On the Fractality of Complex Networks: Covering Problem, Algorithms and Ahlfors Regularity

In this paper, we revisit the fractality of complex network by investigating three dimensions with respect to minimum box-covering, minimum ball-covering and average volume of balls. The first two dimensions are calculated through the minimum box-covering problem and minimum ball-covering problem. For minimum ball-covering problem, we prove its NP-completeness and propose several heuristic algorithms on its feasible solution, and we also compare the performance of these algorithms. For the third dimension, we introduce the random ball-volume algorithm. We introduce the notion of Ahlfors regularity of networks and prove that above three dimensions are the same if networks are Ahlfors regular. We also provide a class of networks satisfying Ahlfors regularity.

In this paper, we revisit the fractality of complex network by investigating three dimensions with respect to minimum box-covering, minimum ball-covering and average volume of balls. The first two dimensions are calculated through the minimum box-covering problem and minimum ball-covering problem. For minimum ball-covering problem, we prove its NP-completeness and propose several heuristic algorithms on its feasible solution, and we also compare the performance of these algorithms. For the third dimension, we introduce the random ball-volume algorithm. We introduce the notion of Ahlfors regularity of networks and prove that above three dimensions are the same if networks are Ahlfors regular. We also provide a class of networks satisfying Ahlfors regularity.
Complex networks arise from natural and social phenomena such as the Internet, the protein interactions, the collaborations in research, and the social relationships. Readers are referred to Watts-Strogatz's 1 small-world network model and Barabási-Albert's 2 scale-free network model, and Newman's review 3 and book 4 , etc.
In this paper, we revisit the fractality of complex network by investigating three dimensions d B 5 , d ball 6 and d f 7 with respect to minimum box-covering, minimum ball-covering and average volume of balls. The compact box burning algorithm (CBB) 8,9 and random ball-covering algorithm 6 are proposed to calculate d B and d ball respectively. However the minimum box-covering problem and minimum ball-covering problem are NP-complete, which are proved rigorously in Theorem 1 and Proposition 2 respectively. The NP-completeness implies that the CBB algorithm and the random ball-covering algorithm do not have high performance, then we suggest some algorithms to improve the random ball-covering algorithm. For the third dimension d f , we obtain an efficient algorithm: the random ball-volume algorithm. When do the three dimensions coincide? To answer this question, we introduce the notion of Ahlfors regularity of networks and prove that d B = d ball = d f (Theorem 2) if networks are Ahlfors regular. Then for Ahlfors regular networks, the random ball-volume algorithm is efficient to obtain the above three fractal dimensions.

Fractal dimensions and covering problems
Song, Havlin and Makse 5 reveal that many real networks have self-similarity and fractality, and Gallos, Song, Havlin and Makse give a review of fractality of complex networks 10 . The algorithms to numerically calculate the fractal dimension of complex networks have been proposed: For example, the CBB algorithm 8,9 is applied to calculate the fractal dimension of complex networks through the minimum box-covering; Kim, Goh, Kahng and Kim 11 improve the CBB algorithm to investigate the fractal scaling property in scale-free networks; Zhou, Jing and Sornette 12 propose the edge-covering box algorithm; Gao, Hu and Di 6 give the minimum ball-covering approach to calculate the fractal dimension of complex networks. Recall some notation. Considering a network as a graph G = (V, E) equipped with geodesic distance d, we let an l-box A denote a subset of V such that the geodesic distance of any two points in the subset is less than l, an l-ball centered at x 0 the subset < y y x l { : d( , ) } 0 . Let N l be the smallest number of l-boxes needed to cover V, and B l the smallest number of l-balls needed to cover V. Suppose that 1 Scientific RepoRts | 7:41385 | DOI: 10.1038/srep41385 where d B is the fractal dimension defined by Song, Havlin and Makse 5 , and d ball is defined by Gao, Hu and Di 6 . For box-covering, Song, Gallos, Havlin and Makse 9 point out that the minimum l-box-covering problem is NP-complete for any l ≥ 2. On the other hand, for ball-covering, which is far from box-covering in graph theory, we have Theorem 1. The minimum l-ball covering problem is NP-complete for any l ≥ 2.

Ball-covering algorithms
Due to the NP-completeness, for finding the feasible solution of minimum ball-covering problem, we can apply the usual random ball-covering algorithm (RBC) 6 : when l is fixed, in each time t, we randomly choose one node x t in the vertex set V t−1 remained in time (t − 1), and obtain V t by cutting all nodes in In the RBC algorithm we give a random sorting for nodes in V t−1 and take the first node. Moreover, given some function , we can sort these nodes according to the values of function f. Given a function  → f V : , suppose we sort nodes according to values of f in nondecreasing order: If f is the degree function, we can obtain degree-order ball-covering algorithm (DOBC); If = f x B x l ( ) # ( , ) and, we obtain volume-order ball-covering algorithm (VOBC).
For a function  → g V : , assume we sort nodes according to values of g in nonincreasing order, we propose the following greedy algorithm: and the sorting of nodes in V t inherits from V 0 = V.
In the point of view on fractal geometry, the box dimension is independent of the geometric shapes of covering, such as ball or box. It is easy to check that . By the above estimate, when the diameter of network is large enough to insure that l can be taken large enough, we have However, for real networks with small-world effect, we can not take l large enough, and the upper bound ⋅ d l ball log 2 log of error is not small enough. On the other hand, we only find the feasible solutions of minimum covering problems due to their NP-completeness. See the following example. Example 1. Through above 5 algorithms (Fig. 1), we calculate d ball for the WWW network (Table 1).
In Table 1, the value of the RBC algorithm is exactly the value d ball = 4.2 by Gao, Hu and Di 6 . Note that Song, Havlin, and Makse 5 obtain that d B = 4.1.
For the WWW network, we also compare the above 5 algorithms (Fig. 2). It seems that the VGBC algorithm is the best and the performance of the RBC is the worst and close to the VOBC.

Random ball-volume algorithm
Based on Shanker's work 13 , Guo and Cai 7 investigate the power law between the average volume of balls and the their radii. Given a network, let p(l) be the average cardinality of nodes in a ball with radius l, suppose that We call d f the volume dimension. Please also see generalized volume dimension 14 by Wei et al. We will discuss the volume dimension d f related to average ball-volume and propose the random ball-volume algorithm for networks. Compared with the minimum box-covering algorithm and the minimum ball-covering algorithm, we have the following algorithm to calculate the average volume of ball with size l approximately.
Random ball-volume algorithm (RBV) (for fixed size l): Step2. Randomly take a node x in the network.
Step3. Repeat the steps 1-2 and obtain average volume of random l-balls.
For the WWW network, using the RBV algorithm we obtain d f = 5.833 (Fig. 3).

Ahlfors regularity of networks
Fractal geometry and fractal network have deep connection. We can generate complex network models from self-similar fractals. For example Andrade et al. 15 and Zhou et al. 16 discuss Apollonian networks generated from Apollonian fractal, Zhang et al. [17][18][19] construct evolving networks modeled from Sierpinski gasket by taking the line segments as nodes. Besides Zhang et al. 20 construct the networks produced from Vicsek fractals, Liu et al. 21 and Chen et al. 22 explore some Koch networks related to Koch curves, Song et al. 23 study complex networks modeled on Platonic solids, Chen et al. 24 investigate networks generated by Sierpinski tetrahedron.   In this paper, we try to find out the farther connection between the fractal networks and fractal geometry. Recall some classical result on fractal dimension. We find out that many dimension results have measure versions. Suppose μ is a Borel (finite) measure supported on a compact subset E, denoted by spt µ ⊂ E. For any ∈ x E, let the lower local dimension of μ at point x be defined by , we obtain packing dimension dim P (·) 25,26 . We always have and there is a suitable measure μ such that = µ → s lim r B x r r 0 log ( ( , )) log , or we can pose the Ahlfors regularity assumption on the measure We give a natural measure on a graph G = (V, E). For Ω ⊂ V , we let ν(Ω) be the cardinality of Ω, which is called the volume of Ω. We say that {G t } t is a family of growing networks, i.e., ⊂ + G G t t 1 , which means the node set of G t+1 contains node set of G t , and neighbors of G t are still neighbors of G t+1 . When {G t } t is growing, we let

Remark 1. When taking ν Ω
( ) as the sum of degrees of nodes in Ω, Wei et al. 14 obtain the generalized volume dimension.

we call the network an Ahlfors s-regular network. When {G t } t is growing, we call {G t } t Ahlfors s-regular networks, if there is an independent constant c such that for all
When the diameter of network is large enough, we have When the networks are regular, we can use RBV algorithm to obtain their fractal dimensions efficiently.

Ahlfors regular trees
Now, we obtain a rule (rule 1) of generating Ahlfors s-regular networks and growing trees in Figs 4 and 5. We have Fig. 6. By embedding the self-similar tree into the self-similar fractal in  2 , we find that the volume of ball in the tree is comparable with the (self-similar) measure of ball in plane, then we can obtain Theorem 3. The growing self-similar trees defined above are Ahlfors s-regular with s = log 5/log 3. Therefore, we have We also have rule 2 and growing trees in Figs 7 and 8. For this self-similar tree with respect to rule 2, we have We can construct a family of growing networks as follows by induction: for time t, we take rule 1 if x t = 1, else take rule 2. For example, if the sequence is  211 , we obtain our growing networks G 1 , G 2 , G 3 as in Fig. 9. This is a family of deterministic growing networks.
Then we can generate a Moran tree with mixed rules. For this Moran tree without self-similarity, we have We also obtain random growing networks, for each time t, we can choose rule 1 in probability p and rule 2 in probability 1 − p.
The rest of paper is organized as follows. Section 2 is devoted to the rigourous proofs on the NP-completeness of minimum ball-covering problem (Theorem 2) and minimum box-covering problem (Proposition 2). Section 3 is the preliminary on the Ahlfors regularity of fractal geometry, including covering inequality and self-similar fractal. In this section, we also recall the fact that the open set condition of self-similar fractal implies the Ahlfors regularity of fractal measure. Replacing the fractal measures by the cardinalities of subsets of networks, we obtain the Ahlfors regularity of networks. In Section 4, we prove Theorem 2 by using covering inequality shown in Section 2, and obtain Ahlfors regularity of a class of self-similar network (Theorem 3) by constructing bilipschitz mappings from a self-similar fractal, satisfying the open set condition, to self-similar networks, and estimating the cardinalities of balls of graph from the Ahlfors regularity of the fractal measure.

NP-completeness of minimum covering problems
Recall some notation of computer science. For an alphabet Σ, let Σ ⁎ be the set of finite strings of elements of Σ, and Π the class of functions from Σ ⁎ into Σ ⁎ defined by one-tape Turing machine which operate in polynomial time.

Definition 2. Let L and M be languages. Then
. We say that some language ∈ M NP is NP-complete, if ∝ L M for all ∈ L NP. The concept of NP-completeness was introduced in 1971 by Cook 27 . In Cook's theorem, he proved that the Boolean satisfiability problem is NP-complete.
In 1972, Karp 28 proved that several other problems were also NP-complete. For example, we give the following two in Karp's 21 NP-complete problems.
(1) Clique covering problem Input: graph G = (V, E), positive integer k Property: V is the union of k or fewer cliques, where a clique is a subset of vertices of G such that its induced subgraph is complete. (2) Set covering problem Input: universe U and a family S of subsets of U, positive integer k Property: there is a set covering of size k or less, where a set covering is a subfamily ⊆ C S of sets whose union is U. In 1992, Kann 29 proved that the set covering problem, which is NP-complete, can be reduced to the following dominating set problem (hence it's also NP-complete).  Step I. For any ∼ ∈ x y V, we insert a median point z (in red) in the edge ∈ x y E ( , ) with degree 2 in G, i.e., in G we have x ~ z, z ~ y and x, y are not neighbors in G.    Step II. We add a Hub (in blue) to connect all median points.
Step III. Insert sub-median-point (in yellow) for every edge between one median point (in Step I) and Hub.
Step IV. We construct a leaf node (in pink) and the median point (in green) between leaf node and the Hub.
We have the following

Claim 1. There is a dominating set of k or fewer nodes in G if and only if V is the union of (k + 1) or fewer 3-balls.
To verify this claim, we notice the following facts.

(a) For any nodes
The subset of all nodes not in V is a 3-ball centered at the Hub. (c) The geodesic distance between the pink node and any node in V is 5, that means any 3-ball can not contain the pink node and any node of V simultaneously.  On the other hand, considering the minimum 3-ball covering Since the pink point must belong to some ball D i 0 , by fact (c), we have ∉ Λ i 0 . Therefore we have Then (1) follows from (2) and (3). Then Theorem 1 is proved for l = 3. For l ≥ 4, we have the similar construction during reduction. In fact, we insert (l − 2) median points into each edge of G, add a Hub to connect all median points, insert (l − 2) sub-median-point for every edge between one median point and Hub. Finally, we construct the leaf node and connect it to the Hub, insert (l − 2) the median point between leaf node and the Hub. See Fig. 11 for l = 4.

Remark 2.
To prove one problem is NP-complete, we always find a reduction from a known NP-complete problem to our problem. On the other hand, we can always construct a reduction from our (NP) problem to a known NP-complete problem due to the definition of NP-completeness.
We give a proof of the following fact which is pointed out by Song, Gallos, Havlin and Makse 9 .

Proposition 2.
For any fixed size l, the l-box-covering problem is NP-complete. Proof. If l = 2, l-box-covering problem is exactly the clique covering problem, which is NP-complete. If l = 3, given a undirected graph G = (V, E), as in Fig. 12, we construct a new graph G′ = (V′ , E′ ) in polynomial time with respect to the size of G.
Step 1. For any ∼ ∈ x y V, we insert a median point z (in red) in the edge (x, y) with degree 2, i.e., x ~ z, z ~ y and x, y are not neighbors in G′ .
Step 2. We add a Hub (in blue) to connect all median points.
Step 3. We construct a leaf node (in pink) adjacent to the Hub.
We have the following

Claim 2. V is the union of k or fewer cliques if and only if V′ is the union of (k + 1) or fewer 3-boxes.
To verify this claim, we notice the following facts.

For any nodes
2. The subset of nodes not in V is a 3-box. 3. The geodesic distance between leaf node (in pink) and any node in V is 3.  On the other hand, it follows from fact (i) that We also notice that if the pink point belongs to some B i 0 , by fact (iii), we have ∉ Λ i 0 . Therefore we have Then (4) follows from (5) and (6). , and a δ-cube a cube of Euclidean space with side length δ, a δ-box B is a subspace of X such that its diameter less than δ, i.e., d( : the smallest number of balls needed to cover , : the smallest number of boxes needed to cover , : the smallest number of cubes needed to cover ( ), : the maximal number of balls pairwise disjoint, n We recall an elementary inequality 26 which is important in this paper. We give the proof for the self-containedness of this paper.
1 is a packing family of δ-balls, we conclude that On the other hand, every x i must be contained in some δ/2-ball. Therefore, we obtain P δ ≤ B δ/2 . □ We also have ( on ) n n n /2 2 By the above inequalities, the classical result 25,26 on box dimension is that In fact, in the above formula, we take upper box dimension dim or lower box dimension dim when the limit does not exist.

Self-similar set on Euclidean space.
be a self-similar set 30 on a Euclidean space  n , where S i is a similarity with ratio r i , i.e., n and R i is orthogonal. That means any similarity is the compositions of homothety, translation and orthogonal transformation.
We A compact set E is said to be Ahlfors s-regular 26 , if there is a Borel measure μ supported on E satisfying (8). That means the self-similar set satisfying the OSC is Ahlfors regular.
Then the OSC also holds. Let E 1 , E 2 be the self-tree of models 1 and 2 respectively. Then An interesting fact is that this is a deterministic fractal without self-similarity. This is a Moran fractals 31 .
An alternative is a random fractal such that for each time t, we can choose model 1 in probability p and model 2 in probability 1 − p. Then we obtain the above dimension almost surely.

Ahlfors regularity of networks
Proof of Theorem 2. By the definition of Ahlfors regularity, we have d f = s.
Since the network is covered by B l balls of radius l, that means On the other hand, we have P l/2 packing balls of radius l/2, which implies and ( . We also notice that each node has three codings at most.
is non-empty, we may assume that without loss of generality.
For D = (1/3, 0), let θ = ∠y Dy  On the other hand, t t t 1 2 1 2 by the tree structure. It follows from (10) and (11) that we only need to verify (9) for the pairs (y 1 , D) and (D, y 2 ). By the self-similarity, now we only need to prove the case when ∈ ( 1) then (9) follows.
It follows from the above lemma and Remark 3 that Notice that the constant in (12) is independent of t. Now, the growing networks {G t } t are Ahlfors s-regular.

Conclusion
We focus on the NP-completeness of minimum ball-covering problem, propose some heuristic ball-covering algorithms such as GOBC, GDBC, VOBC and VGBC, and compare these algorithms with usual RBC algorithm. Inspired by the notion of measure on fractal, a natural measure on the finite graph is obtained such that the measure of every subset is the cardinality of subset. Based on this measure, we revisit the volume dimension d f and propose the random ball-volume algorithm, which has performance better than the above five minimum covering algorithms due to the NP-completeness. Applying the notion of Ahlfors regularity from fractal geometry, we prove that d B = d ball = d f = s if the network is Ahlfors s-regular. Finally, we investigate the Ahlfors regularity of a class of self-similar trees and random trees which come from the self-similar fractals and Moran fractals respectively. Although we only prove Theorem 3 for self-similar tree of model 1, but our approach can be applied to many self-similar trees, Moran tree and random trees. Essentially, our approach is to embed our networks into a self-similar (or Moran) fractal (on Euclidean space) satisfying the open set condition, using the Ahlfors regularity of corresponding self-similar (or Moran) measure, we can estimate the volume of balls in networks.