Structural Sparsity of Complex Networks:Bounded Expansion in Random Models and Real-World

Graphs

Erik D. Demaine∗ Felix Reidl† Peter Rossmanith†

Fernando Sánchez Villaamil† Somnath Sikdar† Blair D. Sullivan‡

May 24, 2019

Abstract

This research aims to identify strong structural features of real-world complex networks,sufficient to enable a host of graph algorithms that are much more efficient than what ispossible for general graphs (and currently used for network analysis). Specifically, we studythe property of bounded expansion—roughly, that any subgraph has bounded average degreeafter contracting disjoint bounded-diameter subgraphs—which formalizes the intuitive notionof “sparsity” well-observed in real-world complex networks.

On the theoretical side, we analyze many previously proposed models for random net-works and characterize, in very general scenarios, which produce graph classes of boundedexpansion. We show that, with high probability, (1) Erdos–Rényi random graphs, generalized tohave non-uniform edge probabilities and start from any bounded-degree graph, have boundedexpansion; (2) the Molloy–Reed configuration model—matching any given degree sequenceincluding “scale-free” networks given by a power-law degree sequence—results in graphs ofbounded expansion; and (3) the Kleinberg model and the Barabási–Albert model, in fairlytypical setups, do not result in graphs of bounded expansion.

On the practical side, we give experimental evidence that many complex networks havebounded expansion, by measuring a closely related “p-centered coloring number” on a corpusof real-world data.

On the algorithmic side, we develop new network analysis algorithms for the followingproblems on graphs of bounded expansion (or more generally, with bounded p-centered coloringnumber): for a fixed graph H, we obtain the fastest-known algorithm for counting the number ofinduced H-subgraphs and the number of H-homomorphisms; and we design linear algorithmsfor computing several centrality measures.

∗Computer Science and Artificial Intelligence Laboratory, Massachusetts Institute of Technology, Cambridge, MA,edemaine@mit.edu

†Theoretical Computer Science, Department of Computer Science, RWTH Aachen University, Aachen, Ger-many, reidl,rossmani,fernando.sanchez,sikdar@cs.rwth-aachen.de, Research funded by DFG Project RO 927/13-1“Pragmatic Parameterized Algorithms.”

‡Department of Computer Science, North Carolina State University, Raleigh, NC, blair_sullivan@ncsu.edu

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Keywords: complex networks · social networks · sparsity · bounded expansion ·Molloy-Reed ·treedepth · parameterized algorithms

1 IntroductionComplex networks vs. structural graph algorithms. Social networks (such as Facebook orphysical disease propagation), biological networks (such as gene interactions or brain networks),and informatics networks (such as autonomous systems) are all examples of complex networks,which have been the attention of much study in recent years, given the surge of available networkdata. Modeled as graphs, complex networks seem to share several structural properties. Perhapsmost famous is the small-world property (“six degrees of separation”): typical distances betweenvertex pairs are small compared to the size of the network. Another important property is scale-freeness: the empirical degree distribution is independent of the size of the graph. In many cases,this degree distribution is a power law: the fraction of vertices of degree k is proportional to k−γ ,for some constant γ , typically between 2 and 3. Finally, complex networks are sparse: the ratio ofedges to vertices (edge density) is usually small (in our collection of networks the highest is 89).

On the other hand, the field of structural graph algorithms has led to impressively efficientand precise algorithms (efficient PTASs, subexponential fixed-parameter algorithms, linear kernel-izations, etc.) for increasingly general families of graphs; see, e.g., [11–15, 18, 22]. Many suchresults proved initially for planar graphs have since been extended to bounded-genus graphs, graphsof bounded local treewidth, and graphs excluding a fixed minor; yet such results are known tobe impossible for general graphs. Can we apply these powerful algorithms to analyze complexnetworks?

Unfortunately, the structural properties of complex networks mentioned above seem too weakto enable better algorithms, and the graph classes mentioned above seem too restrictive to apply tocomplex networks. The goal of this paper is to bridge this gap, by identifying a more general graphclass that simultaneously enables better algorithms and includes many complex networks.

Bounded expansion. In general, complex networks seem to be globally sparse and locallydense (e.g. due to clusters or communities). How can this notion be captured formally? If wecontract disjoint small-diameter subgraphs (representing potentially dense local substructures inthe network), then the resulting graph (representing the global connectivity of these substructures)should be sparse. This is the notion of an r-shallow minor, where r is the maximum diameter of thesubgraphs that were contracted to construct the minor. (For formal definitions, see Section 2.) Tomeasure the sparsity of r-shallow minors, we use edge density. We cannot expect the edge densityof all r-shallow minors to be constant (then r would not play a role), but we require it to grow as anyfunction of r, and thus be independent of the size of the graph. A graph class for which this propertyholds has bounded expansion, a concept introduced by Nešetril and Ossona de Mendez in [36].

Experimental results. We present an experimental study suggesting that important real-worldnetworks have small expansion. Interestingly, the algorithmic tools that become efficient when theexpansion of a graph is small can be directly applied without knowing the actual expansion of thegraph. One of the main such tools is low-treedepth coloring, which colors the graph with f (p)colors such that every induced graph on p colors has bounded treedepth—a measure more restrictivethan treewidth—where f is a function depending only on the expansion. Generally the running timeof algorithms based on low-treedepth colorings depends heavily on the number of colors f (p). In

Section 3, we present experimental results in which a simple algorithm for computing low-treedepthcolorings actually finds such colorings using few colors, in a variety of real-world networks.

Algorithmic results. The definition of bounded expansion is rather abstract, and applies toa graph class instead of an individual graph, making it hard to measure directly. With thesepreliminaries results in hand, we try to exploit both this tools in Section 6 to solve typical problemsfor complex networks: First we develop a better algorithm than the one presented in [38] to countsubgraphs bases on low treedepth colorings. Counting subgraphs is fundamental to motif counting, awidely used concept to analyze networks. Then we develop an algorithm which computes localizedversions of several centrality measures in linear time.

We develop several new algorithms for network analysis. Specifically, we develop:

1. A linear-time algorithm to count the number of times a fixed subgraph appears as an (induced)subgraph/homomorphism in a bounded-expansion graph. We do this by improving thepreviously best known algorithm to count the appearances of a structure of size h on graphsof treedepth t from O(2htht ·n) to O(th23hh2 ·n), thus removing the exponential dependencyon t.

2. A linear-time algorithm to compute localized variants (i.e., computed in a constant-radiusneighborhood around each vertex) of the closeness centrality measure and two other relatedmeasures. The constant in the running time depends on the radius and the expansion functionof the graph class.

Theoretical results. Since the definition of bounded expansion applies to graph classes insteadof individual graphs, it is impossible to settle this question empirically. To ground our hypothesis intheory, we analyze several random graphs model which were designed to mimic the properties ofspecific types of real world networks. Although not perfect, random graph models play a centralrole in network analysis, both to guide our understanding of complex networks, and as a convenientsource of synthetic data for algorithm testing and validation. In our case, random graph modelsallow us to determine whether (synthetic) complex networks have bounded expansion with highprobability. (By contrast, it is unclear how to compute this property in real-world data.) We analyzefor popular random graph models: a significant generalization of Erdos–Rényi graphs (which allowsthe network to be built on top of an arbitrary base graph of bounded degree, so the randomness mayrepresent realistic noise), the Configuration Model with specified asymptotic degree sequence, theKleinberg Model and the Barabasi-Albert Model. We will show that both the Configuration Modeland our extension of Erdos–Rényi graphs have bounded expansion, while the Kleinberg Model andthe Barabasi-Albert Model do not (actually they are not even nowhere dense).

Previous results. There is substantial empirical work studying structural properties of complexnetworks, so we focus here on work closest to structural graph theory. In general, large real-worldcomplex networks are not easily classified as either low- or high-dimensional. In particular, toolswhich implicitly assume low dimensionality (such as singular value decomposition) produce modelsand results incompatible with observed structure and dynamics, yet traditional high-dimensionaltools (like sampling) often fail to achieve measure concentration due to the extreme sparsity ofthe networks. Adcock et al. [1] recently empirically established that, when compared with a suiteof idealized graphs1, realistic large social and informatics networks exhibit meaningful “tree-like”

1representing low-dimensional structures, common hierarchical models, constant-degree expanders, etc.

1

structure at an intermediate scale. Their work related the k-core structure (whose extremal statisticis the degeneracy) to the networks’ Gromov hyperbolicity and tree decompositions. Unfortunately,they showed that straightforward applications of these measures are often insufficient to revealmeaningful structure because of noisy/random edges in the network which contradict the strictstructural requirements. (For example, several families of popular random graph models have beenshown to have very large treewidth [19].)

2 Graphs of Bounded ExpansionThe graph classes that we are interested in this paper are the so-called graphs of bounded expansion2.These graphs satisfy a very general notion of sparsity: new graphs formed by contracting smalldiameter subgraphs cannot significantly inflate the density (it remains no more than some functionof this diameter). To formalize the notion of bounded expansion, we must first wade through sometechnical definitions, starting with that of a shallow topological minor (for those familiar withtopological minors, this just adds the condition that the distance between the “nails” of the minor isbounded).

Definition 1 (Shallow topological minor, nails, subdivision vertices) A graph M is an r-shallowtopological minor if a (≤ 2r)-subdivision of M is isomorphic to a subgraph G′ of G. We callG′ a model of M in G. For simplicity, we assume by default that V (M) ⊆ V (G′) such that theisomorphism between M and G′ is the identity when restricted to V (M). The vertices V (M) arecalled nails and the vertices V (G′)\V (M) subdivision vertices. The set of all r-shallow topologicalminors of a graph G is denoted by G Or.

We also define an associated density measure for shallow topological minors (the grad).

Definition 2 (Topological grad) For a graph G and an integer r ≥ 0, the topological greatestreduced average density (grad) at depth r is defined as

∇r(G) = maxH∈G Or

|E(H)||V (H)|

.

For a graph class G , define ∇r(G ) = supG∈G ∇r(G).

We can now define what it means for a class to have bounded expansion.

Definition 3 (bounded expansion) A graph class G has bounded expansion if and only if thereexists a function f such that for all r, we have ∇r(G )< f (r).

When introduced in [36], bounded expansion was originally defined using an equivalent char-acterization based on the notion of shallow minors: H is a r-shallow minor of G if H can beobtained from G by contracting disjoint r-balls and then taking a subgraph. An r-ball in a graphG is a subgraph G′ ⊆ G with the property that there exists v ∈ V (G′) such that for all u ∈ V (G′),

2Not to be confused with the notion of expansion related to expander graphs.

2

dG′(u,v)≤ r. In the context of our paper, however, the topological shallow minor variant provesmore useful, and we restrict our attention to this setting.

We already see that graphs excluding a topological minor—in particular planar graphs andbounded-degree graphs—have bounded expansion. The condition also generalizes graphs excludinga minor (and thus those of bounded tree-width). Finally, we point out although that boundedexpansion obviously implies bounded degeneracy, the reverse does not hold.

Another important alternate characterization of bounded expansion uses a special coloringnumber.

Definition 4 (p-centered coloring) Given a graph G, let c : V (G)→ [r] be a proper vertex coloringof G with r ≥ χ(G) colors. We say that the coloring c is p-centered, for p ≥ 2, if any connectedsubgraph of G either receives at least p colors or contains some color exactly once. Define χp(G)to be the minimum number of colors needed for a p-centered coloring.

While this definition looks rather cryptic, the fact that there exist proper colorings that are p-centeredis trivial to show: simply assign a distinct color to each vertex of the graph. Note that p-centeredcolorings are proper for p ≥ 2. Typically, the number of colors r is much larger than p and oneis interested in the smallest r for which a p-centerd coloring with r exists. Let us point out theconnection with graph classes with bounded expansion. Suppose that c is indeed a p-centeredcoloring of G using r colors. Then note that any i < p color classes induces a graph of treedepth atmost i. For those unfamiliar with this notion, the treedepth of a graph G is the lowest rooted forestwhose closure contains G as a subgraph. A treedepth decomposition of G is simply a tree T on thevertex set V (G) which witnesses the fact. To put this width measure into perspective, we point outthat a graph of treedepth at most t cannot contain a path of length 2t and has pathwidth at mostt−1.

Nešetril and Ossona de Mendez show that graph classes of bounded expansion are preciselythose for which there exists a function f such that every member of the graph class satisfiesχp ≤ f (p) (see Theorem 7.1 in [36]). In [37], the authors also showed how to obtain a p-centeredcoloring with at most f (p) colors for each fixed p in linear time. We will make use of this fact inSections 3 and 6.

When working with random graphs, we will make heavy use of the following alternativecharacterization of graphs of bounded expansion:

Proposition 1 ([38, 39]) A class C of graphs has bounded expansion if and only if there existreal-valued functions f1, f2, f3, f4 : R+→ R such that the following two conditions hold:

(i) For all ε > 0 and for all G ∈ C with |G|> f1(ε), it holds that

1|V (G)|

· |v ∈V (G) : deg(v)≥ f2(ε)| ≤ ε.

(ii) For all r ∈ N and for all H ⊆ G ∈ C with ∇r(H)> f3(r), it follows that

|V (H)| ≥ f4(r) · |V (G)|.

Intuitively, Proposition 1 characterizes classes of graphs with bounded expansion as those where:

3

(i) all sufficiently large members of the class have a small fraction of vertices of large degree;

(ii) all subgraphs of G ∈ C whose shallow topological minors are sufficiently dense must neces-sarily span a large fraction of the vertices of G.

We sometimes use the term nowhere dense in the following: this is a generalization of boundedexpansion in which we measure the clique number instead of the edge density of shallow minors.See [38] for a definition and many equivalent notions. In particular, a class is somehwere dense iffor every r there exists an arbitrarily large clique as an r-shallow (topological) minor.

3 Experimental MotivationAlthough it has been established that real-world networks are sparse, and tend to have low degeneracy(relative to the size of the network), it is natural to ask whether there is empirical evidence that theysatisfy the stronger conditions of bounded expansion. Unfortunately, since expansion itself is aproperty of graph classes and not single graphs, it is impossible to determine whether individualinstances are bounded expansion or not. One natural proxy would be to evaluate the grad of thesegraphs, calculating the maximum density of an r-shallow minor for each r ∈ N (obviously stoppingwhen r is the diameter of the network), but it is not known how to find such minors in reasonabletime.

In order to get around these difficulties, we calculate upper bounds on χp (the p-centered coloringnumber), a good proxy, since it and the grad are both related to each other by factors independentof the graph size. This is further justified because p-center colorings are directly employed foralgorithmic purposes, where the complexity depends heavily on the number of required colors (aswe will show in Section 6). Since it is very time-consuming to obtain a good p-centered coloringfor large p (this is analogous to determining a reasonable bound for the maximum density of anr-shallow minor for large r), we evaluate this property for small values only. This is also roughlythe range of p which is relevant to the algorithms presented below if applied to practical settings.

To obtain upper bounds on χp, we implemented the transitive-fraternal augmentation procedureof Nešetril and Ossona de Mendez [37], along with simple heuristic improvements (e.g. for mergingcolor classes), and ran it to find p-centered colorings on a small corpus of well-known complexnetworks (see Table 1). We note that the column corresponding to p = 2 contains the degeneracy ofthe graph.

The results show that some networks clearly have a moderately growing grad; in particular thelarger networks netscience, both cpan- networks and diseasome. Other networks, like twittercrawl,have such quickly growing p-centered coloring numbers that we did not invest the time to determinethe value for larger p. Since graphs of bounded crossing number (as infrastructure networks tendto be) and bounded degree have bounded expansion, we are not surprised at the small numberof colors needed by power and hex. Finally some networks, like cond-math and hep-th, start outreasonably well for small p but show a sudden jump at p = 3. At present we do not know whetherthis is an artifact of the procedure we use to obtain the coloring or whether the networks have indeedhighly dense minors from a certain depth on. As we will see in Section 4, some complex networksmodels predict such an occurence already for depth at most two. The growth behavior for small pas depicted in Table 1 might therefore serve as a property to distinguish types of networks, meritingfuture research.

4

Graph Vertices Edges p=2 p=3 p=4 p=5 p=6 treedepth

karate 35 78 6 9 9 10 11 8dolphins 62 159 7 12 17 19 20 24lesmiserables 77 254 10 16 17 17 17 16polbooks 105 441 8 17 22 38 41 30word_adjacencies 112 425 8 19 28 36 42 48football 115 613 9 23 34 67 76 69airlines 235 1297 13! 28! 39 63 67 64sp_data_school 238 5539 31 123 151 174 179 171celegans 306 2148 12 41 74 103 148 153hex 331 930 4 9 21! 22 26 69codeminer 724 1017 5 11 17 20 21 51cpan-authors 839 2212 9 29 40 44! 62 224diseasome 1419 2738 12 18 23 35 39 30polblogs 1491 16715 30 153 294 457 458 603netscience 1589 2742 20 20 28 28 28 20yeast 2284 6646 12 38 254 254 – 408twittercrawl 3656 154824 89 587 – – – –power 4941 6594 6 12 21 27 – 95as20000102 6474 13895 12 32 103 316 317 357hep-th 8361 15751 24 25 137 137 – 558p2p-Gnutella 4 10876 39994 8 49 705 – – –cond-mat 16726 47594 18 47 255 1839 – 1310cond-mat pub 23133 93497 26 89 665 – – –cpan-dist. 2719 5016 7 18 43 60 66 224

Table 1: Number of colors in p-centered colorings computed on real world networks and upperbounds of their respective treedepth.

In the last column, we provide heuristic upper bounds for the treedepth of these graphs. Noticethat the number of colors needed for a p-centered coloring will be fewer than the treedepth of thethe graph for any p≤ n: Given a treedepth decomposition of a graph of depth t we can color everynode by its depth in the treedepth decomposition. Since the treedepth is a hereditary property, thegraph induced by any subset of nodes will have treedepth at most t. Since the running time of ourcounting algorithm in Section 6 for a fixed pattern of size h is O(23h · th ·h2 ·n) it should be directlyapplicable to some of these graphs. Notice that the simple coloring algorithm we used sometimescolors the graph with more than t colors. This might be an indication that a better coloring algorithmexists.

We also computed low treedepth colorings for synthetically generated configuration graphs,using a power-law degree sequence with exponent 2. The results can be seen in Figure 1. Theyindicate that, as expected by the result which will be presented in Section 4.2, the number of colorsneeded for a low treedepth coloring has a horizontal asymptote.

4 Expansion of Random Graph ModelsOften, analytic methods are selected based on their behavior when applied to a class of randomgraphs that is believed to mimic characteristics of the particular application networks under consid-

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Figure 1: Size of p-centered colorings of configuration graphs. Each datapoint is the median of ten graphsgenerated randomly with a power-law degree distribution with exponent 2. The right plot shows the observedminimum and maximum alongside the median.

erations. We will not digress into the arguments for and against this practice, but simply note that itis a de facto part of the standard practice at this point in time, and thus we must be able to establishwhether or not graphs generated by such models form classes of bounded expansion. For moreinformation on random graph models, we refer the readers to several survey papers [34, 35, 41, 42].In this section, we determine whether several such models generate classes of bounded expansion.

Prior work [39] has shown that the Erdos-Rényi model asymptotically almost surely generatesgraphs that fall in a class of bounded expansion. Unfortunately, empirical analysis of real-worldnetworks (including friendships/social networks, telephone/communication networks, and biologi-cal/neural networks) has shown that typical degree distributions are measurably different from thePoisson distribution exhibited by Erdos-Rényi (see [42] and the references therein). We start byextending the results of Nešetril et al. to a model which allows the inclusion of an arbitrary boundeddegree graph in addition to probabilistically generated edges (with non-identical probabilities, sub-ject to a uniform bound) and showing this model to have bounded expansion with high probability(strengthening the previous result which held only asymptotically almost surely). Including a basegraph and allowing non-uniform edge probabilities drastically increases the structural variabilityin the model’s output. We will further argue that there is no stronger extension of Erdos-Rényigraphs that has this property. Pushing these techniques further, we show that the popular configura-tion model [34, 35], which allows a user-specified expected degree sequence, generates classes ofbounded expansion with high probability.

We also consider the Kleinberg and Barabási-Albert models, which were designed to replicate“small-world" properties and heavy-tailed (power-law) degree distributions commonly observedin complex networks. We show that both these models (with typical parameters) generate graphclasses that do not have bounded expansion, and in fact are somewhere dense3.

3A graph is somewhere dense precisely when it is not nowhere dense.

6

4.1 A Generalization of the Erdos-Rényi ModelOne of the most well-studied models for random graphs was introduced by Erdos and Rényi.Often denoted Gn(p), this model has only two input parameters — the number of nodes n and anedge-probability p ∈ [0,1]. An instance is created by starting with a set of n isolated vertices andadding each of the

(n2

)possible edges independently with probability p. Prior work has shown that

the expected vertex degree µ in Gn(p) is p(n−1) and the probability pi that a vertex has degreeexactly i is

pi =

(n−1

i

)pi(1− p)n−1−i ≈ λ ie−λ

i!,

(where the approximate equality becomes exact as n→ ∞), giving a Poisson degree distribution.To model graphs with non-Poisson degree distributions, we will define a new class of graphs

consisting of the union of an Erdos-Rényi-like graph with a bounded-degree graph (which is triviallyof bounded expansion on its own). Specifically, for a ∆ ∈N, define C (∆) to be the set of all (simpleundirected) graphs of maximum degree at most ∆. We modify this class as follows: fix µ ∈ R+; forevery member G ∈ C (∆), let pG : V (G)×V (G)→ [0,1] be such that the following hold:

(i) for all u,v ∈V (G), pG(u,v) = pG(v,u);

(ii) if u = v or if u,v ∈ E(G), then pG(u,v) = 0;

Define G (G, pG) be a random graph that is chosen uniformly at random from among all graphsobtained from G by adding edges with edge probabilities pG. Call G the base graph of the randomgraph G (G, pG). In the Erdos-Rényi model, the base graph G would be the empty graph and the edgeprobability pG(u,v) = p for all distinct u,v∈V (G). We will show that the set of all graphs G (G, pG)with G ∈ C (∆) is of bounded expansion, provided that pG(u,v) ≤ µ/n, where µ ∈ R+ is someconstant. This is tight in the sense that a base graph of arbitrary degree does not result in a classof bounded expansion: take as a simple example the class of graphs consisting of

√n stars each

of degree√

n. Adding Erdos-Renyi edges to this graph will, with high probability, connect thesecomponents pairwise by edges and thus create a 1-shallow minor whose density grows with n.

For the remainder of this section, we will abuse notation by writing G (G,µ/n) for a randomgraph whose edge probabilities are given by an unspecified pG which is bounded above by µ/n.

We now state the main result of this section and give a sketch of the proof.

Theorem 1 Let ∆ ∈ N and µ ∈ R+ be fixed constants and let C (∆) be an infinite set of graphsof maximum degree at most ∆. Moreover assume that every G ∈ C (∆) has an associated edgeprobability function pG such that pG(e)≤ µ/n, for every edge e. Then the class of graphs G (G, pG)obtained by uniformly sampling the random graphs obtained from each base graph G has boundedexpansion with high probability.

Proof: To show that G (G,µ/n) for G ∈ C (∆) is a graph class of bounded expansion, we verify theconditions stated in Proposition 1.

The first condition requires that the fraction of vertices with large degree is vanishingly small.Specifically, in Lemma 3, we prove that the probability there are εn vertices of degree at leastf2(ε) = 2∆+8µα , where ε = 4eα−4αµ , tends to zero as n tends to infinity (so there exists f1(ε)such that for all n > f1(ε), condition (i) is satisfied with high probability).

7

To show that G (G,µ/n) satisfies condition (ii), we prove all subgraphs whose shallow topo-logical minors are sufficiently dense span a large fraction of the vertices. We start with Lemma 1,which bounds the probability of a creating a uv-path in G (G,µ/n) that decreases d(u,v) in termsof a function g(r,∆,µ) and n. Lemma 2 considers a shallow topological minor and shows that theprobability that enough new short paths were formed to make it dense using random edges vanishesin the limit. In the notation of Proposition 1, we let f3(r) = ∆+2 and f4(r) := 1/(2e3g(r,∆,µ)2).

Taken together, this shows that with high probability G (G,µ/n) for G ∈ C (∆) has boundedexpansion. 2

Lemma 1 Let G be a graph with maximum degree ∆ and let r ∈ N. Suppose u and v are verticesin G with dG(u,v)> 2r+1. Then the probability that there exists a path in G (G,µ/n) with at most2r+1 edges between u and v is at most

g(r,∆,µ)n

:=(2r+1)e

√2r ·(∆4r ·µ

)2r+1

n.

Proof: Recall that the length of a path is the number of edges it contains. Since, by hypothesis,dG(u,v) > 2r+ 1, any path from u to v in G (G,µ/n) with at most 2r+ 1 edges must contain atleast one random edge. Let P be such a path and suppose that it contains exactly i random edges.We can view this path as consisting of “segments” Pu,P1, . . . ,Pi−1,Pv where each segment isa sequence of (at most 2r) vertices that forms a path by itself in the base graph G. If the path hasexactly i random edges then there are i−1 internal segments that connect Pu and Pv (the segmentsthat contain the endpoints).

We first wish to estimate an upper bound on the probability that two segments are connectedby a random edge. This probability is not simply µ/n, as each segment can have several vertices.Since the base graph has maximum degree ∆, and since each segment has at most 2r vertices, it iscontained in some 2r-ball in the base graph G. A 2r-ball in the base graph has at most ∆2r verticesand the probability that two such balls are connected by a random edge is at most ∆4r ·µ/n. Thisis also an upper bound on the probability that two segments, each with at most 2r vertices, areconnected by a random edge.

To estimate the overall probability, we sum over all values of i. If there are exactly i randomedges then our path has i−1 internal segments. The number of ways of choosing these segments isat most

( ni−1

), as we can view this process as selecting i−1 representatives, one for each segment.

Moreover, these i−1 internal segments can be ordered in (i−1)! ways. Therefore an upper boundon the probability that there exists a path from u to v with at most 2r+ 1 edges is given by theexpression:

2r+1

∑i=1

(n

i−1

)· (i−1)! ·

(∆4r ·µ

n

)i

. (1)

8

The expression within the sum can be written as:(n

i−1

)· (i−1)! ·

(∆4r ·µ

n

)i

≤(

nei−1

)i−1

· (i−1)! ·(

∆4r ·µn

)i

=(i−1)! · ei−1

(i−1)i−1 · (∆4r ·µ)i

n

≤ e√

i−1 · (∆4r ·µ)i

n.

The last inequality follows from the fact that (err!)/rr ≤ e√

r (using Stirling’s approximation).Thus the sum in Equation 1 is at most:

(2r+1) · e√

2r · (∆4r ·µ)2r+1

n.

2

Lemma 2 Let G ∈ C (∆) and let H be an r-shallow topological minor of G (G,µ/n) with at mostf4(r) ·n vertices. If f3(r) := ξ +2, for any ξ ≥ ∆, and f4(r) := 1/((2r+1)2 · e5r ·ξ 16r2+8r ·µ4r+2),

Pr[|E(H)||V (H)|

> f3(r)]≤ 1/nξ+1.

Proof: Using the terminology from Definition 1, let q := |V (H)| so that the model of H (which isa subgraph of G (G,µ/n)) has q nails and at most q · f3(r) · (2r+ 1) subdivision points. Given aset v1, . . . ,vq of q vertices in G (G,µ/n), let P denote the set of all paths of length at most 2r+1between any two of these vertices. Since the maximum degree of G is ∆ ≤ ξ , at most q · ξ ofthese paths could have existed in G. In other words, at least q · ( f3(r)−ξ ) of these paths exist inG (G,µ/n) because of random edges.

Therefore the probability that we wish to estimate is simply the probability that a set of q verticeschosen uniformly at random have at least q · ( f3(r)−ξ ) new paths of length at most 2r+1 betweenthem in G (G,µ/n). By Lemma 1, this probability is given by:

n/c

∑q=1

(nq

)·( (q

2

)q ·h(r)

)·(

g(r,ξ ,µ)n

)q·h(r), (2)

where we have used h(r) = f3(r)− ξ = 2 and c = 1/ f4(r). The expression in the sum of (2) isbounded from above by:(

neq

)q

·(qe

2

)2q·(

g(r,ξ ,µ)n

)2q

=

(e ·(

e ·g(r,ξ ,µ)2

)2

· qn

)q

=(

α(r,ξ ,µ) · qn

)q, (3)

where we have used α(r,ξ ,µ) to denote the expression (e3/4) ·g(r,ξ ,µ)2. One can verify that

α(r,ξ ,µ) = (e5/2) · r ·ξ 16r2+8r ·µ4r+2,

9

and that we have defined f4(r) so that f4(r) = ((2r+1)2 ·2α(r,ξ ,µ))−1.Notice that a r-shallow topological minor with q nails cannot have a density larger than q.

Therefore, we can start the sum in Equation 2 at ξ +2 instead of 1. We claim that

n/c

∑q=ξ+2

(α(r,ξ ,µ) · q

n

)q≤ 1

nξ+1(4)

The first term in this sum is(α(r,ξ ,µ) · ξ +2

n

)ξ+2

=(α(r,ξ ,µ)(ξ +2))ξ+2

nξ+2<

12nξ+1

.

The remaining terms decrease geometrically by a factor of(α(r,ξ ,µ)q−1

n

)q−1(α(r,ξ ,µ)q

n

)q =n

α(r,ξ ,µ)q

(q−1

q

)q−1

≥ nα(r,ξ ,µ)q

· 1e≥ n

α(r,ξ ,µ)(n/c)· 1

e

=c

eα(r,ξ ,µ).

When we set c≥ 2eα(r,ξ ,µ), this factor is ≥ 2. Hence the overall sum is at most twice the firstterm (1/nξ+1), for a sufficiently large choice of c. The upshot is that

Pr[|E(H)||V (H)|

> ∆+2]≤ 1/nξ+1,

for any ξ ≥ ∆. Hence the edge density of any r-shallow minor is at most ∆+2 w.h.p. 2

Lemma 3 For any 0 < ε < 1, let α be such that ε = 4eα−4αµ . Let G ∈ C (∆) and let A be a set ofs := 2d(εn)/2e vertices in G (G,µ/n) of largest degree. Then

limn→∞

Pr[

minx∈A

deg(x)≥ 2∆+8αµ

]= 0.

Proof: This proof closely follows that of Lemma 4.3 in [39]. Let A be as in the statement of thelemma and let δ := minx∈A deg(x). Then there are at least sδ/2 edges with at least one endpointin A. If A′ is a random subset of A of size s/2, then every edge that has an endpoint in A has aprobability of at least 1/2 of having exactly one endpoint in A′. Therefore there exists a subsetA′ ⊆ A of size s/2 such that the number of edges crossing the cut (A′,V \A′) is at least sδ/4.However if δ ≥ 2∆+8αµ , then it implies that there exists A′ ⊆ A of size s/2 such that the numberof edges crossing the cut (A′,V \A′) is at least (∆+4αµ)s/2.

To show that Pr [δ ≥ 2∆+8αµ] is small, we will show that the probability that there exists aset A′ with s/2 vertices such that the cut (A′,V \A′) has at least (∆+4αµ)s/2 edges is o(1). Fix a

10

set A′ with s/2 vertices. The probability that q edges were added to the cut (A′,V \A′) is given by:

Pr[∣∣E(A′,V \A′)

∣∣= q]≤( s

2 ·(n− s

2

)q

)·(

µ

n

)q

≤(

ens2q

)q

·(

µ

n

)q

≤(

2sµ

q

)q

.

If the total number of edges crossing the cut is (∆+4αµ)s/2 then q must be at least 2αµs. Theprobability that at least 2αµs edges were added to the cut is at most(

2sµ

q

)q

≤(

2sµ

2αµs

)q

≤ α−q ≤ α

−2αµs,

where the last inequality holds since α > 1. Therefore the probability that there exists a set A′ withs/2 vertices such that |E(A′,V \A′)| ≥ (∆+4αµ)s/2 is:

α−2αµs ·

(n

s/2

)< α

−2αµs ·(

2ens

)s/2

≤(

eα−4αµns/2

)s/2

≤(

s/4s/2

)s/2

=1

2s/2 .

Since s = εn (roughly), this probability approaches zero as n→ ∞, which is what we wished toshow. 2

Finally, since Erdos-Rényi random graphs are a special case of our model (when the boundeddegree graph is empty, and all the edge-probabilities are equal), we have a strengthening of thepreviously best known result of Nešetril and Ossona de Mendez (that bounded expansion holdsasymptotically almost surely in this model).

Corollary 1 The class of Erdos-Rényi random graphs with edge probability p≤ µ/n for a constantµ ∈ R+ has bounded expansion with high probability.

4.2 The Configuration ModelSince empirical measurements of real-world graphs typically reveal non-Poisson degree distributions(in particular, they are often heavy tailed, or power-law), one might wish to consider a random graphmodel which generates graphs with a given (graphical) degree sequence uniformly at random. Thegoal of this section is show that asymptotically, the random graphs with a degree sequence that iswell-behaved in a certain sense, is a class of bounded expansion.

In order to consider the question of bounded expansion, we must define classes of these randomgraphs where the number n of vertices in the graph is any natural number. The formalization we givein Definitions 5– 6 appears in work of Molloy and Reed [34] and has been discussed extensively inthe literature [35, 40–42].

Definition 5 (asymptotic degree sequence) An asymptotic degree sequence is a sequence of integer-valued functions D = d1(n),d2(n), . . . such that

11

(i) di(n) = 0 for all i≥ n;

(ii) ∑n−1i=0 di(n) = n.

Informally, di(n) denotes the number of vertices of degree exactly i in an n-vertex graph. Con-dition (1) above simply states that no vertex in an n-vertex graph that has degree n or more.Condition (2) states that the sum over all i of the number of vertices of degree i equals the totalnumber of vertices.

Given an asymptotic degree sequence D , we let Dn denote the vertex degrees c1, . . . ,cnarranged so that:

(i) c j ≥ c j+1, for all 1≤ j ≤ n−1;

(ii) 0≤ c j ≤ n−1, for all 1≤ j ≤ n; and

(iii) | j : c j = i|= di(n), for all i≥ 0.

We let ΩDn denote the set of all graphs with vertex set 1, . . . ,n and degree sequence Dn. Arandom graph Gn(D) on n vertices with degree sequence D is a uniformly random member ofΩDn . Note that the average degree of a vertex in Gn(D) is ∑

n−1i=1 idi(n)/n. Since we will be using

asymptotic properties of random graphs with degree sequence D , we want the sequences Dn to besimilar in the sense that the proportion of vertices of degree i is roughly the same in each sequence.

Definition 6 An asymptotic degree sequence D is

(i) feasible if ΩDn 6=∅ for all n≥ 1.

(ii) smooth if there exist constants λi such that limn→∞ di(n)/n = λi.

(iii) sparse if there exists a constant µ ∈ R+ such that ∀n ∈ N, ∑n−1i=0 idi(n)/n = µ + o(1), and

∑∞i=0 iλi = µ .

Further, Molloy and Reed point out in that one major difficulty in studying random graphs onfixed degree sequences is the difficulty to generate such graphs directly [34]. As such, they proposea different model known as the “configuration model” that is much easier to work with. Given adegree sequence D , a random configuration with n vertices on D is generated as follows:

(i) Create a set L containing deg(v) distinct copies of each vertex v

(ii) Choose a random matching of the elements of L.

Each configuration represents a multigraph whose edges are defined by the pairs in the matching. Inorder to translate results from the configuration model to the random graph model on fixed degreesequences, the degree sequence must be sparse and “well-behaved”:

Definition 7 (well-behaved degree sequence) An asymptotic degree sequence D is well-behavedif:

(i) D is feasible and smooth.

12

(ii) i(i−2)di(n)/n tends uniformly to i(i−2)λi for all n

(iii) L(D) = limn→∞ ∑n−1i=1 i(i−2)di(n)/n exists and the sum approaches the limit uniformly.

More specifically, one can transfer results from the configuration model on a degree sequenceD to random graphs with D when the so-called Molloy-Reed conditions hold [34]:

(i) D is well-behaved and sparse.

(ii) There exists ε > 0 such that for all n and i > n1/4−ε , di(n) = 0.

A word about the definition of well-behaved and the conditions under which one can workwith the configuration model. Molloy and Reed in [34] define the quantity Q(D) = ∑

∞i=1 i(i−2)λi

and show (among other things) that the random graph model on the degree sequence D shows aphase transition similar to the Erdos-Rényi model. If Q(D)> 0, then a random graph with degreesequence D almost surely has a giant component, while if Q(D)< 0, then all components of such arandom graph are almost surely small, provided that the maximum degree is at most n1/4. For easeof mathematical analysis, one requires that the quantities in conditions (i) and (ii) in this definitionconverge uniformly to their respective limits.

We should also point out that the configuration model seems to have been discussed severaltimes in the literature. Molloy and Reed in [34] credit Bender and Canfield [5] to be the first to haveput forward this idea, which was then refined by Bollobás [6] and Wormald [54].

For the rest of this section, we let D = di(n)∞i=1 denote an asymptotic degree sequence that

satisfies the Molloy-Reed conditions. For a positive integer n, the degree sequence D specifies thefunctions di(n), for 0≤ i≤ n−1, and where di(n) is the number of vertices of degree exactly i. Inthe configuration model, each vertex v of degree i is replaced by i distinct copies v1, . . . ,vi and anedge uv between vertices u and v of degrees i and j, respectively, is represented by an edge betweena copy of u and a copy of v. Now it is easy to see that a graph with the degree sequence di(n)n−1

i=0is represented in the configuration model by a matching with the following properties: (i) eachvertex is incident to exactly one matching edge; (ii) no two endpoints of a matching edge are copiesof the same vertex. Conversely, any matching in the configuration model that satisfies the above twoproperties is a graph with the degree sequence di(n)n−1

i=0 . However, as we mentioned, if the degreesequence satisfies the Molloy-Reed conditions, then sampling a matching in the configuration modelwith degree distribution D uniformly at random corresponds to sampling a graph with the samedegree distribution asymptotically almost surely.

We can now state our main result, that Molloy-Reed random graphs are also of boundedexpansion:

Theorem 2 Let D be an asymptotic degree sequence that satisfies the Molloy-Reed conditionsThen the class C (D) of graphs obtained by uniformly sampling graphs with degree sequence D isa.a.s. of bounded expansion.

Proof: As before, we verify the two properties stated in Proposition 1.Property (i) directly follows from the sparseness and smoothness-conditions: suitable functions

f1, f2 are implied by the condition that ∑n−1i=0 idi(n)/n = µ +o(1) with the limit sum∞

i=0iλi = µ .Using Lemma 5, proving condition (ii) works analogous to the proof of Lemma 2 (replacing q

by h(r) · p yields equation 2, up to constants). 2

13

Lemma 4 Let D = di(n)∞i=1 be an asymptotic degree sequence that satisfies the Molloy-Reed

conditions. Fix a constant c > 1 and let F ⊆ [n]× [n] be a set of at most n/c edges chosen uniformlyat random. Then it holds that:

Pr [F ⊆ E(Gn(D))] = O((β/n)|F |),

where β is a constant that depends on the sequence D .

Proof: We prove this on the configuration model. Since, by hypothesis, the degree sequencesatisfies the Molloy-Reed conditions, any result on the configuration model holds for graphs thatare uniformly sampled with the same degree distibution a.a.s.

Fix an integer n and create a set of 2m = ∑n−1i=0 idi(n) vertices for the configuration model.

Denote by Mn(D) a random perfect matching on the vertices of the configuration model. Let Fbe a set of k = bn/cc vertex pairs u1v1, . . . ,ukvk in the configuration model such that no vertexoccurs more than once. The interpretation is that no two potential edges share a vertex. Partition thevertices in F into sets L and R such that every potential edge has exactly one vertex in each of thesesets. For calculating the probability that F appears in a random matching, we need only considerthe degrees of the endpoints. Therefore for 1≤ i≤ k, let deg(ui) = li and deg(vi) = ri where weassumed that the vertices u1, . . . ,uk are in the left set L while the remaining vertices are in R.

We first want to estimate the probability that F appears as an edge set in Mn(D) under thecondition that the degrees of its endpoints is given by the fixed sequence (l1,r1), . . . ,(lk,rk). Wedenote this probability by:

ρ := Pr

[F ⊂Mn(D)

∣∣∣∣∣deg(F) =k∧

i=1

(li,ri)

], (5)

where deg(F) =∧k

i=1(li,ri) represents the event that a randomly chosen matching F of size k hasas endpoints vertices with degrees (li,ri), 1≤ i≤ k. If ei ∈ F has endpoints with degrees li and ri,then it can be chosen in li · ri ways. There are

(2mm

)·m! matchings in total and of these there are

k

∏i=1

(li · ri) ·(

2m−2km− k

)· (m− k)!

matchings that generate F . Therefore the expression for ρ is:

ρ =∏

ki=1(li · ri) ·

(2m−2km−k

)· (m− k)!(2m

m

)·m!

.

Since

Pr

[F ⊂Mn(D) ∧ deg(F) =

k∧i=1

(li,ri)

]= ρ ·Pr

[deg(F) =

k∧i=1

(li,ri)

],

we next consider the second term on the right hand side. This term expresses the probability that arandomly chosen matching with edges e1, . . . ,ek is such that edge ei has as its endpoints verticeswith degrees li and ri. A simple upper bound is the following:

Pr

[deg(F) =

k∧i=1

(li,ri)

]≤

k

∏i=1

dli(n) ·dri(n)(n−2k)2 .

14

Recall that by the sparseness-condition of the degree sequence D , we have that ∑n−1i=0 idi(n)/n =

µ + o(1) := α , from which it follows that 2m = αn. Using the union bound, we sum over allpossible choices of degrees l1, . . . , lk,r1, . . . ,rk to upper bound Pr [F ⊂Mn(D)] by:

∑(l1,r1),··· ,(lk,rk)

∏ki=1(li · ri) ·

(2m−2km−k

)· (m− k)!(2m

m

)·m!

·∏ki=1 dli(n) ·dri(n)(n−2k)2k , (6)

which simplifies to

m!(2m−2k)!(2m)!(m− k)!

· 1(n−2k)2k · ∑

(li,ri)

k

∏i=1

(lidli(n) · ridri(n)). (7)

Let us next consider the sum-of-products expression which may be written as:

∑l1

· · ·∑lk

∑r1

· · ·∑rk

l1dl1(n) · · · lkdlk(n) · r1dr1(n) · · ·rkdrk(n).

Since 2m = ∑n−1i=0 idi(n) = αn, we may upper-bound the above expression by (2m)2k. Using the

bound (n/e)n < n! < n(n/e)n, we may upper-bound expression (7) by:

m · (2m−2k) ·(e

4

)k· (m− k)m−k

mm · 1(n−2k)2k · (2m)2k, (8)

which simplifies to

2m2 · ek ·(

1− km

)m−k

·

(m

n2(1− 2k

n

)2

)k

. (9)

Writing m as αn, gives us:

2m2 ·(

1− km

)m−k

·

(αe

n(1− 2k

n

)2

)k

= O((β/n)k

), (10)

where we defined β to be αe. 2

Corollary 2 Let D = di(n)∞i=1 be an asymptotic degree sequence that satisfies the Molloy-Reed

conditions. There exists µ ∈ R+ so that if u and v are two distinct randomly chosen vertices ofGn(D),

Pr [uv ∈ E(Gn(D))]≤ µ

n.

Lemma 5 Let D = di(n)∞i=1 be an asymptotic degree sequence that satisfies the Molloy-Reed

conditions. Let c > 1 be a constant and for positive n, let k be any integer at most n/c. Finally, let rbe an integer that is independent of n and k. Then the probability that k vertices chosen uniformly atrandom from Gn(D) are connected by at least q > 2k internally vertex-disjoint paths each of lengthat most 2r+1 is bounded by

O

(((k2

)q

)((2r+1) · γ

n−2rq

)q)

where γ is some constant.

15

Proof: Suppose that k randomly chosen vertices had q vertex-disjoint paths between them of lengthsl1, . . . , lq such that 1≤ li ≤ 2r+1 for all 1≤ i≤ q. For a path of length li to exist, there exist li−1internal vertices that are strung together by edges to two of the k vertices chosen uniformly atrandom. Using Lemma 4, the probability that there exist q vertex-disjoint paths between k randomlychosen vertices is given by:((k

2

)q

)· ∑

i : 1≤li≤2r+1O

((β

n

)∑

qi=1 li)·T (n; l1, . . . , lq), (11)

where we used T (n; l1, . . . , lq) to denote:(n

l1−1

)(l1−1)!

(n− l1 +1

l2−1

)(l2−1)! · · ·

(n−∑

q−1i=1 (li−1)lq−1

)(lq−1)!.

Now T (n; l1, . . . , lq) simplifies ton!

(n+q−∑qi=1 li)!

.

Using the bound (n/e)n < n! < n(n/e)n, one can upper-bound the above as follows:

n!(n+q−∑

qi=1 li)!

≤ nn+1 · eq(n+q−∑

qi=1 li

)(n+q−∑qi=1 li)

≤ nn+1 · eq · (n+q)(2r+1)q

(n+q−q(2r+1))n+q

≤ nn+1 · eq · (n+q)(2r+1)q

(n−2rq)n+q . (12)

Using (12), we can upper-bound expression (11) by:((k2

)q

)· (2r+1)q · n

n+1 · eq · (n+q)(2r+1)q

(n−2rq)n+q ·(

β

n

)q(2r+1)

which can be upper-bounded by:((k2

)q

)·((2r+1) · e ·β 2r+1

n−2rq

)q

·(

1+qn

)q(2r+1)· n(

1− 2rqn

)n ,

which is

O

(((k2

)q

)·((2r+1) · γ

n−2rq

)q). (13)

where γ depends on r and β (the constant in Lemma 4). 2

16

4.3 The Kleinberg ModelMany social networks exhibit a property that is commonly referred to as the “small-world phe-nomenon.” This property asserts that any two people in a network are likely to be connected bya short chain of aquaintances. This was first observed by Stanley Milgram in a study publishedin 1967 [32]. Milgram’s study suggested that individuals in a social network who only knew thelocations of their immediate aquaintainces are collectively able to construct short chains betweentwo points in the network. More recently, Kleinberg proposed a family of network models to explainthe success of decentralized algorithms in finding short paths in social networks [23].

Kleinberg’s model starts with a n×n grid as the base graph and allows edges to be directed. Fora universal constant p≥ 1, a node u has a directed edge to every other node within lattice distance p.These are the local neighbors of u. For universal constants q ≥ 0 and r ≥ 0, node u has q longrange neighbors chosen independently at random. The ith directed outarc from u has endpoint vwith probability d(u,v)−r/∑x d(u,x)−r.

When r = 0, the long-range contacts are uniformly distributed throughout the grid and one canshow that there exist paths between every pair of nodes of length bounded by a polynomial in logn,exponentially smaller than the number of nodes. Kleinberg shows that in this case, the expecteddelivery time of every decentralized algorithm (one that uses only local information) is Ω(n2/3).When p = q = 1 and r = 2, then short chains continue to exist between the nodes of the network,but in this case there is a decentralized algorithm for transmitting messages that takes O(log2 n)time in expectation between any two randomly chosen points.

What Kleinberg’s model shows is that if the long-range contacts are formed independently ofthe geometry of the grid, then short chains exist between every pair of nodes, but nodes workingwith local knowledge are unable to find them. If the long-range contacts are formed by taking intoaccount the grid structure in a specific way, then short chains exist and nodes working with localknowledge are able to discover them. We show that when greedy routing is efficient, not only doesthe class not have bounded expansion, it is in fact, somewhere dense.

Theorem 3 The Kleinberg model with parameters p = q = 1 and r = 2 is somewhere dense w.h.p.In particular, it does not have bounded expansion.

Proof: Let Γn be an n× n grid. For p = q = 1 and r = 2, the probability that a node u has vas its long-range contact is proportional to dΓn(u,v)

−1 and the normalizing factor in this case isO(1/ logn). This can be easily seen by summing up 1/dΓn(u,x)

2 for all x and noticing that in thegrid, there are 4d neighbors that are at a distance of d from u.

∑x

1dΓn(u,x)2 =

n

∑d=1

4dd2 ≈ 4logn.

To show that the model is somewhere dense, we show that 2-subvidisions of cliques of a certainsize g(n) occur with high probability. Later we will see that g(n) = Ω(log logn). To this end, letΓ′c·g(n) denote some fixed c ·g(n)× c ·g(n) subgrid of Γn, where c is some constant that we will fixlater. Choose V ′ and E ′ to be, respectively, a set of g(n) nodes and a set of g(n)2 edges from thesubgrid Γ′c·g(n) with the following properties: (i) the endpoints of the edges in E ′ are different fromthe nodes in V ′; (ii) no two edges in E ′ share an endpoint.

17

Given any pair of vertices u,v ∈V ′ and an edge e ∈ E ′ with endpoints a,b, the probability that ahas u as its long-range neighbor is Ω((dΓn(a,u) · logn)−1). Similarly, the probability that b has v asits long-range neighbor is Ω((dΓn(b,v) · logn)−1) The probability of both these events happening is

1dΓn(a,u)2dΓn(b,v)2 ·

1log2 n

≥ 1c4g(n)4 log2 n

, (14)

where we upper-bounded distances dΓn(x,y) by c · g(n). Thus the probability that there exists a2-subdivided g(n)-clique in Γ′c·g(n) is at least:(

1c4g(n)4 log2 n

)g(n)2

=: f (n,c). (15)

The probability that there does not exist a 2-subdivided g(n)-clique in Γ′c·g(n) is at most 1− f (n,c).Hence the probability that there does not exist a 2-subdivided g(n)-clique in any c ·g(n)× c ·g(n)subgrid is at most

(1− f (n,c))n

c2g(n)2 ≤ exp

− n

c2 ·g(n)2 ·(c4 ·g(n)4 · log2 n

)g(n)2

:= en

h(n) .

This follows from the inequality (1− x/p)p ≤ e−x.Choose g(n) = log logn and c = 3 (actually any c ≥ 3 works). Then it is easy to show that

h(n)<√

n. Thus the probability of a 2-subdivided (log logn)-clique not existing is at most e−√

n

(which goes to zero as n→ ∞), and we conclude that the graph class described by the model issomewhere dense. 2

4.4 The Barabási-Albert modelThe Barabási-Albert model uses a preferential attachment paradigm to produce graphs with a degreedistribution that mimics the power-law observed in many real-world networks [4]. This modeluses a random graph process that works as follows (see [4]): Start with a small number n0 ofnodes and at every time step, add a new node and link it to n ≤ n0 nodes already present in the“system.” To model preferential attachment, we assume that the probability with which a new nodeu is connected to node v already present in the system is proportional to the degree of v, so thatPr[u→ v] = deg(v)/∑x deg(x), where the sum in the denominator is over all vertices x that arealready in the system. After t time steps, the model leads to a random network with t +n0 verticesand nt edges.

Barabási and Albert suggested that such a network evolves into one in which the fraction P(d)of nodes of degree d is proportional to d−γ . They observed experimentally that γ = 2.9±0.1 andsuggested that γ is actually 3. This model was rigorously analyzed by Bollobás, Riordan, Spencerand Tusnády in [8] who showed that it is indeed the case that the fraction of vertices of degree d falloff as d−3 as d→ ∞. In a different paper, Bollobás and Riordan showed that the diameter of thegraphs generated by this model is asymptotically logn/ log logn [7].

We first provide a formal restatement of the Barabási-Albert model. Note that this is slightlydifferent from the formalization of Bollobás et al. in [8]. We start with a “seed” graph G0 with n0

18

nodes u1, . . . ,un0 with degrees d1, . . . ,dn0 . The number of edges in the seed graph is denoted bym0 is 1

2 ∑n0i=1 di. At each time step t = 1,2, . . ., we create a graph Gt by adding a new node vt and

linking it to q nodes in Gt−1. These q nodes are picked independently and with a probability that isproportional to their degrees in Gt−1. That is, we choose u ∈V (Gt−1) to link to with probability

Pr [vt ,u ∈ E(Gt)] =degGt−1

(u)

2 · |E(Gt−1)|=

degGt−1(u)

2(m0 +q(t−1)).

Note that in this model, all edges between vt and nodes of Gt−1 are assumed to be added simulta-neously (so that the increasing degrees of nodes which receive edges from vt do not influence theprobabilities for this time step). In this restated version, n0, d1, . . . ,dn0 , and q are the parameters ofthe model.

Theorem 4 Given any fixed r, a graph Gn generated by the preferential attachment model withparameters n0, d1, . . . ,dn0 and q has a 1-subdivided Kr with probability at least (4(m0

q + r+ r2))−r2,

provided n≥ r+ r2.

Proof: Choose any r ∈ N. We will show that there exists a 1-subdivided Kr in the graph withprobability that depends only on m0, q, and r.

Consider the graph after the first r+ r2 time steps. Let v1, . . . ,vr be the nodes that were added inthe first r time steps and fix two nodes vi,v j from among these. The probability that a new node vk(for any r+1≤ k ≤ r2) is connected to these two fixed nodes is at least(

q2(m0 +q(r+ r2))

)2

=: f (m0,r,q),

where the denominator is the sum of the vertex degrees after r+ r2 time steps. Now if vr+1 islinked to v1,v2 and vr+2 is linked to v1,v3 and so on such that the nodes added after time stepr connect the first r nodes in a pairwise fashion, we would have a 1-subdivided Kr in the graphGr+r2 . The probability of this happening is at least f (m0,r,q)r2

. Thus, for every r, the probability of1-subdivided Kr existing is non-zero if the graph is large enough. Hence the graph class generatedby the model is somewhere dense. 2

5 Clustering and bounded expansionThe mixture of positive and negative results on random graphs seems to suggest that modelswhich are more reflective of real-world social networks have unbounded expansion. One possibleexplanation is that they exhibit clustering, while models like Erdos-Rényi do not. Clusteringmeasures the extent to which edges are more likely between nodes that already share a commonneighbor (so one would expect to see a higher number of triangles than in a random graph of asimilar degree distribution).

Both the Kleinberg and Barabási-Albert graphs exhibit clustering, meaning their clusteringcoefficient is constant independent of the graph size, whereas the configuration model does not,except under special circumstances. Our variant of the Erdos-Rényi model falls somewhere in

19

between: the clustering can be adjusted using the underlying bounded-degree graph, but it is not asatisfactory social network model.

Which leads us to investigate the following question: can a graph class of bounded expansionexhibit clustering? We will show that for any class of bounded expansion we can force a constantclustering coefficient in such a way that the class will remain of bounded expansion.

Recall that the average clustering coefficient of a graph G was introduced by Watts and Stro-gatz [53] as

C(G) =1n ∑

v∈V (G)

C(v) =1n ∑

v∈V (G)

λ (v)τ(v)

where λ (v) denotes the number of triangles in G that contain v and τ(v) the number of inducedP3’s in which v is the middle vertex. Note that if G is a graph with bounded average degree d,we can obtain a constant clustering as follows: for every vertex in the graph, add d2 edges to itsneighbourhood. To see that this results in a constant clustering, note that every vertex with degree atmost d now has a clique as its neighbourhood—thus in the resulting graph G′, the local clusteringof such a vertex is necessarily equal to 1. Since the number of vertices that have degree at most thetotal average degree makes up a constant fraction of the whole vertex set, the resulting clusteringcoefficient will be bounded from below by a constant.

However, we need to show that such an operation will not destroy the property of havingbounded expansion. We formalize and prove this fact in the following theorem.

Theorem 5 Let G be a graph class. For any constant t, let Ct(G ) be the class of graphs obtainedfrom any graph of G by adding up to t edges to the neighbourhood of every vertex. Then theexpansion of Ct(G ) depends only on the expansion of G and t. In particular, if G has boundedexpansion, so does Ct(G ).

Proof: Take any graph G ∈ G and for each v ∈ G, let Ev be a set of at most t edges to be added tothe neighborhood of v. Let G′ be the graph obtained from G by adding the edge set

⋃v∈G Ev. We

demonstrate how to create G′ through operations that increase the grad of the graph only by somefunction depending on the old grad and some constants.

Let d = 2∇0(G) denote the degeneracy of G. We color the vertices of with d + 1 colors andobtain a partition V1, . . . ,Vd+1 of the vertex set. This also naturally partitions the edge set

⋃v∈G Ev

into sets E1, . . . ,Ed+1, where Ei =⋃

v∈ViEv for 1≤ i≤ d +1.

Let us demonstrate how the edges from a single set Ei can be added to G through operationsthat do not increase the grad too much; this sequence of operations can then be simply executedfor every 1≤ i≤ d +1 to obtain the complete edge set. Thus, fix one set Ei in the following. Wefirst take the lexicographic product G•Kt+1, which is equivalent to copying the graph t +1 timesand making a clique out of every t +1 copies of a vertex. Let us denote these sets of copies of avertex by Cv for each v ∈ G. By Lemma 5.1 and Lemma 5.2 in [36], this increases the grad of thegraph only by some function of t and the old grad. Now, consider some vertex v ∈ Vi in whoseneighborhod the edge set Ev ⊆ Ei should be added. To this end, for every edge ab ∈ Ev we takesome vertex v′ ∈ Cv and contract it into an edge between Ca and Cb (by the choice of v′, thosevertices are exactly determined). Since |Ev| ≤ t, we can do this without exhausting Cv. Furthermore,since the vertices of Vi are independent, no vertex will both create and receive an edge. Therefore,after contracting each sets Cv,v ∈ G in the remaining graph—i.e. excluding the vertices that where

20

contracted into edges—we obtain exactly the graph G with the edges Ei added to it. This is, byconstruction, a 1-shallow minor of G•Kt+1. Thus this last operation can also not increase the gradarbitrarily, but only by a function of t.

Applying the above steps of operations for each edge set Ei yields the graph G with the edges⋃v∈G Ev added to it. Since d is bounded by the grad of G, the resulting graph’s grad only depends

on t and the grad of G. 2

We therefore conclude that clustering is not orthogonal to bounded expansion. Notice that thepresented algorithm is not meant to be used to force a certain lower bound on the clustering inpractice, but to raise the possibility of a random model existing which exhibits both characteristics.

6 AlgorithmsIn this section, we present efficient algorithms for several important problems arising in the studycomplex networks which exploit bounded expansion. Note that in both cases, only the algorithm’srunning time relies on small expansion, not its correctness. The problems that we discuss revolvearound the themes of subgraph counting and centrality estimation.

Computing the frequency of small fixed pattern graphs inside a network is the key algorithmicchallenge in using network motifs and graphlet degree distributions to analyze network data (both ofwhich are described in more detail in refsec:subgraphs). We present a parameterized algorithm forcounting the number of subgraphs with at most h vertices with a running time of 23h · th ·n, where tis the treedepth of the input graph. In a graph class of bounded expansion, we use this algorithm inconjunction with p-centered colorings (which allow us to reduce the problem to pieces of constanttree depth).

Another topic of interest in complex networks is estimating the relative importance of a vertexin the network (for example, how influential a person is inside a social network, which roads arebusiest in a road-network, or which location is most attractive for business). The typical approach isto define/select an appropriate centrality measure (see [24] for a survey of common measures). Wefocus on the closeness centrality, which was introduced by Sabidussi [49], and related extensions.These measures are related in the sense that computing them requires knowledge of all the pairwisedistances between the vertices of the network, which even in sparse networks takes time O(n2)to compute [9]. We introduce localized variants of the closeness-based centrality measures anddesign a linear-time algorithm to compute them in graphs of bounded expansion, together withexperimental data that suggests that these measures are able to recover the topmost central verticesquite well.

Before we begin, we formally state the relationship between bounded expansion and low-treedepth colorings, and give a result of Nešetril and Ossona de Mendez that we will make useof.

Proposition 2 (Low treedepth colorings [37]) Let G be a graph class of bounded expansion.There exists a function f such that for every G ∈ G , r ∈ N, the graph G can be colored withf (r) colors so that any i < r color classes induce a graph of treedepth ≤ i in G. This coloring canbe computed in linear time.

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Proposition 3 (Truncated distances [37]) Let G be a graph of bounded expansion. For every rone can compute in linear time a directed graph ~Gr with in-degree bounded by f (r) – for somefunction f – on the same vertex set as G and an arc labeling ω : ~E(~Gr)→ N such that for everypair u,v ∈ G with dG(u,v)≤ r one of the following holds:

(i) uv ∈ ~Gr and ω(uv) = dG(u,v);

(ii) vu ∈ ~Gr and ω(vu) = dG(u,v);

(iii) there exists w ∈ N−~Gr(u)∩N−~Gr

(v) such that ω(wu)+ω(wv) = dG(u,v).

6.1 Counting graphlets and subgraphsIn the following we highlight three domain-specific applications of computing the frequency ofsmall fixed pattern graphs inside a network. In particular, the concept of network motifs andgraphlets has proven very useful in the area of computational biology.

A network motif is a subgraph (not necessarily induced and possibly labeled) that appears witha significantly higher frequency in a real-world network than one would expect by pure chance.Introduced in [33] under the hypothesis that such frequently occurring structures have a functionalsignificance, motifs have been identified in a plethora of different domains—including protein-protein-interaction networks [2], brain networks [50] and electronic circuits [21]. We point theinterested reader to the surveys of Kaiser, Ribeiro and Silva [48] and Masoudi-Nejad, Schreiber,and Kashani [28] for a more extensive overview.

Graphlets are a related concept, though their application is in an entirely different scope. Whilemotifs are used to identify and explain local structure in networks, graphlets are used to ‘fingerprint’them. Pržulj [47] introduced the graphlet degree distribution as a way of measuring networksimilarity. To compute it, one enumerates all connected graphs up to a fixed size (five in theoriginal paper) and computes for each vertex of the target graph, how often it appears in a subgraphisomorphic to one of those patterns. Since some graphlets exhibit higher symmetry than others, thecomputation takes into account all possible automorphisms. The degree distribution then describesfor each graphlet Gi, how many vertices of the target graph are contained in 0,1,2, . . . subgraphsisomorphic to Gi—more precisely, in how many orbits of the respective automorphism groups itappears in. Note that if the set of graphlets only contains the single-edge graph this computationyields exactly the classical degree distribution.

The application of this distribution is two-fold: On the one hand, it can be used to measuresimilarity of multiple networks, in particular, networks depicting biological data [20]. On the otherhand, the local structure around a vertex can reveal domain-specific functions. This is the case forprotein-protein interaction networks where local structure correlates with biological activity [31],which has been applied to identify cancer genes [30] and construct phylogenetic trees [25]. Graphletshave further been used in analysis of workplace dynamics [51], photo cropping [10] and DoS attackdetection [46].

A third application of subgraph counting was given by Ugander et al. [52]: their empiricalanalysis and subsequent modeling of social networks revealed that there is an inherent bias towardsthe occurrence of certain subgraphs. Thus the frequencies of constant-sized subgraphs seems animportant indicator for the social domain, similar to the role of graphlet frequencies in biologicalnetworks.

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In Theorem 18.9 from [38] it was shown that for a graph class of bounded expansion countingthe number of satisfying assignments of a fixed boolean query is possible in linear time on a labeledgraph. This implies that (labeled) graphlet and motif counting are linear time on a graph classes ofbounded expansion. This result is achieved by using the algorithm presented in Lemma 17.3 in [38]algorithm to count the number of satisfying assignments of a fixed boolean query parameterized bytreedepth. The running time of this algorithm is O(2ht ·ht ·n), where h is the number of nodes inthe graphlet or motif and t is the depth of a treedepth decomposition of the graph. We provide analgorithm with a running time of O(23h · th ·h2 ·n). This achieves a better running time when usedto count on graphs of bounded expansion, since then, as will be explained later, h will equal t, i.e.a constant. This algorithm is presented in such a way that it can be used to also count how manytimes a node appears as a specific node of a specific graphlet or motif.

The tool of choice for applying a counting algorithm designed for bounded-treedepth graphs to aclass of bounded expansion is a p-centered coloring: to compute the frequency of a given pattern ofsize k, we compute a (k+1)-centered coloring of the input graph in linear time as per Proposition 2.We then enumerate all possible choice of i < k colors and count the frequency of the pattern graphin the graph induced by those colors classes. As this induced subgraph has bounded treedepth, wecan focus on counting a fixed subgraph inside a target graph of treedepth at most k. It is then easyto compute the frequency for the original graph using inclusion-exclusion on the color classes.

Central to the dynamic programming we will use to count isomorphisms is the following notionof a k-pattern which is very similar to the well-known notion of boundaried graphs.

Definition 8 A k-pattern of the graph H is a triple M = (W,X ,π) where X ⊆W ⊆ V (H) and|X | ≤ k and π : X → [k] is an injective function. We will call the set X the boundary of M. For agiven k-pattern M we denote the underlying graph by H[M] = H[W ], the vertex set by V (M) =W,the boundary by bd(M) = X and the mapping by πM.

We denote by Pk(H) the set of all k-patterns of H. Note that every k-pattern (W,X ,π) is also a(k+1)-pattern. In the following we denote by |H|= |V (H)|.

Lemma 6 Let H be a graph. Then |Pk(H)| ≤ 22|H|+|H| logk ≤ 22|H| · k|H|.

Proof: The graph H has 2|H| possible vertex subsets, each with at most 2|H| possible choices for aboundary. The number of ways an injective mapping for a boundary of size b≤ |H| into [k] can bechosen is bounded by k|H| = 2|H| logk. In total the size of Pk(H) is always less than 22|H|+|H| logk. 2

The following definition show how k-patterns will be used structural, namely by gluing themtogether or by demoting a boundary-vertex to a simple vertex. These operations will later be usedfor dynamic programming.

Definition 9 (k-pattern join) Let G be a graph and M1 =(W1,X1,π1),M2 =(W2,X2,π2) k-patternsof G. Then the two patterns are compatible if W1∩W2 = X1 = X2 and for all v ∈ X1 it holds thatπ1(v) = π2(v). Their join is defined as the k-pattern M1⊕M2 = (W1∪W2,X1,π1).

Definition 10 (k-pattern forget) Let G be a graph, let M = (W,X ,π) be a k-pattern of G andi ∈ [k]. Then the forget operation is the k-pattern

M i =

(W,X \π−1(i),π|X\π−1(i)) if π−1(i) 6=∅(W,X ,π) otherwise

23

Structurally, the k-pattern’s boundaries will represent vertices from the path of the root vertex tothe currently considered vertex in the treedepth decomposition, while the remaining vertices of thepattern represent vertice somehwere below it. The following two notations help expressing theseproperties.

Definition 11 (Subtree and root path) Let T be a treedepth decomposition of G rooted at r ∈ Gand let v ∈ V (G) be a vertex. Then the subtree of v is the subtree Tv of T rooted at v. The rootpath of v is the unique path Pv from the root r to v in T . We let Pv[i] denote the iþ vertex of the path(starting at the root), so that Pv[1] = r and Pv[ |Pv| ] = v.

We can now state the main lemma. The proof contains the description of the dynamic program-ming which works bottom-up on the vertices of the given treedepth decomposition (i.e. starting atthe leaves and working towards the root of the decomposition).

Lemma 7 Let H be a fixed graph on h vertices. Given a graph G on n vertices and a treedepthdecomposition T of height t, one can compute the number of isomorphisms from H to inducedsubgraphs of G in time O(23h+h log t ·h2 ·n) and space O(22h+h log t ·ht · logn).

Proof: We provide the following induction that easily lends itself to dynamic programming over T .Denote by MH = (V (H),∅,ε) the trivial t-pattern of H, here ε : ∅→∅ denotes the null function.Consider a set of vertices v1,v2, . . . ,v` ∈ G with a common parent v in T with respective subtreesTvi and root paths Pvi for 1≤ i≤ `. Note that the root paths Pv1, . . . ,Pv` all have the same length kand share the path Pv as a common prefix.

Let M1 be a fixed t-pattern of H. We define the mapping ψM1v : bd(M1)→ Pv via ψ

M1v (v) =

Pv[πM1(v)] which takes the pattern’s boundary and maps it to the vertices of the root-path.

For patterns M1 that satisfy that for all u ∈ bd(M1), πM1[u]≤ l, we denote by f [v1, . . . ,v`][M1]the number of isomorphisms φ1 : V (M1)→V (G) such that

(i) φ1|bd(M1) = ψM1v

(ii) φ1(V (M1)\bd(M1))⊆ G[V (Tv1 ∪·· ·∪Tv`)].

In other words we charge subgraphs to patterns whose boundaries lie on the shared root-path Pv,such that the labeling of the boundary coincides with the numbering induced by Pv while the rest ofthe pattern is contained entirely in the subtree below v.

Let r be the root of the treedepth decomposition. By the above definition, f [r][MH ] countsexactly the number of isomorphisms of H into subgraphs of G.

We will show now how we can compute f [r][MH ] recursively. For a leaf v ∈ T and a t-patternM1 = (W1,X1,π1) ∈Pk(H) we compute f [v][M1] as follows: Defined the value pM1

v to be 1 if thefunction ψ : W1→ Pv defined as ψ(w) = Pv[π1[w]] is an isomorphism from H[W1] to G[ψ(W )] and0 otherwise. In particular, pM1

v will be zero if W1 6= X1 or |W1|> Pv. Then for the leaf v, we compute

f [v][M1] = ∑M2|Pv|=M1

pM2v

where M2 ∈Pt(H).

24

The following recursive definitions show how f [·][M1] can be computed for all inner vertices ofT .

f [v][M1] = ∑M2|Pv|=M1

f [v1, . . . ,v`][M2] (forget)

f [v1, . . . ,v j−1,v j][M1] = ∑M2⊕M3=M1

f [v1, . . . ,v j−1][M2] · f [v j][M3] (join)

where M2,M3 ∈Pt(H).We need to prove that the table f correctly reflects the number of isomorphisms to subgraphs

satisfying properties i and ii.Consider the join-case first: fix a pattern M1 ∈Pt(H). By induction, the entries f [v1, . . . ,v j−1][·]

and f [v j][·] correspond to the number of isomorphisms to subgraphs that satisfy properties i and iiwith the node tuples v1, . . . ,v j−1 and v j, respectively. We need to show that f [v1, . . . ,v j][M1] ascomputed above counts the number of isomorphisms from H[M1] to subgraph of G such thatφ1|bd(M1) = ψ

M1v and φ1(V (M1)\bd(M1))⊆ G[V (Tv1 ∪·· ·∪Tv j)].

Consider the set Φ1 of all isomorphisms from H[M1] to subgraphs of G satisfying prop-erties i and ii for the vertex tuple v1, . . . ,v j. For any vertex subset R ⊆ V (M1) \ bd(M1), de-fine the slice Φ1(R) ⊆ Φ1 as those isomorphisms φ that satisfy φ−1(φ(V (H))∩ Tv j) = R. LetL = (V (M1) \ bd(M1)) \ R and define the patterns ML = (L∪ bd(M1),bd(M1),π

M1) and MR =(R∪ bd(M1),bd(M1),π

M1). Then by induction |Φ1(R)| = f [v1, . . . ,v j−1][ML] · f [v j][MR]. SinceM1 = ML⊕MR and clearly ML,MR ∈Pt(H), the sum computes exactly ∑R⊆V (M1)\bd(M1) |φ1(R)|=|φ1|.

Next, consider the forget-case. Again, fix M1 ∈Pt(H) and let u be the parent of v in T . LetΦ1 be the set of those isomorphisms from H[M1] to subgraphs of G for which φ1|bd(M1) = ψ

M1u and

φ1(V (M1)\bd(M1))⊆ G[V (Tv)]. We partition Φ1 into Φ1 = Φ1,v∪Φ1,v where Φ1,v contains thoseisomorphisms φ for which φ−1(v) 6=∅ and Φ1,v the rest. Since |Φ1,v|= f [v1, . . . ,v`][M1] we focuson Φ1,v in the following. For w ∈V (M1)\bd(M1), define Φ1,v(w) as the set of those isomorphismsφ for which φ(w) = v. Clearly, Φ1,v(w) | w ∈V (M1)\bd(M1) is a partition of Φ1,v. Define thepattern Mw = (V (M1),bd(M1)∪w,πM1

w ) where πM1w is πM1 augmented with the value π

M1w (v) =

|Pv|. Note that by construction M1 = Mw|Pv|. By induction, |Φ1,v(w)| = f [v1, . . . ,v`][Mw] andtherefore

|Φ1|= |Φ1,v|+ ∑w∈V (M1)\bd(M1)

|Φ1,v(w)|= ∑M2|Pv|

f [v1, . . . ,v`][M2]

It remains to prove the claimed running time. Initialization of f for a leaf takes time O(|Pt(H)|h2)since we need to test whether the function ψ defined above is an isomorphism for each patternM1 ∈Pt(H).

For the other vertices, a forget operation can be achieved in time O(|Pt(H)|) per vertex byenumerating all t-patterns, performing the forget operation and looking up the count of the resultingpattern in the previous table.

A join operation needs time O(|Pt(H)| ·h ·2h) per vertex, since for a given pattern M1 thosepatterns M2,M3 with M1 =M2⊕M3 are uniquely determined by partitions of the set V (M1)\bd(M1).

In total the running time of the whole algorithm is O(|Pt(H)| ·2h ·h2 ·n). Note that we onlyhave to keep at most O(t) tables in memory, each of which contains the occurrence of up to |Pt(H)|

25

patterns stored in numbers up to nh. Thus in total the space complexity is O(|Pt(H)| · t · log(nh)) =O(|Pt(H)| ·ht · logn). 2

To count the occurrences of H as an induced subgraph instead the number of subgraph isomor-phisms, one can simply determine the number of automorphisms of H in time 2O(

√h logh) [3, 29]

and divide the total count by this value (since this preprocessing time is dominated by our runningtime we will not mention it in the following). Counting isomorphism to non-induced subgraphs canbe done in the same time and space by changing the initialization on the leaves, such that it checksfor an subgraph instead of an induced subgraph. Dividing again by the number of automorphismsgives the number of subgraphs. By allowing the mapping of the patterns to map several nodesto the same value, we can use them to represent homomorphisms. Testing the leaves accordinglythe same algorithm can be used to count the number of homomorphisms from H to subgraphs ofG. By keeping all tables in memory, thus sacrificing the logarithmic space complexity, and usingbacktracking we can also label every node with the number of times it appears as a certain vertex ofH.

From these observations and Lemma 6 we arrive at the following theorem:

Theorem 6 Given a graph H on h vertices, a graph G on n vertices and a treedepth decompo-sition of G of height t, one can compute the number of isomorphisms from H to subgraphs of G,homomorphisms from H to subgraphs of G, or (induced) subgraphs of G isomorphic to H in timeO(23h · th ·h2 ·n) and space O(22h · th ·ht · logn).

Note that for graphs of unbounded treedepth the running time of the algorithm degenerates toO(23h ·h2 ·nh+1), which is comparable to the running time of 2O(

√h logh) ·nh of the trivial counting

algorithm.

Theorem 7 Given a graph H and a a graph G belonging to a class of bounded expansion, thereexists an algorithm to count the appearances of H as a subgraph of G in time

O((

f (h)h

)·23h+h logh ·h2 ·n

)where f is a function depending only on the graph class.

6.2 Localized CentralityCentrality is a notion used to ascribe the relative importance of a vertex in the network. A centralitymeasure is a real-valued function that assigns each vertex of the network some value with theunderstanding that higher values correspond to more central vertices. Depending on the application,“central" vertices need not be those with high degree (for example, a cut-vertex may have highcentrality as it is the only way for information to flow between two large subgraphs). There havebeen a wide variety of centrality scores introduced in the literature, including degree centrality,closeness centrality, eigenvector centrality, betweenness centrality and others. For a comprehensiveintroduction to centrality measures in social networks see, for instance, [16, 17]. There are severalrecent articles devoted to the topic of centrality measures in general [24,45]. In this section, we con-sider localized variants of measures similar to the closeness centrality introduced by Sabidussi [49]

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Measure Definition Localized

Closeness centrality [49] cC(v) = ( ∑u∈V (G)

d(v,u))−1 crC(v) = ( ∑

u∈Nr(v)d(v,u))−1

Harmonic centrality [43] cH(v) = ∑u∈V (G)

d(v,u)−1 crH(v) = ∑

u∈Nr(v)d(v,u)−1

Lin’s index [26] cL(v) =|v | d(v,v)< ∞|2

∑u∈V (G):d(v,u)<∞ d(v,u)cr

L(v) =|Nr[v]|2

∑u∈Nr[v] d(v,u)

Table 2: Distance-based centrality measures with localized variants that can be computed in lineartime on graphs of bounded expansion.

(see Table 2). The global versions of all these measures require one to compute the distance betweenall vertex pairs in the network, a sub-routine where the fastest known algorithm (due to Brandes [9])is O(n(n+m)) which in the context of sparse networks reduces to quadratic time.

In these localized variants, we compute the measure of a vertex with respect to its rth neighbor-hood rather than with respect to the whole graph. We give linear time algorithms for computingthese measures on graphs of bounded expansion for every constant r. As the value of r increases,the value of the measure computed for a vertex approaches its unlocalized variant at the expenseof an increase in running time.4 The measures in question and their localized variants are listed inTable 2.

One natural question is the utility of localized variants (and their accuracy in reflecting theglobal measure). We remark that Marsden demonstrated that for some networks, calculating themeasure for a vertex v inside its closed neighborhood G[N[v]] can be used as a viable substitute forthe full measure [27]. In the context of computer networks, Pantazopoulos et al. [44] consider localvariants which lend themselves to distributed computing and found a close correlation to the fullmeasures on a sample of networks. We can show experimentally that our variants reliably capturethe top ten percent in (arbitrarily) selected networks of our real-world corpus. To that end, wecompare the top 10 percent as identified by our localized variants to those 10 percent identified bythe respective full centrality measure. The results in Figure 2 suggtest that already a value of r equalto about half the diameter yields very good results across all three measures. Note that we do notcompare rankings; but rather only the difference between the sets of the identified top vertices—ourexperiments rank ordering is not preserved reliably. Furthermore, there seems to be a slight positivetendency towards the localized version being better in larger networks, though it is hard to draw anyconclusions on a small experiment like that.

While the localized Lin’s index and the localized harmonic closeness work as depicted in Table 2,the localized closeness needs a small normalization tweak in order to yield good results: this isachieved by treating the r+1-th neighbourhood of very vertex as if it would contain all remainingvertices and adding this value accordingly (this obviously does not change the running time of thealgorithm).

4For general graphs, we can compute these localized variants in time O(n(n+m)), by performing a breadth-firstsearch from every vertex, for instance. We do not know whether a better running time is possible.

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netscience 379 17codeminer 667 19diseasome 1419 15cpan-distributions 2719 9hep-th 5835 19cond-mat 13861 18

Figure 2: Quality of localized centrality measures in terms of similiarity of the top ten percentvertices against the full centrality measure. Measurements were taken only in the giant componentthe networks displayed in the table.

Let G be a graph from a class of bounded expansion and let ~Gr be the directed graph within-degree bounded by f (r), for some function f , that is obtained from G as by Proposition 3(given at the beginning of Section 6. In the following, we let N−r (v) denote the in-neighborhoodof the vertex v in the directed graph ~Gr. We assume that the vertices of G are ordered so thatevery vertex set has a unique representation as a tuple and, by slight abuse of notation, we useboth representations interchangeably. For v ∈V (G) and A = (a1,a2, . . . ,ap)⊆ N−r (v), define thedistance vector from v to A as dist(v,A) := (ω(a1v),ω(a2v), . . . ,ω(apv)), where ω denotes thearc-labeling from Proposition 3. Since ai ∈ N−r (v), ω(ai,v) = dG(ai,v).

Definition 12 Let v ∈ ~Gr, ∅ 6= X ⊆ N−r (v), α : V (G)→ R a vertex weighting and let d ∈ [r]|X | bea distance vector. We define

N(v,X , d) := v 6= u ∈V (G) | N−r (v)∩N−r (u) = X and dist(u,X) = d

as those vertices whose in-neighborhood in ~Gr overlap with the in-neighborhood of v in exactly X

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and whose distance-vector to X is exactly d. Then the query-function cα is defined as

cα(v,X , d) := ∑u∈N(v,X ,d)

α(u).

Lemma 8 Given ~Gr, one can compute a data structure in time O(n) such that queries cα(v,X , d)as in Definition 12 can be answered in constant time.

Proof: We define an auxiliary dictionary R indexed by vertex sets X ⊆ N−r(v), for some vertexv. At each entry v ∈ ~Gr, we will store another dictionary indexed by distance vectors which inturn stores a simple counter. We initialize R as follows: for every v ∈ ~Gr,X ⊆ N−r (v) and everydistance vector d ∈ [r]|X |, set R[X ][d] = 0. Note that in total, R contains only O(n) entries sinceall in-neighborhoods in ~Gr have constant size. We can implement R as a hash-map to achieve thedesired (expected) constant-time for insertion and look-up, though this would yield a randomizedalgorithm. A possible way to implement R on a RAM deterministically is the following: We storethe key X = x1,x2, . . . ,xp at address x1 +n · x2 + . . .+np · xp. This uses addresses up to size nc,for some constant c, but since we only insert O(n) keys the setup takes only linear time. Our laterqueries to R will be restricted to keys that are guaranteed to be contained in the dictionary, thus wewill never visit a register that has not been initialized.

Now, for every v ∈ ~Gr, every X ⊆ N−r (v), increment the counter R[X ][dist(v,X)] by α(v). Wenow claim that queries of the form cα(v,X , d) can be computed using inclusion-exclusion as follows:

cα(v,X , d) = ∑X⊆Y⊆N−r (v)

(−1)|Y\X | ∑d′:d′|X=d

R[Y ][d′].

The computation of the sum clearly takes constant time5. We now prove that it indeed computes thequantity cα(v,X , d).

Consider a vertex u ∈ ~Gr, such that N−r (u)∩N−r (v) = X and dist(u,X) = d. We argue that α(u)is counted once by the above sum: α(u) is not counted by any R[Y ][·] with Y ) X , therefore onlythe term where X =Y counts α(u) and does so exactly once. It remains to be shown that the weightof vertices that do not conform with Definition 12 are either not counted by the sum or cancel out.

Consider a vertex w ∈ ~Gr with such that dist(w,X) 6= d. The weight of such a vertex is notcounted by the above sum, since α(w) is only counted in entries of R that do not occur as summands.

Finally, consider a vertex w′ ∈ ~Gr with N−r (w′)∩N−r(v) = Z where X ( Z ⊆ N−r (v) and suchthat dist(w′,X) = d. The weight of this vertex is counted in each term of

∑X⊆Y⊆Z

(−1)|Y\X |R[Y ][dist(w′,Z)|Y ]

since

∑X⊆Y⊆Z

(−1)|Y\X | = ∑0≤k≤n

(−1)k(

nk

)= 0

we know that the signs cancel out and thus α(w′) does not contribute to cα(v,X , d). Hence theabove sum computes exactly the query cα(v,X , d). 2

5We tacitly assume that the weights α only assign numbers polynomially bounded by the size of the graph.

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Theorem 8 Let G be a graph class of bounded expansion, G ∈ G a graph and r ∈ N an integer.Then one can compute the quantities αd(v) = ∑w∈Nd(v)α(w) for all v ∈ G,d ≤ r in linear time.

Proof: By Theorem 3 we can compute ~Gr in linear time, thus we can employ Lemma 8 toanswer queries as defined in Definition 12 in constant time. To compute the quantity αd(v) for all0 < d ≤ r and v ∈ V (G), we proceed as follows. Initialize an array C by setting C[v][d] = 0 forevery v ∈ G,1 < d ≤ r.

Now for every v ∈V (G), every X ⊆ N−r (v) and every distance vector d ∈ [r]|X |, update C via

C[v][min(d +dist(v,X))]←C[v][min(d +dist(v,X))]+ cα(v,X , d)

with the convention that we dismiss entries where min(d +dist(v,X)) > r. At this point, C[v][d]contains the sum of weights of vertices u for which min(dist(v,X)+ dist(u,X)) = d where X =N−r (v)∩N−r (u) 6=∅. This follows directly from the definition of cα .

By Theorem 3, every pair of vertices of distance < r in G either is connected by an arc or theyshare a common in-neighbor in ~Gr. Accordingly, we update the values of C as follows: for everyuv ∈ ~E(~Gr)

• if N−r (u)∩N−r (v) =∅, the weights of the vertices u and v were not counted in C[v][·],C[u][·]respectively, thus we update C via

C[v][ω(uv)]←C[v][ω(uv)]+α(u)C[u][ω(uv)]←C[u][ω(uv)]+α(v)

• if X = N−r (u)∩N−r (v) 6= ∅, the weights of the vertices u and v were counted in C[v][d′]and C[u][d′] for d′ = min(dist(u,X)+dist(v,X)), respectively. Since d′ might be larger thanω(uv) (but cannot be smaller), we update C via

C[v][d′]←C[v][d′]−α(u)C[u][d′]←C[u][d′]−α(v)

C[v][ω(uv)]←C[v][ω(uv)]+α(u)C[u][ω(uv)]←C[u][ω(uv)]+α(v)

where we again ignore the update of C[·][d′] if d′ > r.

Note that this procedure is problematic if both uv and vu are present in the graph, since then the thiscorrection would (wrongly) be applied twice. The simple solution is that in the case of both arcsbeing present we only apply the above update for that arc where the start vertex is smaller than theend vertex, i.e. to uv if u < v and vu otherwise.At this point, C[v][d] contains the sum of weights of vertices u for which either

• the value d = min(dist(v,X)+dist(u,X)) where X = N−r (v)∩N−r (u) 6=∅ and uv 6∈ ~E(~Gr),

• or d = ω(uv) and uv ∈ ~E(~Gr).

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Thus by Theorem 3 we have that C[v][d] = αd(v) for d < r and v ∈ G. Since all of the aboveoperations take time linear in |V (G)|, the claim follows. 2

If we take α(·) = 1, the above algorithm counts exactly the sizes of the dth neighborhoods ofeach vertex, for d < r. Thus it can be used to compute the r-centric centrality measures presented inTable 2.

Corollary 3 Let G be a graph class of bounded expansion, G ∈ G a graph and r ∈ N an integer.Then the r-centric closeness, harmonic centrality and Lin’s index can be computed for all verticesof G in total time O(|V (G)|).

7 Open ProblemsWhat other models of complex networks have (or do not have) bounded expansion? Is there a modelwhich achieves high clustering coefficients and still has bounded expansion? The results presentedin Section 5 indicate that such a model could exist. Of separate theoretical interest would be to finda model with unbounded expansion but which is nowhere dense (as it might have a unique methodfor achieving clustering). Finally, are there natural extensions of non-sparse models like Watts andStrogatz [53] that exhibit bounded expansion?

On the algorithmic side, there are several key challenges remaining. Does there exist a betteralgorithm/heuristic to obtain low treedepth colorings? Can we compute or approximate lowerbounds for either χp or ∇r? Both would likely improve our current empirical understanding ofthe expansion of networks. We would also like to investigate whether the grad for small depthsis a reliable measure to differentiate networks; both our empirical and theoretical results seem toindicate so.

Finally, low-grad algorithms should be tested extensively via computational experiments, tohelp ascertain the feasibility of applying these techniques to real-world networks.

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