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Laplacian spectra of a class of small-world networks and their applications

One of the most crucial domains of interdisciplinary research is the relationship between the dynamics and structural characteristics. In this paper, we introduce a family of small-world networks, parameterized through a variable d controlling the scale of graph completeness or of network clustering. We study the Laplacian eigenvalues of these networks, which are determined through analytic recursive equations. This allows us to analyze the spectra in depth and to determine the corresponding spectral dimension. Based on these results, we consider the networks in the framework of generalized Gaussian structures, whose physical behavior is exemplified on the relaxation dynamics and on the fluorescence depolarization under quasiresonant energy transfer. Although the networks have the same number of nodes (beads) and edges (springs) as the dual Sierpinski gaskets, they display rather different dynamic behavior.

One of the most crucial domains of interdisciplinary research is the relationship between the dynamics and structural characteristics. In this paper, we introduce a family of small-world networks, parameterized through a variable d controlling the scale of graph completeness or of network clustering. We study the Laplacian eigenvalues of these networks, which are determined through analytic recursive equations. This allows us to analyze the spectra in depth and to determine the corresponding spectral dimension. Based on these results, we consider the networks in the framework of generalized Gaussian structures, whose physical behavior is exemplified on the relaxation dynamics and on the fluorescence depolarization under quasiresonant energy transfer. Although the networks have the same number of nodes (beads) and edges (springs) as the dual Sierpinski gaskets, they display rather different dynamic behavior.
O ne of the most major problems in the study of networks is to understand the relations between their topology and the dynamics 1 . For instance, in the framework of generalized Gaussian structures (GGSs) [2][3][4][5] , the dynamics of polymer networks is fully described through the Laplacian eigenvectors and eigenvalues. In the field of GGSs and dynamical processes, the investigation of Laplacian eigenmodes has a paramount importance for the relaxation dynamics, the fluorescence depolarization by quasiresonant energy transfer [6][7][8] , the mean first-passage time problems [9][10][11] , and so on. Laplacian eigenvalues and eigenvectors play an irreplaceable role and they are also relevant to multi-aspects of complex network structures, like spanning trees 12 , resistance distance 13 and community structure 14 . However, it is a challenging task to derive exact Laplacian eigenvalues or eigenvectors for a complex system and based on them to describe its dynamics. We remark that for this the use of deterministic structures is of much help [15][16][17][18][19] . Although the structural disorder leads in case of many real networks like hyperbranched polymers to smoothing-out and averaging, the topological features are still reflected in the typical scaling behaviors 20 . Furthermore, recently a striking development of chemistry made possible the synthesis of the hierarchical, fractal Sierpinski-type compounds 21 . Undoubtedly, this new achievement will keep the interest of the theorists on the regular structures, especially on those with loops.
The study of Laplacian eigenvalues has exhibited its activity during the past few decades, among extensive subjects and researches. The works from last century had solved the Laplacian eigenvalues for considerable amount of famous networks, like dual Sierpinski gaskets (in 2 or higher dimensions) 15,16 , dendrimers 17 , and Vicsek fractals 18,19 . Another type of model structures, which often arise in the complex systems or polymer networks, are the so-called small-world networks (SWNs) [22][23][24][25] . Recent studies have also suggested that SWNs play a notable role in real life 26,27 .
In this report we introduce a new kind of SWNs. Their construction is based on complete graphs consisting of d nodes and they have the same number of nodes and of edges as the dual Sierpinski gaskets embedded in (d 2 1)dimension. A complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge. It has been widely used in quantum walks 28,29 , tensor networks 30 , social networks 31 , and explosive percolation problem 32 . While the SWNs introduced here are based on complete graph, their clustering coefficient shows that the SWNs are similar to complete graphs only in the limit d R '. As we proceed to show, also in this limit they have similar behavior as the dual Sierpinski gaskets embedded in to d R ' dimensions. On the other hand, for finite d, the SWNs display a macroscopically distinguishable behavior.
The report is organized as follows: First, we present the construction of SWNs, analyze their properties and their Laplacian spectra (the derivation of the recursive equations for the eigenvalues is given in Methods). Then, based on the spectra we consider the dynamics of networks, namely, the structural average of the mean monomer displacement under applied constant force and the mechanical relaxation moduli, and the dynamics on networks, exemplified through the fluorescence depolarization. Finally, we summarize and discuss our results.

Results
Model structures. We start with a brief introduction to a family of small-world networks (SWNs) V d g characterized by two parameters d and g, where d stands for the number of nodes of complete graph and g for the current generation. Figure 1 shows a construction process from V 3 1 to V 3 3 : At first, V 3 1 is a simple triangle, that is, a complete graph with 3 nodes. At the next stage, each node in V 3 1 is replaced by a new complete graph. Thus each of the newly appeared complete graphs contains exactly one node of V 3 1 and we get the network at second generation V 3 2 . The growth process to the next generation continues in a similar way: Connecting a complete graph to each of the node of V 3 2 one gets V 3 3 . In general, we have d g21 nodes at generation g 2 1. By attaching d 2 1 nodes to each existing node, increases their total number from d g21 to d g . In this way, we get immediately the number of nodes in this network, N d To give evidence of the small-world property, we consider another characteristics, the diameter of the network. For a network, the diameter means the maximum of the shortest distances between all pairs of nodes in it 1 . At each iteration g $ 1, new complete graphs are added to each vertex. Let us define the two nodes with longest distance in the existing network as M A and M B . It is easy to see that these two nodes belong to the complete graphs attached to M A and M B , respectively. Hence, at any iteration, the diameter of the network increases by 2 at most. Then the diameter of V g is just equal to 2g 2 1, a result irrelevant to parameter d. The value can be presented by another form 2 log d N g 2 1, which grows logarithmically with the network size indicating that the networks V d g are smallworld 1 .
Now we turn to the clustering coefficient of any node i, which is given by , where e i is the number of existing links between all the k i neighbors of node i 34 . From the network construction, we come to a simple conclusion that if node x exists for h generations, external (d 2 1)h nodes will be attached to it. That is, k x 5 (d 2 1)h. Among the (d 2 1)h neighbors, d 2 1 nodes that belong to the same complete graph are connected to each other, leading to the total number of links e x 5 h[(d 2 1)(d 2 2)/2]. Thus, the C x is given by Based on equation (1) we can list the correspondence between each kind of clustering coefficient and the corresponding amount of nodes: where the last situation represents the center of the whole network. Then we can obtain the average clustering coefficient of all the nodes, Figure 2 shows AECae as a function of g for d going from 3 to 6. As one can infer from the figure, AECae decreases very rapidly at small generations to a some constant value, which depends on d. In fact, one can find from equation (3) , an inherent property of a complete graph.

Recursion formulae for the Laplacian spectrum. Let
To get a solution for the eigenvalues of L V d g , we have to concentrate our attention on its characteristic polynomial, P d g l ð Þ. Here we just give a result and put off the proof and details in Methods: The recursion relation provided in equation (4) determines the eigenvalues of Laplacian matrix for V d g . Note that P d g has a factor   It is evident that V d g has d g Laplacian eigenvalues, denoted by l g 1 , l g 2 , …, l g d g , the set of which is represented by L g , i.e., L g~l É . In addition, without loss of generality, we assume that l On the basis of above analysis, L g can be divided into two subsets L 1 g includes the remaining eigenvalues. Thus, : ð5Þ The remaining 2d g21 eigenvalues belonging to L 2 or from where i runs from 1 to N g21 5 d g21 .
Solving the quadratic equation (7), we obtain two roots and respectively. Thus, each eigenvalue l g{1 i of L g21 gives rise to two new eigenvalues in L 2 ð Þ g by inserting each Laplacian eigenvalue of V g21 into equations (8) and (9). Considering the initial value g, by recursively applying equations (8) and (9) and accounting for L 1 ð Þ g , the Laplacian eigenvalues of V g are fully determined.
It is simple matter to check that equations (8) and (9) have the following behaviors: and x for x?d: In this way equation (10) produces only small eigenvalues, r 1 (x) g [0, 1) and equation (11) the large ones, r 2 (x) g [d, '). Thus, the eigenvalue spectrum has always a gap [1, d), which is bigger for networks V d g with larger d.
Now, it is interesting to examine the behavior of the small eigenvalues, i.e. to consider equation (10) for 0ƒx=1. Our goal is to obtain the spectral dimensiond (also known as fracton dimension 35 ). For this we use the methods of Ref. 36. Under equation (10) for x=1, the n eigenvalues in the interval [l g , l g 1 Dl g ] go over in n eigenvalues in the interval [l g11 , l g11 1 Dl g11 /d], while the total number of modes increases from N to dN. Hence, the density of states (modes) r(l) for l=1 obeys Nr l ð ÞDl~dNr l=d ð ÞDl=d, i:e: r l ð Þ~r l=d ð Þ: ð12Þ Using now the relation between r(l) and the spectral dimensiond 35 , leads to This means that the spectral dimension of the networks V d g isd~2 andd is independent on d. We note that for the dual Sierpinski gasket embedded in (d 2 1)-dimension the spectral dimension is d~2 ln d ð Þ=ln dz2 ð Þ, see e.g. Refs. 37, 38, i.e. it is similar to that of V d g only in the limit d R '.
Dynamics of polymer networks under external forces. We are going to study the networks V d g under the framework of generalized Gaussian structures (GGS) [3][4][5] , an extension of the classical Rouse beads-springs model 2,39-41 . Here we let all N beads of the GGS to be assigned to the same friction constant, f. The beads are connected to each other by elastic springs with spring constant K. The Langevin equation of motion for the mth bead in a system reads where R m (t) 5 (X m (t), Y m (t), Z m (t)) is the position vector of the mth bead at time t, L describing the Laplacian matrix of the V d g . Moreover, f m (t) is the thermal noise that is assumed to be Gaussian with zero mean value AEf m (t)ae 5 0 and AEf ma (t)f nb (t9)ae 5 2k B Td ab d mn d(t 2 t9), where k B is the Boltzmann constant, T is the temperature, a and b represent the x, y, and z directions; F m (t) is the external force acting on bead m.
First, we consider a quantity which is related to the micromanipulations with the polymer networks 42 . We put a constant external force F k (t) 5 FH(t)d mk e y (;k), started to act at t 5 0 (H(t) is the Heaviside step function) on a single bead m of the V d g in the y direction. After averaging over all possibilities of choosing this monomer randomly, the displacement reads 4,5,39 where s 5 K/f is the bond rate constant, and l i is the eigenvalues of matrix L with l 1 being the unique smallest eigenvalue 0. Another example is the response to harmonically applied forces (strain fields), i.e. F m (t) 5 c 0 e ivt Y m (t)e x . The related response function is the so-called complex dynamic modulus G*(v), or equivalently, its real G9(v) and imaginary G0(v) components (the storage and the loss moduli 41,43 ). In the GGS model (for very dilute thetasolutions) the G9(v) and G0(v) are given by 3 and www.nature.com/scientificreports where n denotes the number of polymer segments (beads) per unit volume. We start by focusing on the averaged displacement AEY(t)ae, equation (16), where we set s 5 1 and F f~1 . Figure 3 displays in double logarithmic scales the AEY(t)ae for the networks V 4 g consisting of 4 7 up to 4 10 beads. As is known 4,5,39 , the AEY(t)ae in such GGS at very long times reaches the domain AEY(t)ae , Ft/(Nf) and at very short times obeying AEY(t)ae , Ft/f. However, in intermediate regime the network's beads move for several decades of time very slowly (logarithmic behavior 5 ), up to the times t , N related to the diffusive motion of the whole structure. This differs from the corresponding patterns for the dual Sierpinski gaskets (embedded in (d 2 1)-dimension) 37 Fluorescence depolarization. We are now embarking on the dynamics of energy transfer over a system of chromophores [6][7][8] . As a usual way, we assume that the nodes (beads) only transfer their energy with their nearest neighbors. Under these conditions the dipolar quasiresonant energy transfer among the chromophores obeys the following equation [6][7][8] : where P i (t) represents the probability that node i is excited at time t and T ij is the transfer rate from node j to node i. Following the framework of Refs. 6-8, we separate the radiative decay (equal for all chromophores) from the transfer problem, which can be included by the multiplication of all the P i (t) by exp(2t/t R ), where 1/t R corresponding to the radiative decay rate. Under the assumption that all microscopic rates are equal to each other, fixed on a valuẽ k, equation (19) becomes where L ij is the ijth entry of Laplacian matrix L g . In equation (20) we used that for L g the relation L ii~{ X j=i L ji holds. The solution of equation (20) requires diagonalization of L g . The result for a given P i (t) depends both on the eigenvalues and on the eigenvectors of L g [6][7][8] . However, by averaging over all sites (a procedure fully justified when the dipolar orientations are independent of the beads' position in the system), the probability of finding the excitation at time t on the originally excited chromophore depends only on the eigenvalues of L g and is given by 6 Measuring the time in units of 1=k, we can obtain the AEP(t)ae with k~1. In Fig. 5 we display in double logarithmic scales the average probability AEP(t)ae that an initially excited chromophore of the network V d g is still or again excited at time t. As for the previous figures, we choose d 5 4 and change the generation g from 7 to 10, which means that the number of beads varies from 4 7 to 4 10 . From Fig. 5 a waviness superimposed at early times can be observed immediately. Such waviness has been predicted in the regular hyperbranched  For the sake of comparison, in Fig. 6 we display the AEP(t)ae for dual Sierpinski gaskets embedded into 3-dimensional space for generations g as those in Fig. 5. What is clear from the figure, the curves also scale in the intermediate time domain, but have a smaller scaling exponent a 5 0.78 compared to that of the networks introduced in this paper. Moreover, the four curves saturate to a constant value later than those of Fig. 5, while the plateau values AEP(')ae are exactly the same for both figures and equal to 1/N g 6,7 . This indicates that the equipartition of the energy over all beads is reached faster for the V d g networks than for the dual Sierpinski gaskets with the same number of nodes and edges.

Discussion
In summary, we have introduced a class of small-world networks constructed based on complete graphs. First, we have calculated the full Laplacian spectrum obtained from recursion formulae and proved its completeness. The corresponding analytic expressions allowed us to analyze the eigenvalues in detail and to calculate the related spectral dimensiond. Using the eigenvalues, we have discussed the dynamics of such polymer networks in the GGSs framework, as well as the energy transfer through fluorescence depolarization. The ensuing spectral dimensiond~2 leaves its fingerprints in all quantities considered in the paper. In the intermediate time or frequency domain they follow the asymptotic relations [5][6][7]35,36 : which were proven here by the numerical calculations. The networks introduced here are deterministic and highly structured, however, in case of a possible weak disorder leading to smoothing out of the curves the conclusions will still hold. We believe that recent advances in the synthesis of fractal supramacromolecular polymers 21 will open new perspectives for the compounds constructed based on the symmetric small-world networks presented in the report. Finally, we remark that we expect to find more applications of the networks considered here; in particular, the analytic expressions for the Laplacian eigenvalues determined here will be of much help. .

Methods
and The characteristic polynomial of the L V d g is determined as: The matrix lI{L V d g can be rewritten as: we obtain Figure 5 | The average probability AEP(t)ae, equation (21), for V 4 g , where g runs from 7 to 10. Thus, where becomes where vector v i (1 # i # d) are components of v. Equation (34) leads to the following equations: Then we know that v 1 is the eigenvector corresponding to the eigenvalue 0 in The set of all solutions to any of the above equations consists of vectors of the following form where k 1,j , k 2,j , …, k d22,j are arbitrary real numbers. In Eq. (37), the solutions for all the vectors v i (2 # i # d) can be rewritten as  where vector v i (1 # i # d) are components of v. Eq (39) leads to the following equations: