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Medium worlds theories I

Authors:
  • IICSE University

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 vertices 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 analysis, and general algebraic procedures.
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,njijj) 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
nðjiÞ6ðjiÞ() n
26ðjiÞ
ðjiÞ<nðjiÞ() n
2>ðjiÞ

:
Therefore
ðjiÞPn
2() lsnðjiÞ
ðjiÞ<n
2() lsðjiÞ

:
But
ðjiÞPn
2)nðjiÞ6n
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;vgN
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
þnðh1Þð2kþ1Þ. 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
hð2kþ1Þð2kþ1ÞPX
n
m¼1
nm
dm

þn2ð2kþ1Þ
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
þnðh1Þð2kþ1Þ6
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
dmnð2k1Þðh1Þ6ð2kþ1Þis 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

þnð2k1Þðh1Þ62kþ1;
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
3nð2k1Þðh1Þ62kþ1
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
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