Abstract
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 smallworld 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.
Introduction
One 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 firstpassage time problems^{9,10,11} and so on. Laplacian eigenvalues and eigenvectors play an irreplaceable role and they are also relevant to multiaspects 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 smoothingout 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 Sierpinskitype 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 socalled smallworld 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 − 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 → ∞. As we proceed to show, also in this limit they have similar behavior as the dual Sierpinski gaskets embedded in to d → ∞ 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 smallworld networks (SWNs) 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 to : At first, is a simple triangle, that is, a complete graph with 3 nodes. At the next stage, each node in is replaced by a new complete graph. Thus each of the newly appeared complete graphs contains exactly one node of and we get the network at second generation . The growth process to the next generation continues in a similar way: Connecting a complete graph to each of the node of one gets . In general, we have d^{g}^{−1} nodes at generation g − 1. By attaching d − 1 nodes to each existing node, increases their total number from d^{g}^{−1} to d^{g}. In this way, we get immediately the number of nodes in this network, and the number of edges, . It has to be mentioned that the dual Sierpinski gaskets embedded in (d − 1)dimension have exactly the same number of nodes and of edges^{33}.
To give evidence of the smallworld 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}. Let be the diameter of network . It is clearly that at generation g = 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 Ω_{g} is just equal to 2g − 1, a result irrelevant to parameter d. The value can be presented by another form 2 log_{d}N_{g} − 1, which grows logarithmically with the network size indicating that the networks are smallworld^{1}.
Now we turn to the clustering coefficient of any node i, which is given by C_{i} = 2e_{i}/[k_{i}(k_{i} − 1)], 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 − 1)h nodes will be attached to it. That is, k_{x} = (d − 1)h. Among the (d − 1)h neighbors, d − 1 nodes that belong to the same complete graph are connected to each other, leading to the total number of links e_{x} = h[(d − 1)(d − 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 〈C〉 as a function of g for d going from 3 to 6. As one can infer from the figure, 〈C〉 decreases very rapidly at small generations to a some constant value, which depends on d. In fact, one can find from equation (3) that for the average clustering coefficient is given by 〈C〉_{∞}(d) = ((d − 1)/d)_{2}F_{1}[(d − 2)/(d − 1), 1; (2d − 3)/(d − 1); 1/d], where _{2}F_{1}[…] is the hypergeometric function, i.e. 〈C〉_{∞}(3) ≈ 0.76, 〈C〉_{∞}(4) ≈ 0.84, 〈C〉_{∞}(5) ≈ 0.88 and 〈C〉_{∞}(6) ≈ 0.9. For very large d (d → ∞), equation (3) converges to value , an inherent property of a complete graph.
Recursion formulae for the Laplacian spectrum
Let denote the adjacency matrix of , where A_{ij} = A_{ji} = 1 if nodes i and j are adjacent, A_{ij} = A_{ji} = 0 otherwise, then the degree of node i is . Let denote the diagonal degree matrix of , then the Laplacian matrix of is defined by .
To get a solution for the eigenvalues of , we have to concentrate our attention on its characteristic polynomial, . 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 . Note that has a factor λ − d with exponent (d − 2)d^{g}^{−1}, i.e. equation (4) has the root λ = d with multiplicity at least (d − 2)d^{g}^{−1}.
It is evident that has d^{g} Laplacian eigenvalues, denoted by , , …, , the set of which is represented by Λ_{g}, i.e., . In addition, without loss of generality, we assume that . On the basis of above analysis, Λ_{g} can be divided into two subsets and satisfying , where contains all eigenvalues equal to d, while includes the remaining eigenvalues. Thus,
The remaining 2d^{g}^{−1} eigenvalues belonging to are determined by . Let the 2d^{g}^{−1} eigenvalues be , , …, , respectively. That is, . Given that the is the characteristic polynomial of leading to N_{g}_{−1} eigenvalues , the set follows from
or from
where i runs from 1 to N_{g}_{−1} = d^{g}^{−1}.
Solving the quadratic equation (7), we obtain two roots and , where r_{1}(x) and r_{2}(x) are
and
respectively. Thus, each eigenvalue of Λ_{g}_{−1} gives rise to two new eigenvalues in by inserting each Laplacian eigenvalue of Ω_{g}_{−1} into equations (8) and (9). Considering the initial value , by recursively applying equations (8) and (9) and accounting for , the Laplacian eigenvalues of Ω_{g} are fully determined.
It is simple matter to check that equations (8) and (9) have the following behaviors:
and
In this way equation (10) produces only small eigenvalues, r_{1}(x) ∈ [0, 1) and equation (11) the large ones, r_{2}(x) ∈ [d, ∞). Thus, the eigenvalue spectrum has always a gap [1, d), which is bigger for networks with larger d.
Now, it is interesting to examine the behavior of the small eigenvalues, i.e. to consider equation (10) for . Our goal is to obtain the spectral dimension (also known as fracton dimension^{35}). For this we use the methods of Ref. 36. Under equation (10) for , the n eigenvalues in the interval [λ^{g}, λ^{g} + Δλ^{g}] go over in n eigenvalues in the interval [λ^{g}^{+1}, λ^{g}^{+1} + Δλ^{g}^{+1}/d], while the total number of modes increases from N to dN. Hence, the density of states (modes) ρ(λ) for obeys
Using now the relation between ρ(λ) and the spectral dimension ^{35},
leads to
This means that the spectral dimension of the networks is and is independent on d. We note that for the dual Sierpinski gasket embedded in (d − 1)dimension the spectral dimension is , see e.g. Refs. 37, 38, i.e. it is similar to that of only in the limit d → ∞.
Dynamics of polymer networks under external forces
We are going to study the networks under the framework of generalized Gaussian structures (GGS)^{3,4,5}, an extension of the classical Rouse beadssprings model^{2,39,40,41}. Here we let all N beads of the GGS to be assigned to the same friction constant, ζ. 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) = (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 . Moreover, f_{m}(t) is the thermal noise that is assumed to be Gaussian with zero mean value 〈f_{m}(t)〉 = 0 and 〈f_{mα}(t)f_{nβ}(t′)〉 = 2k_{B}Tδ_{αβ}δ_{mn}δ(t − t′), where k_{B} is the Boltzmann constant, T is the temperature, α and β 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) = FΘ(t)δ_{mk}e_{y} (∀k), started to act at t = 0 (Θ(t) is the Heaviside step function) on a single bead m of the in the y direction. After averaging over all possibilities of choosing this monomer randomly, the displacement reads^{4,5,39}
where σ = K/ζ is the bond rate constant and λ_{i} is the eigenvalues of matrix L with λ_{1} being the unique smallest eigenvalue 0.
Another example is the response to harmonically applied forces (strain fields), i.e. F_{m}(t) = γ_{0}e^{iωt}Y_{m}(t)e_{x}. The related response function is the socalled complex dynamic modulus G*(ω), or equivalently, its real G′(ω) and imaginary G″(ω) components (the storage and the loss moduli^{41,43}). In the GGS model (for very dilute thetasolutions) the G′(ω) and G″(ω) are given by^{3}
and
where ν denotes the number of polymer segments (beads) per unit volume.
We start by focusing on the averaged displacement 〈Y(t)〉, equation (16), where we set σ = 1 and . Figure 3 displays in double logarithmic scales the 〈Y(t)〉 for the networks consisting of 4^{7} up to 4^{10} beads. As is known^{4,5,39}, the 〈Y(t)〉 in such GGS at very long times reaches the domain 〈Y(t)〉 ~ Ft/(Nζ) and at very short times obeying 〈Y(t)〉 ~ Ft/ζ. 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 − 1)dimension)^{37,38}, which show a slow subdiffusive behavior 〈Y(t)〉 ~ t^{α} with α ≈ 0.23 for d = 4.
While the 〈Y(t)〉 of do not scale in the intermediate domain, the mechanical relaxation functions show in the related frequency domain a scaling behavior, see the results for storage moduli G′(ω) presented in Fig. 4. Here we plot them in dimensionless units by setting σ = 1 and . The networks are the same as for 〈Y(t)〉 of Fig. 3. The G′(ω) behaves commonly at very small and very high frequencies as ω^{2} and ω^{0}, respectively. The inbetween region of G′(ω) (related to the intermediate time domain of 〈Y(t)〉) the curves give in doublelogarithmic scales the slopes around 1. This result is bigger than that in the same region of the corresponding dual Sierpinski gaskets embedded into 3dimensional space (there one has slopes near 0.77)^{37}. For a better visualization, we plot in the inset of Fig. 4 the effective slopes for the same curves of Fig. 4. As expected, the limiting behaviors for very low and very high frequencies hold for slope 2 and slope 0. But in the intermediate frequency region, all of the four curves become wavy. Such a waviness reflects typically^{36,37,38} a very symmetric, hierarchical character of the structures. In case of real polymer systems, the inherit structural disorder smooths out such wavy patterns, while keeping the characteristic intermediate scaling^{20}. Finally, the curves cross each other at the slope 1, keeping a short stable period and then falling into a value of 0.5.
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(−t/τ_{R}), where 1/τ_{R} corresponding to the radiative decay rate. Under the assumption that all microscopic rates are equal to each other, fixed on a value , 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 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^{6,7,8}
Measuring the time in units of , we can obtain the 〈P(t)〉 with . In Fig. 5 we display in double logarithmic scales the average probability 〈P(t)〉 that an initially excited chromophore of the network is still or again excited at time t. As for the previous figures, we choose d = 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 fractals^{6} and it is related to high symmetry (regularity) of the network, i.e. the averaging due to possible disorder will smooth out the curves. Besides, in the intermediate time domain the decays show a powerlaw behavior, i.e. 〈P(t)〉 ~ t^{−α}. In Fig. 5 the α float around 0.98 for all four generations, a very high value among similar kinds of networks.
For the sake of comparison, in Fig. 6 we display the 〈P(t)〉 for dual Sierpinski gaskets embedded into 3dimensional 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 α = 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 〈P(∞)〉 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 networks than for the dual Sierpinski gaskets with the same number of nodes and edges.
Discussion
In summary, we have introduced a class of smallworld 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 dimension . 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 dimension 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 smallworld 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
Characteristic polynomial for the Laplacian eigenvalues of
Following from the construction of , the adjacency matrix , the degree matrix and the Laplacian matrix can be expressed as
and
The characteristic polynomial of the is determined as:
The matrix can be rewritten as:
Now, using the matrix determinant lemma, see e.g. Ref. 44
we obtain
Thus,
where
Laplacian Eigenvectors of
Analogous to the eigenvalues, the eigenvectors of can also be derived directly from those of . Assume that λ is an eigenvalue of Laplacian matrix for , the corresponding eigenvector of which is v ∈ R^{dg}, where R^{dg} is the d^{g}dimensional vector space. Then the eigenvector v can be determined by solving equation . We distinguish two cases: and , which will be separately treated as follows.
For the case of , in which all λ = d, equation 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 , that is, . Let , then, Eq. (35) is equivalent to the following equations:
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_{d}_{−2,j} are arbitrary real numbers. In Eq. (37), the solutions for all the vectors v_{i}(2 ≤ i ≤ d) can be rewritten as
where k_{i}_{,j}(1 ≤ i ≤ d − 2; 1 ≤ j ≤ d^{g}^{−1}) are arbitrary real numbers. Using Eq. (38), we can obtain the eigenvector v associated with the eigenvalue d. Furthermore, we can easily check that the dimension of the eigenspace of matrix corresponding to eigenvalue d is (d − 2)d^{g}^{−1}.
We proceed to address the case of . For this case, equation can be rewritten as
where vector v_{i}(1 ≤ i ≤ d) are components of v. Eq (39) leads to the following equations:
Resolving Eq. (40) yields
As demonstrated in the first subsection of Methods, if λ is an eigenvalue of , then is an eigenvalue of . When i ≤ d^{g}^{−1}, we have , while in the situation d^{g}^{−1} < i ≤ 2d^{g}^{−1}, . From Eq. (41), vector v_{1} is the eigenvector of corresponding to the eigenvalue . Applying the v_{1} into Eq. (41), we will get all of the v_{i}(2 ≤ i ≤ d) and finally the eigenvector of corresponding to . In this way, we have completely determined all eigenvalues and their corresponding eigenvectors of .
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Acknowledgements
This work was supported by the National Natural Science Foundation of China under Grant No. 11275049. M.D. acknowledges DFG through Grant No. Bl 142/111 and through IRTG “Soft Matter Science” (GRK 1642/1).
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H.L., M.D. and Z.Z.Z. designed the research. H.L. and Y.Q. performed the research. H.L., M.D. and Z.Z.Z. wrote the manuscript.
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Liu, H., Dolgushev, M., Qi, Y. et al. Laplacian spectra of a class of smallworld networks and their applications. Sci Rep 5, 9024 (2015). https://doi.org/10.1038/srep09024
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