aa r X i v : . [ c ond - m a t . s t a t - m ec h ] J un Random walks on complex trees
Andrea Baronchelli, Michele Catanzaro, and Romualdo Pastor-Satorras
Departament de F´ısica i Enginyeria Nuclear, Universitat Polit`ecnica de Catalunya, Campus Nord B4, 08034 Barcelona, Spain (Dated: November 20, 2018)We study the properties of random walks on complex trees. We observe that the absence of loopsreflects in physical observables showing large differences with respect to their looped counterparts.First, both the vertex discovery rate and the mean topological displacement from the origin presenta considerable slowing down in the tree case. Second, the mean first passage time (MFPT) displays alogarithmic degree dependence, in contrast to the inverse degree shape exhibited in looped networks.This deviation can be ascribed to the dominance of source-target topological distance in trees. Toshow this, we study the distance dependence of a symmetrized MFPT and derive its logarithmicprofile, obtaining good agreement with simulation results. These unique properties shed light onthe recently reported anomalies observed in diffusive dynamical systems on trees.
PACS numbers: 89.75.Hc, 05.40.Fb, 05.60.Cd
I. INTRODUCTION
Diffusion problems on tree structures pop up in awide range of scientific domains, such as theoreticalphysics [1, 2, 3], computer science [4, 5], phylogeneticanalysis [6] and cognitive science [7]. Moreover, dynam-ics in tree structures have gained a renewed interest in thephysics community as a spin-off of the attention devotedto the structural properties of complex networks [8, 9]and dynamical processes taking place on top of them [10].Thus, along with the widely explored scale-free (SF) net-works [11], also SF trees have started to be used as under-lying topologies for dynamical processes. Interestingly,the absence of loops in trees turns out to have a strongimpact on the considered dynamics, and relevant differ-ences between looped networks and tree topologies havebeen recently reported in several dynamical models, suchas the voter model [12], the naming game [13], the ran-dom walk and the pair-annihilation processes [14], and amodel for norm spreading [15]. The properties of mostdynamical processes on looped network can be reason-ably accounted for by annealed mean-field theories [10],which rely only on information about the degree distribu-tion and degree correlations [16], and consider the net-work as maximally random at all other respects. Thebehavior observed in trees, different from the annealedmean-field predictions, must thus be explained in termsof the non-local constrain of absence of loops imposedin this kind of graphs, which is hard to implement intheoretical approaches.In this paper we explore the peculiarities induced in dy-namical processes by the absence of loops by consideringthe simplest possible example, namely the uncorrelatedrandom walk [17, 18]. Several works have been devotedin the past to the study of random walks on complexnetworks, showing in general a good agreement betweentheory and simulations on looped networks, while differ-ences were reported in tree networks in Ref. [14]. Here,we find that the global constraint of lack of loops inducesa general slowing down of diffusion, as measured by thenetwork coverage and the mean topological displacement from the origin. As well, it profoundly alters the degreedependence of the mean-first passage time. This is dueto the fact that the source-target distance is dominatingin trees. In order to account for this features, we studythe mean round trip time versus distance and find ananalytic expression of its dependence on degree.
II. RANDOM WALKS ON COMPLEXNETWORKS AND TREES
We consider random walks on general networks definedby a walker that, located on a given vertex of degree k at time t , hops with probability 1 /k to one of the k neighbors of that vertex at time t + 1. We have mea-sured the properties of random walks on growing SF treescreated with the linear preferential attachment (LPA)algorithm [11, 19]: at each time step s , a new vertexwith m edges is added to the network and connectedto an existing vertex s ′ of degree k s ′ with probabilityΠ s → s ′ = ( k s ′ + a ) / (2 m + a ) s . This process is iterateduntil reaching the desired size N . The resulting networkhas degree distribution P ( k ) ∼ k − γ with tunable expo-nent γ = 3 + a/m , with γ < a <
0. For m = 1the LPA model yields a strict tree topology. Degree cor-relations, measured by the average degree of the nearestneighbors of the vertices of degree k [20], are given by¯ k nn ( k ) ∼ N (3 − γ ) / ( γ − k − γ [21]. Therefore, only for γ = 3 ( a = 0) we expect to obtain uncorrelated networks.In order to explore the intrinsic properties of a treetopology, disregarding SF effects, we have also consid-ered homogeneous networks. In the growing exponentialnetwork model (EM) [9], at each time step s a new vertexwith m edges is added to the network, and it is connectedto m randomly chosen other vertices. In the continuousdegree approximation (i.e., considering the degree as acontinuous variable and substituting sums by integrals),this models leads to networks with an exponential degreedistribution, P ( k ) = e − k/m /m . Again, homogeneoustrees are generated by selecting m = 1. The randomCayley tree (RC), on the other hand, is generated byadding z neighbors to a randomly selected leaf (i.e. avertex whose degree is k = 1) at each time step s ( z + 1neighbors are added to the first vertex). The resultingtree contains only vertices with degree k = z + 1, andleaves with k = 1.To check our results against looped structures, wehave considered the uncorrelated configuration model(UCM) [22], yielding uncorrelated networks with any pre-scribed SF degree distribution. The model is defined asfollows: (1) Assign to each vertex i in a set of N ini-tially disconnected vertices a degree k i , extracted fromthe probability distribution P ( k ) ∼ k − γ , and subject tothe constraints m ≤ k i ≤ N / and P i k i even. (2) Con-struct the network by randomly connecting the verticeswith P i k i / m ≥
2, we generate connected networkswith probability almost 1. The effect of correlations inlooped structures can be checked by means of the config-uration model (CM) [9], which is analogous to the UCM,but allows degrees to range in the interval m ≤ k i ≤ N [23]. In all present simulations, we set for looped neworks m = 4, tree networks corresponding to m = 1 ( z = 4 forthe RC tree). III. RANDOM WALK EXPLORATION
We start by studying two properties of a random walkthat quantify the speed at which it explores its neigh-borhood in the network. The first one is the coverage S ( t ), defined as the number of different vertices visitedby a walker at time t , averaged for different random walksstarting from different sources. For looped networks, thecoverage reaches after a short transient the functionalform [24, 25] S L ( t ) ∼ t [43], in accordance with theoreti-cal calculations for the Bethe lattice [26], and eventuallysaturates to S L ( ∞ ) = N , due to finite size effects. Ascaling form for the coverage has been proposed [24] tobe S L ( t ) = N f ( t/N ), with f ( x ) ∼ x for x ≪ f ( x ) ∼ x ≫ ρ k ( t ) as the proba-bility that a vertex of degree k hosts the random walkerat time t . During the evolution of the random walk, thisprobability satisfies, in a general network with a correla-tion pattern given by the conditional probability P ( k ′ | k )that a vertex of degree k is connected to another vertexof degree k ′ [16], the mean-field equation ∂ρ k ( t ) ∂t = − ρ k ( t ) + k X k ′ P ( k ′ | k ) k ′ ρ k ′ ( t ) . (1)In the steady state, ∂ t ρ k ( t ) = 0, the solution of this equa-tion, for any correlation pattern, is given by the normal- -6 -4 -2 s k ( t ) N=10 N=10 N=10 kt / (
1, we recover the exact result S ( t ) ∼ t [26]. For SF networks, we obtain within thecontinuous degree approximation S ( t ) N = 1 − ( γ − m γ − Z ∞ m k − γ exp (cid:18) − kt h k i N (cid:19) dk = 1 − ( γ − E γ (cid:18) mt h k i N (cid:19) , (7) -6 -4 -2 t / N -8 -6 -4 -2 S ( t ) / N CM γ =2.5, N=10 CM γ =2.5, N=10 CM γ =2.5, N=10 UCM γ =3.0, N=10 UCM γ =3.0, N=10 UCM γ =3.0, N=10 EM, N=10 EM, N=10 EM, N=10 FIG. 2: (Color online) Rescaled coverage for looped com-plex networks. We plot in full lines the analytical predictionsEqs. (7) and (8), corresponding to SF and EM networks. Re-sults for CM, UCM, and EM networks have been shifted (topto bottom) in the vertical axis for clarity. where E γ ( z ) is the exponential integral function [28]. ForEM networks, on the other hand, we find S ( t ) N = 1 − em Z ∞ m e − k/m exp (cid:18) − kt h k i N (cid:19) dk = 1 − e − mt/ h k i N mt h k i N . (8)In Fig. 1, we can observe that the scaling predictedby Eq. (5) for the coverage spectrum s k ( t ) is very wellsatisfied in looped complex networks, independently oftheir homogeneous or SF nature, and in this last case,of the degree exponent and the presence or absence ofcorrelations. In Fig. 2, on the other hand, we plot thetotal coverage S ( t ) /N , which can be fitted quite correctlyby the analytical expressions Eqs. (7) and (8) for SF andEM networks.On tree networks we find a different scenario. In Fig. 3we can see that the coverage spectrum does not scale aspredicted by our mean-field argument. While we do nothave theoretical predictions for the correct scaling form,a numerical analysis of the total coverage, Fig. 4, showsthat, at short times, it grows in trees as S T ( t ) ∼ t/ ln( t ),preserving an approximate scaling form S T ( t ) = N f (cid:18) t ln( t ) N (cid:19) , (9)with a scaling function f ( x ) that depends slightly onthe network details (degree exponent, correlations, etc.).This observation indicates the presence of a general slow-ing down mechanism in the random walk dynamics intrees: the dynamics turns out to be more recurrent andtherefore it is more costly to find new vertices during thewalk. It is easy to see that this situation will correspondto a walker deep in the leaves of a subtree that has other-wise completely explored. In order to find new vertices,the walker must first find the exit to the subtree. This -4 -2 s k ( t ) N=10 N=10 N=10 kt / (
91 for LPAtrees with γ = 2 . α ≃ .
04 for LPA trees with γ = 3 . α ≃ .
31 for EM trees and α ≃ .
62 for the RC tree. where the exponent α depends on the details of the net-work. The whole function ¯ d T ( t ) is also observed to fulfillthe scaling form ¯ d T ( t ) = h ¯ d i f (cid:18) ln t h ¯ d i /α (cid:19) . (17)This form implies that the characteristic time to es-cape from the neighborhood of the origin scales as t c ∼ exp( h d i /α ) ∼ exp[(ln N ) /α ], which means that the ex-ploration process is much more slower in trees, with thewalker spending large amounts of time exploring the closevicinity of the origin of the walk. We remark that herethe scaling function f ( x ) displays some further depen-dences on degree exponent, average degree and degreecorrelations in both looped and tree networks.The fact that the presence of a tree-like structure slowsdown the distance explored by a random walker on a net-work, allows to interpret the results presented in Ref. [29],in particular the power-law behavior at initial times of¯ d ( t ). In fact, in Ref. [29] the substrate for the ran-dom walk simulations were SF networks generated withthe CM model with minimum degree m = 1. In thiscase, simulations were performed on the giant compo-nent. Apart from the possible effect of degree correla-tions for γ <
3, the point is that, for m = 1, tracesof tree-like structure are still present in the network, inthe form of chains of small degree vertices [32]. Thus,a remnant slowing down effect of the tree component isobserved, see Fig. 9, leading to an MTD that, at shorttimes, scales as ¯ d ( t ) ∼ t . for the data at m = 1 shownin this graph, in excellent agreement with the observationin [29], namely ¯ d ( t ) ∼ t . for the RMSTD.A further remark concerns the relation between ourresults and the above mentioned analytical calculationsfor Bethe lattices [26], according to which these struc-tures exhibit a behavior analogous to the one observedin looped networks. The apparent incongruity vanishes t d ( t ) m=1m=2m=3m=4m=10 FIG. 9: (Color online) MTD as a function of time for UCMlooped networks ( γ = 3 . N = 10 ) with varying mini-mum degree m . when noticing that, while Bethe lattices are infinite hier-archical structures, we have focused on complex (i.e. dis-ordered) finite trees. To recover numerically the Bethelattice behavior, indeed, it is necessary to adopt specialalgorithms in order to simulate an infinite hierarchicaltree [33, 34]. IV. MEAN FIRST-PASSAGE TIME
More information about the dynamics of random walkscan be extracted from the analysis of the mean first pas-sage time (MFPT) [35] τ ( i → j ), defined as the averagetime that a random walker takes to arrive for the firsttime at vertex j , starting from vertex i [27]. In networkswith no translation symmetry, the MFPT from a source i to a target j needs not be equal to the MFTP fromsource j to target i . Therefore, different reduced MF-PTs can be considered. We can thus define the directMFTP τ → ( k ) as the MFPT on a target vertex of degree k , starting from a randomly chosen source vertex, andthe inverse MFPT τ ← ( k ) as the MFTP on a randomlytarget vertex, starting from a source vertex of degree k ,namely τ → ( k ) = 1 N X i N k X j ∈V ( k ) τ ( i → j ) , (18) τ ← ( k ) = 1 N X j N k X i ∈V ( k ) τ ( i → j ) , (19)where V ( k ) is the set of vertices of degree k and N k isthe number of such vertices. A simple argument canpredict the form of the MFPTs for random uncorrelatednetworks. In this case, the probability for the walker toarrive at a vertex i , in a hop following a randomly chosenedge, is given by q ( i ) = q ( k i ) = k i / h k i N [36]. Therefore,the probability of arriving at vertex i for the first timeafter t hops is P a ( i ; t ) = [1 − q ( i )] t − q ( i ). The direct k -2 τ → ( k ) / ( N < k > ) CM γ =2.5, N=10 CM γ =2.5, N=10 UCM γ =3.0, N=10 UCM γ =3.0, N=10 EM, N=10 EM, N=10 k τ ← ( k ) / ( N < k >< k - > ) FIG. 10: (Color online) Reduced MFPTs as a function ofthe degree k for looped complex networks. We recover thesimple mean-field predictions τ → L ( k ) ≃ h k i N/k (main figure)and τ ← L ( k ) ≃ h k ih k − i N (inset). MFTP to vertex i can thus be estimated as the average τ → ( k i ) = X t tP a ( i ; t ) = h k i Nk i . (20)For the inverse MFPT, we notice that, in a random net-work, after the first hop, the walker loses completely thememory of its source degree, therefore we can approxi-mate τ ← ( k ) = X k P ( k ) τ → ( k ) = h k ih k − i N. (21)Less trivial approaches [27, 37] show in fact that theMFPT from a source vertex i to target vertex j dependson the degree of the target vertex as τ ( i → j ) ∼ /k j ,but has a residual dependence on the source vertex andit is actually asymmetric, τ ( i → j ) = τ ( j → i ). This factcould in principle affect the form of the reduced MFPTsin real networks, defined in Eqs. (18) and (19). Fig. 10,however, shows that for looped networks the behaviorpredicted for random uncorrelated networks turns out tobe extremely robust with respect to changes in the topo-logical properties of the network: homogeneous or het-erogenous nature, degree exponent, presence or absenceof correlations, etc. [29, 37, 38].In trees, on the other hand, we find a completely dif-ferent picture, see Fig. 11. Now, the direct MFPT in SFtrees decays with k much slower than in looped networks.In fact, we can fit it numerically to the form τ → T ( k ) = C N ln N − C N ln( k + C ) , (22)where C , C and C fitting parameters that depend onlyslightly on the network size. The N ln N dependencecan be directly observed by plotting τ → T (1) for differentsystem sizes, as shown in Fig. 12. For homogeneous EMnetworks, on the other hand, the direct MFPT can befitted to the form τ → T ( k ) = D N ln N − D N k, (23) k × × × × × τ → ( k ) LPA γ =2.5LPA γ =3.0 τ → (k) = C Nln(N) - C Nln(k+C ) k × × τ → ( k ) EM Tree τ → (k) = D Nln(N) - D Nk FIG. 11: (Color online) Direct MFPT as a function of thedegree k for SF tree networks ( N = 10 ). Dashed lines cor-respond to nonlinear fittings to the empirical form Eq. (22).Inset: Direct MFPT as a function of the degree k for homo-geneous EM tree networks. The dashed line corresponds to afitting to the empirical form Eq. (23). see inset in Fig. 11. The scaling of τ → T (1) in this caseis also checked in Fig. 12. With respect to the inverseMFTP, it is again constant, but now scales with systemsize as τ ← T ( k ) ∼ N ln N for all kinds of trees (inset inFig. 12).The topological structure of the trees can explain theunusual form of the MFPTs. While in looped networksthe number of access paths to the target vertex is re-lated to its degree, on the tree the path is unique, andis given by the one-dimensional set of links and verticesconnecting the starting vertex to the target. In this case,the degree of the target is much less important from thepoint of view of the walker, since finding the target cor-responds to finding a particular leaf (i.e. a k = 1 vertex)of the sub-tree the random walker is exploring. Thisobservation suggests that while in looped networks theMFPT into a vertex is dominated by its degree (becausethe latter is related the multiplicity of the entry paths tothe vertex), in trees the distance between the source andthe target can be much more relevant, and thus induce alarger MFPT.We therefore consider the MFPT as a function of thetopological distance d ij between the starting vertex i andthe target j [37, 39]. Since the distance between twovertices is by definition a symmetric quantity, it seemsnatural to re-define the MFPT in terms of the symmetricmean round trip time (MRTT)¯ τ ( d ij ) = τ ( i → j ) + τ ( j → i ) , (24)i.e. the average time to go from i to j and back or vice-versa. It has been recently proved [39] that, for complexscale-invariant networks, the MRTT averaged for all ver-tices at the same distance scales as¯ τ ( d ) ≃ N d D w − D b , (25)where D b is the box dimension of the network, and D w its walk exponent [40]. For a class of scale-invariant net- N τ → ( ) / l n ( N ) LPA γ = 2.5LPA γ = 3.0EM TreeRC k τ ← ( k ) / N l n ( N ) FIG. 12: (Color online) Direct MFPT on leaves in complextrees as a function of the network size N . The observed scalingis τ → T (1) ∼ N ln N . Inset: Inverse MFPT on complex treesfor different network sizes ( N = 10 full colored points, N =3 × light colored points, N = 10 empty points). Theobserved scaling is again τ ← T ( k ) ∼ N ln N . d τ ( d ) / N LPA γ =3.0, N=10 LPA γ =3.0, N=10 LPA γ =2.5, N=10 LPA γ =2.5, N=10 EM Tree, N=10 EM Tree, N=10 RC, N=10 RC, N=10 τ (d) / N = d FIG. 13: (Color online) MRTT ¯ τ ( d ) as a function of thesource-target topological distance d in trees. Different curvescollapse perfectly on ¯ τ T ( d ) ∼ Nd . works [40] corresponding to a tree structure, for which D w − D b = 1, the authors of Ref. [39] obtained corre-spondingly a linear scaling ¯ τ T ( d ) ≃ N d . We have checkedthat this linear form holds for different SF, EM and RCtrees, see Fig. 13, a result that leads us to conjecturethat, for any complex tree, D w − D b = 1.We can use the result in Eq. (25) to gain insight onthe behavior of the anomalous reduced MFPTs in treenetworks. Considering an average over all vertices withthe same degree, we have that ¯ τ ( d kk ′ ) = τ ( k → k ′ ) + τ ( k ′ → k ). Averaging now over k ′ , we can consider thereduced MRTT¯ τ ( k ) = X k ′ P ( k ′ )[ τ ( k → k ′ )+ τ ( k ′ → k )] = τ ← ( k )+ τ → ( k ) , (26)defined as the average time to go from a randomly chosenvertex to a given vertex of degree k , and back (or vice-versa, since the MRTT is symmetric). Now, since ¯ τ ( d kk ′ ) -2 -1 N τ ( k ) / τ ( )- N=1000N=3000N=10000 k -2 -1 N τ ( k ) / τ ( )- γ =2.5 γ =3.0 FIG. 14: (Color online) Rescaled MRTT ¯ τ T ( t ) as a functionof the source degree k and for randomly chosen targets in SFtrees. Predictions of Eq. (30), i.e. N τ ( k ) /τ (1) − ∼ k (1 − γ ) / ,are plotted as dashed lines. is linear in d kk ′ for tree networks, we have¯ τ ( k ) ≃ X k ′ P ( k ′ ) N d kk ′ = N d k . (27)Assuming the scaling of d k as given by Eqs. (12) and (13),we obtain ¯ τ T ( k ) ≃ N A ln (cid:18) Nk ( γ − / (cid:19) (28)for SF networks and¯ τ T ( k ) ≃ N ( A ′ ln N − B ′ k ) (29)for EM networks. The unknown constant in Eq. (28)can be reabsorbed in the the value of ¯ τ T (1), to obtain ascaling form with system size for SF networks that reads¯ τ T ( k )¯ τ T (1) ∼ N ln (cid:18) Nk ( γ − / (cid:19) . (30)In Fig. 14 we show that this scaling form is very wellsatisfied by the MRTT in SF trees, independently of thedegree exponent and correlation patterns, at least for in-termediate values of k . The observed bending at smalldegrees can be ascribed to the presence of a constant inthe logarithm analogous to empirical parameter C inEq. (22), that does not follow from our argument. Finitesize effects, on the other hand, are responsible for thedeviations present at large degrees, that are indeed moreevident in SF trees with smaller values of γ .This observations allow us to interpret the anomalousfunctional form of the reduced MFTPs observed in trees.From Eq. (26), we have τ → T ( k ) = ¯ τ T ( k ) − τ ← T ( k ) . (31)Writing τ ← T ( k ) ∼ CN ln N , from Eq. (28) we obtain, forSF networks, τ → T ( k ) ∼ ( A − C ) N ln N − A ( γ − N ln k (32)while for homogeneous EM networks, we have τ → T ( k ) ∼ ( A ′ − C ) N ln N − B ′ N k, (33)in agreement with the empirical fitting found in Eqs. (22)and (23).This argument cannot be extended to looped networks,since here ¯ τ ( d kk ′ ) is not linear in d kk ′ . The k depen-dence of the MRTT can be however trivially obtainedfrom the reduced MFPTs as ¯ τ L ( k ) = τ → L ( k ) + τ ← L ( k ) ≃h k i N ( h k − i + 1 /k ). V. CONCLUSIONS
In this paper we have shown that complex tree-liketopologies heavily affect the behavior of a random walkperformed on top of them, with a global slowing down ofthe dynamics and a logarithmic dependence of the firstpassage time properties in SF networks. These featuresare intrinsically connected with the complex tree struc-ture and cannot be attributed to the mere presence ofleaves, while they are radically different from the onesexhibited by Bethe lattices, i.e. infinite and hierarchicaltree structures.We have studied the random walk exploration prop-erties and we have shown that complex trees inducea slower dynamics, compared to looped networks, for both the coverage and the mean topological displace-ment problems. Moreover, by means of the analysis ofthe symmetrized MFPT (the MRTT), we have been ableto recognize the different role played by the degree k ofthe target vertex in looped and tree structures. In theformer, a larger degree corresponds to a larger numberof access ways to the target vertex. In the latter, on theother hand, the target vertex is always seen as a leaf bythe random walker, and its degree k affects the MFPTonly through the dependence of the average distance d k between it and the rest of the vertices. These results pro-vide important insights into diffusion problems on trees,and help explaining the characteristic slow dynamics ob-served on diffusive processes taking place on top of treenetworks [12, 13, 14]. Moreover, they are also interestingin the study of dynamics in real-world networks, in whichthe so-called border trees motifs [41] have been recentlyshown to be significantly present. Acknowledgments
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