Cycle-Star Motifs: Network Response to Link Modifications

Understanding efficient modifications to improve network functionality is a fundamental problem of scientific and industrial interest. We study the response of network dynamics against link modifications on a weakly connected directed graph consisting of two strongly connected components: an undirected star and an undirected cycle. We assume that there are directed edges starting from the cycle and ending at the star (master-slave formalism). We modify the graph by adding directed edges of arbitrarily large weights starting from the star and ending at the cycle (opposite direction of the cutset). We provide criteria (based on the sizes of the star and cycle, the coupling structure, and the weights of cutset and modification edges) that determine how the modification affects the spectral gap of the Laplacian matrix. We apply our approach to understand the modifications that either enhance or hinder synchronization in networks of chaotic Lorenz systems as well as R\"ossler. Our results show that the hindrance of collective dynamics due to link additions is not atypical as previously anticipated by modification analysis and thus allows for better control of collective properties.


Introduction
Many systems in nature are modelled as networks of interacting units with examples ranging from neuroscience [1] to engineering [2].Recent work has revealed that the network interaction structure plays a crucial role in the network emergent dynamics [3,4,5,6,7].Predicting the impact of the network structure on the dynamics is an intricate nonlinear problem that leads to many unexpected results.Indeed, in some situations improving the network structure may lead to functional failures such as Braess's paradox [8] and synchronization loss [9,10].In large networks depending on the interaction function and isolated dynamics of the nodes, a topological hub may fail to be a functional hub [11,12].
The effects of network topology on dynamical phenomena, such as synchronization, diffusion and random walks can be related to spectral properties of the graph, see for instance [13,3].Indeed, to predict the consequences of network modification on the dynamics, one needs to investigate the highly nonlinear changes in the spectrum of the graph Laplacian [14,9,15] Although certain correlations between network structure and dynamics have been observed in experimental [16] and theoretical [9,17] investigations, most of these results are concerned with small modifications to the network.There is a lack of rigorous results to determine the relationship between the network structure and its dynamic properties for arbitrary size modifications.Most of the results in this direction rely on the modification theory of eigenvalues to determine which structural changes are detrimental to the network dynamics.However, previous results relying on perturbation theory suggest that desynchronizing the network by adding new links is unusual [14].To understand this problem, we need to unveil the full nonlinear picture and deal with large changes in the topology.
Networks are a combination of motifs that dictate dynamical behavior and provide resilience to the overall system [18,19].We focus on two motifs of complex networks -a cycle and a star -since they are the main constituents of important networks.Indeed, cycles are typical components in the nervous system [20] and orientation tuning in visual cortex [21].Also, in the context of neuroscience highly connected nodes, called hubs, play a fundamental role in the network [22].These networks with hubs are modeled as a collection of star motifs, and each star motif is capable of generating intricate dynamics [23], as well as their overall interaction [24].
Although both cycle and star motifs were investigated for noteworthy network dynamics such as collective behavior [25,26,27,28,29] and both motifs have a fully developed spectral theory [30] their eigenvectors and eigenvalues can be fully described (as in the case of rings where the matrices are circulant), when these motifs are coupled, the eigenvalues problem becomes an intricate nonlinear problem that remains open.
In this paper, we consider models of networks consisting of cycles and stars coupled in a master-slave topology.Although our problem is dynamics-motivated, we state our main results in a graph theoretic form and consider the synchronization as an application.This is because, in a broader sense, the spectral properties of the graph Laplacian are important in the study of graph connectedness and, hence, any phenomena related to this concept [31].

Informal statements of our results
We consider three models illustrated in Figures 1, 2, and 3.All these three models have a master-slave structure, a cycle C n , a star S m , and cutset edge(s) starting from the cycle and ending at the hub of the star.We modify these networks and break the master-slave structure by adding directed links from the star to the cycle (red-color edges in the figures).We add links from nodes of the star to the one cutset node of the cycle.The weight of each of the black-color edges is one while the weights of the red-color edges (modification edges) are arbitrary.

Adjacency matrices and graph Laplacians
Let G be a weighted directed graph (digraph) whose nodes are labelled by 1, . . ., n.We define the adjacency matrix of G by A G = (A ij ), where A ij ≥ 0 is the weight of the directed edge starting from node j and ending at node i.The in-degree of a node is the sum of the weights of the edges that the node receives from other nodes, i.e., the in-degree of the node i is j A ij .We define the Laplacian matrix of G by , where D G is a diagonal matrix whose (i, i)-entry is the in-degrees of the node i of G. Let L G and L Gp represent the Laplacians of the unmodified and modified graphs, respectively.Let λ 2 (L G ) and λ 2 (L Gp ) be the associated second minimum eigenvalues, so-called spectral gap.Our results explain how the modification affects the spectral gap of the Laplacian matrices of these models.We provide more details in Section 4.

Results (informal version)
Assume δ 0 ≥ 0 is the weight of the modification edge starting from the hub and δ ≥ 0 is the sum of the weights of all the modification edges.In model I, we have δ 0 = δ, and in the other two models, δ 0 ≤ δ.Let m and n be the sizes of the star and cycle, respectively.The term o(1) in the informal statements of the following Theorems A1 and B1 (resp.Theorem C1) stands for a function of m (resp.(m, w)) that converges to 0 as m → ∞ (resp.m w → ∞).When the weight of the modification is small, we will call this modification local.This is because the results follow from local analysis of the eigenvalues.If the weight of the modification is large we called it global, as the analysis require global techniques to gain insights on the eigenvalues.These models are discussed precisely in Section 3. Here, we give an informal version of our main results.
Theorem A1 (Informal statement).Consider model I illustrated in Figure 1.Let the modification δ > 0 be arbitrary (it does not need to be sufficiently small).We have 1.Although L Gp is not necessarily symmetric, all of its eigenvalues are real.

There exists a critical cycle size n
We illustrate Theorem A in Figure 4. To give the informal statement of Theorem B2, let ρ := δ 0 δ .This ratio can be seen as a measure of the modification that the cycle receives from the hub of the star relative to the modification it receives from the leaves of the star.We have ρ ≤ 1, and by setting ρ = 1, the model II reduces to model I.
Theorem B1 (Informal statement).Consider model II illustrated in Figure 2. We have 1.Under a local modification, the statement of Theorem A1 is valid for model II.When δ > 0 is sufficiently small, all the eigenvalues of L Gp are real and, the modification decreases the spectral gap if and only if the size of the cycle is larger than the critical value n c (m).
The Laplacian matrix L G of the unmodified graph in all the models I, II, and III is a block lower-triangular matrix, see the form (4.1).However, adding a modification in the opposite direction of the cutset breaks the triangular structure of L G , which turns the analysis of its spectral gap into a non-trivial problem.Our approach to analyzing the changes in the spectral gap consequent to the graph modification is to investigate a secular equation of the Laplacian matrix and its roots.Our analysis is not restricted to the local modification1 ; we indeed analyze the change in the spectral gap under modification of arbitrary size.This requires further work on not only analyzing the modification of spectral gap but also understanding the modifications and distribution of the whole spectrum of the Laplacian matrix.

Applications to synchronization
We consider synchronization in networks of diffusively coupled oscillators as an application.Consider a triplet G = (G, f, H), where G is a weighted digraph, and f, H ∈ C 1 (R l ) for l ≥ 1.
The triplet G defines a system of the form where Θ ≥ 0 is called the coupling strength.Each variable x i represents the state of the ith node of G, the function f describes the isolated dynamics at each node, and the function H is called the coupling function.We call the triplet G or its associated system (2.1) a network of diffusively coupled (identical) systems.
We define the synchronization manifold as We say that a network G synchronizes if there exists an open neighborhood V of M such that the forward orbit of any point in V converges to M .It is shown [33] that for a network G with a coupling strength Θ, if 1. the graph G has a spanning diverging tree, and 2. there exists an inflowing open ball U ⊂ R l which is invariant with respect to the flow of the isolated system ẋ = f (x), and we have ∥Df (x)∥ ≤ K for some K > 0 and all x ∈ U , and 3. we have H(0) = 0; moreover, all the eigenvalues of DH(0) are real and positive, then there exists Θ c ≥ 0 such that when Θ ≥ Θ c , G synchronizes.We call the critical coupling strength where ρ = ρ(f, DH(0)) is a constant.Note that if the third assumption is not fulfilled, the synchronization condition (2.3) may no longer be valid.However, in this case, new synchronization conditions may be obtained under the framework of master stability function formalism [3].Relation (2.3) with assumptions stated above gives us a criterion to compare synchronizability in networks.More precisely, that satisfy the assumptions above.Let Θ c (G 1 ) and Θ c (G 2 ) be the critical coupling strengths of G 1 and G 2 , respectively.We say that G 1 is more synchronizable than Having Θ c (G 1 ) < Θ c (G 2 ) means that G 1 synchronizes for a larger range of Θ than G 2 .Let us now consider the case that two networks G 1 and G 2 only differ in their topology, i.e., having the same isolated dynamics and coupling functions while the graph structures can be different.In this case, following (2.3), the spectral gaps of the underlying graphs of the networks determine which one is more synchronizable.

Synchronization of coupled Lorenz oscillators
We consider the following settings for model II given in Figure 2: Two networks G = (G, f, H) and G p = (G p , f, H) are generated where G and G p are the unmodified and the modified graphs, respectively.The chosen isolated dynamics f is the Lorenz oscillator where σ = 10, γ = 28, β = 8/3.Here, H is the identity function on R 3 .For the described setting, equation (2.3) can be written as Θ c = κ Re(λ 2 ) , where κ is defined as in Section 5 of [3].We numerically find that κ ≈ 0.9.So, the expected values of Θ c (G) and Θ c (G p ) are calculated accordingly.We examine two experiments to reveal how link addition can lead to synchronization in the network G p or break the synchronization in the initial network G (see Figures 5 and 6).The network of coupled Lorenz oscillators in model II is simulated to show the synchronization error It is worth mentioning that the same simulations that we have done for the Lorenz system can be done for other systems as well.Indeed, similar results hold as long as the initial conditions lead to an attractor in the synchronization manifold that is contained in a compact set.

Hindering synchronization
To examine the hindrance of synchronization due to link addition, the overall coupling constant Θ is selected such that Θ c (G) < Θ < Θ c (G p ) (see Figure 5).Note that such Θ values only exist when λ 2 (G) > λ 2 (G p ) due to the order relations of synchronizability stated above.When the selected Θ is above the Θ c (G), the trajectories synchronize for the network G.Then, the system is modified by adding links at a given time t.Since the selected Θ is below the Θ c (G p ), the system loses its synchronization thereafter.
In model II, the sizes of the cycle and star subgraphs are set to n = 15 and m = 15.The weights of the cutset and modification edges are w 0 = 1 and δ i = 1, where i = 0, 1, . . ., m−1.
All initial states are randomly selected from the uniform distribution over [3.5, 5).We consider H = I as the coupling function.We choose initial conditions randomly selected from the uniform distribution over [3.5, 5), and integrate the network until time t = 2500s.The system goes into synchronization after some transient.At t = 2500s, we add the red links to the system, i.e., δ i = 1, where i = 0, 1, . . ., m − 1, and perturb the system by adding noise randomly selected from the uniform distribution over [0.01, 0.02) to each state, then the synchronization loss occurs and the system doesn't return into synchronization after transient time.Note that, α 1 = 0.04, β − 15,1 = 0.06 in Theorem B.

Enhancing synchronization
To examine the enhancement of synchronization due to link addition, the overall coupling constant Θ is selected such that Θ c (G p ) < Θ < Θ c (G) (see Figure 6).Note that such Θ values only exist when λ 2 (G p ) > λ 2 (G).When the selected Θ is below the Θ c (G), the trajectories cannot synchronize for the network G.Then, the system is modified by adding links at a given time t.Since the selected Θ is above the Θ c (G p ), the system synchronizes.We consider H = I as the coupling function.We choose initial conditions randomly selected from the uniform distribution over [3.5, 5), and integrate the network until time t = 2500s.After the red links are added to the system at t = 2500s, i.e., δ i = 1, where i = 0, 1, . . ., m − 1, the synchronization occurs where the mean error ⟨E⟩ goes to zero.Note that, α 1 = 0.12, β − 15,1 = 0.06 in Theorem B.
In model II, the sizes of the cycle and star subgraphs are set to n = 9 and m = 15.The weights of the cutset and modification edges are w 0 = 1 and δ i = 1, where i = 0, 1, . . ., m−1.
All initial states are randomly selected from the uniform distribution over [3.5, 5).Therefore, hindrance and enhancement of synchronization due to link addition manifest themselves in simulations, and it perfectly agrees with the findings of our theorems.

Problem setting and results
Let G be a weighted directed graph (digraph) whose nodes are labelled by 1, . . ., n.We assume that G is unilaterally connected (a digraph is unilaterally connected if for any two arbitrary nodes i and j, there exists a directed path from i to j or j to i).This implies that the zero eigenvalue of L G is simple [34].It is also an easy consequence of Gershgorin theorem that the real parts of all the non-zero eigenvalues of L G are positive.Let λ 1 , . . ., λ n be the eigenvalues of L G , ordered according to their real parts, i.e.
The second minimum (with respect to the real-part ordering) eigenvalue, i.e. λ 2 , is called the spectral gap of G.In this paper, we are interested in how modifying G can affect its spectral gap for models I, II and III.
Let us start with model I (see Figure 1a).We define Definition 2. Consider arbitrary integers n ≥ 3 and m ≥ 4, and an arbitrary real number w ≥ 0. Then 1. for any integer 0 ≤ l ≤ n, we define 2. we define β − m,w and β + m,w as the roots of the quadratic polynomial λ 2 − (m + w) λ + w, i.e.
Remark 1.By virtue of Taylor's theorem, we can approximate α l for sufficiently small l n by Before we proceed to our first result, let us give some intuition about this definition.The parameter w in β ± m,w stands for the sum of the weights of all the cutset edges starting from the cycle and ending at the star.In the case of model I and II, we assume w = 1, but for model III, we deal with arbitrary w.As it is shown later (see Proposition 1), the spectrum of the unmodified Laplacian L G is {α l : where 0 ≤ l ≤ n and l is even} ∪ {β − m,w , 1, β + m,w }.Thus, the spectral gap of L G is given by min{α 2 , β − m,w }.Although the α l s for odd l do not appear as the eigenvalues of L G , they play an important role in our theory.In particular, α 1 appears in the formulation of all the three main results of this paper.
Here is our main result on model I: Consider an arbitrary modification δ 0 > 0 and the corresponding Laplacian L Gp = L G (δ 0 ).Then, all the eigenvalues of L Gp are real.Moreover, we have The assumption β − m,1 / ∈ {α l : 0 ≤ l < n} in this theorem (and also in the next theorem) typically holds for arbitrary m and n.
Let us now discuss model II (see Figure 2a).Let δ i ≥ 0 be the weight of the edge starting from node i (see Figure 2b).Thus, model II is reduced to model I by setting δ i = 0 for i = 1, . . ., m − 1.In this strand, we define Definition 3. Let δ i ≥ 0, i = 0, . . ., m − 1, be the weight of the modification edge starting from node i of the star and ending at node 0 of the cycle.We define δ := (δ 0 , . . ., δ m−1 ), and Obviously, δ = 0 if and only if δ = 0. Note also that δ = 0 corresponds to the unmodified graph G.We now state our next main result: Consider a modification δ ̸ = 0 and let L Gp = L G δ be the corresponding Laplacian.Then, the following hold.
(i) (Local modification) Let δ ̸ = 0 be a sufficiently small modification.Then, all the eigenvalues of L Gp are real, and Then, all the eigenvalues of L Gp are real, and the statements (ia) and (ib) of this theorem also hold for the modification δ.
This theorem is proved in Section 5.3.Let us mention a few remarks.
Remark 3. Note that, by setting δ = δ 0 , Theorem A2 directly follows from Theorem B2.Remark 4. In spite of Theorem A2 for which the main statements hold for a modification of arbitrary size, in Theorem B2, we require a condition on the modification, i.e. δ < δ 0 β + m,1 , to make the statements for modifications of arbitrary size.Roughly speaking, this is due to the possibility of the emergence of non-real eigenvalues.Indeed, as it is shown in the proof of Theorem B2, for small modification δ ̸ = 0, the modified Laplacian L Gp has two real eigenvalues in the interval (α n−1 , ∞).However, as δ varies and gets larger in size, these two real eigenvalues may collide and become a pair of complex conjugates.In this case, we can think of the scenario in which the real part of these eigenvalues decreases such that for some sufficiently large modification δ, these eigenvalues become the spectral gap of L Gp .By assuming δ < δ 0 β + m,1 , we indeed avoid this scenario.We now discuss model III (see Figure 3a).Let w i ≥ 0, where i = 0, . . ., n − 1, be the weight of the edge starting from node i of the cycle.Without loss of generality, assume w 0 > 0. We also define Definition 4. Let w i be as mentioned above.We define w = (w 0 , . . ., w n−1 ) and w = We show later that λ 2 (L G ) = min{α 2 , β − m,w }.Regarding the modification in the case of model III, we consider the same family of modifications as we considered in model II: for every 0 ≤ i ≤ m − 1, there exists a modification edge with weight δ i ≥ 0 starting from node i of the star and ending at node 0 of the cycle (see Figure 3b).Let δ and δ be as in Definition 3.For given m, n, w, δ 0 and δ, in the case that α 2 ̸ = β − m,w , we also define As it is shown later, the sign of S determines if the characteristic polynomial of L Gp , i.e. det L Gp − λI , decreases or increases at the point λ = α 2 .Our last main result is as follows.
(i) (Local modification) Let δ ̸ = 0 be sufficiently small.Then, all the eigenvalues of L Gp are real, and we have (ii) (Global modification) Let δ ̸ = 0 be an arbitrary modification and assume 4 The Laplacian matrices 4.1 The Laplacian L G of the unmodified graph and its spectrum In this section, we investigate the spectrum of the unmodified Laplacian matrix L G .Denote the Laplacian matrices of the cycle C n and the star S m by L Cn and L Sm , respectively.Then where Moreover, for models I and II, we have and and for model III, we have The block triangular form of ).Thus, to study σ(L G ), we need to investigate each of σ(L Cn ) and σ(L Sm + D C ) individually.In this strand, we have the following lemmas.
Lemma 1. Recall Definition 2. We have σ(L Cn ) = {α l : where 0 ≤ l ≤ n and l is even}.Moreover, the multiplicity of all the eigenvalues except for 0 and 4 (the eigenvalue 4 appears only when n is even) is 2.
Proof.See [30].Proof.This lemma is a special case of Lemma 18, which is proved in B.
The previous two lemmas give the spectrum of the unmodified Laplacian L G : Proposition 1.We have σ(L G ) = {α l : where 0 ≤ l ≤ n and l is even} ∪ {β − m,w , 1, β + m,w }.Remark 5. We assume that m ≥ 4, i.e. the star S m has at least four nodes.It is straightforward to show that for any m ≥ 4 and w > 0, we have β − m,w < 1 and 4 < β + m,w .On the other hand, 0 ≤ α l = 2 1 − cos lπ n ≤ 4, for all 0 ≤ l ≤ n.This means that β + m,w is a simple eigenvalue of L G .

The Laplacian L G p of the modified graph
Consider model III and observe that the modified Laplacian matrix L Gp is given by where C and D C are as in (4.3), . (4.5) Notation 1.For the sake of convenience, we set Using this notation, Laplacian (4.4) is written as The Laplacian L Gp of the modified graph of model II is of the form (4.6), where C and D C are as in (4.2), and ∆ and D ∆ are given by (4.5).
The Laplacian L Gp of the unmodified graph of model I is also of the form (4.6), where C and D C are as in (4.2), and ∆ and D ∆ are given by and Here (model I), we have δ 0 = δ.
Notice that, in all these three models, despite the unmodified Laplacian L G , the modified Laplacian L Gp does not have a triangular form.Due to this reason, analysis of the spectrum of L Gp requires further work.We deal with this analysis in the next section.

Proofs of the main results
In this section, we prove our main results: Theorems B2 and C2 (Theorem A2 follow from Theorem B2).Note that model II can be considered as a special case of model III.Thus, it is reasonable to introduce the main concepts and notations of the proofs in this section mainly based on model III.This section is organized as follows.We first discuss some preliminaries, definitions and notations in Section 5.1.In Section 5.2, we discuss the techniques that are used in the proofs of the theorems.We then prove Theorem B2 in Section 5.3.Finally, we prove Theorem C2 in Section 5.4.

Preliminaries, definitions and notations
In this section, we discuss some preliminaries, and introduce some concepts and notations which are used throughout the proofs.
Notation 2. Throughout, 1 k stands for the k-dimensional vector whose entries are all 1.
We may drop k when it is clear from the context.
Definition 5. Let w, δ 0 and δ be real, and m and k be positive integers.Consider λ ∈ R.
(ii) the matrix R = Q −1 k exists for θ ̸ = lπ k+1 (l = 1, . . ., k), and is given by k exists for all λ ≥ 4, and is given by (5.5) Recall α l defined by (3.1).By Lemmas 3 and 4, and a straightforward calculation, we have 5.2 Our approach for investigating the spectrum of the modified Laplacian L G p In this section, we discuss the method we use to investigate the spectrum of the modified Laplacian L Gp .We directly apply this method to study model III and then use the results to investigate models I and II.
Recall that the modified Laplacian of model III is given by where L 1 and L 2 are as in Notation 1, and the matrices C and ∆ are given by (4.3) and (4.5), respectively.Our study of the eigenvalues of L Gp is based on the following lemma.
Lemma 6.Consider the modified Laplacian L Gp given by (5.6).For λ ∈ R, we have and d k P i dλ k (λ 0 ) ̸ = 0. Remark 6. Lemma 6 allows us to count the multiplicity of λ 0 ∈ σ(L Gp ) when λ 0 / ∈ σ(L 1 ) ∩ σ(L 2 ).However, this lemma may give information about the multiplicity of λ 0 when λ 0 ∈ σ(L 1 ) ∩ σ(L 2 ) as well.This is important for us since we have such eigenvalues in our models.Let λ 0 be such an eigenvalue.Since λ 0 ∈ σ(L 1 ), the matrix (L 1 − λ 0 I) −1 does not exist.However, depending on the matrices C and ∆, the expression lim λ→λ 0 Y (λ), where Y (λ) := C(L 1 − λ 0 I) −1 ∆, may exist.This allows us to define M 1 and P 1 at λ = λ 0 by taking the limit λ → λ 0 .Now, if Y (λ) at λ = λ 0 is smooth enough, then the multiplicity of λ 0 as an eigenvalue of L Gp is l + k, where l is the multiplicity of λ 0 as an eigenvalue of L 1 and k is the integer that satisfies P 1 (λ 0 is well-defined and smooth enough at λ = λ 0 . According to Lemma 6, an eigenvalue λ of L Gp that is not in σ(L 1 ) ∩ σ(L 2 ) must satisfy P 1 (λ) = 0 or P 2 (λ) = 0.The proofs of our results are based on the analysis of these two equations.Sections 5.2.1 and 5.2.2 are dedicated to this analysis.
Before we proceed further, let us show that λ = 1 is an eigenvalue of L Gp for any arbitrary δ.

Proof. Recall that L
It follows from the proof of Lemma 18 (see relation (B.4)) that there exist m − 2 linearly independent left eigenvectors v such that v ⊤ L 2 = v ⊤ .Moreover, any such a vector v is of the form v = (0, v 1 , • • • , v m−1 ) ∈ R m (the first entry is zero).Consider the vector u := (0, v) ∈ R n+m .Taking into account that, except for the first row, all the entries of C are zero (see (4.3)), we obtain This means that for such vs, the corresponding vectors u are left eigenvectors of L Gp associated with the eigenvalue 1.This proves the lemma.

Analysis of P 2
In this section, we investigate the matrix M 2 (λ) and the function P 2 (λ) := det(M 2 (λ)) introduced in Lemma 6 for model III.We first need to analyze the matrix L 2 − λI and its inverse: Lemma 8. Recall µ from (5.1).We have (5.7) Proof.Item (i) is straightforward.Item (ii) follows from Lemma 17.
We now start to calculate By a straightforward calculation and using relation (5.7), for λ / ∈ σ(L 2 ), we have ∆ (L 2 − λI) −1 C = yC, where y = y(λ) is given by (5.2).Note that y, and therefore yC, is well-defined and smooth at λ = 1.In other words, although (L 2 − λI) −1 is not defined at λ = 1 (because 1 ∈ σ(L 2 )), the expression ∆ (L 2 − λI) −1 C can be defined at λ = 1, and so do the matrix M 2 and the function P 2 .This was discussed earlier in Remark 6.We give the following lemma to emphasize this property.

Proof of Theorem B2
In this section, we prove Theorem B2.Throughout this section, we assume that w 0 = 1 and w i = 0, where 1 ≤ i ≤ n − 1.Moreover, we have that δ ≥ δ 0 ≥ 0. Note that, to adapt this proof for the case of Theorem A2, it is sufficient to assume δ = δ 0 .We start with the following definition.Definition 6. Recall Definition 2. Assume β − m,1 / ∈ {α l : 0 ≤ l ≤ n} and let κ ≥ 2 be the even integer such that (ii) Let 2 ≤ l ≤ n − 2 be even.We define (iv) For the sake of convenience, we define the set of indices I := {β − , β + } ∪ {l : 0 < l < n and l is even}.
Remark 8.Note that when κ = 2, there does not exists J l for 2 ≤ l < κ.
Remark 9. Notice that β + m,1 > m ≥ 4, and so β + m,1 ∈ J β + .Considering eigenvalues with their multiplicities, the modified Laplacian L Gp has n + m eigenvalues.The next lemma describes where these n + m eigenvalues are located.Lemma 14.Let δ ̸ = 0 be an arbitrary modification that satisfies δ < δ 0 β + m,1 .Then, all the n + m eigenvalues of the modified Laplacian L Gp of model II are real and given by the union of the following four disjoint groups (see also Remark 11).
(iii) Recall the set I. Each interval J γ for γ ∈ I and γ ̸ = β + contains exactly one real eigenvalue of L Gp (except possibly for the m − 2 eigenvalues 1 counted in item (ii)).
We have ⌊ n 2 ⌋ of these intervals, and so L Gp has ⌊ n 2 ⌋ real eigenvalues given by these intervals.
(iv) The interval J β + contains two real eigenvalues of the modified Laplacian L Gp .Thus, L Gp has 2 eigenvalues given by J β + .
Remark 10.Observe that The sets of the eigenvalues given by items (i) and (ii) might not be disjoint, i.e. α l = 1 for some even l.The same may happen for (ii) and (iii), i.e. the eigenvalue in J γ given by item (iii) equals to 1.The eigenvalue 1 in such scenarios are counted separately from the m − 2 eigenvalues 1 given in item (ii).In such scenarios, the multiplicity of eigenvalue The proof of Lemma 14 is postponed to Section 5.3.1.We now prove Theorem B2.
We first show that λ 2 (δ) > α 1 for all δ.Assume the contrary; there exists δ † and correspondingly δ † and δ † 0 for which λ 2 (δ † ) = α 1 .However, lim δ→δ † ξ = ∞.On the other hand, the numerator of (5.20) converges to δ † − δ † 0 α 1 > 0. Therefore, as δ → δ † , expression (5.20) First, we show that when δ ̸ = 0 is sufficiently small, the interval J β + has exactly two real eigenvalues.For even n, this is obvious since at δ = 0, we have two eigenvalues λ = 4 and λ = β + m,1 , and so, as δ varies and remains sufficiently small, these two eigenvalues might move but they remain in J β + and do not collide (so, they remain real).The case of odd n is similar; since for δ ̸ = 0, we have p ′ (α n−1 ) > 0 and p(4) < 0, the intermediate value theorem implies that there is a root in the interval (α n−1 , 4).On the other hand, the eigenvalue β + m,1 ∈ (4, ∞) of L G might move as δ varies but as far as δ is sufficiently small, it does not collied with the eigenvalue that we just found in the interval (α n−1 , 4).Therefore, we have that for small δ ̸ = 0, the interval J β + contains exactly two real eigenvalues of L Gp .

Proof of Theorem C2
We first prove that for sufficiently small δ, all the eigenvalues of L Gp are real.It is known that the roots of a polynomial (in our case, the characteristic polynomial of L Gp ) depend continuously on the coefficients of that polynomial.Therefore, if {λ i δ : i = 1, . . ., n + m} is the spectrum of L Gp , then λ i δ is a continuous function of δ.It is a direct consequence of the implicit function theorem that if λ i (0) is a simple eigenvalue of L G , then for sufficiently small δ, we have that λ i δ is real.Thus, to prove our statement, we need to investigate how multiple eigenvalues of the unmodified Laplacian L G behave as δ varies.
Recall Proposition 1.According to Remark 5 and the assumption β − m,w / ∈ {α l : 0 ≤ l < n} of the theorem, we have that β − m,w and β + m,w are simple eigenvalues of L G .Note that 1 ∈ {α l : 0 ≤ l ≤ n, and l is even} if and only if n 6 is an integer.First, assume n 6 / ∈ Z.In this case, the multiplicity of all the eigenvalues α l except for 0 and 4 (the eigenvalue 4 appears only when n is even) is 2.However, it follows from Lemma 11 that, for each even l, as δ varies, one of the two eigenvalues α l remains as an eigenvalue of L Gp for small arbitrary δ, and the other eigenvalue moves continuously.This means that from each of the multiple eigenvalues α l , two real eigenvalues get born.On the other hand, following Lemma 7 and its proof, the eigenvalue 1 remains an eigenvalue of L Gp with multiplicity m − 2. This implies that when n 6 / ∈ Z and δ is sufficiently small, all the eigenvalues of L Gp are real.The case of n 6 ∈ Z is similar.With the same conclusion, except for l = n 6 , i.e. α l = 1, two real eigenvalues get born from each eigenvalue α l , where l = 1, . . ., n − 1. Regarding α l = 1 (note that the multiplicity of 1 as an eigenvalue of L G in this case is m), we have that α l = 1 remains an eigenvalue of L Gp with multiplicity m − 1 and a new real eigenvalue gets born from it.This proves that when δ is sufficiently small, all the eigenvalues of L Gp are real.
Proof of part (ii) of Theorem C2.Since α 2 < β − m,w , we have λ 2 (L G ) = α 2 (see Proposition 1).On the other hand, by Lemma 16, S and p ′ 2π n have the same sign.If S < 0, since p π n < 0 (see Lemma 16), the intermediate value theorem implies that L Gp has an eigenvalue (p has a root) smaller than α 2 .Denote this eigenvalue by λ δ .When the modification is small, this eigenvalue is indeed the spectral gap of L Gp , as discussed in the proof of part (ia).However, for large modification, there is the possibility of the emergence of non-real eigenvalues of L Gp .In such a scenario, there might be complex conjugates eigenvalues of L Gp whose real part decreases and becomes smaller than λ δ .This means that the spectral gap is not necessarily a real number, however, since λ δ < α 2 , we always have Re λ 2 L Gp < λ 2 (L G ).This proves part (iia) of the theorem.
The proof of part (iia) is similar.For small modifications, as discussed in the proof of part (ib) of the theorem, we have that λ 2 L Gp = α 2 .In fact, α 2 is an eigenvalue of L Gp for arbitrary δ.However, as we discussed above, there is a possibility of the emergence of nonreal eigenvalues for L Gp when the modification δ is large.Thus, this might be the case that the real parts of these non-real eigenvalues reduce, and they become the spectral gap of L Gp .In any case, the property Re λ 2 L Gp ≤ λ 2 (L G ) always holds.This proves part (iib) of the theorem.

Conclusions
This paper investigates how modifying a network affects the Laplacian spectral gap and, in turn, the collective dynamics (synchronization).We considered modifications of three networks with master-slave topology, where the master is a cycle, and the slave is a star.The cycle and star were chosen since they are common motifs in real-world networks.The considered modifications are of arbitrary size and not necessarily (sufficiently) small.Our investigation was based on the spectral analysis of the Laplacian matrices of these networks.Our results are rigorous and accompanied by simulations of networks of coupled Lorenz oscillators that admit our mathematical results.
One particular interest of this paper was a paradoxical scenario known as Braess's paradox in which improving network connectivity leads to functional failure, such as synchronization loss.We explored and classified such scenarios in these three network models.We have shown that this counter-intuitive scenario in our models is not rare at all.For instance, a critical value exists (proportional to the square root of the size of the star) for which Braess's paradox happens in models I and II if the cycle size exceeds this value.

A.2 Enhancing synchronization
Again, we consider model II with n = 10 and m = 20 along with H = I as the coupling function as before.
We randomly choose initial conditions from the uniform distribution over [0.5, 1), and integrate the network until time t = 1000s.In this master-slave configuration, the network does not synchronize, as can be observed in the the mean synchronization error ⟨E⟩.At time t = 1000s, we break the master-slave configuration by adding a cutset and modification edges are w 0 = 1 and δ i = 1, where i = 0, 1, . . ., m − 1.This is represented as the red edges on the right upper panel of Figure 8.After this modification, the synchronous state becomes stable, and the synchronization error converges to zero.

B Technical lemmas
Proof.See, e.g., [36].initially, and subgraphs are connected via a directed link from the cycle to the star where w 0 = 1.We integrate the network until time t = 1000s and find the synchronization is unstable.After the red links are added to the system at t = 1000s, the synchronization mean error ⟨E⟩ goes to zero, indicating that the modification leads to stable synchronous dynamics.
Lemma 18.For m ≥ 4, let X 0 , X 1 ,..., X m−1 be real-valued functions defined on a subset of R, and consider the m × m matrix Let λ 0 ∈ R, and assume that all the functions X i are defined at λ 0 .If M (1)v = 0, for λ ̸ = 1, has a solution v, then it follows from (B.3) that v 0 ̸ = 0 and v i = v 0 1−λ , for all 1 ≤ i ≤ m − 1. Substituting this into (B.2), and multiplying the derived equation by (a) The unmodified graph G. (b) The modified graph G p .

Figure 1 :
Figure 1: Model I: Breaking the master-slave through hub coupling.We add a directed link from the hub of the star to the cutset node (the red-color edge) where the cutset node refers to the node which the cutset edge starts from and the weakly connected graph becomes strongly connected.The weight of each of the black-color edges is one while the weight of the red-color edge (modification edge) is arbitrary.

( a )
The unmodified graph G. (b) The modified graph G p .

Figure 2 :
Figure 2: Model II: Breaking the master-slave through multiple couplings.We add links from some nodes of the star to the cutset node of the cycle.The weight of each of the black-color edges is one while the weights of the red-color edges (modification edges) are arbitrary.

Figure 3 :
Figure 3: Model III: Breaking the generalized master-slave through multiple couplings.We add links from nodes of the star to the one cutset node of the cycle.The weight of each of the black-color edges is one while the weights of the red-color edges (modification edges) are arbitrary.

Figure 4 :
Figure 4: A comparison of the cases in Theorem A and computations of λ 2 (G p ) − λ 2 (G).We create a graph G as described in model 1a with cycle C n and star S m subgraphs whose sizes are n and m, respectively.Then we modify the network as shown in model 1b where δ 0 = 1 and calculate the difference λ 2 (G p ) − λ 2 (G) to characterize the behavior of the second minimum eigenvalue after modification.The red color of grids corresponds to a decrease of the second minimum eigenvalue after modification, and the blue color of grids corresponds to an increase of the second minimum eigenvalue after modification, where the intensity of the color at each grid shows the size of the difference λ 2 (G p ) − λ 2 (G).Simultaneously, the blue curve given by n c = π √ m + 1 is shown.Thus, regions separated by the blue curve manifest the signature of λ 2 (G p ) − λ 2 (G) and it corresponds to the critical transition between the cases stated in Theorem A, i.e., decreasing or increasing behavior of the second minimum eigenvalue after modification.

Figure 5 :
Figure 5: Hindrance of synchronization due to link addition: Networks of coupled Lorenz oscillators in model II are simulated to show the synchronization error.The sizes of the cycle and star subgraphs are set to n = 15 and m = 15, and subgraphs are connected via a directed link from the cycle subgraph to the star subgraph where w 0 = 1.We consider H = I as the coupling function.We choose initial conditions randomly selected from the uniform distribution over[3.5,5), and integrate the network until time t = 2500s.The system goes into synchronization after some transient.At t = 2500s, we add the red links to the system, i.e., δ i = 1, where i = 0, 1, . . ., m − 1, and perturb the system by adding noise randomly selected from the uniform distribution over [0.01, 0.02) to each state, then the synchronization loss occurs and the system doesn't return into synchronization after transient time.Note that, α 1 = 0.04, β − 15,1 = 0.06 in Theorem B.

Figure 6 :
Figure 6: Enhance of synchronization due to link addition: Networks of coupled Lorenz oscillators in model II are simulated to show the synchronization error.The sizes of the cycle and star subgraphs are set to n = 9 and m = 15, and subgraphs are connected via a directed link from the cycle subgraph to the star subgraph where w 0 = 1.We consider H = I as the coupling function.We choose initial conditions randomly selected from the uniform distribution over[3.5,5), and integrate the network until time t = 2500s.After the red links are added to the system at t = 2500s, i.e., δ i = 1, where i = 0, 1, . . ., m − 1, the synchronization occurs where the mean error ⟨E⟩ goes to zero.Note that, α 1 = 0.12, β − 15,1 = 0.06 in Theorem B.
(i) The expressions S and p ′ 2π n have the same sign.(ii) Assume α 2 < β − m,w .Then, for any arbitrary δ, we have p π n < 0. (iii) For sufficiently small δ, we have p 3π n < 0. Proof.The first part follows from the relation p ′ 2π n = n sin 2π n S (see relation (5.13)).For the other two parts, note that by Lemma 11 and for odd 1 ≤ l ≤ n − 1, we have p( lπ n sin ilπ n .Regarding the case l = 1, assumption α 2 < β − m,w implies that α 2 < 1 (see Remark 5) and therefore α 1 < 1.Thus, δ − δ 0 α 1 > 0, and therefore, y = y(α 1 ) = δ−δ 0 α 1 α 2 1 −(m+w)α 1 +w > 0. On the other hand, sin iπ n ≥ 0 for i = 0, . . ., n − 1 and so n−1 i=0 w i sin ilπ n ≥ 0. This implies p π n < 0 for any δ.The proof of the last part follows from 2y sin 3π n n−1 i=0 w i sin 3iπ n ≪ 4 which holds when δ is small enough.Proof of parts (ia) and (ib) of Theorem C2.Since α 2 < β − m,w , the spectral gap of L G is α 2 .It follows from Lemma 11 that p 2π n = 0 for all δ.On the other hand, p π n and p 3π n are both negative for sufficiently small δ.Therefore, by intermediate value theorem, the eigenvalue that gets born from α 2 as δ varies is located in the interval (α 2 , α 3 ) if p ′ 2π n > 0, and is located in (α 1 , α 2 ) if p ′ 2π n < 0. On the other hand, by part (i) of Lemma 16, we have that p ′ 2π n and S have the same sign.This proves parts (ia) and (ib) of Theorem C2.

Figure 7 :
Figure 7: Hindrance of synchronization due to link addition in networks of coupled Rössler oscillators.The figure shows the synchronization error over time with link addition occurring at time t = 1000s.The sizes of the cycle and star subgraphs are set to n = 15 and m = 15,and subgraphs are connected via a directed link from the cycle subgraph to the star subgraph where w 0 = 1.At t = 1000s, we add the red links to each system with unit weight.After perturbation, the system doesn't return to synchronization.

Lemma 17 .
Consider the block matrix ( A B C D ) and assume D is invertible.Define E := A − BD −1 C. Then (i) det ( A B C D ) = det (D) × det (E).

Figure 8 :
Figure8: Enhancement of synchronization due to link addition in a network of coupled Rössler oscillators.The sizes of the cycle and star subgraphs are n = 10 and m = 20, initially, and subgraphs are connected via a directed link from the cycle to the star where w 0 = 1.We integrate the network until time t = 1000s and find the synchronization is unstable.After the red links are added to the system at t = 1000s, the synchronization mean error ⟨E⟩ goes to zero, indicating that the modification leads to stable synchronous dynamics.