Cheeger constants, structural balance, and spectral clustering analysis for signed graphs
aa r X i v : . [ m a t h . C O ] M a r CHEEGER CONSTANTS, STRUCTURAL BALANCE,AND SPECTRAL CLUSTERING ANALYSIS FORSIGNED GRAPHS
FATIHCAN M. ATAY AND SHIPING LIU
Abstract.
We introduce a family of multi-way Cheeger-type con-stants { h σk , k = 1 , , . . . , n } on a signed graph Γ = ( G, σ ) such that h σk = 0 if and only if Γ has k balanced connected components.These constants are switching invariant and bring together in aunified viewpoint a number of important graph-theoretical con-cepts, including the classical Cheeger constant, those measures ofbipartiteness introduced by Desai-Rao, Trevisan, Bauer-Jost, re-spectively, on unsigned graphs, and the frustration index (origi-nally called the line index of balance by Harary) on signed graphs.We further unify the (higher-order or improved) Cheeger and dualCheeger inequalities for unsigned graphs as well as the underlyingalgorithmic proof techniques by establishing their correspondingversions on signed graphs. In particular, we develop a spectralclustering method for finding k almost-balanced subgraphs, eachdefining a sparse cut. The proper metric for such a clustering is themetric on a real projective space. We also prove estimates of theextremal eigenvalues of signed Laplace matrix in terms of numberof signed triangles (3-cycles). Contents
1. Introduction 22. Preliminaries 52.1. Harary’s balance theorem and bipartition 52.2. Switching equivalence 62.3. Laplace matrices and basic spectral theory 73. (Multi-way) Signed Cheeger constants 93.1. Definition of the signed Cheeger constant 93.2. Switching invariant property 103.3. Relations to other graph-theoretic concepts 113.4. Multi-way signed Cheeger constants 154. Signed Cheeger inequality 165. Higher-order signed Cheeger inequalities 195.1. The lower bound estimate 20 λ and 2 − λ n Introduction
Given a graph, there are two basic properties often of interest: (i)Is it connected ? (ii) Is it bipartite ? We know from spectral graph the-ory that the two properties can be characterized by eigenvalues of theLaplace matrix of the graph. The second smallest eigenvalue of theLaplace matrix vanishes if and only if the graph is disconnected (recallthat the smallest eigenvalue is always zero). A quantitative character-ization is given by the
Cheeger inequality : the
Cheeger constant wasintroduced as a measure of connectedness, and gives upper and lowerbounds of the second smallest eigenvalue [16, 21, 4, 3, 43, 17]. (In fact,such constants have been introduced in graph theory earlier by Poly´aand Szeg¨o [45], though without the connection to eigenvalues.) On theother hand, the bipartiteness property turns out to be closely relatedto the largest eigenvalue of the Laplace matrix. More precisely, theeigenvalues of the normalized Laplace matrix lie in the interval [0 , non-bipartiteness parameter , Trevisan [52] intro-duced a bipartiteness ratio and Bauer and Jost [6] introduced a dualCheeger constant , as measures of bipartiteness. They also establishedthe corresponding eigenvalue estimates, which we will refer to as dualCheeger inequalities , following the terminology of Bauer and Jost.Although the connectedness and bipartiteness are two quite differentproperties, the proofs of Cheeger and dual Cheeger inequalities sharequite a number of comparable ideas. The similarity of the spectralapproaches of connectedness and bipartiteness is further observed inthe proofs of higher order Cheeger inequalities of Lee, Oveis Gharanand Trevisan [35] and the higher order dual Cheeger inequalities of Liu[38], and in the proofs of improved Cheeger inequalities and bounds onbipartiteness ratio of Kwok, Lau, Lee, Oveis Gharan and Trevisan [34].It is then natural to ask for the reason of such similarity. In this pa-per, we present a systematic unified approach to the above-mentionedresults in the framework of signed graphs. This clarifies the reason of HEEGER CONSTANTS FOR SIGNED GRAPHS 3 the similarity and provides a deeper understanding of the relationshipbetween Cheeger and dual Cheeger inequalities.In fact, we more generally study the relation between the spectraand the structural balance theory of signed graphs, which is interestingin its own right. Signed graphs and the idea of balance, introducedby Frank Harary [24] in 1953, and rediscovered since then in differentcontexts many times, are important models and tools for various re-search fields. The concepts were motivated and suggested by problemsin social psychology [24, 25, 14] and have stimulated new methods foranalyzing social networks [32, 54, 51], biological networks [50], logicalprogramming [18], and so on. Signed graphs also play important rolesin various branches of mathematics, such as group theory, root systems(see [12] and the references therein), topology [11, 13], and even physics[9]. By relating signed graphs with 2-lifts of a graph, Bilu and Linial[10] reduce the problem of constructing expander graphs to finding asignature with small spectral radius. In a recent breakthrough, Marcus,Spielman, and Srivastava [40, 41] show the existence of infinite familiesof regular bipartite Ramanujan graphs of every degree larger than 2,by proving a variant conjecture of Bilu and Linial about the existenceof the signature of a given graph with very small spectral radius.Mathematically, a signed graph Γ = (
G, σ ) is an undirected graph G = ( V, E ) with a signature σ : E → { +1 , − } on the edge set E . Onecan think of the vertex set V as a social group, where a positive (resp.,negative) edge between two vertices indicates that the two membersare friends (resp., enemies). The sign of a cycle in G is defined asthe product of the signs of all edges in it. A signed graph Γ is called balanced if all cycles in G are positive. This is a crucial concept due toHarary [24], and plays a central role in our unification of Cheeger anddual Cheeger inequalities in the framework of signed graphs.The properties of being balanced can be characterized by the eigen-values of the signed normalized Laplace matrix. More precisely, thesmallest eigenvalue of the matrix vanishes if and only if the graphΓ has a balanced connected component. In this paper, we define aCheeger-type constant h σ (see Definition 3.1) based on Harary’s bal-ance theorem (Theorem 2.1) such thatΓ has a balanced connected component ⇐⇒ h σ = 0 . (1.1)In the following, we will refer to this Cheeger-type constant of a signedgraph as a signed Cheeger constant for short. Similarly, we will alsospeak of signed inequalities and signed algorithms. We will then provea signed Cheeger inequality (see Theorem 4.1), which is an estimate FATIHCAN M. ATAY AND SHIPING LIU of the smallest eigenvalue from below and above in terms of h σ . (Forprevious results in this aspect, see [8, 28].)When the signature σ on Γ is switching equivalent (see Definition2.1) to all-positive signature, having a balanced connected componentis simply equivalent to the trivial property of having a connected com-ponent. When σ is switching equivalent to all-negative signature, hav-ing a balanced connected component is equivalent to having a bipartiteconnected component. Indeed, we show that the signed Cheeger con-stant h σ and its multi-way versions (Definition 3.2) provide a commonextension of the classical Cheeger constant [16, 21, 4, 3, 43, 17], thenon-bipartiteness parameter of Desai and Rao [20] (after a modifica-tion; see (3.15)), the bipartiteness ratio of Trevisan [52], and the dualCheeger constant of Bauer and Jost [6].The introduction of the signed Cheeger constant further enables usto develop a corresponding spectral clustering method on signed net-works. We propose an algorithm for finding k almost-balanced sub-graphs of a signed graph Γ = ( G, σ ). The novel point is that, afterembedding the graph into the Euclidean space R k via eigenfunctions,we find the proper metric for clustering points is a metric on the realprojective space P k − R (see (5.3)) studied in [38]. Interestingly, thisalgorithm unifies the traditional spectral clustering for finding k sparsecuts (e.g. [44, 39]) and the recent one for finding k almost-bipartitesubgraphs proposed in [38].We further explore the related theoretical analysis of this algorithm.Indeed, we unify the higher-order Cheeger [35], the higher-order dualCheeger [38], and the improved Cheeger inequalities, the improvedbounds of bipartiteness ratio [34] of unsigned graphs as higher or-der signed Cheeger inequalities (Theorem 5.1) and improved signedCheeger inequalities (Theorem 6.1), in terms of our signed Cheegerconstants.Harary [25] defined a signed graph Γ = ( G, σ ) to be antibalanced ifits negation − Γ := ( G, − σ ) is balanced. Thus, Γ is antibalanced if andonly if every odd cycle in it is negative and every even cycle is positive.It is known that a connected signed graph is antibalanced if and only ifthe largest eigenvalue of the signed normalized Laplace matrix equals2 (see [37]). We obtain similar results concerning antibalance and thespectral gap between 2 and the largest eigenvalue via an antitheticaldual signed Cheeger constant (see (3.20)).Finally, we prove estimates for the smallest and largest eigenvaluesof the signed normalized Laplace matrix in terms of signed 3-cycles(we will speak of signed triangles in the following). By definition, thepresence of positive (resp., negative) triangles implies that Γ cannot be HEEGER CONSTANTS FOR SIGNED GRAPHS 5 antibalanced (resp., balanced). Therefore, the number of signed trian-gles relate naturally to the spectral gaps between 0 and the smallesteigenvalue, and, between 2 and the largest eigenvalue. We also estab-lish an upper bound estimate for the largest eigenvalue of the signednon-normalized Laplace matrix, in terms of vertex degrees and numberof positive triangles. This result improves the estimate by Hou, Li, andPan [29], which is only in terms of vertex degrees. In fact, our estimateanswers the question asked in their paper [29, remark after Theorem3.5]. 2.
Preliminaries
Let Γ = (
G, σ ) be a signed graph, where G = ( V, E ) is an undirectedgraph and σ : E → { +1 , − } is a signature function on the edges. Wecollect some basic results from structural balance theory and spectraltheory of signed graphs in this section.We first introduce some notation. We say u, v ∈ V are neighborswhen e = { u, v } ∈ E , and write u ∼ v . For ease of notation wewrite σ ( uv ) := σ ( { u, v } ) for the sign of an edge. In addition to thesign, we also assign a positive symmetric weight w uv to every edge e = { u, v } ∈ E , and set w uv = 0 if e = { u, v } / ∈ E . We say the graphis unweighted if w uv ≡ , ∀ { u, v } ∈ E . The degree d u of a vertex u isdefined as d u = P v ∈ V w uv . We will restrict ourselves to signed simplegraphs, i.e., the case when the underlying graph G has no self-loops ormulti-edges. We also consider a general positive measure µ : V → R > on the vertex set.2.1. Harary’s balance theorem and bipartition.
Recall that Γ iscalled balanced if all cycles in G is positive. The following structuretheorem for balance was proved in [24]. Theorem 2.1 (Harary’s Balance Theorem) . A signed graph Γ is bal-anced if and only if there exists a bipartition of the entire set V intotwo disjoint subsets V and V (one of which may be empty) such thateach positive edge connects two vertices of the same subset and eachnegative edge connects two vertices of different subsets. By reversing the signature, Harary gave the antithetical dual resultfor antibalance. Recall that Γ is called antibalanced if every odd cycleis negative and every even cycle is positive.
Theorem 2.2. [25]
A signed graph Γ is antibalanced if and only ifthere exists a bipartition of the entire set V into two disjoint subsets V and V (one of which may be empty), such that each negative edge FATIHCAN M. ATAY AND SHIPING LIU connects two vertices of the same subset and each positive edge connectstwo vertices of different subsets.
Switching equivalence.
In this subsection, we discuss an im-portant operation of signatures on G , called switching. Definition 2.1 (Switching) . The operation of reversing the signs of alledges connecting a subset S ⊆ V and its complement is called switchingthe subset S . Two signatures σ : E → { +1 , − } and σ ′ : E → { +1 , − } are said to be switching equivalent if there exists a subset S ⊆ V suchthat σ ′ can be obtained from σ by switching the subset S . We write σ ′ ≈ σ in this case.Switching equivalence is an equivalence relation on signatures of afixed underlying graph. We call the corresponding equivalent classesthe switching classes , and denote the switching class of σ by [ σ ].By definition, the sign of any cycle is preserved by the switchingoperation. Therefore, we have the following basic property. Proposition 2.1.
Being balanced is a switching invariant property.
There is a different but equivalent way to describe switching oper-ations, which is more convenient for our later discussions. A function θ : V → { +1 , − } is called a switching function . Switching the sig-nature of Γ = ( G, σ ) by θ refers to the operation of changing σ to σ θ via σ θ ( uv ) := θ ( u ) σ ( uv ) θ ( v ) , ∀ { u, v } ∈ E. Equivalently speaking, switching σ by θ means reversing the signs of alledges between the set V − θ := { u ∈ V : θ ( u ) = − } and its complement.Therefore, this operation is just switching the subset V − θ of V . Given v ∈ V , define θ v ( u ) = − u = v and +1 otherwise. A vertex switchingat v , i.e. switching the vertex v , means switching σ by θ v . Note that θ = Q v ∈ V − θ θ v ; thus, switching a subset of V is equivalent to switchingevery vertex in it one after another.Zaslavsky [55] proved the following useful characterization. Theorem 2.3 (Zaslavsky’s switching lemma) . A signed graph
Γ =(
G, σ ) is balanced if and only if σ is switching equivalent to the all-positive signature, and it is antibalanced if and only if σ is switchingequivalent to the all-negative signature. For more details and history about switching, we refer to [56] andthe references therein.
HEEGER CONSTANTS FOR SIGNED GRAPHS 7
Laplace matrices and basic spectral theory.
In this section,we discuss the spectra of signed graphs, which are in fact switchinginvariant (see, e.g., [56]). Let A σ be the signed adjacency matrix. Thatis, we have for any u, v ∈ V ,( A σ ) uv = (cid:26) σ ( uv ) , if u ∼ v ;0 , otherwise.Given a switching function θ : V → { +1 , − } , we denote by D ( θ )the diagonal matrix with D ( θ ) uu = θ ( u ) , ∀ u ∈ V . It is then straight-forward to check that A σ θ = D ( θ ) − A σ D ( θ ) . (2.1)Therefore, the spectrum of the matrix A σ is switching invariant. Definition 2.2 (signed Laplace matrices) . Let D be the diagonal de-gree matrix. We call L σ := D − A σ the signed non-normalized Laplacematrix , and ∆ σ := I − D − A σ the signed normalized Laplace matrix ,where I denotes the | V | × | V | identity matrix. Remark 2.1.
In the literature, the normalized Laplace matrix of anunsigned graph is defined either as the matrix I − D − AD − [17] or I − D − A [6], where A is the unsigned adjacency matrix. The latterone is also called the random walk Laplacian. The two matrices areunitarily equivalent and hence share the same spectrum. The signednormalized Laplace matrix ∆ σ we use here is an extension of the lattermatrix. it has the same spectrum as the matrix I − D − A σ D − . Thematrix ∆ σ also appears naturally in the context of graph drawing andelectrical networks [33].The following property follows directly from our discussion of thespectrum of A σ . Proposition 2.2 ([56]) . The spectrum of ∆ σ or L σ for a signed graph Γ = (
G, σ ) is switching invariant. It is well-known that the eigenvalues of ∆ σ lie in the interval [0 , ≤ λ (∆ σ ) ≤ λ (∆ σ ) ≤ · · · ≤ λ n (∆ σ ) ≤ , where n is the cardinality of V . Moreover, the cases when 0 and 2 areeigenvalues are characterized as below; see, e.g., [55, 29, 37].Γ has a balanced connected component ⇔ λ (∆ σ ) = 0 , (2.2)Γ has an antibalanced connected component ⇔ λ n (∆ σ ) = 2 . (2.3) FATIHCAN M. ATAY AND SHIPING LIU
Let µ d denote the degree measure on V , i.e. µ d ( u ) = d u , ∀ u ∈ V .The operator form of ∆ σ can be expressed by its action on any function f : V → R and any u ∈ V as∆ σ f ( u ) = 1 µ d ( u ) X v,v ∼ u w uv ( f ( u ) − σ ( uv ) f ( v )) . (2.4)Replacing µ d above by the constant measure µ ≡ L σ . For a general measure µ , we denote the correspondinginner product of two functions f, g : V → R by( f, g ) µ = X u ∈ V µ ( u ) f ( u ) g ( u ) . The signed Rayleigh quotient of a map Φ : V → R k is given by R σ (Φ) = P u ∼ v w uv k Φ( u ) − σ ( uv )Φ( v ) k P u ∈ V µ ( u ) k Φ( u ) k . (2.5)We also define a dual version of the Rayleigh quotient of Φ by e R σ (Φ) = P u ∼ v w uv k Φ( u ) + σ ( uv )Φ( v ) k P u ∈ V µ ( u ) k Φ( u ) k . (2.6)The Courant-Fisher-Weyl min-max principle says that the k -th eigen-value λ k of ∆ σ (or L σ ) satisfies λ k = min f ,f ,...,f k f i ,f j ) µ =0 , ∀ i = j max f f ∈ span { f ,f ,...,f k } R σ ( f ) . (2.7)In particular, we have λ (∆ σ ) = min f R σ ( f ) , and 2 − λ N (∆ σ ) = min f e R σ ( f ) . (2.8) Lemma 2.1.
For any ≤ k ≤ n , it holds that − λ N − k +1 (∆ σ ) = λ k (∆ − σ ) . This follows immediately from the fact that e R σ ( f ) = R − σ ( f ). Thesupport of a map Φ : V → R is defined assupp(Φ) := { u ∈ V : Φ( u ) = 0 } . By (2.7), one can derive the following lemma (see, e.g., [34]).
Lemma 2.2.
For any k disjointly supported functions f , f , . . . , f k : V → R , λ k ≤ ≤ i ≤ k R σ ( f i ) . (2.9)We refer to [28, 36, 23, 5, 8, 46] for more results in the spectral theoryof signed graphs. HEEGER CONSTANTS FOR SIGNED GRAPHS 9 (Multi-way) Signed Cheeger constants In this section, we define the signed Cheeger constant h σ mentionedin the introduction and prove that h σ is switching invariant. We relate h σ to several existing graph-theoretic concepts. We also define thecorresponding multi-way signed Cheeger constants.3.1. Definition of the signed Cheeger constant.
We first definesome notation. Given two subsets V , V of V , denote the set of edgeslying between V and V by E ( V , V ) := {{ u, v } ∈ E : u ∈ V , v ∈ V } . We distinguish the set of positive edges and the set of negative edgesby E + ( V , V ) := {{ u, v } ∈ E : u ∈ V , v ∈ V , σ ( uv ) = +1 } , and E − ( V , V ) := {{ u, v } ∈ E : u ∈ V , v ∈ V , σ ( uv ) = − } . We further define their weighted cardinality as | E ( V , V ) | = X u ∈ V X v ∈ V , { u,v }∈ E ( V ,V ) w uv , | E + ( V , V ) | = X u ∈ V X v ∈ V , { u,v }∈ E + ( V ,V ) w uv , | E − ( V , V ) | = X u ∈ V X v ∈ V , { u,v }∈ E − ( V ,V ) w uv . When V = V , we write | E ( V ) | := | E ( V , V ) | and | E + ( V ) | := | E + ( V , V ) | , | E − ( V ) | := | E − ( V , V ) | for short. Keep in mind that, in this case, the weighted cardinalitydefined above counts every edge weight twice. We adopt this definitionof weighted cardinality for the convenience of later discussions.For a subset S ⊆ V , we define its volume asvol µ ( S ) = X u ∈ S µ ( u ) . Let ( V , V ) denote a sub-bipartition of V , that is, V and V aresubsets of V satisfying ∅ 6 = V ∪ V ⊆ V and V ∩ V = ∅ . We define the signed bipartiteness ratio of ( V , V ) as β σ ( V , V ) = 2 | E + ( V , V ) | + | E − ( V ) | + | E − ( V ) | + | E ( V ∪ V , V ∪ V ) | vol µ ( V ∪ V ) , (3.1)where V ∪ V is the complement of V ∪ V in V .Observe that β σ ( V , V ) can be considered as a modification of theexpansion | E ( V ∪ V , V ∪ V ) | vol µ ( V ∪ V )of the set V ∪ V . If the additional quantity in the numerator2 | E + ( V , V ) | + | E − ( V ) | + | E − ( V ) | vanishes, then, by Harary’s balanced theorem (Theorem 2.1), we con-clude that the induced signed subgraph of V ∪ V in Γ = ( G, σ ) isbalanced. Moreover, V and V give a partition of the induced sub-graph satisfying the properties described in Theorem 2.1. Definition 3.1 (Signed Cheeger constant) . For a signed graph Γ =(
G, σ ), the signed Cheeger constant h σ ( µ ) is defined as h σ ( µ ) = min ( V ,V ) β σ ( V , V ) , (3.2)where the minimum is taken over all possible sub-bipartitions of V .With this definition, we observe directly that h σ ( µ ) = 0 if and onlyif there exists a sub-bipartition ( V , V ) of V with β σ ( V , V ) = 0. Fromthe above discussion about the signed bipartiteness ratio, we have that h σ ( µ ) = 0 if and only if Γ has a balanced connected component. Thatis, the statement (1.1) mentioned in the introduction holds.3.2. Switching invariant property.
We prove that h σ ( µ ) is switch-ing invariant. Proposition 3.1.
Let
Γ = (
G, σ ) be a signed graph. For any switchingfunction θ : V → { +1 , − } , h σ ( µ ) = h σ θ ( µ ) . (3.3)This property is a direct corollary of the following lemma. Lemma 3.1.
For any switching function θ : V → { +1 , − } and anysub-bipartition ( V , V ) , there exists a sub-bipartition ( V ′ , V ′ ) , such that V ′ ∪ V ′ = V ∪ V , and β σ θ ( V ′ , V ′ ) = β σ ( V , V ) . (3.4) HEEGER CONSTANTS FOR SIGNED GRAPHS 11
Moreover, when σ θ can be obtained from σ via switching a subset S ⊆ V ,we have β σ θ ( V \ S, V ∪ S ) = β σ ( V , V ) . (3.5) Proof.
We only need to prove the lemma for a vertex switching at u ∈ V , that is, a switching of σ by θ u . (Recall that θ u ( v ) = − v = u and +1 otherwise, and σ θ u ( uv ) = − σ ( uv ) for all v ∼ u .)Now, if u ∈ V ∪ V , the vertex switching at u does not change thesigned bipartiteness ratio; hence V ′ := V and V ′ := V satisfy theequality (3.4).If u ∈ V ∪ V , suppose without loss of generality that u ∈ V . Afterthe vertex switching at u , we have( β σ θu ( V , V ) − β σ ( V , V )) vol µ ( V ∪ V )= 2 X v ∈ V σ ( uv )= − w uv − X v ∈ V σ ( uv )=+1 w uv + 2 X v ∈ V σ ( uv )=+1 w uv − X v ∈ V σ ( uv )= − w uv . In this case, we set V ′ := V \ { u } and V ′ := V ∪ { u } , that is, we movethe vertex u from V into V . Then we observe that( β σ θu ( V ′ , V ′ ) − β σ θu ( V , V ))vol µ ( V ′ ∪ V ′ )= − X v ∈ V σ θu ( uv )= − w uv + 2 X v ∈ V σ θu ( uv )=+1 w uv − X v ∈ V σ θu ( uv )=+1 w uv + 2 X v ∈ V σ θu ( uv )= − w uv . Adding the two equalities above, we arrive at (3.4). (cid:3)
Relations to other graph-theoretic concepts.
We use Propo-sition 3.1 to explain the relations of the signed Cheeger constant toseveral other graph-theoretic concepts.We denote by σ + (resp., σ − ) the all-positive (resp., all-negative)signature. We consider the signed Cheeger constant of two particularclasses of signatures: (i) σ ∈ [ σ − ], and (ii) σ ∈ [ σ + ].Recall that the bipartiteness ratio of Trevisan [52] is defined as β := min ( V ,V ) | E ( V , V ) | + | E ( V ) | + | E ( V ) | + | E ( V ∪ V , V ∪ V ) | vol µ ( V ∪ V ) . Furthermore, Bauer and Jost [6] introduced the dual Cheeger constant h := max ( V ,V ) | E ( V , V ) | vol µ ( V ∪ V ) . These two concepts are closely related, because, due to the fact thatvol µ ( V ∪ V ) = | E ( V ) | + | E ( V ) | + | E ( V ∪ V , V ∪ V ) | , one has β = 1 − h. Observe that, by definition, h σ − ( µ ) = β = 1 − h. Therefore, we obtain the following fact by Proposition 3.1: If σ ∈ [ σ − ],that is, if Γ = ( G, σ ) is antibalanced, then h σ ( µ ) = β = 1 − h .We proceed to consider h σ ( µ ) when σ ∈ [ σ + ]. To this end, we firstprepare the following reformulation of the signed Cheeger constant.Recall that the expansion (or conductance) of a subset S ⊆ V is definedas ρ ( S ) := | E ( S, S ) | vol µ ( S ) . (3.6)We define a signed expansion of S ⊆ V in Γ to be ρ σ ( S ) := | E − ( S ) | + | E ( S, S ) | vol µ ( S ) . (3.7)We have the following relation between h σ ( µ ) and signed expansions. Corollary 3.1.
Let
Γ = (
G, σ ) be a signed graph. Then, h σ ( µ ) = min σ ′ ∈ [ σ ] min ∅6 = S ⊆ V ρ σ ′ ( S ) . (3.8) Proof.
We denote by ( V , V ) S a bipartition of S , i.e., V ∪ V = S , V ∩ V = ∅ . We claim thatmin ( V ,V ) S β σ ( V , V ) = min σ ′ ∈ [ σ ] ρ σ ′ ( S ) . (3.9)Let V , V be the bipartition of S that achieves the minimum on theleft hand side of (3.9). Suppose that σ can be obtained from σ viaswitching the subset V . Then by Lemma 3.1, β σ ( V , V ) = β σ ( S, ∅ ) = ρ σ ( S ) ≥ min σ ′ ∈ [ σ ] ρ σ ′ ( S ) . (3.10)Moreover, the inequality above can only be an equality, because other-wise there would exist a σ ′ ∈ [ σ ] such that β σ ′ ( S, ∅ ) = ρ σ ′ ( S ) < β σ ( V , V ) . By Lemma 3.1, we could then find a bipartition V ′ , V ′ of S , such that β σ ′ ( S, ∅ ) = β σ ( V ′ , V ′ ) < β σ ( V , V ) , which is a contradiction. Hence (3.9) holds. Then (3.8) follows directly. (cid:3) HEEGER CONSTANTS FOR SIGNED GRAPHS 13
Using Corollary 3.1, we observe the following: When σ ∈ [ σ + ], thatis, when Γ = ( G, σ ) is balanced, we have h σ ( µ ) = min ∅6 = S ⊆ V ρ σ + ( S ) = min ∅6 = S ⊆ V | E ( S, S ) | vol µ ( S ) . This is the one-way Cheeger constant [42, 35], which trivially vanishes.Next, we compare the signed Cheeger constant with a constant dueto Desai and Rao [20]. On unweighted graphs, Desai and Rao [20]introduced the non-bipartiteness parameter α := min ∅6 = S ⊆ V e min ( S ) + | E ( S, S ) | vol µ ( S ) , (3.11)where e min ( S ) is the minimum number of edges that need to be removedfrom the induced subgraph of S in order to make it bipartite. Hou [28]extended this notion to a signed graph Γ = ( G, σ ) as α σ := min ∅6 = S ⊆ V e σ min ( S ) + | E ( S, S ) | vol µ ( S ) , (3.12)where e σ min ( S ) is the minimum number of edges that need to be removedfrom the induced subgraph of S in order to make it balanced. Bydefinition, ( G, σ − ) is balanced if and only if G has no odd cycles, i.e., G is bipartite. Therefore, α σ − = α .We notice that the quantity e σ min ( V ) coincides with the line indexof balance of Γ introduced by Harary [26] (see also [1]), which waslater called the frustration index (suggested to Harary by Zaslavskyin private communication from the work of Toulouse [53]) and studiedextensively, e.g. [27, 2, 48, 8]. For a recent empirical approach of thisindex, see [22].For a subset S , we define a weighted version of the index e σ min ( S ) asfollows. Let Γ S denote the induced signed graph of S , and E | Γ S denotethe edge set of Γ S . Define e σ min ( S ) :=min E ⊆ E | Γ S { X { u,v }∈ E w uv | Γ S becomes balanced after deleting edges in E } . (3.13)We modify the above notions (3.11) and (3.12) to give the followingparameter associated to S ⊆ V : α σ ( S ) := 2 e σ min ( S ) + | E ( S, S ) | vol µ ( S ) . (3.14) This leads to the following reformulation of the signed Cheeger con-stant.
Corollary 3.2.
Let
Γ = (
G, σ ) be a signed graph. Then h σ ( µ ) = min ∅6 = S ⊆ V α σ ( S ) . (3.15)Thus, the constant h σ ( µ ) is larger than the parameter α σ of Hou [28]in general. Corollary 3.2 follows directly from the next lemma. Lemma 3.2.
For any ∅ 6 = S ⊆ V , we have min ( V ,V ) S β σ ( V , V ) = min σ ′ ∈ [ σ ] ρ σ ′ ( S ) = α σ ( S ) . (3.16) Proof.
The first equality follows from (3.9). To prove the second equal-ity, let Γ S denote the induced signed graph of S and let σ be thesignature that achieves min σ ′ ∈ [ σ ] ρ σ ′ ( S ). It is easy to see that2 e σ min ( S ) ≤ | E − ( S ) | ( σ ) . Therefore, α σ ( S ) ≤ min σ ′ ∈ [ σ ] ρ σ ′ ( S ).Now let Γ ′ S be the balanced graph obtained from Γ S by deletingedges in E , which is the subset of E | Γ S that attains the minimumin the definition (3.13) of e σ min ( S ). By Theorem 2.1, there exists abipartition V , V of S such that | E + | Γ ′ S ( V , V ) | = | E −| Γ ′ S ( V ) | = | E −| Γ ′ S ( V ) | = 0 . This implies2 e σ min ( S ) = 2 | E + | Γ S ( V , V ) | + | E −| Γ S ( V ) | + | E −| Γ S ( V ) | . Hence α σ ( S ) ≥ min ( V ,V ) S β σ ( V , V ). This proves the second equality. (cid:3) Remark 3.1.
The (unweighted) equality 2 e σ min ( S ) = min σ ′ ∈ [ σ ] | E − ( S ) | ( σ ′ )seems to be folklore among graph theorists; see [57, Theorem 3.3]. Weinclude a proof here for completeness.We compare the signed Cheeger constant with the degree of balance b (Γ) of a signed graph Γ introduced by Cartwright and Harary [14].In [14], they also aimed at quantifying the deviation of a signed graphfrom being balanced. Their constant b (Γ) is defined as b (Γ) := the number of positive cycles of Γthe number of cycles of Γ . (3.17)Observe b (Γ) ∈ [0 , − b (Γ) imply that Γ is closerto being balanced. Consider the signed graph Γ = ( C n , σ ) where C n isthe unweighted cycle graph on n vertices and σ is the signature such HEEGER CONSTANTS FOR SIGNED GRAPHS 15 that Γ has exactly one negative edge. Intuitively, Γ is close to beingbalanced. Actually, we have1 − b (Γ) = 1 and h σ ( µ d ) = 1 n . (3.18)This shows that the constant h σ ( µ d ) is finer than b (Γ).3.4. Multi-way signed Cheeger constants.
Extending Definition3.2 in the spirit of [42, 35, 38], we can define a family of multi-waysigned Cheeger constant { h σk ( µ ) , k = 1 , , . . . n } in a natural way. Definition 3.2.
For 1 ≤ k ≤ n , the k -way signed Cheeger constant h σk ( µ ) of a signed graph Γ = ( G, σ ) is defined as h σk ( µ ) := min { ( V i − ,V i ) } ki =1 max ≤ i ≤ k β σ ( V i − , V i ) , (3.19)where the minimum is taken over the space of all possible k pairs ofdisjoint sub-bipartitions ( V , V ) , ( V , V ) , . . . , ( V k − , V k ). To ease thenotation, we denote this space by Pair( k ) and call every element ofPair( k ) a k -sub-bipartition of V .The quantity h σ ( µ ) defined in (3.2) is the first one of this family.We have the monotonicity h σk ( µ ) ≤ h σk +1 ( µ ). Moreover, Theorem 2.1implies the following property. Proposition 3.2.
For a signed graph
Γ = (
G, σ ) and ≤ k ≤ n , itholds that h σk ( µ ) = 0 if and only if Γ has k balanced connected compo-nents. Roughly speaking, the k -way signed Cheeger constant is a “mixture”of the k -way Cheeger constant h k ( µ ) introduced by Miclo [42] (see also[35]) and the k -way dual Cheeger constant h k ( µ ) in [38] for unsignedgraphs.By Lemma 3.1, h σk ( µ ) is switching invariant for any 1 ≤ k ≤ n .Furthermore, Lemma 3.2 implies the following equivalent expressionsfor h σk ( µ ): h σk ( µ ) = min σ ′ ∈ [ σ ] min { S i } ki =1 max ≤ i ≤ k ρ σ ′ ( S i )= min { S i } ki =1 max ≤ i ≤ k α σ ( S i ) , where the minimum min { S i } ki =1 is taken over the space of all possible k -subpartitions, S , S , . . . , S k , where S i = ∅ for any 1 ≤ i ≤ k . In par-ticular, we observe that h σ ( µ ) reduces to the classical Cheeger constant when σ ∈ [ σ + ]. Remark 3.2.
The (multi-way) signed Cheeger constants provide newinsights into existing constants reflecting connectivity (e.g., Cheegerconstants) or bipartiteness (e.g., dual Cheeger constants, bipartitenessratio of Trevisan, non-bipartiteness parameter of Desai and Rao) ofunsigned graphs in the language of switching within the framework ofsigned graphs, thus giving a unified viewpoint about connectivity andbipartiteness of the underlying graph via assigning signatures.We can also define a natural family of antithetical dual signed Cheegerconstants { e h σk ( µ ) , k = 1 , , . . . , n } by e h σk ( µ ) := h − σk ( µ ). Dually, we havethatΓ has an antibalanced connected component ⇐⇒ e h σ ( µ ) = 0 . (3.20)4. Signed Cheeger inequality
In this section, we prove the following signed Cheeger inequality.
Theorem 4.1.
Given a signed graph
Γ = (
G, σ ) , we have λ (∆ σ )2 ≤ h σ ( µ d ) ≤ p λ (∆ σ ) . (4.1)The lower bound estimate in (4.1) is easier. Proof of the lower bound estimate of Theorem 4.1.
For any ( V , V ), con-sider the particular function given by, f ( u ) = , if u ∈ V ; − , if u ∈ V ;0 , otherwise.We calculate R σ ( f ) = 4 | E + ( V , V ) | + 2 | E − ( V ) | + 2 | E − ( V ) | + | E ( V ∪ V , V ∪ V ) | vol µ ( V ∪ V ) ≤ β σ ( V , V ) . Then (2.8) implies λ (∆ σ ) ≤ h σ ( µ ). (cid:3) The upper bound estimate in (4.1) is more difficult. The proof isbased on the crucial observation that the estimate of λ (∆ σ ) should beconsidered as a “mixture” of the estimates of the smallest and largesteigenvalues of unsigned graphs (for which the smallest eigenvalue triv-ially equals 0). This can be seen more clearly from the correspondingRayleigh quotients. One can appeal either to the techniques for prov-ing the Cheeger inequality for unsigned graphs [4, 3, 21] or to those forproving the dual Cheeger inequality [52, 6]. For the former strategy,one needs first to switch the signature to be the one achieving the first HEEGER CONSTANTS FOR SIGNED GRAPHS 17 minimum in (3.8). We adopt here the latter strategy, for which we donot need to switch the signature to a proper one first. In fact, we willuse the idea of Trevisan [52] for proving the dual Cheeger inequalityfor unsigned graphs.Given a non-zero function f : V → R and a real number t ≥ V : V f ( t ) := { u ∈ V : f ( u ) ≥ t } , V f ( − t ) := { u ∈ V : f ( u ) ≤ − t } . Note that for t = 0, we have the special definition V f (0) := { u ∈ V : f ( u ) ≥ } and V f ( −
0) := { u ∈ V : f ( u ) < } . Supposemax u ∈ V f ( u ) = 1. For any t ∈ [0 , Y f ( t ) ∈ {− , , } V as follows: For each u ∈ V ,( Y f ( t )) u := , if u ∈ V f ( √ t ); − , if u ∈ V f ( −√ t );0 , otherwise.The following lemma is crucial for our purposes. Lemma 4.1.
For any { u, v } ∈ E , Z | ( Y f ( t )) u − σ ( uv )( Y f ( t )) v | dt ≤ | f ( u ) − σ ( uv ) f ( v ) | ( | f ( u ) | + | f ( v ) | ) . (4.2) Proof.
It is enough to prove the estimate Z | ( Y f ( t )) u − ( Y f ( t )) v | dt ≤ | f ( u ) − f ( v ) | ( | f ( u ) | + | f ( v ) | ) (4.3)for any non-zero functions f : V → [ − , g : V → [ − ,
1] satisfying g ( u ) = f ( u ) and g ( v ) = σ ( uv ) f ( v ).Now we prove (4.3). Without loss of generality, suppose | f ( u ) | ≥| f ( v ) | . Case 1: f ( u ) and f ( v ) have different signs. Then, | ( Y f ( t )) u − ( Y f ( t )) v | = , if t ≤ f ( v ) ;1 , if f ( v ) < t ≤ f ( u ) ;0 , if t > f ( u ) .Therefore, Z | ( Y f ( t )) u − ( Y f ( t )) v | dt = f ( u ) + f ( v ) ≤ ( | f ( u ) | + | f ( v ) | ) = | f ( u ) − f ( v ) | ( | f ( u ) | + | f ( v ) | ) . Case 2: f ( u ) and f ( v ) have the same sign. In this case, | ( Y f ( t )) u − ( Y f ( t )) v | = , if t ≤ f ( v ) ;1 , if f ( v ) < t ≤ f ( u ) ;0 , if t > f ( u ) .Therefore, Z | ( Y f ( t )) u − ( Y f ( t )) v | dt = f ( u ) − f ( v ) = | f ( u ) − f ( v ) | ( | f ( u ) | + | f ( v ) | ) . This completes the proof of (4.3). (cid:3)
Remark 4.1.
In the proof of Lemma 4.1, we used local level dualitiesto bring the two extremal cases together. We will use this principleagain in the proofs of Lemma 5.1, Lemma 6.1, Lemma 6.2(ii), andClaim 6 in later sections.In the following, we denote d wµ := max u (cid:26) P v,v ∼ u w uv µ ( u ) (cid:27) . Lemma 4.2.
For any non-zero function f : V → R , there exists a t ′ ∈ [0 , max u ∈ V f ( u )] such that β σ ( V f ( √ t ′ ) , V f ( −√ t ′ )) ≤ q d wµ R σ ( f ) . (4.4) Proof.
Without loss of generality, we can assume max u ∈ V f ( u ) = 1.Consider the ratio I f := R P u ∼ v w uv | ( Y f ( t )) u − σ ( uv )( Y f ( t )) v | dt R P u ∈ V µ ( u ) | ( Y f ( t )) u | dt . By the definition of the vector Y f ( t ), we check that X u ∼ v w uv | ( Y f ( t )) u − σ ( uv )( Y f ( t )) v | = 2 | E + ( V f ( √ t ) , V f ( −√ t )) | + | E − ( V f ( √ t )) | + | E − ( V f ( −√ t )) | + | E ( V f ( √ t ) ∪ V f ( −√ t ) , V f ( √ t ) ∪ V f ( −√ t )) | , and X u µ ( u ) | ( Y f ( t )) u | = vol µ ( V f ( √ t ) ∪ V f ( −√ t )) . Therefore, there exists t ′ ∈ [0 ,
1] such that β σ ( V f ( √ t ′ ) , V t ( −√ t ′ )) ≤ I f . (4.5) HEEGER CONSTANTS FOR SIGNED GRAPHS 19
On the other hand, we estimate I f ≤ P u ∼ v w uv | f ( u ) − σ ( uv ) f ( v ) | ( | f ( u ) | + | f ( v ) | ) P u µ ( u ) f ( u ) ≤ pP u ∼ v w uv | f ( u ) − σ ( uv ) f ( v ) | pP u ∼ v w uv ( | f ( u ) | + | f ( v ) | ) P u µ ( u ) f ( u ) . In the first inequality above, we used Lemma 4.1 and the fact that Z | ( Y f ( t )) u | dt = f ( u ) . Observing that X u ∼ v w uv ( | f ( u ) | + | f ( v ) | ) ≤ X u X v,v ∼ u w uv (2 | f ( u ) | + 2 | f ( v ) | ) ≤ d wµ X u µ ( u ) f ( u ) , (4.6)we arrive at I f ≤ q d wµ R σ ( f ) . (4.7)Combining (4.5) and (4.7) proves the lemma. (cid:3) Proof of the upper bound estimate of Theorem 4.1.
Applying Lemma 4.2to the first eigenfunction φ and choosing µ = µ d , the upper bound es-timate of (4.1) is proved. (cid:3) For the signed non-normalized Laplace matrix L σ , we have the fol-lowing Cheeger-type estimate. Theorem 4.2.
Let
Γ = (
G, σ ) be a signed graph. Then λ ( L σ )2 ≤ h σ ( µ ) ≤ p d max λ ( L σ ) . (4.8) where d max := max u ∈ V d u .Proof. Applying Lemma 4.2 to the first eigenfunction of L σ and choos-ing µ = µ yield the upper bound estimate. The lower bound estimatefollows similarly as in Theorem 4.1. (cid:3) Higher-order signed Cheeger inequalities
In this section, we prove higher-order versions of the signed Cheegerinequality (4.1).
Theorem 5.1.
There exists an absolute constant C such that for anysigned graph Γ = (
G, σ ) and any k ∈ { , , . . . , N } , λ k (∆ σ )2 ≤ h σk ( µ d ) ≤ Ck p λ k (∆ σ ) . (5.1)This is a generalization of the higher-order Cheeger and dual Cheegerinequalities for unsigned graphs by Lee, Oveis Gharan, and Trevisan[35] and the second named author [38].5.1. The lower bound estimate.
Again, the lower bound estimatein (5.1) is easier to show.
Proof of the lower bound estimate of Theorem 5.1.
For any { ( V i − , V i ) } ki =1 ∈ Pair( k ) , that is, any k -sub-bipartition of V , define for each i a function f i ( u ) := , if u ∈ V i − ; − , if u ∈ V i ;0 , otherwise.Let f = P ki =1 a i f i , where a , . . . , a k ∈ R , be a function in the spacespan { f , . . . , f k } . Consider the signed Rayleigh quotient R σ ( f ) = P u ∼ v w uv ( f ( u ) − σ ( uv ) f ( v )) P u µ ( u ) f ( u ) of f . For the denominator, we have X u µ ( u ) f ( u ) = k X i =1 a i X u µ ( u ) f i ( u ) = k X i =1 a i vol µ ( V i − ∪ V i ) . And for the numerator, we estimate X u ∼ v w uv ( f ( u ) − σ ( uv ) f ( v )) ≤ k X i =1 a i (cid:0) | E + ( V i − , V i ) | + | E − ( V i − ) | + | E − ( V i − ) | + | E ( V i − ∪ V i , V i − ∪ V i ) | (cid:1) . Hence, we arrive atmax a ,...,a k R σ ( f ) ≤ ≤ i ≤ k β σ ( V i − , V i ) . (5.2)By the min-max principle (2.7), this implies λ k ≤ h σk ( µ ). (cid:3) The upper bound estimate.
We next prove the remaining up-per bound estimate of (5.1).
HEEGER CONSTANTS FOR SIGNED GRAPHS 21
Idea of the proof: a signed spectral clustering algorithm.
Theo-rem 5.1 is a mixture of the higher-order Cheeger [35] and dual Cheeger[38] inequalities, the proofs of which utilize spectral clustering algo-rithms via metrics on spheres and real projective spaces, respectively.One might anticipate at first that the proper metrics for proving theupper bound estimate in Theorem 5.1 are a mixture of those two kindsof metrics. It is somewhat surprising that the latter metrics [38] them-selves are competent for the proof and provide unified spectral cluster-ing algorithms.We describe the signed spectral clustering algorithm in detail. Thisalgorithm aims at finding k subsets whose induced subgraphs are nearlybalanced. The connections among those k subsets, regardless of theirsigns, are very sparse.Let { φ , φ , . . . , φ n } be an orthonormal system of eigenfunctions cor-responding to λ (∆ σ ) , λ (∆ σ ) , . . . , λ n (∆ σ ).(1) Spectral embedding.
Using the first k eigenfunctions, we obtaina coordinate system for the vertices via the mapΦ : V → R k , v ( φ ( v ) , φ ( v ) , . . . , φ k ( v )) . (2) Normalization.
We further map e V Φ := { v : Φ( v ) = 0 } to theunit sphere,Φ nor : e V Φ → S k − , v Φ( v ) k Φ( v ) k . (3) Clustering the points.
We use the following pseudometric d Φ on e V Φ studied in [38], d Φ ( u, v ) := min {k Φ nor ( u ) + Φ nor ( v ) k , k Φ nor ( u ) − Φ nor ( v ) k} , (5.3)where k · k stands for the Euclidean norm in R k .Recall that the projective space P k − R is obtained from S k − by iden-tifying the antipodal points, P r : S k − → P k − R : x, − x [ x ] , where x are the unit vectors in R k . The metric (5.3) is induced fromthe following metric on P k − R , d ([ x ] , [ y ]) := min {k x + y k , k x − y k} , ∀ [ x ] , [ y ] ∈ P k − R . If σ = σ + , we have λ (∆ σ ) = 0 and φ is the constant function φ ≡ / p vol µ d ( V ). Our algorithm reduces to the traditional cluster-ing for finding k subsets, each defining a sparse cut [39, 44]. A differ-ence is that, traditionally, the spherical metric (or the radial projectiondistance) d sphereΦ ( u, v ) := k Φ nor ( u ) − Φ nor ( v ) k is used for clustering the points, as verified by Lee, Oveis Gharan, and Trevisan [35]. Here inour case, we use the metric (5.3) instead.If, on the other hand, σ = σ − , our algorithm reduces to finding k almost-bipartite subgraphs, since ( G, σ − ) is balanced if and only if G is bipartite. This is exactly the one proposed in [38].Theorem 5.1 and the proof presented below provide the worst-caseperformance guarantee of the algorithm described above. Remark 5.1.
If we use the last k eigenfunctions φ N − k +1 , φ N − k +2 , . . . , φ N instead of the first k eigenfunctions in the step of spectral embedding,we will obtain an algorithm for finding k subsets whose induced sub-graphs are nearly antibalanced, each defining a sparse cut. Remark 5.2.
We comment here about further related works on signedspectral algorithms. For any signed graph (or subgraph), one can con-tinue to do the next-level clustering. Roughly speaking, the objectiveis to find two subsets whose signed bipartiteness ratio is small. Theheuristics of the spectral method for such clustering was discussed in[33, 31, 15]. Actually, the proof of Theorem 4.1 (especially Lemma 4.2)provides a theoretical guarantee for their heuristic arguments. We canachieve this clustering by the threshold sets V f ( t ) := { u ∈ V : f ( u ) ≥ t } and V f ( − t ) := { u ∈ V : f ( u ) ≤ − t } of a certain function f .There are studies about another kind of multi-way clustering ofsigned networks, called the correlation clustering. It aims at finding k non-trivial disjoint subsets V , V , . . . , V k such that edges connectingtwo vertices from the same subset are almost all positive and edgesconnecting two vertices from different subsets are almost all negative.Heuristic spectral algorithms for such clustering were studied in, e.g.,[32, 33, 31, 51, 50, 15]; for non-spectral algorithms, see, e.g., [49].5.2.2. The proof.
Let φ , φ , . . . , φ n and Φ be defined as above. Since λ i = R σ ( φ i ), i = 1 , , . . . , k , we have λ k ≥ P u ∼ v w uv k Φ( u ) − σ ( uv )Φ( v ) k P u ∈ V µ ( u ) k Φ( u ) k = R σ (Φ) . (5.4)In the following, we will find k disjointly supported maps { Ψ i } ki =1 bylocalizing Φ, such that R σ (Ψ i ) can be bounded from above by R σ (Φ i )(up to a polynomial in k ).Recall the pseudometric space ( e V Φ , d Φ ) from the algorithm describedin the last subsection. In order to localize Φ, we consider the following HEEGER CONSTANTS FOR SIGNED GRAPHS 23 cut-off function: For any S i ⊆ V and ǫ >
0, define θ i ( v ) = , if Φ( v ) = 0;max ( , − d Φ ( v, S i ∩ e V Φ ) ǫ ) , otherwise.Then we can localize Φ asΨ i := θ i Φ : V → R k . Observe that Ψ i | S i = Φ | S i andsupp(Ψ i ) ⊆ N ǫ ( S i ∩ e V Φ , d Φ ) := { v ∈ e V Φ : d Φ ( v, S i ∩ e V Φ ) < ǫ } . The following crucial lemma is an extension of [38, Lemma 5.3] and[35, Lemma 3.3].
Lemma 5.1.
For any given < ǫ < , define Ψ i as above. Then forany { u, v } ∈ E , k Ψ i ( u ) − σ ( uv )Ψ i ( v ) k ≤ (cid:18) ǫ (cid:19) k Φ( u ) − σ ( uv )Φ( v ) k . (5.5) Proof.
If either Φ( u ) or Φ( v ) vanishes, (5.5) follows from the fact that | θ i | ≤
1. So we only need to prove (5.5) for u, v ∈ e V Φ . In this case, wecalculate k Ψ i ( u ) − σ ( uv )Ψ i ( v ) k = k θ i ( u )Φ( u ) − σ ( uv ) θ i ( v )Φ( v ) k≤ | θ i ( u ) | k Φ( u ) − σ ( uv )Φ( v ) k + | θ i ( u ) − θ i ( v ) | k Φ( v ) k . (5.6)We claim that | θ i ( u ) − θ i ( v ) | k Φ( v ) k ≤ ǫ k Φ( u ) − σ ( uv )Φ( v ) k . (5.7)Note that (5.5) follows immediately from (5.6) and (5.7). Hence, theremaining task is to prove (5.7). Similarly to the beginning of the proofof Lemma 4.1, it is enough to show | θ i ( u ) − θ i ( v ) | k Φ( v ) k ≤ ǫ k Φ( u ) − Φ( v ) k (5.8)for any two vectors Φ( u ) , Φ( v ) ∈ R k \ { } . Case 1:
The edge { u, v } satisfies d Φ ( u, v ) = (cid:13)(cid:13)(cid:13)(cid:13) Φ( u ) k Φ( u ) k − Φ( v ) k Φ( v ) k (cid:13)(cid:13)(cid:13)(cid:13) . In this case h Φ( u ) , Φ( v ) i ≥
0, where h· , ·i stands for the innerproduct of R k . We estimate | θ i ( u ) − θ i ( v ) |k Φ( v ) k ≤ ǫ d Φ ( u, v ) k Φ( v ) k = 1 ǫ (cid:13)(cid:13)(cid:13)(cid:13) k Φ( v ) kk Φ( u ) k Φ( u ) − Φ( v ) (cid:13)(cid:13)(cid:13)(cid:13) ≤ ǫ k Φ( u ) − Φ( v ) k + 1 ǫ (cid:13)(cid:13)(cid:13)(cid:13) k Φ( v ) kk Φ( u ) k Φ( u ) − Φ( u ) (cid:13)(cid:13)(cid:13)(cid:13) ≤ ǫ k Φ( u ) − Φ( v ) k + 1 ǫ |k Φ( v ) k − k Φ( u ) k|≤ ǫ k Φ( u ) − Φ( v ) k . Case 2:
The edge { u, v } ∈ E satisfies d Φ ( u, v ) = (cid:13)(cid:13)(cid:13)(cid:13) Φ( u ) k Φ( u ) k + Φ( v ) k Φ( v ) k (cid:13)(cid:13)(cid:13)(cid:13) . In this case h Φ( u ) , Φ( v ) i ≤
0. Thus, | θ i ( u ) − θ i ( v ) | k Φ( v ) k ≤ k Φ( v ) k ≤ k Φ( u ) − Φ( v ) k . This completes the proof of (5.8). (cid:3)
In fact, we can find k disjoint subsets of e V Φ with good properties. Lemma 5.2.
There exist k non-empty, mutually disjoint subsets S , S , . . . , S k ⊆ e V Φ and an absolute constant C > , such that • for any ≤ i = j ≤ k , d Φ ( S i , S j ) ≥ C k ; (5.9) • for any ≤ i ≤ k , X u ∈ S i µ ( u ) k Φ( u ) k ≥ k X u ∈ V µ ( u ) k Φ( u ) k . (5.10)The proof of Lemma 5.2 employs the padded random partition theoryon ( e V Φ , d Φ ). Since the signature plays no role in this lemma, we referto [38, Section 6] for the proof. Remark 5.3.
In fact, from the arguments in [38, Proof of Theorem6.1], we can set C = 768 (cid:0) log π − (cid:1) .Combining Lemma 5.1 and Lemma 5.2 leads to the following result. HEEGER CONSTANTS FOR SIGNED GRAPHS 25
Lemma 5.3.
For any k ∈ { , , . . . , n } , there exist k disjointly sup-ported functions ψ , ψ , . . . , ψ k : V → R such that for each ≤ i ≤ k , R σ ( ψ i ) ≤ Ck R σ (Φ) , (5.11) where C is an absolute constant.Proof. Let { θ i } ki =1 be the k cut-off functions corresponding to { S i } ki =1 obtained in Lemma 5.2, and set ǫ = 1 / (2 C k ). For each i , defineΨ i = θ i F . By Lemma 5.1, X u ∼ v w uv k Ψ i ( u ) − σ ( uv )Ψ i ( v ) k ≤ (cid:18) ǫ (cid:19) X u ∼ v w uv k Φ( u ) − σ ( uv )Φ( v ) k . By Lemma 5.2, X u ∈ V µ ( u ) k Ψ i ( u ) k ≥ k X u ∈ V µ ( u ) k Φ( u ) k . Therefore, R σ (Ψ i ) ≤ k (1 + 2 C k ) R σ (Φ) ≤ Ck R σ (Φ) . (5.12)Let us write Ψ i = (Ψ i , Ψ i , . . . , Ψ ki ) : V → R k . Notice that we canalways find j ∈ { , , . . . , k } such that R σ (Ψ j i ) ≤ R σ (Ψ i ) . Setting ψ i := Ψ j i , 1 ≤ i ≤ k yields (5.11). (cid:3) Proof of the upper bound estimate in Theorem 5.1.
Let us assign µ = µ d . Then combining Lemma 4.2, (5.4) and Lemma 5.3 together leadsto the estimate h σk ( µ d ) ≤ √ Ck p λ k (∆ σ ) , where C is the constant in (5.11). This proves the upper bound esti-mate in (5.1). (cid:3) Remark 5.4.
By (5.12), we observe that we can set the constant C inin (5.11) to be C = 2(2 C + 1) . Then the proof above actually leadsto h σk ( µ d ) ≤ C + 1) k p λ k (∆ σ ) . Inserting the constant from Remark 5.3, we have h σk ( µ d ) ≤ k p λ k (∆ σ ) . (5.13)In this way, we can also get explicit constants in Theorem 5.2, Theorem6.3, Corollary 6.2, in terms of C from Remark 5.3. As those constantare apparently not optimal, we will not pursue that in this paper. If we instead assign µ = µ in the above proof, we obtain the follow-ing estimate for L σ : Theorem 5.2.
There exists an absolute constant C such that for anysigned graph Γ = (
G, σ ) and any k ∈ { , , . . . , N } , λ k ( L σ )2 ≤ h σk ( µ ) ≤ Ck p d max λ k ( L σ ) . (5.14)6. Signed Cheeger constants and higher order spectralgaps
A natural question is, whether the order of λ (∆ σ ) on the right handside of the signed Cheeger inequality (4.1) can be improved to be 1.Extending the ideas of Kwok et al. [34], we answer this question by thefollowing theorem. Theorem 6.1.
For any signed graph
Γ = (
G, σ ) and any k ∈ { , , . . . , n } , h σ ( µ d ) < √ k λ (∆ σ ) p λ k (∆ σ ) . (6.1)In other words, when there exists a k such that the gap between λ and λ k is large, one can improve the order of λ on the right hand sideof (4.1) to be 1. Actually, a slightly stronger version of (6.1) can beproved: Theorem 6.2.
Given a signed graph
Γ = (
G, σ ) and k ∈ { , , . . . , N } ,at least one of the following holds,(i). h σ ( µ d ) ≤ kλ (∆ σ ); (ii). h σ ( µ d ) < √ k λ (∆ σ ) p λ k (∆ σ ) . (6.2)Notice that Theorem 6.1 is a direct corollary of this theorem and thefact that 0 ≤ λ k (∆ σ ) ≤ Lemma 6.1.
For any non-zero function f : V → R , there exists a t ′ ∈ [0 , max u ∈ V | f ( u ) | ] such that β σ ( V f ( t ′ ) , V f ( − t ′ )) ≤ P u ∼ v w uv | f ( u ) − σ ( uv ) f ( v ) | P u µ ( u ) | f ( u ) | . (6.3) Proof.
We can assume max u ∈ V | f ( u ) | = 1 without loss of generalitysince the right hand side of (6.3) in invariant under scaling of f . For HEEGER CONSTANTS FOR SIGNED GRAPHS 27 t ∈ [0 , X f ( t ) ∈ {− , , } V by( X f ( t )) u = , if f ( u ) ≥ t ; − , if f ( u ) ≤ − t ;0 , otherwise. (6.4)We claim that, for any { u, v } ∈ E , Z | ( X f ( t )) u − σ ( uv )( X f ( t )) v | dt = | f ( u ) − σ ( uv ) f ( v ) | . (6.5)Similarly to the proof of Lemma 4.1, we only need to show that Z | ( X f ( t )) u − ( X f ( t )) v | dt = | f ( u ) − f ( v ) | (6.6)for any function f : V → [ − , | f ( u ) | ≥ | f ( v ) | . If f ( u ) and f ( v ) have different signs, then | ( X f ( t )) u − ( X f ( t )) v | = , if t ≤ | f ( v ) | ;1 , if | f ( v ) | < t ≤ | f ( u ) | ;0 , if t > | f ( u ) | , (6.7)and so, Z | ( X f ( t )) u − ( X f ( t )) v | dt = | f ( u ) | + | f ( v ) | = | f ( u ) − f ( v ) | . If, on the other hand, f ( u ) and f ( v ) have the same sign, | ( X f ( t )) u − ( X f ( t )) v | = , if t ≤ | f ( v ) | ;1 , if | f ( v ) | < t ≤ | f ( u ) | ;0 , if t > | f ( u ) | , (6.8)and so, Z | ( X f ( t )) u − ( X f ( t )) v | dt = | f ( u ) | − | f ( v ) | = | f ( u ) − f ( v ) | . This completes the proof of (6.6) and establishes the claim (6.5).By the definition of the vector X f ( t ), we observe that X u ∼ v w uv | ( X f ( t )) u − σ ( uv )( X f ( t )) v | = 2 | E + ( V f ( t ) , V f ( − t )) | + | E − ( V f ( t )) | + | E − ( V f ( − t )) | + | E ( V f ( t ) ∪ V f ( − t ) , V f ( t ) ∪ V f ( − t )) | , and X u µ ( u ) | ( X f ( t )) u | = vol µ ( V f ( t ) ∪ V f ( − t )) . Therefore, there exists a t ′ ∈ [0 ,
1] such that β σ ( V f ( t ′ ) , V f ( − t ′ )) ≤ R P u ∼ v w uv | ( X f ( t )) u − σ ( uv )( X f ( t )) v | dt R P u µ ( u ) | ( X f ( t )) u | dt . Applying the claim (6.5) and the fact R | ( X f ( t )) u | dt = | f ( u ) | , thelemma is proved. (cid:3) For any non-zero function f : V → R , and for any k ∈ N , let0 = t ≤ t ≤ · · · ≤ t k (6.9)be a sequence of real numbers with t k = max u ∈ V | f ( u ) | . We define astep function approximation g to f as g ( u ) = ψ − t k ,..., − t , ,t ,...,t k ( f ( u )) := arg min t ∈{− t k ,..., ,...,t k } | f ( u ) − t | . (6.10)In other words, g : V → {− t k , . . . , , . . . , t k } is a function such that g ( u ) equals the one of {− t k , . . . , , . . . , t k } which is closest to f ( u ).We further construct an auxiliary function F : V → R . First, wedefine a function η : [ − t k , t k ] → R via η ( x ) := | x − ψ − t k ,..., − t , ,t ,...,t k ( x ) | . (6.11)Note that η ( − x ) = η ( x ). Then for each u ∈ V , we assign F ( u ) := Z f ( u )0 η ( x ) dx. (6.12) Lemma 6.2.
The function F defined in (6.12) has the following prop-erties: (i) For any u ∈ V , | F ( u ) | ≥ k | f ( u ) | . (6.13)(ii) For any { u, v } ∈ E , | F ( u ) − σ ( uv ) F ( v ) |≤ | f ( u ) − σ ( uv ) f ( v ) |× (cid:16) | f ( u ) − σ ( uv ) f ( v ) | + | f ( u ) − g ( u ) | + | f ( v ) − g ( v ) | (cid:17) . (6.14) Proof. (i). First observe that F ( u ) and f ( u ) share the same sign. With-out loss of generality, assume f ( u ) >
0. Then the proof can be doneas in [34, Claim 3.3]. For the readers’ convenience, we recall it here.
HEEGER CONSTANTS FOR SIGNED GRAPHS 29
Suppose f ( u ) ∈ [ t i , t i +1 ] for some i . Then by the Cauchy-Schwarzinequality, f ( u ) = i =1 X j =0 ( t j +1 − t j ) + ( f ( u ) − t i ) ! ≤ k i =1 X j =0 ( t j +1 − t j ) + ( f ( u ) − t i ) ! . By the definition of F , F ( u ) = i − X j =0 Z t j +1 t j η ( x ) dx + Z f ( u ) t i η ( x ) dx ≥ i − X j =0
14 ( t j +1 − t j ) + 14 ( f ( u ) − t i ) = 18 k f ( u ) . (ii). Observe that Z σ ( uv ) f ( v )0 η ( x ) dx = σ ( uv ) Z f ( v )0 η ( x ) dx. Hence, we only need to prove (6.14) for the case σ ( uv ) = +1. With-out loss of generality, suppose | f ( u ) | ≥ | f ( v ) | . If f ( u ) and f ( v ) havedifferent signs, say, f ( u ) ≥ f ( v ) ≤
0, then, recalling the fact η ( − x ) = η ( x ), we have | F ( u ) − F ( v ) | = Z f ( u )0 η ( x ) dx + Z − f ( v )0 η ( x ) dx ≤ Z f ( u )0 x dx + Z − f ( v )0 x dx = 12 ( f ( u ) + f ( v ) ) ≤ | f ( u ) − f ( v ) | . (6.15)If, on the other hand, f ( u ) and f ( v ) have the same sign, we can assume f ( u ) > f ( v ) > η ( − x ) = η ( x ) and the step functionapproximation of − f is − g . Then, | F ( u ) − F ( v ) | = Z f ( u ) f ( v ) η ( x ) dx ≤ | f ( u ) − f ( v ) | × max f ( v ) ≤ x ≤ f ( u ) η ( x ) . (6.16) Using the definition of η , we estimate η ( x ) ≤ min {| x − g ( u ) | , | x − g ( v ) |} ≤
12 ( | x − g ( u ) | + | x − g ( v ) | ) ≤
12 ( | x − f ( u ) | + | f ( u ) − g ( u ) | + | x − f ( v ) | + | f ( v ) − g ( v ) | )= 12 ( | f ( u ) − f ( v ) | + | f ( u ) − g ( u ) | + | f ( v ) − g ( v ) | ) . (6.17)Combining (6.15), (6.16), and (6.17) leads to (6.14). (cid:3) With Lemma 6.2 and Lemma 6.1 at hand, we derive the followinglemma.
Lemma 6.3.
For any non-zero function f : V → R and any stepfunction approximation g of f constructed from t ≤ t , . . . , t k asabove, there exists a t ′ ∈ [0 , max u ∈ V | f ( u ) | ] such that β σ ( V f ( t ′ ) , V − f ( t ′ )) ≤ k R σ ( f ) + 4 √ k k f − g k µ k f k µ q d wµ R σ ( f ) , (6.18) where k f k µ := P u ∈ V µ ( u ) f ( u ) . We point out that the notation k · k µ = k · k reduces to the Euclideannorm when µ = µ . Proof.
Applying Lemma 6.1 to the function F , we find t ∈ [0 , max u | F ( u ) | ]such that β σ ( V F ( t ) , V F ( − t )) ≤ P u ∼ v w uv | F ( u ) − σ ( uv ) F ( v ) | P u µ ( u ) | F ( u ) |≤ k R σ ( f )+ 4 k P u ∼ v w uv | f ( u ) − σ ( uv ) f ( v ) | ( | f ( u ) − g ( u ) | + | f ( v ) − g ( v ) | ) P u µ ( u ) f ( u ) ≤ k R σ ( f ) + 4 k p R σ ( f ) pP u ∼ v w uv | ( | f ( u ) − g ( u ) | + | f ( v ) − g ( v ) | ) pP u µ ( u ) f ( u ) . In the above, we used Lemma 6.2 for the second inequality and theCauchy-Schwarz inequality for the last one. Notice that X u ∼ v w uv | ( | f ( u ) − g ( u ) | + | f ( v ) − g ( v ) | ) ≤ X u X v,v ∼ u w uv (cid:0) | f ( u ) − g ( u ) | + 2 | f ( v ) − g ( v ) | (cid:1) ≤ d wµ k f − g k µ . (6.19) HEEGER CONSTANTS FOR SIGNED GRAPHS 31
Inserting (6.19) into the above calculations we obtain β σ ( V F ( t ) , V F ( − t )) ≤ k R σ ( f ) + 4 √ k k f − g k µ k f k µ q d wµ R σ ( f ) . Noting that f ( u ) ≥ f ( v ) if and only if F ( u ) ≥ F ( v ) completes theproof. (cid:3) We are now ready to prove Theorem 6.2.
Lemma 6.4.
For any non-zero function f : V → R and any ≤ k ≤ n , there exists t ′ ∈ [0 , max u ∈ V | f ( u ) | ] , such that at least one of thefollowing estimates holds: (i) β σ ( V f ( t ′ ) , V f ( − t ′ )) ≤ k R σ ( f ) ; (ii) there exists k disjointly supported functions f , f , . . . , f k : V → R such that for each ≤ i ≤ k , R σ ( f i ) < d wµ k R σ ( f ) β σ ( V f ( t ′ ) , V f ( − t ′ )) . (6.20) Proof.
Let M := max u | f ( u ) | . We construct 2 k + 1 real numbers t ≤ t ≤ · · · ≤ t k ≤ M as follows: Set t = 0. Suppose that we havealready fixed t , t , . . . , t i − . Now we look for t i ∈ [ t i − , M ] such that X u : − t i ≤ f ( u ) < − t i − µ ( u ) | f ( u ) − ψ − t i , − t i − ( f ( u )) | + X u : t i − If, on the other hand, the procedure fails, that is, if we have t k < M ,then we define 2 k disjointly supported functions as f i ( u ) := −| f ( u ) − ψ − t i − ,t i ( f ( u )) | , if − t i ≤ f ( u ) ≤ − t i − ; | f ( u ) − ψ t i − ,t i ( f ( v )) | if t i − < f ( u ) ≤ t i ;0 otherwise, (6.23)for 1 ≤ i ≤ k . Recall that (6.21) ensures k f i k = C . Next we estimatethe Rayleigh quotients of these functions. Claim. For any { u, v } ∈ E , k X i =1 | f i ( u ) − σ ( uv ) f i ( v ) | ≤ | f ( u ) − σ ( uv ) f ( v ) | . (6.24)Similarly to the proof of Lemma 4.1, we only need to prove the claimwhen σ ( uv ) = +1. Case 1: u and v lie in the support of the same function f i . Inthis case, k X i =1 | f i ( u ) − f i ( v ) | = | f i ( u ) − f i ( v ) | . If f ( u ) and f ( v ) have the same sign, say f ( u ) ≥ f ( v ) ≥ | f i ( u ) − f i ( v ) | = (cid:12)(cid:12) | f ( u ) − ψ t i − ,t i ( f ( u )) | − | f ( v ) − ψ t i − ,t i ( f ( v )) | (cid:12)(cid:12) ≤ | f ( u ) − f ( v ) | . If f ( u ) and f ( v ) have different signs, say f ( u ) > f ( v ) < | f i ( u ) − f i ( v ) | = (cid:12)(cid:12) | f ( u ) − ψ t i − ,t i ( f ( u )) | + | f ( v ) − ψ − t i , − t i − ( f ( v )) | (cid:12)(cid:12) ≤ (cid:12)(cid:12) | f ( u ) − ψ t i − ,t i ( f ( u )) | + | − f ( v ) − ψ t i − ,t i ( − f ( v )) | (cid:12)(cid:12) ≤ ( | f ( u ) − t i − | + | − f ( v ) − t i − | ) ≤ | f ( u ) − f ( v ) | . Case 2: u ∈ supp( f i ) , v ∈ supp( f j ), where i = j . We can assume j > i . Then, k X i =1 | f i ( u ) − f i ( v ) | = | f i ( u ) | + | f j ( v ) | . If f ( u ) and f ( v ) have the same sign, say f ( u ) ≥ f ( v ) ≥ | f i ( u ) | + | f j ( v ) | = | f ( u ) − ψ t i − ,t i ( f ( u )) | + | f ( v ) − ψ t j − ,t j ( f ( v )) | ≤ | f ( u ) − t i | + | f ( v ) − t i | ≤ | f ( u ) − f ( v ) | . HEEGER CONSTANTS FOR SIGNED GRAPHS 33 If f ( u ) and f ( v ) have different signs, say f ( u ) ≥ f ( v ) ≤ | f i ( u ) | + | f j ( v ) | = | f ( u ) − ψ t i − ,t i ( f ( u )) | + | f ( v ) − ψ − t j , − t j − ( f ( v )) | = | f ( u ) − ψ t i − ,t i ( f ( u )) | + | − f ( v ) − ψ t j − ,t j ( − f ( v )) | ≤ | f ( u ) − t i − | + | − f ( v ) − t j − | ≤ | f ( u ) − f ( v ) | . This completes the proof of Claim.Using the Claim, we calculate k X i =1 R σ ( f i ) k X i =1 R σ ( f i ) = 1 C k X i =1 X u ∼ v w uv ( f i ( u ) − σ ( uv ) f i ( v )) ≤ C X u ∼ v w uv | f ( u ) − σ ( uv ) f ( v ) | = 256 k d wµ R σ ( f ) β σ ( V f ( t ′ ) , V f ( − t ′ )) . Let us abbreviate the above estimate as P ki =1 R σ ( f i ) ≤ C k , where C := 256 d wµ R σ ( f ) β σ ( V f ( t ′ ) , V f ( − t )) . Then we can find k functions from the set { f , f , . . . , f k } , relabelingthem as f , f , . . . , f k if necessary, such that R σ ( f i ) < C k for 1 ≤ i ≤ k .This is true since otherwise there exist at least k +1 of {R σ ( f i ) : 1 ≤ i ≤ n } with the property R σ ( f ) ≥ C k which leads to the contradiction that P ki =1 R σ ( f ) ≥ C k ( k + 1) ≥ C k . That is, the estimate (ii) holds. (cid:3) Proof of Theorem 6.2. Let φ be the eigenfunction of ∆ σ correspondingto λ (∆ σ ). We have λ (∆ σ ) = R σ ( φ ). Moreover, by the definition(3.2) of h σ ( µ d ), h σ ( µ d ) ≤ β σ ( V φ ( t ′ ) , V φ ( − t ′ )) , for any t ′ ∈ (cid:20) , max u ∈ V φ ( u ) (cid:21) . Combining the above two observations with Lemma 6.4, we obtain thefollowing property of φ : For any 1 ≤ k ≤ n , at least one of thefollowing two estimates holds:(i) h σ ( µ d ) ≤ kλ (∆ σ ); (ii) λ k (∆ σ )2 < k λ (∆ σ ) h σ ( µ d ) . Note that in (ii), we have used Lemma 2.2 and the fact that d wµ d = 1for the degree measure µ d . Now rearranging the estimate in (ii) andtaking the square root of both sides, Theorem 6.2 is proved. (cid:3) We further have the following higher-order estimates. Theorem 6.3. There exists an absolute constant C such that for anysigned graph Γ = ( G, σ ) and any ≤ k ≤ l ≤ n , h σk ( µ d ) < Clk λ k (∆ σ ) p λ l (∆ σ ) . (6.25)This generalizes the corresponding results for unsigned graphs givenin [34] and [38]. Proof of Theorem 6.3. Combining Lemma 5.3 and the fact (5.4), weobtain the following property: For any k ∈ { , , . . . , n } , there exist k disjointly supported functions ψ , ψ , . . . , ψ k : V → R such that R σ ( ψ i ) ≤ Ck λ k (∆ σ ) , for each 1 ≤ i ≤ k, (6.26)where C is an absolute constant.Lemma 6.4 tells the following fact: For any 1 ≤ i ≤ k and any k ≤ l ≤ n , there exists a t i,l ∈ [0 , max u ∈ V | ψ i ( u ) | ], such that at leastone of the following two estimates holds:(i) β σ ( V ψ i ( t i,l ) , V ψ i ( − t i,l )) ≤ l R σ ( ψ i );(ii) λ l (∆ σ )2 < l R σ ( ψ i ) β σ ( V ψi ( t i,l ) ,V ψi ( − t i,l )) .Note that in (ii), we have used Lemma 2.2 and the fact that d wµ d = 1.Let i ∈ { , . . . , k } be the index satisfying β σ ( V ψ i ( t i ,l ) , V ψ i ( − t i ,l )) = max ≤ i ≤ k β σ ( V ψ i ( t i,l ) , V ψ i ( − t i,l )) . By the definition (3.19) of h σk ( µ d ), we obtain h σk ( µ d ) ≤ β σ ( V ψ i ( t i ,l ) , V ψ i ( − t i ,l )) . (6.27)Inserting (6.26) and (6.27) into the estimates (i) and (ii), we obtain thefollowing fact: For any 1 ≤ k ≤ l ≤ n , at least one of the two estimatesholds:(i’) h σk ( µ d ) ≤ Clk λ k (∆ σ ); (ii’) λ l (∆ σ )2 < l ( Ck λ k (∆ σ )) h σk ( µ d ) . Rearranging the estimate in (ii’) and taking the square root of bothsides, we obtain h σk ( µ d ) < √ Clk λ k (∆ σ ) p λ l (∆ σ ) . Using the fact λ l (∆ σ ) ≤ 2, (i’) implies h σk ( µ d ) ≤ √ Clk λ k (∆ σ ) p λ l (∆ σ ) . HEEGER CONSTANTS FOR SIGNED GRAPHS 35 This completes the proof. (cid:3) Remark 6.1. Theorems 6.1 and 6.3 suggest that the well-known eigen-gap heuristic [39, 34] for the traditional spectral clustering algorithmstill holds for signed networks. That is, in case that λ k (∆ σ ) is small and λ k +1 (∆ σ ) is large, it is better to cluster the data into k almost-balancedsubgraphs.By setting µ = µ , we obtain the following results for L σ . Theorem 6.4. Given any signed graph Γ = ( G, σ ) and any ≤ k ≤ n ,at least one of the following holds:(i). h σ ( µ ) ≤ kλ ( L σ ); (ii). h σ ( µ ) < p d max k λ ( L σ ) p λ k ( L σ ) . (6.28)Recalling that λ k ( L σ ) ≤ d max , we further obtain the following corol-laries. Corollary 6.1. For any signed graph Γ and any ≤ k ≤ n , h σ ( µ ) < p d max k λ ( L σ ) p λ k ( L σ ) . (6.29) Corollary 6.2. There exists an absolute constant C such that for anysigned graph Γ and ≤ k ≤ l ≤ n , h σk ( µ ) < C p d max lk λ k ( L σ ) p λ l ( L σ ) . (6.30)We comment at this point that the previous estimates about h σk ( µ d )and λ k (∆ σ ) can be directly translated into estimates for e h σk ( µ d ) and2 − λ n − k +1 (∆ σ ) by duality. This is due to Lemma 2.1, which says that2 − λ n − k +1 (∆ σ ) = λ k (∆ − σ ) . For example, the dual version of Theorem 4.1 can be stated as follows. Theorem 6.5. Given a signed graph Γ = ( G, σ ) , we have − λ n (∆ σ )2 ≤ e h σ ( µ d ) ≤ p − λ n (∆ σ )) . (6.31)We omit the dual versions of Theorems 5.1, 6.1 and 6.3 here. Ac-tually, these results are good demonstrations of a general antitheticalduality principle discussed by Harary [25]. Signed triangles and the spectral gaps λ and − λ n In this section, we prove an estimate for λ (∆ σ ) and 2 − λ n (∆ σ ) interms of the number of signed triangles. We will also discuss a similarresult for the non-normalized Laplace matrix L σ .We first introduce some notation. For a given edge { u, v } , we dividethe neighborhood N u := { u ′ | u ′ ∼ u } of a vertex u into disjoint parts as N u = N u ∪ N + uv ∪ N − uv , where N u := { u ′ | u ′ ∼ u, u ′ v } ,N + uv := { u ′ | u ′ ∼ u, u ′ ∼ v, σ ( uv ) σ ( vu ′ ) σ ( u ′ u ) = +1 } , and N − uv := { u ′ | u ′ ∼ u, u ′ ∼ v, σ ( uv ) σ ( vu ′ ) σ ( u ′ u ) = − } . Similarly, we have the partition N v = N v ∪ N + uv ∪ N − uv . Let us denote N uv = N + uv ∪ N − uv , and denote the number of positive and negative triangles including anedge { u, v } by ♯ + ( u, v ) and ♯ − ( u, v ), respectively, where ♯ + ( u, v ) := X u ′ ∈ N + uv ♯ − ( u, v ) := X u ′ ∈ N − uv . Note that the quantities ♯ + ( u, v ) , ♯ − ( u, v ) are switching invariant andtheir unsigned counterpart has an interesting close relation with theOllivier-Ricci curvature of the underlying graph G [7, 30]. We provethe following theorem. Theorem 7.1. For a signed graph Γ = ( G, σ ) , w W min u ∼ v ♯ − ( u, v )max u d u ≤ λ (∆ σ ) ≤ · · · ≤ λ N (∆ σ ) ≤ − w W min u ∼ v ♯ + ( u, v )max u d u , (7.1) where w = min u ∼ v w uv and W = max u ∼ v w uv . This result is obtained by considering the iterated matrix ∆ σ [2] (see(7.2) below), extending an idea of Bauer, Jost and the second namedauthor [7] for the unsigned case. Proof of Theorem 7.1. We consider an iterated matrix∆ σ [2] = I − ( D − A σ ) . (7.2)Then, for any function f : V → R and any u ∈ V ,∆ σ [2] f ( u ) = f ( u ) − d u X v X u ′ ∈ N uv w u ′ u w u ′ v d u ′ σ ( u ′ u ) σ ( u ′ v ) f ( v ) . HEEGER CONSTANTS FOR SIGNED GRAPHS 37 Let f n be the corresponding eigenfunction of λ n (∆ σ ). Then,( f n , ∆ σ [2] f n ) µ ( f n , ∆ σ f n ) µ = ( f n , [1 − (1 − λ n (∆ σ )) ] f n ) µ ( f n , λ n (∆ σ ) f n ) µ = 2 − λ n (∆ σ ) . (7.3)Note λ n (∆ σ )( f n , f n ) µ = 0, hence the above expression is proper. Fur-thermore, ( f n , ∆ σ f n ) µ = X u ∼ v ( f n ( u ) − σ ( uv ) f n ( v )) , (7.4)and ( f n , ∆ σ [2] f n ) µ = X u f n ( u ) X v X u ′ ∈ N uv w u ′ u w u ′ v d u ′ ( f n ( u ) − σ ( u ′ u ) σ ( u ′ v ) f n ( v ))= X ( u,v ) X u ′ ∈ N uv w u ′ u w u ′ v d u ′ ( f n ( u ) − σ ( u ′ u ) σ ( u ′ v ) f n ( v )) ≥ X u ∼ v X u ′ ∈ N + uv w u ′ u w u ′ v d u ′ ( f n ( u ) − σ ( u ′ u ) σ ( u ′ v ) f n ( v )) . In the above, P ( u,v ) stands for the summation over unordered pair ofvertices u, v . Inserting the above estimate and the equality (7.4) into(7.3), we obtain2 − λ n (∆ σ ) ≥ P u ∼ v P u ′ ∈ N + uv w u ′ u w u ′ v d u ′ ( f n ( u ) − σ ( u ′ u ) σ ( u ′ v ) f n ( v )) P u ∼ v w uv ( f n ( u ) − σ ( uv ) f n ( v )) ≥ w W min u ∼ v X u ′ ∈ N + uv d u ′ ≥ w W min u ∼ v ♯ + ( u, v )max u d u . Using Lemma 2.1, the lower bound estimate for λ (∆ σ ) follows fromduality. (cid:3) For the signed non-normalized Laplace matrix L σ , we have the fol-lowing estimate. Theorem 7.2. For a signed unweighted graph Γ = ( G, σ ) , λ N ( L σ ) ≤ max u ∼ v { d u + d v − ♯ + ( u, v ) } . (7.5)This result improves the estimate λ N ( L σ ) ≤ max u ∼ v { d u + d v } dueto Hou, Li, and Pan [29]. In fact, Theorem 7.2 answers the questionasked in their paper [29, remark after Theorem 3.5].The techniques we used in the proof of Theorem 7.1 do not workfor L σ . For example, we do not have a clear relation between theeigenvalues of D − A σ and D − ( A σ ) as in (7.3) anymore. Actually, the underlying idea of the proof of Theorem 7.1 is that the normalizedoperator encodes certain random process on the graph, which is nottrue for the non-normalized operator. We will employ different ideas,which are adapted from Das [19] and Rojo [47]. In fact, we shall provethe following result. Theorem 7.3. For a signed graph Γ = ( G, σ ) , λ n ( L σ ) ≤ 12 max u ∼ v { d u + d v + X u ′ ∈ N u w u ′ u + X u ′ ∈ N v w u ′ v + X u ′ ∈ N − uv ( w u ′ u + w u ′ v ) + X u ′ ∈ N + uv | w u ′ u − w u ′ v |} . Observe that Theorem 7.2 is a direct corollary of this theorem, sincewhen Γ is a signed unweighted graph, we have d u = | N u | + | N + uv | + | N − uv | . Proof of Theorem 7.3. Let f n be the eigenfunction corresponding to λ n ( L σ ). Without loss of generality, suppose f n ( u ) = max u ′ ∈ V | f n ( u ′ ) | ,and σ ( uv ) f n ( v ) = min u ′ ∈ N u σ ( uu ′ ) f n ( u ′ ) . First, observe that f n ( u ) − σ ( uv ) f n ( v ) = 0. (Because otherwise wewould have σ ( uu ′ ) f n ( u ′ ) = f n ( u ) for any u ′ ∈ N u , which would imply λ n ( L σ ) f n ( u ) = L σ f n ( u ) = 0, a contradiction.) Then we calculate λ n ( L σ )( f n ( u ) − σ ( uv ) f n ( v ))= L σ f n ( u ) − σ ( uv ) L σ f n ( v )= d u f n ( u ) − X u ′ ∈ N u w u ′ u σ ( uu ′ ) f n ( u ′ ) − d v σ ( uv ) f n ( v ) + X u ′ ∈ N v w u ′ v σ ( uv ) σ ( u ′ v ) f n ( u ′ ) . HEEGER CONSTANTS FOR SIGNED GRAPHS 39 Using the facts that σ ( uu ′ ) f n ( u ′ ) ≥ σ ( uv ) f n ( v ) for any u ′ ∈ N u and σ ( uv ) σ ( u ′ v ) ≤ u ′ ∈ N v , we continue to estimate: λ n ( L σ )( f n ( u ) − σ ( uv ) f n ( v )) ≤ d u f n ( u ) − σ ( uv ) f n ( v ) X u ′ ∈ N u ∪ N − uv w u ′ u − d v σ ( uv ) f n ( v )+ f n ( u ) X u ′ ∈ N v ∪ N − uv w u ′ v + X u ′ ∈ N + uv ( w u ′ v − w u ′ u ) σ ( u ′ u ) f n ( u ′ )= 12 ( f n ( u ) − σ ( uv ) f n ( v )) d u + d v + X u ′ ∈ N u ∪ N − uv w u ′ u + X u ′ ∈ N v ∪ N − uv w u ′ v + 12 ( f n ( u ) + σ ( uv ) f n ( v )) d u − X u ′ ∈ N u ∪ N − uv w u ′ u − 12 ( f n ( u ) + σ ( uv ) f n ( v )) d v − X u ′ ∈ N v ∪ N − u,v w u ′ v + X u ′ ∈ N + uv ( w u ′ v − w u ′ u ) σ ( u ′ u ) f n ( u ′ ) . Using the fact that d u − P u ′ ∈ N u ∪ N − uv w u ′ u = P u ′ ∈ N + uv w u ′ u , we have λ n ( L σ )( f n ( u ) − σ ( uv ) f n ( v )) ≤ 12 ( f n ( u ) − σ ( uv ) f n ( v )) d u + d v + X u ′ ∈ N u ∪ N − uv w u ′ u + X u ′ ∈ N v ∪ N − uv w u ′ v + 12 X u ′ ∈ N + uv ( w u ′ v − w u ′ u ) ( f n ( u ) + σ ( uv ) f n ( v ) − σ ( u ′ u ) f n ( u ′ )) . (7.6) For the latter term above, we further estimate X u ′ ∈ N + uv ( w u ′ v − w u ′ u ) ( f n ( u ) + σ ( uv ) f n ( v ) − σ ( u ′ u ) f n ( u ′ )) ≤ X u ′ ∈ N + uv | w u ′ v − w u ′ u | ( f n ( u ) − σ ( u ′ u ) f n ( u ′ ))+ X u ′ ∈ N + uv | w u ′ v − w u ′ u | ( σ ( u ′ u ) f n ( u ′ ) − σ ( uv ) f n ( v ))=( f n ( u ) − σ ( uv ) f n ( v )) X u ′ ∈ N + uv | w u ′ v − w u ′ u | . Inserting the above estimate into (7.6), we arrive at λ n ( L σ )( f n ( u ) − σ ( uv ) f n ( v )) ≤ 12 ( f n ( u ) − σ ( uv ) f n ( v )) d u + d v + X u ′ ∈ N u ∪ N − uv w u ′ u + X u ′ ∈ N v ∪ N − uv w u ′ v + 12 ( f n ( u ) − σ ( uv ) f n ( v )) X u ′ ∈ N + uv | w u ′ v − w u ′ u | . This completes the proof. (cid:3) Acknowledgements The authors are very grateful to Norbert Peyerimhoff for suggestingthe crucial idea leading to Proposition 3.1. The authors thank ThomasZaslavsky for his interest and valuable comments. The main part of thiswork was conceived when both authors were visiting the Zentrum f¨urinterdisziplin¨are Forschung (ZiF) of Bielefeld University. The authorsthank the hospitality of ZiF and the financial support from the ZiFcooperation group “Discrete and Continuous Models in the Theory ofNetworks”. Finally, the authors acknowledge many useful commentsof the anonymous referees. SL was partially supported by the EPSRCGrant EP/K016687/1. 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