ON THE CONTINUITY OF GRAPH PARAMETERS

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1 ON THE CONTINUITY OF GRAPH PARAMETERS M. HURSHMAN AND J. JANSSEN Abstract. I this paper, we explore the mathematical properties of a distace fuctio betwee graphs based o the maximum size of a commo subgraph. The otio of distace betwee graphs has prove useful i may areas ivolvig graph based structures such as chemistry, biology ad patter recogitio. Graph distace has bee used as a way of determiig similarity betwee graphs. The distace fuctio studied here forms a metric o isomorphism classes of graphs. We show that this metric iduces the discrete topology. We also show that the distace betwee two graphs i the Erdös-Reyi probability space G(, p) almost always is ear the maximum attaiable value. Fially, we defie a otio of cotiuity of graph parameters ad relate it to a property of graphs that ca be easily verified. We also determie whether the ormalized versios of some commo graph parameters are cotious i this framework. 1. Itroductio The problem of comparig graphs has received a lot of attetio over the past 30 years. I this paper, we will call the problem of comparig of graphs the graph similarity problem. I may other papers, the problem is called the graph matchig problem [5] [8]. The term graph matchig idicates that the mai iterest of the applicatio is to fid correspodig regios i graphs which match. I the graph similarity problem, the mai iterest is to assig a overall similarity score to idicate the level of similarity betwee two graphs. Ay method used for graph matchig ca be tured ito a method for graph similarity by assigig a score to the similarity of the two graphs. That is, the two problems are the same, oly their focus is differet. I graph similarity, we refer to the fuctio d that assigs the similarity score as the distace fuctio. This is because we thik of the similarity score betwee two graphs as the distace betwee them. Though differet similarity methods lead to differet distace fuctios, we would like all distace fuctios to satisfy the three followig properties. d(g, H) = 0 iff G = H d(g, H) = d(h, G) 1

2 2 M. HURSHMAN AND J. JANSSEN Icreasig values of d(g, H) idicate a decreasig amout of similarity betwee G ad H. The first coditio states that if two graphs are isomorphic the the distace betwee them should be 0. The secod requiremet is that the distace fuctio is symmetric. If the distace fuctio additioally satisfies the triagle iequality, as is the case with the similarity method based o the maximum commo subgraph [3], the distace fuctio forms a metric o the set of isomorphism classes of graphs. The mai force drivig the study of this problem has bee its umerous applicatios i may areas such as biology, chemistry ad patter recogitio [8] [15]. There have bee may proposed methods such as graph ad subgraph isomorphism, maximum commo subgraph methods ad edit distace [6]. More recetly, the focus has shifted to the use of machie learig techiques ad graph kerels [23]. The mai iterest i graph similarity has bee to develop methods to determie graph similarity ad efficiet algorithms to compute the distace fuctio. This has left may questios of a more mathematical flavour uexplored. The purpose of this paper is to explore some of these questios usig the distace metric of Buke ad Shearer based o the maximum commo subgraph. The first cotributio of this paper will be to show that the maximum commo subgraph distace fuctio iduces the discrete topology o the set of isomorphism classes of graphs. We will also show that the distace betwee two graphs from the Erdös-Reyi probability space G(, p) almost always have a distace ear 1, which is the maximum value. The implicatio is that two radomly chose graphs are ot likely to be similar. The mai cotributio of this paper will be to itroduce the idea of cotiuity of graph parameters. There are may graph parameters which we commoly assig to graphs such as the diameter, chromatic umber, umber of vertices etc. We might expect that for some of these parameters, a small distace betwee graphs would idicate a small differece betwee their graph parameters as well. Such a otio has a flavour of cotiuity ad i Sectio 5 we will defie both a poitwise ad a uiform versio of graph parameter cotiuity. We will show that all graph parameters are trivially cotiuous with respect to the maximum commo subgraph distace fuctio i the poitwise sese ad give a coditio o which graph parameters are cotiuous i the uiform sese.

3 ON THE CONTINUITY OF GRAPH PARAMETERS 3 2. Graph Distace Metric This metric itroduced by Buke ad Shearer i [3] is based o the size of the largest commo subgraph betwee two graphs. Defiitio 2.1. Give G, G 1 ad G 2, we say that G is a commo subgraph of G 1 ad G 2 if there exist isomorphisms from G to a subgraph of G 1 ad from G to a subgraph of G 2. A maximum commo subgraph of G 1 ad G 2 is a commo subgraph of maximum size. We use the otatio mcs(g 1, G 2 ) to deote the maximum size of ay commo subgraph of G 1 ad G 2. It is importat to ote that a maximum commo subgraph of two graphs is ot ecessarily uique. Note also that maximum commo subgraphs do ot eed to be coected. Defiitio 2.2. The distace betwee two graphs G 1, G 2 G is defied as (1) d mcs (G 1, G 2 ) = 1 mcs(g 1, G 2 ) max( G 1, G 2 ) where max( G 1, G 2 ) is the maximum size of the graphs G 1 ad G 2. Clearly, d mcs is symmetric; Buke ad Shearer showed i [3] that d mcs also satisfies the triagle iequality. Whe d mcs (G 1, G 2 ) = 0, it implies that mcs(g 1, G 2 ) = G 1 = G 2, ad thus G 1 ad G 2 are isomorphic, but ot ecessarily equal. Thus, d mcs is ot a metric over all graphs, but it is a metric over all isomorphism classes of graphs. I the followig, we will use G to deote the space of all isomorphism classes of oempty graphs. We will use G to deote the set of isomorphism classes of graphs o vertices. To avoid cumbersome otatio, we will use the expressio G G to deote both the graph itself, ad its isomorphism class. 3. (G, d mcs ) Iduces The Discrete Topology As d mcs forms a metric o the set G, the pair (d mcs, G) forms a metric space. I this sectio we will show that this metric space is actually the discrete topology. We will show that d mcs iduces the discrete topology o G by showig that all ope balls i the metric space are also ope sets i the topology. We have the followig defiitio for a ope ball i G. Defiitio 3.1. If G G ad r > 0, the a ope ball cetered at G with radius r is the set B(G, r) = {H G : d mcs (G, H) < r}

4 4 M. HURSHMAN AND J. JANSSEN To cosider what ope balls look like, we cosider what possible distaces ca exist betwee a fixed graph G ad all other graphs i G. By defiitio, d mcs (G, H) [0, 1) Q. Sice d mcs is a metric, d mcs (G, H) = 0 precisely whe G = H. A distace of 1 is ot possible as graphs i G are o-empty, so ay two graphs G ad H always have at least oe vertex i commo so that mcs(g, H) 1. Therefore, for r 1 we have that B(G, r) = G. Sice mcs(g, H) is a iteger, for graphs G ad H with sizes G = ad H = m, d mcs (G, H) := i for some i {1,..., m}. This observatio leads to the followig theorem. Theorem 3.2. Cosider a fixed graph G of size ad let H G. The the miimum o zero value for d mcs (G, H) is Proof. Suppose G ad H are ot isomorphic, so mcs(g, H) mi{ G, H }. Let H = m. If m, the miimum distace betwee G ad H with H G m is 1. If m >, the the miimum distace betwee G ad H with H G m is m. We thus have to cosider the miimum of the set m { m : m > }. The miimum distace occurs whe m = + 1, so the m miimum distace betwee G ad H is the d mcs (G, H) = Thus, if we take r = 1 1+ G we get B(G, r) = {G}. Therefore, for all G G, the set {G} is ope, which proves the followig corollary. Corollary 3.3. The distace fuctio d mcs iduces the discrete topology o G. Theorem 3.2 says that for a fixed graph G, there is a limit o how small d mcs (G, H) ca be for H G. We ca ask whether this is the case for ay G, H G. That is, ca we fid G, H so that d mcs (G, H) is as small as we like? This questio is aswered by the followig theorem. Theorem 3.4. For ay q [0, 1) Q, there exist two graphs G ad H so that d mcs (G, H) = q. Proof. Let q = i/, where ad i are o-egative itegers, ad i <. Let G = K. Let H be the disjoit uio of the clique K i ad the set of isolated vertices K i. as follows. The mcs(g, H) = i, ad d mcs (G, H) = 1 i = i = q

5 ON THE CONTINUITY OF GRAPH PARAMETERS 5 4. Distace betwee radom pairs of graphs I this sectio, we ivestigate the mcs distace betwee two radomly chose graphs i the well-kow Erdös-Reyi radom graph probability space G(, p). The mai theorem below states that almost ay pair of radomly chose graphs i G(, p) are at ear-maximum distace from each other. Namely, the distace betwee these graphs teds to 1 as teds to ifiity, while the defiitio of mcs distace is such that this distace ever ca exceed 1. For the special case where p = 1/2, G(, p) gives the uiform distributio o all graphs with vertices (all labelled graphs). Thus, a corollary of the the theorem is that almost all pairs of graphs of size have mcs distace close to 1. Theorem 4.1. Let G ad H be two graphs chose accordig to G(, p). The almost surely 1 5 log 1/p () < d mcs (G, H) < 1 2 log 1/p (). Proof. Let G ad H be two graphs chose accordig to G(, p). Cosider the graph GH formed as follows: V (GH) = [], ad for each pair i < j i [], i ad j are adjacet i GH precisely whe i ad j are adjacet i both G ad H, or i ad j are o-adjacet i both G ad H. Clearly, ay clique of size k i GH correspods to a commo subgraph of size k i G ad H. Also, the probability of a edge occurrig i GH equals p = p 2 + (1 p) 2 = 2p 2 2p + 1, ad edges still occur idepedetly, so GH ca be cosidered to be chose accordig to G(, p ). It a well kow result i radom graph theory (see for example [1]) that almost surely the clique umber G(, p ) is at least 2 log 1/p. This leads to upper boud o d mcs (G, H). For the lower boud, we compute the probability that G ad H have a commo subgraph of size k. Let X be the umber of commo subgraphs of size k. For every set S, ad every oe-to-oe map f : S [], defie the idicator variable X S,f, where X S,f = 1 if f is a isomorphism from the subgraph of G iduced by S to the subgraph of H iduced by f(s), ad zero otherwise. The X = S,f X S,f, ad by liearity of expectatio, E(X) = S,f EX S,f. For each particular choice of S ad f, EX S,f = P(X S,f = 1) = (p ) (k 2). Namely, for each pair {i, j} S, either i, j are adjacet i G ad f(i), f(j) are adjacet i H, which happes with probability p 2, or i, j are o-adjacet i G ad f(i), f(j) are o-adjacet i H, with probability (1 p) 2.

6 6 M. HURSHMAN AND J. JANSSEN There are ( k) k choices for S, ad, give S, there are ( 1)... ( k + 1) k choices for f. Therefore, EX 2k (p ) (k 2) 2k (1/p ) k2 /2 + o(1). Now let k = 5 log 1/p (), so (1/p ) k2 /2 = 2.5k. The EX = O( 0.5k ) = o(1). By Markov s iequality, P(d mcs (G, H) < k ) = P(X 1) EX, ad EX 0 whe. The reader may ot thik that this result exteds to the ulabelled graphs of G sice the proof assumes we are workig with the labelled graphs of G(, p). This is ot the case. For the upper boud, the maximum commo subgraph betwee two labelled graphs is less tha or equal to the maximum commo subgraph betwee their correspodig ulabelled graphs so that the upper boud still holds. For the lower boud, we cosider all subsets of vertices of both graphs ad all oe-tooe fuctios betwee them so that the lower boud give is precisely the lower boud for the ulabelled graphs as well. 5. Graph Cotiuity I this sectio we itroduce a otio of cotiuity for graph parameters. Our defiitios follow the stadard defiitio of cotiuity of fuctios, but are adapted to (G, d mcs ). Most commoly, cotiuity refers to poitwise cotiuity. I our settig, a graph fuctio f : G R is cotiuous at G 0 G if for all ɛ > 0 there exists a δ > 0 such that d(g, G 0 ) < δ implies that f(g) f(g 0 ) < ɛ. This defiitio is ot very useful for the metric space uder cosideratio, sice all graph fuctios f : G R are trivially poitwise cotiuous. Namely, by Theorem 3.2, if we select δ = 1 1+ G 0, the oly graph H satisfyig d mcs(g, G 0 ) < δ is G = G 0. Thus, every graph fuctio f is poitwise cotiuous at G. A more useful cocept is that of uiform cotiuity. Defiitio 5.1. A graph fuctio f : G R is uiformly cotiuous if for all ɛ > 0 ad G, H G there exists a δ > 0 such that whe d(g, H) < δ we have that f(g) f(h) < ɛ. May commo graph fuctios, such as chromatic umber, diameter, or girth, are iteger valued. Thus, for such a iteger valued fuctio f, f(g 1 ) f(g 2 ) < 1 implies that f(g 1 ) = f(g 2 ), makig

7 ON THE CONTINUITY OF GRAPH PARAMETERS 7 the coditio for uiform cotiuity too restrictive. For this reaso, we cosider fuctios that are graph parameters ormalized by the size of the graph. Defiitio 5.2. Let f : G R be a graph fuctio. The fuctio f give by f(g) = f(g) G is the ormalized graph fuctio for f. The remaider of this sectio explores the relatio betwee a boudedess coditio o a graph parameter, ad uiform cotiuity of the derived ormalized graph fuctio. Defiitio 5.3. Give a graph parameter f, the step chage of f equals the supremum, over all choices of graph G ad vertex v of G, of f(g) f(g v). Sice the maximum is take over the ifiite family of all graphs, the step chage ca be ifiite. Note that the step chage is always o-egative. Lemma 5.4. Give a graph parameter with step chage C. The for ay two graphs G ad H of size H G =, f(g) f(h) 2Cd mcs (G, H). Proof. Let i = d mcs (G, H), ad let K be a maximum commo subgraph of G ad H. Note that, by the defiitio of d mcs, i is a iteger, ad K = i. Let A = {a 1,..., a i } V (G) ad B = {b 1..., b j } V (H) be so that G A ad H B are isomorphic to K. By repeated applicatio of the defiitio of the step chage we obtai that f(g) f(g a 1 a i ) ic. Similarly, f(h) f(h b 1 b j ) Cj. Sice f(g a 1 a i ) = f(k) = f(h b 1 b j ), we ca coclude that f(g) f(h) f(g) f(k) + f(h) f(k) 2Ci. Theorem 5.5. Let f be a graph fuctio that satisfies f(g) G for all G, ad has fiite step chage C. The f is uiformly cotiuous. Proof. Let f be a graph fuctio ad G ad H are graphs with G = ad H = m ad m. Let i = d mcs (G, H). By the defiitio of the step chage, this implies that f(g) f(h) 2Ci. Now let f be the ormalized graph fuctio for f.

8 8 M. HURSHMAN AND J. JANSSEN Cosider f(g) f(h) ad write f(g) = p 1 ad f(h) = p 2. We have that f(g) f(h) p 1 = p 2 m mp 1 p 2 = m mp 1 mp 2 p 2 + mp 2 = m p 1 p 2 + p 2 ( m) m Now p 1 p 2 f(g) f(h) = 2Ci/ = 2Cd mcs (G, H). Cosider the secod piece of our iequality p 2( m). By assumptio, m p 2 = f(h) H = m, so p 2 1 givig m p 2 ( m) m Therefore, m mcs(g, H) f(g) f(h) (1 + 2C)d mcs (G, H). = d mcs (G, H). Fix ɛ > 0, ad let δ = ɛ. The d 1+2C mcs(g, H) < δ implies f(g) f(h) < ɛ. Thus, f is uiformly cotiuous. The followig theorem gives a result that is close to the coverse of the above theorem. Theorem 5.6. Let f be a graph parameter so that f is uiformly cotiuous. The for all graphs G ad H where H G = ad d mcs (G, H) = i, f(g) f(h) α( i + 1)i, Where α : N N is a fuctio such that α() 0 as. Proof. Let α () = max G =, v V (G) f(g) f(g v),

9 ON THE CONTINUITY OF GRAPH PARAMETERS 9 We first show that α ()/ 0 as. Fix ɛ > 0. Sice f is uiformly cotiuous, there exists δ > 0 be such that d mcs (G, H)) < δ implies that f(g) f(h) < ɛ. Take N N such that 1 < δ. Fix G G of size N, ad N H = G v for some vertex v G. The d mcs (G, H) = 1/ 1/N < δ. Thus we have that f(g) f(h) = f(g) ( 1) f(h) f(g) f(h) ɛ. So for all ɛ > 0, there exists N N so that α ()/ ɛ for all N. Now let α() = sup{α (k)/k k }. Clearly, sice α ()/ goes to zero as goes to ifiity, the so does α()/. Fix G ad H with maximum commo subgraph K, ad H G =. Let i = d mcs (G, H). By the defiitio of the step chage it follows that f(g) f(h) α () + α ( 1) + + α ( i + 1) ( ) α () + α ( 1) + + α( i+1) 1 i+1 iα( i + 1) Theorem 5.5 makes it easy to fid examples of graph parameters whose ormalized fuctios are uiformly cotiuous. For example, the size of a graph, the chromatic umber of a graph, ad the domiatio umber of a graph all have step chage 1, ad thus their ormalized fuctios are uiformly cotiuous. We coclude with a example of a graph fuctio which is ot cotiuous. Theorem 5.7. The diameter is ot a uiformly cotiuous graph fuctio with respect to d mcs Proof. Cosider the graphs P ad C ad let f be the ormalized graph fuctio for the diameter. For all, we have that d(p, C ) = 1 ad f(c ) = 2 ad f(p ) = 1 = 1. For all the differece i the ormalized diameters is the f(c ) f(p ) = 2 ( 1) 1 2 1

10 10 M. HURSHMAN AND J. JANSSEN for 1. Fix ɛ = 1. Fix ay δ > 0, ad let N max{4, 1/(N + 1))}. 4 The d mcs (P N, C N ) = 1/N < δ, but f(c N ) f(p N ) N 1 4. So, for every choice of δ > 0, there exists a pair of graphs that violates the uiformity coditio. Refereces [1] N. Alo, J.H. Specer, The Probabilistic Method, Wiley, Secod Editio, [2] A.T. Balaba, Chemical Applicatios of Graph Theory, Academic Press Ic, [3] H. Buke, K. Shearer, A graph distace metric based o the maximum commo subgraph, Patter Recogitio Letters 19, , [4] H. Buke, X. Jiag, A. Kadel, O the Miimum Commo Supergraph of two Graphs, Computig 65, No.1, 13-25, [5] H. Buke, Graph Matchig: Theoretical Foudatio, Algorithms, ad Applicatios, i Proc. Visio Iterface 2000, Motreal, 82-88, [6] H. Buke, Error Correctig Graph Matchig: O the Ifluece of the Uderlyig Cost Fuctio, IEEE Tras. Patter Aalysis ad Machie Itelligece, Vol 21, No.1, , [7] H. Buke, O a relatio betwee graph edit distace ad maximum commo subgraph, Patter Recogitio Letters, 18, , [8] D. Cote, P.Foggia, C. Sasoe ad M.Veto, Thirty Years of Graph Matchig i Patter Recogitio, Iteratioal Joural of Patter Recogitio ad Artificial Itelligece, Vol. 18, No.3, , [9] R. Diestel, Graph Theory, Spriger, Third Editio, [10] M.L. Ferádez, G. Valiete A graph distace metric combiig maximum commo subgraph ad miimum commo supergraph, Patter Recogitio Letters, 22, , [11] M.R. Garey, D.S. Johso, Computers ad Itractability: A Guide to the Theory of NP-Completeess, Series of Books i Mathematics Scieces, [12] G. Grimmett, D. Stirzaker, Probability ad Radom Processes, Pretice Hall, 3rd Editio, [13] S. Jaso, T. Luczak, A. Ruciski, Radom Graphs, Wiley, First Editio, [14] I. Koch, Eumeratig all coected maximum subgraphs i two graphs, Theoretical Computer Sciece, 1-30, [15] S.Q. Le, T.B. Ho, T.T.H. Pha, A Novel Graph-Based Similarity Measure for 2D Chemical Structures, Geome Iformatics, 15, 82-91, [16] B.D. McKay, Practical graph isomorphism, Cogressus Numeratium 30: 4587, 10th. Maitoba Coferece o Numerical Mathematics ad Computig (Wiipeg, 1980) [17] J.R. Mukres, Topology, Pretice Hall, 2d Editio, [18] E.M. Palmer, Graphical Evolutio: A Itroductio to the Theory of Radom Graphs, Wiley, First Editio, [19] P. Papadimitriou, A. Dasda, H. Garcia-Molia, Web Graph Similarity for Aomaly Detectio WWW 2008, Beijig, Chia, April

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