Influence of assortativity and degree-preserving rewiring on the spectra of networks

Transcription

1 Eur. Phys. J. B 76, DOI: 1.114/epjb/e1-19-x Regular Article THE EUROPEA PHYSICAL JOURAL B Influence of assortativity and degree-preserving rewiring on the spectra of networks P. Van Mieghem 1,H.Wang 1,a,X.Ge,S.Tang 1,andF.A.Kuipers 1 1 Delft University of Technology, Faculty of Electrical Engineering, Mathematics and Computer Science, 68 CD, Delft, The etherlands ortheastern University, College of Information Science and Engineering, 14, Shenyang, China Received 4 May 1 Published online July 1 c EDP Sciences, Società Italiana di Fisica, Springer-Verlag 1 Abstract. ewman s measure for disassortativity, the linear degree correlation coefficient ρ D,isreformulated in terms of the total number k of walks in the graph with k hops. This reformulation allows us to derive a new formula from which a degree-preserving rewiring algorithm is deduced, that, in each rewiring step, either increases or decreases ρ D conform our desired objective. Spectral metrics eigenvalues of graph-related matrices, especially, the largest eigenvalue λ 1 of the adjacency matrix and the algebraic connectivity µ 1 second-smallest eigenvalue of the Laplacian are powerful characterizers of dynamic processes on networks such as virus spreading and synchronization processes. We present various lower bounds for the largest eigenvalue λ 1 of the adjacency matrix and we show, apart from some classes of graphs such as regular graphs or bipartite graphs, that the lower bounds for λ 1 increase with ρ D.Anewupperbound for the algebraic connectivity µ 1 decreases with ρ D.Applyingthedegree-preservingrewiringalgorithm to various real-world networks illustrates that a assortative degree-preserving rewiring increases λ 1,but decreases µ 1, evenleadingtodisconnectivityofthenetworksinmanydisjointclustersandthatb disassortative degree-preserving rewiring decreases λ 1,butincreasesthealgebraicconnectivity,atleastin the initial rewirings. 1 Introduction Mixing in complex networks [1] refers to the tendency of network nodes to connect preferentially to other nodes with either similar or opposite properties. Mixing is computed via the correlations between the properties, such as the degree, of nodes in a network. etworks, where highdegree nodes preferentially connect to other high-degree nodes, are called assortative, whereas networks, where high-degree nodes connect to low-degree nodes, are called disassortative. The degree correlation is widely studied after it was realized that the degree distribution alone provides a far from sufficient characterization of complex networks. etworks with a same degree distribution may still differ significantly in various topological features as we will also show in this paper. A stronger confinement than a same degree distribution are networks with a same apartfromapossiblenoderelabelling degreevector d T = [d 1,d,...,d ], where d j is the degree of node j. Also same degree vector networks may possess widely different topological properties. Consequently, degree correlation related investigations have been performed along various axes: a models to generate networks with given a h.wang@tudelft.nl degree correlation have been developed [,3]; b the effect of degree correlation on topological properties is studied in [4,5], and c the influence of degree correlation in dynamic processes on networks such as the epidemic spreading [6] andonpercolationphenomena[7] havebeen targeted. Relations between degree correlation and other topological or dynamic features are examined experimentally [5] or in a specific network model [6,7]. Local assortativity is proposed in [8]. Analytic insight in degree correlations in an arbitrary network remains far from well understood. In this work, we explore the influence of the degree correlation on spectra of networks, which well capture both topological properties [9] and dynamic processes [1] onthenetwork. Let G be a graph or a network and let denote the set of = nodes and L the set of L = L links. An undirected graph G can be represented by an symmetric adjacency matrix A, consistingofelementsa ij that are either one or zero depending on whether there is alinkbetweennodei and j. Theadjacencyspectrumof agraphisthesetofeigenvaluesoftheadjacencymatrix, λ λ 1... λ 1,whereλ 1 is called the spectral radius. The Laplacian matrix of G with nodes is an symmetric matrix Q = A, where =diagd i andd i is the degree of node i.thesetof eigenvalues of

2 644 The European Physical Journal B the Laplacian matrix µ = µ 1... µ 1 is called the Laplacian spectrum of G. Thetheoryofthespectra of graphs provides many beautiful results [9]. Recently, modern network theory has been integrated with dynamic system s theory to understand how the network topology can predict dynamic processes such as synchronization or virus spread taking place on networks. The SIS susceptible-infected-susceptible virus spreading [11] and the Kuramoto type of synchronization process of coupled oscillators [1] havebeencharacterizedonagiven,but general, network topology. Both these dynamic and nonlinear processes feature a phase transition that specifies the onset of a remaining fraction of infected nodes and of locked oscillators, respectively. The more curious aspect is that each of the phase transitions in these different processes occurs at an effective spreading rate τ c and coupling strength g c respectively, that is proportional to 1/λ 1.Inaddition,dynamicprocessesongraphsconverge towards their steady-state, in most cases, exponentially fast in time and with a time-constant related to the spectral gap difference between λ 1 and λ. Connectivity and the number of disjoint clusters in G follows from the algebraic connectivity second-smallest eigenvalue of the Laplacian and the number of smallest Laplacian eigenvalues that are zero [9]. Hence, these spectral metrics eigenvalues of graph-related matrices, especially, the largest eigenvalue λ 1 of the adjacency matrix and the algebraic connectivity µ 1,arepowerfulcharacterizersofdynamic processes on graphs. The present paper starts with a reformulation of the linear degree correlation coefficient ρ D, introduced by ewman [1], equation 1, in terms of the total number k of walks in the graph with k hops. This reformulation allows us to derive a new formula from which a degree-preserving rewiring algorithm is deduced, that, in each rewiring step, either increasesordecreasesρ D conform the desired objective. Thus, we construct a sequence of degree-preserving rewirings that monotonously in- or decreases the linear degree correlationcoefficientρ D,or equivalently, that increases the assortativity or disassortativity of G. ext,wepresentvariouslowerboundsfor λ 1 and we show, apart from some classes of graphs such as regular graphs or bipartite graphs, that lower bounds for λ 1 increase with ρ D.Wederiveanupperboundforthealgebraic connectivity µ 1 that decreases with ρ D.Then, as an example, we apply the degree-preserving rewiring algorithm to a real-world network, the USA air transportation network, and compute in each rewiring the entire adjacency and Laplacian spectrum. A major finding, also observed in other real-world networks that we have rewired, is that, increasing λ 1 by increasing the assortativity, relatively rapidly leads to disconnectivity, while increasing disassortativity seems to increase the algebraic connectivity µ 1,thusthetopologicalrobustness. Reformulationofewman sdefinition Here, we study the degree mixing in undirected graphs. Generally, the linear correlation coefficient between two random variables X and Y is defined [13], p. 3 as ρ X, Y = E [XY ] µ Xµ Y σ X σ Y 1 where µ X = E [X] andσ X = Var [X] arethemeanand standard deviation of the random variable X,respectively. ewman [1], equation 1 has expressed the linear degree correlation coefficient of a graph as xy e xy a x b y ρ D = xy σ a σ b where e xy is the fraction of all links that connect the nodes with degree x and y and where a x and b y are the fraction of links that start and end at nodes with degree x and y, satisfying the following three conditions e xy =1,a x = e xy and b y = e xy. xy y x When ρ D >, the graph possesses assortative mixing, a preference of high-degree nodes to connect to other highdegree nodes and, when ρ D <, the graph features disassortative mixing, where high-degree nodes are connected to low-degree nodes. The translation of intothenotationofrandomvariables is presented as follows. Denote by D i and D j the node degree of two connected nodes i and j in an undirected graph with nodes. In fact, we are interested in the degree of nodes at both sides of a link, without taking the link, that we are looking at, into consideration. As ewman [1] pointsout,weneedtoconsiderthenumberof excess links at both sides, hence, the degree D l + = D i 1 and D l = D j 1, where the link l has a start at l + = i and an end at l = j. Thelinearcorrelationcoefficientof those excess degrees is ρ D l +,D l = E [D l +D l ] E [D l +] E [D l ] σ Dl + σ Dl E [D = l + E [D l +] D l E [D l ]] [ E D l + E [D l +] ] E [D l E [D l ] ]. Since D l + E [D l +]=D i E [D i ], subtracting everywhere one link does not change the linear correlation coefficient, provided D i > andsimilarlythatd j >, which is the case if there are no isolated nodes. Removing isolated nodes from the graph does not alter the linear degree correlation coefficient. Hence, we can assume that the graph has no zero-degree nodes. In summary, the linear degree correlation coefficient is ρ D l +,D l =ρ D i,d j = E [D id j ] µ D i E [D i ] µ D i. 3 We now proceed by expressing E [D i D j ], E [D i ]andσ Di in the definition of ρ D l +,D l =ρ D i,d j forundirected

3 P. Van Mieghem et al.: Influence of assortativity and degree-preserving rewiring on the spectra of networks 645 graphs in terms of more appropriate quantities of algebraic graph theory. First, we have that E [D i D j ]= 1 L j=1 d i d j a ij = dt Ad L where d i and d j are the elements in the degree vector d T =[d 1,d,...,d ], and a ij is the element of the symmetric adjacency matrix A, thatexpresses{, 1} connectivity between nodes i and j. Thequadraticformd T Ad can be written in terms of the total number k = u T A k u of walks with k hops see e.g. [9], where u is the all-one vector. Since d = Au, d T Ad equals 3 = u T A 3 u,thetotal number of walks with length equal to 3 hops, which is called the s metric in [4]. The average µ Di and µ Dj are the mean node degree of the two connected nodes i and j, respectively,andnotthemeanofthedegreed of a random node, which equals E [D] = L.Thus, while µ Di = 1 L µ Dj = 1 L = 1 L = 1 L j=1 j=1 d i a ij d i a ij j=1 d i = dt d L d j a ij = µ Di. The variance σ D i =Var[D i ]=E [ D i ] µ Di and E [ Di ] 1 = L j=1 d i a ij = 1 L d 3 i = E [ Dj ]. After substituting all terms into the expression 3 of the linear degree correlation, we obtain, with =L and = d T d,ourreformulationofewman sdefinition in terms of k, ρ D = ρ D i,d j = 3. 4 d 3 i The crucial understanding of disassortativity lies in the total number 3 of walks with 3 hops compared to those with hops,,andonehop, =L. 3Discussionof4 Fiol and Garriga [14] haveshownthatthetotalnumber k = u T A k u of walks of length k is upper bounded by k j=1 d k j with equality only if k and,forallk, onlyifthegraph is regular i.e., d j = r for any node j. Hence, 4 shows that only if the graph is regular, ρ D =1,implyingthat maximum assortativity is only possible in regular graphs 1. Since the variance of the degrees at one side of an arbitrary link σd i = 1 d 3 i 5 the sign of 3 in 4 distinguishesbetweenassortativity ρ D > and disassortativity ρ D <. Using the Laplacian matrix, Fiol and Garriga [14] showthat d 3 i 3 = d i d j = 1 a ij d i d j i j j=1 6 which is the sum over all links of the square of the differences of the degrees at both sides of a link l = i j. Using 6, the degree correlation 4 canberewrittenas ρ D =1 i j d i d j The graph is zero assortative ρ D =if. 7 d 3 i 1 L d i = 3. 8 In Appendix 6, weshowthattheconnectederdős-rényi random graph G p iszero-assortativeforall and link density p > p c,wherep c is the disconnectivity threshold see [13], p Asymptotically for large, the Barabási-Albert power law graph is zero-assortative as shown in [15]. Perfect disassortativity ρ D = 1 in4 implies that = 3 + d 3 i. 9 For a complete bipartite graph K m,n,wehavethat d i d j = mn n m, d 3 i = nm n + m 1 i j and d i = nm n + m otice that the definition 4 isinadequateduetoazero denominator and numerator for a regular graph with degree r because k regular graph = r k. For regular graphs where d 3 i = 3,theperfectdisassortativitycondition9 becomes = 3,whichisequaltothezeroassortativitycondition 8. Of course, ρ regular graph = 1, since all degrees are equal and thus perfectly correlated. The complete bipartite graph K m,n consists of two sets M and with m = M and n = nodes respectively, where each node of one set is connected to all other nodes of the other set. There are no links between nodes of a same set. More properties are deduced in [9].

4 646 The European Physical Journal B such that 7 becomesρ D = 1, provided m n. Hence, any complete bipartite graph K m,n irrespective of its size and structure m, n,exceptfortheregulargraphvariant where m = n isperfectlydisassortative. The perfect disassortativity of complete bipartite graphs is in line with the definition of disassortativity, because each node has only links to nodes of a different set with different properties. evertheless, the fact thatallcompletebipartite graphs K m,n with m n have ρ D = 1, even those with nearly the same degrees m = n ± 1andthusclosetoregular graphs typified by ρ D =1,showsthatassortativity and disassortativity of a graph is not easy to predict. It remains to be shown that the complete bipartite graphs K m,n with m n are the only perfect disassortative class of graphs. There is an interesting relation between the linear degree correlation coefficient ρ D of the graph G and the variance of the degree of a node in the corresponding line graph l G. The line graph l G ofthegraphg,l has as set of nodes the links of G and two nodes in l G are adjacent if and only if they have, as links in G, exactly one node of G in common. The l-th component of the L 1degreevectorinthelinegraphl G see[9] is dlg l = d i + d j, where nodes i and j are connected by link l = i j. ThevarianceofthedegreeD lg of a random node in the line graph equals Var [ ] [ D lg = E D i + D j ] E [D i + D j ] which we rewrite as Var [ D lg ] = E [ D i ] µ Di + E [D i D j ] µ D i. Using 3, we arrive at Var [ ] D lg =1+ρD E [ Di ] µ Di =1+ρ D Var[D i ] 1 =1+ρ D d 3 i. 1 Curiously, the expression1 showsforperfectdisassortative graphs ρ D = 1 that Var [ ] D lg =.Thelatter means that l G isthenaregulargraph,butthisdoesnot imply that the original graph G is regular. Indeed, if G is regular, then l G isalsoregularasfollowsfromthel-th component of the degree vector, d lg l = d i + d j. However, the reverse is not necessarily true: it is possible that l G isregular,whileg is not, as shown above, for complete bipartite graphs K m,n with m n that are not regular. In summary, in both extreme cases ρ D = 1 and ρ D =1,thecorrespondinglinegraphl G isaregular graph. 4Relationbetweengraphspectraandρ D The largest eigenvalue λ 1 of the adjacency matrix A of a graph as well as the algebraic connectivity µ 1,introduced by Fiedler [16], are important characterizers of a graph. Here, we present a new lower bound for λ 1 and upper bound of µ 1 in terms of the linear degree coefficient ρ D. In [9], Chapter 3, we show, for all integers k 1, that λ 1 k 1/k 1/k k from which lim k 1/k k = λ1. We obtain the classical lower bound for k =1, and for k =, λ 1 λ 1 = L = 1 = L = E [D] 11 k=1 1+ d k For k =3andusing4, we obtain λ = 1 ρ D d 3 i Var [D] E [D] The inequality 13 with5 showsthatthelowerbound for the largest eigenvalue λ 1 of the adjacency matrix A is strictly increasing in the linear degree correlation coefficient ρ D except for regular graphs. Given the degree vector d is constant, inequality 13 showsthatthelargest eigenvalue λ 1 is obtained in case we succeed to increase the assortativity of the graph by degree-preserving rewiring, which is discussed in Section 5. Arelatedbound,deducedfromRayleigh sinequality λ 1 yt Ay y T y by choosing the vector y = A m u,whereuis the all-one vector and m is a non-zero integer, is λ 1 ut A m+1 u u T A m u = m+1 m. 14 The case m =1in14, λ 1 3,alreadyappearedasan approximation in [17] ofthelargestadjacencyeigenvalue λ 1,which,againinviewof4, is a perfect linear function of ρ D.Finally,wepresentthenew,optimizedbound, derived in [9], Chapter 3, where λ R 1 15 R =

5 P. Van Mieghem et al.: Influence of assortativity and degree-preserving rewiring on the spectra of networks exact bound 13 bound 14 bound 15 which is an upper bound for the algebraic connectivity µ 1 in terms of the linear correlation coefficient of the degree ρ D.Indegree-preservingrewiring, the fraction in 16, that is always positive, is unchanged and we observe that the upper bound decreases linearly in ρ D. λ Degree-preservingrewiring Fig. 1. Color online The largest eigenvalue λ 1 of the Barabási-Albert power-law graph with = 5 nodes and L = 196 links versus the linear degree correlation coefficient ρ D.Variouslowerboundsareplotted:bound13, bound 15 and bound 14 form =1.Thecorrespondingclassicallower bound 11 is7.84,whilethelowerbound1 is Figure 1 illustrates how the largest eigenvalue λ 1 of the Barabási-Albert power-law graph evolves as a function of the linear degree correlation coefficient ρ D,that can be changed by degree-preserving rewiring. The lower bound 15clearlyoutperformsthelowerbound Especially in degree-preserving rewiring, where, and are constant, the complex looking, but superior formula 15 becomesmanageable,becauseonly 3 changes with ρ D. The Rayleigh principle see e.g. [9,16] provides an upper bound for the second-smallest eigenvalue µ 1 of the Laplacian Q =diagd j A as µ 1 ρ D l L g l+ g l n g n 1 u g u where g n isanynon-constantfunction.letg n =d n, the degree of a node n, then l L µ 1 d l + d l j=1 d j 1 j=1 d j l L = d l + d l. Var [D] After introducing 7, we find for any non-regular graph d 3 i 1 L d i µ 1 1 ρ D j=1 d j 1 16 j=1 d j 3 =1 ρ D E [D] E [ D 3] E [ D ] E [D]Var[D] 17 Especially for strong negative ρ D,wefound veryrarely though that 15 canbeslightlyworsethan1. In degree-preserving rewiring, links in a graph are rewired while maintaining the node degrees unchanged. This means that the degree vector d is constant and, consequently, that = d i, = and do not d i d 3 i change during degree-preserving rewiring, only 3 does, and by 4, also the disassortativity ρ D. Adegree-preservingrewiringchangesonlytheterm i j d i d j in 7, which allows us to understand how adegree-preservingrewiringoperation changes the linear degree correlation ρ D.Eachstepinadegree-preserving random rewiring consists of first randomly selecting two links i j and k l associated with the four nodes i, j, k, l. ext, the links can be rewired either into i k and j l or into i l and j k. Lemma 1. Given a graph in which two links are degreepreservingly rewired. We order the degree of the four involved nodes as d 1 d d 3 d 4.Thetwolinks are associated with the 4 nodes n d1,n d,n d3 and n d4 only in one of the following three ways: a n d1 n d, n d3 n d4, b n d1 n d3, n d n d4, and c n d1 n d4, n d n d3. The corresponding linear degree correlation introduced by these three possibilities obeys ρ a ρ b ρ c. Proof. In these three ways of placing the two links, the degree of each node remains the same. According to definition 7, the linear degree correlation is determined only by ε = i j d i d j.thus,therelativedegreecorrelation difference between a and b is ε a ε b = d 1 d d3 d 4 + d 1 d 3 + d d 4 =d d 3 d 1 d 4 since the rest of the graph remains the same in all three cases. Similarly, ε a ε c =d d 4 d 1 d 3 ε b ε c =d 1 d d 3 d 4. These three inequalities complete the proof. A direct consequence of Lemma 1 is that we can now design a rewiring rule that increases or decreases the linear degree correlation ρ D of a graph. We define degreepreserving assortative random rewiring as follows: randomly select two links associated with four nodes and then

6 648 The European Physical Journal B rewire the two links such that as in a the two nodes with the highest degree and the two lowest-degree nodes are connected. If any of the new links exists before rewiring, discard this step and a new pair of links is randomly selected. Alternatively, we could select two links for which i j d i d j is highest and rewire them such that the highest-degree nodes are connected and the lowest-degree nodes are connected. Similarly, the procedure for degreepreserving disassortative random rewiring is: randomly select two links associated with four nodes and then rewire the two links such that as in c the highest-degree node and the lowest-degree node are connected, while also the remaining two nodes are linked provided the new links do not exist before rewiring. Lemma 1 shows that the degreepreserving assortative disassortative rewiring operations increase decrease the degree correlation of a graph. Degree-preserving rewiring is an interesting tool to modify a graph in which resources of the nodes are constrained. For example, the number of outgoing links in a router [18] aswellasthenumberofflightsatmanyairports per day are almost fixed. Randomdegree-preserving rewiring may be considered as an evolutionary process in nature. 5.1 Algorithmic considerations In this subsection, we concentrate on the following problem: Problem 1. Given a degree vector d consisting of elements, find a not necessarily connected simple 4 graph such that the assortativity ρ D is maximum minimum. Before presenting the solution, we evaluate two intuitive approaches. The first method, used in [19] andcoined the stochastic approach, consists of repeating degreepreserving assortative random rewiring long enough. The stochastic approach stabilizes, after long enough rewiring, to some level ρ D.extwecheckallpossible L pairs of links whether we still can rewire a pair to increase ρ D. If we cannot rewire any pair of links to increase ρ D,we have definitely found a local maximum. However, this local maximum is not guaranteed to be the overall or global maximum. This can be verified by executing the stochastic approach on a same graph a number of times: each realization including the check over all possible L pairs of links does not necessarily achieve the same maxρ D. The second deterministic approach attempts to find an matrix A that maximizes 3 = d T Ad, while maintaining the degree vector d unchanged. Without loss of generality we can assume that the components of the degree vector d are ordered as d 1... d.theelements a ij of A should be determined such that 3 = j=1 a ijd i d j is maximum under the condition that j=1 a ij = d i, a ij = a ji, and a ii = for all i. Let 4 A simple graph is an unweighted, undirected graph containing no self-loops links starting and ending at the same node nor multiple links between the same pair of nodes. us denote the vector c = Ad, which has non-negative components. Recall that d = Au, then 3 = d T c and u T c = c i = d T d is constant because d must be unchanged. Due to the ordering d 1... d, d T c = c id i is maximum if c 1... c.weshouldtherefore shift as much weight as possible of the total c i to the left side of the vector c T.Thegraphconstruction method in [4] thattriestooptimize j=1 a ijd i d j, belongs to the second class. This method ranks all possible links 1 l according to dl +d l from the highest to the lowest resulting in l 1,l,...,l.ext,thegraph is constructed by including sequentially links with increasing index, but links that violate the degree vector, are excluded. Both the stochastic and deterministic approach as deployed by Li et al. [4] are, however, heuristic, while the problem is polynomially solvable. Problem 1 is, in fact, an instance of the maximumweight degree-constrained subgraph problem 5, which is polynomially solvable e.g., see [] and[1]. The degreeconstrained subgraph problem is defined as follows: Definition 1. Degree-constrained subgraph problem: given a graph G, L with nodes and L links, and a degree vector d = d 1,...,d,findasubgraphH, L H, where L H L,andeachnodei has precisely d i neighbors adjacent nodes. The problem instance that we need to solve is to find a maximum-weight degree-constrained subgraph in the complete graph K,whereeachlinkfromanodei to a node j is assigned a weight d i d j.byfindingamaximumweight degree-constrained subgraph H, whereeachnode i has precisely d i neighbors, we obtain a subgraph of the complete graph for which all nodes obey the degree sequence, and for which 3 = j=1 a ijd i d j is maximum corresponding to a graph with highest 3 and hence assortativity. We end the section by considering two additional and related problems. The first problem considers the difference max ρ D min ρ D that may be regarded as a metric of a given degree vector d and that reflects the adaptivity in disassortativity under degree-preserved rewiring. As shown earlier, for some graphs like regular graphs, that difference maxρ D minρ D =, while max ρ D min ρ D. Let A max be the matrix corresponding to max ρ D and A min is the matrix corresponding to min ρ D,thenmaxρ D min ρ D = 1dT A max A mind. 1 d3 i ow, R = A max A min is an matrix with elements r ij {, 1, 1}. The1 sorequivalently 1 s indicate 5 The degree-constrained subgraph problem on its turn falls under the umbrella of b-matching. A perfect b-matching is a set of links subgraph that span all nodes and for which each node i has precisely or at most bi adjacent links in the matching. If links are assigned weights, then the maximum-weight perfect b-matching problem is to find a perfect b-matching, for which the sum of the link weights in the matching is highest among all possible perfect b-matchings. The problem is also known as f-matching or b/f-factor and has many variations.

7 P. Van Mieghem et al.: Influence of assortativity and degree-preserving rewiring on the spectra of networks 649 Adjacency eigenvalues ρ D Fig.. Color online USA air transportation network, with =179andL =3136. where a link is present in A max and not in A min or vice versa for 1 s.consequently,giventhatforanundirected graph R = R T is symmetric, the maximum number of links that would need to be rewired in A min to get A max equals the total amount of 1 s or equivalently 1 s divided by. The rewiring of a link in A min to a link in A max corresponds to rewiring an element in R with a 1 and with a 1 in the same row or column due to symmetry of R. Through appropriate relabeling of nodes, the number of rewirings may decrease. Unfortunately, finding the minimum number of rewirings is an P-complete problem [18]. The problem of finding a connected graph of minimum or maximum weight given a degree sequence is P-complete, since by setting all degrees to, the problem reduces to the P-complete Traveling Salesman Problem []. However, Bienstock and Günlük have proved that If two connected graphs have the same degree sequence, then there exists a sequence of connected intermediate graphs transforming one of them to the other [18]. Even though we cannot efficiently compute a connected graphof highest disassortativity, we can use a rewiring approach to increase disassortativity, while maintaining connectivity. Hence, we would like to possess a criterion to check if a rewiring will lead to disconnectivity. In the field of dynamic graph algorithms, Eppstein et al. [3] havepro- posed a technique to check for connectivity in O time for each link update four updates per rewiring, which naturally beats any standard way of checking for graph connectivity. 5. Application to the USA air transport network As an example, we consider degree-preserving rewiring in the USA air transportation network displayed in Figure, where each node is an airport and each link is a flight connection between two American airports. We are interested in an infection process, where viruses are spread via airplanes from one city to another. From a topological point of view, the infection threshold τ c = 1 λ 1 is the critical de- -5 Fig. 3. Color online The ten largest and five smallest eigenvalues of the adjacency matrix of the USA airport transport network versus the percentage of rewired links. The insert shows the linear degree correlation coefficient ρ D as function of the assortative degree-preserving rewiring. sign parameter that we would like to have as high as possible, because an effective infection rate τ>τ c translates into a certain percentage of people that remains infected after sufficiently long time see for details [11]. Since most airports operate near to full capacity, the number of flights per airport should hardly change during the re-engineering to modify the largest eigenvalue λ 1.Hence,thedegreevector d should not change, which makes degree-preserving rewiring a desirable tool. Figure 3 shows how the adjacency eigenvalues of the USA air transport network change with degree-preserving assortative rewiring. In each step of the rewiring process, only four one elements i.e., two links in the adjacency matrix change position. If we relabel the nodes in such a way that the link between 1 and andbetween3and4caseainlemma1isrewired to either case b or c, then only a 4 4submatrixA 4 of the adjacency matrix A in [ ] A4 C A = C T A c is altered. The Interlacing Theorem [9], Chapter 3 states that λ j+4 A λ j A c λ j A for1 j 4, which holds as well for A r just after one degree-preserving rewiring step. Thus, apart from a few largest and smallest eigenvalues, most of the eigenvalues of A and A r are interlaced, as observed from Figure 3. The large bulk of the 179 eigenvalues not shown in Fig. 3 nor in Fig. 6 remains centered around zero and confined to the almost constant white strip between λ 1 and λ 5. As shown above, assortative rewiring increases λ 1.Figure 3 illustrates, in addition, that the spectral width or range λ 1 λ increases as well, while the spectral gap λ 1 λ remains high, in spite of the fact that the algebraic connectivity µ 1 is small. In fact, Figure 4 shows that µ 1 decreases, in agreement with 16, and vanishes after about 1% of the link rewirings, which indicates [9], Chapter 3 that the graph is then disconnected. 3 4

8 65 The European Physical Journal B Laplacian eigenvalues Pr[D = k] Pr[D = k] ~ k degree k Adjacency eigenvalues Fig. 4. Color online The twenty smallest eigenvalues of the Laplacian matrix of the USA airport transport network versus the percentage of rewired links. The insert shows the degree distribution that is maintained in each degree-preserving rewiring step. Laplacian eigenvalues Fig. 5. Color online The twenty smallest eigenvalues of the Laplacian versus the percentage of rewired links. The insert shows the disassortative degree-preserving rewiring. Figure 4 further shows that by rewiring all links on average once %, assortative degree-preserved rewiring has dissected the USA airport network into disconnected clusters. Increasing the assortativity implies that high-degree and low-degree nodes are linked increasingly more to each other, which, intuitively, explains why disconnectivity in more and more clusters starts occurring during the rewiring process. The opposite happens in disassortative rewiring as shown in Figure 5: the algebraic connectivity µ 1 increases during degree-preserving rewiring up to roughly 15% rewired links from about.5 to almost 1, which is the maximum possible due to µ 1 d min,theminimum degree, and d min =1asfollowsfromtheinsertinFigure4. Finally, Figure 6 plots the disassortative counter part of Figure 3: thespectralgapλ 1 λ reduces with the percentage of rewired links, while the spectral range λ 1 λ does not significantly change. The maximum difference ρ D Fig. 6. Color online The disassortative counterpart of Figure 3. maxρ D minρ D is deduced from the inserts in Figures 6 and 3 and appears to be slightly more than one, such that the adaptivity ratio in assortativity, maxρd minρd,is about 5%. Summarizing, in order to suppress virus propagation via air transport while guaranteeing connectivity, disassortative degree-preserving rewiring is advocated, which, in return, enhances the topological robustness. 5.3 Generalizing the observations Degree-preserving rewirings on various other real-world complex networks confirm the above observations: a assortative degree-preserving rewiring increases λ 1,bbut also decreases the algebraic connectivity µ 1,evenleading to disconnectivity of the network into many clusters. c Disassortative degree-preserving rewiring decreases λ 1, but d increases the algebraic connectivity µ 1 initially roughly up to % and thus strengthens the topological connectivity structure of the network. Often, the value of a network lies in the number of its links L relations between items, which grows as L = O in terms of the nodes at most, and in its connectivity, the ability that each node can reach each other node. A second value pillar of networking lies in positive synergetic coupling: when two nodes interact, their total impact on the network s functioning is larger than the sum of their individual functioning. Sometimes, even additional functionality is created. In order to establish positive synergy, the properties in the nodes often complement each other, which reflects disassortativity. We have shown that disassortativity decreases λ 1,implyingthatdynamicprocesses such as epidemic information spread and synchronization of coupled oscillators are slowed-down as their phasetransition threshold increases. In return, disassortativity increases the algebraic connectivity µ 1,thustheease to tear the network apart is lowered. Consequently, we argue that in most biological, infrastructural, or collaborative complex networks, disassortativity is the natural mode, because nodes with different properties connect to 3 4

9 P. Van Mieghem et al.: Influence of assortativity and degree-preserving rewiring on the spectra of networks 651 each other to create a network with win-win properties. Moreover, disassortativity favors good connectivity. Assortative networks, where nodes of the same type seek to interconnect, are less natural: either these networks are very regular and regularity is their distinguishing strength, or the nodes are selfish and only exchange with those that reward them equally, thereby excluding the lesser ones to participate in the networking. It is interesting to mention that these inferences agree with ewman s observations [1]: most of the biological and technical networks are disassortative, while social networks are found to be assortative. 6Conclusions We have reformulated the linear degree correlation coefficient ρ D,introducedbyewman[1], equation 1 as ameasureforassortativity,intermsofthetotalnumber k of walks in the graph with k hops. This reformulation allows us to derive a new formula from which a degree-preserving rewiring algorithm is deduced, that, in each rewiring step, either increasesordecreasesρ D.We have investigated the degree-preserving rewiring problem in networks with a same degree vector, and presented a solution that finds the maximum assortativity ρ D.Various lower bounds for the largest eigenvalue λ 1 of the adjacency matrix have been presented that all increase with ρ D.Thelargesteigenvalueλ 1 characterizes the effect of the topology on dynamic processes on graphs. The degree-preserving rewiring algorithm is applied to a real-world network, the USA air transportation network, with the aim to make this network more robust against virus spread, given that resources, the number of flights in each airport, are constant. A general observation is that, increasing λ 1 by increasing the assortativity, relatively rapidly leads to disconnectivity, while increasing disassortativity seems to increase the algebraic connectivity µ 1,thusthetopologicalrobustness.Thelatter agrees with the upper bound 16 onµ 1,thatindeed decreases with ρ D. Appendix A: Erdős-Rényi random graph As mentioned in Section, we need to compute ρ D l +,D l, where D l + = D i 1 and node i is connected to node j. The fact that G p is connected restricts p > p c log, where p c is the disconnectivity threshold. We first compute the joint probability Pr[D i =k, D j =m a ij =1],wherenodei and node j are random nodes in G p. Given the existence of the direct link a ij =1,thedirectlinkiscountedboth in D i and in D j such that Pr[D i =k, D j =m a ij =1] = Pr[D i 1 = k 1] Pr[D j 1 = m 1]. Introducing the binomial density of Pr[D i =k], we obtain Pr[D i =k, D j =m a ij =1] The joint expectation is E [D i D j a ij =1] = ext, 1 1 k= m= = p k 1 1 p 1 k k 1 p m 1 1 p 1 k. m 1 mkp r [D i =k, D j =m a ij =1] 1 = k p k 1 1 p 1 k k 1 k= 1 m p m 1 1 p 1 k m 1 m= =1+ p. E [D i a ij =1]=1+ p such that, for all and p>p c, Cov [D i,d j a ij =1]=E [D i D j ] E [D i ] E [D j ] = and, hence, ρ D =:theconnectederdős-rényi random graph G p iszero-assortative. References 1. M.E.J. ewman, Phys. Rev. E 67, J. Berg, M. Lässig, Phys. Rev. Lett. 89, M.E.J. ewman, Phys. Rev. Lett. 89, L. Li, D. Alderson, J.C. Doyle, W. Willinger, Internet Mathematics,, J. Zhao, L. Tao, H. Yu, J. Luo, Z. Cao, Y. Li, Chinese Phys. 16, M. Boguñá, R. Pastor-Satorras, A. Vespignani, Phys. Rev. Lett. 9, J.D. oh, Phys. Rev. E 76, M. Piraveenan, M. Prokopenko, A.Y. Zomaya, Europhys. Lett. EPL 84, P. Van Mieghem, Graph Spectra for Complex etworks, Cambridge University Press Cambridge, U.K., to appear 1 1. M. Boguñá, R. Pastor-Satorras, Phys. Rev. E 66, P. Van Mieghem, J. Omic, R.E. Kooij, IEEE/ACM Transactions on etworking 17, 9

10 65 The European Physical Journal B 1. J.G. Restrepo, E. Ott, B.R. Hunt, Phys. Rev. E 71, P. Van Mieghem, Performance Analysis of Communications Systems and etworks Cambridge University Press, Cambridge, U.K., M.A. Fiol, E. Garriga, Disc. Math. 39, Z. ikoloski,. Deo, L. Kucera, Discrete Mathematics and Theoretical Computer Science DMTCS, proc. AE, EuroComb 5 5, pp M. Fiedler, Czech. Math. J. 3, J.G. Restrepo, E. Ott, B.R. Hunt, Phys. Rev. E 76, D. Bienstock, O. Günlük, etworks 4, S. Zhou, R.J. Mondragón, ew J. Phys. 9, Y. Shiloach, Inf. Process. Lett. 1, H.. Gabow, An efficient reduction technique for degreeconstrained subgraph and bidirected network flow problems, Proc. of the 15th Annual ACM Symposium on the Theory of Computing, 1983,pp M.R. Garey, D.S. Johnson, Computers and Intractability: A Guide to the Theory of P-Completeness W.H. Freeman, D. Eppstein, Z. Galil, G.F. Italiano, A. issenzweig, Journal of the ACM 44,