On the second largest eigenvalue of networks

From predicting the epidemic threshold of a disease outbreak to anticipating the stability of a complex system, analysis of spectra of the adjacency matrices of the underlying networks play a pivotal role. Despite spectra of networks considered as fingerprints of the corresponding complex systems, most works and review articles have circumscribed around the largest eigenvalue (λ1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _1$$\end{document}) only. The second largest eigenvalue of a network that admits many applications in diverse fields, including mathematics and computer science, has not been thoroughly contemplated. This article first reviews existing literature on λ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _2$$\end{document}, predominantly confined to the random regular graphs, followed by the results for various popular model networks. We emphasize the aspect that λ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _2$$\end{document} shows an entirely different behavior than λ1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _1$$\end{document}.


Introduction
Since the resolution of seven bridges of Königsberg problem (Euler 1741), the theory of networks, which deals with the study of complex interacting units represented as graphs, has demonstrated remarkable applications in various real-world systems (Albert and Barabási 2002). With an accelerated development in network science, various communities belonging to different branches of science, economics, biology, and sociology have witnessed growing applications of results and techniques of networks in their respective fields. The spectra of the adjacency matrix of a network have been shown to provide information about various structural properties of the network, as well as the dynamical behavior of the corresponding complex system. For example, the degeneracy of the zero and one eigenvalues provide a clue to the structural symmetries of the underlying networks (Yadav and Jalan 2015). The largest eigenvalue, in addition to capturing the information of the largest degree, is related to the synchronization phenomena of diffusively coupled dynamical units on the network (Restrepo et al. 2005). Further, spectra plays a pivotal role in determining the epidemic threshold of a disease outbreak (Van Mieghem et al. 2008), stability of a system (May 1972), drug identification in complex diseases (Rai et al. 2018), for studying international trade networks (Alhomaidhi et al. 2019) etc. Coupled dynamical evolution on networks can be better understood by the spectra of corresponding Laplacian matrices (Liu et al. 2015). Further, spectra of complex networks have also been studied under the Random matrix theory framework (Jalan and Bandyopadhyay 2007;Erdős et al. 2012). Recently, in Susca et al. (2020), a technique has been proposed to compute the statistics of the second largest eigenvalue and the components of the corresponding eigenvector for sparse symmetric matrices. It is to be noted that various structural properties of networks are computationally hard to determine nevertheless, often, the spectral measures cater good insights into a network structure, and these spectral measures are computationally easier to calculate. For example, calculating various expansion properties of networks is computationally NP-hard in nature. However, fortunately, these properties are closely related to the second largest eigenvalue that we can calculate very fast (in O(n 3 ), where n is the number of nodes in the network) (Alon 1995). Note that, though calculating spectra analytically is a troublesome task, there exist analytical approaches to calculate spectra for a few specific networks (Newman et al. 2019;Peixoto 2013;Newman 2019). Further, the expansion properties of graphs have found their applications in computational complexity theory (com 1980), for the determination of robustness of computer networks (Motwani and Raghavan 1995), graph pebbling, parallel sorting algorithms, and theory of error-correcting codes (Hoory et al. 2006), derandomization of random algorithms, analysis of algorithms in computational group theory, quantum cryptography, etc. Further, the second eigenvalue of the Laplacian matrix of a network is equivalent to its algebraic connectivity, also known as the Fiedler eigenvalue (Bapat 2010). Random regular networks form one very important class of networks and have been investigated intensively (Bender 1974;Bender and Canfield 1978;Bollobás 1980;McKay 1984;Wormald 1981), and for these networks, the second largest eigenvalue of the adjacency matrices is equivalent to the algebraic connectivity. If the second eigenvalues of these networks is lesser than their degrees, these networks are connected (Fiedler 1973). To get an overview of second largest eigenvalue, it is worth to see Dehmer andEmmert-Streib (2009), Van Mieghem (2010), Chung (1994).
Most of the studies on the second largest eigenvalue ( 2 ) are done for random regular graphs, with sporadic studies for other networks. All these studies suggest that 2 can yield a good insight into properties of the underlying network structure (Marcus et al. 2013). Particularly, the suitability of a network for a specific application depends on 2 , and often a small value of 2 is desirable (Hoory et al. 2006). Several classes of networks (graphs) are defined by the value the second largest eigenvalue take, for instance, reflexive graphs or hyperbolic graphs ( 2 ≤ 2 ), Salem graphs ( 2 ∈ [−2, 2] ), k-regular Ramanujan graphs ( 2 ≤ 2 √ d − 1 ) (Hoory et al. 2006;Maxwell 1978). Further, the second eigenvalue is crucial in the dynamics of the Markov chain. It is found that networks with small 2 rapidly mix for reversible Markov chain yielding a high rate of information diffusion in large networks (Desai and Rao 1993).
The article is organized as follows. "Relationship of spectra with structure of networks" section presents results for various analytical relationships between the network's structural properties and their spectral measures. "Results on various network models" section focuses on the results on few popular model networks, namely, 1D lattice, Erdös-Renýi (ER) random, Scale-free networks and small-world networks. Exact analytical results are presented for 1D lattice model and k−random regular graphs, and numerical results are for other model networks. "Conclusion" section concludes the article.

Relationship of spectra with structure of networks
Let G = (V , E) be a network on N nodes, where V, E are the vertex-set and edge-set, respectively. If two distinct vertices i, j ∈ V are adjacent, we write i ∼ j , otherwise, i ∼ j . We represent network G by its adjacency matrix A = (a ij ) N ×N defined as follows.
The eigenvalues of the adjacency matrix A are denoted by { 1 , 2 , 3 , . . . , N } where 1 ≥ 2 ≥...≥ N . For un-directed and unweighted networks, A is a symmetric matrix with all eigenvalues being real. The set of the eigenvalues { 1 , 2 , . . . , N } is known as the spectra of the network G. It is important to note that 1 for adjacency matrix having all non-negative entries is always positive, and for connected networks, 1 > 2 due to Perron-Frobenius Theorem. Let k denote the average degree of G, and k max denote the maximum degree of G. The largest eigenvalue 1 satisfies �k� ≤ 1 ≤ k max (Sarkar and Jalan 2018). Also 1 ≤ 2M(N −1) N 1 2 , where M, N are the number of edges and nodes in the graph, respectively. If G is connected, then χ(G) ≤ 1 + 1 (G), where χ(G) is the chromatic number of G (Bapat 2010). The chromatic number χ(G) of graph G is the minimum number of colors required to color the vertices such that adjacent vertices get distinct colors. However, finding the chromatic number of a graph is an NP-hard problem.
Next, we see that along with 1 , other eigenvalues of A implicate the following information about the structure of graph G (Bapat 2010).
where T is the set of all the cycles of length 3 in G (also referred as triangles). In general, the number of closed walks of length In particular when G is random regular graph, 2 shares an interesting relationship with the expansion coefficient of G and its diameter. The expansion coefficient of G, denoted as h(G) is defined as follows: where S ⊂ V , and E(S,S) are the set of the edges with one end in S and other in S . Alon and Milman (Hoory et al. 2006) proved that if G is a k-random regular graph, This immediately gives the following bounds on 2 .
Nilli (Nilli 1991) proved that if a k-random regular graph has a diameter D, then

Results on various network models
The relation (Eqs. 2-4) is valid for random regular networks, i.e., for networks with each node having the same degree. However, the second largest eigenvalue of the adjacency matrices of non-regular networks has not been given much attention. We numerically investigated the behavior of 2 for various model networks, namely, Erdös-Renýi (ER) random, scale-free, 1D lattice (for completeness), and small-world networks. ER random networks are constructed using the ER model ? as follows. Starting with N nodes, each pair of the nodes is connected with a probability p = �k�/N . Our analysis has considered connected networks only without self-loops and multiple edges. The degree distribution of ER random networks follows a binomial distribution. Scale-free networks are constructed using BA preferential attachment method (Barabási and Albert 1999) in which each node prefers to connect with the existing higher degree nodes. Starting with a network with m 0 nodes, we start adding a new node at each time step having m connections such that m ≤ m 0 . The new nodes can be connected to any existing node, and to incorporate preferential attachment, the probability of it getting connected to the ith nodes is proportional to k i where k i is the degree of the ith node. Thus, after t time steps, the network will have t + m 0 nodes and mt edges. The network evolves into a scalefree network with degree distribution following a power law, that is, P(k) ∼ k −γ where γ = 3 . The 1D lattice considered here consists of a ring lattice with each node connecting exactly with the same number of neighbors on both sides. We construct small-world networks using the Watts and Strogatz algorithm (Watts and Strogatz 1998) as follows.
Starting with a 1D lattice in which all the nodes have an equal degree, we rewire each edge of the network with a probability p r . This procedure of the rewiring allows to transform a 1D lattice with p r = 0 , to a random network with p r = 1 . In the intermediate p r values, the network manifests the small-world behavior, which is quantified by a very small average shortest path length. First, we discuss available analytical results for 1 and 2 , followed by numerical results for different model networks for various network parameters. The variation on 2 is compared with that of 1 . As discussed earlier that 1 is bounded below by the average degree, and above by k max , hence, an increase in the average degree will result in an increase in the lower bound of 1 . In particular, for a k-random regular network 1 increases linearly with the average degree as �k� = k max = 1 . However, 2 behaves quite differently. Figure 1 presents the behavior of 1 and 2 with respect to the average degree for various model networks. The value of 1 increases with the average degree for all the networks. In particular, for 1D lattice in (Fig. 1a) 1 = �k� , it is easy to see that the all-one vector 1 = [1, 1, . . . , 1] T of order N × 1 satisfy A1 = �k�1 . Since A is an non-negative matrix and 1 is an non-negative vector, using Perron-Frobenius theorem, 1 is the largest eigenvalue which is equal to k . The second largest eigenvalue of a 1D lattice G on N nodes and having degree k is given by k 2 j=1 2 cos 2π j N . This can be seen as follows. Each node in G has an edge with each of immediate previous, next k 2 nodes. The first row of the adjacency matrix A of G is the following The i-th row of A is the row-vector resulting after i − 1 right shift of the first row. Thus A is a circulant matrix (for more detail on circulant matrices see (Gray 2006)). The eigenvalues of circulant matrix of order N with the first row as c 0 , c 1 , . . . , c n−1 are given by N , for i = 0, 1, . . . , N − 1. The imaginary part cancels out because the adjacency matrix is symmetric, hence it will have only real eigenvalues. So we are left with only cos terms, which implies, i = k 2 j=1 2 cos 2π ji N , for i = 0, 1, . . . , N − 1. Thus 2 = k 2 j=1 2 cos 2π j N , which follow the trend as is seen in Fig. 1b shown in blue. For k-random regular networks, 1 = k . For ER random networks, 1 ≈ [1 + o(1)]�k� (Chung et al. 2004). For scale-free networks, 1 ≈ √ k max if γ > 2.5 and 1 ≈ k 2 if 2 < γ < 2.5 (Sarkar and Jalan 2018). With increase in the average degree, 1 exhibits a consistent increase for all these networks. However, by keeping the network size fixed, as we increase the average degree of a network, 2 first exhibits an increase until �k� ∼ N /2 , which is not surprising as 1 also exhibits an increase and one can expect of a similar trend followed by the second largest eigenvalue. However, rather intriguingly, if we increase the average degree further, 2 reflects a decreasing behaviour with k . The trend of 2 is symmetric with respect to �k� = N /2 (Fig. 1b). We consider k-random regular networks for analytically substantiating this behaviour observed numerically for ER and SF networks. Let G be a k-random regular network on N nodes and M edges. Let { 1 , 2 , 3 , . . . , N } be the eigenvalues of the adjacency matrix of G such that k = 1 ≥ 2 ≥ · · · ≥ N hold. Since Dividing both the sides by N we get 0, k 2 1, . . . , 1, 0, . . . , 0, k 2 1, . . . , 1.
2 1 + 2 2 + · · · + 2 N = 2M. We have Since N is constant, the quantity k − k 2 N depends on the higher-order terms in L.H.S of the Eq. 6. The quantity k − k 2 N first increases till k becomes N 2 , followed by a decrease, and so does the higher-order terms. Therefore, 2 after reaching the threshold value starts decreasing.
For any other random network, the explanation is similar to what we deduced for random regular networks. Let G be any random network, for which Eq. 6 becomes Since �k� ≤ 1 ≤ , where is the largest degree in G. The RHS of Eq. 7 first takes increasing value till k becomes N 2 , followed by a decrease, and so does 2 and the higher order terms.
Further, in Fig. 2, the plots are shown for 2 N vs k N for various network models. The behaviour is similar to what it is for 2 vs k as shown in Fig. 1. The ratio 2 N increases when k N increases from 0 to 0.5 and after that it shows a decreasing trend from 0.5 to 1. In every iteration, for 1D lattice, we have plot for double the size of previous network instance, but keeping k N fixed. We see that these curves overlaps which can be explained as follows. The second largest eigenvalue of 1D lattice with degree k is 2 = k 2 j=1 2 cos 2π j N . For a given N and k, expanding the summation, we get the following expression Now let us change the parameters N, k to N ′ , k ′ , respectively such that k N = k ′ N ′ . Let the corresponding second largest eigenvalue for these parameters be ′ 2 . We can express ′ 2 in terms of N and k as follows. We have N ′ = 2N and so k ′ = 2k , then Next, we have to prove that ′ 2 N ′ = 2 N , that is, R.H.S of Eq. 9 should be equal to 2 times R.H.S of Eq. 8 as N ′ = 2N . The consecutive terms of Eq. 9 (enclosed in parentheses) can be written as cos π j N + cos π(j−1) N where j = 2, 4, 6, . . . , k . The second term can be further written as cos( π j N − π N ) . Since cos θ is a continuous function and cos(θ − δ) ≈ cos θ when δ → 0 , we have cos( π j N − π N ) ≈ cos( π j N ) for N → ∞ . Thus, R.H.S of Eq. 9 becomes 4 × (cos 2π N + cos 4π N + cos 6π N + · · · + cos π k N ), which proves that

Expansion coefficient h(G) of random regular graphs and 2
The expansion coefficient h(G) of a network G reveals how well the network is connected. For a disconnected network h(G) = 0 , and higher the expansion coefficient better well-connected a network is. This can be directly spotted from the expression for the expansion coefficient This means that for any subset S, the number of edges from S to the rest of the vertices is at least h(G) × min(|S|, |S|) . Therefore, the higher the value of h(G) more well-connected the network is. However, in general calculating h(G) is an intractable problem (NP-hard) (Hoory et al. 2006). Nevertheless, for a 1D lattice, we can easily calculate the expansion coefficient as follows. Consider a 1D lattice of degree k on N nodes. By Eq. 10, the minimum ratio is achieved for |S| = N 2 . It is easy to observe that the total number of edges from S to rest of the vertices are Hence, Figure 3 plots 2 of 1D lattice against the lower and upper bounds in terms of h(G) (Expression 3). It can be seen that the lower bound almost matches the behaviour of 2 given by the Inequality 2 (8) 2 = 2 cos 2π N + cos 4π N + cos 6π N + · · · + cos π k N (9) |S| .
by Definition 1, in general for any k-random regular network G, the expansion coefficient h(G) will increase as k increases, thus the quantity, k − 2 (it is present in both the sides of inequality, also known as spectral gap) must also increase with k. However, in general, calculation of h(G) is NP-hard, it will be interesting to check how 2 behaves with respect to h(G) for other networks (Hoory et al. 2006).

Conclusion
This article reviews the properties of the second largest eigenvalue for 1D lattice, random regular graphs and networks capturing properties of real-world complex systems. The intriguing conclusion of future interests is that 2 behaves in a completely different demeanor from the largest eigenvalues. The largest eigenvalue 1 is more dependent on the highest degree of the corresponding networks. The larger the value of k max , the higher will be the value of 1 , however 2 though having an upper bound of k max turns out to be independent of it. For model networks, an increase in the average degree leads to an increase in the value of 1 . However, contrarily 2 first manifests an increase till �k� = N 2 followed by a decreasing trend. Further, 1 of the scale-free networks is always higher than the 1D lattice and random network since it has higher k max while 2 is higher for 1D lattice than that of the random and scale-free networks. We have analytically obtained the lower bound of 2 for 1D lattice using its relation with the expansion coefficient and found that the numerical value of 2 follows the same behavior as its lower bound.