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NORMALIZED GRAPH LAPLACIANS FOR DIRECTED GRAPHS 2 1 FRANK BAUER 0 2 Abstract. We consider the normalized Laplace operator for di- n rectedgraphswithpositiveandnegativeedgeweights. Thisgener- a J alizationofthe normalizedLaplaceoperatorforundirectedgraphs is usedto characterizedirectedacyclicgraphs. Moreover,we iden- 1 3 tify certain structural properties of the underlying graph with extremal eigenvalues of the normalized Laplace operator. We prove ] comparison theorems that establish a relationship between the O eigenvaluesofdirectedgraphsandcertainundirectedgraphs. This C relationship is used to derive eigenvalue estimates for directed . h graphs. Finally we introduce the concept of neighborhood graphs t for directed graphs and use it to obtain further eigenvalue esti- a m mates. [ 2 To appear in: Linear Algebra and its Applications v 7 4 Contents 8 4 1. Introduction 2 . 7 2. Preliminaries 3 0 1 3. Basic properties of the spectrum 7 1 4. Spectrum of ∆ and isolated components of Γ 9 : v 5. Directed acyclic graphs 12 i X 6. Extremal eigenvalues 13 r 7. k-partite graphs and anti-k-partite graphs 16 a 7.1. k-partite graphs 16 7.2. Anti-k-partite graphs 22 7.3. Special cases: Bipartite and anti-bipartite graphs 24 8. Bounds for the real and imaginary parts of the eigenvalues 26 8.1. Comparison theorems 26 8.2. Further eigenvalue estimates 34 9. Neighborhood graphs 36 References 39 Key words and phrases. directed graphs, normalized graph Laplace operator, eigenvalues, directed acyclic graphs, neighborhood graph. 2010 Mathematics Subject Classification. 05C20,05C22, 05C50. 1 2 FRANKBAUER 1. Introduction For undirected graphs with nonnegative weights, the normalized graph Laplace operator ∆ is a well studied object, see e. g. the mono- graph [8]. In addition to its mathematical importance, the spectrum of the normalized Laplace operator has various applications in chemistry and physics. However, it is not always sufficient to study the normalized Laplace operator for undirected graphs with nonnegative weights. In many biological applications, one naturally has to consider directed graphs with positive and negative weights [3]. For instance, in a neuronal network only the presynaptic neuron influences the postsynaptic one, but not vice versa. Furthermore, the synapses can be of inhibitory or excitatory type. Inhibitory and excitatory synapses enhance or sup- press, respectively, the activity of the postsynaptic neuron and thus the directionality of the synapses and the existence of excitatory and inhibitory synapses crucially influence the dynamics in neuronal networks [3]. Hence, a realistic model of a neuronal network has to be a directed graph with positive and negative weights in which the neurons correspond to the vertices and the excitatory and inhibitory synaptic connections are modelled by directed edges with positive and negative weights, respectively. In contrast to undirected graphs not much is known about normalized Laplace operators for directed graphs. In [9] Chung studied a normalized Laplace operator for strongly connected directed graphs with nonnegative weights. This Laplace operator is defined as a self- adjoint operator using the transition probability operator and the Per- ron vector∗. For our purposes, however, this definition of the normalized Laplace operator is not suitable since by the above considerations we are particularly interested in graphs that are neither strongly connected nor have nonnegative weights. In this article, we define a novel normalized Laplace operator that can in particular be defined for directed graphs that are neither strongly connected nor have nonnegative weights. In contrast to Chung’s normalized Laplace operator our normalized Laplace operator is in general neither self-adjoint nor nonnegative. Moreover, our definition of the normalized Laplace operator is motivated by the observation that it has already found applications in the field of complex networks, see [2, 3]. The paper is organized as follows. In Section 2 we define the normalized Laplace operator for directed graphs and in Section 3 and Section 4 we derive its basic spectral properties. In Section 5 we characterize ∗A similar construction is used in [25] to study the algebraic connectivity of the Laplace operator L=D W defined on directed graphs. − NORMALIZED GRAPH LAPLACIANS FOR DIRECTED GRAPHS 3 directed acyclic graphs by means of their spectrum. Extremal eigenvalues of the Laplace operator are studied in Section 6 and Section 7. In Section 8 we prove several eigenvalues estimates for the normalized Laplace operator. Finally in Section 9 we introduce the concept of neighborhood graphs and use it to derive further eigenvalue estimates. 2. Preliminaries Unless stated otherwise, we consider finite simple loopless graphs. Let Γ = (V,E,w) be a weighted directed graph on n vertices where V denotes the vertex set, E denotes the edge set, and w : V V R × → is the associated weight function of the graph. For a directed edge e = (i,j) E, we say that there is an edge from i to j. The weight ∈ of e = (i,j) is given by w † and we use the convention that w = 0 ji ji if and only if e = (i,j) / E. The graph Γ = (V,E,w) is an undirected ∈ weighted graph if the associated weight function w is symmetric, i.e. satisfies w = w for all i and j. Furthermore, Γ is a graph with non- ij ji negative weights if the associated weight function w satisfies w 0 ij ≥ for all i and j. For ease of notation, let G denote the class of weighted directedgraphsΓ. Furthermore, letGu,G+ andGu+ denotetheclassof weighted undirected graphs, the class of weighted directed graphs with non-negative weights and the class of weighted undirected graphs with non-negative weights, respectively. The in-degree and the out-degree of vertex i are given by din := w and dout := w , respectively. i j ij i j ji A graph is said to be balanced if din = dout for all i V. Since every P i i P∈ undirected graph is balanced, the two notions coincide for undirected graphs. Thus, we simply refer to the degree d of an undirected graph. i A graph Γ is said to have a spanning tree if there exists a vertex from which all other vertices can be reached following directed edges. A directed graph Γ is weakly connected if replacing all of its directed edges with undirected edges produces a connected (undirected) graph. A directed graph Γ is strongly connected if for any pair of distinct vertices i and j there exists a path from i to j and a path from j to i. An undirected graph is weakly connected if and only if it is strongly connected. Hence, we do not distinguish between weakly and strongly connected undirected graphs. We simply say that the undirected graph is connected if it is weakly (strongly) connected. Definition 2.1. Let C(V) denote the space of complex valued func- tionson V. The normalized graphLaplace operator fordirected graphs †We use this convention instead of denoting the weight of the edge e = (i,j) by w , since it is more appropriate if one studies dynamical systems defined on ij graphs, see for example [2]. 4 FRANKBAUER Γ G is defined as ∈ ∆ : C(V) C(V), → v(i) 1 w v(j) if din = 0. (1) ∆v(i) = − diin j ij i 6 0 else. (cid:26) P If din = 0 for all i V, then ∆ is given by i 6 ∈ ∆ = I D−1W, − where D : C(V) C(V) is the multiplication operator defined by → (2) Dv(i) = dinv(i) i and W : C(V) C(V) is the weighted adjacency operator → Wv(i) = w v(j). ij j∈V X Whenrestrictedtoundirectedgraphswithnonnegativeweights, Defini- tion 2.1 reduces to the well-known definition of the normalized Laplace operator for undirected graphs with nonnegative weights, c.f.[19]. The choice of normalizing by the in-degree is to some extend arbi- trary. One could also consider the operator ∆ : C(V) C(V), → v(i) 1 w v(j) if dout = 0. (3) ∆v(i) = − doiut j ji i 6 0 else. (cid:26) P Note however, that both operators ∆ and ∆ are equivalent to each other in the sense that ∆(Γ) = ∆(Γ), where Γ is the graph that is obtained from Γ by reversing all edges. Since we consider a normalized graph Laplace operator, i. e. we nor- malize the edge weights w.r.t. the in-degree, vertices with zero in- degree are of particular interest and need a special treatment. We define the following: Definition 2.2. We say that vertex i is in-isolated or simply isolated if w = 0 for all j V. Similarly, vertex i is said to be in-quasi-isolated ij ∈ or simply quasi-isolated if din = w = 0. i j ij Note that every isolated vertePx is quasi-isolated but not vice versa. These definitions can be extended to induced subgraphs: Definition 2.3. LetΓ = (V,E,w) GbeagraphandΓ′ = (V′,E′,w′) ∈ be an induced subgraph of Γ, i.e. V′ V, E′ = E (V′ V′) E, and w′ : V′ V′ R, w′ := w . We⊆say that Γ′ is i∩solate×d if w⊆= 0 E′ ij × → | NORMALIZED GRAPH LAPLACIANS FOR DIRECTED GRAPHS 5 for all i V′ and j / V′. Similarly, Γ′ is said to be quasi-isolated if ∈ ∈ w = 0 for all i V′. j∈V\V′ ij ∈ PWe do not exclude the case where V′ = V. Thus, in particular, every graph Γ is isolated. It is useful to introduce the reduced Laplace operator ∆ . R Definition 2.4. Let V V be the subset of all vertices that are not R ⊆ quasi-isolated. The reduced Laplace operator ∆ : C(V ) C(V ) is R R R → defined as 1 (4) ∆ v(i) = v(i) w v(j) i V , R − din ij ∈ R i jX∈VR where din is the in-degree of vertex i in Γ. i As above ∆ can be written in the form ∆ = I D−1W where R R R − R R I is the identity operator on V . R R It is easy to see that the spectrum of ∆ consists of the eigenvalues of ∆ and V V times the eigenvalue 0, i. e. R R | \ | (5) spec(∆) = ( V V times the eigenvalue 0) spec(∆ ). R R | \ | ∪ We remark here that ∆ can be considered as a Dirichlet Laplace R operator. The Dirichlet Laplace operator for directed graphs is defined as in the case of undirected graphs, see e. g. [16]. Let Ω V and denote by C(Ω) the space of complex valued functions v :⊆Ω C. → The Dirichlet Laplace operator ∆ on C(Ω) is defined as follows: First Ω extend v to the whole of V by setting v = 0 outside Ω and then ∆ v = (∆v) , Ω Ω | i. e. for any i Ω we have ∈ 1 1 ∆ v(i) = v(i) w v(j) = v(i) w v(j) Ω − din ij − din ij i j∈V i j∈Ω X X since v(j) = 0 for all j V Ω. Hence, ∆ = ∆ if we set Ω = V . R Ω R ∈ \ As already mentioned in the introduction, we are particularly interested in graphs that are not strongly connected. However, every graph that is not strongly connected can uniquely be decomposed into its strongly connected components [6]. Using this decomposition, the Laplace operator ∆ can be represented in the Frobenius normal form [6], i. e. either Γ is strongly connected or there exists an integer z > 1 6 FRANKBAUER s.t. ∆ ∆ ... ∆ 1 12 1z 0 ∆ ... ∆ 2 2z (6) ∆ =  ... ... ... ... ,  0 0 ... ∆   z    where∆ ,...∆ aresquare matrices corresponding tothestrongly con- 1 z nected components Γ ,...,Γ of Γ. In the following, the vertex set of 1 z Γ is denoted by V . Then the off-diagonal elements of ∆ are of the k k k form wij for all i,j V if din = 0 and zero otherwise and the di- diin ∈ k i 6 agonal elements are either zero (if the in-degree of the corresponding vertex is equal to zero) or one (if the in-degree of the corresponding vertex is nonzero). If V does not contain a quasi-isolated vertex, then k ∆ is irreducible. Furthermore, the submatrices ∆ , 1 k < l z k kl ≤ ≤ aredetermined by the connectivity structure between different strongly connected components. For example, ∆ contains all elements of the kl form wij for all i V and all j V . A simple consequence of (6) is diin ∈ k ∈ l that z (7) spec(∆) = spec(∆ ). i i=1 [ Note that ∆ , i = 1,...,z, is a matrix representation of the Dirichlet i Laplace operator of the strongly connected component Γ , i.e. ∆ = i i ∆ for Ω = V . To sum up our discussion, the spectrum of the Laplace Ω i operator of a directed graph is the union of the spectra of the Dirichlet Laplace operators of its strongly connected components Γ . i We conclude this section by introducing the operator P := I ∆. − We have P : C(V) C(V), → 1 w v(j) if din = 0. (8) Pv(i) = diin j ij i 6 v(i) else. (cid:26) P For technical reasons, it is sometimes convenient to study P instead of ∆. Clearly, the eigenvalues of ∆ and P are related to each other by (9) λ(∆) = 1 λ(P), − i. e. if λ is an eigenvalue of P then 1 λ is an eigenvalue of ∆. When − restricted to graphs Γ G+, P(Γ) is equal to the transition probability ∈ operator of the reversal graph Γ. Furthermore, we define the reduced operator P = I ∆ = D−1W . R R − R R R NORMALIZED GRAPH LAPLACIANS FOR DIRECTED GRAPHS 7 3. Basic properties of the spectrum In this section, we collect basic spectral properties of the Laplace operator ∆. Proposition 3.1. Let Γ G then following assertions hold: ∈ (i) The Laplaceoperator∆hasalways an eigenvalueλ = 0 andthe 0 corresponding eigenfunction is given by the constant function. (ii) The eigenvalues of ∆ appear in complex conjugate pairs. (iii) The eigenvalues of ∆ satisfy n−1 n−1 λ = (λ ) = V . i i R ℜ | | i=0 i=0 X X (iv) The spectrum of ∆ is invariant under multiplying all weights of the form w for some fixed i and j = 1,..,n by a non-zero ij constant c. (v) The spectrum of ∆ is invariant under multiplying all weights by a non-zero constant c. (vi) The Laplace operator spectrum of a graph is the union of the Laplace operator spectra of its weakly connected components. Proof. (i) This follows immediately from the definition of ∆ since 1 w (v(i) v(j)) if din = 0. ∆v(i) = diin j ij − i 6 0 else. (cid:26) P (ii) Since ∆ can be represented as a real matrix, the characteristic polynomial is given by det(∆ λI) = a +a λ+...+a λn−1, 0 1 n−1 − with a R for all i = 0,1,...,n 1. Consequently, det(∆ i ∈ − − λI) = 0 if and only if det(∆ λI) = 0. − (iii) The equality n−1λ = n−1 (λ ) follows from (ii). By con- i=0 i i=0 ℜ i sidering the trace of ∆, one obtains n−1λ = V . P P i=0 i | R| (iv), (v) and (vi) follow directly from the definition of ∆. P (cid:3) From Proposition 3.1(v) it follows that it is equivalent to study the spectrumofgraphswithnonnegativeornonpositiveweights. Moreover, because of Proposition 3.1(vi), we will restrict ourselves to weakly connected graphs in the following. Proposition 3.2. The spectrum of ∆ satisfies spec(∆) (1,r ) 0 (1,r ) 0 (1,r) 0 , 1 2 ⊆ D ∪{ } ⊆ D ∪{ } ⊆ D ∪{ } 8 FRANKBAUER where (c,r) denotes the disk in the complex plane centered at c with D radius r and w r := max max j∈VR,p| ij|, 1 p=1,...,zi∈VR,p P |diin| w r := max j∈VR | ij|, 2 i∈VR P |diin| and (10) r := maxr(i), i∈V where r(i) = Pj∈V |wij|. Here, V ,...,V are the strongly connected |din| R,1 R,z i components of the induced subgraph Γ whose vertex set is given by V . R R We use the convention that r ,r and r are equal to zero if din = 0. 1 2 i Proof. Clearly, r r r and the proof follows from Gersgorin’s 1 2 ≤ ≤ (cid:3) circle theorem (see e. g. [18]) and (5)-(7). For undirected graphs with nonnegative weights Proposition 3.2 reduces to the well-known result [8], that all eigenvalues of ∆ are contained in the interval [0,2]. The radius r in Proposition 3.2 has the following properties: r 1 ≥ if and only if V = and r = 0 if and only if V = . R R 6 ∅ ∅ Lemma 3.1. Let Γ be a graph without quasi-isolated vertices and let r(i) = r = 1 for all i V. Then there exists a graph Γ+ G+ that is ∈ ∈ isospectral to Γ. Proof. Since r = 1 it follows from the definition of r that for every vertex i V the sign sgn(w ) is the same for all j V. By Proposition ij ∈ ∈ 3.1 (iv) the graph Γ+ G+ that is obtained from Γ by replacing the ∈ associated weight function w by its absolute value w is isospectral to | | (cid:3) Γ. In the following, Γ+ is called the associated positive graph of Γ. Corollary 3.1. For graphs Γ G the nonzero eigenvalues satisfy ∈ V R (11) 1 r min (λ ) | | max (λ ) 1+r, i i − ≤ i:λi6=0ℜ ≤ n m0 ≤ i:λi6=0ℜ ≤ − where m denotes the multiplicity of the eigenvalue zero. In particular, 0 we have 1 max (λ ). i ≤ i:λi6=0ℜ Proof. This estimate follows from Proposition 3.1 (iii) and Proposition 3.2. The last statement follows from the observation that n m 0 V . − ≤(cid:3) R | | NORMALIZED GRAPH LAPLACIANS FOR DIRECTED GRAPHS 9 1-e2πi1n- 0 2 1-e-2πi1n- Figure 1. For a graph Γ G+ with n vertices, all ∈ eigenvalues of ∆ are contained in the shaded region. Later,inCorollary7.4,wecharacterizeallgraphsforwhichmax (λ ) = i:λi6=0ℜ i 1+r. Similarly, in Corollary 7.7, we characterize all graphs for which min (λ ) = 1 r, provided that r > 1. i:λi6=0ℜ i − For graphs with nonnegative weights, Proposition 3.2 can be further improved. Proposition 3.3. Let Γ G+, then all eigenvalues of the Laplace ∈ operator ∆ are contained in the shaded region in Figure 1. Proof. This follows from the results in [14], see [22] for further discus- (cid:3) sion. We close this section by considering the following example. Example 1. In [8] it is shown that the smallest non-trivial eigenvalue λ of non-complete undirected graphs Γ Gu+ with nonnegative 1 ∈ weightssatisfiesλ 1. Itistemptingtoconjecturethatmin (λ ) 1 i6=0 i ≤ ℜ ≤ 1 for all non-complete undirected graphs with positive and negative weights and for all non-complete directed graphs with nonnegative weights. However, the two examples in Figure 2 show that this is, in general, not true. For both, the non-complete graph Γ Gu in 1 ∈ Figure 2 (a) and the non-complete graph Γ G+ in Figure 2 (b) we 2 ∈ have min (λ ) > 1. Thus, there exist non-complete graphs Γ Gu i6=0 i 1 ℜ ∈ and Γ G+ for which the smallest non-zero real part of the eigenval- 2 ∈ ues is larger than the smallest non-zero eigenvalue of all non-complete graphs Γ Gu+. This observation has interesting consequences for the ∈ synchronization of coupled oscillators, see [1]. 4. Spectrum of ∆ and isolated components of Γ We have the following simple observation: 10 FRANKBAUER a) b) 1/2 1 5 1 -1 1/2 1 1 1/2 1 Figure 2. a) The eigenvalues of ∆ are 1.45 ± 0.46i,1.10,0. b) The eigenvalues of ∆ are 1.65,1.18 ± 0.86i,0. Lemma 4.1. Consider a graph Γ G and let Γ ,1 i r be its i ∈ ≤ ≤ strongly connected components. Furthermore, let the Laplace operator ∆ be represented in Frobenius normal form (6). Then, (i) If Γ is isolated then ∆ = 0 for all j > i. i ij (ii) If Γ is quasi-isolated then the row sums of ∆ ...∆ add i i,(i+1) ir up to zero. Moreover, if Γ G+ then ∈ (iii) Γ is isolated if and only if ∆ = 0 for all j > i. i ij (iv) Γ is quasi-isolated if and only if the row sums of ∆ ...∆ i i,(i+1) ir add up to zero. Lemma 4.2. Every graph Γ G contains at least one isolated strongly ∈ connected component. Furthermore, Γ G contains exactly one iso- ∈ lated strongly connected component if and only if Γ contains a spanning tree. Proof. This follows immediately from the Frobenius normal form of (cid:3) ∆. In particular, every undirected graph Γ Gu is strongly connected ∈ and isolated. In general, it is not true that the spectrum of an induced subgraph Γ′ of Γ is contained in the spectrum of the whole Γ, i. e. spec(∆(Γ′)) * spec(∆(Γ)). However, we have the following result: Proposition 4.1. Let Γ G and Γ′ be an induced subgraph of Γ. If ∈ one of the following conditions is satisfied (i) Γ′ consists of 1 p r strongly connected components of Γ ≤ ≤ and is quasi-isolated, (ii) Γ′ is isolated, then spec(∆(Γ′)) spec(∆(Γ)). ⊆