Graph fractal dimension and structure of fractal networks: a combinatorial perspective

12/23/2019
by   Pavel Skums, et al.
0

In this paper we study self-similar and fractal networks from the combinatorial perspective. 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 (or Prague or Nešetřil-Rödl) 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 colorings and graph Kolmogorov complexity, and analyze 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 to evolutionary studies by revealing the growth of self-organization of heterogeneous viral populations over the course of their intra-host evolution, thus suggesting mechanisms of their gradual adaptation to the host's environment. As far as the authors know, the proposed approach is the first theoretical framework for study of network fractality within the combinatorial paradigm. The obtained results lay a foundation for studying fractal properties of complex networks using combinatorial methods and algorithms.

READ FULL TEXT VIEW PDF

Authors

page 1

page 2

page 3

page 4

08/17/2017

Nowhere Dense Graph Classes and Dimension

Nowhere dense graph classes provide one of the least restrictive notions...
04/02/2021

Some Combinatorial Problems in Power-law Graphs

The power-law behavior is ubiquitous in a majority of real-world network...
07/17/2018

The Simplex Geometry of Graphs

Graphs are a central object of study in various scientific fields, such ...
02/12/2019

Analysis of Coauthorship Network in Political Science using Centrality Measures

In recent era, networks of data are growing massively and forming a shap...
03/02/2022

Graph Rewriting and Relabeling with PBPO+: A Unifying Theory for Quasitoposes

We extend the powerful Pullback-Pushout (PBPO) approach for graph rewrit...
12/04/2020

Asymptotic Dimension of Minor-Closed Families and Assouad-Nagata Dimension of Surfaces

The asymptotic dimension is an invariant of metric spaces introduced by ...
03/05/2018

The morphospace of language networks

Language can be described as a network of interacting objects with diffe...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.

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 [falconer2004fractal]. 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 [falconer2004fractal].

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 [song2005self, shanker2007defining, dorogovtsev2013evolution, newman2003structure]. Although such studies are usually based on geniune 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 [li2005towards, willinger2009mathematics]. 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 [song2005self, shanker2007defining]. 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 [song2005self]. 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 [song2005self]. 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

[smyth2010topological, evako1994dimension]. 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 paper 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 [palla2005uncovering, ahn2010link]. 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 [berge1984hypergraphs], 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 [berge1984hypergraphs] and product (or Prague or Nešetřil-Rödl) dimension [hell2004graphs]. 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 [berge1984hypergraphs], 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.

  • Rank dimension defines a decomposition of a graph into its own images under stronger versions of graph homomorphisms, and can be interpreted as a measure of a graph self-similarity.

2) Fractal graphs naturally emerge as graphs whose Lebesgue dimensions are strictly smaller than Hausdorff dimensions. We analyze 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 colorings, 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 [vizing1964estimate] (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 [gardner1977mathematical, chladny2010factorisation]. 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 [li2009Kolmogorov]

. 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 analyzed 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 explored in this paper will facilitate study of fractal properties of graphs and complex networks. One of its possible applications is the possibility to represent various computational 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 3 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

(1)

Lebesgue and Hausdorff dimension of are related as follows:

(2)

The set is a fractal (by Mandelbrot’s definition) [edgar2007measure], if the inequality (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 , i.e. 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 -colorable, if one can assign colors from the set to its vertices in such a way that vertices of every edge receive different colors. The vertices of the same color form a color class. Strongly -colorable simple graphs are called bipartite. The edge -coloring and edge color classes of a hypergraph are defined analogously, with the condition that the edges that share a vertex receive different colors. Chromatic number and edge chromatic number are minimal numbers of colors required to color 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 1.

[berge1984hypergraphs] A graph is an intersection graph of a hypergraph of rank if and only if it has a clique -cover.

Rank dimension [metelsky2003] of a graph is the minimal such that satisfies Theorem 1. In particular, graphs with are disjoint unions of cliques (such graphs are called equivalence graphs [alon1986covering] or -graphs [tyshkevich1989matr]).

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 or Nešetřil-Rödl in different sources) is the minimal integer such that is an induced subgraph of a categorical product of complete graphs [hell2004graphs].

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 -colorable. 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.

[hell2004graphs, babaits1996kmern] The following statements are equivalent:

1) ;

2) there exists a separating equivalent -cover of ;

3) is an intersection graph of strongly -colorable 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, i.e. by cliques. Thus in light of Theorem 1 we define Lebesgue dimension of a graph through its rank dimension:

(3)

An analogy between Lebesgue dimension of a metric space and rank dimension of a graph is reinforced by Theorem 3. 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 1.

[edgar2007measure] 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.

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, i.e. . 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 1 there is such that , i.e. . 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 1.

. 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 the complement product dimension is associated with a graph measure analogous to the Hausdorff measure of topological spaces. Second, we demonstrate how this dimension is related to graph self-similarity.

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 [hell2004graphs]. Without loss of generality we may assume that , i.e. 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 , i.e. for every pair of non-adjacent vertices there exists such that . We define - volume of a graph as

(4)

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. 3 (left) demonstrates that the 2-dimensional volume of the path is equal to 6.

(a)
(b)
Figure 3: Left: Embedding of into a 2-dimensional space and its rectangle co-cover by a hyper-rectangle of volume . Right: 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 color 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:

(5)

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

Theorem 4.

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

(6)
Proof.

It can be shown [babai1992linear], that can be embedded into a categorical product of complete graphs if and only if both and have such embeddings. Therefore the relation (6) holds, if some of the terms are equal to . If all ,, can be embedded into , we proceed with the series of claims.

Let . We write , if every vertex from is adjacent to every vertex from . Denote by the -th projection of , i.e. the set of all -coordinates of vertices of : In particular, . The following claim follows directly from the definition of .

Claim 1.

if and only if for every .

Assume that is a minimal rectangle co-cover of a minimal embedding , i.e. . We will demonstrate, that has a rather simple structure.

Claim 2.

Let . Then for every , and every .

Proof.

First note, that for every hyper-rectangle , every coordinate and every there exists such that . Indeed, suppose that it does not hold for some . If , then it means that . Thus, is a rectangle co-cover, which contradicts the minimality of . If , consider a hyper-rectangle . The set is a rectangle co-cover, and the -volume of is smaller than the -volume of . Again it contradicts minimality of .

Now assume that for some distinct and we have . Then there exist and such that . So, , and therefore by the definition is covered by some . The hyper-rectangle intersects both and , which contradicts the definition of a rectangle co-cover. ∎

Claim 3.

Let , . Then the set coincides with the set of co-connected components of .

Proof.

Claims 1 and 2 imply, that vertices of distinct hyper-rectangles from the co-cover are pairwise adjacent. So, for every , .

Let be the set of co-connected components of (thus ). Consider a component and the sets , . We have and for all . Therefore, due to co-connectedness of , exactly one of the sets is non-empty.

So, we have demonstrated, that every co-connected component is contained in some of the sets . Now, let some consists of several components, i.e. without loss of generality , . Let , . By Proposition 1 we have for all , . Consider hyper-rectangles ,…,. Those hyper-rectangles are pairwise vertex-disjoint, and for all . Since every pair of non-connected vertices of is contained in some of its co-connected components, we arrived to the conclusion, that the set is a rectangle co-cover. Moreover, , and therefore , which contradicts the minimality of . ∎

Claim 4.

Let be the set of co-connected components of . Then .

Proof.

By Claim 3, , and every component is contained in a unique hyper-rectangle , . Every pair of non-adjacent vertices of belong to some of its co-connected components. This fact, together with the minimality of , implies that is the minimal hyper-rectangle that contains . Thus . ∎

Claim 5.

If is connected, than is the minimal volume of all -dimensional spaces , where can be embedded.

Claim 6.

Let be the set of co-connected components of . Then .

Proof.

Suppose that is the set of co-connected components of , and . Claims 3 and 4 imply that is a rectangle co-cover of an embedding of . Therefore we have and thus . So, it remains to prove the inverse inequality.

Let be a minimal embedding of into . By Claim 3, every minimal hyper-rectangle co-cover of consists of a single hyper-rectangle or, in other words, is embedded into . Now we can construct an embedding of into and its hyper-rectangle co-cover. It can be done as follows. Consider , , and let . Obviously, . Now embed into . Let be those embeddings. By Claim 1, , so is indeed an embedding of .

All hyper-rectangles are pairwise disjoint. Since every pair of non-adjacent vertices belong to some subgraph , we have that is hyper-rectangle co-cover of . Therefore . ∎

Now the equality (6) follows from Corollary 6. In concludes the proof of Theorem. ∎

Following the analogy with Hausdorff dimension of topological spaces (1), we define a Hausdorff dimension of a graph as

(7)

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

According to Theorem 2, graph Hausdorff dimension is defined by the existence of a separating equivalent clique cover. In a typical case the coloring 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, i.e. they have the same closed neighborhoods. 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.

Self-similarity

The self-similarity of compact metric space is defined using the notion of a contraction [edgar2007measure]. 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 [hell2004graphs] is a mapping which maps adjacent vertices to adjacent vertices, i.e. for every . A homomorphism is a similarity mapping, if inverse images of adjacent vertices are also adjacent, i.e. 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. 3).

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. 3).

Proposition 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, i.e. . 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 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, i.e. by its Hausdorff dimension. Smaller number of similarity mappings indicates the denser packing of a graph by its contraction subgraphs, i.e. the higher self-similarity degree. In particular, the normalized Hausdorff dimension could serve as a measure of self-similarity.

5 Fractal graphs: theoretical study

In this section, we consider only connected graphs. Importantly, the relation (2) between Lebesgue and Hausdorff dimensions of topological spaces remains true for graphs.

Proposition 3.

For any graph ,

Proof.

Let product dimension of a graph is equal to . Then by Theorem 2 is an intersection graph of strongly -colorable hypergraph. Since rank of every such hypergraph obviously does not exceed , Theorem 1 implies, that . ∎

Proposition 3 allows us to define graph fractals analogously to the definition of fractals for topological spaces: a graph is a fractal, if , i.e. . In particular, we say that a fractal graph is -fractal, if . For example, the graph on Fig. 3 (right) is self-similar, but not fractal, since . In contrast, Fig. 5 demonstrates that Sierpinski gasket graph is 1-fractal (these graphs are studied in detail in the following section). The only connected 0-fractals are complete graphs . Next, we study fractal graphs of higher dimensions.

Triangle-free graphs

Let denotes the edge chromatic number of a graph . Classical Vizing’s theorem [vizing1964estimate] states that , i.e. the set of all graphs can be partitioned into two classes: graphs, for which (class 1) and graphs, for which (class 2).

By Proposition 1, , if contains no triangles. For such graphs we have

Proposition 4.

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 colors to the cliques is an edge coloring. 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 [konig1916graphen]. 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.

Scale-free graphs

Recall that scale-free networks

are graphs whose degree distribution (asymptotically) follows the power-law, i.e. 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 [janson2010large, chung2002connected], we will consider a more formal probabilistic model. Assume without loss of generality that [janson2010large]. 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 [brandstadt1999graph]. 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.

For graphs with and , with high probability and .

Proof.

It has been proved in [janson2010large] that the clique number of a graph with the scaling constant is either 2 or 3 with high probability, i.e. as for -vertex scale-free graphs that have power-law degree distribution with the exponent . The following lemma complements this fact:

Lemma 2.

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

(8)

Let be the number of diamonds in

. For the expected value of this random variable we have

(9)

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

(10)

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 center we have

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

(11)