^{1}

^{*}

^{2}

^{*}

We propose a new approach to the investigation of deterministic self-similar networks by using contractive iterated multifunction systems (briefly IMSs). Our paper focuses on the generalized version of two graph models introduced by Barabási, Ravasz and Vicsek ([1] [2]). We generalize the graph models using stars and cliques: both algorithm construct graph sequences such that the next iteration is always based on n replicas of the current iteration, where n is the size of the initial graph structure, being a star or a clique. We analyze these self-similar graph sequences using IMSs in function of the size of the initial star and clique, respectively. Our research uses the Cantor set for the description of the fixed set of these IMSs, which we interpret as the limit object of the analyzed self-similar networks.

The aim of this paper is to help connect the results on IMSs ( [

Let us consider the

Let us introduce the

The study of graph limits is well known by testing homomorphisms in graphs sequences (see [

and

respectively.

Based on results known on IMSs (see [

If the functions

is called the fractal operator generated by f. A fixed point of

Moreover, if

Moreover, for any nonempty compact subset

If the multivalued operators

is called the fractal operator generated with the IMS F.

An element

We call a nonempty compact subset

Let X be the compact set

We define IFSs on

We add a simple generalization for two self-similar network models introduced by Barabási, Ravasz and Vicsek ( [

Firstly, we present the modified version of the algorithm from [

Algorithm 1. Let us note the graph given after the

・ Step 0: We init the algorithm from an

・ Step 1: We add

・ Step 2: We add

These rules can be easily generalized, so the

・ Step k: Generally, the creation of

Secondly, we introduce a simple generalization of the Hierarchical Network Model.

Algorithm 2. We init with a

・ Step 0: We init the algorithm from an

・ Step 1: We add

・ Step 2: We create

These iterations can be also easily generalized, so the

・ Step k: We add

After the

We have n nodes at the initial step, which are indexed in the following way: the initial root is indexed with 0 and the other nodes are marked with

in each replica the

Let us look to

as graph sequences. The aim is to characterize these two sequences using IMSs.

Both algorithms constructs the

Thus, the presented algorithms generate self-similar networks based on stars and cliques.

Our paper focuses on two IMSs constructed with set operations of IFSs. We construct these IMSs such that their image will correspond to adjacency matrices projected to

Let us consider

undirected

The aim is to construct IMSs such that their

exists if and only if

In this section we define those iterated function systems, whose will be used for the characterization of the presented self-similar network. We use these mappings in function of the parameter

Let

and let

be the IFS constructed by these functions.

The corresponding

Let

and let

be the IFS constructed by these functions.

Let

and let

be the IFS constructed by these functions.

We construct the iterated function system corresponding to self-similar networks using the presented

Theorem 1. The

where

Proof. We use mathematical induction for showing that

This means that

Moreover, we show that

edges between the initial root and the new peripheral nodes.

Thus, after a reindexig (

On the other hand, we use a simple generalization of the Cantor set for showing that

It is well known that the iterations of the Cantor set can be easily described using the ternary numeral system:

we note the unit segment with 0 and after the first iteration we note the remaining

segments with 0 and 2, respectively. The second iteration generates four segments, whose can be marked with

with induction that the

contain neither the number 1.

Based on the presented construction of the Cantor set, we describe the set generated by the

We refer to the

an unique value in the context of the numeral system based on the integer n. Based on the definition of

second iteration of the IMS is constructed with the union of

checked that

from

If we transform the values of i from the second iteration to the numeral system based on the integer n, then we get the following forms of the values:

We suppose that

the adjacency matrix with side length

these peripheral nodes have indexes between

Last, we show that

rected edges whose connect the current peripheral nodes with the initial root. Based on the construction using

the numeral system based on the integer n, an iteration of the

If we have a little box with side length

node in

Based on the presumption and the definitions of

means, that

in

As a conclusion,

Theorem 2. The

where

Proof. We base the proof on Theorem 1: it is obviously, that the proof of on the IMS

Firstly, we use the IFS ^{th} iteration of the selected IMS,

will include the same iteration of the IFS

usage of open sets at the set minuses causes that the included sets remain closed as we fixed the condition of existence of the edges.

We also proof the usage of the IFS

squares to the adjacency matrix, whose will correspond for the generation of all of the edges and loops in

On the other hand, using the set minus of

matrix corresponding with a clique with n nodes.

Based on these, we suppose that

clique with n nodes. This means, that

representing a clique with n nodes. By definition, the IMS

sented above. Therefore,

to

Based on Theorem 1,

This means, that

Moreover, we also note that if the parameter j of the first set union in (4) goes just to

Thus, the IMS

Firstly, it can be easily checked that

Secondly, as we showed in Theorem 1, the

It is well known that the iterations of the classical Cantor set can be described with the ternary numeral

system and we characterized the iterations of

the integer n.

We showed that the

While the construction of the classical Cantor set adds little segments, which forms in the ternary numeral system don’t contain neither the number 2 our construction adds little squares whose form in the numeral system based on the integer n doesn’t contain neither the number 0. Based on these analogy we refer to

So, let us note unique the fixed set of

On the other hand, the fixed set of

Based on the presented modification of the Cantor set based on the numeral system based on the integer n we can characterize the fixed set of the IMSs

Theorem 3.

Proof. We know that

We constructed the IMS

Theorem 4.

Proof. On

Thus, the fixed set of

We characterized the fixed set of IMSs in function of parameter n. We used the Cantor set for describing these fixed sets, which we interpret as limit objects of graph sequences corresponding to self-similar networks.

We acknowledge the support of Collegium Talentum. This work was possible due to the financial support of the Sectorial Operational Program for Human Resources Development 2007-2013, co-financed by the European Social Fund, under the project number POSDRU/187/1.5/S/155383 with the title “Quality, excellence, transnational mobility in doctoral research”.

LeventeSimon,AnnaSoós, (2016) Cantor Type Fixed Sets of Iterated Multifunction Systems Corresponding to Self-Similar Networks. Applied Mathematics,07,365-374. doi: 10.4236/am.2016.74034