{'fraction_non_alphanumeric': 0.04653810097763493, 'fraction_numerical': 0.022900763358778626, 'mean_word_length': 3.947002318648559, 'pattern_counts': {'":': 0, '<': 1, '<?xml version=': 0, '>': 8, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 3, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Localization and Spreading of Diseases in Complex Networks

Using the SIS model on unweighted and weighted networks, we consider the disease localization phenomenon. In contrast to the well-recognized point of view that diseases infect a finite fraction of vertices right above the epidemic threshold, we show that diseases can be localized on a finite number of vertices, where hubs and edges with large weights are centers of localization. Our results follow from the analysis of standard models of networks and empirical data for real-world networks.

Using the SIS model on unweighted and weighted networks, we consider the disease localization phenomenon. In contrast to the well-recognized point of view that diseases infect a finite fraction of vertices right above the epidemic threshold, we show that diseases can be localized on a finite number of vertices, where hubs and edges with large weights are centers of localization. Our results follow from the analysis of standard models of networks and empirical data for real-world networks. Survey of infectious diseases reveals that before an outbreak, often, if not typically, a disease is localized within a small group of individuals. Changes in environmental conditions or increase in the frequency of external contacts result in an epidemic outbreak. In the present paper we propose an approach that enables us to describe quantitatively this important localization-delocalization phenomenon. Our approach is based on the SIS model [1,2] of spreading of diseases in weighted and unweighted networks, where the weights of edges encode frequency of contacts between vertices. It is widely accepted that in uncorrelated networks the epidemic threshold λ c of the infection rate λ is λ MF = q / q 2 , where q and q 2 are the first and second moments of the degree distribution [2]. So in networks with a finite q 2 the threshold should be non-zero, while it is zero if q 2 diverges. One should stress however that all these well-known results were obtained only within a mean-field theory, actually within an annealed network approximation in which a random network is substituted for its fully connected weighted counterpart [2]. Contrastingly, one can show exactly for an arbitrary graph that λ c is actually determined by the largest eigenvalue Λ 1 of the adjacency matrix A ij of the graph, and λ c = 1/Λ 1 [3][4][5][6][7][8]. For uncorrelated networks, it was found that Λ 1 is determined by the maximum degree q max , Λ 1 ∝ √ q max [3]. Then, if in the infinite size limit, q max tends to infinity, as, e.g., in the Erdős-Rényi graphs, this leads to an amazing conclusion that the epidemic threshold is absent even in (infinite) networks with a finite q 2 in contrast to the mean-field result.
In the present paper we develop a spectral approach to the SIS model on complex networks. We show that the contradiction between the mean-field approximation and the exact result can be resolved if we take into account localization of diseases. It turns out that, in contrast to the mean field theory, in which a finite fraction of vertices are infected at λ > λ c , there are actually two scenarios of the spreading of diseases. If Λ 1 corresponds to a localized eigenstate, then, at λ right above λ c = 1/Λ 1 , disease is mainly localized on a finite number of vertices, i.e., the fraction of infected vertices is negligibly small in large networks. With further increase of λ, disease gradually infects more and more vertices until it will infect a finite fraction of vertices. In the second scenario, Λ 1 corresponds to a delocalized state. Then already at λΛ 1 −1≪1, disease infects a finite fraction of vertices. Analysing network models and real-world networks, we show that hubs, edges with large weights, and other dense subgraphs can be centers of localization.
We consider the standard SIS model of disease spreading in a complex network of size N having the adjacency matrix with arbitrary entries A ij ≥ 0. Infected vertices become susceptible with unit rate, and each susceptible vertex becomes infected by its infective neighbor with the infection rate λ. The probability ρ i (t) that vertex i is infected at time t is described by the evolution equation In the steady state, at t → ∞, the probability ρ i ≡ ρ i (∞) is determined by a non-linear equation, which has a non-zero solution ρ i > 0 if λ is larger than a so-called epidemic threshold λ c . In this case, the prevalence ρ ≡ N i=1 ρ i /N is also non-zero. This transition contrasts to a "thermodynamic phase transition", since the critical feature at λ c is present even for finite nets.
Spectral approach.-To solve the SIS model, we use the spectral properties of the adjacency matrix A. The eigenvalues Λ and the corresponding eigenvectors f with components f i are solutions of the equation Λf = Af . Since the matrix A is real and symmetric, its N eigenvectors f (Λ) (Λ max ≡ Λ 1 ≥ Λ 2 ≥ . . . Λ N ) form a complete orthonormal basis. According to the Perron-Frobenius theorem, the largest eigenvalue Λ 1 and the corresponding principal eigenvector f (Λ 1 ) of a real nonnegative symmetric matrix are nonnegative [9]. The probabilities ρ i can be written as a linear superposition, The coefficients c(Λ) are projections of the vector ρ on f (Λ). Substituting Eq. (3) into Eq. (2), we obtain In order to find the epidemic threshold λ c and ρ(λ) near λ c , it is enough to take into account only the principal eigenvector f (Λ 1 ) in Eqs. (3) and (4) This expression is asymptotically exact if there is a gap between Λ 1 and Λ 2 (see also Ref. [10]). Thus, at τ ≪ 1, ρ is determined by the principal eigenvector. Contributions of other eigenvectors are of the order of τ 2 . Considering two largest eigenvalues in Eq. (4), Λ 1 and Λ 2 , and their eigenvectors, we obtain ρ(λ) ≈ α 1 τ + α 2 τ 2 and so on. The usual point of view is that α 1 is of the order of O(1), and so a finite fraction of vertices is infected right above λ c . To learn if another behavior is possible, we study whether Λ 1 corresponds to a localized or delocalized state. We introduce the inverse participation ratio is delocalized, then ρ is of the order of O(1), and disease infects a finite fraction of vertices right above λ c . These two contrasting scenarios are shown in Fig. 1 for the SIS model on the karate-club network [11] and the weighted collaboration networks of scientists posting preprints on the astrophysics archive at arXiv.org, 1995-1999, and the condensed matter archive at January 1, 1995 -March 31, 2005 [12]. The astro-ph and karate-club nets have delocalized principal eigenstates while the cond-mat-2005 net has a localized principal eigenstate. Numerical solution of Eq. (2) gives α 1 = 1.8×10 −3 for the astro-ph net and α 1 = 1.5×10 −4 for the cond-mat-2005 net. There is an iterative method to find Λ 1 and IP R(Λ 1 ), where g (n+1) = Ag (n) and g (0) is a positive vector. Λ 1 (n) is the lower bound of Λ 1 . For unweighted networks, i.e., A ij = 0, 1, choosing g (0) =1, we obtain that the first iteration n = 1 gives g where Λ MF ≡ q 2 / q and r is the Pearson coefficient [13] (see also Ref. [14]). Eq. (9) shows the influence of degreedegree correlations on Λ 1 . The first iteration also gives the mean-field result IP R = q 4 /[N q 2 2 ] ∼ O(1/N ). Already a few iterations give good approximations for Λ 1 and IP R if the principal eigenstate is delocalized but more iterations are needed if this eigenstate is localized. Bethe lattice.-What structural elements of a network are responsible for the localization of the principal eigenstate? Let us consider a Bethe lattice (see Fig. 2). The adjacency matrix of an unweighted regular Bethe lattice with vertices of degree k has the largest eigenvalue Λ 1 =k.
The principal eigenvector f i (Λ 1 )=N −1/2 is delocalized. Other eigenstates are not important for us. Let us introduce a hub of degree q>k connected to the neighbors by edges with a weight w ≥ 1 [see Fig. 2(b)]. Other edges have weight 1. Solving the equation Λf = Af , we find

FIG. 2. (a) Regular Bethe lattice with degree k = 3. (b)
Bethe lattice with one hub of degree q > k. This hub is connected to neighbors by edges having the same weight w ≥ 1 (red lines). (c) Bethe lattice with two vertices of degrees q1 and q2 connected by an edge with a weight w ≥ 1 (red line).
Here B ≡ k − 1 is the branching coefficient of the graph, a ≡ [qw 2 − B] 1/2 , and f 0 (Λ 1 ) is the component of the principal eigenvector f (Λ 1 ) at the hub. f i (Λ 1 ) depends only on distance n from the hub to vertex i, i.e., f i (Λ 1 ) = f n (Λ 1 ), and exponentially decays with n ≥ 1, This exponential decay leads to a finite IP R, so this eigenstate is localized. If qw 2 ≫ B, then IP R → (1 + 1/q)/4. The second eigenstate with Λ 2 = k is delocalized. The criterion Λ 1 > k, leads to the condition that q must be larger than q loc ≡ (B 2 + B)/w 2 . For the SIS model on this Bethe lattice, Eqs. (5) and (13) give α 1 ∝ 1/N . Now we consider the Bethe lattice with two hubs of degrees q 1 and q 2 connected by an edge with a weight w ≥ 1 [see Fig. 2(c)]. Other edges have weight 1. We find that two eigenstates with eigenvalues Λ 1 and Λ 2 can appear above the delocalized eigenstate with Λ 3 = k, The signs ± correspond to Λ 1 and Λ 2 , respectively, and (2) ) decrease with increasing distance n from vertex i to the hubs as Ψ 1(2) /a n ± if vertex i belongs to branches connected to hub 1 or hub 2, respectively. Ψ 1 (2) are the components of f (Λ 1 ) at hubs 1 and 2, respectively. Their ratio is Ψ 2 /Ψ 1 = (a 2 ± −Q 1 )/(wa ± ). The criterion for localization is given by the condition Λ 1 , Λ 2 > k. If q 1 = q 2 and w ≫ 1, IP R(Λ 1 ) reaches the maximum value 0.5 that means localization on two hubs, Ψ 1 = Ψ 2 → 1/ √ 2. Scale-free networks.-To study appearance and properties of localized eigenstates in uncorrelated complex networks, we use the static model [15] that generates unweighted scale-free networks with degree distribution P (q) ∝ Cq −γ at q ≫ 1. Using software OCTAVE, for each realization of a random network of size N with the mean degree q and γ = 4, we calculated eigenvalues, eigenvectors, and IP R(Λ) of the adjacency matrix. In networks of size N = 10 5 , we found that several (typically, from one to three for different realizations) eigenstates appear above the upper delocalized eigenstate. These states are localized at hubs and their properties are described well by Eqs. (10)-(13) with w = 1 if the branching coefficient B in these equations is replaced by the averaged branching coefficient B = q 2 / q −1. In these scale-free graphs, the upper delocalized eigenstate Λ d is slightly above the mean-field value Λ MF = q 2 / q . The maximum degree q max fluctuates from realization to realization. In realizations with a small q max , the principal eigenvector is delocalized and Λ 1 = Λ d . Therefore, the criterion for localization of the principal eigenstate at a vertex with degree q max can be written as The equality here gives the threshold degree q loc . For N = 10 5 , q = 10, and γ = 4, our numerical calculations give q 2 / q ≈ 14.1 and Λ d ≈ 15.1. According to Eq. (15), a localized state appears above Λ d if q max is larger than q loc ≈ 214. Since the averaged value of q max depends on N , at small N the probability to generate a graph with q max > q loc is small [16]. Only sufficiently large graphs can have a localized principal eigenstate. Fig. 3 represents results of our numerical solution of Eq. (4) for the SIS model on a scale-free network of size N = 10 5 with the principal eigenvector localized at hub with q = 323. Equations (10)- (13) and (5) give Λ 1 = 18.35, IP R = 0.23, and α 1 ≃ 1.4×10 −3 . These values agree well with the measured values Λ 1 = 18.47, IP R = 0.21, and α 1 ≃ 1.7×10 −3 . The eigenvector with Λ 2 is also localized at another hub with q = 254. The third eigenvalue Λ 3 corresponds to a delocalized eigenvector. The first two eigenstates allow to describe ρ(λ) close to λ c =1/Λ 1 . Accounting for the delocalized eigenstate Λ 3 gives better results in a broader range of λ (see Fig. 3). I. Characteristics of real-world networks. N is size, γ is the degree distribution exponent, qmax is the maximum degree, q loc is the localization threshold found from Eq. (15), Λ1 is the largest eigenvalue and Λ1(1) is its lower bound, Eq. (9), respectively. D and A stand for assortative and disassortative mixing. Two last columns represent weighted networks. Real networks.-The largest eigenvalue Λ 1 , IP R(Λ 1 ), and other parameters of a few weighted and unweighted real-world networks are given in Table I. Note first that in all these unweighted real networks the inverse participation ratio IP R(Λ 1 ) is small that evidences a delocalized Λ 1 . Second, in unweighted networks, Λ 1 differs strongly from the mean-field value Λ MF = q 2 / q . Λ 1 < Λ MF in networks with disassortative mixing and Λ 1 > Λ MF in assortative networks. Thus, degree-degree correlations strongly influence Λ 1 . A similar observation has been made in [7]. Qualitatively, this result agrees with the lower bound of Λ 1 determined by Eq. (9). So we suggest that assortative mixing decreases the epidemic threshold λ c = 1/Λ 1 compared to the mean-field value 1/Λ MF while disassortative mixing increases λ c . Table I shows that in contrast to the unweighted hepth and cond-mat-2005 networks, their weighted versions have a localized principal eigenvector with a large IP R. Localization occurs at vertices linked by edges with large weights. In the cond-mat-2005 network, localization occurs at vertices of degrees 37 and 28 connected by an edge with weight 34.3 that is much larger than the averaged weight w = 0.51. Components of the principal eigenvectors decrease exponentially from the center of localization in agreement with the solution Eq. (14). In the hep-th network, this strong edge has weight 34 larger than w = 0.97 and connects vertices of degrees 34 and 33. In unweighted networks localization does not occur because the localization threshold q loc obtained from the criterion Eq. (15) exceeds q max . The prevalence ρ(λ) in two weighted networks is represented in Fig. 1(a).
In conclusion, based on a spectral approach to the SIS model, we showed that if the principal eigenvector of the adjacency matrix of a network is localized, then at the infection rate λ right above the threshold 1/Λ 1 , disease is mainly localized on a finite number of vertices. Importantly, a strict epidemic threshold in this case is actually absent, and a real epidemic affecting a finite fraction of vertices occurs after a smooth crossover, at higher values of λ. On the other hand, if the principal eigenvector is delocalized, the epidemic occurs in the whole region above λ c = 1/Λ 1 . We suggest that further investigations of real-world networks will give many new examples of the disease localization-delocalization phenomena.