Cantor Type Fixed Sets of Iterated Multifunction Systems Corresponding to Self-Similar Networks

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.


Preliminaries and Notations
The aim of this paper is to help connect the results on IMSs ([3] [4]) and on self-similar networks ([1] [2]).We add a generalization the self-similar models introduced by Barabási, Ravasz and Vicsek and we describe these using IMSs constructed by contractions.We generate self-similar networks from 1 n S − stars and n K cliques, where n is the number of the nodes initially.
Let us consider the ( ) , G V E pair as a simple graph, where V denotes the finite set of the nodes and such, that for all ( ) , x y E ∈ there exists the ( ) , y x E ∈ edge too, where x y ≠ and , x y V ∈ .Moreover, the paper doesn't let the existence of loops and multiegdes, so there exists at least one edge between all pair of different nodes.Let us note with V n = the number of the nodes in a given graph.Let us introduce the ( ) notation for a graph with n nodes and 1 n − edges such that one of the nodes will be connected to all of the others and there will not exist any other edges between the others.Let us refer to this 1 n S − as a star with n nodes.On the other hand, let us call the ( ) graph a clique when all of the possible edges exist in a graph.
The study of graph limits is well known by testing homomorphisms in graphs sequences (see [5]).The purpose is to study the limit self-similar networks using IMSs.We note the iterations of the self-similar networks generated by 1 n S − stars and n K cliques the following: ( ) ( ) ( ) , , , , , , , , , Based on results known on IMSs (see [3] and [4]) let us introduce the following notations: If the functions i f are singlevalued continuous self operators on a complete metric space X, then ( ) is called an iterated function system (IFS).Moreover, the operator ( ) ( ) , for each  is called the fractal operator generated by f.A fixed point of f T is called a self-similar set of f and if it has a non-integer Hausdorff dimension, then it is called fractal.
Moreover, if 1 , , m f f  are contractions, then f T is a contraction and its unique fixed point ( ) are similarity contractions (see [6]), then ( ) A P X ∈ is a fractal.Moreover, for any nonempty compact subset A X ∈ , the sequence If the multivalued operators , , , : are defined on the X metric space, then we call as an iterated multifunction system (IMS).If the ( ) F X P X → multivalued operators are upper semicontinuous, then the operator ( ) ( )  is called the fractal operator generated with the IMS F. An element x X ∈ is a fixed point for T if and only if ( )

T x X
∈ .Let us note the set of the fixed points with , which we also call as fixed set in the next.In the case of multivalued contractions the same fixed point results hold (see [7]).
We call a nonempty compact subset * A X ⊆ self-similar corresponding to the iterated multifunction system F if and only if it is a fixed set for the associated IMS, so ( ) Let X be the compact set [ ] 2 0,1 .
We define IFSs on [ ] 2 0,1 such that their combination using set operations will give us IMSs such that the mappings on [ ] 2 0,1 correspond to the adjacency matrices of the self-similar networks presented in the following.
We add a simple generalization for two self-similar network models introduced by Barabási, Ravasz and Vicsek ( [1] [2]).Based on the algorithm introduced in [2], we create deterministic scale free networks from a 1 n S − star formed by a root and ( ) leaves.Moreover, based on [1] we create hierarchical networks constructed on n K clique with n nodes.Firstly, we present the modified version of the algorithm from [1].We eliminate the th 0 step and we generate deterministic scale free networks from a S − star formed by a root and ( ) • Step 1: We add ( ) S − stars, each unit identical to the network created in the previous step, and we connect each of the new leaves of these units to the initial root (see Figure 1 for ( ) G S ).
• Step 2: We add ( ) G S and connect all ( ) bottom nodes of the new units to the initial root (see also Figure 1 for ( )

G S
).These rules can be easily generalized, so the th k step does the following: • Step k: Generally, the creation of ( ) copies identical to the network created in the previous iteration (step ( ) and we connect each ( ) bottom nodes of these units to the initial root of the network.Secondly, we introduce a simple generalization of the Hierarchical Network Model.Algorithm 2. We init with a n K clique with n nodes.Therefore, the steps of the modified algorithm are the following: • Step 0: We init the algorithm from an n K clique, which will be noted as ( ) We also fix a node from the clique, which will be noted as the initial root in the next.
• Step 1: We add ( ) nodes from all of the new cliques to a initial root.We refer to the gotten graph as ( ) with adding ( ) G K and connecting the new peripheral nodes to the initial root (see Figure 2 for ( ) G K ).These iterations can be also easily generalized, so the th k step does the following: • Step k: We add ( ) G K .We also connect the new peri- pheral nodes to the initial root.This iteration can be continued indefinitely.After the th k iteration the graphs generated by a n K clique or an 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 1, 2, and 1 n − .The first iteration is followed by indexing the nodes of the 1 n − replicas: the nodes of the th i replica will be indexed with ( ) We note that in each replica the ( ) th in j + node will be the node corresponding to the th j node in the previous iteration.This numbering can be easily generalized to the th k step: we create ( ) replicas of the previous network with k n nodes.The indexing the nodes of the 1 n − replicas follows the rule that: the nodes of the th i replica will be indexed with ( ) ( ) ( )( ) , where 1, , i n =  .We also note that in each replica the ( ) ( ) node will be the node corresponding to the th j node in the previous iteration.
Let us look to , The aim is to characterize these two sequences using IMSs.

G S and
( ) ( ) G K and ( ) G K from [2] (note that the diagonal nodes in the cliques are also connected, but some links are not visible).
Both algorithms constructs the th k iteration from n replicas of the graph gotten at the ( ) iteration, where n is the parameter of the initial star or the initial clique.The application of Algorithm 1 on 2 S gives us the self-similar deterministic scale-free network introduced in [1] and the application of Algorithm 2 on 5 K construct the self-similar Hierarchial Network Model from [2], respectively.
Thus, the presented algorithms generate self-similar networks based on stars and cliques.

The Graphs' Adjacency Matrices Generated with Iterated Multifunction Systems
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 [ ] , G V E be a simple graph, where n V = .Our construction says that an undirected ( ) The aim is to construct IMSs such that their th k iteration will correspond to same iteration of the self-similar networks' presented above.This means that in the th k iteration of an IMS an undirected ( ) , respectively.

Construction of the IMSs Corresponding to the Self-Similar Networks
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 , which notes the number of nodes of the star and in the clique, respectively.The definitions are followed by the construction of the self-similar networks generated from stars and cliques using these IFSs.Last, we show that the construction corresponds to these networks. Let : 0,1 0,1 i f * → be functions, where for each 0,1   be the IFS constructed by these functions.The corresponding n *  iterated function system is defined as follows.Let : 0,1 0,1 i f * → be functions, where

[ ] [ ]
for each 0,1 : 0,1 0,1 ii f → be functions, where ( ) ( ) , for each 0,1   be the IFS constructed by these functions.We construct the iterated function system corresponding to self-similar networks using the presented , can be constructed as the followings: ( ( ) where n *  and n *  are the iterated function systems defined above.
Proof.We use mathematical induction for showing that ( ) .
This means that ( ) STAR n corresponds to the adjacency matrix of ( ) [ ] ( ( ) ( ) ) , ( ) Thus, after a reindexig ( 1 j j ′ = − ) and a reordering the set operations above we get that the ( )


generate the edges which connect the initial root with the peripheral nodes at the th k step.
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 1 0, 3  segments with 0 and 2, respectively.The second iteration generates four segments, whose can be marked with ( ) ( ) ( )  with induction that the th k iteration generates all of the segments with 1 3 k length, whose ternary form don't contain neither the number 1.
Based on the presented construction of the Cantor set, we describe the set generated by the th


as it's pair) using the numeral system based on the integer n.
We refer to the 1 1 0, , , 1, , 1 sets with the same i values.Thus, STAR n generate the undirected edges between the initial root and the peripheral nodes.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 n − + and k n .Moreover, we also suppose that for of the indexes in the numeral system based on the integer n don't contain neither the number 0.
Last, we show that [ ] ( ) ( ) where , nn n *   and n *  are the iterated function systems defined above.
Proof.We base the proof on Theorem 1: it is obviously, that the proof of on the IMS ( )


adds little squares to the adjacency matrix, whose will correspond for the generation of all of the edges and loops in n K .
On the other hand, using the set minus of ( ) ( ) too, this gives the connection between the initial root and the peripheral nodes at the th k step (for example, see Figure 5 for ( ) ( )

G K ).
This means, that Moreover, we also note that if the parameter j of the first set union in (4) goes just to 1 k − (and not till to k) the IMS generates the same adjacency matrix.It can be easly checked that: Thus, the IMS ( )

Fixed Sets of the IMSs Corresponding to the Self-Similar Networks
Firstly, it can be easily checked that i f * , i f * and ii f are Banach-type contractions in 2  and there exists a unique fixed point for all of these functions, where [ ] 2 0,1 x ∈ .Secondly, as we showed in Theorem 1, the th k iteration of i f * and i f * can be easily described using a bit modified version of the Cantor set.
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 such that it's form in the numeral system based on the integer n doesn't contain neither the number 0.
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 On the other hand, the fixed set of nn  is the whole segment between the ( ) 0, 0 and (1,1) points in 2  .Thus, we note this segment as
. It is easy to check that:

Figure 4 .
Figure 4.The description of Cantor set using the ternary numeral system.
with the 1, , 1 n −  integers, whose note an unique value in the context of the numeral system based on the integer n.Based on the definition of n *  , the second iteration of the IMS is constructed with the union of

1 k
undirected edges (thus, the little boxes in the adjacency matrix with side length 1 k n ) between the initial root and the new peripheral nodes.Basically, these peripheral nodes have indexes between 1


undirected 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 connect the initial root with the peripheral nodes with index i′ , where connect the initial root and the peripheral nodes whose i index is in { } 1 1 2 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 nn  with mathematical induction: if 0

Figure 5
Figure 5. ( ) [ ] ( ) ( 1 connects the initial root and the peripheral nodes whose i index is in { } iterated multifunction systems.So, let us note unique the fixed set of the the mirror of n Cantor * projected to the first bisector of the xOy plane.

xOyFB∠
, referring to the first bisector.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 ( ) STAR n and ( ) will include the same iteration of the IFS nn  , but it won't include the the set generated by n