The Power of D-hops in Matching Power-Law Graphs
aa r X i v : . [ c s . D S ] F e b The Power of D -hops in Matching Power-Law Graphs Liren Yu, Jiaming Xu, and Xiaojun Lin ∗ February 26, 2021
Abstract
This paper studies seeded graph matching for power-law graphs. Assume that two edge-correlated graphs are independently edge-sampled from a common parent graph with a power-law degree distribution. A set of correctly matched vertex-pairs is chosen at random and revealedas initial seeds. Our goal is to use the seeds to recover the remaining latent vertex correspondencebetween the two graphs. Departing from the existing approaches that focus on the use ofhigh-degree seeds in 1-hop neighborhoods, we develop an efficient algorithm that exploits thelow-degree seeds in suitably-defined D -hop neighborhoods. Specifically, we first match a set ofvertex-pairs with appropriate degrees (which we refer to as the first slice) based on the number oflow-degree seeds in their D -hop neighborhoods. This significantly reduces the number of initialseeds needed to trigger a cascading process to match the rest of graphs. Under the Chung-Lurandom graph model with n vertices, max degree Θ( √ n ), and the power-law exponent 2 < β < D > − β − β , by optimally choosing the first slice, with high probabilityour algorithm can correctly match a constant fraction of the true pairs without any error,provided with only Ω((log n ) − β ) initial seeds. Our result achieves an exponential reduction inthe seed size requirement, as the best previously known result requires n / ǫ seeds (for anysmall constant ǫ > Given two edge-correlated graphs, graph matching aims to find a bijective mapping between theirvertex sets so that their edge sets are maximally aligned. It is a fundamental problem with numer-ous applications in a variety of fields, including social network de-anonymization [NS09], machinelearning [CSS07, FSV + [NS08, NS09]. Using seeds, we can then measure thesimilarity of a vertex-pair by its “witnesses”. More precisely, let G and G denote two graphs. Foreach pair of vertices ( u, v ) with u in G and v in G , a seed ( w, w ′ ) is called a for( u, v ) if w is a neighbor of u in G and w ′ is a neighbor of v in G . Since G and G are graphs with ∗ L. Yu and X. Lin are with School of Electrical and Computer Engineering, Purdue University, West Lafayette,USA, [email protected], [email protected] . J. Xu is with The Fuqua School of Business, Duke University,Durham, USA, [email protected] . L. Yu and J. Xu are supported by the NSF Grant IIS-1932630. For example, in social network de-anonymization, some users provide identifiable information in their serviceregistrations or explicitly link their accounts across different social networks. k decays as k − β +1 for some exponent β > . As aconsequence, we expect to see very large degree fluctuations, with some nodes having very highdegrees (so-called hubs) and some other sparsely-connected nodes with small degrees. Intuitively,this degree fluctuation may confuse witness-based vertex matching, e.g., a fake pair with highdegrees may have many more witnesses than a true pair with low degrees, which foils the existingseeded algorithms designed for matching Erd˝os-R´enyi graphs.There have been several attempts to design seeded graph matching algorithms for power-lawgraphs [KL14, CGL16, BFK18]. However, they tend to require a larger number of seeds thanErd˝os-R´enyi graphs. Note that to address the above-mentioned degree variations, a common ideais to first partition graphs into slices consisting of vertex-pairs with similar degrees. Then, thevertices are matched slice-by-slice, starting from the highest-degree slice to lower-degree slices. Acascade process is triggered, in the sense that the matched vertices in the current slice is used asnew seeds to match the next slice. Intuitively, it is critical to correctly match the first slice in orderto successfully trigger the cascading matching process for the later slices. [KL14, CGL16, BFK18]all use this idea and match the first slice based on 1-hop witnesses. Unfortunately, they also requirea large number of correct seeds to match the first slice successfully. Specifically, [KL14] assumespreferential-attachment graphs with n vertices [BA99] and their algorithm requires Ω( n/ log( n ))seeds to match a constant fraction of all vertices correctly. [CGL16, BFK18] instead assume theChung-Lu graph model [CCG +
06] (cf. Section 2). When all seeds are chosen from the high-degreevertices, [CGL16, BFK18] show that their algorithm require only n ǫ seeds to correctly match aconstant fraction of the vertices. However, if the seeds are chosen uniformly from all vertices,the number of high-degree seeds will be much smaller than n ǫ . In that case, the degree-drivengraph matching (DDM) algorithm in [CGL16] requires n / ǫ seeds to match a constant fractionof vertices correctly.In this paper, we propose a new algorithm for matching power-law graphs that only requiresΩ((log n ) − β ) initial seeds chosen randomly, to correctly match a provably constant fraction of allvertices. Our key departure from [KL14, CGL16, BFK18] is to use “witnesses” in larger D -hopneighborhoods. More precisely, a seed ( w, w ′ ) is a D -hop witness for ( u, v ) if w is a D -hop neighborof u in G and w ′ is a D -hop neighbor of v in G . To see why using D -hop witnesses is crucial, notethat, under the Chung-Lu model of [CCG +
06] (cf. Section 2), even the highest degree vertices onlyhave a 1-hop neighborhood of size at most O ( √ n ). Since seeds are uniformly chosen, it is clear thatat least Ω( √ n ) seeds are needed to ensure that a true pair in the first slice can have Ω(1) 1-hopwitnesses. In contrast, as D increases, the size of the D -hop neighborhoods grows rapidly, and thusthere are substantially more seeds that can serve as D -hop witnesses for true pairs, which provideshope to significantly reduce the number of initial seeds.The idea of D -hop witnesses has also been used for matching Erd˝os-R´enyi graphs in [MX19,YXL21]. However, as can be seen in the rest of the paper, the application of D -hop witnesses topower-law graphs is highly non-trivial. Specifically, due to the power-law degree variations, the D -hop neighborhoods of some high-degree vertices may become so large that even a fake pair canhave many D -hop witnesses. Therefore, a key challenge is to properly control the size of the D -hopneighborhoods. This size depends not only on the degrees of the vertex pair to be matched, but2lso that of the intermediate nodes (to reach D -hop) and that of the seeds. To overcome thischallenge, our algorithm design (to be explained in Section 3) (i) carefully chooses the first slice ofvertices to be matched. (ii) carefully chooses the intermediate vertices when constructing the D -hopneighborhoods, and (iii) carefully avoids high-degree seeds in order to eliminate the confusion forfake pairs. These three ideas altogether ensure that the true pairs in the first slice have many more D -hop witnesses than the fake pairs, and thus can be correctly matched to trigger the cascadingprocess to match the rest of the graphs. See Section 3 for more detailed discussions.To fully realize the power of D -hops, we further need to carefully construct overlapping slicesto account for the potential mismatch in the vertex slicing of graphs G and G , and to designeffective ways to match the remaining slices. Assembling all these pieces together enables us toachieve an exponential reduction in the required number of seeds compared to state-of-art resultsin [CGL16]. Specifically, under the Chung-Lu model with power-law exponent 2 < β < √ n ), we prove the following performance guarantee of our algorithm, stated informallyhere and formally in Section 5. Theorem 1 (Summary of main result) . Suppose
D > − β − β . If there are Ω((log n ) − β ) initialseeds chosen independently at random, by optimally choosing the first slice, our algorithm correctlymatches Ω( n ) vertex-pairs without any error with high probability. This reduces the seed size requirement exponentially, as the best previously known result [CGL16]requires n / ǫ seeds. To prove Theorem 1, there are several key innovations in our analysis in par-ticular to address the difficult dependency issues across edges and slices. First, note that when wedefine the D -hop neighborhoods, we use vertex degrees to construct the slices and to select the seedsand intermediate nodes. This degree-based slicing unfortunately brings dependency issues. In par-ticular, if we condition on the vertex degrees, then the edges are no longer independently generatedaccording to the Chung-Lu model. To circumvent this dependency issue, we first show that thedegree-guided construction and selection can be closely approximated by the weight-guided coun-terparts with high probability. Then we restore the independence by studying the weight-guidedconstruction and selection, since the edges are independently generated according to the Chung-Lumodel given the weights. Second, as we use the matched pairs in the current slice as new seeds tomatch the next slice, the matching results are correlated across different slices. To deal with thesecorrelations, we carefully construct sets of matched pairs that only depend on vertex weights to“sandwich” the original set of matched pairs at each slice, but are not correlated any more, whichallows us to eliminate the slice-dependency issue. Last but not least, to derive the optimal choiceof the first slice and attain the smallest seed size requirement, we tightly bound the sizes of thecommon D -hop neighborhoods for both true pairs and fake pairs. Compared to the Erd˝os-R´enyigraphs, this requires much more sophisticated lines of analysis of the neighborhood explorationprocess in the power-law graphs due to the heterogeneous vertex weights.In the literature, the idea of D -hop witnesses has been used in Erd˝os-R´enyi graphs [MX19,YXL21]. However, there is a significant difference in our results for power-law graphs. Specifically,in the Erd˝os-R´enyi graphs with average degree d , the sizes of the D -hop neighborhoods are highlyconcentrated on d D . Moreover, when the average degree d is a constant, the size of D -hop neighbor-hoods is always O (1) for any constant D . Thus, unless D increases with n , at least Ω( n ) seeds arestill needed to ensure that there are enough D -hop witnesses for true pairs. In stark contrast, thepower of the D -hop becomes much more significant for matching power-law graphs. In particular,for power-law graphs with constant average degrees, by properly using the D -hop witnesses, wedramatically reduce the seed requirement to Ω((log n ) − β ), as soon as D exceeds − β − β . Further,we note that the algorithms in [MX19, YXL21] do not need to worry about controlling the D -hopneighborhood, as they do not face the challenge of power-law degree variations.3inally, we conduct extensive experiments on both synthetic and real power-law graphs tocorroborate our theoretical analysis. In particular, we compare our algorithm with five other state-of-the-art seeded graph matching algorithms. Numerical results demonstrate that our algorithmdrastically boosts the matching accuracy and requires substantially fewer seeds to correctly matcha large fraction of vertices. Further, although our analysis focuses on matching two graphs of thesame number of vertices, our algorithm can be readily applied to match two graphs of differentsizes and return an accurate matching between vertices in the common subgraph of the two graphs.Indeed, our experiments on real networks in Section 6.3.2 and Section 6.3.3 show that our algorithmstill achieves outstanding matching performance, even when two graphs are of very different sizes. Following [CGL16, BFK18], we adopt the Chung-Lu random graph model [CCG +
06] to generate theunderlying parent graph with a power-law degree distribution. Here, [ n ] denotes the set { , , ..., n } . Definition 1.
Given parameters w > , w ≪ w max ≤ √ nw, and β >
2, the Chung-Lu graph isa random graph G ([ n ] , E ) generated as follows. Each vertex i ∈ [ n ] is associated with a positiveweight w i = w β − β − (cid:16) ni + i (cid:17) β − , where i = n (cid:16) w ( β − w max ( β − (cid:17) β − . For any pair of two vertices i, j ∈ [ n ]with i = j , they are connected independently by an edge with probability p ij = w i w j nw .Note that i is chosen such that w = w max , which is the largest weight among all vertices.Further, w approximates the average weight as follows. Since w ≪ w max , it follows that i ≪ n .It can then be verified that n P ni =1 w i → w and n P ni =1 { w i ≥ w } ∝ w − β +1 as n → ∞ . Thus,the degree of vertex i is expected to be close to w i , which admits a power-law distribution withexponent β. The Chung-Lu model is convenient for modelling the degree variations in real-world networks.In these real-world networks, while the average degree is often a constant, a small but non-negligiblefraction of the vertices has very large degrees (the so-called hubs) [B + w = Θ(1) and 2 < β <
3. Empirical studies have shownthat the vertex degrees of many real-world networks indeed follow a power-law distribution with2 < β < +
16, CSN09, New03]. Note that if 0 < β ≤
2, the average degree diverges and thenetwork cannot be sparse; if β ≥
3, the degree variance is bounded and no large hub can appear[B + G by sampling each edge of G into G independently withprobability s , which is a constant independent of n . To construct another subgraph G , repeatthe same sub-sampling process independently and relabel the vertices according to an unknown permutation π : [ n ] → [ n ]. Throughout the paper, we denote a vertex-pair by ( u, v ), where u ∈ G and v ∈ G . For each vertex-pair ( u, v ), if v = π ( u ), then ( u, v ) is a true pair; if v = π ( u ), then( u, v ) is a fake pair.Finally, there is an initial seed set S consisting of true pairs. Each true pair is added into S with probability θ independently. Our goal is to recover π based on the observation of G , G and S . To see the first part of the statement, let f ( x ) = w x /n . Then R n +11 f ( x ) dx ≤ n P ni =1 w i ≤ f ( n ) + R n f ( x ) dx .Moreover, R n f ( x ) dx = wn − ββ − (cid:18) ( n + i + 1) β − β − − ( i + 1) β − β − (cid:19) → w, in view of i ≪ n due to w max ≫ w . Further,we can verify the second part of the statement by n P ni =1 { w i ≥ w } = (cid:16) ( β − w ( β − w (cid:17) β − − i n → (cid:16) ( β − w ( β − w (cid:17) β − . otation We use standard asymptotic notation: for two positive sequences { a n } and { b n } , wewrite a n = O ( b n ) or a n . b n , if a n ≤ Cb n for some an absolute constant C and for all n ; a n = Ω( b n )or a n & b n , if b n = O ( a n ); a n = Θ( b n ) or a n ≍ b n , if a n = O ( b n ) and a n = Ω( b n ); a n = o ( b n ) or b n = ω ( a n ), if a n /b n → n → ∞ . As we discussed in Section 1, previous graph matching algorithms that use 1-hop witnesses tomatch power-law graphs [KL14, CGL16, BFK18] require at least n / ǫ seeds if the seeds arechosen uniformly from all vertices. In order to significantly reduce the number of seeds, it is thencrucial to use D -hop witnesses. However, for power-law graphs, the use of D -hop witnesses ishighly non-trivial because, as the size of the D -hop neighborhood increases, fake pairs may alsohave many D -hop witnesses and thus could be confused as true pairs. Therefore, it is importantto carefully control the D -hop neighborhood. In this section, we elaborate on our three designchoices to properly control the D -hop neighborhood sizes: the weight of the seeds, the weight ofthe candidate vertex-pairs, and the weight of the intermediate vertices.First, it is important to utilize low-weight seeds while avoiding high-weight seeds. Due to thepower-law degree distribution, when seeds are uniformly chosen, there are many more low-weightseeds than high-weight seeds. Thus, the D -hop neighborhoods need to be large enough to reachsufficiently many low-weight seeds. However, for fake pairs, their large D -hop neighborhoods mayalso overlap. This implies that high-weight seeds may easily become witnesses for fake pairs, whichcan appear in many D -hop neighborhoods. Therefore, in order to avoid having too many witnessesfor fake pairs, it is important to eliminate the high-weight seeds.Second, for a given D , we need to carefully choose the first slice of candidate vertex-pairs tobe matched using the D -hop witnesses. Recall that [KL14, CGL16, BFK18] also use this idea ofslicing the vertices according to their degrees, and focus on matching the first slice with vertices ofhigh degree. However, we find that the degree range of this first slice needs to be carefully chosen.On the one hand, if the weight of the candidate vertex-pairs is too small, the common D -hopneighborhoods of a true pair are too small to produce enough witnesses. On the other hand, if theweight of the candidate vertex-pairs is too large, the D -hop neighborhoods of a fake pair wouldintersect a lot, leading to too many D -hop witnesses.Third, the high-weight vertices are not suitable to be the intermediate vertices in D -hop neigh-borhoods when D is large. This is because, when D is large, there exist some high-weight verticeswith very large d -hop ( d < D ) neighborhoods. If these high-weight vertices become ( D − d )-hopneighbors of the candidate vertices, the D -hop neighborhoods of the fake pairs would become toolarge. Thus, we should avoid using the high-weight vertices as the intermediate vertices.Prompted by the above three ideas, we partition the graph into “perfect” slices P k = { u : w u ∈ [ α k , α k − ] } where α k = n γ / k for k ≥ , and α − = ∞ , (1)for some γ ∈ (0 , log n w max ]. In particular, the first slice P is the set of vertices with weight in[ n γ / , n γ ], which is the first set of the vertices that we wish to match. We will show in (8) thatfor a vertex in the first slice P , its number of Θ(1)-weight D -hop neighbors is on the order of n γ ((3 − β )( D − . Hence, we optimally choose γ close to − β )( D − so that its number of Θ(1)-weight D -hop neighbors is close to Θ( n ) . Under this optimal choice, we prove that sufficiently manyvertex-pairs in the first slice are correctly matched so that they can be used as new seeds to triggerthe cascading process to match the rest of the graphs slice-by-slice. In fact, for slice k ≥ k = k ∗ for some k ∗ , since the earlier slices provide so many new seeds, it turns out that using5-hop witnesses suffices. When k > k ∗ , the slice-by-slice matching process stops, as there are notenough 1-hop witnesses to correctly match the slices with low-weight vertices. Fortunately, for thefake pairs with such low-weights, there are very few 1-hop witnesses as well. Thus we treat allthe low-weight vertices as a single slice and apply the PGM algorithm in [KHG15] to match them.Finally, we use all the matched vertex-pairs as new seeds to match the zero slice P with very highweights.For the above ideas to work, however, it is important that the earlier slices do not producewrong matches; otherwise, the wrong matches will propagate errors to the subsequent slices. Assuch, we only match pairs with the number of witnesses larger than a threshold, as we will see nextin the detailed algorithm. D -hop (PLD) Algorithm In this section, we present our Power-Law D-hop (PLD) algorithm, shown in Algorithm 1 andprovide the intuition why it works.
We first introduce some notations regarding D -hop neighborhoods. Given any graph G and twovertices u, v in G , we denote the length of the shortest path from u to v in G by dist G ( u, v ). Foreach vertex u ∈ G , the d -hop neighbors of u is denoted by Γ Gd ( u ) = { v ∈ G : dist G ( u, v ) = d } . Theneighbors within d -hop of u is denoted by N Gd ( u ) = S dj =1 Γ Gj ( u ).Our PLD algorithm carefully incorporates the key algorithmic ideas described in Section 3.At a high-level, we first slice the vertices according to their degrees. We then apply the D -hopalgorithm to the first slice (which is carefully chosen). Afterwards, we apply the 1-hop algorithmto the lower-degree slices 2 to k ∗ , until the vertex degrees are about poly-logarithmic in n , in whichcase we apply the PGM algorithm to the last slice with the lowest-degree vertices. Finally, wereturn to slice 0 of vertices with very high degrees.The full algorithm is presented in Algorithm 1. We now describe the details.In line 2, we construct a subset of low-weight seeds to use a future witnesses. Recall fromSection 3 that we aim to utilize low-weight seeds while avoiding high-weight seeds. Specifically, inour algorithm we wish to use seeds with Θ(1) weights. However, since we do not have access tothe vertex weights directly, we have to estimate vertex weights by vertex degrees. Therefore, weconstruct a seed subset b S that contains seeds with degrees no larger than 5 log n to ensure that allseeds with Θ(1) weights are included.In line 3, we eliminate the vertices with degrees larger than (1 + δ ) n γ and their adjacent edges,because we do not want to use the high-weight vertices as the intermediate vertices.In line 4, we partition the graphs G and G into slices. Recall that the “perfect” slices P k in(1) described in Section 3 are defined with the vertex weights. Again, since we can not observethe vertex weight directly, we need to use the vertex degree as an estimate of the vertex weight.However, using vertex degree to slice vertices creates new technical difficulties. Specifically, fortwo vertices corresponding to a true pair, their actual degrees in G and G may differ becausethe edges are sub-sampled from the parent graph randomly. As a result, it is possible that thesetwo vertices are assigned to two slices of different indices in G and G . This becomes problematicbecause, if we only match slices with the same index, such a true pair would never be matched.(This problem does not occur for the “perfect” slices since they are based on the weight of thevertex in the original parent graph.) Fortunately, the actual degrees of the vertices corresponding6 lgorithm 1
The Power-Law D-hop (PLD) Algorithm. Input:
Graphs G and G , initial seed set S , parameters D, γ, τ , τ , k ∗ Construct a subset of low-degree seeds b S = n ( u, v ) ∈ S : (cid:12)(cid:12)(cid:12) Γ G ( u ) (cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12) Γ G ( v ) (cid:12)(cid:12)(cid:12) ≤ n o . Let b G i denote the subgraph of G i induced by the vertex set V i = n u : (cid:12)(cid:12)(cid:12) Γ G i ( u ) (cid:12)(cid:12)(cid:12) ≤ (1 + δ ) n γ s o for i = 1 , . Partition the graph G i into slices b P G i k for i = 1 , ≤ k ≤ k ∗ , according to (2). In b G and b G , for candidate vertex-pairs in b Q , count their D -hop witnesses in b S and useGMWM to match pairs with more than τ D -hop witnesses ( τ is given in (3)). The set ofmatched pairs is R . for k = 2 to k ∗ do For candidate vertex-pairs in b Q k , count their 1-hop witnesses in R k − and use GMWM tomatch pairs with more than τ ( k ) 1-hop witnesses ( τ ( k ) is given in (5)). The set of matchedpairs is R k . end for Let G ′ i denote the subgraph of G i induced by the vertex set V ′ i = n u : (cid:12)(cid:12)(cid:12) Γ G i ( u ) (cid:12)(cid:12)(cid:12) ≤ (1 + δ ) α k ∗ − s o , for i = 1 , . Apply
PGM to G ′ and G ′ , with the seed set R k ∗ and the threshold r = 3. The set of matchedpairs is denoted by R k ∗ +1 . For candidate vertex-pairs in b Q , count their 1-hop witnesses in b R , S k ∗ +1 k =1 R k and match pairswith GMWM. The set of matched pairs is R . Output:
All matched pairs R = b R ∪ R ∪ S to a true pair should not differ too much (assuming a common sub-sampling probability s for bothgraphs). Thus, to address the above difficulty, we enlarge the slices a little bit, so that with highprobability the two vertices corresponding to a true pair can fall into slices with the same index, andtherefore have the opportunity to be matched. More precisely, for k ≥
0, we define the imperfectslice as b P Gk = (cid:8) u : (1 − δ ) α k s ≤ (cid:12)(cid:12) Γ G ( u ) (cid:12)(cid:12) ≤ (1 + δ ) α k − s (cid:9) , for k ≥ , (2)where δ = throughout this paper. Here, α k are the same as (1), and the parameters γ and D willbe set to satisfy (10) in Theorem 2. The imperfect slice-pair is then defined as b Q k = b P G k × b P G k = { ( u, v ) : u ∈ b P G k , v ∈ b P G k } . However, these enlarged imperfect slices also create a new problem ofmatching fake pairs, which will be discussed next.In line 5, we count the D -hop witnesses for all vertex-pairs in the first slices b P G and b P G , andthen use Greedy Maximum Weight Matching (GMWM) [Avi83] to find the vertex correspondencesuch that the total number of witnesses is large. Here, we note that our earlier idea of enlarging theimperfect slices b P k creates a new problem. That is, the imperfect slices with neighboring indicesnow have some overlap. As a result, it is possible that a slice pair contains a fake pair ( u, π ( v )),but does not contain the true pairs ( u, π ( u )) and ( v, π ( v )) . When that happens, the fake pairs( u, π ( v )) may have the most witnesses among all the candidate vertex-pairs containing either u or π ( v ). Thus, the fake pair ( u, π ( v )) may be matched by GMWM. Fortunately, the number ofwitnesses of these fake pairs is still expected to be smaller than that of any true pair. Therefore, This phenomenon does not contradict the idea of enlarging the slices. Enlarging the slices only guarantees thetrue pairs ( u, π ( u )) and ( v, π ( v )) are assigned into some slice-pairs. However, for other slice-pairs that contain thefake pair ( u, π ( v )), it is still possible that the two true pairs are not included.
7o resolve this difficulty and to ensure that only the true pairs are matched, for the first slice wematch only the vertex-pairs with no less than τ D -hop witnesses, where τ is set to be a constantfraction of the expected number of the D -hop witnesses for true pairs, i.e., τ = 310 (cid:18) Cs w (cid:19) D n γ ((3 − β )( D − θ, (3)where C , (2 β − − (cid:16) ( β − w ( β − (cid:17) β − . Similar thresholds are also used in the following steps whenwe match other slices.In line 6-8, we use the matched pairs from the previous slice as new seeds, and use the 1-hopalgorithm to match the vertices in slices k = 2 , ..., k ∗ , where k ∗ = $ log n γ (cid:18) Cs w log n (cid:19) − β !% . (4)In other words, we match the vertices with degrees larger than (1 − δ ) α k ∗ , where α k ∗ ≥ (cid:16) w log nCs (cid:17) − β .Again to ensure that only the true pairs are matched for each slice, we only match the vertex-pairswith at least τ ( k ) 1-hop witnesses, where τ ( k ) is set to be half of the expected number of the1-hop witnesses of the true pairs, i.e., τ ( k ) = Cα − βk − s w . (5)In line 9-10, we apply the PGM algorithm [YG13, Section 3] to match the remaining verticeswith degrees no larger than (1 + δ ) α k ∗ . Note that when the vertex weight is this small, estimatingthe vertex weight based on its degree is not accurate anymore. Thus, it is difficult to use thevertex degree to distinguish which slices should these vertices fall into. Instead, we treat all ofthese low-weight vertices as one slice. Further, for such low-degree vertices, using 1-hop algorithmbased on the seeds from earlier slices will lead to poor performance, because even the true pairsin this slice have too few 1-hop witnesses. Fortunately, there are even fewer witnesses for the fakepairs with such low degrees. Thus, we can use the PGM algorithm, which iteratively generates newseeds as new correct matches are found. In this way, PGM can match a constant fraction of therest of vertex-pairs, while avoiding matching fake pairs.Finally, in line 11, the algorithm uses all vertex-pairs matched above as new seeds and matchesthe vertices in b Q via the 1-hop algorithm.The total complexity of our algorithm is O ( n − γ ( β − ). The proof can be found in AppendixA. Before we present the main results, we explain the intuition why the above algorithm will work onlywith Ω((log n ) − β ) seeds. For the purpose of explaining this intuition, we ignore the inaccuracy ofestimating the weights by the vertex degrees and assume that the graphs can be partitioned intoperfect slices P k . We further assume that the true mapping π is the identity permutation. Also,when we write ≈ , we ignore the constant factors that are non-essential.The key to the success of Algorithm 1 is appropriately choosing the first slice to apply the D -hop algorithm. We first calculate the probability that a vertex of Θ(1) weight lies in the D -hopneighborhood of a vertex in the first slice. Specifically, given a vertex u in the first slice P and8nother vertex v of weight 1, we want to compute the probability q D that v is a D -hop neighborof u , i.e., q D , P (cid:26) v ∈ Γ b G j D ( u ) (cid:27) , where j is either 1 or 2. Note that if v is a D -hop neighbor of u , then v is connected to some ( D − i of u . Therefore, q D satisfies the followingrecursion: q D ≈ X i ∈ b G j P (cid:26) v ∈ Γ b G j ( i ) (cid:27) × P (cid:26) i ∈ Γ b G j D − ( u ) (cid:27) ( a ) ≈ c Z n γ nw − β · wnw · wq D − dw = c q D − w Z n γ w − β dw = cn γ (3 − β ) w (3 − β ) q D − . (6)In step ( a ), we integrate over the degree w of the ( D − i . Thus, P (cid:26) i ∈ Γ b G j D − ( u ) (cid:27) is wP D − by our definition. Further, w/n ¯ w is the probability that v (with degree 1) is connected to i ,and number of such vertices i with degree in [ w, w + dw ] is about P ni =1 { w ≤ w i ≤ w + dw } → cnw − β dw with c = (cid:16) ( β − w ( β − (cid:17) β − ( β − . By the Chung-Lu model, q ≈ n γ nw . Iterating (6) over D , it followsthat q D ≈ c n γ (3 − β ) w (3 − β ) ! D − q ≈ c D − n γ ((3 − β )( D − nw D (3 − β ) D − . (7)As explained in Section 3, for the success of the D -hop algorithm, there are two key considerations.On the one hand, we need to ensure that the fake pairs in Q , P × P have very few D -hopwitnesses. As such, we want to prevent the fake pairs in Q from having too many common neighborsof small weight. Therefore, we require q D ≪ n γ ((3 − β )( D − ≪ n and is close to the condition (10) (stated later in Theorem 2). On the other hand, we need toensure that the true pairs in Q have sufficiently many Θ(1)-weight D -hop witnesses. Indeed, for u ∈ P , its number of common D -hop neighbors of Θ(1)-weight is at least (cid:12)(cid:12)(cid:12) { v : w v = Θ(1) } ∩ Γ b G ∧ b G D ( u ) (cid:12)(cid:12)(cid:12) ≈ nq D ≈ n γ ((3 − β )( D − , (8)where the first approximation holds because there are about Θ( n ) vertices with Θ(1) weight basedon the power-law weight distribution. Therefore, under condition (11) stated in Theorem 2, whichis roughly θ = Ω (cid:16) log nn γ ((3 − β )( D − (cid:17) , all the true pairs have at least Ω(log n ) low-degree D -hopwitnesses. The above choices thus ensure that all true pairs (but no fake pairs) are matched.Interestingly, after matching the first slice, it triggers a cascading process, where the newmatches at one slice can be used as new seeds to match the subsequent slice by the 1-hop al-gorithm. To see why using 1-hop witnesses is sufficient, recall that the weight of vertices in P k satisfies α k ≤ w i ≤ α k − ⇐⇒ n (cid:16) ( β − α k − ( β − w (cid:17) β − − i ≤ i ≤ n (cid:16) ( β − α k ( β − w (cid:17) β − − i . According to the index range of these vertices, we get that the number of vertices in P k isΘ (cid:16) nα − βk − (cid:17) . Since the vertices in P k and the vertices in P k +1 are connected independently with9robability at least α k α k +1 nw , it follows that, for a vertex in P k +1 , its number of 1-hop neighbors in P k is about nα − βk − × α k α k +1 nw = α − βk − α k α k +1 w ≥ α − βk w . (9)Note that for the 1-hop algorithm to succeed, the true pairs need to have more than log n < β <
3, we have α − βk w > log n , as long as α k > α k ∗ ≈ (log n ) − β .Therefore, assuming that the true pairs in Q k , P k × P k are correctly matched, we expect that the1-hop algorithm can correctly match the true pairs in Q k +1 as long as k < k ∗ . However, when k ≥ k ∗ , for a vertex in P k +1 , its number of 1-hop neighbors in P k becomessmaller than log n , and thus the 1-hop algorithm can no longer match the vertices in P k +1 correctly.Even worse, the vertex degrees become inaccurate to distinguish the vertices with at most poly-logarithmic weights, and hence the 1-hop algorithm can not even match the vertices slice by slice.As discussed in Section 3, we instead resort to the PGM algorithm to match a constant fraction ofthe rest of low-weight vertices. Note that the key to the success of the PGM is that the numberof witnesses for a fake pair is no more than 2 [KHG15]. To see why this condition holds for theremaining low-weight vertices, note that the probability that a low-weight seed (with weight nolarger than α k ∗ ) becomes a 1-hop witnesses for a fake pair with weight no larger than α k ∗ is atmost (cid:0) α k ∗ α k ∗ nw (cid:1) = α k ∗ n w . Since there are at most n seeds and the majority of them are low-weight,the number of witnesses for any fake pair with low-weights is about α k ∗ nw . (log n ) − β nw ≪
1. Thus,we can use the PGM algorithm with threshold r = 3 to match a constant fraction of the low-weightvertex-pairs without errors.Finally, the number of vertices with weight less than α is Θ( n ). If most true pairs with weightless than α are matched, we can use them as new seeds to exactly match the remaining vertex-pairsin Q . The following theorem provides a sufficient condition for our algorithm to correctly match a constantfraction of nodes without any errors. We define C , (2 β − − (cid:16) ( β − w ( β − (cid:17) β − and κ , (1+2 δ ) − β C (2 − β − w throughout this paper. Theorem 2.
Suppose γ > and the positive integer D are chosen such that γ ≤ log n w max , n γ = o ( n ) , and n γ ((3 − β )( D − ≤ Cs (2 − β − · − β (cid:18) Cs κ · w (cid:19) D n (log n ) − β . (10) If the fraction θ of seeds satisfies θ ≥
320 log n (cid:16) Cs · w (cid:17) D n γ ((3 − β )( D − , (11) then for all sufficiently large n , Algorithm 1 with τ in (3) and τ ( k ) in (5) outputs Θ( n ) true pairsand zero fake pairs with probability at least − n − o (1) . n γ ((3 − β )( D − is roughly the size of the D -hop neighborhood of a vertex(with weight around n γ ) in the first slice P . Therefore, on the one hand, (10) ensures that fortwo distinct vertices ( u, v ) in the first slice, the intersection of their D -hop neighborhoods is muchsmaller than the two neighborhoods, so that the fake pairs have much fewer D -hop witnesses thanthe true pairs. On the other hand, (11) ensures that the true pairs have at least Ω(log n ) D -hopwitnesses.Assuming w max = Θ( √ n ), if we set D = 1 and γ = − ǫ for a small constant ǫ >
0, thenTheorem 2 recovers the seed requirement n / ǫ for the 1-hop algorithm which is comparable tothe result in [CGL16]. Surprisingly, for larger D , if we optimally choose n γ in (13), then the seedrequirement can be dramatically reduced to Ω((log n ) − β ), as shown by the following corollary. Corollary 1 (The formal version of Theorem 1) . Suppose D ≥ − β (cid:18) log n log( w max ) − (cid:19) + 1 and D > − β − β . (12) Choose n γ ((3 − β )( D − = cn (log n ) − β , (13) for a sufficiently small constant c so that (10) is satisfied, and τ , τ ( k ) according to (3) and (5),respectively. If the fraction of seeds satisfies θ ≥ C (log n ) − β n for a sufficiently large constant C , then for all sufficiently large n , Algorithm 1 outputs Ω( n ) truepairs and zero fake pairs, with probability at least − n − . According to (13), we choose γ asymptotically equal to − β )( D − . Condition (12) is imposedto ensure that this choice satisfies γ < / γ ≤ log n ( w max ) in Theorem 2. Theorem 1 is aspecial case of Corollary 1, where w max = Θ( √ n ) so that (12) reduces to D > − β − β . In this section, we conduct numerical experiments to verify our theoretical findings and the effec-tiveness of the PLD algorithm. For all experimental results, we calculate the accuracy rate as themedian of the proportion of vertices that are correctly matched, taken over 10 independent runs. D and γ In this section, we simulate our PLD algorithm with different D and γ to investigate the impactof the two parameters. We generate the underlying parent graph G according to the Chung-Lumodel with n = 10000, β = 2 . w = 10. Then, we construct G and G by sampling each edgeof G twice independently with probability s = 0 .
8. The seeds are selected such that each true pairbecomes a seed with probability θ independently.In Fig. 1, we first plot the accuracy rates of our PLD algorithm with D = 3 and different γ , when θ varies from 0 to 0 .
01. We observe that for a given accuracy rate, when γ = 1 / [(3 − β )( D −
1) + 1],the PLD algorithm requires the smallest number of seeds. This is consistent with the theoreticalprediction in Corollary 1, i.e., the optimal choice of γ approaches 1 / [(3 − β )( D −
1) + 1] as n → ∞ .11 A cc u r a cy R a t e = 1/(3- )(D-1)= 1/((3- )(D-1)+1)= 1/((3- )(D-1)+2)= 1/((3- )(D-1)+3)= 1/((3- )(D-1)+4) Figure 1: The performance of the PLD algorithm with D = 3 and varying γ .Then, in Fig. 2, we plot the accuracy rates of our PLD algorithm with different choices of D byfixing γ = 1 / [(3 − β )( D −
1) + 1]. We can see that the curves for different D align well with eachother, showing that the PLD algorithm with different D requires a comparable number of seeds tosucceed when γ is optimally chosen, as suggested by Corollary 1. A cc u r a cy R a t e PLD D=2PLD D=3PLD D=4PLD D=5
Figure 2: The performance of the PLD algorithm with different D and γ = − β )( D − . For our experiments on synthetic data, we still use the graphs generated in Section 6.1 accordingto the Chung-Lu model. Then, our PLD algorithm is simulated and compared with the otherfive state-of-the-art seeded graph matching algorithms, namely DDM [CGL16], Y-test [BFK18],User-Matching [KL14], 2-hop [MX19] and PGM [KHG15] algorithms. For the PLD algorithm, weselect D = 2 , , γ = 1 / ((3 − β )( D −
1) + 1) as suggested in Corollary 1. In Fig. 3, we plot the12erformance comparison when θ varies from 0 to 0 .
03. We observe that our PLD algorithm withdifferent D achieves similar performance, and it significantly outperforms all the other algorithms.Specifically, our PLD algorithm only requires around 50 seeds to match almost all vertices, whilethe User-Matching algorithm requires at least 150 seeds, and the DDM requires at least 220 seeds.The other algorithms perform even worse. Note that roughly 5% of vertices have degree at most 1in both graphs; thus we do not expect to correctly match them. That is why the accuracy rates ofour PLD algorithm saturated around 95%.Note that the 2-hop and PGM algorithms have been known to work well for matching Erd˝os-R´enyi graphs [MX19, KHG15]. However, we see that they are brittle to the power-law degreevariations. The DDM, Y-test, and User-Matching algorithms perform slightly better. However,since they all rely on the 1-hop witnesses, they still require a large number of seeds to succeed. A cc u r a cy R a t e PLD D=2PLD D=3PLD D=4DDMY-testUser Matching2-hopPGM
Figure 3: Performance comparison of our PLD algorithm and five other algorithms on the Chung-Lumodel with different θ . We see that the performance of our PLD algorithm is outstanding on synthetic graphs. To furtherdemonstrate the power of D -hops, we investigate its performance in matching real graphs. However,our algorithm based on the Chung-Lu model requires several parameters, which are unknown forreal graphs. As such, in this section, we describe our method to estimate the key model parametersbefore implementing our algorithm.First and foremost, we estimate the power-law exponent of real graphs by fitting them to theChung-Lu model using the maximum-likelihood estimation given in [CSN09]: b β = 1 + N X d i ≥ d min ln (cid:18) d i d min − / (cid:19) − , (14)where d i is the degree of vertex i , N is the number of vertices with degree at least d min , and d min is some lower bound on the vertex degrees to be specified. It is suggested in [CSN09] to estimate13 min using the Kolmogorov-Smirnov approach, which minimizes the maximum distance betweenthe empirical CDF and the theoretical CDF of vertex degrees. More precisely, d min = arg min d max d i ≥ d (cid:12)(cid:12)(cid:12) b F d ( d i ) − F d ( d i ) (cid:12)(cid:12)(cid:12) , where b F d ( x ) is the CDF of the observed vertex degrees with values at least d , and F ( x ) is the CDFof the power-law vertex distribution restricted to [ d, + ∞ ). Numerical experiments in [CSN09] show b β is accurate to 1% or better if d min is set to be around 6. Thus, we fix d min = 6 throughout ourreal-data experiments.Next, we estimate the subsampling probability s , which characterizes the edge correlation be-tween the two observed graphs. Let G j [ S ] denote the subgraph of G j induced by vertices in S = { i : ( i, i ) ∈ S} , where S is the initial seed set. Note that under our subsampling model,given an edge in one graph, it appears in the other graph with probability s. Thus we estimate thesampling probability s by b s = 2 | E [ G [ S ] ∧ G [ S ]] || E [ G [ S ]] | + | E [ G [ S ]] | , (15)where E [ G ] denotes the edge set of graph G .Based on b s , we can further estimate the average weight w . Recall that w is close to the averagedegree under the Chung-Lu model. Thus, we estimate w by d ( G )+ d ( G )2 b s , where d ( G ) is the averagedegree in graph G. Finally, for the fraction of seeds θ , if it is unknown, we can simply estimate itby |S| n . Note that since w max will not be used by our algorithm, we do not need to estimate it.Based on the estimated model parameters, we can then determine the input parameters of ourPLD algorithm. Since we optimally choose γ = 1 / ((3 − β )( D −
1) + 1), the threshold τ in (3) canbe simplified to τ = (cid:16) Cs w (cid:17) D nθ. Further, the threshold τ ( k ) can be set according to (5). We use a Facebook friendship network (provided in [TMP12]) of 63392 students and staffs fromUniversity of Oregon as the parent graph G . There are 1633772 edges in G . The power-lawexponent of the Facebook social network is estimated as 2.09 by (14). To obtain two edge-correlatedsubgraphs G and G of different sizes, we independently sample each edge of G twice withprobability s = 0 . G twice with probability 0 .
8. Then, we relabel thevertices in G according to a random permutation π : [ n ] → [ n ], where n is the number of nodesin G . Let m denote the number of common vertices that appear in both G and G . The initialseed set is constructed by including each true pair independently with probability θ . We treat G as the public network and G as the private network, and the goal is to de-anonymize the nodeidentities in G by matching G and G . In Fig. 4, we show the performance of our PLD algorithmand five other algorithms, when the fraction of initial seeds θ varies from 0 to 0 .
05. We can observethat our PLD algorithm significantly outperforms the other algorithms.
Following [FMWX20], we use the Autonomous Systems (AS) data set from [LK14] to furthertest the graph matching performance on power-law graphs. The data set consists of 9 graphs ofAutonomous Systems peering information inferred from Oregon route-views between March 31,2001, and May 26, 2001. Since some vertices and edges are changed over time, these nine graphs14 A cc u r a cy R a t e PLD D=2PLD D=3PLD D=4DDMY-testUser Matching2-hopPGM
Figure 4: Performance comparison of the PLD algorithm and five other algorithms applied to theFacebook networks.can be viewed as correlated versions of each other. The number of vertices of the 9 graphs rangesfrom 10,670 to 11,174 and the number of edges from 22,002 to 23,409. We aim to match eachgraph to that on March 31, with vertices randomly permuted. The initial seed set is obtained byincluding each true pair independently with probability θ = 0 . s according to (15).The performance comparison of the six algorithms is plotted in Fig. 5 for θ = 0 .
1. We observethat our PLD algorithm again significantly outperforms the other algorithms. Note that the ac-curacy rates for all algorithms decay in time, because over time the graphs become less correlatedwith the initial one on March 31.
Date A cc u r a cy R a t e PLD D=2PLD D=3PLD D=4DDMY-testUser Matching2-hopPGM
Figure 5: Performance comparison of the PLD algorithm and five other algorithms applied to theAutonomous Systems graphs when θ = 0 .
1. 15
Analysis
In this section, we present the proof for Theorem 2. In Section 7.1, we describe the dependencyissue in our analysis and how we deal with it. In Section 7.2, we prove that all the true pairs inthe first slice Q are matched error-free by the D -hop algorithm. Using the matched vertices in theprevious slice as new seeds, we show in Section 7.3 that all the true pairs in slice Q k are matchederror-free by the 1-hop algorithm for 2 ≤ k ≤ k ∗ . Further, Section 7.4 proves that using the matchpairs in slice k ∗ as new seeds, the PGM algorithm correctly matches a constant fraction of truepairs with low weights. Finally, in Section 7.5, we come back to Q and prove that using all thematched pairs as seeds, all the true pairs in Q are matched error-free by the 1-hop algorithm.Theorem 2 readily follows by combining these results. The proofs of auxiliary lemmas can be foundin Appendix B.2.For ease of presentation, throughout the analysis, we assume without loss of generality that thetrue mapping π is the identity permutation. We further assume γ > D are suchthat γ ≤ log n w max , n γ = o ( n ), and (10) holds. In Algorithm 1, we use degrees as guidance to define the imperfect slice b P G j k for j = 1 , b G , b G . However, if we condition on the degrees, then the edges are no longerindependently generated with probability p ij as defined in the Chung-Lu model. To deal with thisdependency issue, we construct slices based on vertex weight that “sandwich” b P G j k . Recall thatthe perfect slices defined as P k = { u : w u ∈ [ α k , α k − ] } . By construction and the concentrationof vertex degrees, we expect that P k ⊂ b P G j k . We also need another weight-guided slice to contain b P G j k . Specifically, define P k = { u : w u ∈ [(1 − δ ) α k , (1 + 2 δ ) α k − ] } , where δ = . We also define Q k , P k × P k . The following lemma shows that with high probability, P k ⊂ b P G j k ⊂ P k and hence Q k ⊂ b Q k ⊂ Q k . Similarly, we define two different subsets of verticesthat “sandwich” V j : V = { u : w u ∈ [0 , n γ ] } and V = { u : w u ∈ [0 , (1 + 2 δ ) n γ ] } . Further, let G j and G j denote the subgraph of G j induced by the vertex set V and V , respectively,for j = 1 ,
2. The following lemma shows that with high probability, V ⊂ V j ⊂ V and hence G j ⊂ b G j ⊂ G j . Lemma 1.
For any ≤ k ≤ k ∗ , P n Q k ⊂ b Q k ⊂ Q k o ≥ − n − o (1) , and P n Q ≥ k ∗ ⊂ b Q ≥ k ∗ ⊂ Q ≥ k ∗ o ≥ − n − o (1) . For j = 1 , , P (cid:8) V ⊂ V j ⊂ V (cid:9) = P n G j ⊂ b G j ⊂ G j o ≥ − n − o (1) . .2 Match Pairs in b Q using D -hop Algorithm Recall that we give a heuristic argument of (8), showing that for a true pair in Q , the numberof common D -hop neighbors of Θ(1) weights is on the order of n γ (3 − β )( D − , by ignoring thethe potential dependency between b G j , b Q and graphs G , G . To resolve this dependency, wecrucially exploit the fact that with high probability Q ⊂ b Q and G j ⊂ b G j as shown in Lemma 1.In particular, we consider a true pair ( u, u ) in Q and bound its number of Θ(1)-weight D -hopneighbors in G j . Unfortunately, even when G j ⊂ b G j , the D -hop neighbors of u in G j may containsome vertices that are within the ( D − u in b G j , which means Γ G j D * Γ b G j D .In order to exclude such vertices, we bound the number of Θ(1)-weight vertices in N G j D − ( u ) fromabove. Fortunately, (cid:12)(cid:12) N G j D − ( u ) (cid:12)(cid:12) is close to (cid:12)(cid:12) Γ G j D − ( u ) (cid:12)(cid:12) , which is on the order of n γ (3 − β )( D − andthus is much smaller than (cid:12)(cid:12) Γ G j D (cid:12)(cid:12) . To be more precise, we have the following lemma. Lemma 2.
Fix any vertex u ∈ P and constant c . For all sufficiently large n , P n(cid:12)(cid:12)(cid:12) Γ G ∧ G D ( u ) ∩ { i : w i ≤ c } (cid:12)(cid:12)(cid:12) ≥ Γ min o ≥ − n − o (1) , (16) P n(cid:12)(cid:12)(cid:12) N G j D − ( u ) ∩ { i : w i ≤ c } (cid:12)(cid:12)(cid:12) ≤ N max o ≥ − n − o (1) , for j = 1 , , (17) where Γ min = (cid:16) C · s · w (cid:17) D n γ ((3 − β )( D − and N max = 2 cκ D n γ ((3 − β )( D − . To appreciate the utility of Lemma 2, note that under the high-probability event G j ⊂ b G j ⊂ G j for j = 1 ,
2, we have Γ b G D ( u ) ∩ Γ b G D ( u ) ⊃ Γ G ∧ G D ( u ) \ (cid:16) N G D − ( u ) ∪ N G D − ( u ) (cid:17) . Therefore, combining (16) and (17) implies that with high probability, (cid:12)(cid:12)(cid:12) Γ b G D ( u ) ∩ Γ b G D ( u ) ∩ { i : w i ≤ c } (cid:12)(cid:12)(cid:12) ≥ Γ min − N max ≈ Γ min , (18)where the last approximation holds because Γ min ≫ N max due to 2 < β < . Hence, the last displayyields the desired lower bound (8) to the number of common D -hop neighbors of Θ(1) weights fora true pair ( u, u ) in Q .Next, we adopt a similar strategy to study fake pairs. In particular, for a fake pair in b Q ,we bound from above its number of common D -hop neighbors of weights smaller than s log n . Again, to circumvent the dependency between b G j , b Q and graphs G , G , we consider a fake pair( u, v ) in Q and bound from above its number of Θ(1)-weight neighbors within the common D -hopneighborhood in G and G . Lemma 3.
Fix any two distinct vertices u, v ∈ P . For sufficiently large n , P (cid:26)(cid:12)(cid:12)(cid:12)(cid:12) N G D ( u ) ∩ N G D ( v ) ∩ { i : w i ≤ s log n } (cid:12)(cid:12)(cid:12)(cid:12) ≤ Ψ max (cid:27) ≥ − n − o (1) , (19) where Ψ max = − β κ D n γ ((3 − β )( D − (2 − β − Cn (cid:0) s log n (cid:1) − β + β − β − − κ D − n ( γ (3 − β )( D − (4 + 6 log n ) . The threshold s log n is chosen such that { i : w i ≤ s log n } contains { i : | Γ G ( i ) | ≤ n, | Γ G ( i ) | ≤ n } with high probability. emark 1. To see how (19) follows, note that N G D ( u ) ∩ N G D ( v ) ⊂ (cid:16) Γ G D ( u ) ∪ N D − ( u, v ) (cid:17) ∩ (cid:16) Γ G D ( v ) ∪ N D − ( u, v ) (cid:17) = (cid:16) Γ G D ( u ) ∩ Γ G D ( v ) (cid:17) ∪ N D − ( u, v ) , where N D − ( u, v ) = N G D − ( u ) ∪ N G D − ( v ). We have already obtained an upper bound to (cid:12)(cid:12)(cid:12) N G j D − (cid:12)(cid:12)(cid:12) when proving (17) for j = 1 ,
2. Thus, it remains to bound from above (cid:12)(cid:12)(cid:12) Γ G D ( u ) ∩ Γ G D ( v ) (cid:12)(cid:12)(cid:12) . Asimple yet key observation is that for a vertex i of weight 1, there are two extreme cases in which i becomes a common D -hop neighbor of ( u, v ). One case is that i connects to some vertex inΓ G D − ( u ) \ Γ G D − ( v ), and connects to some other vertex in Γ G D − ( v ) \ Γ G D − ( u ). It can be shown thateach of these two connections happens independently with probability approximately q D and thusthe number of such common D -hop neighbors is about nq D , which roughly gives rise to the firstterm of Ψ max . The other extreme case is that i is a ( D − u, v ). Luckily, the common 1-hop neighborhood of ( u, v ) is typically of a very small size andthus we can bound from above (cid:12)(cid:12)(cid:12) Γ G ( u ) ∩ Γ G ( v ) (cid:12)(cid:12)(cid:12) by approximately log n . Moreover, i becomes a( D − G ( u ) ∩ Γ G ( v ) with probability at most q D − . Thus, thenumber of such common D -hop neighbors is at most around nq D − log n , which gives an expressionclose to the second term of Ψ max . These two extreme cases turn out to be the dominating cases asshown in the proof of Lemma 3.To see the usage of Lemma 3, note that under the high-probability event b G j ⊂ G j for j = 1 , b G D ( u ) ∩ Γ b G D ( v ) ⊂ N G D ( u ) ∩ N G D ( v ) . Therefore, (19) implies that with high probability (cid:12)(cid:12)(cid:12)(cid:12) Γ b G D ( u ) ∩ Γ b G D ( v ) ∩ { i : w i ≤ s log n } (cid:12)(cid:12)(cid:12)(cid:12) ≤ max , (20)which yields the desired upper bound to the number of common D -hop neighbors of Θ(1) weightsfor a fake pair ( u, v ) in b Q .Finally, since we have n γ (3 − β ) ≫ log n and n γ ((3 − β )( D − (log n ) − β = O ( n ) based on thechoice in (11), it follows that Γ min > max . Moreover, (11) ensures that Γ min θ = Ω(log n ).Therefore, combining (18) and (20) implies that the true pairs in Q have more D -hop witnessesthan the fake pairs in b Q . Hence, we can use Algorithm 1 to match pairs in b Q correctly. Moreprecisely, we have the following lemma. Lemma 4.
Under the conditions of Theorem 2, for all sufficiently large n , the set of matched pairsin Step 5 of Algorithm 1, denoted by R , contains all true pairs in Q and no fake pairs in b Q withprobability at least − n − . o (1) . b Q k Slice by Slice using -hop Algorithm Given that all the true pairs in Q are matched error-free, we show that all the true pairs in Q k are matched error-free by the 1-hop algorithm for all 2 ≤ k ≤ k ∗ .Note that when matching pairs in b Q k , we use R k − , the set of matched vertices in b Q k − , asseeds. Suppose slice k − R k − contains all the true pairs in Q k − .Therefore, for a true pair in Q k , to bound from below its number of 1-hop witnesses in R k − , itsuffices to consider its number of 1-hop common neighbors in P k − , which is on the order of α − βk − w as we explained in (9). This is made precise by the following lemma.18 emma 5. Fix any ≤ k ≤ k ∗ and any vertex u ∈ P k . For all sufficiently large n , P n(cid:12)(cid:12)(cid:12) Γ G ( u ) ∩ Γ G ( u ) ∩ P k − (cid:12)(cid:12)(cid:12) ≥ ξ k o ≥ − n − , (21) where ξ k = Cα − βk − s w . Moreover, if slice k − b P k − ⊂ P k − , R k − is contained by the set of true pairs in Q k − , P k − × P k − .Therefore, for a fake pair in b Q k , to bound from above its the number of 1-hop witnesses in R k − ,it suffices to bound its number of 1-hop common neighbors in P k − , which is done in the followinglemma. Note that to resolve the potential dependency between b Q k and graphs G , G , we state thelemma for a fake pair in Q k , which contains b Q k with high probability. Lemma 6.
Fix any ≤ k ≤ k ∗ and any two distinct vertices u, v ∈ P k , Then for all sufficientlylarge n , P n(cid:12)(cid:12)(cid:12) Γ G ( u ) ∩ Γ G ( v ) ∩ P k − (cid:12)(cid:12)(cid:12) ≤ ζ k o ≥ − n − , (22) where ζ k = δ ) Cα − βk − w n + log n. To see how (22) follows, note that a vertex in P k − is a 1-hop common neighbor for the fake pair( u, v ) with probability at most on the order of (cid:0) α k α k − nw (cid:1) = α k − n w . Since there are Θ( nα − βk − ) verticesin P k − , the number of 1-hop common neighbors in P k − is about α − βk − nw on expectation. The extraterm log n in (22) comes from the sub-exponential tail bounds when we apply concentrationinequalities.Recall that we assume n γ = o ( n ) and hence α − βk − ≫ α − βk − n for 2 ≤ k ≤ k ∗ . Moreover, α − βk − ≥ α − βk ∗ ≥ w log nCs for 2 ≤ k ≤ k ∗ . It then can be verified that ξ k > ζ k . Thus, we expect thatthe 1-hop algorithm can match vertex pairs in b Q k correctly. More precisely, we have the followinglemma. Lemma 7.
Under the conditions of Theorem 2, for all sufficiently large n , with probability at least − n − . o (1) , the set of matched pairs in Step 6-8 of Algorithm 1, denoted by R k , contains all truepairs in Q k and no fake pairs in b Q k for all ≤ k ≤ k ∗ . We proceed to match pairs with weight smaller than α k ∗ using the PGM algorithm. As explained inSection 4.2, we expect that the number of common 1-hop neighbors for any fake pair with weightssmaller than α k ∗ is at most 2. Thus, even if all low-weight true pairs are provided as seeds, no fakepair will be matched by the PGM algorithm with threshold r = 3. This is made precise by thefollowing lemma. Lemma 8.
Denote P ≥ k ∗ = { u : w u ∈ [0 , (1 + 2 δ ) α k ∗ − ] } . Fix any two distinct vertices u, v ∈ P ≥ k ∗ +1 . Then for all sufficiently large n, P n(cid:12)(cid:12)(cid:12) Γ G ( u ) ∩ Γ G ( v ) ∩ P ≥ k ∗ (cid:12)(cid:12)(cid:12) ≤ o ≥ − n − . (23)19lthough the PGM algorithm may fail to match some true pairs with very few common 1-hopneighbors, it is expected to match the true pair with at least three 1-hop witnesses. In particular,let us recursively define S = P k ∗ , S h = { u : u ∈ P h + k ∗ , | Γ G ( u ) ∩ Γ G ( u ) ∩ S h − | ≥ } for h ≥ . Note that S = P k ∗ has been correctly matched based on Lemma 7 in the previous step. Also, oncethe true pairs in S h − are added into the set of matched pairs, the PGM algorithm with threshold r = 3 can use the vertices in S h − as new seeds to match vertices in S h correctly. Therefore, all thetrue pairs in S h for any h ≥ S h for h ≤ h ∗ , which is done by the followingtheorem. Lemma 9.
Let e w , (cid:0) w ln 2 Cs (cid:1) / (3 − β ) . Define h ∗ such that e w ≤ α k ∗ + h ∗ < e w. Then for any ≤ h ≤ h ∗ , and all sufficiently large n, P (cid:26) | S h | ≥ n k ∗ + h (cid:27) ≥ − n − o (1) . (24)The proof of Lemma 9 follows by induction. Assume (24) holds for h −
1. Then analogous tothe intuition of (9), for any u in P k ∗ + h , E h(cid:12)(cid:12)(cid:12) Γ G ( u ) ∩ Γ G ( u ) ∩ S h − (cid:12)(cid:12)(cid:12)i ≈ α − βk ∗ + h Cs w ≥ P { u ∈ S h } ≥ , which further implies (24) holds for h by concentration.By Lemma 9, the PGM matches at least half of true pairs in P k ∗ + h ∗ . Note that the number ofvertices in P k ∗ + h ∗ satisfies n k ∗ + h ∗ = Cn ( α k ∗ + h ∗ − ) − β ≥ Cn ( e w ) − β = Θ( n ), as e w = Θ(1). Thus,the set of matched pairs by the PGM contains a constant fraction of true pairs. More precisely, wehave the following lemma. Lemma 10.
Under the conditions of Theorem 2, for all sufficiently large n , with probability at least − n − o (1) , the set of matched pairs in Step 10 of Algorithm 1, denoted by R k ∗ +1 , contains alltrue pairs in S h and no fake pairs in b Q k ∗ + h for all h ≥ . In particular, we have |R k ∗ +1 | = Θ( n ) with probability at least − n − o (1) . b Q using -hop Algorithm Given that a large constant fraction of true pairs with weights smaller than α are matched error-free, we show that all the true pairs in Q are matched error-free by the 1-hop algorithm.When we match vertices in b Q , we use b R , the set of pairs matched in Step 5 −
10 of Algorithm1, as seeds. Note that all true pairs in Q k ∗ have been proved to be matched correctly with highprobability. The number of true pairs in Q k ∗ is Θ( nα − βk ∗ − ) and the vertex in P has weight largerthan n γ . Moreover, a vertex in P connects to a vertex in P k ∗ with probability at least α α k ∗ nw .Therefore, for a true pair in Q , to bound from below its number of 1-hop witnesses in b R , itsuffices to consider its number of 1-hop common neighbors in P k ∗ , which is about nα − βk ∗ − × α α k ∗ nw =Θ( α − βk ∗ n γ ). More precisely, we have the following theorem. Lemma 11.
Fix any vertex u ∈ P . For all sufficiently large n , P ((cid:12)(cid:12)(cid:12) Γ G ( u ) ∩ Γ G ( u ) ∩ P k ∗ (cid:12)(cid:12)(cid:12) ≥ Cα − βk ∗ α s w ) ≥ − n − . (25)20e caution the reader that even though the true pair ( u, u ) may have more 1-hop witnesses in Q k ∗ +1 than Q k ∗ , we cannot consider its number of 1-hop common neighbors in P k ∗ +1 , because thePGM algorithm only matches a subset of the true pairs in Q k ∗ +1 and this subset is random andmay incur dependency issues to the analysis.Next we study fake pairs. Note that with high probability b R contains no fake pair in S k ≥ b Q k .Therefore, on the event that b P k ⊂ P k for all k ≥ , all the matched pairs in b R is contained by theset of true pairs in R × R , where R = S k ≥ P k = { i : w i ∈ [0 , (1 + 2 δ ) n γ ] } . Therefore, for a fakepair in b Q , to bound from above its the number of 1-hop witnesses in b R , it suffices to bound itsnumber of 1-hop common neighbors in R , which is done in the following lemma. Again, to resolvethe potential dependency between b Q and graphs G , G , we state the lemma for a fake pair in Q ,which contains b Q with high probability. Lemma 12.
Denote R = { i : w i ∈ [0 , (1 + 2 δ ) n γ ] } . Fix any two distinct vertices u, v ∈ P . For allsufficiently large n , P n(cid:12)(cid:12)(cid:12) Γ G ( u ) ∩ Γ G ( v ) ∩ R (cid:12)(cid:12)(cid:12) ≤ κn γ (3 − β ) s o ≥ − n − , (26) where κ = (1+2 δ ) − β C (2 − β − w . To see how (26) follows, note that a vertex in P k becomes a common 1-hop neighbor of the fakepair ( u, v ) with probability at most (cid:0) α k w max nw (cid:1) ≤ α k nw . Since there are Θ( nα − βk ) true pairs in Q k ,the number of common 1-hop neighbors in R is on the order of P Kk =1 α − βk w = Θ (cid:0) n γ (3 − β ) (cid:1) .Recall that P ⊂ P ∪ P . Thus for any fake pair ( u, v ) ∈ Q , the two corresponding true pairs( u, u ) , ( v, v ) ∈ Q ∪ Q . If one of them is in Q , then it has already been matched in b Q by Lemma 4.If one of them is in Q , since α − βk ∗ n γ = Θ (cid:0) n γ (log n ) (2 − β ) / (3 − β ) (cid:1) ≫ n γ (3 − β ) in view of 2 < β <
3, ithas more 1-hop witnesses than the fake pair ( u, v ) . Thus, we expect that the 1-hop algorithm canmatch all the true pairs in b Q error-free. More precisely, we have the following lemma. Lemma 13.
Under the conditions of Theorem 2, for all sufficiently large n , with probability atleast − n − . , the set of matched pairs in Step 11 of Algorithm 1, denoted by R , contains all truepairs in Q and no fake pairs in b Q . Due to Lemma 10 and R k ∗ +1 ⊂ R , the set of matched pairs by Algorithm 1 contains Θ( n ) true pairswith probability at least 1 − n − o (1) . Combining Lemma 4, Lemma 7, Lemma 10 and Lemma 13, R contains no fake pairs with probability at least 1 − n − o (1) . In this paper, we propose an efficient seeded algorithm for matching graphs with power-law degreedistributions. Theoretically, under the Chung-Lu model with power-law exponent 2 < β < √ n ), we show that as soon as D > − β − β , by optimally choosing the first slice, ouralgorithm correctly matches a constant fraction of true pairs without any error with high prob-ability, provided with only Ω((log n ) − β ) initial seeds. This achieves an exponential reduction inthe seed size requirement, as the previously best known result requires n / ǫ initial seeds. Em-pirically, numerical experiments in both synthetic and real power-law graphs further demonstratethat our algorithm significantly outperforms the state-of-the-art algorithms. These results uncover21he enormous power of D -hops in seeded graph matching under power-law graphs. An interestingand important future direction is to further investigate the power of D -hops in matching power-lawgraphs without seeds. A Computational Complexity Analysis
We analyze the computational complexity of Algorithm 1 in each step.First, Algorithm 1 checks all the vertex degrees to construct the subgraphs b G i , G ′ i for i = 1 , G and G into slices based on vertex degrees in line 2-4 and line 9.The total time complexity of this step is O ( n ).We then apply the D -hop algorithm in the first slice. Searching for all D -hop neighbors of agiven vertex u in the first slice takes a total of O ( n ) time steps. The number of vertices in the firstslice in Θ( nα − β ). Thus, the complexity of counting D -hop witnesses for all vertices-pairs in thefirst slice-pair is O ( n α − β )1 ) = O ( n − γ ( β − ). Since we have shown that with high probability,all the fake pairs have D -hop witnesses fewer than the threshold, we only need to sort and matchat most n true pairs using GMWM and hence the complexity of the GMWM step is O ( n log n ) . We next apply the 1-hop algorithm in the subsequent slices. We compute the number of 1-hop witnesses via neighborhood exploration. For each matched pair in Q k − , we fetch its 1-hopneighbors of size O ( α k − ) in b G and b G , and then increase the number of 1-hop witnesses by 1 for O ( α k − ) vertex-pairs. Thus, the total complexity of our algorithm to match vertices in P k is about nα − βk − × α k − = O ( n γ (3 − β ) ). Further, we match k ∗ − O ( n γ (3 − β ) log n ).Analogously, the PGM algorithm explores the 1-hop neighbors of each matched pair. There areat most n matched pair, and for each mathced pair, we increase the number of 1-hop witnesses by1 for O ( △ ) vertex-pairs, where △ is the largest degree among G ′ and G ′ . By the definition, △ is O ((log n ) − β ). Therefore, the total complexity in line 10 is O ( n (log n ) − β ).Finally, there are at most n true pairs to serve as 1-hop witnesses for vertex-pairs in b Q . Forany true pair ( i, i ), the complexity of neighborhood exploration is O ( | Γ G ( i ) || Γ G ( i ) | ). Thus, thecomplexity of line 11 is P ni =1 | Γ G ( i ) || Γ G ( i ) | = O ( P ni =1 w i ) = O ( n − β ) / ) as shown in [CL03,page 98].In conclusion, by summing up the complexity for each step, the total computational complex-ity of our algorithm is O (cid:16) ( n − γ ( β − + n log n + n γ (3 − β ) log n + n (log n ) − β + n − β ) / (cid:17) = O (cid:0) n − γ ( β − (cid:1) due to γ < / < β < . B Postponed Proofs
B.1 Supporting Theorems
Theorem 3.
Chernoff Bound ([DP09]): Let X = P i ∈ [ n ] X i , where X i , i ∈ [ n ] , are independentrandom variables taking values in { , } . Then, for η ∈ [0 , , P { X ≤ (1 − η ) E [ X ] } ≤ exp (cid:18) − η E [ X ] (cid:19) , P { X ≥ (1 + η ) E [ X ] } ≤ exp (cid:18) − η E [ X ] (cid:19) . heorem 4. Bernstein’s Inequality ([DP09]): Let X = P i ∈ [ n ] X i , where X i , i ∈ [ n ] , are indepen-dent random variables such that | X i | ≤ K almost surely. Then, for t > , we have P { X ≥ E [ X ] + t } ≤ exp (cid:18) − t σ + Kt/ (cid:19) , where σ = P i ∈ [ n ] var ( X i ) is the variance of X . It follows then for ρ > , we have P (cid:26) X ≥ E [ X ] + p σ ρ + 2 Kρ (cid:27) ≤ exp( − ρ ) . The obtained estimate holds for P n X ≤ E [ X ] − p σ ρ − Kρ o too (by considering − X ), i.e., P (cid:26) X ≤ E [ X ] − p σ ρ − Kρ (cid:27) ≤ exp( − ρ ) . Theorem 5. ([YXL21, Theorem 6]) For r ≥ , every real number x ∈ (0 , and rx ≤ , it holdsthat r log (1 − x ) ≤ log (cid:16) − rx (cid:17) . Theorem 6. ([YXL21, Corollary 1]) Let X denote a random variable such that X ∼ Binom( n, p ) .If n ∈ [ n min , n max ] , then for λ > , P (cid:26) X ≥ n max α + 4 γ (cid:27) ≤ exp( − γ ) (27) B.2 Proof of the Main Result
First, we define some notations related to graph slicing. We count the number of vertices in theslice P k and P k . The vertices in P k satisfies α k ≤ w i ≤ α k − ⇐⇒ n (cid:16) ( β − n γ ( β − w k − (cid:17) β − − i ≤ i ≤ n (cid:16) ( β − n γ ( β − w k (cid:17) β − − i . According to the index range of the vertices, we define n k to be the difference between the twobounds. To be more precise, n k , Cnα − βk − , (28)where C throughout this paper denotes (2 β − − (cid:16) ( β − w ( β − (cid:17) β − . Moreover, we have that n k ≤ | P k | ≤ n k + 1 ≤ n k . (29)Similarly, the vertices in P k satisfies(1 − δ ) α k ≤ w i ≤ (1 + 2 δ ) α k − . Thus, (cid:12)(cid:12) P k (cid:12)(cid:12) ≤ (cid:16) β − (1 + 2 δ ) β − − (1 − δ ) β − (cid:17) n k β − − a ) ≤ (cid:0) (cid:1) β − − (cid:0) (cid:1) β − β − − n k + 1 ≤ n k , (30)where ( a ) follows from δ = .The number of perfect slices, denoted by K , islog ( n γ ) ≤ K ≤ ( n γ ) . B.2.1 Proof of Lemma 1
First, we prove P k ⊂ b P G j k with high probability for 0 ≤ k ≤ k ∗ and j = 1 ,
2. Fix any vertex u in P k . It suffices to show with high probability u ∈ b P G j k . Note that any vertex v connects to u in G j independently with probability p uv s , where j = 1 , p uv = w u w v nw . Thus E h(cid:12)(cid:12)(cid:12) Γ G j ( u ) (cid:12)(cid:12)(cid:12)i = X v ∈ G j p uv s = w u s. Note that α k ≤ w u ≤ α k − and α k ≥ α k ∗ ≥ (cid:16) w log nCs (cid:17) − β ≥
20 log nδ s for the choice of k ∗ in (4) andsufficiently large n , in view of 2 < β < . Then, applying the Chernoff Bound in Theorem 3 with η = δ yields P n(cid:12)(cid:12)(cid:12) Γ G j ( u ) (cid:12)(cid:12)(cid:12) ≥ (1 + δ ) α k − s o ≤ exp (cid:16) − δ α k − s (cid:17) ≤ n − , and P n(cid:12)(cid:12)(cid:12) Γ G j ( u ) (cid:12)(cid:12)(cid:12) ≤ (1 − δ ) α k s o ≤ exp (cid:16) − δ α k s (cid:17) ≤ n − . Combining the last two displayed equation yields that P n u / ∈ b P G j k o ≤ n − . Taking an union bound over u gives P n P k ⊂ b P G j k o ≥ − X u ∈ P k P n u / ∈ b P G j k o ≥ − n − o (1) . (31)Next we show that P ≥ k ∗ ⊂ b P G j ≥ k ∗ with high probability. Fix any vertex u ∈ P k with k ≥ k ∗ .Take a vertex v ∈ P k ∗ with w v = α k ∗ − . Since w u ≤ w v , we have (cid:12)(cid:12)(cid:12) Γ G j ( u ) (cid:12)(cid:12)(cid:12) s.t. ≤ (cid:12)(cid:12)(cid:12) Γ G j ( v ) (cid:12)(cid:12)(cid:12) . Therefore, P n u / ∈ b P G j ≥ k ∗ o = P n(cid:12)(cid:12)(cid:12) Γ G j ( u ) (cid:12)(cid:12)(cid:12) ≥ (1 + δ ) α k ∗ − s o ≤ P n(cid:12)(cid:12)(cid:12) Γ G j ( v ) (cid:12)(cid:12)(cid:12) ≥ (1 + δ ) α k ∗ − s o ≤ n − , Taking a union bound over u gives P n P ≥ k ∗ ⊂ b P G j ≥ k ∗ o ≥ − n − o (1) . (32)Second, we prove that for 0 ≤ k ≤ k ∗ , with high probability b P k ⊂ P k , or equivalently, [ n ] \ P k ⊂ [ n ] \ b P k , Fix any vertex u with w u > (1 + 2 δ ) α k − , applying the Chernoff Bound with η = δ δ yields P n(cid:12)(cid:12)(cid:12) Γ G j ( u ) (cid:12)(cid:12)(cid:12) ≤ (1 + δ ) α k − s o ≤ exp (cid:18) − δ α k − s δ ) (cid:19) ≤ n − . (33)24or any vertex u with w u < (1 − δ ) α k , applying the Chernoff Bound with η = δ − δ yields P n(cid:12)(cid:12)(cid:12) Γ G j ( u ) (cid:12)(cid:12)(cid:12) ≥ (1 − δ ) α k s o ≤ exp (cid:18) − δ α k s − δ ) (cid:19) ≤ n − . (34)Thus, we have P n b P G j k ⊂ P k o = P n [ n ] \ P k ⊂ [ n ] \ b P G j k o ≥ − X u/ ∈ P k P n u ∈ b P G j k o ≥ − n − , (35)where the last inequality holds by combining (33) and (34). Moreover, P n b P G j ≥ k ∗ ⊂ P ≥ k ∗ o = P n [ n ] \ P ≥ k ∗ ⊂ [ n ] \ b P G j ≥ k ∗ o ≥ − X u : w u > (1+2 δ ) α k ∗− P n u ∈ b P G j ≥ k ∗ o ≥ − n − , (36)where the last inequality holds by (33).Then, combining (31) and (35) with the union bound yields that P n Q k ⊂ b Q k ⊂ Q k o ≥ − n − o (1) for 0 ≤ k ≤ k ∗ . Similarly, combining (32) and (36) with a union bound yields that P n Q ≥ k ∗ ⊂ b Q ≥ k ∗ ⊂ Q ≥ k ∗ o ≥ − n − o (1) . Finally, since V = S k ≥ P k , V = S k ≥ b P G j k and V = S k ≥ P k , combining (31), (32), (35),and (36) with the union bound, we have P n G j ⊂ b G j ⊂ G j o = P (cid:8) V ⊂ V j ⊂ V (cid:9) ≥ − n − o (1) . B.2.2 Proof of Lemma 2
Note that G ∧ G , G , and G are graphs that are edge-sampled from G with probability s , s , s ,respectively. Thus, we let G denote a graph obtained by sampling each edge of G independentlywith probability t = Θ(1) and G denote a subgraph of G induced by the vertex set V = { u : w u ∈ [0 , (1 + 2 δ ) n γ ] } . Fix a vertex u ∈ P , we first study its number of d -hop neighbors in each slice in G .Then, we can arrive at Lemma 2 by selecting the corresponding parameters. To be more precise,we define Γ Gd,k ( u ) = Γ Gd ( u ) ∩ P k and N Gd,k ( u ) = S ≤ j ≤ d Γ Gj,k ( u ). We bound Γ Gd,k ( u ) and N Gd,k ( u ) bythe following lemma. Lemma 14.
Fix any vertex u ∈ P , and let Ω d denote the event such that the followings holdsimultaneously for k = 1 , . . . , K : (cid:12)(cid:12)(cid:12) Γ Gd,k ( u ) (cid:12)(cid:12)(cid:12) ≥ ( k − β − (cid:18) (1 − δ ) C · t · w (cid:19) d n γ (3 − β ) d , Γ min ( d, k ) , (37) (cid:12)(cid:12)(cid:12) Γ Gd,k ( u ) (cid:12)(cid:12)(cid:12) ≤ ( k − β − κ d n γ (3 − β ) d , Γ max ( d, k ) , (38) (cid:12)(cid:12)(cid:12) N Gd,k ( u ) (cid:12)(cid:12)(cid:12) ≤ ( k − β − κ d n γ (3 − β ) d , (39) where κ = (1+2 δ ) − β C (2 − β − w . Suppose γ and D are chosen such that condition (10) holds. Then, for all ≤ d ≤ D and sufficiently large n , P { Ω d } ≥ − (4 d − n − . (40)25 emark 2. The intuition behind Lemma 14 is as follows. Recall that q d , the probability that avertex of Θ(1) weight lies in the d -hop neighborhood of a vertex in the first slice, is on the order of n γ [(3 − β )( d − − in view of (7). Note that the weight of vertices in P k is about α k , and the size of P k is Θ( nα − βk − ). Thus, the expected number of vertices in P k that are d -hop neighbors of a givenvertex in the first slice is roughly nq d α − βk − ≈ ( k − β − n γ (3 − β ) d . Hence, we expect (37)-(39) tohold with high probability by concentration.Before proving Lemma 14, we first show how to apply Lemma 14 to prove Lemma 2. By setting δ = 0 and t = s , we have G = G ∧ G . Thus, (37) with k = ⌈ log ( n γ ) ⌉ and d = D leads to thedesired conclusion (16). Moreover, there are at most c slices in { i : w i ≤ c } . By setting δ = , d = D − G = G j (i.e., t = s ), (39) with log ( n γ /c ) ≤ k ≤ K ≤ log ( n γ ) + 1, we have K X k = ⌊ log ( n γ /c ) ⌋ ( k − β − κ D − n γ (3 − β )( D − ≤ cκ D − n γ ((3 − β )( D − = N max , where N max is given in (17). Thus, we prove the desired conclusion (17).We then present the proof of Lemma 14. Proof of Lemma 14.
Fix a vertex u in P , we study its d -hop neighborhood in G from d = 1. For d = 1 : For each vertex i ∈ P k , define an indicator variable x ki = n i ∈ Γ G ( u ) o . In other words, x ki = 1 if i is connected to u in G , and x ki = 0 otherwise. Since u ∈ P , it followsthat p k min = (1 − δ ) α k α nw t ≤ P n x ki = 1 o ≤ (1 + 2 δ ) α k − α nw t = p k max . Then, we have (cid:12)(cid:12)(cid:12) Γ G ,k ( u ) (cid:12)(cid:12)(cid:12) = P i ∈ P k x ki and x ki ’s are independent. Recall that n k = Cnα − βk − inview of (28) and n k ≤ (cid:12)(cid:12) P k (cid:12)(cid:12) ≤ n k in view of (30). Thus n k p k min = (1 − δ ) C α − βk − α w t = (1 − δ ) C n γ (3 − β ) · ( k − − β ) w t,n k p k max = (1 + 2 δ ) C α − βk − α w t = (1 + 2 δ ) C n γ (3 − β ) ( k − − β ) w t. Hence, applying Chernoff Bound in Theorem 3 with η = yields that P ((cid:12)(cid:12)(cid:12) Γ G ,k ( u ) (cid:12)(cid:12)(cid:12) ≤ (1 − δ ) Cn γ (3 − β ) t · ( k − − β ) w ) ≤ P (cid:26) Binom (cid:16) n k , p k min (cid:17) ≤ n k p k min (cid:27) ( a ) ≤ n − , P ((cid:12)(cid:12)(cid:12) Γ G ,k ( u ) (cid:12)(cid:12)(cid:12) ≥ (1 + 2 δ ) 3 Cn γ (3 − β ) t ( k − − β ) w ) ≤ P n Binom (cid:16) n k , p k max (cid:17) ≤ n k p k max o ( b ) ≤ n − , where ( a ) and ( b ) hold because n k p k max ≥ n k p k min ≥ (1 − δ ) Cn γ (3 − β ) t · w ≥
108 log n for sufficientlylarge n .We also have P n(cid:12)(cid:12)(cid:12) N G ,k ( u ) (cid:12)(cid:12)(cid:12) ≥ n k p k max o ≤ n − due to N G ,k ( u ) = Γ G ,k ( u ). Finally, taking theunion bound leads to (40) for d = 1. 26 or ≤ d ≤ D : We first count the d -hop neighbors conditional on the ( d − u such that Ω d − holds. The high-level idea is as follows. After the conditioning, every vertex i outside the ( d − u will become a d -hop neighbor by connecting to at leastone of the ( d − v of u . These edge connections are still independently generatedacross different v and i according to the Chung-Lu model.We first bound (cid:12)(cid:12)(cid:12) Γ Gd,k ( u ) (cid:12)(cid:12)(cid:12) from below. For each vertex i ∈ P k \ (cid:16) N Gd − ,k ( u ) (cid:17) , P ′ k , define anindicator variable y ki = n ∃ v ∈ Γ Gd − ( u ): i ∈ Γ G ( v ) o . In other words, y ki = 1 if i is connected to at least one ( d − u in G , and y ki = 0 otherwise. Thus, we have (cid:12)(cid:12)(cid:12) Γ Gd,k ( u ) (cid:12)(cid:12)(cid:12) = P i ∈ P ′ k y ki , and y ki ’s are independent across different i conditional on Ω d − .Note that Γ Gd − , ( u ) ⊂ Γ Gd − ( u ). Thus, we can bound P (cid:8) y ki = 1 | Ω d − (cid:9) from below by consideringthe possible edge connections between i and vertices in Γ Gd − , ( u ). More precisely, we get that P n y ki = 1 | Ω d − o ≥ P n ∃ v ∈ Γ Gd − , ( u ) : i ∈ Γ G ( v ) | Ω d − o ( a ) ≥ − (1 − p vi ) Γ min ( d − , ≥ − (cid:16) − (1 − δ ) α k α nw t (cid:17) Γ min ( d − , b ) ≥ (1 − δ ) min ( d − , α k α tnw = 32 k Cn (cid:18) (1 − δ ) C · t · w (cid:19) d n γ ((3 − β )( d − , p k,d min . where ( a ) holds because n i / ∈ Γ G ( v ) o are independent across v ; ( b ) follows from Theorem 5.Now, to bound (cid:12)(cid:12)(cid:12) Γ Gd,k ( u ) (cid:12)(cid:12)(cid:12) from below, we also need a lower bound to | P ′ k | , or equivalently anupper bound to (cid:12)(cid:12)(cid:12) N Gd − ,k ( u ) (cid:12)(cid:12)(cid:12) . Since we have conditioned on the ( d − u suchthat event Ω d − holds. It follows from (39) that (cid:12)(cid:12)(cid:12) N Gd − ,k ( u ) (cid:12)(cid:12)(cid:12) ≤ ( k − β − κ d − n γ (3 − β )( d − =2 κ d − n γ ((3 − β )( d − α − βk − a ) ≤ C nα − βk − ≤ n k , where ( a ) holds due to the condition (10). Thus, we have | P ′ k | ≥ | P k | − (cid:12)(cid:12)(cid:12) N Gd − ,k ( u ) (cid:12)(cid:12)(cid:12) ≥ n k .Note that for sufficiently large n ,89 n k p k,d min = 43 · ( k − − β ) (cid:18) (1 − δ ) Ct · w (cid:19) d n γ (3 − β ) d = 43 Γ min ( d, k ) ≥
128 log n. Thus, we apply the Chernoff Bound in Theorem 3 with η = and get P n(cid:12)(cid:12)(cid:12) Γ Gd,k ( u ) (cid:12)(cid:12)(cid:12) ≤ Γ min ( d, k ) | Ω d − o ≤ P (cid:26) Binom (cid:18) n k , p k,d min (cid:19) ≤ Γ min ( d, k ) | Ω d − (cid:27) ≤ n − . (cid:12)(cid:12)(cid:12) Γ Gd,k ( u ) (cid:12)(cid:12)(cid:12) from above. To this end, we bound P (cid:8) y ki = 1 | Ω d − (cid:9) from above andget P n y ki = 1 | Ω d − o ( a ) ≤ K X l =1 P n ∃ j ∈ Γ Gd − ,l ( u ) : i ∈ Γ G ( j ) | Ω d − o ( b ) ≤ (1 + 2 δ ) K X l =1 Γ max ( d − , l ) α k − α l − nw =(1 + 2 δ ) κ d − n γ ((3 − β )( d − k − nw K X l =1 ( l − β − ≤ κ d n γ ((3 − β )( d − k +1 Cn , p k,d max , (41)where ( a ) follow from the union bound; ( b ) holds due to the union bound and event Ω d − ; ( b )follows from (1 + x ) r ≥ rx for every integer r ≥ x ≥ −
2; and the lastinequality follows from the definition of κ = (1+2 δ ) − β C (2 − β − w .Also, note that P ′ k ⊂ P k and thus | P ′ k | ≤ | P k | ≤ n k . For sufficiently large n , we have2 n k p k,d max = 2 ( k − β − − κ d n γ (3 − β ) d = 12 Γ max ( d, k ) . Hence, applying Chernoff Bound in Theorem 3 with η = 1 yields that P n(cid:12)(cid:12)(cid:12) Γ Gd,k ( u ) (cid:12)(cid:12)(cid:12) ≥ Γ max ( d, k ) | Ω d − o ≤ P n Binom (cid:16) n k , p k,d max (cid:17) ≥ Γ max ( d, k ) o ≤ n − . Induction:
Finally, we prove (40) by induction.For d = 1, we have proved that (40) holds. Suppose that (40) holds for d −
1. Then we have P n(cid:12)(cid:12)(cid:12) Γ Gd,k ( u ) (cid:12)(cid:12)(cid:12) ≤ Γ min ( d, k ) o ≤ P (cid:8) Ω cd − (cid:9) + P n(cid:12)(cid:12)(cid:12) Γ Gd,k ( u ) (cid:12)(cid:12)(cid:12) ≤ Γ min ( d, k ) | Ω d − o P { Ω d − } ≤ d − · n − . (42)Similarly, we get P n(cid:12)(cid:12)(cid:12) Γ Gd,k ( u ) (cid:12)(cid:12)(cid:12) ≥ Γ max ( d, k ) o ≤ d − · n − (43)Since (cid:12)(cid:12)(cid:12) N Gd,k ( u ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) N Gd − ,k ( u ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) Γ Gd,k ( u ) (cid:12)(cid:12)(cid:12) , we take an union bound and have P n(cid:12)(cid:12)(cid:12) N Gd,k ( u ) (cid:12)(cid:12)(cid:12) ≥ ( k − β − κ d n γ (3 − β ) d o ≤ (4 d − − · n − + 4 d − · n − = (2 · d − − n − . (44)Combining (42), (43) and (44) with an union bound, we prove that (40) holds for any 1 ≤ k ≤ K and 1 ≤ d ≤ D . 28 .2.3 Proof of Lemma 3 Note that N G D,k ( u ) ∩ N G D,k ( v ) ⊂ (cid:16) Γ G D,k ( u ) ∪ N D − ,k ( u, v ) (cid:17) ∩ (cid:16) Γ G D,k ( v ) ∪ N D − ,k ( u, v ) (cid:17) = (cid:16) Γ G D,k ( u ) ∩ Γ G D,k ( v ) (cid:17) ∪ N D − ,k ( u, v ) , (45)where N D − ,k ( u, v ) = N G D − ,k ( u ) ∪ N G D − ,k ( v ). Since we have already obtained the upper bounds of (cid:12)(cid:12)(cid:12) N G D − ,k ( u ) (cid:12)(cid:12)(cid:12) and (cid:12)(cid:12)(cid:12) N G D − ,k ( v ) (cid:12)(cid:12)(cid:12) by Lemma 14 by letting G to be either G or G , it remains to boundfrom above (cid:12)(cid:12)(cid:12) Γ G D,k ( u ) ∩ Γ G D,k ( v ) (cid:12)(cid:12)(cid:12) , which is done in the following lemma. Lemma 15.
Suppose γ and D are chosen such that condition (10) holds. Fix any two distinctvertices u, v ∈ P , for all ≤ d ≤ D , k = 1 , . . . , K , and sufficiently large n , P n(cid:12)(cid:12)(cid:12) Γ G d,k ( u ) ∩ Γ G d,k ( v ) (cid:12)(cid:12)(cid:12) ≤ Ψ( d, k ) o ≥ − · d · n − , (46) where Ψ( d, k ) = κ Γ ( d − , n γ (5 − β ) ( k − − β ) Cn + 6Γ max ( d − ,
1) log n ( k − − β ) with Γ max ( d − ,
1) = κ d − n γ (3 − β )( d − as defined in (38) and κ = (1+2 δ ) − β C (2 − β − w . Remark 3.
We provide an intuitive explanation on Ψ( d, k ) . Analogous to Remark 1, there are twoextreme cases in which a vertex i in P k becomes a common d -hop neighbor of ( u, v ). One case isthat i connects to some ( d − u and v , respectively. Recall that Γ max ( d − , l ) isan upper bound of its ( d − P l by Lemma 14. Thus, a vertex i in P k connectsto at least one ( d − u with probability at most P Kl =1 Γ max ( d − , l ) α k α l nw ≈ Γ max ( d − , α k n γ − , where the approximation holds because l = 1 is the dominating term in thesummation. Moreover, there are Θ( nα − βk ) vertices in the slice P k . Thus, for a fake pair ( u, v ),its number of such common d -hop neighbors in P k is about Γ ( d − , n γ − α − βk , which givesrise to the first term of Ψ( d, k ) . The other extreme case is that i is a ( d − u, v ). As stated in Remark 1, we can bound from above (cid:12)(cid:12)(cid:12) Γ G ( u ) ∩ Γ G ( v ) (cid:12)(cid:12)(cid:12) by log n . Then, the vertex i connects to at least one ( d − G ( u ) ∩ Γ G ( v ) with probability at most P Kl =1 Γ max ( d − , l ) α k α l nw ≈ Γ max ( d − , α k n γ − . Again,there are Θ( nα − βk ) vertices in the slice P k . Thus, the number of such common d -hop neighborsin P k is about Γ max ( d − , α − βk n γ log n ≈ ( k − β − Γ max ( d − ,
1) log n , which gives rise to thesecond term of Ψ( d, k ) . Before proving Lemma 15, we first show how to apply Lemma 15 to prove Lemma 3. combining(45), (39), and (46) yields that P n(cid:12)(cid:12)(cid:12) Γ G d,k ( u ) ∩ Γ G d,k ( v ) (cid:12)(cid:12)(cid:12) ≤ Ψ( d, k ) + 2 N max ( d − , k ) o > − n − o (1) . Next we set d = D and sum over k for all the slices P k with weight at most s log n , i.e., α k ≤ s log n . In particular, we have k ≥ k , ⌊ log ( n γ s
15 log n ) ⌋ and K X k = k Ψ( D, k ) + 2 N max ( D − , k ) 29 K X k = k κ Γ ( D − , n γ (5 − β ) ( k − − β ) Cn + Γ max ( D − , ( k − − β ) n + 4 κ D − n γ (3 − β )( D − ( k − − β ) ≤ − β κ Γ ( D − , n γ (5 − β ) (2 − β − ( k − − β ) Cn + 2 β − β − − max ( D − , ( K − − β ) n + 2 β − β − − κ D − n γ (3 − β )( D − ( K − − β ) ≤ − β κ D n γ ((3 − β )( D − (2 − β − Cn (cid:18) s log n (cid:19) − β + 2 β − β − − κ D − n ( γ (3 − β )( D − (4 + 6 log n ) = Ψ max , where Ψ max is given in (19). Thus, we prove the desired conclusion (19).Next we present the proof of Lemma 15. Proof of Lemma 15.
Fix two distinct vertices u, v in P , we study their common d -hop neighbor-hood from d = 1. For d = 1 : For each vertex i ∈ P k , define an indicator variable x ki = (cid:26) i ∈ Γ G ( u ) ∩ Γ G ( v ) (cid:27) . In other words, x ki = 1 if i is connected to u in G and v in G , and x ki = 0 otherwise. Then, wehave (cid:12)(cid:12)(cid:12) Γ G ,k ( u ) ∩ Γ G ,k ( v ) (cid:12)(cid:12)(cid:12) = P i ∈ P k x ki . Since w u , w v ∈ [(1 − δ ] α , (1 + 2 δ ) α ], it follows that P n x ki = 1 o ≤ (cid:16) (1 + 2 δ ) α k − α nw (cid:17) , p k max . Hence, we have (cid:12)(cid:12)(cid:12) Γ G ,k ( u ) ∩ Γ G ,k ( v ) (cid:12)(cid:12)(cid:12) s.t. ≤ Binom (cid:16)(cid:12)(cid:12) P k (cid:12)(cid:12) , p k max (cid:17) . Recall n k = Cnα − βk − in view of (28) and (cid:12)(cid:12) P k (cid:12)(cid:12) ≤ n k in view of (30). Hence,2 n k p k max = (1 + 2 δ ) Cα − βk − n γ w n . Hence, we apply Lemma 6 with λ = 4 log n , and get P ((cid:12)(cid:12)(cid:12) Γ G ,k ( u ) ∩ Γ G ,k ( v ) (cid:12)(cid:12)(cid:12) ≥ δ ) Cα − βk − n γ w n + 163 log n ) ≤ n − . Since Γ max (0 ,
1) = 1, we have Ψ(1 , k ) = κ α − βk − n γ Cn + 6 log n . Thus, (46) holds for d = 1. For ≤ d ≤ D : We first count the d -hop neighbors conditional on the ( d − u and v . We use Ω ∗ d to denote the event that Ω d − with G = G , G hold, and for all k = 1 , . . . , K , (cid:12)(cid:12)(cid:12) Γ G d,k ( u ) ∩ Γ G d,k ( v ) (cid:12)(cid:12)(cid:12) ≤ Ψ( d, k ) , with Ψ( d, k ) defined in Lemma 15.Conditioning on Ω ∗ d − , note that there are two possible cases under which each true pair ( i, i )becomes a common d -hop neighbor of ( u, v ). One case is that i connects to some common ( d − u, v ) in both G and G . The other case is that i connects to different ( d − u, v ) in G and G , respectively. 30or each vertex i ∈ P k \ N D − ( u, v ), define two indicator variables y ki = (cid:26) i ∈ Γ G d ( u ) ,i ∈ Γ G d ( v ) (cid:27) ,z ki = (cid:26) ∃ j ∈ Γ G d − ( u ) ∩ Γ G d − ( v ): i ∈ Γ G ( j ) (cid:27) . In other words, y ki = 1 if i is a d -hop neighbor of u in G and v in G , and y ki = 0 otherwise.Similarly, z ki = 1 if i is connected to at least one common ( d − u, v ) in both G and G , and z ki = 0 otherwise. Note that z ki = 1 includes the case that i connects to some common( d − u, v ) in both G and G .We first bound P (cid:8) z ki = 1 | Ω ∗ d − (cid:9) from above by P n z ki = 1 | Ω ∗ d − o ( a ) ≤ K X l =1 P n ∃ j ∈ Γ G d − ( u ) ∩ Γ G d − ( v ) : i ∈ Γ G ( j ) | Ω ∗ d − o ( b ) ≤ (1 + 2 δ ) K X l =1 Ψ( d − , l ) α k − α l − nw ≤ κ Γ ( d − , n γ (7 − β ) k − Cn w + 6Γ max ( d − , n γ log n k − nw ! K X l =1 (1 + 2 δ ) ( l − − β ) ≤ κ d − n γ (3 − β )( d − n γ (7 − β ) k +1 C n + 6 κ d − n γ ((3 − β )( d − log n k +1 Cn = ν , where ( a ) holds due to the union bound; ( b ) follows from the union bound and event Ω ∗ d − .Then, the event { y ki = 1 } \ { z ki = 1 } denotes the event that i connects to some vertex inΓ G d − ,k ( u ) \ Γ G d − ,k ( v ) and connects to some vertex in Γ G d − ,k ( v ) independently. Thus, P (cid:8) { y ki = 1 } \ { z ki = 1 } | Ω ∗ d − (cid:9) can be bounded by P n { y ki = 1 } \ { z ki = 1 } | Ω ∗ d − o ≤ P n ∃ j ∈ Γ G d − ,k ( u ) \ Γ G d − ,k ( v ) : i ∈ Γ G ( j ) | Ω ∗ d − o P n ∃ j ∈ Γ G d − ( v ) : i ∈ Γ G ( j ) | Ω ∗ d − o ≤ P n i ∈ Γ G d ( v ) | Ω ∗ d − o P n i ∈ Γ G d ( v ) | Ω ∗ d − o ( a ) ≤ κ d n γ ((3 − β )( d − k +1 Cn ! ≤ κ d n γ (3 − β )( d − k +1) C n n γ (5 − β ) = ν , where ( a ) follows from a similar proof of (41).When we compare the first term of ν and ν , we have n γ (3 − β )( d − n n γ (7 − β ) ≪ n γ (3 − β )( d − n n γ (5 − β ) , where the inequality follows from n γ (7 − β ) n γ (5 − β ) = n γ ( β − = o (1).Thus, we have P n y ki = 1 | Ω ∗ d − o ≤ ν + ν ≤ ν + 4 κ d n γ ((3 − β )( d − log n k Cn κ d n γ (3 − β )( d − n γ · k − C n + 4 κ d n γ ((3 − β )( d − log n k Cn , µ k . Thus, conditional on Ω ∗ d − , we have (cid:12)(cid:12)(cid:12) Γ G d,k ( u ) ∩ Γ G d,k ( v ) (cid:12)(cid:12)(cid:12) s.t. ≤ Binom (cid:0)(cid:12)(cid:12) P k (cid:12)(cid:12) , µ k (cid:1) . Recall n k = Cnα − βk − in view of (28) and (cid:12)(cid:12) P k (cid:12)(cid:12) ≤ n k in view of (30). Therefore, for sufficientlylarge n , 2 n k µ k = 2 κ d n γ (3 − β )( d − n γ (5 − β ) · ( k − − β ) Cn + 4 κ d − n γ (3 − β )( d − log n ( k − − β ) ≤
23 Ψ( d, k ) . We then apply Chernoff Bound with η = and get P n(cid:12)(cid:12)(cid:12) Γ G d,k ( u ) ∩ Γ G d,k ( v ) (cid:12)(cid:12)(cid:12) ≥ Ψ max ( d, k ) | Ω ∗ d − o ≤ n − . Induction:
Finally, we prove (46) by induction.For d = 1, we have proved that (46) holds.Suppose (46) holds for d −
1, then taking the union bound yields that P (cid:8) Ω ∗ cd − (cid:9) ≤ · P (cid:8) Ω cd − (cid:9) + P n(cid:12)(cid:12)(cid:12) Γ G d − ,k ( u ) ∩ Γ G d − ,k ( v ) (cid:12)(cid:12)(cid:12) ≥ Ψ max ( d − , k ) o ≤ d − − n − + 2 · d − n − = (cid:18) · d − (cid:19) · n − . Thus, we have P n(cid:12)(cid:12)(cid:12) Γ G d,k ( u ) ∩ Γ G d,k ( v ) (cid:12)(cid:12)(cid:12) ≥ Ψ max ( d, k ) o ≤ P (cid:8) Ω ∗ cd − (cid:9) + P n(cid:12)(cid:12)(cid:12) Γ G d,k ( u ) ∩ Γ G d,k ( v ) (cid:12)(cid:12)(cid:12) ≥ Ψ max ( d, k ) | Ω ∗ d − o ≤ (cid:18) · d − (cid:19) · n − + n − ≤ · d · n − . B.2.4 Proof of Lemma 4
The main idea of the proof is to bound the number of D -hop witnesses for both true pairs andfake pairs in the first slice, using the bounds to the number of the D -hop neighbors established inLemma 2 and Lemma 3.Recall that in Algorithm 1, we select the set b S of low-degree seeds. Let b S = { i : ( i, i ) ∈ b S} .To circumvent the dependency between b S and the graphs G and G , we will introduce S and S such that they are independent from graphs and S ⊂ b S ⊂ S with high probability. To this end, wedefine an event E such that { i : w i ≤ c } ⊂ { i : | Γ G ( i ) | ≤ n, | Γ G ( i ) | ≤ n } ⊂ { i : w i ≤ s log n } . For any i with w i ≤ c , E h | Γ G ( i ) | i = cs . Thus, applying Lemma 6 with λ = 3 log n yields P n | Γ G ( i ) | ≥ n o ≤ P n | Γ G i ( i ) | ≥ cs + 4 log n o ≤ n − . i gives P n { i : w i ≤ c } ⊂ { i : | Γ G ( i ) | ≤ n, | Γ G ( i ) | ≤ n } o ≥ − n − o (1) .For any i with w i > s log n , E h | Γ G ( i ) | i = 15 log n . we apply Chernoff Bound in Theorem 3with η = 2 / P n | Γ G ( i ) | ≤ n o ≤ P (cid:26) | Γ G i ( i ) | ≤ (cid:18) − (cid:19)
15 log n (cid:27) ≤ n − . Thus, we have P (cid:26) { i : | Γ G ( i ) | ≤ n, | Γ G ( i ) | ≤ n } ⊂ { i : w i ≤ s log n } (cid:27) = P (cid:26) { i : w i > s log n } ⊂ { i : | Γ G ( i ) | > n, | Γ G ( i ) | > n } (cid:27) = 1 − n − o (1) . Thus, P {E} ≥ − n − o (1) . On event E , we have S , { i : w i ≤ c } ∩ S ⊂ b S ⊂ { i : w i ≤ s log n } ∩ S , S, where S = { i : ( i, i ) ∈ S} denotes the set of vertices selected as the initial seed set S . Note thatcrucially the initial seeds in S are selected among all true pairs with probability θ , independentlyfrom everything else. Thus S and S are independent from graphs. As a consequence, to boundfrom below (resp. above) the number of D -hop witnesses for the true (resp. fake) pair, it suffices toconsider their common D -hop neighbors in S (resp. S ).More specifically, let us first consider the true pairs. Fix any vertex u ∈ P . Let Λ( u ) =Γ G D ( u ) ∩ Γ G D ( u ) \ (cid:16) N G D − ( u ) ∩ N G D − ( u ) (cid:17) . Define event A u = (cid:26) | Λ( u ) ∩ S | >
35 Γ min θ (cid:27) , where Γ min = 12 (cid:18) C · s · w (cid:19) D n γ ((3 − β )( D − . Note that due to assumption (10) and n γ (3 − β ) ≫ log n for sufficiently large n , N max ≤ Γ min .Hence it follows from Lemma 2 that P (cid:26) | Λ( u ) ∩ { i : w i ≤ c | <
45 Γ min (cid:27) ≤ n − o (1) . Because the seeds S are selected among all true pairs with probability θ , independently fromeverything else, we have | Λ( u ) ∩ S | ∼ Binom ( | Λ( u ) ∩ { i : w i ≤ c }| , θ ) . Then, we apply Chernoff Bound in Theorem 3 with η = and get P {A cu } ≤ P (cid:26) | Λ( u ) ∩ { i : w i ≤ c }| <
45 Γ min (cid:27) + P (cid:26) A cu (cid:12)(cid:12)(cid:12)(cid:12) | Λ( u ) ∩ { i : w i ≤ c }| ≥
45 Γ min (cid:27) ≤ n − o (1) + P (cid:26) Binom (Γ min , θ ) ≤
35 Γ min θ (cid:27) ≤ n − o (1) + exp (cid:18) −
140 Γ min θ (cid:19) ( a ) ≤ n − o (1) , a ) holds due to assumption (11). Let A = ∩ u ∈ P A u . It follows from the union bound that P {A} ≤ n − o (1) . We next consider the fake pairs. Fix any two distinct vertices u, v ∈ P . Define an event B uv = (cid:26)(cid:12)(cid:12)(cid:12) N G D ( u ) ∩ N G D ( v ) ∩ S (cid:12)(cid:12)(cid:12) ≤
12 Γ min θ (cid:27) . Note that due to the assumption (10) and n γ (3 − β ) ≫ log n for sufficiently large n ,Ψ max ≤ κ D (cid:16) Cs · w (cid:17) D − β n γ ((3 − β )( D − (2 − β − Cn (cid:18) s log n (cid:19) − β + 2 β − (4 + 6 log n )(2 β − − n γ (3 − β ) ! Γ min ≤
14 Γ min . Hence, it follows from Lemma 3 that P (cid:26)(cid:12)(cid:12)(cid:12)(cid:12) N G D ( u ) ∩ N G D ( v ) ∩ { i : w i ≤ s log n } (cid:12)(cid:12)(cid:12)(cid:12) >
14 Γ min (cid:27) ≤ n − o (1) . Since the seeds S are selected among all true pairs with probability θ independently, we have (cid:12)(cid:12)(cid:12) Γ G D ( u ) ∩ Γ G D ( v ) ∩ S (cid:12)(cid:12)(cid:12) ∼ Binom (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) N G D ( u ) ∩ N G D ( v ) ∩ { i : w i ≤ s log n } (cid:12)(cid:12)(cid:12)(cid:12) , θ (cid:19) . Then, we apply Chernoff Bound in Theorem 3 with η = 1 and get P {B cuv } ≤ P (cid:26)(cid:12)(cid:12)(cid:12)(cid:12) N G D ( u ) ∩ N G D ( v ) ∩ { i : w i ≤ s log n } (cid:12)(cid:12)(cid:12)(cid:12) >
14 Γ min (cid:27) + P (cid:26) E cuv (cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) N G D ( u ) ∩ N G D ( u ) ∩ { i : w i ≤ s log n } (cid:12)(cid:12)(cid:12)(cid:12) ≤
14 Γ min (cid:27) ≤ n − o (1) + P (cid:26) Binom (cid:18)
14 Γ min , θ (cid:19) ≤
12 Γ min θ (cid:27) ≤ n − o (1) + exp (cid:18) −
112 Γ min θ (cid:19) ( a ) ≤ n − o (1) , where ( a ) holds due to assumption (11). Let B = ∩ u,v ∈ P : u = v B uv . It follows from the union boundthat P {B c } ≤ n − o (1) . Finally, we define event C such that G j ⊂ b G j ⊂ G j , ∀ j = 1 , P ⊂ b P ⊂ P . It follows from Lemma 1 that P {C} ≥ − n − o (1) . Taking the union bound, we have P {A ∩ B ∩ C ∩ E} ≥ − n − o (1) − n − o (1) − n − o (1) ≥ − n − o (1) . It remains to verify that on the event
A ∩ B ∩ C ∩ E , R contains all true pairs in Q and no fakepairs in b Q .Recall that we uses seeds in b S and count the D -hop witnesses in b G and b G for all candidatevertex pairs in b Q in Step 4 of Algorithm 1. On event A ∩ C ∩ E , Λ( u ) ⊂ Γ b G D ( u ) ∩ Γ b G D ( u ) and theminimum number of D -hop witnesses among all true pairs ( u, u ) in Q is lower bounded by Γ min θ .On event B ∩ C ∩ E , Γ b G D ( u ) ∩ Γ b G D ( v ) ⊂ N G D ( u ) ∩ N G D ( v ) the maximum number of D -hop witnessesamong all fake pairs ( u, v ) in b Q is upper bounded by Γ min θ . Thus, GMWM with threshold τ = Γ min θ outputs R , which contains all true pairs in Q and no fake pairs in b Q .34 .2.5 Proof of Lemma 5 Fix a vertex u ∈ P k . For each vertex i ∈ P k − , let x i be a binary random variable such that x i = 1if i connects to u both in G and G , and x i = 0 otherwise. Then, (cid:12)(cid:12)(cid:12) Γ G ( u ) ∩ Γ G ( u ) ∩ P k − (cid:12)(cid:12)(cid:12) = P i ∈ P k − x i and x i ’s are independent. Moreover, we have P { x i = 1 } ≥ α k α k − nw s . Therefore, applying Chernoff Bound in Theorem 3 with η = yields that P ((cid:12)(cid:12)(cid:12) Γ G ( u ) ∩ Γ G ( u ) ∩ P k − (cid:12)(cid:12)(cid:12) ≤ Cα − βk − s w ) ≤ P ( Binom (cid:16) n k − , α k α k − nw s (cid:17) ≤ Cα − βk − s β +1 w ) ≤ n − , where the last inequality holds because n k − α k α k − nw s = Cα − βk − s β w ≥
24 log n in view of ( α k ∗ ) − β ≥ w log nCs . B.2.6 Proof of Lemma 6
Fix a pair of two distinct vertices u, v ∈ P k . For each vertex i ∈ P k − , let x i be a binary randomvariable such that x i = 1 if i is connected to u in G and v in G , and x i = 0 otherwise. Since theevent that i is connected to u is independent of the event that i is connected to v , we have P { x i = 1 } ≤ (cid:16) (1 + 2 δ ) α k − α k − nw s (cid:17) = 4(1 + 2 δ ) α k − s n w , p max . Moreover, x i ’s are independent. Therefore, (cid:12)(cid:12)(cid:12) Γ G ( u ) ∩ Γ G ( v ) ∩ P k − (cid:12)(cid:12)(cid:12) s.t. ≤ Binom (cid:0)(cid:12)(cid:12) P k − (cid:12)(cid:12) , p max (cid:1) .Recall n k − = Cnα − βk − in view of (28) and (cid:12)(cid:12) P k − (cid:12)(cid:12) ≤ n k − in view of (30). Thus, we applyLemma 6 with λ = 4 log n , and get P ((cid:12)(cid:12)(cid:12) Γ G ( u ) ∩ Γ G ( v ) ∩ P k − (cid:12)(cid:12)(cid:12) ≥ δ ) Cα − βk − s w n + 163 log n ) ≤ n − . B.2.7 Proof of Lemma 7
The proof is divided into two parts. The first part is to identify a set of “good” events whose inter-section holds with high probability. The second part provides a deterministic argument, showingthat on the intersection of these good events, the 1-hop algorithm successfully matches slice k forall 2 ≤ k ≤ k ∗ . First, we identify a good event under which the number of common 1-hop neighbors of a truepair is large. More precisely, for any vertex u ∈ P k , define event A k ( u ) = n(cid:12)(cid:12)(cid:12) Γ G ( u ) ∩ Γ G ( u ) ∩ P k − (cid:12)(cid:12)(cid:12) ≥ ξ k o , where ξ k , Cα − βk − s w , and A = ∩ ≤ k ≤ k ∗ ∩ u ∈ P k A k ( u ). By Lemma 5 and union bound, we have P {A c } ≤ n − o (1) . Second, we determine a good event under which the number of common 1-hop neighbors of afake pair is small. More formally, for any pair of distinct vertices u, v ∈ P k , define event B k ( u, v ) = n(cid:12)(cid:12)(cid:12) Γ G ( u ) ∩ Γ G ( v ) ∩ P k − (cid:12)(cid:12)(cid:12) ≤ ζ k o , where ζ k , δ ) Cα − βk − s w n + 163 log n, B = ∩ ≤ k ≤ k ∗ ∩ u,v ∈ P k : u = v B k ( u, v ). By Lemma 6 and union bound, we have P {B c } ≤ n − o (1) . Third, we define an event C = ∩ ≤ k ≤ k ∗ n Q k ⊂ b Q k ⊂ Q k o . By Lemma 1 and union bound, wehave P {C c } ≤ n − o (1) . Finally, we let F denote the event that the first slice is successfully matched, i.e., R containsall true pairs in Q and no fake pairs in b Q . By Lemma 4, P {F c } ≤ n − . o (1) . Combining the above, it follows that P {A ∩ B ∩ C ∩ F } ≥ − n − o (1) − n − o (1) − n − o (1) − n − . o (1) ≥ − n − . o (1) . It remains to verify on the event
A ∩ B ∩ C ∩ F , R k contains all true pairs in Q k and no fake pairsin b Q k for all 1 ≤ k ≤ k ∗ . We prove this by induction. The base case with k = 1 follows fromthe definition of F . Assume the induction hypothesis holds for the slice k −
1, we aim to show itcontinues to hold for k. Recall that when matching the slice b Q k , we use R k − as the set of seeds. Since the inductionhypothesis is true for slice k −
1, it follows that R k − contains all the true pairs in Q k − . Thus,the minimum number of 1-hop witnesses among all true pairs ( u, u ) in Q k is lower bounded by ξ k . Moreover, since R k − contains no fake pairs in b Q k − and on event C , b Q k − ⊂ Q k , it follows that R k − is contained by all the true pairs in Q k − . Also, the set of fake pairs in b Q k is contained bythe set of fake pairs in Q k . Thus, the maximum number of 1-hop witnesses among all fake pairs( u, v ) in b Q k is upper bounded by ζ k . Note that ξ k ( a ) ≥ τ ( k ) and ζ k τ ( k ) ( b ) ≤ δ ) n γ wn + 49 ( c ) < , where ( a ) holds by definition of τ ( k ) in (5); ( b ) follows from n γ ≥ α k ≥ α k ∗ ≥ (cid:16) w log nCs (cid:17) − β for2 ≤ k ≤ k ∗ ; ( c ) holds as n is sufficiently large in view of n γ = o ( n ) and w = Θ(1) . Thus, R k contains all true pairs in Q k and no fake pairs in b Q k , completing the induction. B.2.8 Proof of Lemma 8
Fix any two distinct vertices u, v ∈ P ≥ k ∗ +1 . Then w u , w v ≤ (1 + 2 δ ) α k ∗ . For each vertex i ∈ P ≥ k ∗ ,let x i be a binary random variable such that x i = 1 if i connects to u in G and v in G , and x i = 0otherwise. Since the event that i connects to u is independent of the event that i connects to v , wehave P { x i = 1 } ≤ (cid:16) (1 + 2 δ ) α k ∗ α k ∗ − nw s (cid:17) = (1 + 2 δ ) α k ∗ s n w , p max . Moreover, x i ’s are independent. Therefore, (cid:12)(cid:12)(cid:12) Γ G ( u ) ∩ Γ G ( v ) ∩ P ≥ k ∗ (cid:12)(cid:12)(cid:12) s.t. ≤ Binom (cid:0)(cid:12)(cid:12) P ≥ k ∗ (cid:12)(cid:12) , p max (cid:1) s.t. ≤ Binom ( n, p max ) . Thus, we get P n(cid:12)(cid:12)(cid:12) Γ G ( u ) ∩ Γ G ( v ) ∩ P ≥ k ∗ (cid:12)(cid:12)(cid:12) ≥ o ≤ P { Binom ( n, p max ) ≥ } ( a ) ≤ n p ≤ δ ) C α k ∗ s n w ≤ n − o (1) . where ( a ) follows from the union bound. 36 .2.9 Proof of Lemma 9 We first bound | S h | by conditioning on S h − . For any u ∈ P k ∗ + h , let x i be a binary random variablesuch that x i = 1 if i ∈ S h − connects to u , and x i = 0 otherwise. Since S h − is only determinedby the vertex weight and the edges connecting to previous S l , l < h −
1, the event that i and u isconnected is independent across i conditional on S h − . It follows that P { x i = 1 | S h − } ≥ α k ∗ + h α k ∗ + h − nw s . Thus, we have (cid:12)(cid:12)(cid:12) Γ G ( u ) ∩ Γ G ( u ) ∩ S h − (cid:12)(cid:12)(cid:12) s.t. ≥ Binom (cid:0) | S h − | , α k ∗ + h α k ∗ + h − nw s (cid:1) conditional on S h − .Applying Chernoff Bound in Theorem 3 yields that P (cid:26)(cid:12)(cid:12)(cid:12) Γ G ( u ) ∩ Γ G ( u ) ∩ S h − (cid:12)(cid:12)(cid:12) < (cid:12)(cid:12)(cid:12)(cid:12) | S h − | ≥ n k ∗ + h − (cid:27) ≤ P (cid:26) Binom (cid:18) n k ∗ + h − , α k ∗ + h α k ∗ + h − nw s (cid:19) ≤ (1 − η ) µ (cid:27) ≤ exp (cid:18) − η µ (cid:19) , p h ≤ √ , where µ = n k ∗ + h − α k ∗ + h α k ∗ + h − nw s = Cα − βk ∗ + h − s w ≥
12 ln 2 due to α k ∗ + h ≥ (cid:0) w ln 2 Cs (cid:1) / (3 − β ) and η = µ − µ ≥ .Then, the above result implies that: E (cid:2) | S h | | | S h − | ≥ n k ∗ + h − (cid:3) ≥ (1 − p h ) n k ∗ + h . Note thatthe event u ∈ S h only depends on the vertex weight and the edge set E u , { ( u, i ) : i ∈ S h − } . Because E u ’s are disjoint, the event u ∈ S h is independent across u ∈ P k ∗ + h . Thus, we applyChernoff Bound in Theorem 3 with η = − p h − p h ) and have P (cid:26) | S h | < n k ∗ + h (cid:12)(cid:12)(cid:12)(cid:12) | S h − | ≥ n k ∗ + h − (cid:27) ≤ P (cid:26) Binom ( n k ∗ + h , − p h ) < n k ∗ + h (cid:27) ≤ exp (cid:18) − (1 − p h ) n k ∗ + h − p h ) (cid:19) ≤ n − , where the last inequality holds due to n k ∗ + h ≥ n k ∗ ≥ Cn (cid:16) w log nCs (cid:17) − β − β ≥
128 log n due to thechoice of k ∗ in (4) and sufficiently large n .Finally, we prove by induction that P (cid:8) | S h | < n k ∗ + h (cid:9) ≤ h · n − .For h = 0, it is true by definition.For h ≥
1, if P (cid:8) | S h − | ≥ n k ∗ + h − (cid:9) ≥ − ( h − · n − , then P (cid:26) | S h | < n k ∗ + h (cid:27) ≤ P (cid:26) | S h | < n k ∗ + h | | S h − | ≥ n k ∗ + h − (cid:27) + P (cid:26) | S h − | < n k ∗ + h − (cid:27) ≤ n − + ( h − · n − = h · n − . B.2.10 Proof of Lemma 10
First, for any two distinct vertices u, v ∈ P ≥ k ∗ , define event A uv = n(cid:12)(cid:12)(cid:12) Γ G ( u ) ∩ Γ G ( v ) ∩ P ≥ k ∗ (cid:12)(cid:12)(cid:12) ≤ o , A = T u,v ∈ P ≥ k ∗ : u = v A uv . By Lemma 12 and union bound, we have P {A c } ≤ n − o (1) . Second, let B denote the event that all true pairs in P k ∗ are matched successfully. By Lemma 7, P {B} ≥ − n − . o (1) .Third, by Lemma 1 and union bound, we have P n Q ≥ k ∗ ⊂ b Q ≥ k ∗ ⊂ Q ≥ k ∗ o ≤ n − o (1) . Finally, by Lemma 9, we have P (cid:26) | S h ∗ | ≥ n k ∗ + h ∗ (cid:27) ≥ − n − o (1) . Combining the above, it follows that P (cid:26) A ∩ B ∩ { Q ≥ k ∗ ⊂ b Q ≥ k ∗ ⊂ Q ≥ k ∗ } ∩ {| S h ∗ | ≥ n k ∗ + h ∗ } (cid:27) ≥ − n − o (1) . Now, suppose event
A ∩ B ∩ { Q ≥ k ∗ ⊂ b Q ≥ k ∗ ⊂ Q ≥ k ∗ } ∩ {| S h ∗ | ≥ n k ∗ + h ∗ } holds. We aim toshow that R k ∗ +1 contains no fake pair in b Q ≥ k ∗ and all true pairs ( u, u ) with u ∈ S h for h ≥ . We first show R k ∗ +1 contains no fake pair in b Q ≥ k ∗ . Suppose not. Let ( u, v ) denote the firstfake pair in b Q ≥ k ∗ matched by the PGM algorithm. This implies that the PGM only matches truepairs before matching ( u, v ). Since the threshold r of the PGM is set to be 3, it follows that ( u, v )has at least three 1-hop witnesses that are true pairs in b Q ≥ k ∗ . Since b Q ≥ k ∗ ⊂ Q ≥ k ∗ , it follows that (cid:12)(cid:12)(cid:12) Γ G ( u ) ∩ Γ G ( v ) ∩ P ≥ k ∗ (cid:12)(cid:12)(cid:12) ≥
3, which contradicts the fact that event A holds. Thus, R k ∗ +1 containsno fake pairs in b Q ≥ k ∗ .Next, we prove that R k ∗ +1 contains all true pairs in S h for all h ≥ R k ∗ +1 contains the match pairs in the previous slice, that is R k ∗ +1 ⊃ R k ∗ . The base case with h = 0 follows from the definition of B . Assume the induction hypothesis holdsfor h −
1, we aim to show it continues to hold for h. Based on the definition of S h , the true pairsin S h have at least 3 common 1-hop neighbors in S h − . Because all true pairs in S h − have beenmatched and Q ≥ k ∗ ⊂ b Q ≥ k ∗ , the true pairs in S h would be matched by the PGM algorithm withthreshold r = 3. Therefore, R k ∗ +1 contains all true pairs in S h for all h ≥ | S h ∗ | ≥ n k ∗ + h ∗ = C nα − βk ∗ + h ∗ ≥ C n (2 e w ) − β , where e w = (cid:0) w ln 2 Cs (cid:1) / (3 − β ) = Θ(1) and the last inequality holds due to the choice of h ∗ . Thus, R k ∗ +1 has Θ( n ) true pairs. B.2.11 Proof of Lemma 11
Fix a vertex u ∈ P . For each vertex i ∈ P k ∗ , let x i be a binary random variable such that x i = 1 if i connects to u both in G and G , and x i = 0 otherwise. Then, (cid:12)(cid:12)(cid:12) Γ G ( u ) ∩ Γ G ( u ) ∩ P k ∗ (cid:12)(cid:12)(cid:12) = P i ∈ P k ∗ x i and x i ’s are independent. Moreover, we have P { x i = 1 } ≥ α k ∗ α nw s . Recall | P k ∗ | ≥ n k ∗ = Cnα − βk ∗ − in view of (28). Hence, (cid:12)(cid:12)(cid:12) Γ G ( u ) ∩ Γ G ( u ) ∩ P k ∗ (cid:12)(cid:12)(cid:12) s.t. ≥ Binom (cid:16) n k ∗ , α α k ∗ nw s (cid:17) . η = and get P ((cid:12)(cid:12)(cid:12) Γ G ( u ) ∩ Γ G ( u ) ∩ P k ∗ (cid:12)(cid:12)(cid:12) ≤ Cα − βk ∗ α s w ) ≤ P ( Binom (cid:16) n k ∗ , α α k ∗ nw s (cid:17) ≤ Cα − βk ∗ α s β w ) ≤ n − , where the last inequality holds because n k ∗ α k ∗ α nw s = Cα − βk ∗ α s β − w ≥
64 log n , due to the choice of k ∗ in (4). B.2.12 Proof of Lemma 12
Fix two distinct vertices u, v ∈ P . We bound from above the number of their common 1-hopneighbors in R = S k ≥ P k .For each k ≥ i ∈ P k , let y ki be a binary random variable such that y ki = 1 if i is connected to u in G and v in G , and y ki = 0 otherwise. Since the event that i is connected to u is independent of the event that i is connected to v , we have P n y ki = 1 o ≤ (cid:18) (1 + 2 δ ) α k − w max nw s (cid:19) ≤ (1 + 2 δ ) α k − nw s , p k max , ∀ k ≥ . Moreover, y ki ’s are independent. Thus, (cid:12)(cid:12)(cid:12) Γ G ( u ) ∩ Γ G ( v ) ∩ R (cid:12)(cid:12)(cid:12) s.t. ≤ K X k =1 Binom (cid:16)(cid:12)(cid:12) P k (cid:12)(cid:12) , p k max (cid:17) . Recall n k = Cnα − βk − in view of (28), n k ≤ (cid:12)(cid:12) P k (cid:12)(cid:12) ≤ n k , and κ = (1+2 δ ) − β C (2 − β − w . Thus, K X k =1 (cid:12)(cid:12) P k (cid:12)(cid:12) p k max ≤ K X k =1 n k (1 + 2 δ ) α k − nw s = 2 Cn γ (3 − β ) s w K X k =1 (1 + 2 δ ) ( k − − β ) ≤ κn γ (3 − β ) s , K X k =1 (cid:12)(cid:12) P k (cid:12)(cid:12) p k max ≥ n α nw s = Cn γ (3 − β ) w s ≥
64 log n. Then, we apply Chernoff Bound in Theorem 3 with η = 1, and get P n(cid:12)(cid:12)(cid:12) Γ G ( u ) ∩ Γ G ( v ) ∩ R (cid:12)(cid:12)(cid:12) ≥ κn γ (3 − β ) s o ≤ P ( K X k =1 Binom (cid:16)(cid:12)(cid:12) P k (cid:12)(cid:12) , p k max (cid:17) ≥ κn γ (3 − β ) s ) ≤ n − . B.2.13 Proof of Lemma 13
Recall the bound of the number of 1-hop witnesses is provided by Lemma 11 and Lemma 12.First, for any vertex u ∈ P , define event A u = ((cid:12)(cid:12)(cid:12) Γ G ( u ) ∩ Γ G ( u ) ∩ P k ∗ (cid:12)(cid:12)(cid:12) ≥ Cα − βk ∗ α s w ) , and A = T u ∈ P A u . By Lemma 11 and union bound, we have P {A} ≤ n − o (1) . Second, for any two distinct vertices u, v ∈ P , define event B uv = n(cid:12)(cid:12)(cid:12) Γ G ( u ) ∩ Γ G ( v ) ∩ R (cid:12)(cid:12)(cid:12) ≤ κn γ (3 − β ) s o , B = T u,v ∈ P : u = v B uv . By Lemma 12 and union bound, we have P {B c } ≤ n − o (1) . Third, we define an event C = T ≤ k ≤ k ∗ n Q k ⊂ b Q k ⊂ Q k o ∩ n Q ≥ k ∗ ⊂ b Q ≥ k ∗ ⊂ Q ≥ k ∗ o . By Lemma 1and union bound, we have P {C c } ≤ n − o (1) . Finally, we let E denote the event that b R contains all true pairs in Q k ∗ and no fake pairs in b Q k for any k ≥
1. By Lemma 4, Lemma 7 and Lemma 10, P {E c } ≤ n − . o (1) . Combining the above, it follows that P {A ∩ B ∩ C ∩ E} ≥ − n − o (1) − n − o (1) − n − o (1) − n − . o (1) ≥ − n − . o (1) . Suppose
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