Graph fractal dimension and the structure of fractal networks

Abstract

Fractals are geometric objects that are self-similar at different scales and whose geometric dimensions differ from so-called fractal dimensions. Fractals describe complex continuous structures in nature. Although indications of self-similarity and fractality of complex networks has been previously observed, it is challenging to adapt the machinery from the theory of fractality of continuous objects to discrete objects such as networks. In this article, we identify and study fractal networks using the innate methods of graph theory and combinatorics. We establish analogues of topological (Lebesgue) and fractal (Hausdorff) dimensions for graphs and demonstrate that they are naturally related to known graph-theoretical characteristics: rank dimension and product dimension. Our approach reveals how self-similarity and fractality of a network are defined by a pattern of overlaps between densely connected network communities. It allows us to identify fractal graphs, explore the relations between graph fractality, graph colourings and graph descriptive complexity, and analyse the fractality of several classes of graphs and network models, as well as of a number of real-life networks. We demonstrate the application of our framework in evolutionary biology and virology by analysing networks of viral strains sampled at different stages of evolution inside their hosts. Our methodology revealed gradual self-organization of intra-host viral populations over the course of infection and their adaptation to the host environment. The obtained results lay a foundation for studying fractal properties of complex networks using combinatorial methods and algorithms.

1. Introduction

Fractals are geometric objects that are widespread in nature and appear in many research domains, including dynamical systems, physics, biology and behavioural sciences [1]. By Mandelbrot’s classical definition, geometric fractal is a topological space (usually a subspace of an Euclidean space), whose topological (Lebesgue) dimension is strictly smaller than the fractal (Hausdorff) dimension. It is also usually assumed that fractals have some form of geometric or statistical self-similarity [1]. Lately, there was a growing interest in studying self-similarity and fractal properties of complex networks, which is largely inspired by applications in biology, sociology, chemistry and computer science [2–5]. Although such studies are usually based on genuine ideas from graph theory and general topology and provided a deep insight into structures of complex networks and mechanisms of their formation, they are often not supported by a rigorous mathematical framework. As a result, such methods may not be directly applicable to many important classes of graphs and networks [6,7]. In particular, many studies translate the definition of a topological fractal to networks by considering a graph as the finite metric space with the metric being the standard shortest path length, and identifying graph fractal dimension with the Minkowski–Bouligand (box-counting) dimension [4,5].

However, direct applications of the continuous definition to discrete objects such as networks can be problematic. Indeed, under this definition many real-life networks do not have well-defined fractal dimension and/or are not fractal and self-similar. This is in particular due to the fact that these networks have so-called ‘small-world’ property, which implies that their diameters are exponentially smaller than the numbers of their vertices [5]. Moreover, even if the box-counting dimension of a network can be defined and calculated, it is challenging to associate it with graph structural/topological properties. As regards to the phenomenon of network self-similarity, previous studies described it as the preservation of network properties under a length-scale transformation [5]. However, geometric fractals possess somewhat stronger property: they are comprised of parts topologically similar to the whole rather than just have similar features at different scales. Finally, many computational tasks associated with the continuous definitions cannot be formulated as well-defined algorithmic problems and studied within the framework of theory of computational complexity, discrete optimization and machine learning. Thus, it is highly desirable to develop an understanding of graph dimensionsionality, self-similarity and fractality based on innate ideas and machineries of graph theory and combinatorics. There are several studies that translate certain notions of topological dimension theory to graphs using combinatorial methods [8,9]. However, to the best of our knowledge, a rigorous combinatorial theory of graph-theoretical analogues of topological fractals still has not been developed. In this article, we propose a combinatorial approach to the fractality of graphs, which consider natural network analogues of Lebesgue and Hausdorff dimensions of topological spaces from the graph-theoretical point of view. This approach allows to overcome the aforementioned difficulties and provides mathematically rigorous, algorithmically tractable and practical framework for study of network self-similarity and fractality. Roughly speaking, our approach suggests that fractality of a network is more naturally related to a pattern of overlaps between densely connected network communities rather than to the distances between individual nodes. It is worth noting that overlapping community structure of complex networks received considerable attention in network theory and has been a subject of multiple studies [10,11]. Furthermore, such approach allows us to exploit the duality between partitions of networks into communities and encoding of networks using set systems. This duality has been studied in graph theory for a long time [12] and allows for topological and information-theoretical interpretations of network self-similarity and fractality.

The major results of this study can be summarized as follows:

(1) Lebesgue and Hausdorff dimensions of graphs are naturally related to known characteristics from the graph theory and combinatorics: rank dimension [12] and product (or Prague or Nešetřil–Rödl) dimension [13]. These dimensions are associated with the patterns of overlapping cliques in graphs. We underpin the connection between general topological dimensions and their network analogues by demonstrating that they measure the analogous characteristics of the respective objects:

• Topological Lebesgue dimension and graph rank dimension are both associated with the representation of general compact metric spaces and graphs by intersecting families of sets. Such representations have been extensively studied in graph theory [12], where it has been shown that any graph of a given rank dimension encodes the pattern of intersections of a family of finite sets with particular properties. It turned out, that general compact metric spaces of a given Lebesgue dimension also can be approximated by intersecting families of sets with analogous properties.

• Product dimension is a measure of a graph self-similarity, as it defines a decomposition of a graph into its own images under stronger versions of graph homomorphisms.

(2) Fractal graphs naturally emerge as graphs whose Lebesgue dimensions are strictly smaller than Hausdorff dimensions. We analyse in detail the fractality and self-similarity of scale-free networks, Erdös–Renyi graphs and cubic and subcubic graphs. For such graphs, fractality is closely related to edge colourings, and separation of graphs into fractals and non-fractals could be considered as a generalization of one of the most renowned dichotomies in graph theory—the separation of graphs into class 1 and class 2 [14] (i.e. graphs whose edge chromatic number is equal to or , where is the maximum vertex degree of a graph). One of the examples of graph fractals is the remarkable class of snarks [15,16]. Snarks turned out to be the basic cubic fractals, with other cubic fractals being topologically reducible to them.

(3) Lebesgue and Hausdorff dimension of graphs are related to their Kolmogorov complexity—one of the basic concepts of information theory, which is often studied in association with fractal and chaotic systems [17]. These dimensions measure the complexity of graph encoding using so-called set and vector representations. Non-fractal graphs are the graphs for which these representations are equivalent, while fractal graphs possesses additional structural properties that manifest themselves in extra dimensions needed to describe them using the latter representation.

(4) Analytical estimations and experimental results reveal high self-similarity of sparse Erdös–Renyi and Wattz–Strogatz networks, and lower self-similarity of preferential attachment and dense Erdös–Renyi networks. Numerical experiments suggest that fractality is a rare phenomenon for basic network models but could be significantly more common for real networks.

(5) The proposed theory can be used to infer information about the mechanisms of real-life network formation. As an example, we analysed genetic networks representing structures of 323 intra-host Hepatitis C populations sampled at different infection stages. The analysis revealed the increase of network self-similarity over the course of infection, thus suggesting intra-host viral adaptation and emergence of self-organization of viral populations over the course of their evolution.

We expect that the theory of graph fractals developed in this article will facilitate study of fractal properties of graphs and complex networks. One of its possible applications is a representation of various combinatorial tasks as algorithmic problems to be studied within the framework of theory of algorithms and computational complexity.

2. Basic definitions and facts from measure theory, dimension theory and graph theory

Let be a compact metric space. A family of open subsets of is a cover, if . A cover is -cover, if every belongs to at most sets from ; -cover, if for every set its diameter does not exceed ; -cover, if it is both -cover and -cover. Lebesgue dimension  of is the minimal integer such that for every there exists -cover of .

Let be a semiring of subsets of a set X. A function is a measure, if and for any countable collection of pairwise disjoint sets , one has .

Let now be a subspace of an Euclidean space . Hyper-rectangle  is a Cartesian product of semi-open intervals: , where ; the volume of a the hyper-rectangle is equal to . The -dimensional Jordan measure of the set is defined as where the infimum is taken over all finite covers of by disjoint hyper-rectangles. The -dimensional Lebesgue measure of a measurable set is defined analogously, with the infimum taken over all countable covers of by (not necessarily disjoint) hyper-rectangles. Finally, the -dimensional Hausdorff measure of the set is defined as , where and the infimum is taken over all -covers of . These three measures are related: the Jordan and Lebesgue measures of the set are equal, if the former exists, while Lebesgue and Hausdorff measures of Borel sets differ only by a multiplicative constant.

Hausdorff dimension  of the set is the value

(2.1)

Lebesgue and Hausdorff dimension of are related as follows:

(2.2)

The set is a fractal (by Mandelbrot’s definition) [18], if the inequality (2.2) is strict.

Now let be a simple graph. The notation indicates that the vertices are adjacent, and denotes the maximum vertex degree of . We denote by the complement of , that is, the graph on the same vertex set and with two vertices being adjacent whenever they are not adjacent in . Connected components of are called co-connected components of . A graph is biconnected, if there is no vertex or edge (called a bridge), whose removal makes it disconnected.

A graph is a subgraph of , if and . A subgraph is induced by a vertex subset , if it contains all edges with both endpoints in . The complete graph, the chordless path and the chordless cycle on vertices are denoted by , and , respectively. A star  is the graph on vertices with one vertex of the degree and vertices of the degree 1.

A clique of is a set of pairwise adjacent vertices. A clique number  is the number of vertices in the largest clique of . The family of cliques of is a clique cover, if every edge is contained in at least one clique from . The subgraphs forming the cover are referred to as its clusters. A cover is -cover, if every vertex belongs to at most clusters. A cluster  separates vertices , if . A cover is separating, if every two distinct vertices are separated by some cluster.

Now consider a hypergraph (i.e. a finite set together with a family of its subsets called edges). Simple graphs are special cases of hypergraphs. The rank  is the maximal size of an edge of . A hypergraph is strongly -colourable, if one can assign colours from the set to its vertices in such a way that vertices of every edge receive different colours. The vertices of the same colour form a colour class. Strongly -colourable simple graphs are called bipartite. The edge -colouring and edge colour classes of a hypergraph are defined analogously, with the condition that the edges that share a vertex receive different colours. Chromatic number  and edge chromatic number  are minimal numbers of colours required to colour vertices and edges of a hypergraph, respectively.

Intersection graph of a hypergraph is a simple graph with a vertex set in a bijective correspondence with the edge set of and two distinct vertices being adjacent, if and only if . The following theorem establishes a connection between intersection graphs and clique -covers:

Theorem 2.1

[12] A graph is an intersection graph of a hypergraph of rank if and only if it has a clique -cover.

Rank dimension [19] of a graph is the minimal such that satisfies Theorem 2.1. In particular, graphs with are disjoint unions of cliques (such graphs are called equivalence graphs [20] or -graphs [21]).

Categorical product of graphs and is the graph with the vertex set with two vertices and being adjacent whenever and . Product dimension (or Prague dimension or Nešetřil–Rödl in different sources) is the minimal integer such that is an induced subgraph of a categorical product of complete graphs [13].

Equivalent -cover of the graph is a cover of its edges by equivalence graphs. It can be equivalently defined as a clique cover such that the hypergraph is edge -colourable. Relations between product dimension, clique covers and intersection graphs are described by the following theorem that comprises results obtained in several prior studies:

Theorem 2.2

[13,22] The following statements are equivalent:

• (1) ;

• (2) there exists a separating equivalent -cover of ;

• (3) is an intersection graph of strongly -colourable hypergraph without multiple edges;

• (4) there exists an injective mapping , such that whenever for some .

3. Lebesgue dimension of graphs

Lebesgue dimension of a metric space is defined through -covers by sets of arbitrary small diameter. It is natural to transfer this definition to graphs using graph -covers by subgraphs of smallest possible diameter, that is, by cliques. Thus in light of Theorem 2.1, we define Lebesgue dimension of a graph through its rank dimension:

(3.1)

An analogy between Lebesgue dimension of a metric space and rank dimension of a graph is reinforced by Theorem 3.1. This theorem basically extends the analogy from graph theory back to general topology by stating that any compact metric spaces of bounded Lebesgue measure could be approximated by intersection graphs of (infinite) hypergraphs of bounded rank. To prove it, we will use the following fact:

Lemma 3.1

[18] Let be a compact metric space and be its open cover. Then there exists (called a Lebesgue number of ) such that for every subset with there is a set such that .

Theorem 3.1

Let be a compact metric space with a metric . Then if and only if for any there exists a number and a hypergraph on a finite vertex set with an edge set , which satisfies the following conditions:

• (1) ;

• (2) for every such that ;

• (3) for every such that ;

• (4) for every the set is open.

Proof.

Suppose that , and let be the corresponding -cover of . Since is compact, we can assume that a cover is finite, that is, . Let be the Lebesgue number of .

For a point let . Consider a hypergraph with and . Then satisfies conditions (1)–(4). Indeed, , since is -cover. If , then by Lemma 3.1 there is such that , that is, . Condition means that , and so , since . Finally, for every we have , and thus is open.

Conversely, let be a hypergraph with satisfying conditions (1)–(4). Then, it is straightforward to check that is an open -cover of . □

So, whenever for any there is a hypergraph of with edges in bijective correspondence with points of such that two points are close if and only if corresponding edges intersect.

Clique cover consisting of all edges of is a -cover. It implies the following upper bound for :

Proposition 3.2

. The equality holds, if is triangle-free.

4. Hausdorff dimension of graphs

The goal of this section is to demonstrate that the complement product dimension is a graph-theoretical analogue of the Hausdorff dimension. First, we establish a formal connection by proving that this dimension is associated with a graph measure analogous to the Hausdorff measure of topological spaces. Second, we demonstrate that this dimension is associated with a graph self-similarity.

4.1 Graph measure

In order to rigorously define a graph analogue of Hausdorff dimension, we need to define the corresponding measure first. Note that in any meaningful finite graph topology every set is a Borel set. As mentioned above, for measurable Borel sets in Jordan, Lebesgue and Hausdorff measures are equivalent. Thus, further we will consider the graph analogue of Jordan measure. We propose a parameter which is aimed to serve as the graph analogue of the Jordan measure and prove that it indeed satisfies the axioms of measure. Finally, based on this parameter we define the Hausdorff dimension of a graph.

It is known, that every graph is isomorphic to an induced subgraph of a categorical product of complete graphs [13]. Without loss of generality we may assume that , that is, is an induced subgraph of the graph . will be referred to as a space of dimension and as an embedding of into . After assuming that , we may say that every vertex is a vector , and two vertices and are adjacent if and only if for every .

Hyper-rectangle  is a subgraph of , that is defined as follows: , where for every the set is non-empty. The volume of is naturally defined as .

The family of hyper-rectangles is a rectangle co-cover of , if the subgraphs are pairwise vertex-disjoint, and covers all non-edges of , that is, for every pair of non-adjacent vertices there exists such that . We define - volume of a graph as

(4.1)

where the first minimum is taken over all embeddings of into -dimensional spaces and the second minimum—over all rectangle co-covers of . For example, Fig. 1 (left) demonstrates that the two-dimensional volume of the path is equal to 6.

Fig. 1.

Top: Embedding of into a 2-dimensional space and its rectangle co-cover by a hyper-rectangle of volume . Bottom: equivalent -cover defining a self-similarity of a graph . From left to right: the graph ; an equivalent 2-cover of with the clusters of the same colour highlighted in red and green; subgraphs and such that for the contracting family defined by ; contractions and

Based on the definition of -volume, we define a -measure of a graph as follows:

(4.2)

The main theorem of this section confirms that indeed satisfy the axioms of a measure:

Theorem 4.1

Let and be two graphs, and is their disjoint union. Then

(4.3)

The proof of Theorem 4.1 is presented in Supplementary Materials.

Following the analogy with Hausdorff dimension of topological spaces (2.1), we define a Hausdorff dimension of a graph as

(4.4)

Thus, Hausdorff dimension of a graph can be identified with a Prague dimension of its complement.

According to Theorem 2.2, graph Hausdorff dimension is defined by the existence of a separating equivalent clique cover. In a typical case, the colouring requirement is more important than separation requirement. Indeed, two vertices may not be separated by some cluster of a given clique cover only if these two vertices are true twins, that is, they have the same closed neighbourhoods. In most network models and experimental networks presence of such vertices in highly unlikely; besides in most situations they can be collapsed into a single vertex without changing the majority of important network topological properties.

4.2 Self-similarity

The self-similarity of compact metric space is defined using the notion of a contraction [18]. An open mapping is a similarity mapping, if for all , where is called its similarity ratio (such mapping is obviously continuous). If , then it is a contraction. The space is self-similar, if there exists a family of contractions such that .

This definition cannot be directly applied to discrete metric spaces such as graphs, since for them contractions in the strict sense do not exist. To formally and rigorously define the self-similarity of graphs, we proceed as follows. It is convenient to assume that every vertex is adjacent to itself. For two graphs and , a homomorphism [13] is a mapping which maps adjacent vertices to adjacent vertices, that is, for every . A homomorphism is a similarity mapping, if inverse images of adjacent vertices are also adjacent, that is, whenever (it is possible that ). In other words, for a similarity mapping, images and inverse images of cliques are cliques. With a similarity mapping we can associate a subgraph of , which is formed by all edges such that (Fig. 1).

A family of graph similarity mappings , , is a contracting family, if every edge of is contracted by some mapping, i.e. for every there exists such that . The graphs are contractions of . Finally, a graph is self-similar, if (Fig. 1).

Proposition 4.2

Graph is self-similar with a contracting family if and only if there is an equivalent separating -cover of .

Proof.

For a given contracting family and any , the sets , consist of disjoint cliques. By the definition, every edge of is covered by one of these cliques. Therefore is an equivalent -cover of . Furthermore, due to the self-similarity of , for every edge there is a mapping that does not contract it, that is, . Thus, and are separated by the cliques and , and therefore is a separating cover.

Conversely, let be a separating equivalent -cover, where is the set of connected components of the th equivalence graph (some of them may consist of a single vertex). Construct a graph by contracting every clique into a single vertex and the mapping by setting . Then the collection is a contracting family. □

According to Proposition 4.2, all graphs could be considered as self-similar—for example, we can construct trivial similarity mappings by individually contracting each edge. Thus, it is natural to concentrate our attention on non-trivial similarity mappings and measure the degree of the graph self-similarity by the minimal number of similarity mappings in a contracting family, that is, by its Hausdorff dimension. Smaller number of similarity mappings indicates the denser packing of a graph by its contraction subgraphs, that is, the higher self-similarity degree. In particular, the normalized Hausdorff dimension  could serve as a measure of self-similarity.

5. Fractal graphs: analytical study

In this section, we consider only connected graphs. Importantly, the relation (2.2) between Lebesgue and Hausdorff dimensions of topological spaces remains true for graphs.

Proposition 5.1

For any graph ,

Proof.

Let product dimension of a graph is equal to . Then by Theorem 2.2 is an intersection graph of strongly -colourable hypergraph. Since rank of every such hypergraph obviously does not exceed , Theorem 2.1 implies, that . □

Thus, Proposition 5.1 allows us to define graph fractals analogously to the definition of fractals for topological spaces: a graph is a fractal, if , that is, . In particular, we say that a fractal graph is -fractal, if . For example, the graph on Fig. 1 (right) is self-similar, but not fractal, since .

As the first example of a fractal graph, we consider so-called Sierpinski gasket graphs [23]. They are associated with the Sierpinski gasket—well-known topological fractal with a Hausdorff dimension . Edges of are line segments of the th approximation of the Sierpinski gasket, and vertices are intersection points of these segments (Fig. 2). Figure 2 demonstrates that Sierpinski gasket graph is 1-fractal. In fact, all Sierpinski gasket graphs are fractals, as the following theorem indicates (the proof can be found in Supplementary material):

Fig. 2.

Sierpinski gasket graphs and the optimal equivalent separating -cover of . Clusters of the same colour are highlighted in red, green and blue. is a fractal: every vertex is covered by 2 clusters, while the clusters can be coloured using three colours.

Theorem 5.2

For every Sierpinski gasket graph is a fractal with and

In the remaining part of this section, we will study fractality of more substantial classes of networks.

5.1 Triangle-free graphs

Let denotes the edge chromatic number of a graph . Classical Vizing’s theorem [14] states that , that is, the set of all graphs can be partitioned into two classes: graphs, for which (class 1) and graphs, for which (class 2).

By Proposition 3.2, , if contains no triangles. For such graphs, we have

Proposition 5.3

Triangle-free fractals are exactly triangle-free graphs of class 2.

Proof.

The statement holds, if . Suppose that has vertices. For such graphs, every clique cover is a collection of its edges and vertices. However, since is connected, for every pair of vertices there is an edge that separates them. Therefore, we may assume that the clique cover consists only of edges, and a feasible assignment of colours to the cliques is an edge colouring. Thus, it is true that (i.e. ), and the statement of the proposition follows. □

In particular, bipartite graphs are triangle-free graphs of class 1 [24]. Therefore bipartite graphs (and trees in particular) are not fractals, even though some of them may have high degree of self-similarity (e.g. binary trees). It also should be noted that although some known geometric fractals are called trees (e.g. so-called -trees), they are not discrete object, and their fractality is associated with their drawings on a plane; thus our framework does not apply to them.

5.2 Scale-free graphs

Recall that scale-free networks are graphs whose degree distribution (asymptotically) follows the power-law, that is, the probability that a given vertex has a degree could be approximated by the function , where is a constant and is a scaling exponent. There is a number of models of scale-free networks of different degree of mathematical rigour known in the literature, including various modifications of the preferential attachment scheme. Following [25,26], we will consider a more formal probabilistic model. Assume without loss of generality that [26]. For each vertex , we assign a weight . Then, we construct a graph by independently connecting any pair of vertices by an edge with the probability , where and is a constant.

From now on, we will use the following standard nomenclature [27]. An induced subgraph isomorphic to a cycle is a hole, a hole with the odd number of vertices is an odd hole. The star is the claw, the 4-vertex graph consisting of two triangles with a common edge is the diamond and the 5-vertex graph consisting of two triangles with a common vertex is the butterfly.

Theorem 5.4

For graphs with and , with high probability and .

Proof.

It has been proved in [26] that the clique number of a graph with the scaling constant is either 2 or 3 with high probability, that is, as for -vertex scale-free graphs that have power-law degree distribution with the exponent . The following lemma complements this fact:

Lemma 5.1

• (1) For , graphs with high probability do not contain diamonds.

• (2) For , graphs with high probability do not contain butterflies.

Proof.

(1) Let vertices form a diamond, where and are non-adjacent. For the probability of this event, we have

Thus, the total probability that these vertices form a diamond can be estimated as

(5.1)

Let be the number of diamonds in . For the expected value of this random variable, we have

(5.2)

Using an integral upper bound, it is easy to see that

(5.3)

Furthermore, whenever .

Select such that . Then, we have . Thus . Finally, by Markov’s inequality we have as .

(2) Similarly to (1), for the probability that vertices form a butterfly with the centre we have

Thus, for the number of butterflies its expectation satisfy the following chain of inequalities:

(5.4)

 

(5.5)

Like in (1), select a small number such that ). Then by (5.3), we have and . Thus, . After applying Markov’s inequality the claim follows. □

Thus, a typical scale-free graph with the scaling exponent has only cliques of size 2 and 3, and every vertex belongs to at most one triangle. For such graphs, a minimal clique cover defining consists of all triangles and the edges that do not belong to triangles. Let be the number of triangles which include a vertex . Then with high probability , and the statement about follows. By Vizing’s theorem, two-vertex clusters of this cover could be coloured by at most colour, and one additional colour could be used to colour the triangles. Thus, , which proves the statement for .

Note that the similar arguments may be used to prove Theorem 5.4 for more general model, when the weights are identically distributed random variables with power-law-distributed tail (see [26]). In particular, for preferential attachment graph following this model, Theorem 5.4 and the estimations of from [28,29] imply that its Hausdorff dimension is roughly asymptotically equivalent to (up to an arbitrarily slowly growing multiplicative factor).

5.3 Erdös–Renyi graphs

Similar properties of dimensions hold for sparse Erdös–Renyi graphs , where :

Theorem 5.5

For Erdös–Renyi graphs, with and , with high probability and .

Indeed, it is implied by the following simple statement and considerations analogous to the ones in Theorem 5.4:

Lemma 5.2

• (1) For , graphs with high probability do not contain .

• (2) For , graphs with high probability do not contain diamonds.

• (3) For , graphs with high probability do not contain butterflies.

Proof.

Let be the number of diamonds in . Then , and (2) follows from Markov’s inequality. Other statements can be proved analogously. □

Note that for sparse Erdös–Renyi graphs, [30] and thus . For dense Erdös–Renyi graphs, the asymptotics is described by Theorem 6.2 (see Section on information-theoretic connection).

5.4 Cubic and subcubic graphs

A graph is cubic, if all its vertices have the degree three. Cubic graphs arise naturally in graph theory, topology, physics and network theory [31,32], and has been extensively studied. Among cubic graphs, the class of so-called snarks is distinguished in graph theory (the most famous snark—Petersen graph—is shown on Fig. 3). Snark is defined as a biconnected cubic graph of class 2 [15]. Snark is non-trivial, if it is triangle-free [15]. Snarks constitute important class of graphs, which has been studied for more than a century and whose structural properties continue to puzzle researchers to this day [15]. Discovery of new non-trivial snarks is a valuable scientific result¹, not unlike the discovery of new fractals, with many known snarks also possessing high degree of symmetry and being constructed by certain recursive procedures. According to Proposition 5.3, this analogy is well-justified, as non-trivial snarks are indeed fractals according to our definition. As shown by Theorem 5.7, the inverse relation also holds, as cubic fractals could be reduced to snarks.

Fig. 3.

Left: Pendant triple contraction (top) and pendant edge identification (bottom) operations. Identified vertices and edges are highlighted in red. Right: Transformations of a cubic graph . For each transformation, removed edges are highlighted in blue, vertices involved in the pendant triple contraction are highlighted in red, and vertices and edges involved in the pendant edge identification are highlighted in green. The top transformation converts into the Petersen snark, which is a fractal. However, is not fractal, since the bottom transformation converts it into 3-edge-colourable cubic graph.

The connection between fractality and class 2 graphs continues to hold for wider class of subcubic graphs (i.e. the graphs with ). Indeed, consider a graph obtained from by removal of edges of all its triangles. Given that , the following theorem describes subcubic fractals:

Theorem 5.6

Let be a subcubic connected graph with vertices.

• (1) is 1-fractal, if and only if it is claw-free, but contains the diamond or an odd hole.

• (2) is 2-fractal, if and only if it contains the claw and is of class 2.

Proof.

First we will prove (1). By Theorem 2.1, if and only if is a line graph of a multigraph. Such graphs are characterized by a list of seven forbidden induced subgraphs, only one of which () has the maximal degree, which does not exceed 3 [12]. Therefore if and only if is claw-free. By Theorem 2.2, if and only if is a line graph of a bipartite graph. These graphs are exactly (claw,diamond,odd-hole)-free graphs [27]. By combining these facts, we get that , if and only if is claw-free, but contains diamond or odd hole.

Now we will prove (2). Suppose that contains the claw or, synonymously, some vertex of does not belong to a triangle. In this case, , and therefore is either 3- or 4-edge colourable. We will demonstrate that if and only if is of class 1. The necessity is obvious, so it remains to prove the sufficiency. Let be a 3-edge colouring of , and be the clique cover of with all clusters being single edges, whose colours are set by . The cover could be extended to the equivalent -cover of as follows. Given that , there are two possible arrangements between pairs of triangles in .

(1) There are two triangles and which share an edge . Suppose also that and , (it is possible that ). If, without loss of generality , then the edges of and could be covered by single-edge cliques, whose colours could be set as , , , . If, say, , , then we may cover and by the cliques , , with colours , , .

(2) A triangle does not share edges with other triangles. Suppose that , , , . All these vertices are distinct, and and do not belong to any triangles and therefore are present in . If the colours and are distinct, then cover by single-edge cliques with colours , , . If, alternatively, some of these colours are identical, then there is a colour not present among them. In this case cover with the single clique of the colour .

If in the resulting cover some vertex is covered by a single triangle , add the single-vertex clique with an appropriate colour . Thus, the constructed cover is a separating equivalent -cover of . This concludes the proof. □

Theorem 5.6 states that subcubic 1-fractals are reducible to the diamond and odd cycles, while subcubic 2-fractals could be reduced to class 2 graphs. The next theorem will demonstrate, that cubic 2-fractals could be reduced to snarks.

Let be a cubic graph with . In this case every vertex of has the degree , or . Vertices of degree 1 are further referred to as pendant vertices, and edges incident to pendant vertices as pendant edges. We will establish the deeper relation between the topology of general cubic fractals and snarks. By Theorem 5.6, the case of 1-fractals is rather simple, so we will concentrate on 2-fractals. Thus, we will assume that contains a claw. Consider the following graph operations:

• (O1) Pendant triple contraction consists in replacement of pendant vertices , and by a single vertex , which is adjacent to all neighbours of , and (Fig. 3).

• (O2) Pendant edge identification of two edges and with and consists in removal of and and replacement of and with the edge (Fig. 3).

Let be the graph obtained from by removal of isolated vertices and edges. is of class 1 if and only if so is .

Lemma 5.3

Suppose that is of class 1. Let be its 3-edge colouring and , and be the numbers of pendant edges with colours 1, 2 and 3, respectively. Then ,, are either all odd or all even.

Proof.

Let , where . For each colour , consider pairs of -coloured pendant edges, identify the edges from each pair, and assign to each newly added edge the colour . So, the resulting graph is also of class 1, has all vertex degrees equal to 1 or 3 and contains pendant edges of colour .

Now consider a subgraph of formed by edges of colours and . Obviously, is a disjoint union of even cycles and, possibly, a single path with distinctly coloured end-edges. If the path is not present, then , otherwise . □

Theorem 5.7

The cubic graph is 2-fractal if and only if it contains a claw and any cubic graph obtained from by pendant edge identifications and pendant triple contractions either has a bridge or is a snark.

Proof.

It can be easily shown that if a cubic graph has a bridge, then it is of class 2 [15]. Thus, the statement of the theorem is equivalent to the following statement: the cubic graph , which contains a claw, is not 2-fractal if and only if it is possible to construct a cubic graph of class 1 from by pendant edge identifications and pendant triple contractions.

To prove the necessity, suppose that is not 2-fractal, for example, the graph is of class 1. Consider any 3-edge colouring of and identify pendant edges of the same colour, as described in Lemma 5.3. If after this operation all pendant edges are eliminated, then the desired graph of class 1 is constructed. Otherwise, by Lemma 5.3 contains three pendant edges of pairwise distinct colours. Then the desired graph can be constructed by contracting the pendant end-vertices of these edges.

Conversely, suppose that the graph of class 1 is obtained from by pendant edge identifications and pendant triple contractions. Consider any 3-edge colouring of . Obviously, could be transformed into a 3-edge colouring of by assigning the colour to the identified edges and . □

Thus, Theorem 5.7 states that the biconnected cubic graph is 2-fractal whenever any sequence of removal of triangle edges, isolated edges and vertices, pendant triple contractions and pendant edge identifications, which preserve the graph connectivity, transforms it into a snark. Figure 3 provides an example of such transformations.

6. Network fractality and network complexity

Theorems 2.1 and 2.2 allow to interpret graph Lebesgue and Hausdorff dimensions and fractality from the information-theoretical point of view. Indeed, graphs with could be described by assigning to every vertex a set of integer ‘coordinates’ represented by hyperedges of a -uniform hypergraph such that . Importantly, these coordinates are non-ordered, and edges of are defined by a presence of a shared coordinate for their end vertices. In contrast, graphs with are defined by ordered vectors of coordinates (Theorem 2.2,4), and an adjacency of a pair of vertices is determined by a presence of a shared coordinate on the same position. Thus, non-fractal graphs are the graphs for which the set and vector representations are equivalent, while fractal graphs have additional structural properties that manifest themselves in extra dimensions needed to describe them using a vector representation. The whole concept is illustrated on Fig. 4.

Fig. 4.

(A) Sierpinski gasket graph ; (B) Its optimal equivalent separating -cover. Clusters of the same colour are highlighted in red, green and blue. (C) Left: hypergraph such that . The edges of correspond to the vertices of , with two vertices being adjacent if and only if the corresponding edges intersect. Right: table with the corresponding unordered set coordinates encoding the graph . (D) Left: embedding of the graph into 3-dimensional space such that two vertices are adjacent whenever they share a coordinate. Colours highlight different clusters. Right: table with the corresponding ordered vector coordinates encoding the graph . is a fractal, and thus the dimensionalities of encodings (C) and (D) differ.

Relations between graph dimension and information complexity could be analysed using a Kolmogorov complexity. Informally, Kolmogorov complexity of a string could be described as a length of its shortest lossless encoding. Formally, let be the set of all finite binary strings and be a computable function. Kolmogorov complexity  of a binary string with respect to is the minimal length of a string such as . Since Kolmogorov complexities with respect to any two functions differ by an additive constant [17], it can be assumed that some canonical function is fixed. For two strings , a conditional Kolmogorov complexity  is a length of a shortest encoding of , if is known in advance.

Every connected graph can be encoded using the string representation of an upper triangle of its adjacency matrix. Kolmogorov complexity of a graph could be defined as a Kolmogorov complexity of that string [33]. In addition, the conditional graph Kolmogorov complexity is often considered, which is the complexity given that the number of vertices is known. Obviously, and . Alternatively, -vertex connected labelled graph can be represented as a list of edges with ends of each edge encoded using their binary representations concatenated with a binary representation of . It gives estimations , [17,33].

Let and . Then is an induced subgraph of a product

(6.1)

where . Thus, by Theorem 2.2, and could be encoded using a collection of vectors , , . Such encoding could be stored as a string containing binary representations of coordinates using bits concatenated with a binary representations of and , . The length of this string is . Analogously, if and are given, then the length of encoding is . Thus, the following estimations hold:

Proposition 6.1

(6.2)

 

(6.3)

Let . Then we have   By minimality of the representation (6.1), we have . Thus , . So, Hausdorff (Prague) dimension could be considered as a measure of descriptive complexity of a graph.

Relations between Hausdorff (Prague) dimension and Kolmogorov complexity could be used to derive lower bound for Hausdorff dimension of a dense Erdös–Renyi random graph. Formally, let be a graph property and be the set of labelled n-vertex graphs having this property. The property holds for almost all graphs [34], if as , that is, the probability that the sparse Erdös–Renyi random graph has the property converges to 1 as . We will use the following lemma:

Lemma 6.1

[35] For every and , there are at least  -vertex labelled graphs such that .

The following theorem states that almost all sparse Erdös–Renyi graphs have large Hausdorff dimension:

Theorem 6.2

For every , almost all sparse Erdös–Renyi graphs have Hausdorff dimension such that

(6.4)

where is a constant.

Proof.

The upper bound has been proved in [36], so we will prove the lower bound. Let . Consider a graph with . From (6.3), we have . Using the fact, that , it is straightforward to check that . Therefore, we have

(6.5)

Let be the set of all graphs such that

(6.6)

Using Lemma 6.1 with , we conclude that , and so almost all graphs have the property .

Now, it is easy to see that for graphs with the property and with the inequality (6.4) holds. It follows by combining inequalities (6.5)-(6.6) and using the fact that . □

7. Fractality and self-similarity of networks: experimental study

7.1 Calculation of Lebesgue and Hausdorff dimensions

The problems of calculating Hausdorff and Lebesgue dimension of graphs are algorithmically hard. Indeed, the problem of verifying whether is NP-complete for [37] (the complexity for is unknown). It is easy to see that the problems of checking whether and deciding whether a given graph is a fractal are also NP-complete. It follows from Proposition 5.3 and NP-completeness of the edge chromatic number problem for triangle-free cubic graphs [38]. Therefore, we use Integer Linear Programming (ILP) for calculation of Hausdorff and Lebesgue dimensions and detection of fractal graphs. Let us call a clique -cover and a separating equivalent -cover optimal, if and . For Lebesgue dimension, we are looking for an optimal clique cover which consists of a minimal number of clusters. For such cover, every maximal clique of contains at most one cluster (otherwise, we can join the clusters contained in the same clique). Using this fact, we proceed as follows. Let be the list of maximal cliques of found using Bron–Kerbosch algorithm [39]. Then, an optimal clique cover is found by solving the following ILP problem:

(7.1)

(7.2)

(7.3)

(7.4)

(7.5)

Here, is the variable representing the rank dimension of ; the binary variables and indicate whether a vertex and an edge are covered by a cluster contained in ; and are binary constants indicating whether a corresponding vertex/edge belongs to and . The constraints (7.2) state that every vertex is covered by at most cliques; the constraints (7.3) enforce the requirement that every edge is covered by at least one clique and the constraints (7.4) ensure that an edge is covered by a clique if and only if both its ends are covered by it.

As before, we assume that a given graph has no true twins. In this case, the Hausdorff dimension of the graph is found by generating the set of all cliques of and solving the following ILP problem:

(7.6)

(7.7)

(7.8)

(7.9)

(7.10)

Here, is an upper bound on the Hausdorff dimension of the graph . The binary variable indicates whether the clique is coloured by a colour , and the binary variable indicates whether the colour is used; the relation between these variables is enforced by the constraints (7.7). The constraints (7.9) state that every clique receives at most one colour; it is possible that a clique does not have any colour, which means that a clique is not selected as a cluster. By the constraints (7.9), all cliques containing any given vertex receive different colours, and the constraints (7.10) ensure that at least one of the cliques covering any edge receives a colour (i.e. selected as a cluster). If the Lebesgue dimension has been previously estimated, then the calculations could be accelerated by removal from of all cliques that intersect at most other cliques.

For all networks described below, Lebesgue and Hausdorff dimensions were calculated using Gurobi 8.1.1.

7.2 Network models

Three common models have been considered: preferential attachment, Erdös–Renyi and Watts–Strogatz. For each model, 1350 networks with 20–150 vertices have been generated using MIT Matlab Toolbox for Network Analysis [40]. For a given network size, the model parameters were selected in a way resulting in the same network density for all three models.

For preferential attachment and Erdös–Renyi networks, their average Hausdorff dimensions grew as () and (), respectively, just as suggested by the estimations in Section 5 (Fig. 5). Hausdorff dimension of Watts–Strogatz networks showed the behaviour similar to that of the latter (). Importantly, none of the analysed preferential attachment and Erdös–Renyi networks were fractal. In contrast, Watts–Strogatz fractal networks have been observed, although their proportion exponentially decreases with the growth of (Fig. 5). It suggests, that for the analysed models the network fractality is rare. It is known that almost all graphs (in the sense of Erdös–Renyi graphs ) are of class 1 [34]. Thus graph fractality inherits the asymptotic behaviour of edge colourings dichotomy.

Fig. 5.

Top: expected Hausdorff dimensions for Preferential Attachment (left), Erdös-Renyi (center) and Watts-Strogatz (right) networks. Bottom: (left) observed frequency of fractal networks for Wattz-Strogatz model; (right): distributions of normalized Hausdorff dimensions for genetic networks of recent and persistent intra-host HCV populations.

7.2 Real networks with known communities

To calculate Lebesgue and Hausdorff dimensions of a graph , it is required to find the sets of communities of representing clusters of its optimal -cover and equivalent separating -cover. If the communities are known in advance, we may consider restricted Lebesgue dimension and restricted Hausdorff dimension with respect to these communities that can be defined as follows: given a hypergraph with all twin vertices removed, and .

We calculated the restricted dimensions of eight real-life networks with known ground-truth communities from Stanford Large Network Dataset [41]. To calculate Hausdorff dimensions, the standard ILP formulation for the Vertex Colouring problem has been utilized. If the solver was not able to handle the full community dataset, we analysed 5000 communities of highest quality provided by the database’s curators. Three out of eight networks have been found to be fractal. It is significantly higher proportion than suggested by the analysis of network models above, thus suggesting that for real networks the fractality is more prevalent.

7.4 Viral genetic networks

For a given biological population, the vertices of its genetic network [42] are genomes of the members of the population, and two vertices are adjacent if and only if the corresponding genomes are genetically close. Genetic network represents a snapshot of the mutational landscape of the population, whose structure is shaped by selection pressures, epistatic interactions and other evolutionary factors [43].

RNA viruses exist in infected hosts as highly heterogeneous populations of genomic variants or quasispecies. Recently, indications of self-similarity in quasispecies genetic networks were found (D.S. Campo, personal communication). We investigated this phenomenon using the proposed theoretical framework. We considered genetic networks of intra-host Hepatitis C (HCV) populations of infected individuals at early () and persistent () stages of infection [44]. The networks were constructed using high-throughput sequencing data of HCV Hypervariable Region 1 (HVR1), with two HVR1 sequences being adjacent, if they differ by a single mutation. For each network, the dimensions of the largest connected component has been calculated with the time limit of s. Solutions have been obtained for networks with vertices in average.

The normalized Hausdorff dimensions of networks of persistent populations was found to be significantly lower than for recent populations (, Kruskal–Wallis test, Fig. 5), thus indicating significantly higher level of their self-similarity. This finding is biologically significant. Indeed, one of fundamental questions in the study of pathogens is the role of different evolutionary mechanisms in the infection progression. For HCV, the standard assumption, that the major driving force of intra-host viral evolution is the continuous immune escape, has been put into question by the series of observations that suggest high level of intra-host viral adaptation [42,45]. Increase in self-similarity of HCV genetic networks implies the gradual self-organization of viral populations and emergence of structural patterns in population composition and points to the presence of a dynamical mechanism of their formation at later stages of infection, which may be associated with the higher level of adaptation and specialization of viral variants. Thus, it supports the adaptation hypothesis and is consistent with the recently proposed models of viral antigenic cooperation [46,47], which suggests the emergence of complementary specialization of viral variants and their adaptation to the host environment as a quasi-social system.

8. Conclusions

We presented a theoretical framework for study of fractal properties of networks, which is based on the combinatorial and graph-theoretical notions and methods. We anticipate that the proposed framework could be useful for theoretical studies of properties of network models as well as for analysis of experimental networks which arise in biology, epidemiology and social sciences. In particular, this study has been triggered by biological questions raised by studies of genetic and cross-immunoreactivity networks of RNA viruses, such as HIV and Hepatitis C [42,47].

We would like to emphasize that combinatorial Lebesgue and Hausdorff dimensions described in this article should not be considered as approximations of their previously studied variants based on box-counting approach. The reason is that they are based on different theoretical frameworks and reflect different network features. One of the goals of this article was to demonstrate that for finite graphs the combinatorial framework is more appropriate than the straightforward translation of continuous definitions, especially when we are interested in studying graph fractality. Our major arguments in favour of this claim could be summarized as follows:

• Box-counting definition of a fractal dimension involves a limit transition. Therefore strictly speaking it is applicable to graph sequences rather than individual graphs. This makes the reliable estimation of box-counting dimensions of real networks problematic, because their intrinsic finiteness and discreteness prevents the accumulation of a sufficient number of data points to get reliable finite approximations of continuous functions—the fact previously noted in the literature [48,49]. In particular, this applies to biological networks studied in this article. In contrast, we define a fractal dimension as a combinatorial parameter that can be calculated for any finite graph without the need for approximation.

  The conventional topological graph dimension is 1, while the box-counting dimension is usually greater than 1. This makes almost all graphs fractal in terms of Mandelbrot’s definition. Such understanding of fractality is not practically useful. In contrast, the combinatorial definition allows only some graphs to be fractal, and lead to substantial and deep structural properties distinguishing fractal graphs from non-fractal graphs.

  The properties of combinatorial Lebesgue dimension, Hausdorff dimension and fractality of graphs very well agree with the properties of the corresponding notions from general topology: they are naturally related to structural self-similarity of graphs, representations of graph as measurable spaces and by set systems, as well as their descriptive complexity as information systems. To the best of our knowledge, few such connections have been established for box-counting definitions.

  Combinatorial approach approach is also practically useful and reflect the properties of real systems, as demonstrated by the example with viral genetic networks.

All of the above do not mean that combinatorial dimension and box-counting dimension have nothing in common. On the contrary, the example of Sierpinski gasket graph demonstrate that these notions agree with each other, when the scheme for construction of the continuous fractal is essentially combinatorial. In that case, for example, the combinatorial Hausdorff dimension is obtained by rounding of the continuous dimension. We expect that more such examples will be found in the future.

The ideas presented in this article may facilitate study of properties of networks using convergent machineries of graph theory, general topology, algorithmic information theory and discrete optimization. Furthermore, the problems of detection of fractal properties could be now formulated and studied as rigorously defined algorithmic problems. One such problem is the detection of fractal graphs, another—calculation of invariants measuring how close is the given graph from being a fractal (in terms of number of edges or certain modification operations). It should be noted, however, that these problems are likely to be algorithmically hard since, as discussed above, the fractality recognition problem is NP-complete. Thus, approximation algorithms and heuristics for these problems should be developed, and the graph classes where the problems become polynomially solvable should be identified. Another important direction of future research is a deeper understanding of structural properties of graph fractals in general and in particular graph classes, as well as identification of network construction models which produce fractals. In particular, our results suggest that fractality is more common for real-life networks than can be concluded from analysis of standard network models.

Supplementary data

Supplementary data are available at COMNET online.

Funding

The National Institutes of Health (1R01EB025022) ‘Viral Evolution and Spread of Infectious Diseases in Complex Networks: Big Data Analysis and Modeling’.