Hubs-biased resistance distances on graphs and networks
aa r X i v : . [ m a t h . SP ] J a n Hubs-biased resistance distances on graphs and networks
Ernesto Estrada , , and Delio Mugnolo Institute of Mathematics and Applications, University of Zaragoza, Pedro Cerbuna 12,Zaragoza 50009, Spain; ARAID Foundation, Government of Aragon, Spain. Institute forCross-Disciplinary Physics and Complex Systems (IFISC, UIB-CSIC), Campus Universitatde les Illes Balears E-07122, Palma de Mallorca, Spain. Lehrgebiet Analysis, FakultätMathematik und Informatik, FernUniversität in Hagen, D-58084 Hagen, Germany;
Abstract
We define and study two new kinds of “effective resistances” based on hubs-biased–hubs-repelling and hubs-attracting– models of navigating a graph/network.We prove that these effective resistances are squared Euclidean distances be-tween the vertices of a graph. They can be expressed in terms of the Moore-Penrose pseudoinverse of the hubs-biased Laplacian matrices of the graph. Wedefine the analogous of the Kirchhoff indices of the graph based of these resis-tance distances. We prove several results for the new resistance distances andthe Kirchhoff indices based on spectral properties of the corresponding Lapla-cians. After an intensive computational search we conjecture that the Kirchhoffindex based on the hubs-repelling resistance distance is not smaller than thatbased on the standard resistance distance, and that the last is not smaller thanthe one based on the hubs-attracting resistance distance. We also observe thatin real-world brain and neural systems the efficiency of standard random walkprocesses is as high as that of hubs-attracting schemes. On the contrary, infras-tructures and modular software networks seem to be designed to be navigatedby using their hubs.
AMS Subject Classification:
Keywords: graph Laplacians; resistance distances; spectral properties;algebraic connectivity; complex networksCorresponding author: Ernesto Estrada; email: [email protected]
1. Introduction
Random walk and diffusive models are ubiquitous in mathematics, physics,biology and social sciences, in particular when the random walker moves throughthe nodes and edges of a graph G = ( V, E ) [1, 2, 3, 4, 5]. In this scenario arandom walker at the node j ∈ V of G at time t can move to any of the nearestneighbors of j with equal probability at time t +1 [1]. That is, if as illustrated in Preprint submitted to Elsevier 19th January 2021 a) (b) (c)Figure 1.1: Schematic illustration of a normal (a), hubs-attracting (b) and hubs-repellingschemes of a particle hopping on a network.
Fig. 1.1(a) the node j has three nearest neighbors { k, l, m } the random walkercan move to any of them with probability p jm = p jl = p jk = k − j , where k j is thedegree of j . We can figure out situations in which the movement of the randomwalker at a given position is facilitated by the high degree of any of its nearestneighbors. Let us call informally a “ hub ” to a node u with a large degree, i.e., k u ≫ . Then, let us suppose that there are situations in which the probabilitythat the random walker moves to a nearest neighbor of j at time t + 1 increaseswith the degree of the nearest neighbor. This is illustrated in Fig. 1.1(b) where k k > k l > k m , and consequently p jm < p jl < p jk . We will refer hereafter to thisscenario as the “hubs-attracting ” one. Another possibility is that the randomwalker is repelled by high degree nodes, such as for k k > k l > k m , we have p jm > p jl > p jk as illustrated in Fig. 1.1(c). We will refer to this model asthe “ hubs-repelling ” one. These scenarios could be relevant in the context ofreal-world complex networks where nodes represent the entities of the systemand the edges the interrelation between them [6, 7, 8]. An example of hubs-repelling strategies of navigation are some of the diffusive processes in the brainwhere there is a high energetic cost for navigating through the hubs of thesystem [9, 10]. Hubs-attracting mechanisms could be exhibited, for instance, bydiffusive epidemic processes in which hubs are major attractors and propagatorsof the disease [11].From a mathematical perspective, one of the most important aspects of themodel of random walks on graphs is its connection with the graph Laplacianmatrix [12, 13, 14, 15, 16] and with the concept of resistance distance [17, 18, 19,20, 21, 22]. In a simple graph the resistance distance is the effective resistancebetween two nodes v and w , which measures the resistance of the total systemwhen a voltage is connected across v and w . Klein and Randić [18] proved thatthe effective resistance is a squared Euclidean distance between the two nodesof the graph, which can be obtained from the Moore-Penrose pseudoinverse ofthe graph Laplacian. It is also known that the commute time between twonodes v and w of a random walker [17, 19, 21, 22], i.e., the number of steps ofa random walker starting at v , before arriving at w and returning to v again, isproportional to the resistance distance between the two nodes.2trategies for avoiding large/low degree nodes in random-walk processes ongraphs/networks have been proposed in the literature under the general um-brella of ‘degree-biased random walks’ [23, 24, 25, 26]. However, in the currentwork we go beyond the random walk formulation of the problem and expressit in terms of hubs-biased Laplacians [27, 28] and the corresponding resistancedistances matrices on graphs. Thus, we focus here on the algebraic properties ofthese resistance distance matrices. We note in passing that the current concepthas nothing in common with the so-called “degree-resistance” between a pairof nodes, which is nothing else that the resistance distance multiplied by thedifference of degrees of the nodes forming the corresponding edge [29].Due to the relation between resistance distance and commute time of ran-dom walks on graphs, we study here the efficiency of hubs attracting/repellingdiffusive processes on graphs. In closing, in this work we define the hubs-biasedresistance distances between pairs of nodes of a simple graph and study theirmain spectral properties. We also propose analogues of the Kirchhoff index[18, 30, 31, 32], the semi-sum of all resistance distances in the graph, for thehubs-biased resistances. We report here several bounds for the two new kinds ofresistance distances as well as for the corresponding Kirchhoff indices. Finally,we analyze the relative improvement in the commute time of the hubs-attractingrandom walks respect to the normal one. We observe that certain classes of real-world networks, such as brain/neuronal networks and electronic circuits, havenormal random walks as efficient as the hubs-attracting one, while others, likeinfrastructural networks, can reduce their average commuting times by 300%by using the hubs-attracting mechanism.
2. Preliminaries
Here we use interchangeably the terms graphs and networks. In this articlewe consider simple, undirected graphs G = ( V, E ) on n vertices and m edges:we denote the set of vertices and edges by V and E , respectively.We will always consider connected graphs. In the case of weighted graphs G = ( V, E, W, ϕ ) there is a surjective mapping ϕ : E → W assigning elementsof the weight set W to the edges of the graph. Here we consider only positiveweights. Let A be the adjacency matrix of the (weighted) graph G and let k i denotes the degree of the node i ∈ V , i.e., the sum of the i th row or columnof A . In the case of weighted graphs the degree is often referred as strength ,but we will use the general term degree here in all cases. In the case of infinitegraphs we should consider only locally-finite ones, which means that the degreeof their nodes is bounded.Let i ∈ V be a node of G . We will denote by N j = { j ∈ V | ( i, j ) ∈ E } theset of all nearest neighbors of i . A node for which k i ≫ will be ‘informally’called a hub. We will denote by K the diagonal matrix of node degrees. We usethe following condensed notation across this paper.If x α is a number depending on an index α –in the following typically α ∈{− , } –then we will write x α to symbolize both x − and x depending on the3hoice on the index α . Let ℓ ( V ) be the Hilbert space of square-summablefunctions on V with inner product h f, g i = X v ∈ V f ( v ) g ( v ) , f, g ∈ ℓ ( V ) . The graph Laplacian is an operator in ℓ ( V ) which is defined by [33] (cid:0) L f (cid:1) ( v ) := X w ∈ V : ( v,w ) ∈ E ρ vw (cid:0) f ( v ) − f ( w ) (cid:1) , f ∈ dom ( L ) , (2.1)where ρ vw ∈ W . The Laplacian matrix can be obtained as L = K − A [12, 13,14, 15, 16]. We remind that in the case of weighted graphs K is the diagonalmatrix of weighted degrees and A is the weighted adjacency matrix.Here ~ denotes the all-ones column vector of order n , J n the n × n all-onesmatrix and I n the identity matrix of order n .
3. Hubs-biased Laplacians and their spectra
Here we introduce the concepts of hubs-biased Laplacians in the context ofresistive networks.
Definition 1.
A conductance function is a function c : V × V → R + whichdefines adjacency between nodes by ( v, w ) ∈ E iff c ( v, w ) > . Definition 2.
The total conductance at a node v is defined as c ( v ) := X ( v,w ) ∈ E c ( v, w ) . (3.1)We require that c ( v ) < ∞ , i.e., locally finite graph, i.e., the degree of the nodesis bounded, in case of infinite graphs.We observe that the total conductance at v ∈ V can be equivalently definedas c ( v ) := X w ∈ V a vw c ( v, w ) , (3.2)where a vw is the entry of the adjacency matrix A corresponding to the pair ( v, w ) ∈ V × V , i.e., a vw = 1 if v, w are connected by an edge and 0 otherwise.Let us now consider a diffusive particle that not necessarily hops to anynearest neighbor with the same probability. Let us for instance consider that itis ‘not welcomed’ by nodes of high degree, but it is ‘preferred’ by those of lowdegree. That is, let p, q ∈ N i , such that k i ≫ k p > k q . Then, a diffusive particleat the hub i will tend to hop to node q with higher probability than to node p , due to the fact that p, which has a higher degree, is repelling particles awayfrom it. By resuming, for ( i, j ) ∈ E the hopping of a diffusive particle from i to4 is favored by a large degree of i and a low degree of j . On the other hand, thehopping is disfavored by a low degree of i and a large degree of j . Notice thatthese conditions for the hopping from i to j are different from the ones for thehopping from j to i . This will become clear immediately with the following. Definition 3.
Let G be a graph and let α ∈ {− , } E . The hubs-biased Lapla-cian corresponding to α is the operator on ℓ ( V ) defined by (cid:0) L α f (cid:1) ( v ) := X w ∈N v c α ( v, w ) (cid:0) f ( v ) − f ( w ) (cid:1) , f ∈ dom ( L α ) , (3.3)where c α ( v, w ) = (cid:18) k v k w (cid:19) α . If α ≡ and hence c α ( v, w ) = k v k w , then we call L α the hubs-repelling Laplacian.If α ≡ − and hence c α ( v, w ) = k w k v , then we call L α the hubs-attractingLaplacian.Actually, we could easily extend this definition by allowing for vectors α whose entries are either + x or − x for a given x ≥ ; for x = 0 and hence α ≡ , corresponding to the unweighted case, we would then recover the standard(discrete) Laplacian L . However, we will not explore this direction in thispaper and restrict ourselves, in accordance with Definition 3, with hubs-biasedLaplacians L α defined for any vector whose entries are either +1 or − . Inpractice, we will mostly focus on the relevant cases α ≡ and α = − , especiallyin Section 4; the results of this Section 3, though, are remarkably still valid ifwe allow for alternating values of α , see Lemma 4 below.Let e v , v ∈ V be a standard orthonormal basis in ℓ ( V ) consisting of thevectors e v ( w ) := ( if w = v, otherwise . (3.4)Then, L α act on the vectors e v as follows: ( L α e v )( w ) = c α ( v ) if w = v, − c α ( v, w ) if ( v, w ) ∈ E, otherwise , (3.5)where c α ( v ) = P w ∈N v c α ( v, w ) is the α -conductance of the vertex v . Then,the hubs-biased matrices can be expressed as L α = Ξ α − K α AK − α , where Ξ α are the corresponding diagonal matrices whose diagonal entries are Ξ α ( v, v ) = c α ( v ) , K := diag ( k v ) v ∈ V (3.6)is the diagonal matrix of node degrees and A is the unweighted adjacency matrixof the graph. 5or the sake of later reference, let us note the following fact, which is alsofollowed by several new results about the hubs-biased Laplacians. Lemma 4.
Let G be a graph. Then for all α, α ′ ∈ {− , } E the following hold: tr ( L α ) = tr ( L α ′ ) = X v,w ∈ V a vw k v k w . (3.7) This suggests to introduce the notation tr HB ( G ) (3.8) for the trace of any hubs-biased Laplacian on G .Proof. For the first equality, which is easy to prove, let us consider the definitionof total conductance and in view of (3.2),tr ( L α ) = X v ∈ V c α ( v ) = X v,w ∈ V a vw (cid:18) k v k w (cid:19) α . (3.9)Now, because a v,w = 1 if and only if a w,v = 1 , and in this case both addends k v k w , k w k v appear tr ( L α ) = X v ∈ V c α ( v ) = X v,w ∈ V a vw k v k w , (3.10)which is independent of α .Lemma 4 has a few interesting consequences. Let δ and ∆ be the minimumand maximum degree of the vertices of G , respectively. Then, because for all v ∈ V δ ≤ k v ≤ ∆ and P v,w ∈ V a vw = P v ∈ V k v = 2 m by the Handshakinglemma, (3.7) immediately implies that the hubs-biased Laplacian trace satisfies m δ∆ ≤ tr HB ( G ) ≤ m ∆δ . (3.11)This estimate is rough, yet sharp as both inequalities become equalities forcomplete graphs. We should observe that if G is complete then tr HB ( G ) alsoagrees with the trace of the standard discrete Laplacian. Remark . The right hand side of (3.10) and hence tr HB ( G ) of any hub-biasedLaplacian on G can be written in different ways: X v,w ∈ V a vw k v k w = X v ∈ V k v X w ∈ V a vw k w = X v ∈ V k v X w ∈N v k w , (3.12)but also X v,w ∈ V a vw k v k w = X w ∈ V k w X v ∈ V a vw k v ! = X v ∈ V k w X v ∈N w k v . (3.13)6et us elaborate on the representation in (3.12) based on X v,w ∈ V a vw k v = X v ∈ V k v X w ∈ V a vw = X v ∈ V k v = X v,w ∈ E ( k v + k w ) . (3.14)Hence ∆ X v ∈ V k v ≤ tr HB ( G ) ≤ δ X v ∈ V k v . (3.15)Because of the last identity in (3.14), these inequalities imply the estimatesin (3.11), but are clearly sharper if, e.g., G is bi-regular (a bipartite graph with V = V ∪ V where all nodes in each partition has the same degree).More explicit bounds in (3.15) can be obtained using known estimates on P v ∈ V k v . Indeed, it is known that m n ≤ X v ∈ V k v ≤ m (2 m + ( n −
1) ( ∆ − δ )) n + ∆ − δ , (3.16)with the upper inequality becoming an equality iff G is regular, or else a bi-regular graph, whence m n∆ ≤ tr HB ( G ) ≤ m (2 m + ( n −
1) ( ∆ − δ )) δ ( n + ∆ − δ ) . (3.17)The lower estimate in (3.17) is significantly better than the lower estimatein (3.11): indeed, the net effect is like replacing δ by the average degree mn in(3.11). The upper estimate in (3.17) is better than the upper estimate in (3.11)iff m + ( n −
1) ( ∆ − δ ) ≤ ∆ ( n + ∆ − δ ) , (3.18)which is sometimes the case, for instance for any regular graph, and sometimesnot, for instance for any path on more than 2 edges. The equality in (3.18)holds for complete graphs.Studying the behavior of the degree function on neighborhoods–and in par-ticular considering their minima δ N v and maxima ∆ N v –we can deduce from(3.12) and (3.13) the following sharper estimates: X v ∈ V k v ∆ N v ≤ tr HB ( G ) ≤ X v ∈ V k v δ N v , (3.19) X v ∈ V δ N v ≤ tr HB ( G ) ≤ X v ∈ V ∆ N v , (3.20)which shows, for instance, that tr HB ( G ) = 2 pq for the complete bipartite graph K p,q . X w ∈N v k w ≥ (cid:0)P w ∈N v (cid:1) P w ∈N v k w = k v P w ∈N v k w . (3.21)Because Titu’s Lemma is equivalent to the Cauchy-Schwarz inequality, theprevious inequality becomes an equality iff (1) w ∈N v and ( k w ) w ∈N v are linearlydependent, i.e., iff the degree function is constant on each neighborhood, whichis the case for instance in regular and bi-regular graphs. We finally deduce thefollowing estimate: tr HB ( G ) ≥ X v ∈ V k v P w ∈N v k w , (3.22)which of course implies the lower estimate in (3.19).We now resume a series of important properties of the hubs-biased Laplacianmatrices of simple graphs. Theorem 6.
Let G be a fine graph with C connected components. Then, thehubs-biased Laplacian matrices L α have the following properties:(i) their spectra are real;(ii) they are positive semidefinite;(iii) rank L α = n − C ;(iv) they can be diagonalized as L α = ( KU α ) Λ α ( KU α ) − , where Ξ α − A = U α Λ α U − α and K is as in (3.6).Proof. We start by showing that the diagonal matrix of node degrees L α = Ξ α − K α AK − α = K α (cid:0) K − α Ξ α K α − A (cid:1) K − α = K α ( Ξ α − A ) K − α , (3.23)thus, L α are similar to the symmetric matrices ( Ξ α − A ) , and so their eigen-values are real.Now let ~x ∈ R n and ~x = ~ . Then, we can write ~x T ( Ξ α − A ) ~x = X ( i,j ) ∈ E (cid:16)(cid:16) k α/ i k − α/ j (cid:17) x i − (cid:16) k α/ j k − α/ i (cid:17) x j (cid:17) . (3.24)Therefore, because Ξ α − A and L α are similar, we have that ~x T L α ~x ≥ .Let us now prove that the dimension of the null space of L α is C . Let ~z be a vector such that L α ~z = 0 . This implies that for every ( i, j ) ∈ E , z i = z j . Therefore ~z takes the same value on all nodes of the same connectedcomponent, which indicates that the dimension of the null space is C , and so rank L α = n − C . 8inally, we also have that because Ξ α − A is symmetric we can write it as: Ξ α − A = U α Λ α U − α . Thus, L α = ( KU α ) Λ α ( KU α ) − which indicates that allhubs-biased Laplacians are diagonalizable.Let us conclude this section deducing some estimates on the eigenvalues ofthe hubs-biased Laplacians, which are known to be real and positive by Theorem6. Corollary 7.
Let G has n vertices and m edges, minimal, average and maxi-mum degree δ , ¯ k and ∆ , respectively. Then, we have ρ α,n ≥ mn − k∆ , (3.25) and mn − m + ( n −
1) ( ∆ − δ )) δ ( n + ∆ − δ )2 mn − ∆δ ≥ ρ α, . (3.26)We note in passing that on a ( k + 1) -star, the upper bound on Corollary 7reads k k + 1 (cid:27) ≥ ρ α, , (3.27)which does not prevent ρ α, to tend to infinite as the graph grows.We also remark that these estimates are only really interesting for non-regular graphs, because in regular graphs the hubs-biased Laplacians coincidewith the standard discrete Laplacian for which many different bounds are knownfor its eigenvalues. Proof.
Observe that n − tr HB ( G ) yields the arithmetic mean of all non-zeroeigenvalues of L α ; in particular, the lowest non-zero eigenvalue ρ α, cannot belarger than n − tr HB ( G ) while the largest eigenvalue ρ α,n cannot be largerthat n − tr HB ( G ) . Taking into account (3.11) and (3.17) we deduce theclaimed estimate. Remark . The naive upper bound ρ α,n ≤ v ∈ V ( L α ) vv (3.28)on the largest eigenvalue of L α follows from Geršgorin’s Theorem, since by (3.5) ( L α ) vv = P w ∈ V | ( L α ) vw | . Now, ( L − ) vv = X w ∈N v k w k v ≤ ∆ N v k v X w ∈N v ∆ N v , (3.29)9 L ) vv = X w ∈N v k v k w ≤ k v δ X w ∈N v k v δ ≤ ∆ δ , (3.30)which are both sharp for regular graphs. Whence, ρ − ,n ≤ ∆ N v , (3.31)and ρ ,n ≤ ∆ δ . (3.32)In order to show some lower bounds we remark that the standard discreteLaplacian L and L α share the null space. Then, we can quotient it out andstudy the lowest eigenvalue of L and L α on C n / h i . Take a normalized vector f in this space, then one sees that δ∆ ( L f, f ) ≤ ( L α f, f ) . (3.33)In particular, choosing f to be an eigenfunction associated with the lowestnon-zero eigenvalue ρ α, of L α and applying Courant’s characterization of theeigenvalues of the Hermitian matrix we deduce δ∆ ρ ≤ ρ α, , (3.34)where ρ is the second lowest eigenvalue of the discrete Laplacian, i.e., thealgebraic connectivity [36]. Alternatively, for α = − we can use δ ( L norm f, f ) ≤ ( L α f, f ) , (3.35)and deduce δρ , norm ≤ ρ α, , (3.36)where L norm := K − / L K − / is the normalized Laplacian. Now we can eitherapply explicit formulae for ρ and ρ , norm for classes of graphs or use generalestimates like ρ ≥ η (cid:16) − cos πn (cid:17) , (3.37)from [37] or ρ , norm ≥ Dn , − cos πm , (3.38)from [39, 40], respectively, where η ≥ is the edge connectivity and D is thediameter of G . 10 . Hubs-biased Resistance Distance We adapt here some general definitions of resistive networks to the caseof hubs-biased systems, mainly following the classic formulations given in [17,18, 22]. Another approach can be found in [34, 35]. Let us consider G as aresistive network in which every edge ( v, w ) ∈ E has edge resistance r α ( v, w ) := c − α ( v, w ) . Let us consider the connection of a voltage between the vertices v and w , and let i ( v, w ) > be the net current out the source v and into thesink w , such that i ( v, w ) = − i ( w, v ) . Then, according to the first Kirchhoff’slaw we have that P w ∈N v i ( v, w ) = I if v is a source, P w ∈N v i ( v, w ) = − I if v is a sink, or zero otherwise. The application of the second Kirchhoff’slaw, namely that P ( v,w ) ∈ C i ( v, w ) r α ( v, w ) = 0 where C is a cycle with edgeslabeled in consecutive order, implies that a potential V may be associated withany vertex v , such that for all edges i ( v, w ) r α ( v, w ) = V ( v ) − V ( w ) , (4.1)which represents the Ohm’s law, and where i and V depend on the net current I and on the pair of vertices where the voltage source has been placed. Let usnow define formally the hubs-biased effective resistance , which is the resistanceof the total system when a voltage source is connected across a correspondingpair of vertices. Definition 9.
The hubs-biased effective resistance between the nodes v and w of G is Ω α ( v, w ) = V ( v ) − V ( w ) I . (4.2)We now prove the following result.
Lemma 10.
The hubs-biased resistance Ω α ( v, w ) between two nodes v and w of G is a squared Euclidean distance given by: Ω α ( v, w ) = L + α ( v, v ) + L + α ( w, w ) − L + α ( v, w ) − L + α ( w, v ) , (4.3) where L + α stands for the Moore-Penrose pseudoinverse of L α .Proof. First, we will prove that Ω α ( v, w ) = ( ~e v − ~e w ) T L + α ( v, v ) ( ~e v − ~e w ) , (4.4)where ~e v is the vector with all entries equal to zero except the one correspondingto vertex v which is equal to one. Using the second Kirchhoff’s law we have X w ∈N v r ( v, w ) ( V ( v ) − V ( w )) = I − I if v is a sourceif v is a sinkotherwise, (4.5)11hich can also be written as c α ( v ) V ( v ) − n X w =1 r ( v, w ) V ( w ) = I − I if v is a sourceif v is a sinkotherwise. (4.6)Let us write it in matrix-vector form as L α ~ V = I ( ~e v − ~e w ) . (4.7)Due to the fact that the RHS of (4.7) is orthogonal to ~ we can obtain ~ V as V ( v ) − V ( w ) = ( ~e v − ~e w ) T ~ V = I ( ~e v − ~e w ) T L + α ( ~e v − ~e w ) . (4.8)Then, using the definition of the effective resistance we have Ω α ( v, w ) = V ( v ) − V ( w ) I = ( ~e v − ~e w ) T L + α ( ~e v − ~e w ) . (4.9)Now, because ( ~e v − ~e w ) T L + α ( ~e v − ~e w ) = L + α ( v, v )+ L + α ( w, w ) − L + α ( v, w ) − L + α ( w, v ) we only remain to prove that it is a distance. Let L α = V α Λ α V − α ,where V α = KU α . Then L + α = V α Λ + α V − α , with Λ + α being the Moore-Penrosepseudoinverse of the diagonal matrix of eigenvalues of L α , i.e., the diagonal ma-trix whose i th entry is Λ + α ( i, i ) = (cid:26) ρ − α,i if the i th eigenvalue is = 0 if the i th eigenvalue is = 0 .Let us write the right-hand side of (4.3) as ~v α Λ + α ~u Tα + ~w α Λ + α ~w Tα − ~v α Λ + α ~w Tα − ~w α Λ + α ~v Tα , (4.10)where ~v and ~w are the corresponding rows of V α for the nodes u and w , respec-tively. Then, we have L + α ( u, u ) + L + α ( w, w ) − L + α ( u, w ) − L + α ( w, u )= ~v α (cid:0) Λ + α ~v Tα − Λ + α ~w Tα (cid:1) − ~w α (cid:0) Λ + α ~v Tα − Λ + α ~w Tα (cid:1) , = ( ~v α − ~w α ) (cid:0) Λ + α ~v Tα − Λ + α ~w Tα (cid:1) = ( ~v α − ~w α ) Λ + α ( ~v α − ~w α ) T = (cid:16) ( ~v α − ~w α ) p Λ + α (cid:17) (cid:16) ( ~v α − ~w α ) p Λ + α (cid:17) T = ( V α ( v ) − V α ( w )) T ( V α ( v ) − V α ( w ))= k V α ( v ) − V α ( w ) k , (4.11)where V α ( v ) = ~v α p Λ + α is the position vector of the node v in the Euclideanspace induced by the hubs-repelling Laplacian.12 orollary 11. Let G = ( V, E ) has n ≥ and let L α = V α Λ α V − α , where V α = KU α , V α = h ~ψ α, , ~ψ α, , · · · , ~ψ αn i and Λ α = diag ( ρ α,k ) . Then, Ω α ( u, w ) = n X k =2 ρ − α,k ( ψ α,k,u − ψ α,k,w ) . (4.12) Proof.
It is easy to see from Lemma 10 that Ω α ( u, w )= n X k =2 ρ − α,k ψ α,k,u + n X k =2 ρ − α,k ψ α,k,w − n X k =2 ρ − α,k ψ α,k,u ψ α,k,w = n X k =2 ρ − α,k (cid:0) ψ α,k,u + ψ α,k,w − ψ α,k,u ψ { R,A } ,k,w (cid:1) = n X k =2 ρ − α,k ( ψ α,k,u − ψ α,k,w ) . (4.13) Corollary 12.
Let G = ( V, E ) has n ≥ and let ρ α, < ρ α, ≤ · · · ≤ ρ α,n be the eigenvalues of L α . Then, ρ α,n ≤ Ω α ( u, w ) ≤ ρ α, . (4.14) Proof.
Using Corollary 11 and the fact that ρ α, is the smallest eigenvalue of L α we have the upper bound if all the eigenvalues are equal to ρ α, . The resultfollows from the fact that P nk =1 ψ α,k,u = 1 and P nk =1 ψ α,k,u ψ α,k,w = 0 for every u = w . The lower bound is obtained similarly by the fact that ρ α,n is the largesteigenvalue of L α . In full analogy with the definition of the so-called Kirchhoff index by Kleinand Randić [18] we define here the hubs-biased Kirchhoff indices of a graph.
Definition 13.
The total hubs-biased resistance distance, or hubs-biased Kirch-hoff index , of a graph G is defined as R α ( G ) = X u Let us write the sum of the hubs-biased resistance distances as X u =1 X w =1 Ω α ( u, w )= 12 (cid:16) ~ T diag (cid:0) L + α (cid:1) ~ T ~ ~ T ~ (cid:0) diag (cid:0) L + α (cid:1)(cid:1) T ~ − ~ T L + α ~ − ~ T (cid:0) L + α (cid:1) T ~ (cid:17) = 12 (cid:0) n tr (cid:0) L + α (cid:1)(cid:1) = n n X k =2 ρ α,k , (4.17)where tr ( L + α ) is the trace of L + α . Corollary 15. Let ρ α, < ρ α, ≤ · · · ≤ ρ α,n be the eigenvalues of L α for G with n ≥ . Then, n ( n − ρ α,n ≤ R α ( G ) ≤ n ( n − ρ α, . (4.18) Lemma 16. Let G has n ≥ nodes. Then, R α ( G ) ≥ n − , (4.19) with equality if and only if G = K n .Proof. Let S be the set of all column vectors x such that x · x = 1 , x · e = 0.Then, it is known that ρ α, = min x ∈ S x T L α x. (4.20)Then, we can show that the matrix ˜ L α := L α − ρ α, (cid:0) I n − n − J n (cid:1) is also positivesemidefinite. Since ˜ L α e = 0 we have y T ˜ L α y = c x T ˜ L α x = c (cid:0) x T L α x − ρ α, (cid:1) ≥ . (4.21)14hus min v L α ( v, v ) − ρ α, (cid:0) − n − (cid:1) ≥ . (4.22)We can then write, ρ α, ≤ nn − v L α ( v, v ) . (4.23)Now, because min v L α ( v, v ) cannot be bigger than ∆ /δ ≤ n − we have that ρ α, ≤ n and then using the result in Corollary 15 we have R α ( G ) ≥ n − . Theequality is obtained only for the case of the complete graph where ρ α,k =2 = n − ,which proves the final result.We now obtain some bounds for the Kirchhoff index for α = 1 and α = − ,respectively.Using the Geršgorin circle theorem it is also possible to bound the spectralradius of the hubs-repelling Laplacian as follow. Lemma 17. Let ∆ and δ be the maximum and minimum degree of G , respec-tively. Then, R α = ( G ) ≥ n ( n − δ∆ . (4.24) Proof. First using the the Geršgorin circle theorem we have: µ α =1 ,n ≤ max v c α =1 ( v ) ≤ ∆ /δ . Then, c α =1 ( v ) = k v P j ∈ η v k − j ≤ ∆ /δ and using the Corollary 15 weobtain the result. Remark . The four graph with the largest value of R α = ( G ) among all con-nected graphs with 8 nodes are illustrated in Fig. 4.1. Lemma 19. Let G has n nodes. Then, R α = − ( G ) ≥ n ( n − ∆ , (4.25) where ∆ is maximum degree of the graph G .Proof. Using the Geršgorin circle theorem we have that µ α = − ,n is containedin a circle of radius max i c α = − ( i ) . Thus, ˆ µ n ≤ i c α = − ( i ) ≤ ∆ . Then, max i c α = − ( i ) = max i k − i P v ∈ η i k v , the last maximum is obtained when k i = δ ,where δ is the minimum degree, and it is connected to δ nodes v ∈ η i whichall have the maximum degree ∆ . Thus, the result follows using the Corollary15. Remark . The four graph with the largest value of R α = − ( G ) among allconnected graphs with 8 nodes are illustrated in Fig. 4.2.15 igure 4.1: Graphs with the maximum values of R α = ( G ) among all connected graphs with8 nodes.Figure 4.2: Graphs with the maximum values of R α = − ( G ) among all connected graphs with8 nodes. act 21. In d -regular graphs, c α =1 ( v, w ) = c α = − ( v, w ) = 1 for all ( v, w ) ∈ E . Thus, Ω α =1 ( u, w ) = Ω α = − ( u, w ) = Ω ( u, w ) for all ( v, w ) ∈ E , and R α = ( G ) = R α = − ( G ) = R α =0 . Then, we calculated the hubs-repelling R α = ( G ) , attracting R α = − ( G ) andnormal R α =0 ( G ) Kirchhoff indices for all connected graphs with ≤ n ≤ .In general, we observed for these more than 12,000 graphs that: R α = ( G ) ≥ R α =0 ( G ) ≥ R α = − ( G ) . Finally we formulate the following conjecture for theKirchhoff indices of graphs. Conjecture 22. Let G = ( V, E ) has n nodes. Then, R α = ( G ) ≥ R α =0 ( G ) ≥ R α = − ( G ) , (4.26) with equality iff G is regular. 5. Computational results Let us consider a “particle” performing a standard random walk through thenodes and edges of any of the networks studied here. The use of random walksin graphs [1] and networks [5] is one of the most fundamental types of stochasticprocesses, used to model diffusive processes, different kinds of interactions, andopinions among humans and animals [5]. They can also be used as a way toextract information about the structure of networks, including the detection ofdense groups of entities in a network [5]. Then, we consider a random walk on G , which represents the real-world network under consideration. We start at anode v ; if at the r th step we are at a node v r , we move to any neighbor of v r ,with probability k − r , where k r is the degree of the node r . Clearly, the sequenceof random nodes ( v t : t = 0 , , . . . ) is a Markov chain [1].An important quantity in the study of random walks on graphs is the accessor hitting time H ( v, w ) , which is the expected number of steps before node w is visited, starting from node v [1]. The sum C ( v, w ) = H ( v, w ) + H ( w, v ) iscalled the commute time, which is the expected number of steps in a randomwalk starting at v , before node w is visited and then the walker come back againto node v [1]. The connection between random walks and resistance distanceon graphs is then provided by the following result (see for instance [2]). Lemma 23. Let G has node conductances c ( v ) . Then, for any two nodes v and w in G , the expected commute time E C ( v, w ) = 2 V Ω ( v, w ) , (5.1) where V = P nv =1 c ( v ) is volume of the graph, and E denotes expected value. Notice that if the graph is unweighted then V = m , i.e., the number of edgesin G . The expected “efficiency” of a standard random walk process on G canthen be measured by 17 ε ( G ) := 1 / X v,w E C ( v, w ) , (5.2)That is, if a standard random walker on a graph uses small times to commutebetween every pair of nodes in the graph, it is an efficient navigational process.On the contrary, large commuting times between pairs of nodes reveal veryinefficient processes. Obviously, E ε ( G ) := 1 / (cid:16) V P v,w Ω ( v, w ) (cid:17) .We now extend these concepts to the use of hubs-biased random walks andcalculated the Kirchhoff indices R ( G ) , R α = − ( G ) and R α = ( G ) for all thesenetworks. Following a similar reasoning as before we define the expected effi-ciencies of the hubs-biased random walks by: E ε α ( G ) := 1 / V α X v,w Ω α ( v, w ) ! , (5.3)where V α is the volume of the graph with conductances based on α . We are in-terested here in the efficiency of the hubs-biased random walk processes relativeto the standard random walk. We propose to measure these expected relativeefficiencies by E α ( G ) := E ε α ( G ) E ε ( G ) = VV α R ( G ) R α ( G ) . (5.4)When E α ( G ) > the hubs-biased random walk is more efficient than thestandard random walk. On the other hand, when E α ( G ) < the standardrandom walk is more efficient than the hubs-biased one. When the efficiency ofboth processes, hubs-biased and standard, are similar we have E α ( G ) ≈ . We start by analyzing the 11,117 connected graphs with 8 nodes that westudied previously. In Fig. 5.1 we illustrate the results in a graphical way. Ascan be seen• E α = − ( G ) > for 95.7% of the graphs considered (10,640 out of 11,117),indicating that in the majority of graphs a hubs-attracting random walkcan be more efficient than the standard random walk;• E α =1 ( G ) > only in 91 out of the 11,117 graphs, which indicates thathubs-repelling random walks are more efficient than the standard one onlyin a handful of graphs;• All graphs for which E α =1 ( G ) ≥ also have E α = − ( G ) ≥ , with equalityonly for regular graphs;• Only 461 graphs (4.15% of all graphs considered) have simultaneously E α =1 ( G ) < and E α = − ( G ) < . These are graphs for which the standardrandom walk is more efficient than both hubs-biased random walks.18 .5 1 1.500.20.40.60.811.2 Figure 5.1: Plot of the efficiency of hubs-biased random walks relative to the standard one forall 11,117 connected graphs with 8 nodes. α = − ) random walks is explained as follow. A standard randomwalk typically does not follow the shortest topological path connecting a pair ofnon-connected nodes. However, the number of shortest paths crossing a nodeincreases with the degree of that node. For instance, let k v and t v be the de-gree and the number of triangles incident to a node v . The number of shortestpaths connecting pairs of nodes is P ≥ k v ( k v − − t v . Therefore, the hubs-attracting strategy redirect the random walker to navigate the network usingmany of the shortest paths interconnecting pairs of nodes, which obviously de-creases the commute time and increases the efficiency of the process. Mostnetworks can be benefited from this strategy.The case of the hubs-repelling strategies is more subtle. To reveal the detailswe illustrate the four graphs having the minimum efficiency of a hubs-repelling( α = 1 ) random walk in Fig. 5.2. As can be seen all these graphs have star-likestructures, which are the graphs with the largest possible degree heterogeneity[41, 42]. Therefore, in these graphs the use of hubs-repelling strategies makesthat the random walker get trapped in low degree nodes without the possibilityof visiting other nodes, because for such navigation they have to cross the hubsof the graph, a process which is impeded by the hubs-repelling strategy.The previous analysis allows us to consider the reason why so little numberof graphs display E α =1 ( G ) > . Such graphs have to display very low degreeheterogeneity, but without being regular, as for regular graphs E α =1 ( G ) ≡ .The four graphs with the highest value of E α =1 ( G ) among all connected graphswith 8 nodes are illustrated in Fig. 5.3. As can be seen these graphs display“quasi-regular” structures but having at least one pendant node connected toanother low-degree node. Then, in a hubs-repelling strategy this relatively iso-lated node (the pendant one) has larger changes (than in a standard randomwalk) of being visited by the random walker who is escaping from the nodes oflarger degree.In closing, we have observed that hubs-attracting random walks are veryefficient in most of graphs due to the fact that such processes increase the chancesof navigating the graph through their shortest paths. On the other hand, hubs-repelling random walks are efficient only in those quasi-regular graphs havingsome relatively isolated nodes which can be visited by the hubs-repelling walkerwith higher chances than in a standard random walk. Here we study 59 real-world networks representing brain/neuronal systems,electronic circuits, social systems, food webs, protein-protein interaction net-works (PIN), modular software, citation networks, transcription networks andinfrastructure networks. The description of all the networks is in the Appendixof the book [8]. 20 igure 5.2: Illustration of the graphs with the minimum values of E α =1 ( G ) among the 11,117connected graphs with 8 nodes. igure 5.3: Illustration of the graphs with the maximum values of E α =1 ( G ) among the 11,117connected graphs with 8 nodes. ¯ E α =1 ( G ) std ¯ E α = − ( G ) stdbrain 3 0.6365 0.2552 0.9938 0.0489circuits 3 0.5187 0.0277 1.0662 0.0095foodweb 14 0.4134 0.2715 1.1247 0.2545social 12 0.3536 0.1917 1.0103 0.1812citations 7 0.2032 0.1758 0.8199 0.2769PIN 8 0.1385 0.0896 0.7855 0.1864infrastructure 4 0.0869 0.1439 0.4846 0.4638software 5 0.0712 0.0296 0.6148 0.1515transcription 3 0.0711 0.0710 0.5806 0.3218 Table 1: Average values of the relative efficiency of using a hubs-attracting random walk onthe graph respect to the use of the normal random walk ¯ E α = − ( G ) for different networksgrouped in different classes. The number of networks in each class is given in the columnlabeled as “number”. The same for a hubs-repelling random walk, ¯ E α =1 ( G ) . In both casesthe standard deviations of the samples of networks in each class is also reported. The results obtained here for the 59 real-world networks are resumed inTable 1 where we report the average values of the previous indices for groups ofnetworks in different functional classes, i.e., brain networks, electronic circuits,social networks, etc.The main observations from the analysis of these real-world networks are:• E α = − ( G ) > for 50.8% of the networks considered, indicating that inonly in half of the networks a hubs-attracting random walk can be moreefficient than the standard random walk;• E α =1 ( G ) > in none of the networks, which indicates that standardrandom walks are always more efficient than hubs-repelling random walksin all these networks.The first result is understood by the large variability in the degree heterogeneityof real-world networks. In this case, only those networks with skew degreedistributions are benefited from the use of hubs-attracting random walks, whilein those with more regular structures are not.The second result indicates that there are no network with such quasi-regularstructures where some of the nodes are relatively isolated as the graphs displayedin Fig. 5.3. However, as usual the devil is in the details. The analysis of theresults in Table 1 indicates that brain networks, followed closely by electronicflip-flop circuits, are the networks in which the use of hubs-repelling strate-gies of navigation produces the highest efficiency relative to standard randomwalks. This can also be read as that these brain networks have evolved in a wayin which their topologies guarantee random walk processes as efficient as thehubs-attracting ones without the necessity of navigating the brain using suchspecific mechanisms. In addition, the use of hubs-repelling processes do notaffect significantly the average efficiency of brain networks, as indicated by thevalue of ¯ E α =1 ( G ) , which is very close to one. This result indicates that if these23etworks have to use a hubs-repelling strategies of navigation due to certain bi-ological constraints, they have topologies which are minimally affected–in termsof efficiency–when using such strategies.Finally, another remarkable result is that the efficiency of navigational pro-cesses in infrastructural and modular software systems is more than 1000%efficient by using normal random-walk approaches than by using hubs-repellingstrategies. Those infrastructural networks seem to be wired to be navigatedby using their hubs, and avoiding them cost a lot in terms of efficiency. Thisis clearly observed in many transportation networks, such as air transportationnetworks, where the connection between pairs of airports is realized through theintermediate of a major airport, i.e., a hub, in the network. 6. Conclusions We have introduced the concept of hubs-biased resistance distance. TheseEuclidean distances are based on graph Laplacians which consider the edges e = ( v, w ) of a graph weighted by the degrees of the vertices v and w in adouble orientation of that edge. Therefore, the hubs-biased Laplacian matri-ces are non-symmetric and reflect the capacity of a graph/network to diffuseparticles using hubs-attractive or hubs-repulsive strategies. The correspond-ing hubs-biased resistance distances and the corresponding Kirchhoff indicescan be seen as the efficiencies of these hubs-attracting/repelling random walksof graphs/networks. We have proved several mathematical results for both thehubs-biased Laplacian matrices and the corresponding resistances and Kirchhoffindices. Finally we studied a large number of real-world networks representinga variety of complex systems in nature, and society. All in all we have seenthat there are networks which have evolved, or have being designed, to operateefficiently under hubs-attracting strategies. Other networks, like brain ones, arealmost immune to the change of strategies, because the use of hubs-attractingstrategies improve very little the efficiency of a standard random walk, and theefficiency of hubs-repelling strategies is not significantly different than that ofthe classical random walks. Therefore, in such networks the use of the standardrandom walk approach is an efficient strategy of navigation, while infrastruc-tures and modular software networks seem to be designed to be navigated byusing their hubs. Acknowledgment eferences [1] L. Lovász. 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