter is D

G

62k 1+D

H

.

Proof. The proof follows similar reasoning to the one applied in the previous theorem. h

8. Starant graphs

Deﬁnition 3. A Starant graph is a circulant graph with nvertices and degree dwith a star of nvertices inserted

in it where both the vertices from the star graph and the vertices from the circulant graph coincide.

Theorem 13. The diameter of a starant graph with degree up to (n 2) is equal to 2.

Proof. There is a single vertex to which a vertex xis not directly connected to. Let us call it y. It is trivial to

infer that there will be two steps until one can reach vertex y.h

Lemma 5. The only way of decreasing the diameter of a starant graph is having d=n1.

Proof. The worst spaced vertices in a circulant graph are spaced by n

2

. There is only one way of making one

step from vertex mto vertex mþn

2

. This way is having d

2¼n

2

. But this will imply that dPn, what can

only mean that d=nsince dcannot overcome n.h

8.1. Networks formed by one set of starant graphs

8.1.1. String of starants

Theorem 14. In this case, the new diameter is (3n 1).

Proof. If each starant graph has got diameter equal to 2 and we make (n1) insertions of K

2

, the result will

follow naturally. h

1068 M.R. Pinheiro / Applied Mathematics and Computation 188 (2007) 1061–1070

8.1.2. Circle of starant graphs

Theorem 15. In this case, the new diameter is DCSC

n¼3n

2

.

Proof. Trivial. h

8.2. Clustering

The notion of clustering is a little bit intricate in our case since it involves each circulant/star-graph, plus the

clustering of their ring. The same combinatorial point of view from [1] can be adopted here, though.

8.2.1. Circle of starant graphs

Lemma 6. Each Starant graph has clustering parameter equal to 3

4

d2

3

dþ1

.

Proof. Since we now have a circulant graph plus a star-graph inserted in it, each vertex iwill have one more

neighbor and we just have to add one to the vertices in common that vertex iand vertex (i+j) will have. This

way, the sum of all triangles to which vertex ibelongs will give us

X

d

2

j¼1

ðdjÞ¼3

4dd

21

3

:

The number of possible connections between neighbors of vertex iis now ðdþ1Þd

2and the ratio between both of

them will give us the clustering, as deﬁned by Comellas et al. [1], that is CSCn;d¼Pd

2

j¼1ðdjÞ

ðdþ1Þd

2

¼

3

4dd

21

3

ðdþ1Þd2¼3

4

d2

3

dþ1

.h

Theorem 16. The clustering parameter of a circle of starant graphs is

CCSC

n;d¼

3

2Pn

m¼1

dm

21

3

dmþ1þ

d

21

d1

nþ1:

Proof. Considering that each vertex i, from a starant graph, has 3

4dðd

21

3Þtriangles to which it belongs, and

each hub from the circle will obey the rule for circulant graphs, that is, will have 3

4dðd

21Þtriangles to which

it belongs, and that each starant has got ðdþ1Þd

2possible edges amongst its neighbors, whilst our circle will have

dðd1Þ

2possible edges amongst its neighbors, just like the usual circulant graphs, the result for C

v

, as explained

earlier, follows naturally. See: for the circle itself, we have

CC

v¼

3

4dðd

21Þ

dðd1Þ

2

¼3

2

d

21

d1

whilst, for each starant graph, we have:

CSC

V¼

3

2

d

21

3

dþ1

The only information we ﬁnd about the subject in [2] is that C

v

is the fraction of allowable edges that actually

exist and Cis the average of C

v

over all v. This is not enough since we are now dealing with a set of starants in

a circulant arrangement. Since we have to consider all starants in our counting, it seems that taking the aver-

age to be our C

v

makes sense. This way, for the circle of nstarant graphs, we have:

CCSC

n;d¼Pn

m¼1CSC

vmþCC

v

nþ1

M.R. Pinheiro / Applied Mathematics and Computation 188 (2007) 1061–1070 1069

that is

CCSC

n;d¼

3

2Pn

m¼1

dm

21

3

dmþ1þ

d

21

d1

nþ1:

9. Conclusion

In this paper, we have investigated all possible string and ring situations with circle of circulants and sta-

rants. We also intend to discuss the applicability of starant graphs to represent people’s circle of acquain-

tances, plus develop a medium-world theory to account for this, which will include work on the eﬀects of

subtracting edges from a starant graph and having non-regular ones.

References

[1] F. Comellas, J.G. Peters, J. Ozon, Deterministic small-world communication networks, Information Processing Letters 76 (83–90)

(2000).

[2] S.H. Strogatz, D.J. Watts, Collective dynamics of ‘small-world’ networks, Nature (393) (1998).

1070 M.R. Pinheiro / Applied Mathematics and Computation 188 (2007) 1061–1070