Graph Concatenations to Derive Weighted Fractal Networks

Given an initial weighted graph G0, an integer m> 1, and m scaling factors f1, . . . , fm ∈ (0, 1), we define a sequence of weighted graphs Gk 􏼈 􏼉 ∞ k�0 iteratively. Provided that Gk− 1 is given for k≥ 1, we let G (1) k− 1, . . . , G (m) k− 1 be m copies of Gk− 1, whose weighted edges have been scaled by f1, . . . , fm, respectively. )en, Gk is constructed by concatenating G0 with all the m copies. )e proposed framework shares several properties with fractal sets, and the similarity dimension dfract has a great impact on the topology of the graphs Gk (e.g., node strength distribution). Moreover, the average geodesic distance of Gk increases logarithmically with the system size; thus, this framework also generates the small-world property.


Introduction
For the past two decades, various networked systems and their models have been studied intensively by researchers from many different scientific fields. Examples include neural networks, protein networks, metabolic networks, phone call networks, the Internet and the World Wide Web, social networks of acquaintance, and many others. e graph models of networked systems such as these often consist of millions or even billions of nodes and have large-scale statistical properties. Due to the abundance of systems of this sort and the increased computing powers, complex networks are currently very hot and attractive research objects, and they define a powerful framework for describing and analyzing the real-world networks. In comparison with regular networks and random networks, the structure of complex networks is irregular, complex, and usually dynamically evolving in time. Complex networks possess all or part of the remarkable properties including small-world effect, scale-free property, correlations in the node degrees, and community structure, which reproduce topological properties of most of the realworld networks considered. One can refer to [1][2][3][4][5] for a detailed exposition of the theory of complex networks.
In the study of complex networks, one of the topics is to construct models mimicking real-world networks. Some construction algorithms are based on the fixed fractal sets or merely the ideas taken from the fractal construction. In [6], Barabási et al. presented a model that constructs a sequence of networks in a deterministic fashion and proved the model generates the scale-free property. In [7], Zhang et al. constructed a class of networks named incompatibility networks based on the structure of the Sierpiński gasket and showed that the networks are scale-free and small-world. In [8], the authors proposed a method converting the Koch curve into a family of deterministic networks called Koch networks, which also possess general properties observed in real-world networks. Inspired by the works of mapping fractal sets into complex networks and in order to generalize the former constructions, Carletti and Righi defined a class of weighted complex networks via an explicit algorithm in [9]. eir framework shares several properties with fractal sets, and the procedure of constructing the networks is similar to constructing a sequence of sets by the iterated function system [10]. ey hereby called this framework weighted fractal networks, WFNs for short. Recently, in [11], Li et al. introduced a method of generating a family of growing symmetrical tree networks T k ∞ k�0 from a symmetrical tree T 0 , where T k is constructed by replacing each edge of T k− 1 with a reduced scale of T 0 . ey studied the asymptotic behavior of the average geodesic distance of T k . In [12], Xi and Ye did a similar work, whereas the initial graph turns into a directed tree.
In this paper, we introduce a method which generates a class of weighted graphs based on a weighted graph and some parameters. Our framework is different from the construction in [9]; it, however, also shares some properties with fractal sets, hereby named weighted fractal networks. As for the model introduced in [9], the topological properties of the networks are mainly determined by the scaling factor, the number of copies, and the order and size of the initial network, whereas other topologies of the initial network have only a small impact on the properties of their framework. In our case, the initial graph plays a dominant role in the construction and topologies of the weighted graphs. In Section 2, we will introduce the model and show some applications of it. In Section 3, we study several topological properties of the graphs, including the average weighted geodesic distance and the node strength distribution. We found that the average geodesic distance increases logarithmically with the system size; thus, the weighted fractal networks generate the small-world property. Besides, the similarity dimension plays a relevant role in the topologies of the weighted fractal networks.

Weighted Fractal Networks
) be an initial weighted graph satisfying three conditions: G 0 has no loops or parallel edges; G 0 is connected; and the cardinality of nodes in G 0 , denoted by n, satisfies n � #(V(G 0 )) > m. One of the nodes has been labeled as the beginning node and is denoted by s 0 . m other nodes have been labeled as end nodes and are denoted by v 1 , v 2 , . . . , v m , respectively.
We construct a sequence of weighted graphs G k ∞ k�0 iteratively. G 0 is given. We let G (1) 0 , . . . , G (m) 0 be m copies of G 0 . For each i ∈ 1, . . . , m { }, weighted edges of G (i) 0 have been scaled by f i , and the node in the G (i) 0 image of the beginning node s 0 is denoted by s (i) 0 . Concatenate G 0 with G (i) 0 by merging node v i and node s (i) 0 into a single new node, denoted by v (1) i , for i � 1, . . . , m. en, we obtain the new graph G 1 , of which G 0 is a subgraph. s 0 in the new graph is also labeled as a beginning node, and we denote it by s 1 when talking about G 1 .
Suppose G k− 1 is given, we define G k similarly. Let . . , m. We obtain graph G k . s 0 in the new graph G k is labeled as a beginning node and is also denoted by s k . Two examples are stated for illustration (see Figures 1 and 2). e similarity dimension d fract of the WFN is the real number satisfying Our framework supplies a general way of constructing complex networks with respect to plenty of fractal sets, e.g., Sierpinski gasket, uniform Cantor set [13], generalized Koch curves [14], and Cantor dust [15]. Roughly speaking, using this framework, we could construct complex networks with remarkable properties (e.g., small-world effect and scale-free property) from a lot of self-similar sets, for which the iterated function systems satisfy the open set condition. e approach provides an alternative way to study fractals via the theory of complex networks, and vice versa.

Generalized Koch Curve.
Let us fix a positive integer p ∈ N. We define an operation as follows: partition each existing line segment into 2p + 1 segments, which are consecutively numbered 1, . . . , 2p + 1 from one endpoint to the other; replace each even-numbered segment by the other two sides of the equilateral triangle based on the removed segment. Let H p,0 be a line segment of unitary length. For t > 0, we obtain H p,t by performing the operation on H p,t− 1 . e sequence of curves H p,t approaches a limit curve, called the generalized Koch curve. With respect to this fractal set, we set m � 3p + 1 and f 1 � · · · � f 3p+1 � 1/(2p + 1). Obviously, the similarity dimension d fract � log(3p + 1)/log(2p + 1) is equal to the Hausdorff dimension of the generalized Koch curve. e initial weighted graph G 0 is as follows: G 0 consists of 3p + 2 nodes, one of which has been labeled the beginning node, and the others have been labeled end nodes. Each end node and the beginning node are adjacent with an edge of unitary weight. ere is a one-toone mapping from the segments of H p,1 to the end nodes of G 0 . Two end nodes are adjacent if and only if their inverse images in H p,1 are connected. e edges that join two end nodes have a weight 1/(2p + 1). Figure 3 shows the structure of H 2,2 and the corresponding initial graph G 0 .

Cantor Dust.
Define an operation as follows: divide each existing square into 16 identical squares, which are consecutively numbered 1, . . . , 16 from left to right and from top to bottom. Set A � 2, 3,5,8,9,12,14,15 { }. Remove the squares whose numbers belong to A, and we obtain 5 squares. Let S 0 be a unit square. For t > 0, we obtain S t by performing the operation on S t− 1 . e limit set of the sequence S t ∞ t�0 is Cantor dust. With respect to this fractal set, we set m � 5, f 1 � · · · � f 4 � 1/4, and f 5 � 1/2. Once again, the similarity dimension and the Hausdorff dimension of the Cantor dust coincide. e initial weighted graph G 0 is as follows: G 0 consists of 14 nodes, one of which has been labeled the beginning node, and 5 others have been labeled end nodes. Assume V(G 0 ) � s 0 , v 1 , . . . , v 5 , u 1 , . . . , u 8 . e beginning node s 0 and other nodes are adjacent with an edge of unitary weight. For each node x ∈ V(G 0 )\ s 0 , v 5 , x is connected to v 5 with an edge of weight 1/4 + 1/8. ere is a one-to-one mapping ψ from V(G 0 )\ s 0 , v 5 to the squares in S 1 whose numbers belong to A ∪ 1, 4, 13, 16

Complexity
Due to the self-similarity of the weighted fractal networks and the scaling factors f 1 , . . . , f m ∈ (0, 1), our approach may have some applications in the spread of some infections, connection of the social communication, airflows in mammalian lungs, phone call networks, and so on. e framework also provides an alternative to explain the frequent appearance of fractal phenomena in nature and society.  ickness of the lines reproduces edge weights: the thicker the line, the larger the weight. Figure 1: From the left to the right, G 0 , G 1 , and G 2 . G 0 is a 5-cycle, whose edges have weight 5. For each i ∈ 0, 1, 2 { }, the red point of G i denotes the beginning node, and the blue points of G i denote the end nodes. Set f 1 � f 2 � f 3 � 0.6. ickness of the lines reproduces edge weights: the thicker the line, the larger the weight.

Topological Properties
In this section, we characterize the topology of the weighted graphs G k for all k ≥ 0. Let N k be the cardinality of nodes in G k and E k be the cardinality of edges in G k . One can easily verify that e degree of node v in graph G is the cardinality of edges of G incident with v. e sum of degrees of all nodes in G equals to 2#(E(G)) ( [16], eorem 1.1). Note that, for any graph G, we denote the class of all nodes in G by V(G) and denote by E(G) the class of all edges in G. e cardinal numbers of V(G) and E(G) are called the order and size of G, respectively. us, the asymptotic behavior of the average degree of G k satisfies

Average Weighted Geodesic Distance. Let d (k) (x, y)
denote the weighted geodesic distance between nodes x and y in G k . Set d(·, ·) � d (0) (·, ·) for convention. Define us, the average weighted geodesic distance of G k is given by For k ≥ 0, we define

Complexity
i.e., the sum of all weighted geodesic distances of the nodes in G k from the beginning node s k . Define F � m i�1 f i . Set e asymptotic behavior of λ k satisfies Proof. By the construction of G k , we obtain that By direct calculation, we obtain equation (8).
us, in order to prove equation (9), it suffices to prove However, e second equality uses O. Stolz's formula. e proof is complete.
Recall the construction algorithm. In order to obtain G k with k ≥ 1, we concatenate G 0 with G (i) k− 1 for all i ∈ 1, . . . , m { }; thus, it is proper to regard G (i) k− 1 as a subgraph of G k . Set For i � 1, . . . , m, we can decompose the sum and use the scaling mechanism for the edges and with the construction of G k where Similarly, we decompose the sum x∈V * y∈G k d (k) (x, y) into two terms: and use the scaling mechanism for the edges In conclusion, we obtain where Now, we define for k ≥ 0. Hence, we obtain the following key equation: After all the preparations are complete, we are finally ready to study the asymptotic behavior of the average weighted geodesic distance of G k . e result is stated in the following theorem. □ Theorem 1.
e asymptotic behavior of the average weighted geodesic distance of G k satisfies Proof. First, we prove By equation (22), for all k ≥ 1, we have en, e last but one equality uses O. Stolz's formula. So, equation (24) is valid.
By the definition of Γ k , one can verify that By equation (9) in Lemma 1, we have Combining equations (24) and (28), we have the desired result.
In networks G k , let p (k) (x, y) denote the geodesic distance between nodes x and y in G k . Set p(·, ·) � p (0) (·, ·) for convention. Define for k ≥ 0. us, the average geodesic distance of G k is given by We can also compute θ k , formally obtained by setting f 1 � · · · � f m � 1 and replacing weighted geodesic distance d (k) (·, ·) by geodesic distance p (k) (·, ·) in the previous argument. e asymptotic behavior of the average geodesic distance of G k is given in the following theorem, and the theorem shows the average geodesic distance increases logarithmically with the network size.

Node Strength Distribution.
Given node x in G k , we denote by N (k) (x) the set of nodes adjacent to x. If y ∈ N (k) (x), the weight of edge (xy) in G k is w (k) xy . e node strength of x is defined by Node strength is a generalization of the degree to weighted graphs. We are in the place to study the node strength distribution of the weighted graphs G k . Let g k (h) denote the cardinality of nodes in G k that have strength h. Without loss of generality, we assume that n − m − 1 � l > 0 and V(G 0 )\ s 0 , v 1 , . . . , v m � u 1 , . . . , u l .
For Δ ⊂ Π k , we can also define Complexity