Eigenvector-based identification of bipartite subgraphs
EEigenvector-based identification of bipartite subgraphs
Debdas Paul a, ∗ , Dragan Stevanović b a Institute for Systems Theory and Automatic Control, Pfaffenwaldring 9, 70569 Stuttgart, Germany b Mathematical Institute of the Serbian Academy of Sciences and Arts, Knez Mihailova 36, 11001 Belgrade, Serbia
Abstract
We report our experiments in identifying large bipartite subgraphs of simple connected graphs which arebased on the sign pattern of eigenvectors belonging to the extremal eigenvalues of different graph matrices:adjacency, signless Laplacian, Laplacian, and normalized Laplacian matrix. We compare the performanceof these methods to a ‘local switching’ algorithm based on the Erdös’ bound that each graph contains abipartite subgraph with at least half of its edges. Experiments with one scale-free and three random graphmodels, which cover a wide range of real-world networks, show that the methods based on the eigenvectorsof the normalized Laplacian and the adjacency matrix yield slightly better, but comparable results to thelocal switching algorithm. We also formulate two edge bipartivity indices based on the former eigenvectors,and observe that the method of iterative removal of edges with maximum bipartivity index until one obtainsa bipartite subgraph, yields comparable results to the local switching algorithm, and significantly betterresults than an analogous method that employs the edge bipartivity index of Estrada and Gomez-Gardeñes.
Keywords:
Bipartite subgraphs, Eigenvectors, Complex networks
1. Introduction
A graph G ( V, E ) with a set of vertices V and a set of edges E is bipartite if there exists a partition V = X ∪ Y , X ∩ Y = ∅ such that every edge e ∈ E has one end in X and another in Y . It is a classical resultthat a graph is bipartite if and only if it does not contain a cycle of odd length as a subgraph (Asratianet al., 1998).Bipartite graphs have many important applications in various fields of science and technology. Forexample, in modern coding theory, bipartite graphs such as Factor graphs and Tanner graphs are used todecode codewords received from the channel (Moon, 2005). It has also been found that the underlyingstructure of many complex networks (Watts and Strogatz, 1998; Barabási and Albert, 1999; Guillaume andLatapy, 2004, 2006), like biological networks (Pavlopoulos et al., 2011, 2018; Platig et al., 2016; Baker et al.,2014; Moon et al., 2016), social networks (Newman et al., 2002; Schweitzer et al., 2009), and technologicalnetworks is bipartite in nature. In the emerging field of human-robot collaborative system design (Roy andEdan, 2018), bipartite graphs are used to improve the temporal coordination in human-robot teamwork(Chao and Thomaz, 2012). For a more comprehensive review of various applications of bipartite graphs,interested readers are referred to Holme et al. (2003). Recently bipartivity, a quantitative descriptor ofbipartiteness of a graph, has been shown to be inversely correlated with the efficiency of airline transportationin Europe (Estrada and Gómez-Gardeñes, 2016).Here we propose simple eigenvector-based methods aimed at identifying large bipartite subgraphs. Theproblem of identifying the largest bipartite subgraph of a graph is an instance of the MAX-CUT problemin which each edge has the same weight (Michael and David, 1979). From the computability point of view,the MAX-CUT problem is NP-hard (Michael and David, 1979; Goemans and Williamson, 1994) and there ∗ Corresponding author
Email addresses: [email protected] (Debdas Paul), [email protected] (Dragan Stevanović)
Preprint submitted to Elsevier June 4, 2018 a r X i v : . [ m a t h . SP ] J un xists several approximation algorithms whose approximation ratio ranges from . (Mitzenmacher andUpfal, 2005; Motwani, 1995) to . (Goemans and Williamson, 1994) up to . (if P = NP) (Håstad,2001). The simplest approximation algorithm with an approximation ratio . is based on a theorem dueto Erdös (Erdös, 1965), which shows that any graph G ( V, E ) contains a bipartite subgraph with at least | E | edges. This lower bound was later improved by Edwards to | E | + | V |− (Edwards, 1973). We referto the method used in the proof of the Erdös’ result as a local switching algorithm and implemented it ina programmatic manner for comparison purposes. Previously, Bylka et al. (1999) have introduced a localswitching algorithm that guarantees the bound due to Edwards.Motivation to consider eigenvector-based methods comes from a classical result in spectral graph theorywhich claims that a connected graph is bipartite if and only if for the spectral radius λ of the adjacencymatrix A of G , − λ is also an eigenvalue of A , in which case the eigenvector of − λ is obtained from theprincipal eigenvector by changing signs of the components in one part of the bipartition (Sachs, 1964). Asimilar result holds for the normalized Laplacian matrix L = D − / LD / , where D is the diagonal matrixof vertex degrees, which claims that a connected graph is bipartite if and only if the largest eigenvalue of L isequal to 2 in which case the components of the eigenvector of 2 have equal value in one part and the oppositevalue in the other part of the bipartition (Chung, 1996). It is thus natural to expect that, in cases when agraph is close to being bipartite, the sign patterns of the eigenvector corresponding to the smallest eigenvalueof A and the eigenvector corresponding to the largest eigenvalue of L will produce bipartitions of the vertexset containing most edges of the graph. The relation between bipartiteness and the spectral properties of A had been further extended: if B is an essentially non-negative symmetric matrix representing a connected,bipartite graph G with bipartition X ∪ Y , then there exists a non-zero eigenvector z that belongs to theminimum eigenvalue of B such that z i > for i ∈ X and z j < for j ∈ Y (Roth, 1989). In addition to A ,the signless Laplacian matrix Q = D + A is also essentially non-negative symmetric matrix, which partiallyjustifies the use of the sign pattern of the eigenvector of the smallest eigenvalue of Q as well. Further, it isknown that the spectra of Q and the Laplacian matrix L = D − A coincide for a bipartite graph (Groneet al., 1990, prop. 2.2). If G is a connected bipartite graph with bipartition X ∪ Y , let R be the diagonalmatrix indexed by vertices of G such that r j,j = 1 if j ∈ X and r j,j = − if j ∈ Y . Observe that R = R − .It is straightforward to see that R − LR = Q , which yields a relationship between the eigenvectors of Q and L : Qz = λz ⇐⇒ L ( Rz ) = λ ( Rz ) . The eigenvector of the smallest eigenvalue of L is the all-one vector so the eigenvector of the smallest eigenvalue of Q is R ; note that R has entries of one sign on S and theopposite sign on T . The eigenvector of the largest eigenvalue of Q is a positive Perron vector v , so that theeigenvector of the largest eigenvalue of L is Rv ; note that Rv has entries of one sign on S and the oppositesign on T , which partially justifies the use of the sign pattern of the eigenvector of the largest eigenvalue of L as well.To start with, we compare methods based on these particular eigenvectors to the local switching algorithmfor three random graph models and one scale-free graph model, which cover a wide range of real-worldnetworks. We observe that in these cases, methods based on the eigenvectors of A and L yield slightlybetter, but comparable results to the local switching algorithm.Further, Estrada and Rodríguez-Velázquez provided in (Estrada and Rodríguez-Velázquez, 2005) a mea-sure of edge bipartivity that quantifies the contribution of edges toward bipartiteness based on the ratioof the number of closed even walks and the number of all closed walks present in a graph. The measurewas further improved by Estrada and Gomez-Gardeñes (Estrada and Gómez-Gardeñes, 2016). We defineanalogous measures for edge bipartivity based on the eigenvectors that belong to the smallest eigenvalueof A and the largest eigenvalues of L , respectively, and then compare the performance of using these twomeasures in identifying large bipartite subgraphs. Simulation results show that these eigenvector-basedmeasures outperform the analogous method that uses the Estrada and Gomez-Gardeñes’ edge bipartivity.
2. Methods
We describe in this section algorithmic methods for identification of bipartite subgraphs. Notation usedhere and in subsequent sections is given in Table 1. 2 able 1. Notation
Notation Meaning G ( V, E ) or G A simple, connected, unweighted, nonbipartite graph(unless explicitly stated otherwise) with vertex set V and edge set EA , L , Q , L , and D Adjacency, Laplacian, signless Laplacian, and nor-malized Laplacian, and degree matrix of G , respec-tively | E bipart ( G ) | Number of edges remaining when edge-deleted sub-graph of G becomes bipartite | E ( G ) | Number of edges in Gr b | E bipart ( G ) || E ( G ) | | E ext u ( G ) | Number of neighbors of vertex u in the other part ofthe bipartition | E int u ( G ) | Number of neighbors of vertex u in its part of thebipartition { λ Mi } , λ Mi The set of eigenvalues, and the i th largest eigenvalueof matrix Mν λ Mi Eigenvector of the eigenvalue λ Mi of matrix Mν λ Mi k The k th entry of the eigenvector ν λ Mi of MX i −→ Y Movement of vertex i from the set X to Y E-R, W-S, RG, and B-A Erdös-Rényi (Erdös and Rényi, 1959, 1960; Newman,2003), Watts-Strogatz (Watts and Strogatz, 1998),Random Geometric (Gilbert, 1961; Penrose, 2003;Dall and Christensen, 2002; Pržulj et al., 2004), andBarabasi-Albert graph model(Barabási and Albert,1999) respectively 3 .1. Local switching algorithm
The local switching algorithm proceeds by first randomly permuting the vertices and then partitioningthem into sets X and Y ( X ∪ Y = V , X ∩ Y = ∅ ) with sizes j | V | k and l | V | m , respectively. Afterwards, a vertex u is moved to another part if | E int u ( G ) | > | E ext u ( G ) | , or if ×| E int u ( G ) | > | E u ( G ) | since | E int u ( G ) | + | E ext u ( G ) | = | E u ( G ) | . Vertex u is then marked as moved and not considered for movement in subsequent iterations. Theprocess continues alternately between the two parts until no more movements of vertices are possible. Afterthe termination of the process, the ratio r b is calculated. Algorithm 1 summarizes the pseudo-code for thelocal switching algorithm. Remark 2.0.1.
The ratio r b depends on the initial partition. The effect of the initial partition on r b isdemonstrated via Example 2.1. In order to alleviate the problem, the maximum value of r b is taken over anumber of different partitions. Algorithm 1: local switching algorithm
Input : G = ( V, E ) Result : r b Initialize : V moved = ∅ ; Partition randomly X ∪ Y = V ∧ X ∩ Y = ∅ ; Choose randomly P = X, Q = Y or P = Y, Q = X ; T Movement = 0 ; T noMovement = 1 repeat for all u ∈ P do Compute E Xu and E Yu ; if E Pu > E Vu ∧ u / ∈ V moved then Q ← u ; V moved ← u ; T Movement = 1 ; break; end end if T Movement = 0 then T noMovement = T noMovement + 1 ; else T Movement = 0 ; T noMovement = 1 ; end if P = X then P = Y ; Q = X ; else P = X ; Q = Y ; end until T noMovement > ; 4 xample 2.1. Consider a simple, undirected, and connected graph G ( V, E ) with V = { , , , , } and E = { (1 , , (1 , , (1 , , (3 , , (3 , , (4 , } . Consider the following initial partition of V : X = { , } and Y = { , , } . Initially E X = 1 , E X = 1 , E Y = 3 , E Y = 2 , E Y = 2 , while E V = 3 , E V = 1 , E V = 3 , E V = 2 , E V = 2 . If we start with P = X , the process continues as follows: As E Xi > E Vi for i = 2 , therefore X −→ Y , X = X \{ } and Y = Y ∪ { } . Now E Yi > E Vi is satisfied for vertices , and , so that any vertex among them is suitable formovement. Let us pick vertex : Y −→ X , Y = Y \{ } and X = X ∪ { } . In the part X = { , } , no movement of vertices from X to Y is possible as is already been movedand does not satisfy E X > E V . Therefore T noMovement = 1 . In the part Y = { , , } , vertex cannot move, while and do not satisfy E Yi > E Vi . Hence T noMovement becomes 2 and in the next iteration the algorithm terminates.The resulting bipartite subgraph is determined by the bipartition X = { , } and Y = { , , } , which gives r b = . Note that this ratio could be improved to , if the initial partition was chosen so that it ends with X = { , , } and Y = { , } .2.2. Eigenvector-based methods In these methods, the initial partition is determined from the sign pattern of nonzero entries of ν λ iM ,where i corresponds to the smallest eigenvalue for M = A and M = Q , and to the largest eigenvalue for M = L and M = L , while vertices with zero entries are randomly distributed to one of the parts. Afterwards,the movement routine of the local switching algorithm is applied in order to further increase the number ofedges in the resulting bipartite subgraph. Estrada and Rodríguez-Velázquez (Estrada and Rodríguez-Velázquez, 2005) quantified the degree ofbipartivity of a complex network based on the ratio between the numbers of even and all closed walks: β ( G ) = P Vj =1 cosh( λ Aj ) P Vj =1 e λ Aj . (1)The denominator above is the Estrada index (Estrada, 2000; de la Peña et al., 2007) which, through theexpansion e x = P k ≥ x k k ! , results in the weighted series of all closed walks: X j e λ Aj = X k ≥ Tr( A k ) k ! . Similarly, the numerator represents equally weighted series of all even walks: X j cosh( λ Aj ) = X k ≥ Tr( A k )(2 k )! . It holds that < β ( G ) ≤ . If a graph is bipartite, it does not contain odd closed walks, so that β ( G ) = 1 in such case. The lower bound is reached in the limit n → ∞ for the complete graph G = K n .5strada and Rodríguez-Velázquez (Estrada and Gómez-Gardeñes, 2016) improved the lower bound tobecome 0 by introducing a new measure of bipartivity: β ( G ) new = P Vj =1 e − λ Aj P Vj =1 e λ Aj , opening up the possibility to identify bipartite subgraphs with more edges. Nevertheless, both measures failto distinguish among graphs cospectral with respect to A (see Section 4 for discussion of an example). Inthis work, we take into account the improved measure β ( G ) new .The measure β ( G ) new is monotone in the sense that if addition of an edge e increases the proportion ofodd closed walks, then β ( G ) new ≥ β ( G + e ) new . Hence contribution of a particular edge e toward the bipartiteness of G may be given by β ( e ) = 1 − [ β ( G − e ) new − β ( G ) new ] . (2)A bipartite subgraph of G may now be constructed by repeatedly removing an edge minimizing β ( e ) untilthe graph becomes bipartite, as shown in Algorithm 2. Algorithm 2:
Identification of bipartite subgraphs using β ( e ) . Input : G = ( V, E ) Result : r b while do if G is bipartite then compute r b ; exit; else e ← arg min e ∈ E ( G ) β ( e ) ; G ← G − e end end Inspired by the edge bipartivity measure of Estrada and Rodríguez-Velázquez, we propose two newmeasures, denoted as Φ A ( e ) and Φ L ( e ) , based on the smallest eigenvector of A and the largest eigenvectorof L , respectively. For an edge e with the end vertices i and j , Φ A ( e ) is defined as: Φ A ( e ) = ν λ An i ν λ An j ν λ A i ν λ A j + | ν λ An i ν λ An j | , (3)where, as we recall, λ A > λ A ≥ · · · ≥ λ An . Since G is assumed to be connected, by the Perron-Frobeniustheorem (Perron, 1907; Pillai et al., 2005), λ A is a single eigenvalue with a positive eigenvector ν λ A i > for i ∈ V . Addition of the absolute value of the term ν λ An i ν λ An j in the denominator makes the value Φ A ( e ) bounded as − < Φ A ( e ) < . In case of bipartite graph, each edge e has Φ A ( e ) = − , and edges thatso-to-say stay in the way of bipartiteness are those with Φ A ( e ) > . Greedy approach then assumes thatiterative removal of an edge with the maximum value of Φ A ( e ) will make edge-deleted subgraph bipartiteas quickly as possible.The measure Φ L ( e ) may be defined similarly: Φ L ( e ) = ν λ L i ν λ L j ν λ L n i ν λ L n j , (4)6here λ L > λ L ≥ · · · ≥ λ L n . However as the eigenvector of the smallest eigenvalue of L is the all-one vector , Eq. 4 reduces simply to: Φ L ( e ) = ν λ L i ν λ L j . (5)Since | ν λ L i ν λ L j | ≤ (cid:18) ν λ L i (cid:19) + (cid:18) ν λ L j (cid:19) , the value Φ L ( e ) is bounded as well: assuming that max i ν λ L i = 1 we getthat − ≤ Φ L ( e ) ≤ .Similarly as in the case of β ( e ) , bipartite subgraphs of G may be constructed by either repeatedlyremoving an edge that maximizes the value of Φ A ( e ) or repeatedly removing an edge that maximizes thevalue of Φ L ( e ) until the graph becomes bipartite. Pseudo code for the method based on Φ A ( e ) is shown inAlgorithm 3. The method for Φ L ( e ) is obtained by replacing A by L . Algorithm 3:
Identification of bipartite subgraphs using Φ A ( e ) . Input : G = ( V, E ) Result : r b while do if G is bipartite then compute r b ; exit; else e ← arg max e ∈ E ( G ) ν λAni ν λAnj ν λA i ν λA j + | ν λAni ν λAnj | ; G ← ( G − e ) end end For each of the four graph models: E-R, W-S, RG, and B-A, different graphs with vertices aregenerated by uniformly sampling the respective parameter in the range specified in Table 2, and the ratio r b is calculated. To minimize the effect of random partitioning of vertices in the local switching algorithm, themaximum value of r b over different random permutations (of vertices) are considered. For W-S graphmodel, the range is chosen such that the resulting graph will either be a regular graph (when ψ is close to0) or a graph with the small-world property because as ψ approaches to 1, the graph tends to become anE-R type graph.Graph Models Parameters Sampled range [initial value finalvalue]E-R p [0.2 1]W-S ( ψ , k ) ([0 0.3], 8)RG ( l , r) (2, [0.5 1])B-A m [1 10] Table 2.
Parameter values for the graph models. For an E-A type graph, p is the probability of attachment. For a W-S typegraph, ψ represents the probability of rewiring and k is the mean degree. For a random geometric graph, l represents the k . k l norm, while r is the threshold for neighbor joining. For a B-A type graph, m is the number of edges to attach in every step. .5. Software routines All the algorithms described in this paper are implemented in Python. Generations of different graphmodels are carried out using the
NetworkX ( https://networkx.github.io/ ) package. For the purposeof reproducibility, codes are made available at https://github.com/DebCompBio/Eigenvector_based_bipartite_graphs .
3. Results
Figures 1, 2, 3, and 4 summarize the distribution of the r b values for the local switching and eigenvector-based methods on E-R, W-S, RG, and B-A graph models (see Appendix A for a brief overview) respectively.The distribution is obtained for different graphs corresponding to each graph model. Initially, weobserve that for E-R and RG graph models and to some extent for W-S graph model as well, the β ( e ) new based method yields positive values of c pdf ( r b ) much before other methods. This is undesirable as it meansthat there are graph instances for which β ( e ) new yields lower value of r b ; hence, smaller bipartite subgraphs,whereas other methods yield a higher value of r b ; hence, larger bipartite subgraphs. Following the same lineof argument, left-skewed distributions are also undesirable as seen for the method based on the L − matrixfor E-R (Fig. 1) and RG (Fig. 3) graph models.In order to obtain a finer distinction between the methods, we proceed toward obtaining the empiricalcumulative distribution (eCDF) function of the r b values for each graph model. The eCDF tells us about theprobability that a randomly chosen r b value is less than a specified value. In this context, a higher value ofthe cumulative distribution function is undesirable as it indicates higher probability of bipartite subgraphswith lesser number of edges obtained using the respective method. Figure 5 and 6 represent the empiricalcumulative distribution function (eCDF) of the r b values without and with β ( e ) new , Φ A ( e ) , and Φ L ( e ) basedmethods respectively.An even finer quantitative comparison between the methods in discussion has been made based on thefraction of graph instances for which a particular method yields higher and the same r b values to thatof another method. The latter indicates the degree of similarity between two methods. Figures 7 and 8show the heat-maps based on the fractions of graph instances representing superiority ( left with cool colormap) and similarity ( right with copper color map) between methods. For example, consider the followingtwo methods: M i and M j , to obtain bipartite subgraphs. Additionally, consider Frac ( r bM i>j ( i = j ) ) as thefraction of graph instances for which r bM i is higher than (same as) r bM j . Now, the higher is the value ofFrac ( r bM i>j ( i = j ) ) , as represented by the value of the cell ( i, j ) in the heat-map, the better (more similar) isthe method M i compare to (to) the method M j . 8 ig. 1. Distribution of the r b values for the E-R graph model. ig. 2. Distribution of the r b values for the B-A graph model. ig. 3. Distribution of the r b values for the RG graph model. ig. 4. Distribution of the r b values for the W-S graph model with mean degree 8. .5 0.6 0.7 0.8 0.900.20.40.60.81 (a) (b) (c) (d) Fig. 5. Empirical cumulative distribution of the r b values for local switching algorithm, A, Q, L , and L -matrixbased methods. Comparing local switching algorithm,
A, Q, L , and L -matrix based methods for (a) E-R (b) B-A (c) RG (d)W-S graph models by constructing the empirical cumulative distributions corresponding to Figs 1, 2, 3, and 4 respectively. a) (b)(c) (d) Fig. 6. Empirical cumulative distribution of the r b values for (a) E-R (b) B-A (c) RG (d) W-S graph models as inFig. 5 but including β ( e ) new , Φ A ( e ) , and Φ L ( e ) based methods. - R m o d e l (a) (b) B - A m o d e l (c) (d) Fig. 7. Heat-maps to compare different methods for E-R and B-A graph models. (a)-(b) For E-R model, anentry ( i, j ) in the heat-map matrix represents fraction of graph instances out of graphs when r bM i > r bM j and r bM i = r bM j respectively, where r bM i and r bM j corresponds to the value of r b for a method M in the i th row and j th column respectively.(c)-(d) represent the same as (a)-(b) but for the B-A model. G m o d e l (a) (b) W - S m o d e l ( w i t h m e a nd e g r ee k = ) (c) (d) Fig. 8. Heat-maps to compare different methods for RG and W-S graph models.
Same set of figures as in Fig. 7but for RG (a)-(b), and for W-S (with mean degree k = 8 ) (c)-(d) graph models. . Discussion In this work, we propose a set of eigenvector-based methods to identify bipartite subgraphs within asimple, undirected, connected, and non-trivial graph. At first, we compare the method based on sign-basedpartitioning to that of a local switching algorithm (see Algorithm 1) based on Erdös’ bound. Subsequently,analogous to the notion of edge bipartivity index of Estrada and Gomez-Gardeñes denoted here by β ( e ) new ,we propose two new measures for edge bipartivity index, Φ A ( e ) and Φ L ( e ) (see Eqs 3 and 4 respectively),based on the eigenvectors of A and L matrices relying on their superior performances as evident from Figs (1- 4) and Fig. 5. Experimental results over four different graph models: E-R, B-A, RG, and W-S, revealthat the local switching algorithm along with the Φ A ( e ) and the Φ L ( e ) based methods outperform the rest.In fact, for graph models like RG and W-S, the local switching algorithm has been found to exhibit betterperformance than rest of the methods (see Figs 5(c) and 5(d)).A finer distinction between the methods are illustrated in Figs 7 and 8 through heat-maps. The heat-maps quantitatively describe the degree of superiority ( left ) of a particular method over others as well as thedegree of similarity ( right ) of that method with others. For example, in Fig. 7(a), for the E-R graph model,the cells in the row corresponding to the L -matrix mostly belongs to the lighter region of the spectrumin the color bar. It indicates that, out of graph instances, the L -matrix based method mostly yieldslower r b values than the other methods for a significantly higher number (the number is represented asthe fraction of the total number of graph instances) of graph instances and therefore towards its worstperformance. It should be noted that Fig. 6(a) is in agreement with this observation as well. On theother hand, the respective rows corresponding to the local switching algorithm and the Φ L ( e ) -based methodcontain comparatively higher number of cells that belong to the darker region of the spectrum indicatingtowards their superior performances. Next, while comparing the local switching algorithm and Φ L ( e ) -basedmethod, we observe that for around . % of graph instances, the later yields higher r b values whereasfor around . % of graph instances, the former yields higher r b values; hence, performs slightly better.Next, according to the similarity matrix (Fig. 7(b)) for the E-R graph model, the local switching algorithmexhibits higher similarity with A -matrix, L -matrix, Φ A ( e ) , and Φ L ( e ) based methods. In addition, oneinteresting observation is that the β ( e ) new -based method neither exhibit a good performance nor exhibitgood similarities to other methods. A concrete reasoning for this behavior is still lacking and can be posedas an open problem for the future.In summary, we observe that the local switching algorithm outperforms rest of the methods for all thegraph models. Moreover, as the runtime complexity of the switching algorithm is linearly dependent on thenumber of vertices, it is time-efficient for denser graphs as well unlike the other methods which are primarilyedge-based. On the contrary, a potential drawback of the switching algorithm is that the update of thestatus of a vertex is based on its neighbors only, and because of this local updates the performance dependson the nature of the initial partition as illustrated in the Example 2.1. Therefore, one has to consider themaximum value over a number of different partition pairs. The optimum number of different partition pairsrequired for an arbitrary graph is still unknown; hence, can be posed as another open problem for the futureas well. In this scenario, methods based on Φ A ( e ) and Φ L ( e ) can be suggested as an alternative to the localswitching algorithm that do not suffer from the problem based on the nature of initial partition, and inaddition both the measures yield comparable results for all the graph models.Although we propose two new edge bipartivity indices based on eigenvectors, we are unable to providea measure for overall graph bipartivity unlike β ( G ) new of Estrada and Gomez-Gardeñes. In this context, itis worth noting that construction wise β ( G ) new relies on the { λ Ai } only, and therefore unable to distinguishbetween two co-spectral mates (with respect to A ). For example, consider two co-spectral mates (non-isomorphic with same number of edges): G and G as depicted in the Figure 9. Additionally, considerthat for G it takes at least e edges to be removed in order to make it bipartite. For G , it takes e edgeswhere e > e . Therefore G is closer than G to become a bipartite graph which should be reflected as β ( G ) new > β ( G ) new . But, the values of β new ( G ) will remain same for both the graphs G and G .17 (a) G (b) G Fig. 9. Connected cospectral mates with respect to
A G and G have the same spectrum and number of edges. G requires at least two edges (say (1 − and (5 − , it can be some other pairs) to be removed to become bipartite while G requires only one: (3 − Recall β ( G ) new = P Vj =1 e − λAj P Vj =1 e λAj . As { λ Ai } for G and G are same: − . − . − . . . . , the values of β ( G ) new willbe same for both the graphs. Now, while applying Algorithm 2 on both the graphs, G requires at leasttwo edges to be removed before it becomes bipartite, while G requires only one edge: (3 − to be removed.The resulting bipartition will be X = { , , , } , Y = { , } . The same partition can be obtained from G after removing edges (1 − and (5 − . Therefore, G is more bipartite or nearer to bipartite than G but β ( G ) new is unable to distinguish that. The problem might be alleviated through a measure based oneigenvectors instead of eigenvalues but it depends on further properties whether they will also have the sameeigenvector belonging to the minimal eigenvalue. This is possible without graphs being isomorphic but atthe moment this is a problem worth to be addressed in future.Overall, our study introduces eigenvector-based alternatives to the solution of the MAX-CUT problemwhich is, to the best of our knowledge, has not been addressed yet. Additionally, the study opens up newpossibilities in addressing similar kind of problems based on the spectral graph theory. Appendix A. Graph models
Erdös-Rényi graph model
Erdös and Rényi in their 1959’s paper (Erdös and Rényi, 1959) introduced the concept of a random graphmodel denoted by G ( V, E ) with | V | labeled vertices and | E | edges. The graph is random in the sense that theconfiguration is chosen randomly from one of the (cid:0) ( | V | ) | E | (cid:1) configurations with equal probability. In anothervariant of the same graph model, denoted by G ( V, p ) , the nodes are connected randomly with probability p (Erdös and Rényi, 1960). The connected edges are independent from each other. In this study, we use the G ( V, p ) variant. For G ( V, p ) to be almost surely connected, the condition is p > (1+ (cid:15) ) ∗ ln | V || V | , for < (cid:15) < . Random Geometric graph model
The motivation for a Random Geometric graph lies in communication between two points in a d -dimensional space based on their closeness . In real life context, the points may be stations distributedacross the whole country, connected nerve cells inside the brain cortex etc (Penrose, 2003). To put thedescription in a formal way, given a vertex set V ∈ R d and a norm k·k on R d , where R d is a d -dimensional18pace on the set of real numbers R , the vertex pairs { ( v , v ) : v , v ∈ V } are connected with k v − v k ≤ r ,where r > . Unlike the E-R graph model, the RG graph model does not follow the property of independence on edges, which means that a property defined on the pair of vertices of a RG graph is transitive in nature.This makes the RG graph model more realistic than that of a E-R graph model (Penrose, 2003). Watts-Strogatz graph model
The Watts-Strogatz graph model addresses the first of the two shortcomings of the E-R graph models: • Low clustering coefficient due to a constant, random and independent probability of connection betweentwo nodes. • Inability to account for power-law degree distribution which seems relevant for many biological, social,and technological networks (Watts and Strogatz, 1998).A W-S graph model has a small average path length and high clustering coefficient. The construction of aW-S graph follows the two simple steps:S-1:
Creation of ring lattice : A ring lattice with |V| nodes with mean degree k is created where eachnode is connected to k/ nearest neighbors.S-2: Rewiring of the target node : For each edge in the graph, the target node is rewired with probability ψ Barabasi-Albert graph model
The main concept of the Barabasi-Albert graph model is to make a node richer (in links) if it is alreadyrich - a process called preferential attachment (Barabási and Albert, 1999). A node which is more connectedis more likely to get new connections. This kind of preferential attachment alleviates the problem of nothaving a power law degree distribution which is more prevalent in real-world networks.A shortcoming of the B-A model is that it fails to account for high-level clustering in real-world networks.
Declarations
Funding
This work is a part of a research collaboration between DP and DS initiated during a research visit ofDP for three months at the University of Primorska, Koper, Slovenia. The reserach visit was funded by theMinistry of Education, Science, Culture and Sport of the Republic of Slovenia under bilateral mobility grantfor the foreign nationals for 2010-11 (CMEPIUS).
Author’s contributions
DP performed all the simulations. DS conceived the idea and and designed the study. DP wrote themain part of the manuscript. All authors discussed the results and implications and commented on themanuscript at all stages of the project. All authors read and approved the final manuscript.
Declarations of interest
None. 19 eferences