First-Order Model-Checking in Random Graphs and Complex Networks
aa r X i v : . [ c s . D M ] J un First-Order Model-Checking inRandom Graphs and Complex Networks ∗ Jan Dreier, Philipp Kuinke, Peter Rossmanith
Theoretical Computer Science, RWTH Aachen University {dreier,kuinke,rossmani}@cs.rwth-aachen.de
Abstract
Complex networks are everywhere. They appear for example in the form of biologi-cal networks, social networks, or computer networks and have been studied extensively.Efficient algorithms to solve problems on complex networks play a central role in today’ssociety. Algorithmic meta-theorems show that many problems can be solved efficiently.Since logic is a powerful tool to model problems, it has been used to obtain very generalmeta-theorems. In this work, we consider all problems definable in first-order logic andanalyze which properties of complex networks allow them to be solved efficiently.The mathematical tool to describe complex networks are random graph models. Wedefine a property of random graph models called α -power-law-boundedness. Roughlyspeaking, a random graph is α -power-law-bounded if it does not admit strong clusteringand its degree sequence is bounded by a power-law distribution with exponent at least α (i.e. the fraction of vertices with degree k is roughly O ( k − α )).We solve the first-order model-checking problem (parameterized by the length of theformula) in almost linear FPT time on random graph models satisfying this propertywith α ≥
3. This means in particular that one can solve every problem expressible infirst-order logic in almost linear expected time on these random graph models. Thisincludes for example preferential attachment graphs, Chung–Lu graphs, configurationgraphs, and sparse Erd˝os–R´enyi graphs. Our results match known hardness results andgeneralize previous tractability results on this topic.
Complex networks, as they occur in society, biology and technology, play a central role inour everyday lives. Even though these networks occur in vastly different contexts, they arestructured and evolve according to a common set of underlying principles. Over the lasttwo decades, with the emergence of the field of network science, there has been an explosionin research to understand these fundamental laws. One well observed property is the small-world phenomenon , which means that distances between vertices are very small. This hasbeen verified for the internet and many other networks [1, 55]. Furthermore, many real net-works tend to be clustered . They contain groups of vertices that are densely connected [66].If two vertices share a common neighbor, then there is a high chance that there is also anedge between them. A network can be considered clustered if the ratio between the numberof triangles and the number of paths with three vertices is non-vanishing. This is formal-ized by the clustering coefficient, which is high for many networks [72]. A third importantproperty is a heavy tailed degree distribution . While most vertices have a low number ofconnections, there are a few hubs with a high degree. Experiments show that the degreesfollow for example a power-law or log-normal distribution. In a power-law distribution, the ∗ A short version of this paper appeared in the Proceedings of the 28th Annual European Symposium onAlgorithms (ESA 2020). k is proportional to k − α (usually with α between 2 and 3).This behavior makes complex networks highly inhomogeneous [64, 57, 10, 15].One important goal of theoretical computer science has always been to explore whatkinds of inputs allow or forbid us to construct efficient algorithms. In this context, algo-rithmic meta-theorems [51] are of particular interest. They are usually theorems statingthat problems definable in a certain logic can be solved efficiently on graph classes thatsatisfy certain properties. Logic is a powerful tool to model problems and therefore hasbeen used to obtain very general meta-theorems. A well-known example is Courcelle’s the-orem [16], which states that every problem expressible in counting monadic second-orderlogic can be solved in linear time on graph classes with bounded treewidth. It has beenfurther generalized to graph classes with bounded cliquewidth [17]. To obtain results forlarger graph classes one has to consider weaker logics. The languages of relational databasesystems are based on first-order logic. In this logic, one is allowed to quantify over ver-tices and to test equality and adjacency of vertices. With k existential quantifiers, one mayask for the existence of a fixed graph with k vertices ( k -subgraph isomorphism), a problemrelevant to motif-counting [56, 28]. On the other hand, connectivity properties cannot beexpressed in first-order logic. We define for every graph class G the parameterized first-ordermodel-checking problem p -MC(FO , G ) [44]. p -MC(FO , G ) Input:
A graph G ∈ G and a first-order sentence ϕ Parameter:
The number of symbols in ϕ , denoted by | ϕ | Problem:
Does ϕ hold on G (i.e. G | = ϕ )?The aim is to show for a given graph class G that p -MC(FO , G ) is fixed parameter tractable (FPT), i.e., can be decided in time f ( | ϕ | ) n O (1) for some function f (see for example [18]for an introduction to fixed parameter tractability). Since input graphs may be large, alinear dependence on n is desirable. If one is successful, then every problem expressible infirst-order logic can be solved on G in linear time.For the class of all graphs G , p -MC(FO , G ) is AW[ ∗ ]-complete [25] and therefore mostlikely not fpt. Over time, tractability of p -MC(FO , G ) has been shown for more and moresparse graph classes G : bounded vertex degree [68], forbidden minors [34], bounded localtreewidth [33], and further generalizations [19, 29, 67]. Grohe, Kreutzer and Siebertz provethat p -MC(FO , G ) can be solved in almost linear FPT time f ( | ϕ | , ε ) n ε for all ε > G is a nowhere dense graph class [45]. On the other hand if G is a monotone somewhere densegraph class, p -MC(FO , G ) is AW[ ∗ ]-hard [45]. Nowhere dense graph classes were introducedby Neˇsetˇril and Ossona de Mendez as those graph classes where for every r ∈ N the sizeof all r -shallow clique minors of all graphs in the graph class is bounded by a function of r (Section 4.3). A graph class is somewhere dense if it is not nowhere dense. The tractabilityof the model-checking problem on monotone graph classes is completely characterized witha dichotomy between nowhere dense and somewhere dense graph classes. These very generalresults come at a cost: Frick and Grohe showed that the dependence of the run time on ϕ isnon-elementary [37]. We want to transfer this rich algorithmic theory to complex networks.But what is the right abstraction to describe complex networks?Network scientists observed that the chaotic and unordered structure of real networkscan by captured using randomness . There is a vast body of research using random processesto create graphs that mimic the fundamental properties of complex networks. The mostprominent ones are the preferential attachment model [3, 63], Chung–Lu model [12, 13],configuration model [59, 58], Kleinberg model [49, 50], hyperbolic graph model [52], andrandom intersection graph model [46, 65]. All these are random models. It has been thor-oughly analyzed how well they predict various properties of complex networks [42].When it comes to algorithmic meta-theorems on random graph models “even the mostbasic questions are wide open,” as Grohe puts it [44]. By analyzing which models of complex2etworks and which values of the model-parameters allow for efficient algorithms, we aimto develop an understanding how the different properties of complex networks control theiralgorithmic tractability.In this work we show for a wide range of models, including the well known preferentialattachment model, that one can solve the parameterized first-order model-checking problemin almost linear FPT time. This means in particular that one can solve every problemexpressible in first-order logic efficiently on these models. Our original goal was to obtainefficient algorithms only for preferential attachment graphs, but we found an abstractionthat transfers these results to many other random graph models. Roughly speaking, thefollowing two criteria are sufficient for efficiently solving first-order definable problems on arandom graph model: • The model needs to be unclustered. In particular the expected number of trianglesneeds to be subpolynomial. • For every k , the fraction of vertices with degree k is roughly O ( k − ). In other words,the degree sequence needs to be bounded by a power-law distribution with exponent 3or higher.Models satisfying these properties include sparse Erd˝os–R´enyi graphs, preferential attach-ment graphs as well as certain Chung–Lu and configuration graphs. On the other hand,the Kleinberg model, the hyperbolic random graph model, or the random intersection graphmodel do not satisfy these properties. Our results generalize previous results [43, 22] andmatch known hardness results: The model-checking problem has been proven to be hardon power-law distributions with exponent smaller than 3 [27]. We therefore identify thethreshold for tractability to be a power-law coefficient of 3. It is also a big open questionwhether the model-checking problem can also be solved on clustered random graph models,especially since real networks tend to be clustered. Furthermore, significant engineeringchallenges need to be overcome to make our algorithms applicable in practice. Average-case complexity analyzes the typical run time of algorithms on random instances(see [7] for a survey), based on the idea that a worst-case analysis often is too pessimistic asfor many problems hard instances occur rarely in the real world. Since models of complexnetworks are probability distributions over graphs, we analyze the run time of algorithmsunder average-case complexity. However, there are multiple notions and one needs to becareful which one to choose.Assume a random graph model is asymptotically almost surely (a.a.s.) nowhere dense,i.e., a random graph from the model with n vertices belongs with probability 1 − δ ( n )to a nowhere dense graph class, where lim n →∞ δ ( n ) = 0 (Section 4.3). Then the first-order model-checking problem can be efficiently solved with a probability converging toone [45]. However, with probability δ ( n ) the run time can be arbitrarily high and therate of convergence of δ ( n ) to zero can be arbitrarily slow. These two missing bounds areundesirable from an algorithmic standpoint and the field of average-case complexity hasestablished a theory on how the run time needs to be bounded with respect to the fractionof inputs that lead to this run time.This is formalized by the well-established notion of average polynomial run time , in-troduced by Levin [53]. An algorithm has average polynomial run time with respect to arandom graph model if there is an ε > p such that for every n, t the prob-ability that the algorithm runs longer than t steps on an input of size n is at most p ( n ) /t ε .This means there is a polynomial trade-off between run time and fraction of inputs. Thisnotion has been widely studied [7, 2] and is considered from a complexity theoretic stand-point the right notion of polynomial run time on random inputs. It is closed under invokingpolynomial subroutines. 3n our work, however, we wish to explicitly distinguish linear time. While Levin’s com-plexity class is a good analogy to the class P, it is not suited to capture algorithms withaverage linear run time. For this reason, we turn to the expected value of the run time, astronger notion than average polynomial time. In fact, using Markov’s inequality we see thatif an algorithm has expected linear run time, all previous measures of average tractabilityare also bounded. Their relationship is as follows.expected linear ⇒ expected polynomial ⇒ average polynomial ⇒ a.a.s. polynomialWith this in mind we can present our notion of algorithmic tractability. A labeled graphis a graph where every vertex can have (multiple) labels. First-order formulas can haveunary predicates for each type of label. These predicates test whether a vertex has a labelof a certain type. We define G to be the class of all graphs, and G lb to be the class of allvertex-labeled graphs. A function L : G → G lb is an l -labeling function for l ∈ N if for every G ∈ G , L ( G ) is a labeling of G with up to l classes of labels (see Section 4 for details).Furthermore, a random graph model is a sequence G = ( G n ) n ∈ N , where G n is a probabilitydistribution over unlabeled simple graphs with n vertices. Definition 4.6.
We say p -MC(FO , G lb ) can be decided on a random graph model ( G n ) n ∈ N in expected time f ( | ϕ | , n ) if there exists a deterministic algorithm A which decides p -MC(FO , G lb ) on input G , ϕ in time t A ( G, ϕ ) and if for all n ∈ N , all first-order sentences ϕ and all | ϕ | -labeling functions L , E G ∼ G n (cid:2) t A ( L ( G ) , ϕ ) (cid:3) ≤ f ( | ϕ | , n ) . We say p -MC(FO , G lb ) ona random graph model can be decided in expected FPT time if it can be decided in expectedtime g ( | ϕ | ) n O (1) for some function g .In particular, this definition implies efficient average run time according to Levin’s notion(which is closed under polynomial subroutines). We choose to include labels into our notionof average-case hardness for two reasons: First, it makes our algorithmic results stronger, asthe expected run time is small, even in the presence of an adversary that labels the verticesof the graph. Secondly, it matches known hardness results that require adversary labeling. There have been efforts to transfer the results for classical graph classes to random graphmodels by showing that a graph sampled from some random graph model belongs with highprobability to a certain algorithmically tractable graph class. For most random graph modelsthe treewidth is polynomial in the size of the graph [40, 5]. Therefore, people have consid-ered more permissive graph measures than treewidth, such as low degree [43], or boundedexpansion [22, 32]. Demaine et al. showed that some Chung–Lu and configuration graphshave bounded expansion and provided empirical evidence that some real-world networks dotoo [22]. However, this technique is still limited, as many random graph models (such asthe preferential attachment model [22, 26]) are not known to be contained in any of thewell-known tractable graph classes.The previous tractability results presented in this section all use the following technique:Assume we have a formula ϕ and sample a graph of size n from a random graph model. If thesampled graph belongs to the tractable graph class, an efficient model-checking algorithmfor the graph class can solve the instance in FPT time. If the graph does not belong thegraph class, the naive model-checking algorithm can still solve the instance in time O ( n | ϕ | ).Assume we can show that the second case only happens with probability δ ( n ) converging tozero faster than any polynomial. Then δ ( n ) O ( n | ϕ | ) converges to zero and the expected runtime remains bounded by an FPT function.Let p ( n ) be a function with p ( n ) = O ( n ε /n ) for all ε >
0. Grohe showed that onecan solve p -MC(FO , G lb ) on Erd˝os–R´enyi graphs G ( n, p ( n )) in expected time f ( | ϕ | , ε ) n ε for every ε > O ( n ε ) for every ε > α > p -MC(FO , G lb ) inexpected time f ( | ϕ | ) n on these random graph models.There further exist some average-case hardness results for the model-checking problem.It has been shown that one cannot decide p -MC(FO , G lb ) on Erd˝os–R´enyi graphs G ( n, / G ( n, p ( n )) with p ( n ) = n ε /n for some 0 < ε < ε ∈ Q , in expected FPT time (unlessAW[ ∗ ] ⊆ FPT / poly) [27]. The same holds for Chung–Lu graphs with exponent 2 . < α < α ∈ Q . These hardness results fundamentally require the adversary labeling of Definition 4.6.It is a big open question whether they can be transferred to model-checking without labels.Another thing to keep in mind when considering logic and random graphs [69] are zero-one laws . They state that in many Erd˝os–R´enyi graphs every first-order formula holds in thelimit either with probability zero or one [69, 41, 31]. Not all random graph models satisfya zero-one law for first-order logic (e.g. the limit probability of the existence of a K in aChung–Lu graph with weights w i = p n/i is neither zero nor one). We define a property called α -power-law-boundedness . This property depends on a parameter α and captures many unclustered random graph models for which the fraction of verticeswith expected degree d ∈ N is roughly O ( d − α ). Our main contribution is solving the model-checking problem efficiently on all α -power-law-bounded random graph models with α ≥ α -power-law-bounded. Our results hold for arbitrary labelings of therandom graph and are based on a novel decomposition technique for local regions of randomgraphs. While all previous algorithms work by placing the random graph model with highprobability in a sparse graph class, our technique also works for some a.a.s. somewhere denserandom graphs (e.g. preferential attachment graphs [26]). We start by formalizing our property. Since it generalizes the Chung–Lu model, we definethis model first. A Chung–Lu graph with exponent α and vertices v , . . . , v n is definedsuch that two vertices v i and v j are adjacent with probability Θ( w i w j /n ) where w i =( n/i ) / ( α − [12]. Furthermore all edges are independent, which means that the probabilitythat a set of edges occurs equals the product over the probabilities of each individual edge.In our model the probability of a set of edges can be a certain factor larger than the productof the individual probabilities, which allows edges to be moderately dependent. Definition 2.1.
Let α >
2. We say a random graph model ( G n ) n ∈ N is α -power-law-bounded if for every n ∈ N there exists an ordering v , . . . , v n of V ( G n ) such that for all E ⊆ (cid:0) { v ,...,v n } (cid:1) Pr (cid:2) E ⊆ E ( G n ) (cid:3) ≤ Y v i v j ∈ E ( n/i ) / ( α − ( n/j ) / ( α − n · O ( | E | ) if α > n ) O ( | E | ) if α = 3 O ( n ε ) | E | for every ε > α < . The probability that a set of edges E occurs may be up to a factor 2 O ( | E | ) or log( n ) O ( | E | ) or O ( n ε ) | E | (depending on α ) larger than the probability in the corresponding Chung–Lugraph. For conditional probabilities this means the following: The probability bound for an5dge under the condition that some set of l edges is already present may be up to a factor2 O ( l ) or log( n ) O ( l ) or O ( n ε ) l larger than the unconditional probability. This lets power-law-bounded random graphs capture moderate dependence between edges. The factor undergoesa phase transition at α = 3. The smaller factor 2 O ( | E | ) for α > α >
3. The slightlylarger factor of log( n ) O ( | E | ) for α = 3 was chosen to capture preferential attachment graphswhile still maintaining a quasilinear FPT run time of our algorithm.The parameter α of an α -power-law-bounded random graph model controls the degreedistribution. Note that if a graph class is α -power-law-bounded it is also α ′ -power-law-bounded for all 2 < α ′ < α . It can be easily seen that a vertex v i has expected degree atmost O ( n ε )( n/i ) / ( α − for every ε >
0. This means the expected degree sequence of an α -power-law-bounded random graph model is not power-law distributed with exponent smallerthan α . The gap is often tight: For example, Chung–Lu graphs with a power-law degreedistribution exponent α are α -power-law-bounded and preferential attachment graphs havea power-law degree distribution with exponent 3 and are 3-power-law-bounded. For theinteresting case α = 3, the inequality in Definition 2.1 simplifies toPr (cid:2) E ⊆ E ( G n ) (cid:3) ≤ log( n ) O ( | E | ) Y v i v j ∈ E √ ij . We now present our model-checking algorithm for α -power-law-bounded graphs. We expressits run time relative to the term˜ d α ( n ) = O (1) α > n ) O (1) α = 3 O ( n − α ) α < . This term is related to an established property of degree distributions, namely the second or-der average degree [12]. If a random graph with n vertices has expected degrees w , . . . , w n then the second order average degree is defined as P ni =1 w i / P nk =1 w k . In graphs with apower-law degree distribution α we have w i = Θ(( n/i ) / ( α − ). The second order aver-age degree then equals Θ (cid:0)P ni =1 ( n/i ) / ( α − / P nk =1 ( n/k ) / ( α − (cid:1) . For α >
3, this term isconstant, for α = 3 it is logarithmic, and for α < n [12]. Thus, wecan interpret ˜ d α ( n ) as an estimate of the second order average degree. We prove that themodel-checking problem can be solved efficiently if ˜ d α ( n ) is small. Theorem 8.5.
There exists a function f such that one can solve p - MC(FO , G lb ) on every α -power-law-bounded random graph model in expected time ˜ d α ( n ) f ( | ϕ | ) n . The term ˜ d α ( n ) naturally arises in our proofs and is not a consequence of how we definedthe multiplicative factor (i.e., 2 O ( | E | ) , log( n ) O ( | E | ) , O ( n ε ) | E | ) in Definition 2.1. In factthe dependence goes the other way: We defined the factor for each α as large as possiblesuch that it does not dominate the run time of the algorithm. Next we specify exactly thosevalues of α where the previous theorem leads to FPT run times. (In the third case ε > α to be arbitrarily close to 3.) Theorem 8.6.
Let ( G n ) n ∈ N be a random graph model. There exists a function f such thatone can solve p - MC(FO , G lb ) in expected time • f ( | ϕ | ) n if ( G n ) n ∈ N is α -power-law-bounded for some α > , • log( n ) f ( | ϕ | ) n if ( G n ) n ∈ N is α -power-law-bounded for α = 3 , • f ( | ϕ | , ε ) n ε for all ε > if ( G n ) n ∈ N is α -power-law-bounded for every < α < . every < α < Proposition 2.2 ([27] and Lemma 10.3) . For every < α < there exists an α -power-law-bounded random graph model ( G n ) n ∈ N such that one cannot solve p - MC(FO , G lb ) on ( G n ) n ∈ N in expected FPT time unless
AW[ ∗ ] ⊆ FPT / poly . We observe a phase transition in tractability at power-law exponent α = 3. Also the runtime of our algorithm cannot be linear in n for α ≤ n log( n ) edges in expectation. We discuss the algorithmicimplications of our result for some well-known random graph models in Section 10. Many algorithmic results are based on structural decompositions.
For example, bidimen-sionality theory introduced by Demaine et al. [20, 21] is based on the grid minor theorem,which is itself based on a structural decomposition into a clique-sum of almost-embeddablegraphs developed by Robertson and Seymour [60]. The model-checking algorithm for graphclasses with bounded expansion by Dvoˇrak, Kr´al, and Thomas [29] relies on a structuraldecomposition of bounded expansion graph classes by Neˇsetˇril and Ossona de Mendez calledlow tree-depth colorings [62]. Our algorithms are based on a structural decomposition of α -power-law-bounded random graph models.All algorithms prior to this work rely on showing that a certain graph model is with highprobability contained in a certain well-known tractable graph class (for example boundedexpansion) and then use the structural decompositions [62] of said graph class. However,these decompositions were not originally designed with random graphs in mind and thereforemay not provide the optimal level of abstraction for random graphs. Our algorithms arebased on a specially defined structural decomposition. This direct approach helps us capturerandom graph models that could otherwise not be captured such as the a.a.s. somewheredense preferential attachment model. By focusing on α -power-law-bounded random graphmodels, we obtain structural decompositions for a wide range of models.We observe that α -power-law-bounded random graphs have mostly an extremely sparsestructure with the exception of a part whose size is bounded by the second order averagedegree. However, this denser part can be separated well from the remaining graph. Weshow that local regions consist of a core part, bounded in size by the second order averagedegree, to which trees and graphs of constant size are attached by a constant number ofedges. This decomposition is similar to so called protrusion decompositions , which havebeen used by Bodlaender et al. to obtain meta-theorems on kernelization [6]. Our structuraldecomposition is valid for all graphs that fit into the framework of α -power-law-boundedness,such as preferential attachment graphs or Chung–Lu graphs. We define an approximationof the second order average degree of the degree distribution as ˆ d α ( n ) = 2 for α > d α ( n ) = log( n ) for α = 3 and ˆ d α ( n ) = n − α for α < d α ( n ) without O -notation). Theorem 9.5.
Let ( G n ) n ∈ N be an α -power-law-bounded random graph model. There existconstants c, r such that for every r ≥ r a.a.s. for every r -neighborhood H of G n one canpartition V ( H ) into three (possibly empty) sets X , Y , Z with the following properties. • | X | ≤ ˆ d α ( n ) cr . • Every connected component of H [ Y ] has size at most cr and at most c neighbors in X . • Every connected component of H [ Z ] is a tree with at most one edge to H [ X ∪ Y ] . Removing a few vertices makes the local neighborhoods even sparser:7 orollary 9.4.
Let ( G n ) n ∈ N be an α -power-law-bounded random graph model. There existconstants c, r such that for every r ≥ r a.a.s. one can remove ˆ d α ( n ) cr vertices from G n such that every r -neighborhood has treewidth at most . Corollary 9.4 is a consequence of Theorem 9.3 from Section 9. Further structural resultsthat may be interesting beyond the purpose of model-checking can be found in Section 9.We now discuss how we use the decomposition of Theorem 9.5 for our algorithms and whydecompositions similar to Corollary 9.4 are not sufficient for our purposes.
A first building block of our algorithm is Gaifman’s locality theorem [38]. It implies thatin order to solve the first-order model-checking problem on a graph, it is sufficient to solvethe problem on all r -neighborhoods of the graph for some small r . We can therefore restrictourselves to the model-checking problem on the neighborhoods of random graphs. With thisin mind, we want to obtain structural decompositions of these neighborhoods.One important thing to note is that a decomposition according to Corollary 9.4 is notsufficient. Let us focus on the interesting case α = 3 where efficient model-checking is stillpossible. Corollary 9.4 then states that the removal of polylogarithmically many verticesyields neighborhoods with treewidth at most 26. While we could easily solve the model-checking problem on graphs with treewidth at most 26 via Courcelle’s theorem [16], wecannot solve it on graphs where we need to remove a set X of log( n ) vertices to obtaina treewidth of at most 26. Every vertex not in X may have an arbitrary subset of X asneighborhood. Since there are 2 | X | = n possible neighborhoods, we can encode a largecomplicated structure into this graph by stating that two vertices i, j ∈ N are adjacent ifand only if there is a vertex whose neighborhood in X represents a binary encoding of theedge ij (omitting some details). Because of this, the model-checking problem on this graphclass is as hard as on general graphs. We need the additional requirement that X is onlyloosely connected to the remaining graph. The decomposition in Theorem 9.5 fulfills thisrequirement. Every component of H \ X has at most a constant number of neighbors in X .Let us assume we have decompositions of the neighborhoods of a graph according toTheorem 9.5 where the sets X are chosen as small as possible. We can now use a variantof the Feferman–Vaught theorem [47] for each r -neighborhood to prune the protrusions andthereby construct a smaller graph that satisfies the same (short) first-order formulas as theoriginal graph, We call this smaller graph the kernel . The size of this kernel will be somefunction of | X | . We then use the brute-force model-checking algorithm on the kernel.For the first steps of the algorithm (decomposition into neighborhoods, kernelizationusing Feferman–Vaught) one can easily show that they always take FPT time. However, therun time of the last step requires a careful analysis. One can check a formula ϕ on a graph ofsize x in time O ( x | ϕ | ) by brute force. Thus, checking the formula on the kernel of all n many r -neighborhoods of a random graph takes expected time at most n P nx =1 p x O ( x | ϕ | ) , where p x is the probability that the kernelization procedure on an r -neighborhood of a randomgraph yields a kernel of size x . In order to guarantee a run time of the form log( n ) f ( | ϕ | ) n for some function f , p x should be of order log( n ) f ( | ϕ | ) x −| ϕ | .Earlier, we discussed that the size of the kernel will be some function of | X | and thatwe choose X as small as possible. It is therefore sufficient to bound the probability thatthe set X of the decomposition of a neighborhood exceeds a certain size. Parameterizingthe decomposition by two values (denoted by b and µ later on) gives us enough control toguarantee such a bound on p x . A large part of this work is devoted to proving a goodtrade-off between the size of the set X of the decomposition and the probability that X isof minimal size. Furthermore, computing the set X is computationally hard, so the wholeprocedure has to work without knowing the set X , but only its existence.Our proofs are structured as follows. First, we show in Section 5 that α -power-law-bounded random graph models have the following structure with high probability: They8an be partitioned into sets A , B , C , where A ∪ B is small, B ∪ C is sparse and A and C locally share only few edges. This is done by characterizing this structure by a collectionof small forbidden edge-sets and then excluding these edge-sets using the union bound andDefinition 2.1. Then in Section 6 we show that the partition into A , B , C implies theprotrusion decomposition of Theorem 9.5. In Section 7, we partially recover the protrusiondecomposition from a given input, and use it to kernelize each r -neighborhood into anequivalent smaller graph. At last, in Section 8, we combine Gaifman’s locality theoremwith the previous algorithms and probability bounds to obtain our algorithm and bound itsrun time. Some proofs are quite tedious, but the nature of this problem seems to stop usfrom using simpler methods. Furthermore, in Section 9 give a simpler presentation of ourstructural results and in Section 10, we discuss the algorithmic implications of our resultsfor various random graph models. We use common graph theory notation [23]. The length of a path equals its number of edges.The distance between to vertices u and v (dist( u, v )) equals the length of a shortest pathbetween u and v . For a vertex v let N Gr ( v ) be the set of vertices which have in G distance atmost r to v . The radius of a graph is the minimum among all maximum distances from onevertex to all other vertices. An r -neighborhood in G is an induced subgraph of G with radiusat most r . The order of a graph is | G | = | V ( G ) | . The size of a graph is k G k = | V ( G )+ E ( G ) | .The edge-excess of a graph G is | E ( G ) | − | V ( G ) | .In this work we obtain results for labeled graphs [44]. A labeled graph is a tuple G =( V ( G ) , E ( G ) , P ( G ) , . . . , P l ( G )) with P i ( G ) ⊆ V ( G ). We call P ( G ) , . . . , P l ( G ) the labels of G . We say a vertex v is labeled with label P i ( G ) if v ∈ P i ( G ). A vertex may have multiplelabels. We say the unlabeled simple graph G ′ = ( V ( G ) , E ( G )) is the underlying graph of G and G is a labeling of G ′ . All notion for graphs extends to labeled graphs as expected.The union of two labeled graphs G and H , ( G ∪ H ), is obtained by setting V ( G ∪ H ) = V ( G ) ∪ V ( H ), E ( G ∪ H ) = E ( G ) ∪ E ( H ) and for each label P i ( G ∪ H ) = P i ( G ) ∪ P i ( H ).For a graph class G , we define G lb to be the class of all labelings of G . We define G tobe the class of all simple graphs and G lb to be the class of all labeled simple graphs. We denote probabilities by Pr[ ∗ ] and expectation by E[ ∗ ]. We consider a random graphmodel to be a sequence of probability distributions. For every n ∈ N a random graph modeldescribes a probability distribution on unlabeled simple graphs with n vertices. In orderto speak of probability distributions over graphs we fix a sequence of vertices ( v i ) i ≥ andrequire that a graph with n vertices has the vertex set { v , . . . , v n } . A random graph modelis a sequence G = ( G n ) n ∈ N , where G n is a probability distribution over all unlabeled simplegraphs G with V ( G ) = { v , . . . , v n } . Even though some random processes naturally lead tographs with multi-edges or self-loops, we interpret them as simple graphs by removing allself-loops and replacing multiple edges with one single edge. In slight abuse of notation, wealso write G n for the random variable which is distributed according to G n . This way, wecan lift graph notation to notation for random variables of graphs: For example edge setsand neighborhoods of a random graph G n are represented by random variables E ( G n ) and N G n r ( v ). At first, we define nowhere and somewhere density as a property of graph classes and thenlift the notation to random graph models . There are various equivalent definitions and we9se the most common definition based on shallow topological minors.
Definition 4.1 (Shallow topological minor [62]) . A graph H is an r -shallow topologicalminor of G if a graph obtained from H by subdividing every edge up to 2 r times is isomorphicto a subgraph of G . The set of all r -shallow topological minors of a graph G is denoted by G e ▽ r . We define the maximum clique size over all shallow topological minors of G as ω ( G e ▽ r ) = max H ∈ G e ▽ r ω ( H ) . Definition 4.2 (Nowhere dense [61]) . A graph class G is nowhere dense if there exists afunction f , such that for all r ∈ N and all G ∈ G , ω ( G e ▽ r ) ≤ f ( r ). Definition 4.3 (Somewhere dense [61]) . A graph class G is somewhere dense if for allfunctions f there exists an r ∈ N and a G ∈ G , such that ω ( G e ▽ r ) > f ( r ).Observe that a graph class is somewhere dense if and only if it is not nowhere dense. Welift these notions to random graph models using the following two definitions. Definition 4.4 (a.a.s. nowhere dense) . A random graph model G is a.a.s. nowhere dense ifthere exists a function f such that for all r ∈ N lim n →∞ Pr[ ω ( G n e ▽ r ) ≤ f ( r )] = 1 . Definition 4.5 (a.a.s. somewhere dense) . A random graph model G is a.a.s. somewheredense if for all functions f there is an r ∈ N such thatlim n →∞ Pr[ ω ( G n e ▽ r ) > f ( r )] = 1 . While for graph classes the concepts are complementary, a random graph model canboth be neither a.a.s. somewhere dense nor a.a.s. nowhere dense (e.g., if the random graphmodel is either the empty or the complete graph, both with a probability of 1 / We consider only first-order logic over labeled graphs. We interpret a labeled graph G =( V, E, P , . . . , P l ), as a structure with universe V and signature ( E, P , . . . , P l ). The binaryrelation E expresses adjacency between vertices and the unary relations P , . . . , P l indicatethe labels of the vertices. Other structures can be easily converted into labeled graphs. Wewrite ϕ ( x , . . . , x k ) to indicate that a formula ϕ has free variables x , . . . , x k . The quantifierrank of a formula is the maximum nesting depth of quantifiers in the formula. Two labeledgraphs G , G with the same signature are q - equivalent ( G ≡ q G ) if for every first-ordersentence ϕ with quantifier rank at most q and matching signature holds G | = ϕ if and onlyif G | = ϕ . Furthermore, | ϕ | is the number of symbols in ϕ . There exists a simple algorithmwhich decides whether G | = ϕ in time O ( | G | | ϕ | ). With all definitions in place, we can now properly restate the model-checking problem andwhat it means to solve it efficiently on a random graph model. The model-checking problemon labeled graphs is defined as follows. p -MC(FO , G lb ) Input:
A graph G ∈ G lb and a first-order sentence ϕ Parameter: | ϕ | Problem: G | = ϕ ? 10nder worst-case complexity, p -MC(FO , G lb ) is AW[ ∗ ]-complete [25] (and PSPACE-complete when unparameterized [70]). We want average case algorithms for p -MC(FO , G lb )to be efficient for all possible labelings of a random graph model. A function L : G → G lb is called a l -labeling function for l ∈ N if for every G ∈ G , L ( G ) is a labeling of G with upto l labels. Definition 4.6.
We say p -MC(FO , G lb ) can be decided on a random graph model ( G n ) n ∈ N in expected time f ( | ϕ | , n ) if there exists a deterministic algorithm A which decides p -MC(FO , G lb ) on input G , ϕ in time t A ( G, ϕ ) and if for all n ∈ N , all first-order sentences ϕ and all | ϕ | -labeling functions L , E G ∼ G n (cid:2) t A ( L ( G ) , ϕ ) (cid:3) ≤ f ( | ϕ | , n ) . We say p -MC(FO , G lb ) ona random graph model can be decided in expected FPT time if it can be decided in expectedtime g ( | ϕ | ) n O (1) for some function g . The goal of this section is to partition α -power-law-bounded random graph models. Weshow in Theorem 5.10 that their vertices can with high probability be partitioned into sets A, B, C with the following properties: The sets A and B are small, the graph G [ B ∪ C ]is locally almost a tree, i.e., has locally only a small edge-excess, and the set B almostseparates A from C , i.e., every neighborhood in G [ C ] has only a small number of edges to A . We call ( A, B, C ) an b - r - µ -partition. We state the formal definition. Definition 5.1 ( b - r - µ -partition) . Let b, r, µ ∈ N + . Let G be a graph. A tuple ( A, B, C ) iscalled an b - r - µ -partition of G if1. the sets A, B, C are pairwise disjoint and their union is V ( G ),2. | A | ≤ b and | B | ≤ b µ ,3. every 40 µr -neighborhood in G [ B ∪ C ] has an edge-excess of at most µ , and4. for every 20 µr -neighborhood in G [ C ] there are at most µ edges incident to both theneighborhood and to A .A graph for which an b - r - µ -partition exists is called b - r - µ -partitionable.In summary, B and C are well behaved and the large set C is almost separated from A . Note that the properties of an b - r - µ -partition depend on three parameters b , r , µ . Theresults of this section imply that our random graphs are asymptotically almost surely b - r - µ -partitionable for b = ˜ d α ( n ) Ω(1) and constant r, µ . It therefore helps to assume that b is aslowly growing function in n , such as log( n ) and r, µ are constants. Higher values of µ boostthe probability of a random graph being b - r - µ -partitionable. The parameter µ is thereforecrucial for the design of efficient algorithms.For an α -power-law-bounded random graph model ( G n ) n ∈ N , we always assume the ver-tices of G n to be v , . . . , v n , ordered as in Definition 2.1. We will choose A = { v , . . . , v b } , B = { v b +1 , . . . , v b µ } , C = { v b µ +1 , . . . , v n } and show that the probability is low that ( A, B, C )does not form a b - r - µ -partition. We do this in two steps: In Section 5.1, we define H n ( b, r, µ )to be a set of graphs over the vertex set { v , . . . , v n } . We show that if ( A, B, C ) is not a b - r - µ -partition then the complete edge-set of some graph in H n ( b, r, µ ) is present in thegraph. In Section 5.2 we bound the probability of the edge-set of any graph from H n ( b, r, µ )being present in the random graph model.At last, in Lemma 5.12, Section 5.3, we bound the sum of the expected sizes of all r -neighborhoods in an α -power-law-bounded graph class. This is needed to bound the expectedrun time of an algorithm that iterates over all r -neighborhoods of a graph.11 .1 Forbidden Edge-Sets Characterization Here we show that if a graph is not b - r - µ -partitionable then it contains some forbiddenedge-set. Definition 5.2.
Let G be a graph and H be a set of graphs over V ( G ). We say H ⊑ G iffor some H ∈ H , E ( H ) ⊆ E ( G ). Definition 5.3.
Let b, r, µ, n ∈ N + . We define H n ( b, r, µ ) to be the set of • all graphs with vertex set V ⊆ { v b +1 , . . . , v n } such that | V | ≤ rµ , all vertices havedegree at least two, and the graph has an edge-excess of µ , and • all graphs ( V ∪ V , E ) such that V ⊆ { v , . . . , v b } , V ⊆ { v b µ +1 , . . . , v n } , | V ∪ V | ≤ rµ , | E | ≤ rµ , | V | ≤ µ , all vertices in V have degree at least two, and thesummed degree of V is 2 | V | − µ . Lemma 5.4.
Let b, r, µ, n ∈ N + . If a graph G with vertex set { v , . . . , v n } is not b - r - µ -partitionable, then H n ( b, r, µ ) ⊑ G .Proof. Assume a graph G is not b - r - µ -partitionable. Then the tuple ( A, B, C ) with A = { v , . . . , v b } , B = { v b +1 , . . . , v b µ } , C = { v b µ +1 , . . . , v n } is not a b - r - µ -partition of G . Thismeans ( A, B, C ) either does not satisfy Property 3 or 4 of Definition 5.1.Assume now (
A, B, C ) does not satisfy Property 3. Then there is a 40 µr -neighborhoodin G [ B ∪ C ] with an edge-excess of at least µ . Let T be a breadth-first-search tree of thisneighborhood of depth 40 µr with a root v . There are µ +1 extra edges in this neighborhoodwhich are not contained in T . Let H be the graph constructed by the following procedure:We induce G on all vertices which are either an endpoint of the µ + 1 extra edges or lieon the unique path in T from such an endpoint to the root v . Then we iteratively removeall vertices with degree one. Every vertex in H has degree at least two. In T , each pathstarting at v has length at most 40 µr , and there are at most ( µ + 1) extra edges. Therefore, H consists of at most 2( µ + 1)(40 µr + 1) vertices. Furthermore, H contains µ more edgesthan vertices. This means G [ B ∪ C ] contains a subgraph with an edge-excess of µ and2( µ + 1)(40 µr + 1) ≤ rµ vertices that all have degree at least two. Such a graph iscontained in H n ( b, r, µ ).Assume now ( A, B, C ) does not satisfy Property 4. Then G [ C ] contains a 20 µr -neighbor-hood such that there are µ edges going from this 20 µr -neighborhood to A . Let these edgesbe u w , . . . , u µ w µ with u i ∈ A and w i ∈ C . Let T be a breadth-first-search tree of depth20 µr of this 20 µr -neighborhood with root v . Let V = { u , . . . , u µ } and let V be the setof vertices that lie for each w i on the unique path in T of length at most 20 µr from w i to the root v , including w i and v . Let H be the graph with vertex set V ∪ V and alledges from T [ V ], as well as all edges between V and V in G . Notice that | V | ≤ µ and | V | ≤ (20 µr + 1) µ . Also H [ V ] forms a tree with µ outgoing edges to V . Therefore, thevertices in V have in H a summed degree of 2 | V | − µ . They also have degree at leasttwo in H . This means G contains a subgraph ( V ∪ V , E ) such that V ⊆ A , V ⊆ C , | V + V | ≤ rµ , | V | ≤ µ , | E | ≤ rµ , and the vertices in V have degree at least twoand a summed degree of 2 | V | − µ . Such a graph is contained in H n ( b, r, µ ). In this section we bound for an α -power-law-bounded random graph model ( G n ) n ∈ N theprobability that H n ( b, r, µ ) ⊑ G n , thereby bounding the probability that G n is not b - r - µ -partitionable. 12 efinition 5.5. Let H be a set of graphs over the vertex set { v , v , . . . } , and let E be aset of edges over the same vertex set. We define p α ( E, n ) = ˜ d α ( n ) | E | Y v i v j ∈ E ( n/i ) / ( α − n / ( n/j ) / ( α − n / p α ( H , n ) = X H ∈ H p α ( E ( H ) , n ) . Lemma 5.6.
Let ( G n ) n ∈ N be an α -power-law-bounded random graph model. Let ( H n ) n ∈ N bea sequence of sets of graphs over the vertex set { v , v , . . . } . Then Pr[ H n ⊑ G n ] ≤ p α ( H n , n ) .Proof. Using the union bound and Definition 2.1 we seePr[ H n ⊑ G n ] ≤ X H ∈ H n Pr[ E ( H ) ⊆ E ( G n )] ≤ X H ∈ H n p α ( H, n ) = p α ( H n , n ) . It is therefore sufficient to bound p α ( H n ( b, r, µ )). The following two Lemmas prove sometechnicalities we need to do so. Lemma 5.7.
Let α ≥ . For n ∈ N + n X i =1 ( n/i ) / ( α − n / ≤ ˜ d α ( n ) √ n, n X i =1 ( n/i ) / ( α − n ≤ ˜ d α ( n ) . For b, n ∈ N + , δ ∈ N b X i =1 ( n/i ) δ/ ( α − n δ/ ≤ ˜ d α ( n ) δ b. For b, n, δ ∈ N + , δ ≥ n X i = b +1 ( n/i ) δ/ ( α − n δ/ ≤ ˜ d α ( n ) δ b − δ/ . Proof.
We define γ = 1 / ( α −
1) and ρ γ ( n ) = O (1) γ < / n ) O (1) γ = 1 / O ( n γ − / ) γ > / . For α > γ < / ρ γ ( n ) = ˜ d α ( n ) = O (1). Similarly, for α = 3 holds γ = 1 / ρ γ ( n ) = ˜ d α ( n ) = log( n ) O (1) . For 2 ≤ α < γ − / ≤ − α . Therefore ρ γ ( n ) ≤ ˜ d α ( n ) for all values of α ≥
2. It is now sufficient to show n X i =1 ( n/i ) γ n / = ρ γ √ n, n X i =1 ( n/i ) γ n = ρ γ ( n ) , b X i =1 ( n/i ) δγ n δ/ ≤ ρ γ ( n ) δ b, and for δ ≥ n X i = b +1 ( n/i ) δγ n δ/ ≤ ρ γ ( n ) δ b − δ/ .
13e bound with γ < n X i =1 ( n/i ) γ n / ≤ n γ − / Z n t γ dt = n γ − / n − γ / (1 − γ ) ≤ ρ γ ( n ) √ n and n X i =1 ( n/i ) γ n = n γ − + n γ − n X i γ ≤ O ( n γ − ) Z n t γ dt ≤ ρ γ ( n ) . (1)We further bound b X i =1 ( n/i ) δγ n δ/ ≤ n δ ( γ − / b ≤ ρ γ ( n ) δ b. To prove the last bound, we make a case distinction over δ and γ . At first, assume δ = 2.Then n X i = b +1 ( n/i ) δγ n δ/ ) ≤ ρ γ ( n ) δ = ρ γ ( n ) δ b − δ/ . Assume now that δ ≥
3. We have for γ ≤ / n X i = b +1 ( n/i ) δγ n δ/ ≤ n X i = b +1 ( n/i ) δ/ n δ/ ≤ Z nb t δ/ dt ≤ O (1) b − δ/ ≤ ρ γ ( n ) δ b − δ/ and for γ > / n X i = b +1 ( n/i ) δγ n δ/ ≤ n δ ( γ − / Z nb t δγ dt ≤ O ( n δ ( γ − / ) b − δγ ≤ ρ γ ( n ) δ b − δ/ . Lemma 5.8.
Let b , b , l, k, n ∈ N + with b ≤ b . Let L n ( b , b , l, k ) be the set of allgraphs ( V ∪ V , E ) such that V ⊆ { v , . . . , v b } , V ⊆ { v b +1 , . . . , v n } , | V ∪ V | ≤ l , | E | ≤ l , | V | ≤ k , all vertices in V have degree at least two, and the summed degree of V is | V | − k . Then p α ( L n ( b , b , l, k ) , n ) ≤ ˜ d α ( n ) O ( l ) b k b − k/ .Proof. We can partition the set L n ( b , b , l, k ) into at most 2 l many isomorphism classes.Let L ′ ⊆ L n ( b , b , l, k ) be the isomorphism class which maximizes p α ( L ′ , n ). We have that p α ( L n ( b , b , l, k ) , n ) ≤ l p α ( L ′ , n ). We fix a representative H = ( V ∪ V , E ) ∈ L ′ .Let now γ = | V | and Γ = | V | . We order the sets V and V such that we can speakof the first, second, etc. vertex in each set. Let F be the set of all sequences of integers( x , . . . , x γ , y , . . . , y Γ ) without duplicates and with 1 ≤ x i ≤ b and b + 1 ≤ y i ≤ n . Fora sequence f ∈ F let f ( H ) be the homomorphism of H where the i th vertex from V isassigned to v x i (for 1 ≤ i ≤ γ ) and the i th vertex from V is assigned to v y i (for 1 ≤ i ≤ Γ).Then L ′ = S f ∈ F f ( H ).The vertices in V and V have a degree sequence δ , . . . , δ γ , ∆ , . . . , ∆ Γ . We fix a se-quence f = ( x , . . . , x γ , y , . . . , y Γ ) ∈ F . Then by Definition 5.5 p α ( E ( f ( H )) , n ) = ˜ d α ( n ) ( δ + ··· + δ γ +∆ + ··· +∆ Γ ) / γ Y i =1 ( n/x i ) δ i / ( α − n δ i / Y i =1 ( n/y i ) ∆ i / ( α − n ∆ i / . (2)Observe that ∆ i ≥ ≤ i ≤ Γ , (3) δ + · · · + δ γ + ∆ + · · · + ∆ Γ ≤ l, (4)14 X i =1 (1 − ∆ i /
2) = Γ − Γ X i =1 ∆ i = Γ − (Γ − k/
2) = 1 − k/ . (5)We enumerate all sequences in F , and use (2), (3), (4), (5), and Lemma 5.7 to bound p α ( L n ( b , b , l, k ) , n ) ≤ l p α ( L ′ , n ) = 2 l X f ∈ F p α ( E ( f ( H )) , n ) (4)(2) ≤ l b X x =1 · · · b X x γ =1 n X y = b +1 · · · n X y Γ = b +1 ˜ d α ( n ) O ( l ) γ Y i =1 ( n/x i ) δ i / ( α − n δ i / Y i =1 ( n/y i ) ∆ i / ( α − n ∆ i / ≤ ˜ d α ( n ) O ( l ) b X x =1 ( n/x ) δ / ( α − n δ / · · · b X x γ =1 ( n/x γ ) δ γ / ( α − n δ γ / n X y = b +1 ( n/y ) ∆ / ( α − n ∆ / · · · n X y Γ = b +1 ( n/y Γ ) ∆ Γ / ( α − n ∆ Γ / ≤ ˜ d α ( n ) O ( l ) γ Y i =1 ˜ d α ( n ) δ i b Y i =1 ˜ d α ( n ) ∆ i b − ∆ i / ≤ ˜ d α ( n ) O ( l ) γ Y i =1 b Y i =1 b − ∆ i /
22 ( ) ≤ ˜ d α ( n ) O ( l ) b k b − k/ . Lemma 5.9.
Let b, r, µ, n ∈ N + with µ ≥ . Then p α ( H n ( b, r, µ ) , n ) ≤ ˜ d α ( n ) O ( µ r ) b − µ / . Proof.
We compare the definition of H n ( b, r, µ ) and L n ( b , b , l, k ) and see that H n ( b, r, µ ) ⊆ L n (1 , b, rµ + µ , µ + 2) ∪ L n ( b, b µ , rµ , µ ) . Using Lemma 5.8 and the union bound we compute p α ( H n ( b, r, µ ) , n ) ≤ p α ( L n (1 , b, rµ + µ , µ + 2) , n ) + p α ( L n ( b, b µ , rµ , µ ) , n ) ≤ ˜ d α ( n ) O ( rµ ) b − µ + ˜ d α ( n ) O ( rµ ) b µ ( b µ ) − µ/ ≤ ˜ d α ( n ) O ( r µ ) ( b − µ + b µ ( b µ ) − µ/ ) ≤ ˜ d α ( n ) O ( r µ ) ( b − µ + b − µ / µ ) ≤ ˜ d α ( n ) O ( r µ ) b − µ / µµ ≥ ≤ ˜ d α ( n ) O ( r µ ) b − µ / . Theorem 5.10.
Let ( G n ) n ∈ N be an α -power-law-bounded random graph model and let b, r, µ, n ∈ N + with µ ≥ . The probability that G n is not b - r - µ -partitionable is at most ˜ d α ( n ) O ( µ r ) b − µ / .Proof. Combining Lemma 5.4, 5.6, and 5.9. 15 .3 Expected Neighborhood Sizes
In this section we bound the sum of the expected sizes of all r -neighborhoods of a graphfrom an α -power-law-bounded random graph model under the condition that b ∈ N is theminimal value such that a graph is b - r - µ -partitionable. We start with the simpler conditionthat H n ⊑ G n for some set of graphs H n and then lift this result using Lemma 5.4 from theprevious section. Lemma 5.11.
Let ( G n ) n ∈ N be an α -power-law-bounded random graph model. Let r ∈ N + and let ( H n ) n ∈ N be a sequence of sets of graphs where every graph has size at most h ≥ .Then E (cid:2) X v ∈ V ( G n ) k G n [ N G n r ( v )] k (cid:12)(cid:12) H n ⊑ G n (cid:3) Pr[ H n ⊑ G n ] ≤ h O ( r ) ˜ d α ( n ) O ( r + h ) np α ( H n , n ) . Proof.
We fix an n . Let Q be the set of all paths of length at most r + 1 over V ( G n ). Thenby linearity of expectationE (cid:2) X v ∈ V ( G n ) k G n [ N G n r ( v )] k (cid:3) ≤ X Q ∈ Q Pr[ E ( Q ) ⊆ E ( G n )] . We use this observation and the union bound to computeE (cid:2) X v ∈ V ( G n ) k G n [ N G n r ( v )] k (cid:12)(cid:12) H n ⊑ G n (cid:3) Pr[ H n ⊑ G n ] ≤ X H ∈ H n E (cid:2) X v ∈ V ( G n ) k G n [ N G n r ( v )] k (cid:12)(cid:12) E ( H ) ⊆ E ( G n ) (cid:3) Pr[ E ( H ) ⊆ E ( G n )] ≤ X H ∈ H n X Q ∈ Q Pr[ E ( Q ) ⊆ E ( G n ) | E ( H ) ⊆ E ( G n )] Pr[ E ( H ) ⊆ E ( G n )] ≤ X H ∈ H n X Q ∈ Q Pr[ E ( Q ) ⊆ E ( G n ) , E ( H ) ⊆ E ( G n )] ≤ X H ∈ H n X Q ∈ Q p α ( E ( H ) ∪ ( E ( Q ) \ E ( H )) , n ) ≤ ˜ d α ( n ) O ( r + h ) X H ∈ H n p α ( E ( H ) , n ) X Q ∈ Q p α ( E ( Q ) \ E ( H ) , n ) . (6)We fix a graph H ∈ H n . We want to find a good bound for p α ( E ( Q ) \ E ( H ) , n ) forevery Q ∈ Q . Let Q ∈ Q be a path. We assume the vertices V ( Q ) = { w , . . . , w q } with q ≤ r + 2 to be ordered such that edges are only between consecutive vertices. Let s = ( s , . . . , s q − ) ∈ { , } q − be the unique bit-string with { ( w i , w i +1 ) | s i = 1 , ≤ i < q } = E ( Q ) \ E ( H ) . This means s describes which edges of Q are not present in H . Let Q ′ = ( V ( Q ) , E ( Q ) \ E ( H )). The degree sequence of Q ′ is δ s , . . . , δ sq with δ si = s i − + s i (we assume s = s q = 0).We define W ( δ ) = ( V ( G n ) δ = 2 V ( H ) δ < X ( δ ) = ( { , . . . , n } δ = 2 { i | v i ∈ V ( H ) } δ < . If δ s = 0 then w ∈ V ( H ) = W ( δ s +1). If δ sq = 0 then w q ∈ V ( H ) = W ( δ sq +1). If δ si ∈ { , } w i ∈ V ( H ) = W ( δ si ) for 2 ≤ i ≤ q −
1. Therefore X Q ∈ Q p α ( E ( Q ) \ E ( H )) ≤ r +2 X q =1 X s ∈{ , } q − X w ∈ W ( δ s +1) X w ∈ W ( δ s ) · · · X w q − ∈ W ( δ sq − ) X w q ∈ W ( δ sq +1) p α ( { ( w i , w i +1 ) | s i = 1 , ≤ i < q } , n )= r +2 X q =1 X s ∈{ , } q − X x ∈ X ( δ s +1) X x ∈ X ( δ s ) · · · X x q − ∈ X ( δ sq − ) X x q ∈ X ( δ sq +1) ˜ d α ( n ) O ( q ) q Y i =1 ( n/x i ) δ si / ( α − n δ si / = ˜ d α ( n ) O ( r ) r +2 X q =1 X s ∈{ , } q − X x ∈ X ( δ s +1) ( n/x ) δ s / ( α − n δ s / X x ∈ X ( δ s ) ( n/x ) δ s / ( α − n δ s / . . . X x q − ∈ X ( δ sq − ) ( n/x q − ) δ sq − / ( α − n δ sq − / X x q ∈ X ( δ sq +1) ( n/x q ) δ sq / ( α − n δ sq / . (7)The bound of (7) depends on the degree sequence δ s , . . . , δ sq . Remember that δ s , δ sq ∈ { , } and δ si ∈ { , , } for 1 < i < q . The following five bounds follow from Lemma 5.7. X x ∈ X (0) ( n/x ) / ( α − n / ≤ h ≤ ˜ d α ( n ) h X x ∈ X (1) ( n/x ) / ( α − n / ≤ ˜ d α ( n ) h ≤ ˜ d α ( n ) h X x ∈ X (2) ( n/x ) / ( α − n / ≤ ˜ d α ( n ) ≤ ˜ d α ( n ) h X x ∈ X (1) ( n/x ) / ( α − n / ≤ h ≤ ˜ d α ( n ) h X x ∈ X (2) ( n/x ) / ( α − n / ≤ ˜ d α ( n ) √ n ≤ ˜ d α ( n ) h √ n These five bounds can be used to bound the inner q sums of (7). This yields X Q ∈ Q p α ( E ( Q ) \ E ( H )) ≤ ˜ d α ( n ) O ( r ) r +2 X q =1 X s ∈{ , } q − ˜ d α ( n ) q h q √ n √ n ≤ h O ( r ) ˜ d α ( n ) O ( r ) n. (8)At last, we combine (6) and (8) and getE (cid:2) X v ∈ V ( G n ) k G n [ N G n r ( v )] k (cid:12)(cid:12) H n ⊑ G n (cid:3) Pr[ H ⊑ G n ] ≤ ˜ d α ( n ) O ( r + h ) X H ∈ H n p α ( E ( H ) , n ) X Q ∈ Q p α ( E ( Q ) \ E ( H ) , n ) ≤ p α ( H n , n ) h O ( r ) ˜ d α ( n ) O ( r + h ) n. emma 5.12. Let ( G n ) n ∈ N be an α -power-law-bounded random graph model. Let r, µ, n ∈ N + with µ ≥ . Let A b be the event that b ∈ N + is the minimal value such that G n is b - r - µ -partitionable. Then E (cid:2) X v ∈ V ( G n ) k G n [ N G n r ( v )] k (cid:12)(cid:12) A b (cid:3) Pr[ A b ] ≤ ( rµ ) O ( r ) ˜ d α ( n ) O ( µ r ) b − µ / n. Proof.
We start with a general observation about conditional expected values. Let X be anon-negative random variable and A ⊆ B be events. ThenE[ X | A ] Pr[ A ] ≤ E[ X | B ] Pr[ B ] . (9)Assume b ≥
3. Let G be a graph with V ( G ) = { v , . . . , v n } . If b ∈ N + is the minimalvalue such that G is b - r - µ -partitionable then G is not ( b − r - µ -partitionable. Then byLemma 5.4, H n ( b − , r, µ ) ⊑ G . Using (9), we seeE (cid:2) X v ∈ V ( G n ) k G n [ N G n r ( v )] k (cid:12)(cid:12) A b (cid:3) Pr[ A b ] ≤ E (cid:2) X v ∈ V ( G n ) k G n [ N G n r ( v )] k (cid:12)(cid:12) H n ( b − , r, µ ) ⊑ G n (cid:3) Pr[ H n ( b − , r, µ ) ⊑ G n ] . Every subgraph in H n ( b − , r, µ ) has by Definition 5.3 size at most 200 rµ . Also for b ≥ b − − ≤ b − / . Lemma 5.11 and 5.9 implyE (cid:2) X v ∈ V ( G n ) k G n [ N G n r ( v )] k (cid:12)(cid:12) A b (cid:3) Pr[ A b ] ≤ E (cid:2) X v ∈ V ( G n ) k G n [ N G n r ( v )] k (cid:12)(cid:12) H n ( b − , r, µ ) ⊑ G n (cid:3) Pr[ H n ( b − , r, µ ) ⊑ G n ]5 . ≤ (200 rµ ) O ( r ) ˜ d α ( n ) O ( µ r ) np α ( H n ( b − , r, µ ) , n )5 . ≤ ( rµ ) O ( r ) ˜ d α ( n ) O ( µ r ) n ˜ d α ( n ) O ( µ r ) ( b − − µ / ≤ ( rµ ) O ( r ) ˜ d α ( n ) O ( µ r ) b − µ / n. Assume b ≤
2. By (9) and Lemma 5.11 with H n = {∅} E (cid:2) X v ∈ V ( G n ) k G n [ N G n r ( v )] k (cid:12)(cid:12) A b (cid:3) Pr[ A b ] ≤ E (cid:2) X v ∈ V ( G n ) k G n [ N G n r ( v )] k (cid:3) ≤ ˜ d α ( n ) O ( r ) n ≤ ( rµ ) O ( r ) ˜ d α ( n ) O ( µ r ) b − µ / n. In this section, we show that local neighborhoods of power-law-bounded graph classes arelikely to have the following nice structure: They consist of a (small) core graph to whichso called protrusions are attached. Protrusions are (possibly large) subgraphs with smalltreewidth and boundary. The boundary of a subgraph is the size of its neighborhood inthe remaining graph. Protrusions were introduced by Bodlaender et al. for very generalkernelization results in graph classes with bounded genus [6].Earlier, (Theorem 5.10) we showed that α -power-law-bounded random graph models are(for certain values of α , b , r , µ ) likely to be b - r - µ -partitionable. It is therefore sufficient18o show that r -neighborhoods of b - r - µ -partitionable graphs have such a nice protrusionstructure.However, in general it is not easy to find protrusions in a graph [48]. As we later needto be able to find them, we define special protrusion decompositions, called b - r - µ -local-protrusion-partitions in which (most of) the protrusions can be efficiently identified. Themain and only result of this section is the following theorem. Theorem 6.15.
Let b, r, µ ∈ N + and let G be an b - r - µ -partitionable graph. Let G r be an r -neighborhood in G . Then G r is O ( µ r b ) - r - O ( µ ) -locally-protrusion-partitionable. It remains to define what a b - r - µ -local-protrusion-partition of a graph G r with radius atmost r is. The definition has to strike the right balance: It needs to be permissive enoughsuch that neighborhoods of power-law-bounded graph classes are likely to have this structureand it needs to be restrictive enough to admit efficient algorithms. Informally speaking, a b - r - µ -local-protrusion-partition of a graph G r is a partition ( X, Y, Z ) of the vertices of G r suchthat X has small size and the connected components of G r [ Y ∪ Z ] are protrusions. In orderto be able to efficiently identify the protrusions, we further require that the components of G r [ Y ] have bounded size and the components of G r [ Z ] are trees. This is formalized in thefollowing definition. Definition 6.1 ( b - r - µ -local-protrusion-partition) . Let b, r, µ ∈ N + . Let G r be a graph withradius at most r . A tuple ( X, Y, Z ) is called an b - r - µ -local-protrusion-partition of G r if1. the sets X, Y, Z are pairwise disjoint and their union is V ( G r ).2. | X | ≤ b µ ,3. every connected component of G r [ Y ] has size at most rµ and at most µ neighborsin X ,4. every connected component of G r [ Z ] is a tree with at most one edge to G r [ X ∪ Y ].5. For a subgraph H of G r [ Y ∪ Z ] we say N G r ( V ( H )) ∩ X is the boundary of H .The connected components of G r [ Y ] may have at most b µ distinct boundaries, i.e., |{ N G r ( V ( H )) ∩ X | H connected component of G r [ Y ∪ Z ] }| ≤ b µ ,A graph for which an b - r - µ -local-protrusion-partition exists is called b - r - µ -locally-protrusion-partitionable.Property 3 and 4 enforce that the components of G r [ Y ∪ Z ] are protrusions. Later, wewill transform b - r - µ -local-protrusion-partitions into equivalent graphs of bounded size byreplacing the protrusions with small graphs. Thus, Property 2 and 5 are there to ensurethe resulting kernelized graph will have size roughly b µ (without Property 5 we could onlyguarantee a size of roughly b µ ). To simplify our proofs, we fix some notation which will be valid for this whole section.Let a graph G and b, r, µ ∈ N + be fixed. We further assume G to be b - r - µ -partitionable andwe fix a b - r - µ -partition ( A, B, C ) of G . Let further G r be an r -neighborhood in G and let A r = A ∩ V ( G r ) , B r = B ∩ V ( G r ) , C r = C ∩ V ( G r ) . The O ( µ r b )- r - O ( µ )-local-protrusion-partition ( X, Y, Z ) of G r will be created by build-ing X from A r ∪ B r and some vertices from C r . The remaining vertices from C r will besplit into the sets Y and Z . The remainder of this section will describe how this procedurehappens in detail. b - r - µ -Partitionable Graphs We will start with the straight-forward result that Properties 3 and 4 of a b - r - µ -partition(Definition 5.1) can be transferred to neighborhoods.19 emma 6.2. Every µr -neighborhood in G [ B r ∪ C r ] has an edge-excess of at most µ , andevery µr -neighborhood in G [ C r ] has at most µ edges to A r .Proof. Since C r ⊆ C , an r -neighborhood in G [ C r ] is a connected subgraph of an r -neighbor-hood in G [ C ]. Since G is b - r - µ -partitionable, a 20 µr -neighborhood in G [ C ] has at most µ edges to A and A r ⊆ A . Therefore, a 20 µr -neighborhood of G [ C r ] has at most µ edgesto A r .Similarly, a 40 µr -neighborhood in G [ B r ∪ C r ] is a connected subgraph of a 40 µr -neighbo-rhood in G [ B ∪ C ]. If a connected graph has an edge-excess of at most µ , then so doesevery connected subgraph. Since G is b - r - µ -partitionable, an 40 µr -neighborhood in G [ B ∪ C ]has an edge-excess of at most µ , which bounds the excess of every 40 µr -neighborhood in G [ B r ∪ C r ]. The vertices from A r and B r will all be put into the set X of a O ( µ r b )- r - O ( µ )-local-protrusion-partition. The situation for the C r vertices is more complicated. In this subsec-tion we define so called ties , which we use in the next subsection to distribute the vertices C r to the sets X , Y , and Z of a b - r - µ -local-protrusion-partition. Definition 6.3 (Tie) . Let W ⊆ B r ∪ C r . We say ( u , u , v ) is a W -tie if u , u ∈ W and v lies on a walk p with the following properties: Every inner vertex of p is contained in C r and has at least two neighbors in p ; u and u are contained only as endpoints of p ; and p is contained in a 20 µr -neighborhood in G [ B r ∪ C r ]. We further say V ( p ) is a walk set of( u , u , v ).Ties are triples of vertices that are connected by a walk with certain properties. In thefollowing two lemmas, we bound the size of their walk sets, as well as the number of ties.We need this later to prove the size constraints of a b - r - µ -local-protrusion-partition. Lemma 6.4.
A walk set of a tie has at most size rµ .Proof. Let the walk set of a tie be the vertices on a walk p . By definition, p is contained ina 20 µr -neighborhood in G [ B r ∪ C r ]. Let T be a breadth-first-search spanning tree of sucha neighborhood. According to Lemma 6.2, every 40 µr -neighborhood in G [ B r ∪ C r ] has anedge-excess of at most µ . A tree has an edge-excess of −
1. Therefore, there are at most µ + 1 edges in p which are not contained in T . Also, every path in T contains at most2 · µr + 1 vertices. Thus, p contains at most (2 · µr + 1)( µ + 2) ≤ rµ vertices.Next we take the first step of counting the vertices of C r , by showing that for W ⊆ B r ∪ C r , the number of W -ties in G r is quadratic in | W | . Note that this does not directlylead to a bound for | C r | since it might be that | W | > | C r | . Lemma 6.5.
Let W ⊆ B r ∪ C r . There are at most rµ | W | W -ties in G r .Proof. We fix u , u ∈ W . Let X u ,u = { ( u , u , v ) | v ∈ B r ∪ C r , ( u , u , v ) is a W -tie } be the set of all W -ties for fixed endpoints u , u . There are exactly | W | ways to choose u , u , thus, it is sufficient to show that | X u ,u | ≤ rµ ( µ + 2) ≤ rµ .Assume for contradiction | X u ,u | > rµ ( µ + 2). For every x ∈ X u ,u let V ( x )be a walk set of x . The size of a walk set of a tie is at most 130 rµ (Lemma 6.4). Let l = µ + 3. By a pigeonhole argument, one can choose l -many W -ties x , . . . , x l ∈ X u ,u such that V ( x i ) \ ( V ( x ) ∪ · · · ∪ V ( x i − )) = ∅ for 1 ≤ i ≤ l . We define for 1 ≤ i ≤ l a graph G i = G [ V ( x ) ∪ · · · ∪ V ( x i )]. We show by induction that G l has an edge-excess of at least l − G [ V ( x i )] are connected and all vertices except for u , u have degree at least two in G [ V ( x i )]. That means G = G [ V ( x )] has an edge-excess of atleast −
1. It also means that every vertex in the non-empty set V ( G i ) \ V ( G i − ) has degree20t least two in G i . Since G i is connected, there is at least one edge between V ( G i ) \ V ( G i − )and V ( G i − ) in G i . If every vertex in V ( G i ) \ V ( G i − ) has degree exactly two in G i thereare at least two edges between V ( G i ) \ V ( G i − ) and V ( G i − ) in G i . Thus, in the step from G i − to G i the number of added edges is at least one greater than the number of addedvertices. The edge-excess increases by one.The walk set of each W -tie in X u ,u contains u and is contained in a 20 µr -neighborhoodin G [ B r ∪ C r ]. This means G l is contained in the (2 · µr )-neighborhood in u in G [ B r ∪ C r ].The graph G l has an edge-excess of at least l − µ + 1 and according to Lemma 6.2, thisis a contradiction. C r into C rA , C rB , C rY , and C rZ We use the notion of ties (Definition 6.3) to partition the set C r . We distinguish verticesconnected to A r , vertices connected to B r (but not to A r ), those which are connected toneither but lie on a tie, and the rest. We set • C rA = N ( A r ) ∩ C r , • C rB = ( N ( B r ) \ N ( A r )) ∩ C r , • C rY = { v | v ∈ C r \ ( C rA ∪ C rB ) and there exist u , u ∈ C rA ∪ C rB such that ( u , u , v )is a ( C rA ∪ C rB )-tie } , • C rZ = C r \ ( C rA ∪ C rB ∪ C rY ).We will show that the four previously defined sets have desirable structural properties. ByDefinition 5.1, we know that | C rA | ≤ a and | C rB | ≤ a µ . We use the previously defined ties toshow that G [ C rZ ] is a forest and to bound the size of components of G [ C rA ∪ C rB ∪ C rY ]. Wethen use these properties to construct a b - r - µ -local-protrusion-partition. We start with twoauxiliary lemmas. Lemma 6.6. In G [ C r ] , every vertex has distance at most r to a vertex in C rA ∪ C rB .Proof. We fix a vertex v ∈ C r . The graph G r has radius at most r , thus, v has in G r distance at most 2 r to C rA ∪ C rB . Vertices in C r \ ( C rA ∪ C rB ) are in G r only adjacent to othervertices from C r . This means, the distance from v to the nearest vertex in C rA ∪ C rB is in G [ C r ] the same as in G r . Lemma 6.7.
A connected component of G [ C r ] with at most l ∈ N vertices from C rA ∪ C rB is contained in a rl -neighborhood in G [ C r ] .Proof. Let G ∗ be a connected component of G [ C r ]. By Lemma 6.6, every vertex from( C rY ∪ C rZ ) ∩ V ( G ∗ ) has distance at most 2 r from ( C rA ∪ C rB ) ∩ V ( G ∗ ) in G [ C r ]. Therefore, G ∗ is contained in a (4 r + 1) l -neighborhood in G [ C r ]. We have (4 r + 1) l ≤ rl . G [ C rZ ] are Trees In this section we show the somewhat surprising property that if you take away all verticesthat are connected to A ∪ B and those that lie on a tie, you are left with a forest. Lemma 6.8.
Each connected component of G [ C rZ ] is a tree and has at most one outgoingedge in G r .Proof. We consider a connected component H of G [ C rZ ]. Assume for contradiction thateither H is not a tree, or has more than one outgoing edge in G r . Then there has to exista walk p in G r whose inner vertices are in V ( H ), whose endpoints are in V ( C r ) \ C rZ , andevery inner vertex of p has at least two different neighbors in p . We pick an arbitrary innervertex v ∈ V ( H ) from p .In this proof, we will successively construct walks p ′ , p ′′ and p ∗ w ′ , w ′ ), ( w ′′ , w ′′ ) and ( w ∗ , w ∗ ) which contain v . The final walk p ∗ will besuch that ( w ∗ , w ∗ , v ) is a ( C rA ∪ C rB )-tie. This means by definition that v ∈ C rY and therefore v C rZ (a contradiction). Constructing p ′ : Let w , w be the endpoints of p . Since C rA ∪ C rB separates C rZ from A r ∪ B r in G r , we know that w , w ∈ C rY ∪ C rA ∪ C rB . If all w i ∈ C rA ∪ C rB , we set p ′ = p .If any w i ∈ C rY then, by definition, w i lies on a ( C rA ∪ C rB )-tie walk p i . By definition thewalk p i contains no vertex from V ( H ), since this would imply that said vertex is in C rY . Wemodify p into p ′ as follows: At every endpoint w i ∈ C rY we extend p by traversing p i in anarbitrary direction until we reach an endpoint w ′ i ∈ C rA ∪ C rB and then iteratively removingvertices with degree one that might have been introduced.Now p ′ is a walk from w ′ to w ′ ,that goes over v and where every inner vertex of p ′ has at least two different neighbors in p ′ . Furthermore, w ′ , w ′ ∈ C rA ∪ C rB . However, p ′ is still not necessarily a tie-walk, since itis not guaranteed to be contained in a 20 µr -neighborhood in G [ B r ∪ C r ]. Constructing p ′′ : We construct a sub-walk p ′′ of p ′ by starting at v and traversing p ′ in both directions until we either reach an endpoint in C rA ∪ C rB or a vertex with distanceexactly 2 r + 1 in G [ C r ] to v . The walk p ′′ contains v and every vertex on p ′′ has distanceat most 2 r + 1 in G [ C r ] from v . The endpoints w ′′ , w ′′ of p ′′ are either in C rA ∪ C rB or havedistance exactly 2 r + 1 in G [ C r ] from v . Every inner vertex of p ′′ has at least two differentneighbors in p ′′ . Constructing p ∗ : At last, we extend p ′′ into p ∗ as follows: If w ′′ i ∈ C rA ∪ C rB , we set w ∗ i = w ′′ i . Otherwise, by Lemma 6.6, there exists a vertex w ∗ i ∈ C rA ∪ C rB with distance atmost 2 r in G [ C r ] from w ′′ i . Let q i be the shortest path from w ∗ i to w ′′ i in G [ C r ]. The vertex v has in G [ C r ] distance exactly 2 r + 1 from w ′′ i , thus, v is not contained in q i . We traverse p ′′ from v in both directions. While traversing in direction of w ′′ i , as soon as we reach avertex from q i we continue traversing q i until we reach w ∗ i . The walk p ∗ contains v , andevery inner vertex has at least two neighbors on p ′ . Also, every vertex has distance at most4 r + 1 in G [ C r ] from v . The endpoints w ∗ , w ∗ are contained in C rA ∪ C rB . This means that( w ∗ , w ∗ , v ) is a ( C rA ∪ B r )-tie. G [ C r ] In this subsection we will speak only about connected components of G [ C r ]. While theirnumber is unbounded we show that inside a component the number of vertices that are notfrom C rZ will be bounded. For every component, we first show that if it has few verticesfrom C rB , it has few vertices from C rA (Lemma 6.9) and that a component with few edges to B r has few vertices from C rA ∪ C rB ∪ C rY (Lemma 6.10). Lemma 6.9.
A connected component of G [ C r ] with l ∈ N vertices from C rB contains atmost ( l + 1) µ vertices from C rA .Proof. Let G ∗ be a connected component of G [ C r ] and let C ∗ A = C rA ∩ V ( G ∗ ), C ∗ B = C rB ∩ V ( G ∗ ), C ∗ Y = C rY ∩ V ( G ∗ ), C ∗ Z = C rZ ∩ V ( G ∗ ). We show that | C ∗ A | ≤ ( | C ∗ B | + 1) µ in twosteps: At first we show that if | C ∗ A | > ( | C ∗ B | + 1) µ then there exists a connected subgraph H of G ∗ which contains at least µ + 1 vertices from C ∗ A and at most one vertex from C ∗ B .Second, we show that such a subgraph H cannot exist.Assume that | C ∗ A | > ( | C ∗ B | + 1) µ . If C ∗ B = ∅ we set H = G ∗ . Then H contains at least µ + 1 vertices from C ∗ A and no vertex from C ∗ B . If C ∗ B = ∅ we proceed as follows. For every v ∈ C ∗ B we define A ( v ) to be the set of all vertices from C ∗ A that are reachable from v in G [ C ∗ A ∪ C ∗ Y ∪ C ∗ Z ∪ { v } ]. Since G ∗ is connected and C ∗ B = ∅ , for all w ∈ C ∗ A exists v ∈ C ∗ B with w ∈ A ( v ). This means | C ∗ A | ≤ P v ∈ C ∗ B | A ( v ) | . Since | C ∗ A | > ( | C ∗ B | + 1) µ , there exists v ∈ C ∗ B with | A ( v ) | > µ . Let H be the connected component of v in G [ C ∗ A ∪ C ∗ Y ∪ C ∗ Z ∪ { v } ].The graph H is connected and contains exactly one vertex from C ∗ B . Since | A ( v ) | > µ , italso contains at least µ + 1 vertices from C ∗ A .We now show that such a graph H cannot exist. Let H ′ be a connected subgraph of H which contains exactly µ + 1 vertices from C ∗ A and at most one vertex from C ∗ B (we can22onstruct H ′ by taking a spanning tree of H and iteratively removing leaves until we haveexactly µ + 1 vertices from C ∗ A ). The graph H ′ contains µ + 1 vertices from C rA and at mostone vertex from C rB . According to Lemma 6.7, H ′ is contained in a 5 r ( µ + 2)-neighborhoodin G [ C r ]. Furthermore H ′ has by construction at least µ + 1 edges to A r . Since G is an b - r - µ -partition every 20 µr -neighborhood in G [ C r ] has, by Lemma 6.2, at most µ edges to A r . This is a contradiction, so H cannot exist. Lemma 6.10.
A connected component of G [ C r ] with l ∈ N edges to B r contains at most µ r ( l + 1) vertices from C rA ∪ C rB ∪ C rY .Proof. Let G ∗ be a connected component of G [ C r ] and let C ∗ A = C rA ∩ V ( G ∗ ), C ∗ B = C rB ∩ V ( G ∗ ), C ∗ Y = C rY ∩ V ( G ∗ ), C ∗ Z = C rZ ∩ V ( G ∗ ). Since G ∗ has l edges to B r we have | C ∗ B | ≤ l . According to Lemma 6.9, | C ∗ A | ≤ ( l + 1) µ . Let v ∈ C ∗ Y . By definition, there existsa ( C rA ∪ C rB )-tie x = ( u , u , v ). Since G ∗ is a connected component, x is also a ( C ∗ A ∪ C ∗ B )-tie. By Lemma 6.5, G r contains at most 390 rµ ( C ∗ A ∪ C ∗ B ) ≤ rµ ( l + ( l + 1) µ ) many( C ∗ A ∪ C ∗ B )-ties, which also bounds the number of vertices in C ∗ Y . We add up the bounds forthe number of vertices from C rB , C rA , and C rY and get l + ( l + 1) µ + 390 rµ ( l + ( l + 1) µ ) ≤ µ r ( l + 1) .For components that only have one edge to B r we can directly say how many edges to A r it has. Lemma 6.11.
A connected component of G [ C r ] with at most one edge to B r has at most µ edges to A r .Proof. Let G ∗ be a connected component of G [ C r ] with at most one edge to B r and thereforeat most one vertex from C rB . According to Lemma 6.9, it contains at most 2 µ vertices from C rA . By Lemma 6.7, G ∗ is contained in a 5 r (2 µ + 1)-neighborhood in G [ C r ]. By Lemma 6.2,every 20 µr -neighborhood can only have at most µ edges to A r . G [ C r ] With More Than One Edgeto B r In this subsection we want to look at components that have more than one edge to B r . Westart with a helping lemma, that states that for every vertex there is a close vertex from C rB (or none at all). Lemma 6.12.
Let v ∈ C r . If a vertex u ∈ C rB with u = v is reachable from v in G [ C r ] thenthere also is a vertex w ∈ C rB with w = v that has in G [ C r ] distance at most µr from v .Proof. We can assume that the shortest path from u ∈ C rB to v in G [ C r ] has length at least17 µr (otherwise let w = u ). We pick vertices x , . . . , x µ +2 along this path, such that x i hasdistance 5 ri from v in G [ C r ]. Therefore, x i has distance at least 5 r from x j in G [ C r ] for i = j . For every x i there exists a vertex s i ∈ C rA ∪ C rB with distance at most 2 r in G [ C r ] from x i (Lemma 6.6). Since the vertices x i are spaced sufficiently far apart, we have s i = s j , and v = s i for i = j . Each vertex s i has in G [ C r ] distance at most 5 ri +2 r ≤ r ( µ +2)+2 r ≤ µr from v . If s i ∈ C rB for some i we set w = s i and there is a path in G [ C r ] from v to w oflength at most 17 µr . Assume now s i ∈ C rA for all i ≤ µ + 2. The vertices s i are contained inthe 17 µr -neighborhood of v in G [ C r ] and each vertex s i has one edge to A r . In total, thereare at least µ + 2 edges to A r . According to Lemma 6.2, every 20 µr -neighborhood in G [ C r ]has at most µ edges to A . This is a contradiction.As stated earlier ties are our tool of choice that we use to count vertices. We willestablish this in the following lemma that shows that for every edge a component has to B r one introduces more ties. This will in turn bound the number of vertices in componentswith more than one edge to B r . 23 emma 6.13. Let G ∗ be a connected component of G [ C r ] with l ≥ edges to B r . Thereare at least l many B r -ties of the form ( u , u , v ) with v ∈ V ( G ∗ ) .Proof. Let u v be an edge between B r and G ∗ with u ∈ B r and v ∈ V ( G ∗ ). Since l ≥ u ′ w between B r and G ∗ with u ′ ∈ B r and w ∈ V ( G ∗ ). If w = v , it follows u ′ = u and ( u , u ′ , v ) is a B r -tie. Otherwise, w = v and since G ∗ is aconnected component, w is reachable from v in G ∗ . According to Lemma 6.12, there also isa vertex w ′ ∈ C rB ∩ V ( G ∗ ) with w ′ = v which has in G ∗ distance at most 17 µr from v . Since w ′ ∈ C rB , w ′ also has a neighbor u ∈ B r . There is a path from u to u which contains v , whose inner vertices are contained in G ∗ , and which has length at most 17 µr + 2. Thismeans ( u , u , v ) is a B r -tie.For each of the l edges between G ∗ and B r we can use the technique above to constructa B r -tie. The first and third entry of the tuple correspond to an edge between G ∗ and B r and thus no two edges create the same tie.With the next lemma we show that a connected component in G [ C r ] with many edgesto B r has many paths with certain properties and then show that only b O ( µ ) many verticesfrom C rA ∪ C rB ∪ C rY are in a connected component of G [ C r ] with more than one edge to B r . Lemma 6.14.
The number of vertices in C rA ∪ C rB ∪ C rY which are in a connected componentof G [ C r ] with more than one edge to B r is at most O ( µ r b µ ) .Proof. Let H , . . . , H m be the connected components of G [ C r ] with more than one edge to B r . Let k i be the number of vertices in H i which are from C rA ∪ C rB ∪ C rY . Let l i be thenumber of edges to B r in H i . Let k = P mi =1 k i and l = P mi =1 l i . At first, we show that k ≤ µ rl . Then, we show that l ≤ rµ b µ . Together, this yields k = O ( µ r b µ ).According to Lemma 6.10, each connected component H i contains at most 1600 µ r ( l i +1) vertices from C rA ∪ C rB ∪ C rY . We bound k ≤ P mi =1 µ r ( l i +1) ≤ µ r ( P mi =1 l i ) =6400 µ rl . By Lemma 6.13, for 1 ≤ i ≤ m there are at least l i many B r -ties ( u , u , v ) with v ∈ V ( H i ), so in total, there are at least l many B r -ties. With Lemma 6.5 and | B r | ≤ a µ ,we bound l ≤ rµ | B r | ≤ rµ b µ . Having analyzed the structure of G [ C r ] we can finally show that for every b - r - µ -partitionablegraph G , every r -neighborhood G r is O ( µ r b )- r - O ( µ )-locally-protrusion-partitionable. Theorem 6.15.
Let b, r, µ ∈ N + and let G be an b - r - µ -partitionable graph. Let G r be an r -neighborhood in G . Then G r is O ( µ r b ) - r - O ( µ ) -locally-protrusion-partitionable.Proof. Let A r , B r , C r , C rA , C rB , C rY , C rZ be as defined earlier. We need to define sets( X, Y, Z ) and show all the properties of Definition 6.1. We define X to be the union of A r , B r and all vertices from C rA ∪ C rB ∪ C rY which are in a connected component of G [ C r ] withmore than one edge to B r . Since ( A, B, C ) is an b - r - µ -partition, we know that | A r | ≤ b and | B r | ≤ b µ . Lemma 6.14 bounds the number of vertices from C rA ∪ C rB ∪ C rY which are ina connected component of G [ C r ] with more than one edge to B r by at most O ( µ r b µ ).This implies | X | = O ( µ r b ) O ( µ ) (Property 2)We define Y to be the vertices from C rA ∪ C rB ∪ C rY which are in a connected componentof G [ C r ] with at most one edge to B r . Each connected component of G [ Y ] is containedin a connected component of G [ C r ] with at most one edge to B r . Thus, by Lemma 6.10,connected components of G [ Y ] have size at most 1600 µ r (1+1) = O ( µ r ). Every connectedcomponent of G [ Y ] has at most one edge to B r and by Lemma 6.11 at most µ edges to A r .By construction, every edge from it to X goes either to A r or B r . This means it has atmost µ + 1 neighbors in X (Property 3). Since | A r | ≤ b and | B r | ≤ b µ there are at most b O ( µ ) (choose at most µ of b and at most one of b µ vertices) possible sets of boundaries in X . This satisfies Property 5. 24e define Z = C rZ . According to Lemma 6.8, every connected component of G [ Z ] is atree and has at most one edge to X ∪ Y (Property 4). Finally, the sets X, Y, Z are pairwisedisjoint and their union is V ( G r ) (Property 1). Earlier, (Theorem 5.10, Theorem 6.15) we showed that neighborhoods of α -power-law-bounded random graph models are (for certain values of α , b , r , µ ) likely to be b - r - µ -locally-protrusion-partitionable (Definition 6.1). This means these neighborhoods have thefollowing nice structure: They consist of a (small) core graph to which protrusions are at-tached. Remember that protrusions are (possibly large) subgraphs with small treewidth andboundary and that the boundary of a subgraph is its neighborhood in the remaining graph.In this section, we replace these protrusions by subgraphs with bounded size that retainthe same boundary. This yields a small graph which is q -equivalent to the original graph.The same technique has been used for obtaining small kernels in larger graph classes, e.g.,in graphs that exclude a fixed minor [35]. The main result of this section is the followingtheorem. Theorem 7.9.
There exists an algorithm that takes q, r, µ ∈ N + and a connected labeledgraph G with radius at most r and at most q labels as input, runs in time at most f ( q, r, µ ) k G k for some function f ( q, r, µ ) , and computes a labeled graph G ∗ ≡ q G . If G is b - r - µ -locally-protrusion-partitionable for some b ∈ N + then | G ∗ | ≤ f ( q, r, µ ) b µ . This kernelization procedure and its run time bound is independent in b but the sizeof the output kernel is not: If b is small, then the output is small. The result is obtainedby replacing protrusions with the help of the Feferman–Vaught theorem [47]. However, inorder to replace the protrusions, one first has to identify them. The main complicationin this section lies in partitioning a graph such that the relevant protrusions can be easilyidentified. It is crucial that we obtain the size bound | G ∗ | ≤ f ( q, r, µ ) b µ in Theorem 7.9.Weaker bounds are easier to obtain but would not be sufficient for our purposes. In this subsection we obtain a suitable protrusion replacement procedure (Lemma 7.4).We use a variant of the Feferman–Vaught theorem [47] to replace a protrusion by a q -equivalent boundaried graph of minimal size. This size depends only on q and the size of theboundary. The original Feferman–Vaught theorem states that the validity of FO-formulason the disjoint union or Cartesian product of two graphs is uniquely determined by the valueof FO-formulas on the individual graphs. Makowsky adjusted the theorem for algorithmicuse [54] in the context of MSO model-checking. The following proposition contains theFeferman–Vaught theorem in a very accessible form. There is also a nice and short proofin [44]. The notation is borrowed from [44], too. At first, we need to define so called q -types. Definition 7.1 ([44]) . Let G be a labeled graph and ¯ v = ( v , . . . , v k ) ∈ V ( G ) k , for somenonnegative integer k . The first-order q -type of ¯ v in G is the set tp FO q ( G, ¯ v ) of all first-orderformulas ψ ( x , . . . x k ) of rank at most q such that G | = ψ ( v , . . . , v k ).A q -type could be an infinite set, but one can reduce them to a finite set by syntacticallynormalizing formulas, so that there are only finitely many normalized formulas of fixedquantifier rank and with a fixed set of free variables. These finitely many formulas canbe enumerated. For a tuple ¯ u = ( u , . . . , u k ), we write { ¯ u } for the set { u , . . . , u k } . Thefollowing is a variant of the Feferman–Vaught theorem [47]. Proposition 7.2 ([44, Lemma 2.3]) . Let
G, H be labeled graphs and ¯ u ∈ V ( G ) k , such that V ( G ) ∩ V ( H ) = { ¯ u } . Then for all q ≥ , tp FO q ( G ∪ H, ¯ u ) is determined by tp FO q ( G, ¯ u ) and tp FO q ( H, ¯ u ) .
25e use this proposition in the following two lemmas to introduce a q -type preservingprotrusion replacement procedure. Lemma 7.3.
Let H be a connected labeled graph with treewidth at most t , at most q labels,and ¯ u ∈ V ( H ) k for some k . One can find in time h ( q, t, k ) | H | a connected labeled graph H ′ with { ¯ u } ⊆ V ( H ′ ) ⊆ V ( H ) , such that | H ′ | ≤ h ( q, t, k ) and tp FO q ( H, ¯ u ) = tp FO q ( H ′ , ¯ u ) , forsome function h ( q, t, k ) .Proof. The q -type tp FO q ( H, ¯ u ) can be represented by a set of normalized FO-formulas withquantifier rank at most q and k free variables. The number and length of these representingformulas can be bounded by a function of q and k . Courcelle’s theorem states that for agraph H (with treewidth at most t ) and a formula ψ (with quantifier rank at most q and k free variables) one can decide whether H | = ψ (¯ u ) in time g ( q, t, k ) | H | , for some function g ( q, t, k ). This lets us efficiently compute the q -type tp FO q ( H, ¯ u ) by checking all representingformulas.We now have to find a small graph H ′ with the same q -type as H . We enumerate allconnected graphs whose vertex set is a superset of { ¯ u } and which are labeled using the samelabels as H in ascending order by vertex count. For each graph, we compute the q -type.We finish as soon as we find a graph H ′ with tp FO q ( H, ¯ u ) = tp FO q ( H ′ , ¯ u ). Such a graph H ′ exists. Each q -type of a k -tuple is represented by a subset of normalized formulas of rank atmost q and at most k free variables. This bounds the number of different q -types of k -tuplesby a function of q and k . Thus, the size of H ′ can also be bounded by a function of q and k . Since | H ′ | ≤ | H | , we can rename the vertices of H ′ such that V ( H ′ ) ⊆ V ( H ). Lemma 7.4.
Let
G, H be labeled graphs and ¯ u ∈ V ( G ) k for some k , such that V ( G ) ∩ V ( H ) = { ¯ u } . Let H be connected with treewidth at most t and at most q labels. One can findin time h ( q, t, k ) | H | a connected labeled graph H ′ such that | H ′ | ≤ h ( q, t, k ) , V ( G ) ∩ V ( H ′ ) = { ¯ u } , and G ∪ H ≡ q G ∪ H ′ , for some function h ( q, t, k ) .Proof. We use Lemma 7.3 to construct a connected labeled graph H ′ such that | H ′ | ≤ h ( q, t, k ), tp FO q ( H, ¯ u ) = tp FO q ( H ′ , ¯ u ), and V ( G ) ∩ V ( H ′ ) = { ¯ u } . According to Proposition 7.2,tp FO q ( G ∪ H, ¯ u ) is determined by tp FO q ( G, ¯ u ) and tp FO q ( H, ¯ u ). Therefore, tp FO q ( G ∪ H, ¯ u ) =tp FO q ( G ∪ H ′ , ¯ u ), which implies G ∪ H ≡ q G ∪ H ′ . Let G be an b - r - µ -locally-protrusion-partitionable graph. We want to construct a graphwhich is q -equivalent to G and small if b is small. We know there exists an b - r - µ -local-protrusion-partition ( X, Y, Z ) of G , but it is non-trivial to compute it. In Lemma 7.5 and7.7, we identify Z and parts of Y . In Lemma 7.8 and Theorem 7.9, we replace these partsusing the protrusion-replace technique from Lemma 7.4. Lemma 7.5.
There exists an algorithm that takes r, µ ∈ N + and a graph G with radius atmost r as input, runs in time O ( k G k ) , and computes a set Z ⊆ V ( G ) with the followingproperty: If G is b - r - µ -locally-protrusion-partitionable for some b ∈ N + then there exists an b - r - µ -local-protrusion-partition ( X, Y, Z ) of G .Proof. We construct Z iteratively. At first Z is empty, then in each step we add a vertex v ∈ V ( G ) to Z if it is a degree-one vertex of G [ V ( G ) \ Z ]. We repeat until there are nodegree-one vertices in G [ V ( G ) \ Z ]. This can be done in O ( k G k ) steps. The set Z satisfiesProperty 4 of Definition 6.1. Assume that G is b - r - µ -locally-protrusion-partitionable. Weneed to show that X, Y exist such that (
X, Y, Z ) is an b - r - µ -local-protrusion-partition of G .We consider an arbitrary b - r - µ -local-protrusion-partition ( X ′ , Y ′ , Z ′ ) of G . Every connectedcomponent of G [ Z ′ ] is a tree with at most one edge to V ( G ) \ Z ′ . The set Z was constructedsuch that Z ′ ⊆ Z . We set X = X ′ \ Z and Y = Y ′ \ Z . The sets X , Y , Z are disjoint and26heir union is V ( G ) (Property 1). Since X ⊆ X ′ and Y ⊆ Y ′ , the tuple ( X, Y, Z ) satisfiesProperties 2, and 3.Let H be a connected component of G [ Y ]. By definition of Z , there exists a uniquecomponent H ′ of G [ Y ′ ] such that H = H ′ \ Z . Furthermore, X ⊆ X ′ . This means the numberof distinct boundaries of G [ Y ] in X is not larger than the number of distinct boundaries of G [ Y ′ ] in X ′ . Since ( X ′ , Y ′ , Z ′ ) satisfies Property 5, ( X, Y, Z ) satisfies Property 5 as well.
Definition 7.6 (Heavy boundary) . Let (
X, Y, Z ) be an b - r - µ -local-protrusion-partition of agraph G . We call a set S ⊆ X a heavy boundary if there exist more than rµ + µ connectedcomponents in G [ Y ] whose boundary in X is exactly S . Lemma 7.7.
There exists an algorithm that takes r, µ ∈ N + and a graph G with radiusat most r as input, runs in time O ( k G k ) , and computes sets Z, P ⊆ V ( G ) , S ⊆ V ( G ) withthe following properties: If G is b - r - µ -locally-protrusion-partitionable for some b ∈ N + thenthere exists an b - r - µ -local-protrusion-partition ( X, Y, Z ) of G . The connected components of G [ Y ] with a heavy boundary are connected components of G [ P ] . The set S contains subsetsof X of size at most µ and | S | ≤ min(2 b µ , | G | ) . Every heavy boundary of ( X, Y, Z ) iscontained in S .Proof. Let b ∈ N + such that G is b - r - µ -locally-protrusion-partitionable. We use Lemma 7.5to construct in time O ( | G | ) a set Z such that there exists an b - r - µ -local-protrusion-partition( X, Y, Z ) of G .Let P be the set of all vertices with degree at most rµ + µ in G [ V ( G ) \ Z ]. The set P canbe computed in O ( k G k ). A vertex v ∈ Y is contained in a connected component of G [ Y ] ofsize at most rµ with at most µ neighbors in X (Property 3). Therefore v has degree at most rµ + µ in G [ V ( G ) \ Z ], which implies Y ⊆ P . Let H be a connected component of G [ Y ]with the heavy boundary S . There are more than rµ + µ connected components in G [ Y ]with boundary S . Therefore, every vertex in S has degree more than rµ + µ in G [ V ( G ) \ Z ]and thus S ∩ P = ∅ . This and V ( H ) ⊆ P imply that H is a connected component of G [ P ],i.e., the connected components of G [ Y ] with a heavy boundary are connected componentsof G [ P ].Let S be the set of all subsets of V ( G ) \ ( P ∪ Z ) with size at most µ which are theboundary of some connected component of G [ P ] in V ( G ) \ ( P ∪ Z ). The set S can becomputed in time O ( k G k ). Note that | S | ≤ | G | . Let S ∈ S . Since S ∩ ( P ∪ Z ) = ∅ and Y ⊆ P , we have S ⊆ X . For every heavy boundary of ( X, Y, Z ), there exists a connectedcomponent of G [ P ] with this boundary in V ( G ) \ ( P ∪ Z ). Boundaries of ( X, Y, Z ) have bydefinition (Property 3) size at most µ . Thus, every heavy boundary of ( X, Y, Z ) is containedin S .Since Y ⊆ P and P ∩ Z = ∅ , connected components of G [ P ] are either connectedcomponents of G [ Y ] or contain a vertex from X . Since | X | ≤ b µ , there are at most b µ connected components of G [ P ] that are not connected component of G [ Y ]. Components of G [ Y ] have by definition (Property 5) at most b µ distinct boundaries in V ( G ) \ ( P ∪ Z ). Theremaining at most b µ many connected components of G [ P ] that contain a vertex from X have at most b µ boundaries in G [ V ( G ) \ ( P ∪ Z )]. Together, this gives | S | ≤ b µ . Lemma 7.8.
There exists an algorithm that takes q, r, µ ∈ N + and a connected labeled graph G with radius at most r and at most q labels as input, runs in time at most f ( q, r, µ ) k G k for some function f ( q, r, µ ) , and computes a connected labeled graph G ∗ ≡ q G and a set Z ∗ ⊆ V ( G ∗ ) . In G ∗ , every connected component of G ∗ [ Z ∗ ] is a tree and has at most oneneighbor in V ( G ∗ ) \ Z ∗ . If G is b - r - µ -locally-protrusion-partitionable for some b ∈ N + then | V ( G ∗ ) \ Z ∗ | ≤ f ( q, r, µ ) min( b µ , | G | ) .Proof. Let b ∈ N + such that G is b - r - µ -locally-protrusion-partitionable. We use Lemma 7.7to compute sets Z, P ⊆ V ( G ), S ⊆ V ( G ) . There exists an b - r - µ -local-protrusion-partition( X, Y, Z ) of G . The connected components of G [ Y ] with a heavy boundary are con-nected components of G [ P ]. The set S contains subsets of X of size at most µ and27 S | ≤ min(2 b µ , | G | ). Every heavy boundary of ( X, Y, Z ) is contained in S . For every W ⊆ V ( G ) \ Z , we define Z ( W ) ⊆ Z to be the vertices that are reachable from W in G [ W ∪ Z ].For every S ∈ S we do the following: We compute the set P S of all vertices whichare contained in a connected component of G [ P ] with size at most rµ and boundary S in V ( G ) \ ( P ∪ Z ). We compute H S = G [ S ∪ P S ∪ Z ( P S )] and G S = G [ V ( G ) \ ( P S ∪ Z ( P S ))].Notice that G = H S ∪ G S and V ( H S ) ∩ V ( G S ) = S . Furthermore, | S | ≤ µ and if we remove S from H S , the remaining graph consists of connected components of size at most rµ towhich trees are attached. This means H S has treewidth at most t = µ + rµ . Also, H S hasat most q labels. Lemma 7.4 lets us construct in time at most h ( q, t, µ ) | H S | a graph H ′ S such that | H ′ S | ≤ h ( q, t, µ ), V ( G S ) ∩ V ( H ′ S ) = S , and G ≡ q G S ∪ H ′ S . We now replace H S with H ′ S .This replacement procedure gives us graphs ˆ G = G [ V ( G ) \ S S ∈ S ( P S ∪ Z ( P S ))] andˆ H = S S ∈ S H ′ S with G ≡ q ˆ G ∪ ˆ H . We set G ∗ = ˆ G ∪ ˆ H . Notice that G ∗ is connected, sincethe described construction preserves connectivity.We now bound the run time of this procedure. The underlying graph of G can beextracted in time q k G k . Constructing Z , P , and S by Lemma 7.7 takes time O ( k G k ). Thegraphs ˆ G , and { H S | S ∈ S } can be constructed in time O ( P S ∈ S k H S k ). Constructing ˆ H takes time h ( q, t, µ ) P S ∈ S k H S k .For every vertex v ∈ P ∪ Z there exists at most one S ∈ S such that v ∈ H S . Also | S | ≤ | G | . Furthermore, since a graph H S has treewidth at most t , k H S k ≤ t | H S | . Therefore, X S ∈ S k H S k ≤ t X S ∈ S | H S | = t X S ∈ S | S | + | H S ∩ ( P ∪ Z ) | ≤ tµ | S | + t | P ∪ Z | ≤ tµ | G | + t | G | . In total, the whole algorithm runs in time f ( q, r, µ ) k G k , for some function f ( q, r, µ ).We proceed to show that | V ( G ∗ ) \ Z | is small. Note that | V ( G ∗ ) \ Z | ≤ | X | + | V ( ˆ G ) ∩ Y | + | ˆ H | . (10)We know that | X | ≤ b µ . Furthermore, with | S | ≤ min(2 b µ , | G | ) | ˆ H | ≤ X S ∈ S | H ′ S | ≤ min(2 b µ , | G | ) h ( q, t, µ ) . (11)We further bound the number of vertices from Y in ˆ G . Let X be the set of all boundariesin X of connected components of G [ Y ]. Let S ∈ X be a boundary, we define Y S to be thevertices of all connected components of G [ Y ] which have S as their boundary. We distinguishbetween S being a heavy or non-heavy boundary: Assume S is heavy. Then S ∈ S . Theconnected components of G [ Y ] with a heavy boundary are connected components of G [ P ]and have size at most rµ . This means Y S ⊆ P S . The graph ˆ G was defined such that P S ∩ V ( ˆ G ) = ∅ , and we have | Y S ∩ V ( ˆ G ) | = 0. Assume now S is non-heavy. Thus, Y S consists of at most rµ + µ connected components of G [ Y ] of size at most rµ . This means | Y S | ≤ ( rµ + µ ) rµ .In total, for every S ∈ X , | V ( ˆ G ) ∩ Y S | ≤ ( rµ + µ ) rµ . Note that Y = S S ∈ X Y S and | X | ≤ b µ (Property 5). We can therefore bound | V ( ˆ G ) ∩ Y | ≤ X S ∈ X | V ( ˆ G ) ∩ Y S | ≤ b µ ( rµ + µ ) rµ . (12)Combining (10), (11), (12), and | X | ≤ b µ yields | V ( G ∗ ) \ Z | ≤ b µ + b µ ( rµ + µ ) rµ + 2 b µ h ( q, t, µ ) , which can be bounded by f ( q, r, µ ) b µ for some function f ( q, r, µ ). Furthermore, combining | V ( G ∗ ) \ Z | ≤ | ˆ G | + | ˆ H | , (11), and | ˆ G | ≤ | G | yields | V ( G ∗ ) \ Z | ≤ f ( q, r, µ ) | G | for somefunction f ( q, r, µ ). 28et Z ∗ = Z ∩ V ( G ∗ ). At last, we need to show that in G ∗ , the connected components of G ∗ [ Z ∗ ] are trees with at most one neighbor in V ( G ∗ ) \ Z ∗ . This follows from that fact thatin G the connected components of G [ Z ] are trees with at most one neighbor in X ∪ Y , andthe graph G ∗ introduces no new edges to vertices from Z . Theorem 7.9.
There exists an algorithm that takes q, r, µ ∈ N + and a connected labeledgraph G with radius at most r and at most q labels as input, runs in time at most f ( q, r, µ ) k G k for some function f ( q, r, µ ) , and computes a labeled graph G ∗ ≡ q G . If G is b - r - µ -locally-protrusion-partitionable for some b ∈ N + then | G ∗ | ≤ f ( q, r, µ ) b µ .Proof. Let b ∈ N + such that G is b - r - µ -locally-protrusion-partitionable. We use the algo-rithm of Lemma 7.8 to compute a connected graph G ′ ≡ q G and a set Z ⊆ V ( G ′ ). Let¯ Z = V ( G ′ ) \ Z . We have | ¯ Z | ≤ g ( q, r, µ ) min( b µ , | G | ), for some function g ( q, r, µ ). Also, in G ′ , every connected component of G ′ [ Z ] is a tree and has at most one neighbor in ¯ Z . Since G ′ ≡ q G , G ′ also has at most q labels. If ¯ Z is empty, then G ′ is a tree with at most q labelsand we use Lemma 7.4 to construct in time h ( q, , | G ′ | a graph G ∗ with | G ∗ | ≤ h ( q, , G ∗ ≡ q G ′ ≡ q G . We therefore assume ¯ Z = ∅ .For every v ∈ ¯ Z we do the following: We define Z v to be the set of vertices which arecontained in a connected component of G ′ [ Z ] which has v as its only neighbor. We alsodefine the graph H v = G ′ [ { v } ∪ Z v ], which is a tree with at most q labels and intersects G ′ [ ¯ Z ] only in v . We use Lemma 7.4 to construct in time h ( q, , | H v | a graph H ′ v with | H ′ v | ≤ h ( q, ,
1) and G ≡ q G ′ [ V ( G ′ ) \ Z v ] ∪ H ′ v . We replace the subgraph H v of G ′ with H ′ v .This gives us a graph G ∗ = G ′ [ V ( G ′ ) \ S v ∈ ¯ Z Z v ] ∪ S v ∈ ¯ Z H ′ v with G ∗ ≡ q G ′ ≡ q G . Since¯ Z = ∅ and G ′ is connected S v ∈ ¯ Z Z v = Z . We bound with | ¯ Z | ≤ g ( q, r, µ ) b µ and v ∈ V ( H ′ v )for all v ∈ ¯ Z | G ∗ | = X v ∈ ¯ Z | H ′ v | ≤ g ( q, r, µ ) b µ h ( q, , ≤ f ( q, r, µ ) b µ , for some function f ( q, r, µ ) ≥ g ( q, r, µ ) h ( q, , { H v | v ∈ ¯ Z } aredisjoint and their union is G ′ . The time needed to construct ˆ H therefore is at most X v ∈ ¯ V h ( q, , k H v k ≤ h ( q, , k G ′ k ≤ h ( q, , g ( q, r, µ ) k G k ≤ f ( q, r, µ ) k G k . As in Lemma 7.8, f can be chosen such that the algorithm runs in time f ( q, r, µ ) k G k . In this section, we finally obtain the main result of this paper, namely that for certainvalues of α one can perform model-checking on power-law-bounded random graph modelsin efficient expected time.An important tool in this section is Gaifman’s locality theorem [38]. It states that first-order formulas can express only local properties of graphs. It is a well established tool forthe design of model-checking algorithms (e.g. [43, 44, 36]). We use it to reduce the model-checking problem on a graph to the model-checking problem on neighborhoods of said graph(Lemma 8.2). This technique is described well by Grohe [44, section 5].To illustrate our approach, consider the following thought experiment: Let X be a non-negative random variable with Pr[ X = b ] = Θ( b − ) for all b ∈ N . Assume an algorithmthat gets an integer b ∈ N as input and runs in time t ( b ). Its expected run time on input X is P b ∈ N Θ( b − ) t ( X ). If t ( b ) = b then the expected run time is infinite. If t ( b ) = b thenthe expected run time is Θ(1). Thus, small polynomial differences in the run time can havea huge impact on the expected run time. We notice that the run time on an input has togrow slower than the inverse of the probability that the input occurs.Let us fix a formula ϕ and let r and µ be constants depending on ϕ . In this section weprovide a model-checking algorithm whose run time on a graph G depends on the minimal29alue b ∈ N such that G is b - r - µ -partitionable. This means, we need to solve the model-checking problem on b - r - µ -partitionable graphs faster than the inverse of the probabilitythat b is minimal.Section 5 states that a graph from power-law-bounded graph classes is for some b not b - r - µ -partitionable with probability approximately b − µ (we ignore the terms in r , µ and˜ d α ( n ) for now). Thus, the probability that a value b is minimal is approximately b − µ .Let G be a graph and a be the minimal value such that G is b - r - µ -partitionable. InSection 6 we showed that all its r -neighborhoods are O ( µ r b )- r - O ( µ )-locally-protrusion-partitionable. The kernelization result from Section 7 states that such r -neighborhoodscan be converted in linear time into | ϕ | -equivalent graphs of size approximately b µ (weagain ignore the factors independent of b for now). This means, using the naive model-checking algorithm, one can decide for an r -neighborhood G r of G whether G r | = ϕ in timeapproximately k G k b µ | ϕ | . Thus, one can perform model-checking on all r -neighborhoods of G in time approximately b µ | ϕ | P v k N Gr ( v ) k . Using Gaifman’s locality theorem, this (moreor less) yields the answer to the model checking problem in the whole graph.Let G be a graph from a power-law-bounded random graph model. In summary, we havefor every b ∈ N : • b ∈ N is the minimal value such that a graph is b - r - µ -partitionable with probabilityapproximately b − µ . • If b ∈ N is the minimal value such that G is b - r - µ -partitionable then we can decidewhether G | = ϕ in time approximately b µ | ϕ | P v k N Gr ( v ) k .In this example one may choose µ = | ϕ | such that the run time grows slower than theinverse of the probability. We changed some numbers in these examples to simplify ourarguments. Thus, in reality, µ needs to be chosen slightly differently.This section is structured as follows: In Section 8.1, we introduce the concept of Gaifmanlocality. Then, in Section 8.2, we use Gaifman locality and the kernelization result fromSection 7 to solve the model-checking in b - r - µ -partitionable graphs. At last, in Section 8.3,we prove our main result by showing that the run time of this algorithm grows slower thanthe inverse of the probability that b is minimal. In this section, we present a well-known technique which reduces the model-checking problemto local regions. Lemma 8.2 gives a slightly different version of what can be found inthe literature [44]. Without this modification we would only be able to prove expectedpolynomial time of our model-checking algorithm instead of expected linear time.A formula ω ( x ) is called r -local if G | = ω ( v ) if and only if G [ N Gr ( v )] | = ω ( v ) for alllabeled graphs G and all v ∈ V ( G ). Let dist >r ( x , x ) be the first-order formula denotingthat the distance between x and x is greater than r . Let ω be an r -local formula. A basiclocal sentence is a sentence of the form ∃ x . . . ∃ x s (cid:0) ^ i = j dist > r ( x i , x j ) ∧ ^ i ω ( x i ) (cid:1) . Proposition 8.1 (Gaifman’s locality theorem [38, 44]) . Every first-order sentence is equiv-alent to a boolean combination of basic local sentences. Furthermore, there is an algorithmthat computes a boolean combination of basic local sentences equivalent to a given first-ordersentence.
The following lemma uses Gaifman locality (Proposition 8.1) to reduce model-checkingin graphs to model-checking in neighborhoods of graphs. The proof is similar to [44,Lemma 4.9]. 30 emma 8.2.
Let g be a function such that for every graph G with at most r labels, every r -neighborhood H of G , every v ∈ V ( H ) , and every first-order formula ϕ ( x ) with | ϕ | ≤ r one can decide whether H | = ϕ ( v ) in time g ( v, G, r ) .There exists a function ρ such that for every first-order sentence ϕ and every labeledgraph G with at most | ϕ | labels one can decide whether G | = ϕ in time at most O ( k G k ) + ρ ( | ϕ | ) P v ∈ V ( G ) g ( v, G, ρ ( | ϕ | )) .Proof. We can reduce the first-order sentence ϕ to a boolean combination of basic localsentences Ψ with Proposition 8.1. We will independently evaluate each basic local sentence ψ ∈ Ψ in the graph and use the result to determine whether ϕ is satisfied. Let ψ = ∃ x . . . ∃ x s (cid:0) ^ i = j dist > r ( x i , x j ) ∧ ^ i ω ( x i ) (cid:1) be a basic local sentence, where ω is r -local. Let G be a graph with at most | ϕ | labelsand v ∈ V ( G ). We have G | = ω ( v ) if and only if G [ N r ( v )] | = ω ( v ) for v ∈ V . Wecompute for all v ∈ V whether G | = ω ( v ). By our assumption, this can be done in time P v ∈ V ( G ) g ( v, G, r + | ω | + | ϕ | ). Let now W be the set of all v ∈ V such that G | = ω ( v ).A set of vertices is called an r -scattered set if the r -neighborhoods of all pairs of verticesin this set are disjoint. Notice that G | = ψ if and only if there exists an r -scattered set ofcardinality s which is a subset of W . Therefore, all left to do is to find out whether there isan r -scattered set S ⊆ W with | S | ≥ s .In time O ( k G k ) we do the following: Construct a graph H that consists of all nodesthat have distance at most r from W , and construct the connected components of H . Foreach component H ′ of H pick a vertex v ∈ V ( H ′ ) and perform a breadth-first-search in H ′ ,starting at v . This way, we either find out that the radius of H ′ larger than 12 rs or thatthe diameter of H ′ is at most 12 rs .Let us consider two cases. First, we verified that there is a component of H whosediameter is at least 12 rs . Then this component must contain a shortest path p of length 12 rs .We constructed H such that the r -neighborhoods of every vertex u on p contains a vertexfrom W . Since there are at least s nodes on p whose r -neighborhoods are disjoint and eachof the neighborhoods contains a vertex from W , we know that W contains an r -scatteredset of size at least s .The second case we have to consider is that we verified that all components of H havea radius at most 12 rs . Note that W ⊆ V ( H ). For u, v ∈ W from different components of H , the distance between u and v in G is at least 2 r . Hence, maximal cardinality r -scatteredsubsets of W of the components of H form together a maximal cardinality r -scattered subsetof W in G . A component H ′ of H contains an r -scattered subset of W of size l iff H ′ | = ψ l with ψ l = ∃ x . . . ∃ x l (cid:0) ^ i = j dist > r ( x i , x j ) ∧ ^ i ω ( x i ) (cid:1) , which we can evaluate in time g ( v, G, rs + | ψ l | + | ϕ | ) for some v ∈ V ( H ′ ). We need tocheck whether H ′ | = ψ l for every component H ′ of H and l ∈ { , . . . , s } . In that way wecan compute the maximal size of an r -scattered subset of W in H and therefore in G .The complete procedure has to be repeated for each ψ ∈ Ψ. Note that r , s , | ψ l | , and | ψ | depend only on ϕ . This means we can choose ρ such that all this can be done in time O ( k G k ) + ρ ( | ϕ | ) P v ∈ V ( G ) g ( v, G, ρ ( | ϕ | )). b - r - µ -Locally-Protrusion-PartitionableGraphs We use the kernelization result of Theorem 7.9 to construct a model-checking algorithm forneighborhoods. 31 emma 8.3.
There is a function f ( r, µ ) such that for every r, µ ∈ N + , every graph G withat most r labels, every r -neighborhood H of G , every v ∈ V ( H ) , and every first-order formula ϕ ( x ) with | ϕ | ≤ r one can decide whether H | = ϕ ( v ) in time f ( r, µ ) b O ( µr ) k G [ N G r ( v )] k , where b ∈ N + be the minimal value such that G is b - r - µ -partitionable.Proof. We construct a graph H ′ by adding another label to H that identifies v and constructa sentence ϕ ′ with | ϕ ′ | = O ( | ϕ | ) such that H ′ | = ϕ ′ if and only if H | = ϕ ( v ). Let b ∈ N + be the minimal value such that G is b - r - µ -partitionable. According to Theorem 6.15, H ′ is O ( µ r b )- r - O ( µ )-locally-protrusion-partitionable. We use Theorem 7.9 to construct intime f ′ ( r, µ ) k H ′ k a graph H ∗ with H ∗ ≡ | ϕ ′ | H ′ and | H ∗ | ≤ f ′ ( r, µ ) b O ( µ ) , for some function f ′ . On this smaller structure we can perform the naive model-checking algorithm in time O ( | H ∗ | r ) = O (cid:0) f ′ ( r, µ ) | ϕ | b O ( µr ) (cid:1) . Furthermore, the radius of H ′ is at most r , thus k H ′ k ≤k G [ N G r ( v )] k . We choose f ( r, µ ) accordingly. Lemma 8.4.
Let µ ∈ N + . There exist functions ρ and f such that for every first-ordersentence ϕ and every labeled graph G with at most | ϕ | labels one can decide whether G | = ϕ in time f ( ρ ( | ϕ | ) , µ ) b µρ ( | ϕ | ) P v ∈ V ( G ) k G [ N Gρ ( | ϕ | ) ( v )] k , where b ∈ N + is the minimal value suchthat G is b - ρ ( r ) - µ -partitionable.Proof. By Lemma 8.2 and 8.3, there exist functions ρ ′ and f ′ such that one can decidewhether G | = ϕ in time O ( k G k ) + ρ ′ ( | ϕ | ) X v ∈ V ( G ) f ′ ( ρ ′ ( | ϕ | ) , µ ) b O ( µρ ′ ( | ϕ | )) k G [ N G ρ ′ ( | ϕ | ) ( v )] k , where b ∈ N + is the minimal value such that G is b - ρ ′ ( r )- µ -partitionable. We choose f and ρ sufficiently large. In this section we show that the algorithm from Lemma 8.4 has efficient expected run timeon power-law-bounded random graph models. Our analysis is based upon two results weestablished earlier: First, the run time of the algorithm in Lemma 8.4 depends on theminimal value b such that the input graph is b - r - µ -partitionable. If it is b - r - µ -partitionablefor a small b the run time is fast. Secondly, Theorem 5.10 bounds for our random graphs theprobability that b ∈ N is the minimal value such that a graph is an b - r - µ -partitionable. Forbigger b it is more and more unlikely that b is minimal. In order to have an efficient expectedrun time on our random graphs, the run time of the algorithm needs to grow asymptoticallyslower in b than the inverse of the probability that b is minimal. In Theorem 8.5 we showthat this is the case.The run time of the algorithm from Lemma 8.4 depends not only on b but also on thesum of the sizes of all neighborhoods in a graph, which might be quadratic in the worst case.In order to get almost linear expected run time, we bound the expectation of this value inLemma 5.12. We can now prove our main result. Theorem 8.5.
There exists a function f such that one can solve p - MC(FO , G lb ) on every α -power-law-bounded random graph model in expected time ˜ d α ( n ) f ( | ϕ | ) n .Proof. Let ( G n ) n ∈ N be an α -power-law-bounded random graph model and ϕ be a first-order formula. We fix a | ϕ | -labeling function L and n ∈ N . We consider labeled graphswith vertices V ( G n ) whose underlying graph is distributed according to G n , and analyze theexpected run time of the model-checking algorithm from Lemma 8.4 on these graphs.Let ρ be the function from Lemma 8.4 and let r = ρ ( | ϕ | ) and µ = ρ ( | ϕ | ) + 100. Forevery graph G there exists a value b ∈ N + such that G is b - r - µ -partitionable (i.e., by setting b = | V ( G ) | , A = V ( G )). Let A b be the event that b ∈ N + is the minimal value such that G n b - r - µ -partitionable and let R be the expected run time of the model-checking algorithmfrom Lemma 8.4. The expected run time of the algorithm is exactly P ∞ b =1 E[ R | A b ] Pr[ A b ].We use Lemma 8.4 and 5.12 to bound ∞ X b =1 E[ R | A b ] Pr[ A b ] ≤ ∞ X b =1 E (cid:2) f ′ ( r, µ ) b rµ X v ∈ V ( G n ) k G n [ N G n r ( v )] k | A b (cid:3) Pr[ A b ]= ∞ X b =1 f ′ ( r, µ ) b rµ E (cid:2) X v ∈ V ( G n ) k G n [ N G n r ( v )] k | A b (cid:3) Pr[ A b ] ≤ ∞ X b =1 f ′ ( r, µ ) b rµ (200 rµ ) O ( r ) ˜ d α ( n ) O ( µ r ) b − µ / n = f ′ ( r, µ )(200 rµ ) O ( r ) ˜ d α ( n ) O ( µ r ) n ∞ X b =1 b − µ / rµ . Note that for µ = ρ ( | ϕ | ) +100 and r = ρ ( | ϕ | ) we have P ∞ b =1 b − µ / rµ ≤ P ∞ b =1 b − = O (1).This yields a run time of ˜ d α ( n ) f ( | ϕ | ) n for some function f . Theorem 8.6.
Let ( G n ) n ∈ N be a random graph model. There exists a function f such thatone can solve p - MC(FO , G lb ) in expected time • f ( | ϕ | ) n if ( G n ) n ∈ N is α -power-law-bounded for some α > , • log( n ) f ( | ϕ | ) n if ( G n ) n ∈ N is α -power-law-bounded for α = 3 , • f ( | ϕ | , ε ) n ε for all ε > if ( G n ) n ∈ N is α -power-law-bounded for every < α < .Proof. Assume that ( G n ) n ∈ N is α -power-law-bounded for some α >
3. By Theorem 8.5,there exists a function f ′ such that p -MC(FO , G lb ) can be solved on ( G n ) n ∈ N in expectedtime O (1) f ′ ( | ϕ | ) n . By choosing f ( | ϕ | ) = c f ′ ( | ϕ | ) for a suitable c we get the desired expectedrun time.Assume that ( G n ) n ∈ N is α -power-law-bounded with α = 3. By Theorem 8.5, thereexists a function f ′ such that p -MC(FO , G lb ) can be solved on ( G n ) n ∈ N in expected timelog( n ) O (1) f ′ ( | ϕ | ) n . By choosing f ( | ϕ | ) = cf ′ ( | ϕ | ) for a suitable c we get the desired expectedrun time.Assume that ( G n ) n ∈ N is α -power-law-bounded for every α <
3. According to The-orem 8.5, there exists a function f ′ such that one can solve p -MC(FO , G lb ) in expectedtime O ( n ε ′ f ′ ( | ϕ | ) n ) for all ε ′ >
0. This means, there exists functions c ( ε ′ ) and n ( ε ′ ) suchthat for all ε ′ > n ≥ n ( ε ) the expected time is at most c ( ε ′ ) n ε ′ f ′ ( | ϕ | ) . Thus,we can choose c ′ ( ε ′ ) such that for all ε ′ > n ∈ N the expected time is at most c ′ ( ε ′ ) n ε ′ f ′ ( | ϕ | ) . Let ε >
0. With ε ′ = ε/f ′ ( | ϕ | ), the algorithm runs for all n ∈ N inexpected time c ′ ( ε/f ′ ( | ϕ | )) n ε . We set f ( x, ε ) = c ′ ( ε/f ′ ( x )). The algorithm runs for all n ∈ N in expected time f ( | ϕ | , ε ) n ε . In Section 5 and 6 we analyzed the structure of α -power-law-bounded random graphs. Weobtained decompositions depending on parameters b , r and µ . These parameters are neededfor algorithmic purposes. In this section we substitute the parameters b and µ , which leadsto structural results in a more accessible form.We observe that α -power-law-bounded random graphs have mostly an extremely sparsestructure, with the exception of a part whose size is bounded by the second order averagedegree of the degree distribution. This denser part can be separated well from the remaining33raph. We show that local regions admit a protrusion decomposition consisting of a corepart, bounded in size by the second order average degree, to which trees and graphs ofconstant size are attached. At first, we define a function ˆ d α ( n ) similarly to ˜ d α ( n ) without O -notation. Definition 9.1.
We define ˆ d α ( n ) = α > n ) α = 3 n (3 − α ) α < . We use ˆ d α ( n ) to obtain a good bound on the minimal value b such that α -power-law-bounded graphs are b - r - µ -partitionable. We fix µ = 5 to have one free variable less. Lemma 9.2.
Let ( G n ) n ∈ N be an α -power-law-bounded random graph model. There existconstants c, r such that for every r ≥ r , ( G n ) n ∈ N is a.a.s. ˆ d α ( n ) cr - r - -partitionable.Proof. Assume ( G n ) n ∈ N is α -power-law-bounded. By Theorem 5.10, the probability that( G n ) n ∈ N is not b - r - µ -partitionable is bouned by at most ˜ d α ( n ) O ( µ r ) b − µ / . Let µ = 5, b = ˆ d α ( n ) cr . We bound the probability of not being ˆ d α ( n ) cr - r -5-partitionable by at most˜ d α ( n ) O ( r ) ˆ d α ( n ) − cr . We set c large enough such that the probability converges to zerowith n .Substituting the definition of a b - r - µ -partition into Lemma 9.2 yields the following self-sufficient theorem. Theorem 9.3.
Let ( G n ) n ∈ N be an α -power-law-bounded random graph model. There existconstants c, r such that for every r ≥ r a.a.s. one can partition V ( G n ) into three (possiblyempty) sets A , B , C with the following properties. • | A | , | B | ≤ ˆ d α ( n ) cr . • Every r -neighborhood in G n [ B ∪ C ] has at most more edges than vertices. • Every r -neighborhood in G n [ C ] has at most edges to A .Proof. Direct consequence of Lemma 9.2.Therefore, one can remove a few vertices to make the graph extremely sparse, as observedby the following corollary. This corollary might not have algorithmic consequences by itself,but sheds a lot of light on the structure of such graphs.
Corollary 9.4.
Let ( G n ) n ∈ N be an α -power-law-bounded random graph model. There existconstants c, r such that for every r ≥ r a.a.s. one can remove ˆ d α ( n ) cr vertices from G n such that every r -neighborhood has treewidth at most . In Section 6 we analyze the local structure of α -power-law-bounded graphs. We observethat local regions consist of a core part, bounded in size by the second order average degree,to which trees and graphs of constant size are attached. We obtain a self-contained theorem. Theorem 9.5.
Let ( G n ) n ∈ N be an α -power-law-bounded random graph model. There existconstants c, r such that for every r ≥ r a.a.s. for every r -neighborhood H of G n one canpartition V ( H ) into three (possibly empty) sets X , Y , Z with the following properties. • | X | ≤ ˆ d α ( n ) cr . • Every connected component of H [ Y ] has size at most cr and at most c neighbors in X . • Every connected component of H [ Z ] is a tree with at most one edge to H [ X ∪ Y ] . roof. By Lemma 9.2, G n is a.a.s. ˆ d α ( n ) c ′ r - r -5-partitionable for some constant c ′ . Thus,by Theorem 6.15, every r -neighborhood of G n is O (cid:0) r ˆ d α ( n ) c ′ r (cid:1) - r - O (5)-partitionable. Werefer to Definition 6.1 and choose c large enough such that this statement holds.Using Theorem 9.5, we can make statements about the structural sparsity of a randomgraph model. Note that locally bounded treewidth implies nowhere density [61]. The firstcorollary is based on the fact that X has a.a.s. constant size if α > Corollary 9.6.
Let ( G n ) n ∈ N be an α -power-law-bounded random graph model with α > .Then ( G n ) n ∈ N has a.a.s. locally bounded treewidth. Corollary 9.7.
Let ( G n ) n ∈ N be an α -power-law-bounded random graph model. There existconstants c, r such that for every r ≥ r a.a.s. the size of the largest r -subdivided clique in G n is at most ˆ d α ( n ) cr .
10 Implications for Various Graph Models
A wide range of unclustered random graph models are α -power-law-bounded. In this section,we show that certain Erd˝os–R´enyi graphs, preferential attachment graphs, configurationgraphs and Chung–Lu graphs are α -power-law-bounded and discuss what implications thishas for the tractability of the model-checking problem on these graph models. We alsodiscuss the connections to clustered random graph models, which currently do not fit intoour framework. For convenience, we restate the definition of α -power-law-boundedness. Definition 2.1.
Let α >
2. We say a random graph model ( G n ) n ∈ N is α -power-law-bounded if for every n ∈ N there exists an ordering v , . . . , v n of V ( G n ) such that for all E ⊆ (cid:0) { v ,...,v n } (cid:1) Pr (cid:2) E ⊆ E ( G n ) (cid:3) ≤ Y v i v j ∈ E ( n/i ) / ( α − ( n/j ) / ( α − n · O ( | E | ) if α > n ) O ( | E | ) if α = 3 O ( n ε ) | E | for every ε > α < . The maybe best-known model proposed to mimic the features observed in complex networksare preferential attachment graphs introduced by Barabsi and Albert [3, 63]. They have beenstudied in great detail (see for example [71]). These random graphs are created by a processthat iteratively adds new vertices and randomly connects them to already existing ones,where the attachment probability is proportional to the current degree of a vertex. Themodel depends on a constant m which is the number of edges that are inserted per vertex.The random graph with n vertices and parameter m is denoted by G nm .The preferential attachment process exhibits small world behavior [24] and has beenwidely recognized as a reasonable explanation of the heavy tailed degree distribution ofcomplex networks [8].Recent efficient model-checking algorithms on random graph models only worked onrandom graph models that asymptotically almost surely (a.a.s.) are nowhere dense [43, 22].It is known that preferential attachment graphs are not a.a.s. nowhere dense [22] and evena.a.s. somewhere dense [26], thus previous techniques do not work.Nevertheless, we are able to solve the model-checking problem efficiently on these graphs.Usually, the parameter m of the model is considered to be constant. We obtain efficientalgorithms even if we allow m to be a function of the size of the network. For a function m ( n ) : N → N we define ( G nm ( n ) ) n ∈ N be the corresponding preferential attachment model.The following lemma follows directly from [28].35 emma 10.1 ([28], Lemma 10) . Let m : N → N . The preferential attachment model ( G nm ( n ) ) n ∈ N is • -power-law-bounded if m ( n ) = log( n ) O (1) , • α -power-law-bounded for every < α < if m ( n ) = O ( n ε ) for every ε > . According to Lemma 10.1 and Theorem 8.6 one can therefore solve the model-checkingproblem efficiently on preferential attachment graphs.
Corollary 10.2.
Let m : N → N . There exists a function f such that one can solve p - MC(FO , G lb ) on the preferential attachment model ( G nm ( n ) ) n ∈ N in expected time • log( n ) f ( | ϕ | ) n if m ( n ) = log( n ) O (1) , • f ( | ϕ | , ε ) n ε for every ε > if m ( n ) = O ( n ε ) for every ε > . The Chung–Lu model has been proposed to generate random graphs that fit a certaindegree sequence and has been studied extensively [12, 13, 14]. We completely characterizethe tractability of the model-checking problem on Chung–Lu graphs based on the power-lawexponent α (Corollary 10.4). Previous tractability results were obtained for a non-standardvariant of the model and did not cover the case α = 3.Let W = ( w , . . . , w n ) be a sequence of positive weights with max ni =1 w i ≤ P nk =1 w k . TheChung–Lu random graph to W is a random graph G n with vertices v , . . . , v n such that eachedge v i v j with 1 ≤ i, j ≤ n occurs in G n independently with probability w i w j / P nk =1 w k .Often, the weights are chosen according to a power-law distribution. Let α >
2. Wesay ( G n ) n ∈ N is the Chung–Lu random graph model with exponent α if for every n ∈ N , G n is the Chung–Lu random graph to W n = { w , . . . , w n } with w i = c · ( n/i ) / ( α − where c is a constant depending on α [12]. This model nicely matches our concept of α -power-law-boundedness. Lemma 10.3.
Let α > . The Chung–Lu random graph model with exponent α is α -power-law-bounded.Proof. One can easily verify that P nk =1 w k = Θ( n ) for all α >
2. Thus, the probability ofan edge v i v j in a Chung–Lu graph with exponent α > n is w i w j / n X k =1 w k = ( n/i ) / ( α − ( n/j ) / ( α − Θ( n ) . All edges are independent of each other, therefore the probability that an edge set E iscontained is the product of the probabilities of the individual edges. This yieldsPr (cid:2) E ⊆ E ( G n ) (cid:3) ≤ O ( | E | ) Y v i v j ∈ E ( n/i ) / ( α − ( n/j ) / ( α − n . We can combine Lemma 10.3, Theorem 8.6 and [27] to characterize the tractability ofthe labeled model-checking problem on Chung–Lu graphs.
Corollary 10.4.
Let G be the Chung–Lu random graph model with exponent α . There existsa function f such that one can solve p - MC(FO , G lb ) on G in expected time • f ( | ϕ | ) n if α > , • log( n ) f ( | ϕ | ) n if α = 3 . urthermore, if . ≤ α < , α ∈ Q then one cannot solve p - MC(FO , G lb ) on G in expected FPT time unless
AW[ ∗ ] ⊆ FPT / poly . Previously, the model-checking problem has been known to be tractable on Chung–Lugraphs with exponent α >
3, and hard on Chung–Lu graphs with exponent 2 . ≤ α < α = 3 was open. Furthermore, the previous tractability result assumesthe maximum expected degree of a Chung–Lu graph with exponent α to be at most O ( n /α ),while in the canonical definition of Chung–Lu graphs (stated above) it is Θ( n / ( α − ). Ourresults hold for the canonical definition. The missing case α < . . ≤ α < d of a Chung–Lu graph with weights w , . . . , w n isdefined as P ni =1 w i / P nk =1 w k . After substituting the maximum degree m = Ω( n / ( α − )in [12] one can see for the Chung–Lu graph with exponent α that¯ d = Ω(1) α > n )) α = 3Ω( n (3 − α ) / ( α − ) α < . We can further bound the run time of the model-checking problem in terms of ¯ d . Lemma 10.5.
There exist a function f such that one can solve p - MC(FO , G lb ) on Chung–Lu graphs with exponent α in expected time ( c α ¯ d ) f ( | ϕ | ) n , where ¯ d is the second order averagedegree and c α is a constant depending on α .Proof. According to Theorem 8.5 one can solve p -MC(FO , G lb ) on the Chung–Lu graph withexponent α in expected time µ f ′ ( | ϕ | ) α n α > n ) µ α f ′ ( | ϕ | ) n α = 3 µ α n (3 − α ) f ′ ( | ϕ | ) n α < f ′ is some function and µ α is a constant depending on α . On the other hand, wehave ¯ d = λ α α > λ α log( n ) α = 3 λ α n (3 − α ) / ( α − α < λ α depending on α . Thus, one can solve p -MC(FO , G lb ) in expectedtime (cid:0) ( µ α /λ α ) ¯ d (cid:1) max(2 ,µ ) f ′ ( | ϕ | ) n. The result follows by setting c α = µ α /λ α and f ( | ϕ | ) = max(2 , µ ) f ′ ( | ϕ | ). The configuration model has been proposed to generate random multigraphs whose degreesare fixed [59, 58, 4]. We solve the model-checking problem on configuration graphs with apower-law exponent 3 (Corollary 10.7). Previously, this was only known for those configu-ration graphs with an exponent strictly larger than 3 [22].Let W = ( w , . . . , w n ) be a degree sequence of a multigraph (i.e., a sequence of positiveintegers whose sum is even). The configuration model constructs a random multigraph with n vertices whose degree sequence is exactly W as follows [59]: Let v , . . . , v n be the verticesof the graph. We form a set L of w i many distinct copies of v i for 1 ≤ i ≤ n . We call thecopies of a node v i in L the stubs of v i . We then construct a random perfect matching on L .This describes a multigraph on v , . . . , v n where the number of edges between two vertices37quals the number of edges between their stubs. The degree sequence of this multigraphis exactly W . Since we only consider simple graphs in this work, we turn to the so called erased [71] model. Here self-loops are removed and multi-edges are replaced with singleedges. As self-loops can be expressed by labels, this is no real limitation for the model-checking problem. Let G n be the probability distribution over simple graphs with n verticesdefined by this process. We say G n is the random configuration graph corresponding to W .This defines a random graph with a fixed number of vertices. In order to define a randomgraph model we need to define configuration graphs of arbitrary size. Let ( w i ( n )) i ∈ N bea sequence of functions such that all n ∈ N , ( w ( n ) , . . . , w n ( n )) is a degree sequence ofa multigraph. For n ∈ N let G n be the random configuration graph corresponding to thedegree sequence ( w ( n ) , . . . , w n ( n )). We then say ( G n ) n ∈ N is the random configuration graphmodel corresponding to ( w i ( n )) i ∈ N . For technical reasons, our definition differs slightly fromthe original one by Molloy and Reed [59]. Lemma 10.6.
Let ( G n ) n ∈ N be a random configuration graph model with correspondingsequence ( w i ( n )) i ∈ N . Assume there exists a function p ( n ) with p ( n ) = O ( n ε ) for all ε > such that for all i, n ∈ N , w i ( n ) ≤ p ( n ) p n/i and P nk =1 w k ( n ) ≥ n/p ( n ) . Then ( G n ) n ∈ N is -power-law-bounded.Proof. We consider the configuration model with weight sequence ( w ( n ) , . . . , w n ( n )) andvertices v , . . . , v n . Let E ⊆ (cid:0) { v ,...,v n } (cid:1) . By Definition 2.1, it suffices to show that for every ε > (cid:2) E ⊆ E ( G n ) (cid:3) ≤ O ( n ε ) | E | Y v i v j ∈ E √ ij . We can assume | E | ≤ n / , since for | E | > n / and every ε > E ⊆ E ( G n )] ≤ O ( n ε ) | E | Y v i v j ∈ E n . As described in [71, Lemma 7.6], the perfect matching of the stubs in the configurationmodel can also be generated by a so-called adaptive pairing scheme , where unmatched stubsare taken one-by-one and matched uniformly to the remaining unmatched stubs. Assume atmost l stubs have been matched already in such a scheme. We fix i, j ∈ N with i, j ≤ n and i = j . The probability that a fixed stub of v i is matched with some stub of v j is at most w j ( n ) / (cid:0) ( P nk =1 w k ( n )) − − l (cid:1) . By applying the union bound to a pairing scheme whichmatches the w i ( n ) many stubs of v i we obtainPr[ v i v j ∈ E ( G n )] ≤ w i ( n ) w j ( n )( P nk =1 w k ( n )) − − l . Let d be the maximum of w ( n ) , . . . , w n ( n ). We consider an adaptive pairing scheme whichiteratively matches the stubs of the vertices in E and obtainPr[ E ⊆ E ( G n )] ≤ Y v i v j ∈ E w i ( n ) w j ( n )( P nk =1 w k ( n )) − − | E | d . Since d ≤ p ( n ) √ n , | E | ≤ n / and P nk =1 w k ( n ) ≥ n/p ( n ) we can further bound w i ( n ) w j ( n )( P nk =1 w k ( n )) − − | E | d = O ( p ( n )) w i ( n ) w j ( n ) n = O ( p ( n ) ) 1 √ ij . The final result follows from the fact that p ( n ) = O ( n ε ) for all ε > orollary 10.7. Let ( G n ) n ∈ N be a random configuration graph model with correspondingsequence ( w i ( n )) i ∈ N . Assume there exists a function p ( n ) with p ( n ) = O ( n ε ) for all ε > such that for all i, n ∈ N , w i ( n ) ≤ p ( n ) p n/i and P nk =1 w k ( n ) ≥ n/p ( n ) .Then there exists a function f such that one can decide p - MC(FO , G lb ) on ( G n ) n ∈ N inexpected time f ( | ϕ | , ε ) n ε for every ε > . One of the earliest and most intensively studied random graphs is the Erd˝os–R´enyi model [9,30]. We say G ( n, p ( n )) is a random graph with n vertices where each pair of vertices isconnected independently uniformly at random with probability p ( n ). Many properties ofErd˝os–R´enyi graphs are well studied, including but not limited to, threshold phenomena,the sizes of components, diameter, and length of paths [9]. We classify sparse Erd˝os–R´enyigraphs with respect to α -power-law-boundedness. Lemma 10.8.
Erd˝os–R´enyi graphs G ( n, p ( n )) are • α -power-law-bounded for every < α if p ( n ) = O (1 /n ) , • -power-law-bounded if p ( n ) = log( n ) O (1) /n , • α -power-law-bounded for every < α < if p ( n ) = O ( n ε /n ) for every ε > .Proof. The probability of a set of edges E to exist in G ( n, p ( n )) isPr[ E ⊆ E ( G ( n, p ( n ))] = p ( n ) | E | ≤ (cid:0) np ( n ) (cid:1) | E | Y v i v j ∈ E ( n/i ) / ( α − ( n/j ) / ( α − n , since ( n/i ) / ( α − ≥ ≤ i ≤ n . The rest follows from Definition 2.1.Using the previous Lemma 10.8 and Theorem 8.6 we obtain a fine grained picture overthe tractability of the model-checking problem on sparse Erd˝os–R´enyi graphs. Corollary 10.9.
There exists a function f such that one can solve p - MC(FO , G lb ) on G ( n, p ( n )) in expected time • f ( | ϕ | ) n if p ( n ) = O (1 /n ) , • log( n ) f ( | ϕ | ) n if p ( n ) = log( n ) O (1) /n , • f ( | ϕ | , ε ) n ε for every ε > if p ( n ) = O ( n ε /n ) for every ε > . The third case has been shown previously by Grohe [43]. Furthermore, under reason-able assumptions (AW[ ∗ ] FPT/poly) we know that p -MC(FO , G lb ) cannot be decided inexpected FPT time on denser Erd˝os–R´enyi graphs with p ( n ) = n δ /n for some 0 < δ < δ ∈ Q [27]. α -power-law-bounded random graphs tend to capture unclustered random graphs. One canshow that for the algorithmically tractable values of α close to or larger than three theexpected number of triangles is subpolynomial (via union bound over all embeddings as inlemma 5.8). Random models with non-vanishing clustering coefficient, such as the Kleinbergmodel [49, 50], the hyperbolic random graph model [52, 11], or the random intersection graphmodel [46, 65] generally have a high expected number of triangles. This means these modelsare not α -power-law-bounded for interesting values of α close to three (they may be forsmaller α ). We shall prove a stronger statement for the random intersection graph modelwhich is defined as follows. 39 efinition 10.10 (Random Intersection Graph Model, [32]) . Fix a positive constant δ .Let B be a random bipartite graph on parts of sizes n and ⌊ n δ ⌋ with each edge presentindependently with probability n − (1+ δ ) / . Let V (the vertices) denote the part of size n and A (the attributes) the part of size ⌊ n δ ⌋ . The associated random intersection graph G ( n, δ )is defined on the vertices V : two vertices are connected in G if they share (are in B bothadjacent to) at least one attribute in A .It has been shown that ( G ( n, δ )) n ∈ N has a.a.s. bounded expansion [32] if and only if δ > δ >
1, then one can solve p -MC(FO , G lb ) in expected time f ( | ϕ | ) n [32]. Wenow argue that intersection graphs nevertheless do not fit into our framework of α -power-law-boundedness. Lemma 10.11. ( G ( n, δ )) n ∈ N is not α -power-law-bounded for all values of δ and α .Proof. Assume δ and α such that ( G ( n, δ )) n ∈ N is α -power-law-bounded. For a fixed n letthe vertices of G ( n, δ ) be v , . . . , v n , ordered as in Definition 2.1.If a fixed set of k vertices shares a common attribute then these vertices form a clique.The probability that this happens is at least n − k (1+ δ ) / ≥ n − ck for some constant c . Let E = (cid:0) { v n − k ,...,v n } (cid:1) be the set of all edges between the last k vertices in the ordering. By theprevious argument, Pr (cid:2) E ⊆ E ( G ( n, δ )) (cid:3) ≥ n − ck . By Definition 2.1 there exists a term p ( n ) with p ( n ) = O ( n ε ) | E | for every ε > (cid:2) E ⊆ E ( G ( n, δ )) (cid:3) ≤ p ( n ) Y v i v j ∈ E ( n/i ) / ( α − ( n/j ) / ( α − n . By the definition of p ( n ), there exists a monotone function f such that p ( n ) ≤ ( f (1 /ε ) n ε ) | E | for every ε >
0. By setting ε = 1 / | E | , we obtain p ( n ) ≤ f ( | E | ) | E | n . We consider onlyedges between the last k vertices, and for n ≥ k holds ( n/ ( n − k )) / ( α − ( n/ ( n − k )) / ( α − ≤
4. By assuming n ≥ k we obtainPr (cid:2) E ⊆ E ( G ( n, δ )) (cid:3) ≤ f ( | E | ) | E | n Y v i v j ∈ E n ≤ k f ( k ) k n − ( k ) +1 . Together, this yields n − ck ≤ Pr (cid:2) E ⊆ E ( G ( n, δ )) (cid:3) ≤ k f ( k ) k n − ( k ) +1 . We choose k large enough such that ck < (cid:0) k (cid:1) −
1. Then the previous bound yields acontradiction for sufficiently large n .
11 Conclusion
We define α -power-law-bounded random graphs which generalize many unclustered randomgraphs models. We provide a structural decomposition of neighborhoods of these graphs anduse it to obtain a meta-algorithm for deciding first-order properties in the the preferentialattachment-, Erd˝os–R´enyi-, Chung–Lu- and configuration random graph model.There are various factors to consider when evaluating the practical implications of thisresult. The degree distribution of most real world networks is similar to a power-law distribu-tion with exponent between two and three [15], but our algorithm is only fast for exponentsat least three. This leaves many real world networks where our algorithm is slow. However,it has been shown that the model-checking problem (with labels) becomes hard on thesegraphs if we assume independently distributed edges [27].40o far, we do not know whether the model-checking problem is hard or tractable on clus-tered random graphs. If a random graph model is 3-power-law-bounded then one can showthat the expected number of triangles is polylogarithmic (via union bound of all possibleembeddings of a triangle). Therefore, random models with clustering, such as the Kleinbergmodel [49], the hyperbolic random graph model [52, 11], or the random intersection graphmodel [46], which have a high number of triangles currently do not fit into our framework(see Section 10.5 for a proof that random intersection graphs are not α -power-law-boundedfor any α ). This is unfortunate, since clustering is a key aspect of real networks [72]. Inthe future, we hope to extend our results to clustered random graph models. We observethat some clustered random graph models can be expressed as first-order transductions of α -power-law-bounded random graph models. For example the random intersection graphmodel is a transduction of a sparse Erd˝os–R´enyi graph. We believe this connection can beused to transfer tractability results to clustered random graphs. If we can efficiently com-pute for a clustered random graph model G a pre-image of a transduction that is distributedlike an α -power-law-bounded random graph then we can efficiently solve p -MC(FO , G lb )on G . The same idea is currently being considered for solving the model checking problemfor transductions of sparse graph classes (e.g. structurally bounded expansion classes) [39].In our algorithm, we use Gaifman’s locality theorem to reduce our problem to r -neighbor-hoods of the input graph. In this construction the value of r can be exponential in the lengthof the formula [38]. On the other hand, the small world property states that the radius ofreal networks is rather small. This means, even for short formulas our neighborhood-basedapproach may practically be working on the whole graph instead of neighborhoods. It wouldbe interesting to analyze for which values of r practical protrusion decompositions accordingto Theorem 9.5 exist in the real world.At last, a big problem with all parameterized model-checking algorithms is their largerun time dependence on the length of the formula. Grohe and Frick showed that already ontrees every first-order model-checking algorithm takes worst-case time at least f ( | ϕ | ) n where f is a non-elementary tower function [37]. So far, it is unclear whether this also holds in theaverage-case setting. The results presented in this paper have a non-elementary dependenceon the length of the formula. We are curious whether one can find average-case model-checking algorithms with elementary expected FPT run time. In summary, many moreobstacles need to be to overcome to obtain a truly practical general purpose meta-algorithmfor complex networks. References [1] R´eka Albert, Hawoong Jeong, and Albert-L´aszl´o Barab´asi. Internet: Diameter of theworld-wide web.
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