Master M2 Sciences de la Matière ENS de Lyon 2015-2016 Phase Transitions and Critical Phenomena Epidemics in Complex Networks and Phase Transitions Jordan Cambe January 13, 2016 Abstract Spreading phenomena have been observed in a wide range of domains, going from epidemics to rumor spreading or computer viruses. Along years different kind of models have been setup in order to fit the different characteristics of particular systems. Nowadays one the most efficient way to represent such kind of complex systems (like populations, the Internet, etc) uses the formalism of graph theory. Considering the universality of phase transitions, it does not seem surprising to observe them in the study of networks. Indeed, in the case of epidemiology, we will introduce the concept of epidemic threshold, after which the epidemics grows exponentially fast in the network and we will see that this concept is analogous to the concept of critical point in phase transition theory. In this report by comparing different studies of spreading phenomena, which have been done on different kind of networks, we will show the effects of networks topology on the phase transition characterizing these phenomena. All these studies focus on a epidemics model : the SIS model - Susceptible, Infected, Susceptible - which we will carefully define, along with the different kind of networks presented here. We also won t forget to give concrete applications of this model. By an analytical study and results of a few simulations we will see that networks topology affects the phase transition behavior and we will discuss what are repercutions of this result on the real world interpretation.
1 Definitions Before going through the specific definitions of network theory, a brief point about phase transition : phase transitions are defined as abrupt changes between different states, they are characterized by 2 parameters : a control parameter, which is varying (we will call λ) and an order parameter (we will call ρ). This order parameter is equal to zero in a phase and to a non-zero value in another phase. Then the phase transition takes place for a particular value λ c of λ, the critical point. Hence, we have : { ρ = 0 for λ λc ρ > 0 for λ > λ c (1) In the case of critical phase transition ρ(λ) become a powerlaw and the challenge is then to determine the critical exponent. Let s start, now, with the definition of a network. It is in a general way, a system that can be represented by a graph, i.e. a set V of N nodes connected to each other by E edges, the nodes are the units of the system and the edges represent the pairs of vertices interacting with each other. Though there is a huge amount of definitions characterizing network s properties, we are just going to give one of them, which is essential to understand the following (a complete set of definitions about network s properties can be found in (Barrat, Barthlemy, Vespignani, 2008)[1]). The degree k i of a node i is the number of edges which link i to the others nodes of the network. We can define the average degree of the network as : k = k i (2) N i Where N is the total number of nodes in the network. Having set this, it is possible now to split all network models into 2 kinds. Thus, we call a homogeneous network a network with a degree distribution, which remains close to the average degree. These networks have been the first network models to try to represent real world data, since they are simpler. But this homogeneity assumption can lead to strong biases, especially in the phase transition behavior as we will discuss in the last section. Now it is easy to figure out what is going to be a heterogeneous network. It is simply a network where distributions are broad, they are also called heavy-tail distributions. Here the average is not representative of the network structure, since the heavy-tail brings too high fluctuations. Heterogeneous networks are also called scale-free networks. Most of the time, they are powerlaw distributed, with exponent γ between 2 and 3. In such networks, due to the broad structure, we observe emergence of a few hubs, which are highly connected, among a majority of nodes with low degrees (i.e. with a small number of connections to others nodes). This appearence of hubs brings very deep changes in the behavior of spreading phenomena, as we will discuss later. So, to sum up : homogeneous network s.t. i k i k heterogeneous network s.t. P(k) α k γ 1
Figure 1: Illustration of the SIS model. Now we have everything we need about networks, let s define the SIS model and set the general analytical form of the system of equations, before changing parts and looking at this model applied to homogeneous and heterogeneous networks. The SIS model is an epidemiological model, in which we split the studied population into 2 classes : the Susceptibles who are healty agents and who can be infected with a probability ν, so called the infection rate. Once infected, agents enter into the 2nd class, the Infected one. In this class, agents can recover and become susceptible again with a probability δ, so called the recovery rate. Using these 2 parameters, we define the effective spreading rate : λ = ν δ. Figure 1 summarize the SIS model. Seeing how is define this model, it is easy to imagine the real world phenomena it can be associated with. An example about the spreading of computer viruses in the beginning of the 2000 s is given in (Pastor-Satorras, Vespignani, 2001)[2], where it is stated that once an infected computer get clean using an anti-virus, it immediatly becomes susceptible again. we will expose some of the results later. We can also think of the spreading of a disease where people do not get immunated during long periods of time. Using such kind of model enables to see the proportion of the network a given spreading phenomena (disease, computer virus, financial crisis, rumor) is going to reach, when it emerges. The goal being to act on the network structure to modify the spreading behavior, e.g. which link to remove to make a quarantine more efficient, or decreasing the spreading rate λ using the optimized quantity of insecticide in the case of insect transmitted diseases (an interesting example is given in the case of Chikungunya in (Naowarat, Tawarat, Ming Tang, 2011)[3]), etc. In the next sections, we are going to write the time evolution equation of the number of infected people, starting on a homogeneous network and then comparing with an heterogeneous one. In each of these parts, we will see where the phase transitions take place and what are theirs parameters. NB. : we consider here the simplest network models, i.e. we do not take into account weights or directions, which could be applied to edges, stressing importance of a link or unidirection relationship between nodes. 2
2 SIS Model on Homogeneous Networks When looking at homogeneous networks, we consider that population is uniform among the two different classes S and I. Under this homogeneous assumption presented in the previous part the strength of the infectious process, at each time step, is directly related to the average number of contact with infected people, which is : k i, where i is the normalized number of infected people (i = I N ), in others words it is the average number of infected neighbors. This assumption is only valid for large networks. Thus, it is easily understandable that the probability for a susceptible to get infected is given by : P (S I) = k iν (3) Knowing this, let s write the time evolution equation of the infected population, taking place in the same homogeneous network : di = δi(t) + ν k i(t)(1 i(t)) (4) Where (1 i(t)) is the normalized number of susceptible people, as the number of nodes N is constant and there is only 2 classes of people. From the parameters δ and ν we can deduce the time scales of the process, thus the spreading time scale is 1 ν and the recovery time scale 1 δ. We can isolate 2 cases : 1 δ < 1 ν in this case infected people recover faster than they transmit the disease and the system converges to a healty state. In the other case 1 δ > 1 ν, infected people infect susceptible ones faster than they recover, this leads to an endemic state, where the disease spreads more or less fast. It ends in a steady endemic state with a constant number of infected people. We can see that depending on parameters, the system will follow 2 very different behaviors. We are now going to look the critical point at which this transition happens. To do so, let s look at the early stage of the spreading process, when i(t) is considered very small compared to the total number of people N. In this case, we can use the linear regime assumption to look at what happen. Thus the equation becomes, after neglecting the i 2 term : This leads to a solution of the form : di = (ν k δ)i(t) (5) i(t) = i 0 exp t/τ (6) And 1 τ = ν k δ Looking at the solution, we see the previous phases discussed in last paragraph, depending on ν k and δ. If 1 τ < 0 the virus has no time to spread in the network, the system ends up in the healty state, whereas if it is positive the infected population grows exponentially fast. From this, we set the famous reproduction number R 0, for which the disease spreads exponentially fast if it is larger than 1 : R 0 = ν k (7) δ In the case where R 0 > 1, then comes a time when the linear regime assumption is no more valid, the term in i 2 slows down the spreading until the system ends up in a state, where there is a constant fraction of the population, which is infected. In 3
Figure 2: The density of infected people for the SIS model as a function of time. In the first compartment the system is subject to high statistical fluctuations. Then, after the system reached a threshold an exponential growth takes place (if R 0 > 1). Finally a stable state settles with a non-zero constant density of infected people. Figure has been taken from (Pastor-Satorras, Castellano, Van Mieghem, Vespignani, 2015)[5], page 4. figure 2 we can see the typical evolution in time of the number of infected people for the SIS model. As it has been introduced in the very first paragraph, we can look at the analogy with phase transition formalism. So, we understand that the control parameter is the reproduction number R 0, the epidemic threshold R 0 = 1 is the critical point of the phase transition, previously called λ c in the definitions part. We have seen from the previous equations that for R 0 < 1 the number of infected people remains small and ends up at 0. Whereas when R 0 > 1 i(t) grows exponentially fast and finishes at a constant value. From this behavior we see that the number of infected people corresponds to the order parameter. On figure 3 is the typical phase diagram differencing the 2 states depending on the epidemic threshold. In fact, when thinking about it, the SIS model is simply a generalisation of the contact process model on a lattice 1 (defined by Harris in 1974)[6][4] for graphs where the number of neighbors is not constant. In others words we can see the SIS model as an Ising model, where the nearest neighbors coupling depends on a number of neighbors, which is not constant depending on the spin for which it is computed. 1 Originally, in this model all nodes of a lattice could be occupied or empty and occupied nodes annihilate with rate 1 ; whereas they can only reproduce with rate λ. Where there is a phase transtion at a critical point λ c between an active phase, in which activity lasts forever and another one, in which activity eventually vanishes, ending with an empty system. 4
Figure 3: Typical phase diagram for the phase transition happening in the SIS model case. Here λ = R 0 and λ c = 1. When λ < λ c the order parameter is zero, this is the healthy phase. When λ > λ c the order parameter converges to a non-zero value. Figure has been taken from (Pastor-Satorras, Castellano, Van Mieghem, Vespignani, 2015)[5], page 8. 3 SIS Model on Heterogeneous Networks Scale-free networks correspond to real-world networks. Indeed, in the nature, all networks have broad distrubutions of theirs features (e.g. degree, weight, etc...). This is why it is particularly important to understand how spreading phenomena evolve in such kind of networks. When considering scale-free networks (for large size) the average degree is not relevant, because fluctuations are too important due to the heavy-tailed behavior of distributions. So, in order to set the equation, we cannot use the average degree k, as we did for homogeneous networks. Hence, we are going to use the degreebased mean-field approximation (DBMF) as we neglect correlation between nodes (the DBMF approximation is also called in physics litterature the heterogeneous meanfield approach). The use of the degree-based mean-field theory to apply SIS model on heterogeneous networks has been done in (Pastor-Satorras, Vespignani, 2001)[2]. So, using the DBMF approach we describe the SIS model using the probability ρ k that node with degree k gets infected, in others words it is the density of infected nodes with degree k. Moreover we assume the equivalence of all nodes with the same degree k. Thus, we can write the SIS equation : dρ k (t) = δρ k (t) + νk(1 ρ k (t)) k P (k k)ρ k (8) In order to get only one parameter we divide by δ. dρ k (t) = ρ k (t) + λk(1 ρ k (t)) k P (k k)ρ k (9) 5
Where ρ k (t) is the fraction of infected nodes, who get recovered. The second term is the probability of getting infected, this comes from the probability of being connected to a node with degree k among our k connections, multiplied by the chance that this node is infected, multiplied by the spreading rate λ and this is done for all possible degree k. So, having an uncorrelated network, such that P (k k) = k P (k ) k, we can write from the previous equation : With dρ k (t) = ρ k (t) + λk(1 ρ k (t))θ (10) Θ = k P (k ) k k kp (k)ρ k (t) (11) Θ = k P (k ) ρ k (t) (12) k k In order to find the expression of ρ k, we look at the steady state, by setting dρ k(t) gives us from the equation (10) : = 0, this ρ k = λkθ(λ) 1 + λkθ(λ) Where Θ can be interpreted as the probability for an edge to point to an infected node, finally Θ only depends on the spreading rate λ. We, now, introduce the equation (13) into equation (10), this way we find self-consistently the expression of Θ : Θ(λ) = 1 k λkθ(λ) kp (k) 1 + λkθ(λ) k Moreover, it has been shown in (Pastor-Satorras, Castellano, Van Mieghem, Vespignani, 2015)[5], by computing the eigenvalues of the connectivity matrix (notion which I have not introduced in this report) that the equation (14) admits a non-zero solution only if : (13) (14) λ > λ c = k k 2 (15) As explained in the beginning of the section, the heavy-tail degree distribution make the fluctuations high, i.e. k 2 and thus λ c 0 in the limit of large networks. So, here we see the effect of degree heterogeneity, the heterogeneity makes the epidemic threshold (the critical point of the phase transition) desappear. Now, we know that the phase transition, here, is a 2nd order phase transition. So, in order to find the critical exponent, we define ρ(λ) as being the average over all k of ρ k, i.e. ρ(λ) = k P (k)ρ k. Then, by solving the self-consistent equation (eq. (14)) and introducing the result into the expression of ρ(λ), we find an expression of the form (all details about this last calculation are given in (Pastor-Satorras, Vespignani, 2001)[8]) : and the exponent is of the form : ρ(λ) (λ λ c ) βcrit (16) 6
β crit = 1 3 γ for γ < 3 β crit = 1 γ 3 for 3 < γ 4 β crit = 1 for γ > 4 Where γ is the exponent of the powerlaw degree distribution. (17) To sum up, in the case of heterogeneous networks the critical point desappear and we observe a critical phase transition, where the spreading rate λ is the control parameter, the order parameter is the density of infected nodes in the network and the critical exponent in the order parameter directly depends on the degree distribution exponent. 4 Discussion In this report we have seen that the network topology affects spreading phenomena, by changing in the case of the SIS model the phase transition from a 1st order phase transition (homogeneous networks) to a 2nd order phase transition (heterogeneous networks). Figure 4 illustrates this comparison. It is worth noting that same results have been found for different kind of dynamical processes, such as the SIR model (Susceptible - Infected - Recovered), where here once recovered people are immunized against the virus. On the point of view of epidemiology, real-world networks being scale-free networks the disappearance of epidemic threshold indicates that viruses can proliferate in real-world networks, no matter theirs spreading rates, in other words even infections with small infection rates and bigger recovery rates can reach a stable endemic state. Observation which was not possible when looking at homogeneous networks as a first approximation of real-world networks. Note on finite size effect : after having stated that even the least replicative virus can infect the world, one question should come to mind : why the Internet (or the world itself) has not been yet destroyed by tons of nasty viruses proliferating in them? As we have seen in the previous part the critical point tends asymptotically to zero. But even if real-world networks are very large, they always have a finite number of nodes. This number of nodes induces a very small critical point, which is greater than zero and it turns out that for most of infections, which have been seen until now the spreading rate λ remains under this threshold. Thus, balancing the theoritical desappearance of the critical behavior in scale-free networks. 7
Figure 4: Comparison of the transitions behavior in the density of infected people as a function of the spreading rate λ using SIS model, for heterogeneous networks (solid red line) and homogeneous networks (blue dashed line). Figure has been taken from (Vespignani, 2012)[7]. References [1] A. Barrat, M. Barthlemy and A. Vespignani, Dynamical Processes on Complex Networks, Cambridge University Press Chap. 1, (2008). [2] R. Pastor-Satorras and A. Vespignani, Epidemic Spreading in Scale-Free Networks, Phys. Rev. Lett. 86, 3200 (2001). [3] S. Naowarat, W. Tawarat and I. Ming Tang, Control of the Transmission of Chikungunya Fever Epidemic Through the use of Adulticide, Am. J. Appl. Sci. (2011). [4] J. Marro and R. Dickman, Nonequilibrium Phase Transitions in Lattice Models, Cambridge University Press Chap. 5, (1999). [5] R. Pastor-Satorras, C. Castellano, P. Van Mieghem, A. Vespignani, Epidemic processes in complex networks, (only on ArXiv), (2015). [6] T.E. Harris, Ann. Prob. 2, 969, (1974). [7] A. Vespignani, Modelling dynamical processes in complex socio-technical systems, Nature physics 8, 32-39 (2012). [8] R. Pastor-Satorras and A. Vespignani, Epidemic dynamics and endemic states in complex networks, Phys. Rev. E 63, 066117 (2001). 8