A Local Spectral Method for Graphs: with Applications to Improving Graph Partitions and Exploring Data Graphs Locally

A Lo cal Sp ectral Metho d for Graphs: with Applications to Impro ving Graph P artitions and Exploring Data Graphs Lo cally Mic hael W. Mahoney ∗ Lorenzo Orecc hia † Nisheeth K. Vishnoi ‡ Abstract The second eigen v alue of the Laplacian matrix and its asso ciated eigen vector are fund amen- tal features of an undirected graph, and as such they ha v e found widespread use in scientiﬁc computing, mac hine learning, and data analysis. In many applications, how ev er, graphs that arise hav e sev eral lo c al regions of in terest, and the second eigenv ector will typically fail to pro- vide information ﬁne-tuned to each lo cal region. In this pap er, we in tro duce a lo cally-biased analogue of the second eigenv ector, and w e demonstrate its usefulness at highligh ting lo cal prop erties of data graphs in a semi-sup ervised manner. T o do so, w e ﬁrst view the second eigen vector as the solution to a constrained optimization problem, and w e incorp orate the lo cal information as an additional constraint; we then c haracterize the optimal solution to this new problem and sho w that it can b e interpreted as a generalization of a Personalized P ageRank v ector; and ﬁnally , as a consequence, we show that the solution can b e computed in nearly-linear time. In addition, we sho w that this lo cally-biased v ector can b e used to compute an appro ximation to the b est partition ne ar an input seed set in a manner analo- gous to the wa y in which the second eigenv ector of the Laplacian can b e used to obtain an appro ximation to the b est partition in the entire input graph. Suc h a primitive is useful for iden tifying and reﬁning clusters lo cally , as it allows us to fo cus on a lo cal region of in terest in a semi-sup ervised manner. Finally , we pro vide a detailed empirical ev aluation of our metho d b y showing how it can applied to ﬁnding lo cally-biased sparse cuts around an input vertex seed set in social and information netw orks. 1 In tro duction Sp ectral metho ds are p opular in machine learning, data analysis, and applied mathematics due to their strong underlying theory and their go o d performance in a wide range of applications. In the study of undirected graphs, in particular, spectral techniques pla y an imp ortan t role, as man y fundamental structural prop erties of a graph dep end directly on spectral quantities asso ciated with matrices representing the graph. Tw o fundamental ob jects of study in this area are the second smallest eigen v alue of the graph Laplacian and its asso ciated eigenv ector. These quan tities determine many features of the graph, including the b ehavior of random walks and the presence of sparse cuts. This relationship b et w een the graph structure and an easily-computable quan tity has been exploited in data clustering, comm unity detection, image segmentation, parall el computing, and many other applications. ∗ Departmen t of Mathematics, Stanford Univ ersity , Stanford, CA 94305. mmahoney@cs.stanford.edu . † Computer Science Division, UC Berkeley , Berkeley , CA, 94720. orecchia@eecs.berkeley.edu . ‡ Microsoft Research, Bangalore, India. nisheeth.vishnoi@gmail.com . 1 A potential drawbac k of using the second eigen v alue and its associated eigen vector is that they are inheren tly glob al quan tities, and thus they ma y not b e sensitive to v ery lo c al information. F or instance, a sparse cut in a graph may b e p oorly correlated with the second eigenv ector (and ev en with all the eigenv ectors of the Laplacian) and thus in visible to a metho d based only on eigen vector analysis. Similarly , based on domain knowledge one might ha ve information ab out a sp eciﬁc target region in the graph, in which case one migh t be in terested in ﬁnding clusters only near this presp eciﬁed lo cal region, e.g. , in a semi-sup ervised manner; but this lo cal region might b e essentially invisible to a metho d that uses only global eigen vectors. F or these and related reasons, standard global sp ectral techniques can hav e substantial diﬃculties in semi-supervised settings, where the goal is to learn more ab out a lo cally-biased target region of the graph. In this pap er, we pro vide a metho dology to construct a lo cally-biased analogue of the second eigen v alue and its asso ciated eigenv ector, and w e demonstrate b oth theoretically and empirically that this lo calized vector inherits many of the go o d prop erties of the global second eigen vector. Our approach is inspired b y viewing the second eigen vector as the optimum of a constrained global quadratic optimization program. T o mo del the lo calization step, we mo dify this program b y adding a natural lo calit y constrain t. This locality constraint requires that an y feasible solution ha ve suﬃcient correlation with the target region, which we assume is given as input in the form of a set of no des or a distribution ov er vertices. The resulting optimization problem, which we name Lo calSp ectral and whic h is display ed in Figure 1, is the main ob ject of our w ork. The main adv antage of our form ulation is that an optimal solution to Lo calSp ectral captures man y of the same structural properties as the global eigenv ector, except in a locally-biased setting. F or example, as with the global optimization program, our lo cally-biased optimization program has an in tuitive geometric in terpretation. Similarly , as with the global eigenv ector, an optimal solution to Lo calSp ectral is eﬃciently computable. T o sho w this, we c haracterize the optimal solutions of LocalSp ectral and sho w that such a solution can b e constructed in nearly-linear time b y solving a system of linear equations. In applications where the eigenv ectors of the graph are pre- computed and only a small num b er of them are needed to describ e the data, the optimal solution to our program can be obtained b y p erforming a small num b er of inner product computations. Finally , the optimal solution to LocalSp ectral can b e used to deriv e b ounds on the mixing time of random w alks that start near the lo cal target region as well as on the existence of sparse cuts near the locally-biased target region. In particular, it low er b ounds the conductance of cuts as a function of ho w well-correlated they are with the seed vector. This will allow us to exploit the analogy betw een global eigenv ectors and our localized analogue to design an algorithm for disco vering sparse cuts near an input seed set of vertices. In order to illustrate the empirical b eha vior of our metho d, we will describ e its p erformance on the problem of ﬁnding lo cally-biased sparse cuts in real data graphs. Subsequen t to the dissemination of the initial technical rep ort version of this pap er, our metho dology was applied to the problem of ﬁnding, given a small num b er of “ground truth” labels that correspond to kno wn segmen ts in an image, the segments in whic h those lab els reside [24]. This computer vision application will be discussed brieﬂy . Then, we will describe in detail ho w our algorithm for discov ering sparse cuts near an input seed set of vertices ma y be applied to the problem of exploring data graphs lo cally and to identifying locally-biased clusters and comm unities in a more diﬃcult-to-visualize so cial netw ork application. In addition to illustrating the p erformance of the metho d in a practical application related to the one that initially motiv ated this work [20, 21, 22], this so cial graph application will illustrate how the v arious “knobs” of our metho d can b e used in practice to explore the structure of data graphs in a lo cally-biased manner. Recen t theoretical w ork has fo cused on using spectral ideas to ﬁnd goo d clusters nearby an input seed set of nodes [30, 1, 10]. These metho ds are based on running a num ber of lo cal random w alks around the seed set and using the resulting distributions to extract information ab out 2 clusters in the graph. Recent empirical work has used P ersonalized P ageRank, a particular v arian t of a lo cal random walk, to characterize very ﬁnely the clustering and communit y structure in a wide range of very large so cial and information netw orks [2, 20, 21, 22]. In contrast with previous metho ds, our lo cal sp ectral metho d is the ﬁrst to b e derived in a direct wa y from an explicit optimization problem inspired b y the global sp ectral problem. Interestingly , our characterization also shows that optimal solutions to Lo calSpectral are generalizations of Personalized PageRank, pro viding an additional insight to why lo cal random w alk metho ds work w ell in practice. In the next section, w e will describ e relev ant background and notation; and then, in Section 3, w e will presen t our form ulation of a locally-biased sp ectral optimization program, the solution of which will provide a lo cally-biased analogue of the second eigenv ector of the graph Laplacian. Then, in Section 4 w e will describ e how our metho d ma y b e applied to identifying and reﬁning lo cally-biased partitions in a graph; and in Section 5 we will provide a detailed empirical ev aluation of our algorithm. Finally , in Section 6, w e will conclude with a discussion of our results in a broader context. 2 Bac kground and Notation. Let G = ( V , E , w ) be a connected undirected graph with n = | V | v ertices and m = | E | edges, in which edge { i, j } has weigh t w ij . F or a set of v ertices S ⊆ V in a graph, the volume of S is v ol( S ) def = P i ∈ S d i , in whic h case the volume of the gr aph G is vol( G ) def = vol( V ) = 2 m . In the follo wing, A G ∈ R V × V will denote the adjacency matrix of G , while D G ∈ R V × V will denote the diagonal degree matrix of G , i.e. , D G ( i, i ) = d i = P { i,j }∈ E w ij , the weigh ted degree of v ertex i . The Laplacian of G is deﬁned as L G def = D G − A G . (This is also called the combinatorial Laplacian, in which case the normalized Laplacian of G is L G def = D − 1 / 2 G L G D − 1 / 2 G .) The Laplacian is the symmetric matrix ha ving quadratic form x T L G x = P ij ∈ E w ij ( x i − x j ) 2 , for x ∈ R V . This implies that L G is p ositive semideﬁnite and that the all-one vector 1 ∈ R V is the eigenv ector corresp onding to the smallest eigen v alue 0. F or a symmetric matrix A , we will use A  0 to denote that it is positive semi-deﬁnite. Moreo ver, giv en t w o symmetric matrices A and B , the expression A  B will mean A − B  0. F urther, for t wo n × n matrices A and B , w e let A ◦ B denote T r ( A T B ). Finally , for a matrix A, let A + denote its (uniquely deﬁned) Mo ore-P enrose pseudoinv erse. F or t wo vectors x, y ∈ R n , and the degree matrix D G for a graph G , w e deﬁne the degree- w eighted inner pro duct as x T D G y def = P n i =1 x i y i d i . Given a subset of vertices S ⊆ V , w e denote b y 1 S the indicator vector of S in R V and by 1 the vector in R V ha ving all entries set equal to 1. W e consider the following deﬁnition of the complete graph K n on the vertex set V : A K n def = 1 vol( G ) D G 11 T D G . Note that this is not the standard complete graph, but a weigh ted version of it, where the weigh ts dep end on D G . With this scaling we ha ve D K n = D G . Hence, the Laplacian of the complete graph deﬁned in this manner b ecomes L K n = D G − 1 vol( G ) D G 11 T D G . In this pap er, the c onductanc e φ ( S ) of a cut ( S, ¯ S ) is φ ( S ) def = vol( G ) · | E ( S, ¯ S ) | vol( S ) · vol( ¯ S ) . A sparse cut, also called a go o d-conductance partition, is one for which φ ( S ) is small. The c onductanc e of the gr aph G is then φ ( G ) = min S ⊆ V φ ( S ). Note that the conductance of a set S , or equiv alen tly a cut ( S, ¯ S ), is often deﬁned as φ 0 ( S ) = | E ( S, ¯ S ) | / min { vol( S ) , v ol( ¯ S ) } . This notion is equiv alent to that φ ( S ), in that the v alue φ ( G ) thereb y obtained for the conductance of the graph G diﬀers b y no more than a factor of 2 times the constant v ol( G ), dep ending on which notion we use for the conductance of a set. 3 min x T L G x s.t. x T D G x = 1 ( x T D G 1) 2 = 0 x ∈ R V min x T L G x s.t. x T D G x = 1 ( x T D G 1) 2 = 0 ( x T D G s ) 2 ≥ κ x ∈ R V Figure 1: Global and lo cal spectral optimization programs. Left: The usual sp ectral program Sp ectral ( G ). Righ t: Our new lo cally-biased sp ectral program LocalSp ectral ( G, s, κ ). In b oth cases, the optimization v ariable is the vector x ∈ R n . 3 The Lo calSp ectral Optimization Program In this section, w e introduce the lo cal sp ectral optimization program Lo calSpectral ( G, s, κ ) as a strengthening of the usual global sp ectral program Sp ectral ( G ). T o do so, w e will augmen t Sp ectral ( G ) with a locality constrain t of the form ( x T D G s ) 2 ≥ κ , for a seed v ector s and a corre- lation parameter κ . Both these programs are homogeneous quadratic programs, with optimization v ariable the v ector x ∈ R V , and thus any solution v ector x is essentially equiv alen t to − x for the purp ose of these optimizations. Hence, in the following w e do not diﬀerentiate b et ween x and − x , and we assume a suitable direction is c hosen in eac h instance. 3.1 Motiv ation for the Program Recall that the second eigenv alue λ 2 ( G ) of the Laplacian L G can b e viewed as the optim um of the standard optimization problem Sp ectral ( G ) describ ed in Figure 1. In matrix terminology , the corresp onding optimal solution v 2 is a generalized eigen vector of L G with resp ect to D G . F or our purp oses, how ev er, it is b est to consider the geometric meaning of this optimization form ulation. T o do so, suppose w e are op erating in a vector space R V , where the i th dimension is stretched b y a factor of d i , so that the natural iden tity op erator is D G and the inner pro duct b et ween t wo v ectors x and y is given b y P i ∈ V d i x i y i = x T D G y . In this represen tation, Sp ectral ( G ) is seeking the vector x ∈ R V that is orthogonal to the all-one vector, lies on the unit sphere, and minimizes the Laplacian quadratic form. Note that such an optimum v 2 ma y lie an ywhere on the unit sphere. Our goal here is to modify Spectral ( G ) to incorp orate a bias to w ards a target region which w e assume is given to us as an input vector s . W e will assume (without loss of generality) that s is prop erly normalized and orthogonalized so that s T D G s = 1 and s T D G 1 = 0. While s can b e a general unit v ector orthogonal to 1, it may b e helpful to think of s as the indicator v ector of one or more vertices in V , corresp onding to the target region of the graph. W e obtain Lo calSpectral ( G, s, κ ) from Spectral ( G ) b y requiring that a feasible solution also hav e a suﬃciently large correlation with the v ector s . This is ac hieved b y the addition of the constraint ( x T D G s ) 2 ≥ κ , whic h ensures that the pro jection of x onto the direction s is at least √ κ in absolute v alue, where the parameter κ is also an input parameter ranging b etw een 0 and 1. Thus, we would lik e the solution to b e well-connected with or to lie near the se e d v ector s . In particular, as display ed pictorially in Figure 2, x must lie within the spherical cap cen tered at s that con tains all vectors at an angle of at most arccos( √ κ ) from s . Thus, higher v alues of κ demand a higher correlation with s and, hence, a stronger lo calization. Note that in the limit κ = 0, the sph erical cap constituting the feasible region of the program is guaranteed to include v 2 and Lo calSp ectral ( G, s, κ ) is equiv alen t to Sp ectral ( G ). In the rest of this pap er, we refer to s as the se e d ve ctor and to κ as the c orr elation 4 p ar ameter for a giv en Lo calSpectral ( G, s, κ ) optimization problem. Moreo ver, we denote the ob jective v alue of the program Lo calSpectral ( G, s, κ ) b y the num b er λ ( G, s, κ ). 1 s v 2 √κ Figure 2: (Best seen in color.) Pictorial representation of the feasible regions of the optimization programs Sp ectral ( G ) and Lo calSpect ral ( G, s, κ ) that are deﬁned in Figure 1. See the text for a discussion. 3.2 Characterization of the Optimal Solutions of Lo calSp ectral Our ﬁrst theorem is a c haracterization of the optimal solutions of Lo calSp ectral . Although Lo- calSp ectral is a non-con vex program (as, of course, is Sp ectral ), the follo wing theorem states that solutions to it can b e expressed as the solution to a system of linear equations which has a nat- ural interpretation. The pro of of this theorem (which may be found in Section 3.4) will inv olv e a relaxation of the non-con v ex program Lo calSp ectral to a con vex semideﬁnite program (SDP), i.e. , the v ariables in the optimization program will b e distributions ov er v ectors rather than the v ectors themselv es. F or the statement of this theorem, recall that A + denotes the (uniquely deﬁned) Mo ore-P enrose pseudoinv erse of the matrix A . Theorem 1 (Solution Characterization) L et s ∈ R V b e a se e d ve ctor such that s T D G 1 = 0 , s T D G s = 1 , and s T D G v 2 6 = 0 , wher e v 2 is the se c ond gener alize d eigenve ctor of L G with r esp e ct to D G . In addition, let 1 > κ ≥ 0 b e a c orr elation p ar ameter, and let x ? b e an optimal solution to Lo calSp ectral ( G, s, κ ) . Then, ther e exists some γ ∈ ( −∞ , λ 2 ( G )) and a c ∈ [0 , ∞ ] such that x ? = c ( L G − γ D G ) + D G s. (1) There are several parameters (such as s , κ , γ , and c ) in the statement of Theorem 1, and un- derstanding their relationship is imp ortan t: s and κ are the parameters of the program; c is a normalization factor that rescales the norm of the solution vector to b e 1 (and that can b e com- puted in linear time, given the solution vector); and γ is implicitly deﬁned b y κ , G , and s . The correct setting of γ ensures that ( s T D G x ? ) 2 = κ, i.e. , that x ? is found exactly on the b oundary of the feasible region. A t this p oin t, it is imp ortan t to notice the b eha vior of x ? and γ as κ c hanges. As κ go es to 1, γ tends to −∞ and x ? approac hes s ; conv ersely , as κ go es to 0, γ goes to λ 2 ( G ) and x ? tends tow ards v 2 , the global eigenv ector. W e will discuss how to compute γ and x ? , giv en a sp eciﬁc κ , in Section 3.3. Finally , we should note that there is a close connection b et w een the solution v ector x ? and the p opular P ageRank pro cedure. Recall that P ageRank refers to a metho d to determine a global rank or global notion of imp ortance for a no de in a graph suc h as the web that is based on the link structure of the graph [8, 19, 5]. There hav e b een several extensions to the basic P ageRank 5 concept, including T opic-Sensitive P ageRank [14] and Personalized PageRank [15]. In the same w ay that PageRank can b e viewed as a wa y to express the quality of a w eb page ov er the entire w eb, Personalized PageRank expresses a link-based measure of page quality around user-selected pages. In particular, giv en a v ector s ∈ R V and a telep ortation constant α > 0, the P ersonalized P ageRank vector can b e written as pr α,s =  L G + 1 − α α D G  − 1 D G s [1]. By setting γ = − 1 − α α , the optimal solution to Lo calSpect ral is prov ed to b e a generalization of Personalized PageRank. In particular, this means that for high v alues of the correlation parameter κ , for which the corresp onding γ in Theorem 1 is negative, the optimal solution to Lo calSp ectral tak es the form of a P ersonalized P ageRank vector. On the other hand, when γ ≥ 0 , the optimal solution to Lo calSpectral pro vides a smo oth wa y of transitioning from the Personalized P ageRank v ector to the global second eigen vector v 2 . 3.3 Computation of the Optimal Solutions of Lo calSp ectral In this section, w e discuss how to compute eﬃciently an optimal solution for Lo calSp ectral ( G, s, κ ), for a ﬁxed c hoice of the parameters G , s , and κ . The following theorem is our main result. Theorem 2 (Solution Computation) F or any ε > 0 , a solution to Lo calSpectral ( G, s, κ ) of value at most (1 + ε ) · λ ( G, s, κ ) c an b e c ompute d in time ˜ O ( m / √ λ 2 ( G ) · log( 1 / ε )) using the Conjugate Gr adient Metho d [13]. Alternatively, such a solution c an b e c ompute d in time ˜ O ( m log( 1 / ε )) using the Spielman-T eng line ar-e quation solver [30]. Pr o of: By Theorem 1, we kno w that the optimal solution x ? m ust b e a unit-scaled version of y ( γ ) = ( L G − γ D G ) + D G s, for an appropriate choice of γ ∈ ( −∞ , λ 2 ( G )) . Notice that, giv en a ﬁxed γ , the task of computing y ( γ ) is equiv alent to solving the system of linear equations ( L G − γ D G ) y = D G s for the unkno wn y . This op eration can b e p erformed, up to accuracy ε, in time ˜ O ( m / √ λ 2 ( G ) · log( 1 / ε )) using the Conjugate Gradien t Metho d, or in time ˜ O ( m log( 1 / ε )) using the Spielman-T eng linear-equation solver. T o ﬁnd the correct setting of γ , it suﬃces to p erform a binary search ov er the p ossible v alues of γ in the in terv al ( − v ol( G ) , λ 2 ( G )) , un til ( s T D G x ) 2 is suﬃcien tly close to κ.  W e should note that, depending on the application, other methods of computing a solution to Lo calSpectral ( G, s, κ ) might b e more appropriate. In particular, if an eigen vector decomp osition of L G has b een pre-computed, as is the case in certain machine learning and data analysis appli- cations, then this computation can b e mo diﬁed as follows. Given an eigenv ector decomp osition of L G as L G = P n i =2 λ i D 1 / 2 G u i u T i D 1 / 2 G , then y ( γ ) must tak e the form y ( γ ) = ( L G − γ D G ) + D G s = n X i =2 1 λ i − γ ( s T D 1 / 2 G u ) 2 , for the same choice of c and γ , as in Theorem 1. Hence, giv en the eigen v ector decomp osition, eac h guess y ( γ ) of the binary searc h can b e computed b y expanding the ab ov e series, which requires a linear n umber of inner pro duct computations. While this may yield a worse running time than Theorem 2 in the w orst case, in the case that the graph is w ell-approximated by a small num b er k of dominan t eigenv ectors, then the computation is reduced to only k straigh tforward inner pro duct computations. 3.4 Pro of of Theorem 1 W e start with an outline of the pro of. Although the program Lo calSp ectral ( G, s, κ ) is not con vex, it can b e relaxed to the conv ex semideﬁnite program SDP p ( G, s, κ ) of Figure 3. Then, one 6 minimize L G ◦ X s.t. L K n ◦ X = 1 ( D G s )( D G s ) T ◦ X ≥ κ X  0 maximize α + κβ s.t. L G  αL K n + β ( D G s )( D G s ) T β ≥ 0 α ∈ R Figure 3: Left: Primal SDP relaxation of Lo calSp ectral ( G, s, κ ): SDP p ( G, s, κ ); for this primal, the optimization v ariable is X ∈ R V × V suc h that X is symmetric and p ositiv e semideﬁnite. Righ t: Dual SDP relaxation of Lo calSpectral ( G, s, κ ): SDP d ( G, s, κ ); for this dual, the optimization v ariables are α, β ∈ R . Recall that L K n def = D G − 1 vol( G ) D G 11 T D G . can observe that strong duality holds for this SDP relaxation. Using strong dualit y and the related complementary slackness conditions, one can argue that the primal SDP p ( G, s, κ ) has a rank one unique optimal solution under the conditions of the theorem. This implies that the optimal solution of SDP p ( G, s, κ ) is the same as the optimal solution of Lo calSp ectral ( G, s, κ ). Moreo ver, combining this fact with the complemen tary slac kness condition obtained from the dual SDP d ( G, s, κ ) of Figure 3, one can derive that the optimal rank one solution is of the form promised by Theorem 1. Before proceeding with the details of the pro of, we pause to mak e sev eral p oin ts that should help to clarify our approach. • First, since it may seem to some readers to b e unnecessarily complex to relax Lo calSp ectral as an SDP , we emphasize that the motiv ation for relaxing it in this wa y is that we would lik e to pro ve Theorem 1. T o prov e this theorem, we must understand the form of the optimal solutions to the non-conv ex program Lo calSpectral . Thus, in order to ov ercome the non-con vexit y , w e relax Lo calSp ectral to SDP p ( G, s, κ ) (of Figure 3) by “lifting” the rank-1 condition implicit in Lo calSpectral . Then, strong duality applies; and it implies a set of suﬃcien t optimality conditions. By combining these conditions, w e will b e able to establish that an optimal solution X ? to SDP p ( G, s, κ ) has rank 1, i.e. , it has the form X ? = x ? x ?T for some vector x ? ; and thus it yields an optimal solution to Lo calSp ectral , i.e. , the vector x ? . • Second, in general, the v alue of a relaxation lik e SDP p ( G, s, κ ) ma y b e strictly less than that of the original program ( LocalSp ectral , in this case). Our c haracterization and proof will imply that the relaxation is tight, i.e. , that the optimum of SDP p ( G, s, κ ) equals that of Lo calSp ectral . The reason is that one can ﬁnd a rank-1 optimal solution to SDP p ( G, s, κ ), whic h then yields an optimal solution of the same v alue for LocalSp ectral . Note that this also implies that strong dualit y holds for the non-con vex Lo calSpectral , although this observ ation is not needed for our pro of. That is, although it ma y b e p ossible to prov e Theorem 1 in some other w ay that do es not in volv e SDPs, w e chose this pro of since it is simple and intuitiv e and correct; and w e note that App endix B in the textb ook of Boyd and V andenberghe [7] prov es a similar statemen t by the same SDP-based approac h. Returning to the details of the pro of, we will proceed to prov e the theorem by establishing a sequence of claims. First, consider SDP p ( G, s, κ ) and its dual SDP d ( G, s, κ ) (as sho wn in Figure 3). The following claim uses the fact that, giv en X = xx T for x ∈ R V , and for any matrix A ∈ R V × V , w e hav e that A ◦ X = x T Ax . In particular, L G ◦ X = x T L G x , for an y graph G , and ( x T D G s ) 2 = x T D G ss T D G x = D G ss T D G ◦ X . 7 Claim 1 The primal SDP p ( G, s, κ ) is a r elaxation of the ve ctor pr o gr am Lo calSpect ral ( G, s, κ ) . Pr o of: Consider a v ector x that is a feasible solution to LocalSp ectral ( G, s, κ ), and note that X = xx T is a feasible solution to SDP p ( G, s, κ ).  Next, we establish the strong duality of SDP p ( G, s, κ ). (Note that the feasibility conditions and complemen tary slackness conditions stated b elo w may not suﬃce to establish the optimalit y , in the absence of this claim; hence, without this claim, we could not pro ve the subsequent claims, whic h are needed to pro ve the theorem.) Claim 2 Str ong duality holds b etwe en SDP p ( G, s, κ ) and SDP d ( G, s, κ ) . Pr o of: Since SDP p ( G, s, κ ) is con v ex, it suﬃces to v erify that Slater’s constraint qualiﬁcation condition [7] is true for this primal SDP . Consider X = ss T . Then, ( D G s )( D G s ) T ◦ ss T = ( s T D G s ) 2 = 1 > κ .  Next, we use this result to establish the following tw o claims. In particular, strong duality allows us to prov e the follo wing claim showing the KKT-conditions, i.e. , the feasibility conditions and complemen tary slac kness conditions stated b elo w, suﬃce to establish optimalit y . Claim 3 The fol lowing fe asibility and c omplementary slackness c onditions ar e suﬃcient for a primal-dual p air X ? , α ? , β ? to b e an optimal solution. The fe asibility c onditions ar e: L K n ◦ X ? = 1 (2) ( D G s )( D G s ) T ◦ X ? ≥ κ (3) L G − α ? L K n − β ? ( D G s )( D G s ) T  0 (4) β ? ≥ 0 , (5) and the c omplementary slackness c onditions ar e: α ? ( L K n ◦ X ? − 1) = 0 (6) β ? (( D G s )( D G s ) T ◦ X ? − κ ) = 0 (7) X ? ◦ ( L G − α ? L K n − β ? ( D G s )( D G s ) T ) = 0 . (8) Pr o of: This follows from the conv exit y of SDP p ( G, s, κ ) and Slater’s condition [7].  Claim 4 These fe asibility and c omplementary slackness c onditions, c ouple d with the assumptions of the the or em, imply that X ? must b e r ank 1 and β ? > 0 . Pr o of: Plugging in v 2 in Equation (4), w e obtain that v T 2 L G v 2 − α ? − β ? ( v T 2 D G s ) 2 ≥ 0 . But v T 2 L G v 2 = λ 2 ( G ) and β ? ≥ 0 . Hence, λ 2 ( G ) ≥ α ? . Suppose α ? = λ 2 ( G ) . As s T D G v 2 6 = 0 , it m ust b e the case that β ? = 0 . Hence, by Equation (8), we m ust hav e X ? ◦ L ( G ) = λ 2 ( G ) , whic h implies that X ? = v 2 v T 2 , i.e. , the optim um for LocalSp ectral is the global eigen v ector v 2 . This corresp onds to a choice of γ = λ 2 ( G ) and c tending to inﬁnit y . Otherwise, w e may assume that α ? < λ 2 ( G ) . Hence, since G is connected and α ? < λ 2 ( G ) , L G − α ? L K n has rank exactly n − 1 and kernel parallel to the vector 1 . F rom the complementary slac kness condition (8) we can deduce that the image of X ? is in the kernel of L G − α ? L K n − β ? ( D G s )( D G s ) T . If β ? > 0 , w e hav e that β ? ( D G s )( D G s ) T is a rank one matrix and, since s T D G 1 = 0 , it reduces the rank of L G − α ? L K n b y one precisely . If β ? = 0 then X ? m ust b e 0 which is not 8 p ossible if SDP p ( G, s, κ ) is feasible. Hence, the rank of L G − α ? L K n − β ? ( D G s )( D G s ) T m ust b e exactly n − 2 . As w e ma y assume that 1 is in the k ernel of X ? , X ? m ust be of rank one. This pro ves the claim.  No w we complete the pro of of the theorem. F rom the claim it follo ws that, X ? = x ? x ?T where x ? satisﬁes the equation ( L G − α ? L K n − β ? ( D G s )( D G s ) T ) x ? = 0 . F rom the second complementary slac kness condition, Equation (7), and the fact that β ? > 0 , we obtain that ( x ? ) T D G s = ± √ κ. Th us, x ? = ± β ? √ κ ( L G − α ? L K n ) + D G s, as required. 4 Application to P artitioning Graphs Lo cally In this section, we describ e the application of LocalSp ectral to ﬁnding lo cally-biased partitions in a graph, i.e. , to ﬁnding sparse cuts around an input seed vertex set in the graph. F or simplicity , in this part of the pap er, we let the instance graph G b e unw eigh ted. 4.1 Bac kground on Global Sp ectral Algorithms for P artitioning Graphs W e start with a brief review of global sp ectral graph partitioning. Recall that the basic global graph partitioning problem is: giv en as input a graph G = ( V , E ), ﬁnd a set of no des S ⊆ V to solv e φ ( G ) = min S ⊆ V φ ( S ) . Sp ectral methods approximate the solution to this intractable global problem by solving the relaxed problem Sp ectral ( G ) presented in Figure 1. T o understand this optimization problem, recall that x T L G x counts the n um b er of edges crossing the cut and that x T D G x = 1 enco des a v ariance constrain t; th us, the goal of Sp ectral ( G ) is to minimize the n umber of edges crossing the cut sub ject to a given v ariance. Recall that for T ⊆ V , w e let 1 T ∈ { 0 , 1 } V b e a v ector whic h is 1 for v ertices in T and 0 otherwise. Then for a cut ( S, ¯ S ), if we deﬁne the vector v S def = q vol( S ) · vol( ¯ S ) vol( G ) ·  1 S vol( S ) − 1 ¯ S vol ¯ S  , it can b e chec k ed that v S satisﬁes the constrain ts of Sp ectral and has ob jective v alue φ ( S ). Thus, λ 2 ( G ) ≤ min S ⊆ V φ ( S ) = φ ( G ). Hence, Sp ectral ( G ) is a relaxation of the minimum conductance problem. Moreov er, this program is a go o d relaxation in that a go o d cut can b e reco v ered b y considering a truncation, i.e. , a sw eep cut, of the v ector v 2 that is the optimal solution to Sp ectral ( G ). (That is, e.g. , consider eac h of the n cuts deﬁned by the v ector v 2 , and return the cut with minim um conductance v alue.) This is captured b y the following celebrated result often referred to as Cheeger’s Inequality . Theorem 3 (Cheeger’s Inequalit y) F or a c onne cte d gr aph G , φ ( G ) ≤ O ( p λ 2 ( G )) . Although there are man y pro ofs known for this theorem (see, e.g. , [9]), a particularly interesting pro of was found by Mihail [25]; this pro of inv olv es rounding any test ve ctor (rather than just the optimal vector), and it achiev es the same guaran tee as Cheeger’s Inequality . Theorem 4 (Sw eep Cut Rounding) L et x b e a ve ctor such that x T D G 1 = 0 . Then ther e is a t for which the set of vertic es S := SweepCut t ( x ) def = { i : x i ≥ t } satisﬁes x T L G x x T D G x ≥ φ 2 ( S ) / 8 . It is the form of Cheeger’s Inequality provided b y Theorem 4 that w e will use b elo w. 9 4.2 Lo cally-Biased Sp ectral Graph P artitioning Here, we will exploit the analogy betw een Sp ectral and Lo calSpectral b y applying the global approac h just outlined to the follo wing lo cally-biased graph partitioning problem: given as input a graph G = ( V , E ), an input node u , and a positive in teger k , ﬁnd a set of nodes T ⊆ V ac hieving φ ( u, k ) = min T ⊆ V : u ∈ T , vol( T ) ≤ k φ ( T ) . That is, the problem is to ﬁnd the b est conductance set of nodes of v olume no greater than k that contains the input no de v . As a ﬁrst step, w e show that w e can c ho ose the seed set and correlation parameters s and κ suc h that Lo calSp ectral ( G, s, κ ) is a relaxation for this lo cally-biased graph partitioning problem. Lemma 1 F or u ∈ V , Lo calSp ectral ( G, v { u } , 1 /k ) is a r elaxation of the pr oblem of ﬁnding a minimum c onductanc e cut T in G which c ontains the vertex u and is of volume at most k . In p articular, λ ( G, v { u } , 1 /k ) ≤ φ ( u, k ) . Pr o of: If w e let x = v T in Lo calSp ectral ( G, v { u } , 1 /k ), then v T T L G v T = φ ( T ), v T T D G 1 = 0, and v T T D G v T = 1. Moreo ver, we hav e that ( v T T D G v { u } ) 2 = d u (2 m − vol( T )) vol( T )(2 m − d u ) ≥ 1 /k , which establishes the lemma.  Next, w e can apply Theorem 4 to the optimal solution for Lo calSp ectral ( G, v { u } , 1 /k ) and obtain a cut T whose conductance is quadratically close to the optimal v alue λ ( G, v { u } , 1 /k ). By Lemma 1, this implies that φ ( T ) ≤ O ( p φ ( u, k )). This argument pro ves the following theorem. Theorem 5 (Finding a Cut) Given an unweighte d gr aph G = ( V , E ) , a vertex u ∈ V and a p ositive inte ger k , we c an ﬁnd a cut in G of c onductanc e at most O ( p φ ( u, k )) by c omputing a swe ep cut of the optimal ve ctor for Lo calSpectral ( G, v { u } , 1 /k ) . Mor e over, this algorithm runs in ne arly-line ar time in the size of the gr aph. That is, this theorem states that w e can p erform a sw eet cut ov er the vector that is the solution to Lo calSp ectral ( G, v { u } , 1 /k ) in order to obtain a lo cally-biased partition; and that this partition comes with quality-of-appro ximation guarantees analogous to that provided for the global problem Sp ectral ( G ) by Cheeger’s inequality . Our ﬁnal theorem sho ws that the optimal v alue of Lo calSp ectral also pro vides a lo wer b ound on the conductance of other cuts , as a function of ho w w ell-correlated they are with the input seed vector. In particular, when the seed vector corresp onds to a cut U , this result allows us to lo wer b ound the conductance of an arbitrary cut T , in terms of the correlation b etw een U and T . The pro of of this theorem also uses in an essen tial manner the duality prop erties that were used in the pro of of Theorem 1. Theorem 6 (Cut Impro vemen t) L et G b e a gr aph and s ∈ R n b e such that s T D G 1 = 0 , wher e D G is the de gr e e matrix of G. In addition, let κ ≥ 0 b e a c orr elation p ar ameter. Then, for al l sets T ⊆ V such that κ 0 def = ( s T D G v T ) 2 , we have that φ ( T ) ≥  λ ( G, s, κ ) if κ ≤ κ 0 κ 0 / κ · λ ( G, s, κ ) if κ 0 ≤ κ . Pr o of: It follo ws from Theorem 1 that λ ( G, s, κ ) is the same as the optimal v alue of SDP p ( G, s, κ ) whic h, b y strong dualit y , is the same as the optimal v alue of SDP d ( G, s, κ ). Let α ? , β ? b e the 10 optimal dual v alues to SDP d ( G, s, κ ) . Then, from the dual feasibility constraint L G − α ? L K n − β ? ( D G s )( D G s ) T  0 , it follo ws that s T T L G s T − α ? s T T L K n s T − β ? ( s T D G s T ) 2 ≥ 0 . Notice that since s T T D G 1 = 0, it follows that s T T L K n s T = s T T D G s T = 1. F urther, since s T T L G s T = φ ( T ) , we obtain, if κ ≤ κ 0 , that φ ( T ) ≥ α ? + β ? ( s T D G s T ) 2 ≥ α ? + β ? κ = λ ( G, s, κ ) . If on the other hand, κ 0 ≤ κ, then φ ( T ) ≥ α ? + β ? ( s T D G s T ) 2 ≥ α ? + β ? κ ≥ κ 0 / κ · ( α ? + β ? κ ) = κ 0 / κ · λ ( G, s, κ ) . Note that strong dualit y w as used here.  Th us, although the relaxation guarantees of Lemma 1 only hold when the seed set is a single v ertex, w e can use Theorem 6 to consider the following problem: giv en a graph G and a cut ( T , ¯ T ) in the graph, ﬁnd a cut of minim um conductance in G which is w ell-correlated with T or certify that there is none. Although one can imagine many applications of this primitive, the main application that motiv ated this work was to explore clusters nearb y or around a given se e d set of no des in data graphs. This will be illustrated in our empirical ev aluation in Section 5. 4.3 Our Geometric Notion of Correlation Bet w een Cuts Here we pause to mak e explicit the geometric notion of correlation b et ween cuts (or partitions, or sets of no des) that is used by Lo calSpectral , and that has already b een used in v arious guises in previous sections. Giv en a cut ( T , ¯ T ) in a graph G = ( V , E ), a natural v ector in R V to asso ciate with it is its c haracteristic v ector, in which case the correlation b et w een a cut ( T , ¯ T ) and another cut ( U, ¯ U ) can b e captured by the inner pro duct of the characteristic vectors of the tw o cuts. A somewhat more reﬁned vector to asso ciate with a cut is the v ector obtained after removing from the c haracteristic vector its pro jection along the all-ones vector. In that case, again, a notion of correlation is related to the inner pro duct of tw o such vectors for t wo cuts. More precisely , given a set of no des T ⊆ V , or equiv alen tly a cut ( T , ¯ T ), one can deﬁne the unit vector s T as s T ( i ) =  p vol( T )vol( ¯ T ) / 2 m · 1 / vol( T ) if i ∈ T − p vol( T )vol( ¯ T ) / 2 m · 1 / vol( ¯ T ) if i ∈ ¯ T . That is, s T def = q vol( T )vol( ¯ T ) 2 m  1 T vol( T ) − 1 ¯ T vol( ¯ T )  , which is exactly the v ector deﬁned in Section 4.1. It is easy to chec k that this is well deﬁned: one can replace s T b y s ¯ T and the correlation remains the same with an y other set. Moreo v er, sev eral observ ations are immediate. First, deﬁned this w ay , it immediately follows that s T T D G 1 = 0 and that s T T D G s T = 1. Thus, s T ∈ S D for T ⊆ V , where w e denote by S D the set of v ectors { x ∈ R V : x T D G 1 = 0 } ; and s T can b e seen as an appropriately normalized version of the vector consisting of the uniform distribution ov er T minus the uniform distribution ov er ¯ T . 1 Second, one can introduce the following measure of correlation b et w een t wo sets of no des, or equiv alen tly b et w een t wo cuts, say a cut ( T , ¯ T ) and a cut ( U, ¯ U ): K ( T , U ) def = ( s T D G s U ) 2 . 1 Notice also that s T = − s ¯ T . Thus, since we only consider quadratic functions of s T , we can consider b oth s T and s ¯ T to b e representativ e vectors for the cut ( T , ¯ T ) . 11 The pro ofs of the following simple facts regarding K ( T , U ) are omitted: K ( T , U ) ∈ [0 , 1]; K ( T , U ) = 1 if and only if T = U or ¯ T = U ; K ( T , U ) = K ( ¯ T , U ); and K ( T , U ) = K ( T , ¯ U ). Third, although we ha ve describ ed this notion of geometric correlation in terms of v ectors of the form s T ∈ S D that represen t partitions ( T , ¯ T ), this correlation is clearly well-deﬁned for other v ectors s ∈ S D for which there is not such a simple in terpretation in terms of cuts. Indeed, in Section 3 we considered the case that s w as an arbitrary vector in S D , while in the ﬁrst part of Section 4.2 we considered the case that s was the seed set of a single no de. In our empirical ev aluation in Section 5, we will consider b oth of these cases as well as the case that s enco des the correlation with cuts consisting of m ultiple no des. 5 Empirical Ev aluation In this section, w e pro vide an empirical ev aluation of Lo calSp ectral by illustrating its use at ﬁnding and ev aluating lo cally-biased low-conductance cuts, i.e. , sparse cuts or go od clusters, around an input seed set of no des in a data graph. W e start with a brief discussion of a v ery recen t and pictorially-comp elling application of our metho d to a computer vision problem; and then w e discuss in detail how our metho d can b e applied to iden tify clusters and communities in a more heterogeneous and more diﬃcult-to-visualize so cial netw ork application. 5.1 Semi-Sup ervised Image Segmen tation Subsequen t to the initial dissemination of the technical rep ort version of this pap er, Ma ji, Vishnoi, and Malik [24] applied our metho dology to the problem of ﬁnding lo cally-biased cuts in a computer vision application. Recall that image segmen tation is the problem of partitioning a digital image in to segments corresp onding to signiﬁcant ob jects and areas in the image. A standard approach consists in conv erting the image data into a similarity graph ov er the the pixels and applying a graph partitioning algorithm to identify relev an t segmen ts. In particular, sp ectral metho ds hav e b een p opular in this area since the work of Shi and Malik [29], which used the second eigenv ector of the graph to approximate the so-called normalized cut (whic h, recall, is an ob jectiv e measure for image segmentation that is practically equiv alen t to conductance). How ev er, a diﬃcult y in applying the normalized cut me thod is that in many cases global eigenv ectors may fail to capture imp ortan t lo cal segmen ts of the image. The reason for this is that they aggressively optimize a global ob jective function and thus they tend to combine m ultiple segmen ts together; this is illustrated pictorially in the ﬁrst ro w of Figure 4. This diﬃcult y can b e ov ercome in a semi-sup ervised scenario by using our Lo calSp ectral metho d. Sp eciﬁcally , one often has a small n umber of “ground truth” lab els that corresp ond to known segments, and one is interested in extracting and reﬁning the segmen ts in which those lab els reside. In this case, if one considers an input seed corresp onding to a small n umber of pixels within a target ob ject, then Lo calSp ectral can recov er the corresp onding segment with high precision. This is illustrated in the second row of Figure 4. This computer vision application of our methodology w as motiv ated b y a preliminary v ersion of this paper, and it was describ ed in de- tail and ev aluated against competing algorithms b y Ma ji, Vishnoi, and Malik [24]. In particular, they sho w that Lo calSp ectral ac hieves a p erformance superior to that of other semi-sup ervised segmen tation algorithms [32, 11]; and they also show ho w Lo calSp ectral can be incorp orated in an unsup ervised segmentation pip eline by using as input seed distributions obtained by an ob ject-detector algorithm [6]. 12 Figure 4: The ﬁrst row sho ws the input image and the three smallest eigen vectors of the Laplacian of the corresp onding similarity graph computed using the in tervening con tour cue [23]. Note that no sweep cut of these eigen vectors rev eals the leopard. The second ro w sho ws the results of Lo calSpectral with a setting of γ = − 10 λ 2 ( G ) with the seed pixels highlighted b y crosshairs. Note ho w one can to reco ver the leopard b y using a seed v ector representing a set of only 4 pixels. In addition, note ho w the ﬁrst seed pixel allo ws us to capture the head of the animal, while the other seeds help rev eal other parts of its b o dy . 5.2 Detecting Comm unities in So cial Net w orks Finding lo cal clusters and meaningful lo cally-biased comm unities is also of in terest in the analysis of large so cial and information netw orks. A standard approach to ﬁnding clusters and comm unities in many net work analysis applications is to formalize the idea of a go o d comm unity with an “edge counting” metric such as conductance or mo dularit y and then to use a sp ectral relaxation to optimize it appro ximately [26, 27]. F or man y v ery large so cial and information net works, ho wev er, there simply do not exist go o d large global clusters, but there do exist small meaningful lo cal clusters that ma y b e thought of as b eing nearby presp eciﬁed seed sets of no des [20, 21, 22]. In these cases, a local v ersion of the global sp ectral partitioning problem is of interest, as was sho wn b y Lesk ov ec, Lang, and Mahoney [22]. Typical netw orks are very large and, due to their expander-lik e prop erties, are not easily-visualizable [20, 21]. Th us, in order to illustrate the empirical behavior of our Lo calSpec tral metho dology in a “real” net work application related to the one that motiv ated this w ork [20, 21, 22], w e examined a small “coauthorship netw ork” of scien tists. This net work was previously used by Newman [26] to study communit y structure in small so cial and information net works. The corresp onding graph G is illustrated in Figure 5 and consists of 379 no des and 914 edges, where each no de represen ts an author and each un weigh ted edge represents a coauthorship relationship. The sp ectral gap λ 2 ( G ) = 0 . 0029; and a sweep cut of the eigen vector corresp onding to this second eigenv alue yields the globally-optimal spectral cut separating the graph into tw o w ell-balanced partitions, corresp onding to the left half and the right half of the net w ork, as shown in Figure 5. Our main empirical observ ations, describ ed in detail in the remainder of this section, are the following. • First, w e sho w ho w v arying the telep ortation parameter allows us to detect lo w-conductance cuts of diﬀeren t volumes that are lo cally-biased around a presp eciﬁed seed vertex; and ho w this information, aggregated ov er m ultiple choices of telep ortation, can improv e our understanding of the net work structure in the neigh b orho od of the seed. 13 P e r i p h e r y n o d e C o r e n o d e I n t e r m e d i a t e n o d e Figure 5: [Best viewed in color.] The coauthorship net work of Newman [26]. This lay out w as obtained in the P a jek [4] visualization softw are, using the Kamada-Kaw ai metho d [16] on each comp onen t of a partition provided by Lo calCut and tiling the lay outs at the end. Boxes show the t wo main global components of the netw ork, which are displa y ed separately in subsequen t ﬁgures. • Second, we demonstrate the more general usefulness of our deﬁnition of a gener alize d Per- sonalized P ageRank vector (where the γ parameter in Eqn. (1) can b e γ ∈ ( −∞ , λ 2 ( G )) b y displa ying sp eciﬁc instances in whic h that v ector is more eﬀective than the usual P er- sonalized P ageRank (where only p ositiv e telep ortation probabilities are allo wed and th us where γ m ust b e negativ e). W e do this by detecting a wider range of low-conductance cuts at a given v olume and b y interpolating smo othly b et ween very lo cally-biased solutions to Lo calSpectral and the global solution provided by the Sp ectral program. • Third, we demonstrate how our metho d can ﬁnd low-conductance cuts that are well- correlated to more general input seed v ectors by demonstrating an application to the de- tection of sparse p eripheral regions, e.g. , regions of the netw ork that are well-correlated with lo w-degree no des. This suggests that our metho d may ﬁnd applications in lev eraging feature data, whic h are often associated with the v ertices of a data graph, to ﬁnd interesting and meaningful cuts. W e emphasize that the goal of this empirical ev aluation is to illustrate ho w our prop osed metho d- ology can b e applied in real applications; and th us we w ork with a relatively easy-to-visualize example of a small so cial graph. This will allow us to illustrate ho w the “knobs” of our prop osed metho d can b e used in practice. In particular, the goal is not to illustrate that our metho d or heuristic v arian ts of it or other sp ectral-based metho ds scale to muc h larger graphs—this latter fact is by no w w ell-established [2, 20, 21, 22]. 5.2.1 Algorithm Description and Implemen tation W e refer to our cut-ﬁnding algorithm, whic h will b e used to guide our empirical study of ﬁnding and ev aluating cuts around an input seed set of no des and which is a straightforw ard extension of 14 the algorithm referred to in Theorem 5, as Lo calCut . In addition to the graph, the input param- eters for LocalCut are a seed v ector s ( e.g. , corresponding to a single v ertex v ), a telep ortation parameter γ , and (optionally) a size factor c . Then, Lo calCut p erforms the follo wing steps. • First, compute the vector x ? of Eqn. (1) with seed s and telep ortation γ . • Second, either p erform a sw eep of the v ector x ? , e.g. , consider each of the n cuts deﬁned b y the vector and return the the minimum conductance cut found along the sweep; or consider only sweep cuts along the vector x ? of volume at most c · k γ , where k γ = 1 /κ γ , that contain the input vertex v , and return the minimum conductance cut among suc h cuts. By Theorem 1, the v ector computed in the ﬁrst step of Lo calCut , x ? , is an optimal solution to Lo calSpectral ( G, s, κ γ ) for some choice of κ γ . (Indeed, by ﬁxing the abov e parameters, the κ parameter is ﬁxed implicitly .) Then, b y Theorem 5, when the vector x ? is rounded (to, e.g. , {− 1 , +1 } ) by p erforming the sw eep cut, pro v ably-go od approximations are guaranteed. In addi- tion, when the seed v ector corresponds to a single v ertex v , it follows from Lemma 1 that x ? yields a low er b ound to the conductance of cuts that con tain v and hav e less than a certain v olume k γ . Although the full sweep-cut rounding do es not give a sp eciﬁc guarantee on the volume of the output cut, empirically we ha ve found that it is often p ossible to ﬁnd small lo w-conductance cuts in the range dictated by k γ . Thus, in our empirical ev aluation, w e also consider volume- constrained sw eep cuts (which departs slightly from the theory but can be useful in practice). That is, w e also introduce a new input parameter, a size factor c > 0, that regulates the maxim um v olume of the sweep cuts considered when s represents a single v ertex. In this case, Lo calCut do es not consider all n cuts deﬁned b y the v ector x ? , but instead it considers only sw eep cuts of volume at most c · k γ that contain the vertex v . (Note that it is a simple consequence of our optimization c haracterization that the optimal vector has sweep cuts of volume at most k γ con taining v .) This new input parameter turns out to be extremely useful in exploring cuts at diﬀerent sizes, as it neglects sweep cuts of low conductance at large v olume and allo ws us to pic k out more lo cal cuts around the seed v ertex. In our ﬁrst tw o sets of exp eriments, summarized in Sections 5.2.2 and 5.2.3, w e used single- v ertex seed v ectors, and we analyzed the eﬀects of v arying the parameters γ and c , as a function of the location of the seed vertex in the input graph. In the last set of exp erimen ts, presen ted in Section 5.2.4, we considered more general seed vectors, including b oth seed v ectors that cor- resp ond to multiple no des, i.e. , to cuts or partitions in the graph, as well as seed v ectors that do not ha ve an obvious interpretation in terms of input cuts. W e implemen ted our co de in a combi- nation of MA TLAB and C++, solving linear systems using the Stabilized Biconjugate Gradien t Metho d [31] provided in MA TLAB 2006b. On this particular coauthorship net w ork, and on a Dell Po w erEdge 1950 machine with 2.33 GHz and 16GB of RAM, the algorithm ran in less than a few seconds. 5.2.2 V arying the T elep ortation P arameter Here, we ev aluate the eﬀect of v arying the telep ortation parameter γ ∈ ( −∞ , λ 2 ( G )), where recall λ 2 ( G ) = 0 . 0029. Since it is known that large so cial and information netw orks are quite hetero- geneous and exhibit a very strong “nested core-p eriphery” structure [20, 21, 22], we p erform this ev aluation by considering the b eha vior of Lo calCut when applied to three t yp es of seed no des, examples of which are the highlighted v ertices in Figure 5. These three no des were c hosen to rep- resen t three diﬀerent types of no des seen in larger netw orks: a p eriphery-like no de , whic h b elongs to a lo wer-degree and less expander-like part of the graph, and which tends to b e surrounded b y 15 lo wer-conductance cuts of small volume; a c or e-like no de , which b elongs to a denser and higher- conductance or more expander-like part of the graph; and an interme diate no de , whic h b elongs to a regime b et w een the core-like and the p eriphery-lik e regions. F or each of the three representativ e seed no des, we executed 1000 runs of Lo calCut with c = 2 and γ v arying by 0 . 001 increments. Figure 6 displa ys, for each of these three seeds, a plot of the conductance as a function of volume of the cuts found by each run of Lo calCut . W e refer to this t yp e of plot as a lo c al pr oﬁle plot since it is a sp ecialization of the network c ommunity pr oﬁle plot [20, 21, 22] to cuts around the sp eciﬁed seed v ertex. In addition, Figure 6 also plots several other quantities of interest: ﬁrst, the volume and conductance of the theoretical low er b ound yielded by eac h run; second, the volume and conductance of the cuts deﬁned by the shortest-path balls (in squares and n umbered according to the length of the path) around each seed (whic h should and do provide a sanity-c heck upp er b ound); third, next to each of the plots, we present a color-co ded image of representativ e cuts detected b y Lo calCut ; and fourth, for eac h of the cuts illustrated on the left, a color-co ded triangle and the numerical v alue of − γ is shown on the right. Sev eral p oin ts ab out the b eha vior of the Lo calCut algorithm as a function of the lo cation of the input seed no de and that are illustrated in Figure 6 are w orth emphasizing. • First, for the core-like no de, whose proﬁle plot is shown in Figure 6(a), the volume of the output cuts grows relatively smoothly as γ is increased ( i.e. , as − γ is decreased). F or small γ , e.g. , γ = − 0 . 0463 or γ = − 0 . 0207, the output cuts are forced to b e small and hence display high conductance, as the region around the no de is somewhat expander-like. By decreasing the telep ortation, the conductance progressiv ely decreases, as the rounding starts to hit no des in p eripheral regions, whose inclusion only improv es conductance (since it increases the cut volume without adding many additional cut edges). In this case, this phenomena ends at γ = − 0 . 0013 , when a cut of conductance v alue close to that of the global optim um is found. (After that, larger and slightly b etter conductance cuts can still b e found, but, as discussed b elow, they require γ > 0.) • Second, a similar in terpretation applies to the proﬁle plot of the intermediate no de, as sho wn in Figure 6(b). Here, how ev er, the global comp onen t of the netw ork containing the seed has smaller volume, around 300, and a very low conductance (again, requiring γ > 0). Th us, the proﬁle plot jumps from this cut to the m uch larger eigenv ector sweep cut, as will b e discussed below. • Third, a more extreme case is that of the p eriphery-lik e no de, whose proﬁle plot is display ed in Figure 6(c). In this case, an initial increase in γ do es not yield larger cuts. This vertex is con tained in a small-volume cut of lo w conductance, and thus diﬀusion-based metho ds get “stuc k” on the small side of the cut. The only cuts of lo wer conductance in the netw ork are those separating the global components, which can only be accessed when γ > 0. Hence, the telep ortation must b e greatly decreased b efore the algorithm starts outputting cuts at larger v olumes. (As an aside, this b eha vior is also often seen with so-called “whiskers” in m uch larger so cial and information netw orks [20, 21, 22].) In addition, several general p oints that are illustrated in Figure 6 are worth emphasizing ab out the b eha vior of our algorithm. • First, LocalCut found low-conductance cuts of diﬀerent volumes around eac h s eed v ertex, outp erforming the shortest-path algorithm (as it should) by a factor of roughly 4 in most cases. How ev er, the results of Lo calCut still lie aw a y from the lo wer b ound, which is also a factor of roughly 4 smaller at most volumes. 16 Pajek ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0 200 400 600 800 0.0 0.1 0.2 0.3 0.4 0.5 Volume Conductance ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 1 2 3 4 0.0463 0.0207 0.0063 0.0013 −2e−04 −0.0028 ● ● Theoretical Lower Bound Algorithm Output (Cut Shown) Algorithm Output (Cut Not Shown) Shortest Path Algorithm Output (a) Selected cuts and proﬁle plot for the c or e-like no de . Pajek ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 200 400 600 800 0.00 0.05 0.10 0.15 0.20 Volume Conductance ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 1 2 3 4 5 6 0.0852 0.0396 0.0333 −0.0018 −0.0028 ● ● Theoretical Lower Bound Algorithm Output (Cut Shown) Algorithm Output (Cut Not Shown) Shortest Path Algorithm Output (b) Selected cuts and proﬁles plot for the interme diate no de . Pajek ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0 200 400 600 800 0.00 0.05 0.10 0.15 Volume Conductance ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 1 5 6 7 8 0.081 −0.0024 −0.0025 −0.0026 −0.0027 −0.0028 ● ● Theoretical Lower Bound Algorithm Output (Cut Shown) Algorithm Output (Cut Not Shown) Shortest Path Algorithm Output (c) Selected cuts and proﬁle plot for the p eriphery-like no de . Figure 6: [Best viewed in color.] Selected cuts and lo cal proﬁle plots for v arying γ . The cuts on the left are displa yed by assigning to each vertex a color corresp onding to the smallest selected cut in which the v ertex w as included. Smaller cuts are darker, larger cuts are lighter; and the seed vertex is sho wn slightly larger. Each proﬁle plot on the right sho ws results from 1000 runs of Lo calCut , with c = 2 and γ decreasing in 0 . 001 incremen ts starting at 0 . 0028. F or eac h color-co ded triangle, corresp onding to a cut on the left, − γ is also listed. 17 • Second, consider the range of the telep ortation parameter necessary for the Lo calCut algo- rithm to discov er the well-balanced globally-optimal spectral partition. In all three cases, it was necessary to mak e γ p ositiv e ( i.e. , − γ negativ e) to detect the w ell-balanced global sp ectral cut. Importantly , how ev er, the quantitativ e details dep end strongly on whether the seed is core-lik e, in termediate, or periphery-like. That is, by formal ly allowing “negative telep ortation” probabilities, which corresp ond to γ > 0, the use of gener alize d Personal- ized P ageRank vectors as an exploratory to ol is m uch stronger than the usual Personalized P ageRank [1, 2], in that it p ermits one to ﬁnd a larger class of clusters, up to and including the global partition found b y the solution to the global Sp ectral program. Relatedly , it pro vides a smo oth interpolation b etw een P ersonalized PageRank and the second eigenv ec- tor of the graph. Indeed, for γ = 0 . 0028 ≈ λ 2 ( G ), Lo calCut outputs the same cut as the eigen vector sw eep cut for all three seeds. • Third, recall that, given a teleportation parameter γ , the rounding step selects the cut of smallest conductance along the sw eep cut of the solution v ector. (Alternatively , if v olume- constrained sw eeps are considered, then it selects the cut of smallest conductance among sw eep cuts of v olume at most c · k γ , where k γ is the low er b ound obtained from the opti- mization program.) In either case, increasing γ can lead Lo calCut to pick out larger cuts, but it do es not guar ante e this will happ en. In particular, due to the local top ology of the graph, in many instances there may not b e a w ay of slightly increasing the volume of a cut while slightly decreasing its conductance. In those cases, Lo calCut may output the same small sweep cut for a range of telep ortation parameters until a m uch larger, muc h lo wer-conductance cut is then found. The presence suc h horizon tal and v ertical jumps in the lo cal proﬁle plot con veys useful information ab out the structure of the netw ork in the neigh b orho od of the seed at diﬀerent size scales, illustrating that the practice follows the theory quite well. 5.2.3 V arying the Output-Size P arameter Here, we ev aluate the eﬀect of v arying the size factor c , for a ﬁxed choice of telep ortation parameter γ . (In the previous section, c w as ﬁxed at c = 2 and γ w as v aried.) W e hav e observ ed that v arying c , like v arying γ , tends to ha ve the eﬀect of producing lo w-conductance cuts of diﬀerent volumes around the seed vertex. Moreov er, it is p ossible to obtain lo w-conductance large-volume cuts, ev en at lo wer v alues of the telep ortation parameter, by increasing c to a suﬃciently large v alue. This is illustrated in Figure 7, whic h shows the result of v arying c with the core-like no de as the seed and − γ = 0 . 02. Figure 6(a) illustrated that when c = 2 , this setting only yielded a cut of v olume close to 100 (see the red triangle with − γ = 0 . 0207); but the y ellow crosses in Figure 7 illustrate that by allowing larger v alues of c , better conductance cuts of larger volume can b e obtained. While many of these cuts tend to hav e conductance slightly w orse than the b est found by v arying the telep ortation parameter, the observ ation that cuts of a wide range of volumes can b e obtained with a single v alue of γ leav es op en the p ossibilit y that there exists a single choice of teleportation parameter γ that produces go od low-conductance cuts at all v olumes simply b y v arying c . (This would allow us to only solve a single optimization problem and still ﬁnd cuts of diﬀerent volumes.) T o address (and rule out) this possibility , we selected three c hoices of the telep ortation parameter for each of the three seed no des, and then we let c v ary . The resulting output cuts for the core-like no de as the seed are plotted (in blue, green, and yello w) in Figure 7. (The plots for the other seeds are similar and are not display ed.) Clearly , no single telep ortation setting dominates the others: in particular, at volume 200 the low est-conductance 18 Pajek 0 200 400 600 800 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Volume Conductance ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● Theoretical Lower Bound Teleportation 0 Teleportation 0.01 Teleportation 0.02 Figure 7: [Best viewed in color.] Selected cuts and lo cal proﬁle plots for v arying c with the core- lik e no de as the seed. The cuts are display ed by assigning to each v ertex a color corresp onding to the smallest selected cut in whic h the v ertex w as included. Smaller cuts are dark er, larger are ligh ter. The seed v ertex is sho wn larger. The proﬁle plot sho ws results from 1000 runs of Lo calCut , with v arying c and − γ ∈ { 0 , 0 . 01 , 0 . 02 } . cut was produced with − γ = 0 . 02; at volume 400 it was produced with − γ = 0 . 01; and at volume 600 with it was pro duced with γ = 0. The highest choice of γ = 0 p erformed marginally b etter o verall, recording low est conductance cuts at b oth small and large v olumes. That b eing said, the results of all three settings roughly track eac h other, and cuts of a wide range of volumes w ere able to b e obtained b y v arying the size parameter c . These and other empirical results suggest that the b est results are ac hieved when w e v ary b oth the telep ortation parameter and the size factor. In addition, the use of multiple teleportation c hoices ha ve the side-eﬀect adv an tage of yielding multiple lo wer b ounds at diﬀeren t v olumes. 5.2.4 Multiple Seeds and Correlation Here, we ev aluate the b eha vior of Lo calCut on more general seed vectors. W e consider tw o examples—for the ﬁrst example, there is an interpretation as a cut or partition consisting of m ultiple no des; while the second example does not hav e any immediate interpretation in terms of cuts or partitions. In our ﬁrst example, we consider a seed vector representing a subset of four no des, lo cated in diﬀeren t peripheral branc hes of the left half of the global partition of the the netw ork: see the four slightly larger (and darker) vertices in Figure 8(a). This is of in terest since, depending on the size-scale at which one is interested, such sets of no des can b e though t of as either “nearby” or “far apart.” F or example, when viewing the entire graph of 379 nodes, these four no des are all close, in that they are all on the left side of the optimal global sp ectral partition; but when considering smaller clusters suc h as w ell-connected sets of 10 or 15 no des, these four no des are m uch farther apart. In Figure 8(a), we display a selection of the cuts found by v arying the telep ortation, with c = 2. The smaller cuts tend to contain the branches in whic h eac h seed no de is found, while larger cuts start to incorp orate nearby branches. Not shown in the color-co ding is that the optimal global sp ectral partition is even tually recov ered. Iden tifying p eripheral areas that are well-separated from the rest of the graph is a useful primitiv e in studying the structure of so cial net works [20, 21, 22]; and thus, this shows ho w LocalCut ma y b e used in this con text, 19 Pajek (a) Seed set of four seed no des. Pajek (b) A more general seed vector. Figure 8: [Best viewed in color.] Multiple seeds and correlation. 8(a) shows selected cuts for v arying γ with the seed v ector corresp onding to a subset of 4 v ertices lying in the p eriphery-like region of the netw ork. 8(b) sho ws selected cuts for v arying γ with the seed v ertex equal to a normalized v ersion of the degree v ector. In b oth cases, the cuts are display ed by assigning to eac h vertex a color corresp onding to the smallest selected cut in which the vertex w as included. Smaller cuts are dark er, larger are lighter. when some p eriphery-lik e seed no des of the graph are known. In our second example, w e consider a seed vector that represents a feature vector on the v ertices but that do es not hav e an in terpretation in terms of cuts. In particular, w e consider a seed vector that is a normalized v ersion of the degree distribution vector. Since no des that are p eriphery-lik e tend to hav e lo wer degree than those that are core-like [20, 21, 22], this choice of seed vector biases Lo calCut tow ards cuts that are well-correlated with p eriphery-lik e and low- degree vertices. A selection of the cuts found on this seed v ector when v arying the telep ortation with c = 2 is display ed in Figure 8(b). These cuts partition the netw ork naturally into three well- separated regions: a sparser p eriphery-lik e region in darker colors, a ligh ter-colored in termediate region, and a white dense core-like region, where higher-degree vertices tend to lie. Clearly , this approac h could b e applied more generally to ﬁnd low-conductance cuts that are well-correlated with a known feature of the node vector. 6 Discussion In this ﬁnal section, we provide a brief discussion of our results in a broader context. Relationship to lo cal graph partitioning. Recent theoretical work has fo cused on using sp ectral ideas to ﬁnd goo d clusters nearb y an input seed set of no des [30, 1, 10]. In particular, lo cal graph partitioning—roughly , the problem of ﬁnding a lo w-conductance cut in a graph in time dep ending only on the v olume of the output cut—w as introduced by Spielman and T eng [30]. They used random w alk based methods; and they used this as a subroutine to give a nearly linear-time algorithm for outputting balanced cuts that matc h the Cheeger Inequalit y up to p olylog factors. In our language, a lo cal graph partitioning algorithm w ould start a random walk at a seed no de, truncating the w alk after a suitably c hosen num b er of steps, and outputting the no des visited by the walk. This result was improv ed b y Andersen, Chung and Lang [1] 20 b y performing a truncated P ersonalized PageRank computation. These and subsequent pap ers building on them were motiv ated b y local graph partitioning [10], but they do not address the problem of discov ering cuts near general seed vectors, as do we, or of generalizing the second eigen vector of the Laplacian. Moreov er, these approac hes are more op erationally-deﬁned, while ours is axiomatic and optimization-based. Relationship to empirical w ork on comm unity structure. Recent empirical work has used P ersonalized P ageRank, a particular v arian t of a lo cal random walk, to c haracterize v ery ﬁnely the clustering and communit y structure in a wide range of very large so cial and informa- tion netw orks [2, 20, 21, 22]. In particular, Andersen and Lang used local spectral methods to iden tify comm unities in certain informatics graphs using an input set of no des as a seed set [2]. Subsequen tly , Lesko v ec, Lang, Dasgupta, and Mahoney used related metho ds to characterize the small-scale and large-scale clustering and communit y structure in a wide range of large so cial and information netw orks [20, 21, 22]. Our optimization program and empirical results suggest that this line of work can b e extended to ask in a theoretically principled manner m uch more reﬁned questions ab out graph structure near presp eciﬁed seed vectors. Relationship to cut-improv ement algorithms. Many recen tly-p opular algorithms for ﬁnd- ing minimum-conductance cuts, such as those in [17, 28], use as a crucial building blo c k a prim- itiv e that takes as input a cut ( T , ¯ T ) and attempts to ﬁnd a lo w er-conductance cut that is wel l c orr elate d with ( T , ¯ T ). This primitiv e is referred to as a cut-impr ovement algorithm [18, 3], as its original purp ose w as limited to post-pro cessing cuts output by other algorithms. Recently , cut-impro vemen t algorithms hav e also b een used to ﬁnd lo w conductance cuts in sp eciﬁc regions of large graphs [22]. Giv en a notion of correlation b et ween cuts, cut-impro v ement algorithms t ypically produce appro ximation guaran tees of the following form: for an y cut ( C, ¯ C ) that is ε -correlated with the input cut, the cut output by the algorithm has conductance upp er-bounded b y a function of the conductance of ( C, ¯ C ) and ε . This line of w ork has typ ically used ﬂow-based tec hniques. F or example, Gallo, Grigoriadis and T arjan [12] w ere the ﬁrst to sho w that one can ﬁnd a subset of an input set T ⊆ V with minim um conductance in p olynomial time. Similarly , Lang and Rao [18] implement a closely related algorithm and demonstrate its eﬀectiveness at reﬁning cuts output b y other metho ds. Finally , Andersen and Lang [3] give a more general al- gorithm that uses a small n umber of single-commo dit y maximum-ﬂo ws to ﬁnd low-conductance cuts not only inside the input subset T , but among all cuts whic h are well-correlated with ( T , ¯ T ). View ed from this p ersp ectiv e, our w ork may b e seen as a spectral analogue of these ﬂow-based tec hniques, since Theorem 6 pro vides lo wer b ounds on the conductance of other cuts as a function of how well-correlated they are with the seed v ector. Alternate interpretation of our main optimization program. There are a few in teresting w ays to view our lo cal optimization problem of Figure 1 whic h w ould like to point out here. Recall that Lo calSp ec tral may b e in terpreted as augmenting the standard sp ectral optimization program with a constrain t that the output cut b e w ell-correlated with the input seed set. T o understand this program from the p ersp ectiv e of the dual, recall that the dual of LocalSp ectral is giv en b y the following. maximize α + β κ s.t. L G  αL K n + β Ω T β ≥ 0 , 21 where Ω T = D G s T s T T D G . Alternativ ely , by subtracting the second constrain t of Lo calSp ectral from the ﬁrst constrain t, it follows that x T  L K n − L K n s T s T T L K n  x ≤ 1 − κ. It can b e shown that L K n − L K n s T s T T L K n = L K T v ol( ¯ T ) + L K ¯ T v ol( T ) , where L K T is the D G -w eighted complete graph on the vertex set T . Thus, Lo calSpectral is clearly equiv alent to minimize x T L G x s.t. x T L K n x = 1 x T  L K T v ol( ¯ T ) + L K ¯ T v ol( T )  x ≤ 1 − κ. The dual of this program is given by the following. maximize α − β (1 − κ ) s.t. L G  αL K n − β  L K T v ol( ¯ T ) + L K ¯ T v ol( T )  β ≥ 0 . F rom the persp ectiv e of this dual, this can b e viewed as “em b edding” a com bination of a complete graph K n and a weigh ted combination of complete graphs on the sets T and ¯ T , i.e. , K T and K ¯ T . Dep ending on the v alue of β , the latter terms clearly discourage cuts that substan tially cut into T or ¯ T , thus encouraging partitions that are well-correlated with the input cut ( T , ¯ T ). Bounding the size of the output cut. Readers familiar with the sp ectral metho d may recall that giv en a graph with a small balanced cut, it is not p ossible, in general, to guarantee that the sweep cut pro cedure of Theorem 4 applied to the optimal of Spectral outputs a balanced cut. 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A Local Spectral Method for Graphs: with Applications to Improving Graph Partitions and Exploring Data Graphs Locally

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