The effects of evolutionary adaptations on spreading processes in complex networks

Significance

In this article, we bridge the disconnect between how spreading processes propagate and evolve in real life and the current mathematical and simulation models that ignore evolutionary adaptations. We propose a mathematical theory that reveals the effects of evolutionary adaptations on spreading processes in complex networks and highlights the shortcomings of classical epidemic models that do not capture evolution. Our work provides substantive developments to the classical theory of network epidemics and paves the way for exploring depths and revealing insights on the emergence phenomenon.

Abstract

A common theme among previously proposed models for network epidemics is the assumption that the propagating object (e.g., a pathogen [in the context of infectious disease propagation] or a piece of information [in the context of information propagation]) is transferred across network nodes without going through any modification or evolutionary adaptations. However, in real-life spreading processes, pathogens often evolve in response to changing environments and medical interventions, and information is often modified by individuals before being forwarded. In this article, we investigate the effects of evolutionary adaptations on spreading processes in complex networks with the aim of 1) revealing the role of evolutionary adaptations on the threshold, probability, and final size of epidemics and 2) exploring the interplay between the structural properties of the network and the evolutionary adaptations of the spreading process.

A common theme among previously proposed models for network epidemics is the assumption that the propagating object (e.g., a virus or a piece of information) is transferred across the nodes without going through any modification or evolution (1–11). However, in real-life spreading processes, pathogens often evolve in response to changing environments and medical interventions (12–16), and information is often modified by individuals before being forwarded (17, 18). In fact, 60% of the (approximately) 400 emerging infectious diseases that have been identified since 1940 are zoonotic (19, 20). A zoonotic disease is initially poorly adapted, poorly replicated, and inefficiently transmitted (21); hence, its ability to go from animal-to-human transmissions to human-to-human transmissions depends on the pathogen evolving to a strain that is well adapted to the human host.

Similar patterns of evolution are observed in the way that information propagates among individuals. Needless to say, one observes on a daily basis how information mutates unintentionally or perhaps, intentionally by an adversary on social media platforms (17). At a high level, an individual may mutate the information by exaggeration, hoping for her variant to go viral. Mutations may also occur unintentionally. In particular, Dawkins (22) argued that ideas and information spread and evolve between individuals with patterns similar to genes in a sense that they self-replicate, mutate, and respond to selective pressure as they interact with their host.

In this article, we aim to bridge the disconnect between how spreading processes propagate and evolve in real life and the current mathematical and simulation models that do not capture evolution. In particular, we investigate the effects of evolutionary adaptations on spreading processes with the aim of 1) revealing the role of evolutionary adaptations on the threshold, probability, and final size of epidemics and 2) understanding the interplay between the structural properties of the network and the evolutionary adaptations of the spreading process. Throughout, we use the term strain to denote a pathogen strain in the context of infectious disease propagation or a particular variation of the information in the context of information propagation.

In modeling the evolution of spreading processes, we adopt the multiple-strain model that was introduced by Alexander and Day in ref. 13. We develop a mathematical theory that fully characterizes the process and accurately predicts the epidemic threshold, expected epidemic size, and the expected fraction of individuals infected by each strain (all at steady state). In addition to the mathematical theory, we perform extensive simulations on random graphs with arbitrary degree distributions [generated by the configuration model (23–25)] as well as on real-world networks [obtained from the SNAP (Stanford Network Analysis Platform) dataset (26, 27)] to verify our theory and reveal the significant shortcomings of the classical mathematical models that do not capture evolution. In particular, we show that the classical, single-type bond percolation models (28–30) may accurately predict the threshold and final size of epidemics, but their predictions on the probability of emergence are significantly inaccurate on both random and real-world networks. This inaccuracy highlights a fundamental disconnect between the classical single-type, bond percolation models and real-life spreading processes that entail evolutionary adaptations.

Results

A Model for Evolutionary Adaptations.

In ref. 13, Alexander and Day proposed the multiple-strain model that accounts for evolution. Their model is captured by two matrices, namely the transmissibility matrix $\mathit{T}$ and the mutation matrix $\mathit{\mu }$, both with dimensions $m\times m$ for a finite integer $m\ge 2$ denoting the number of possible strains. The transmissibility matrix $\mathit{T}$ is a $m\times m$ diagonal matrix, with $\left[T_i\right]$ representing the transmissibility of strain $i$, while the mutation matrix $\mathit{\mu }$ is a $m\times m$ matrix with ${\mu }_{ij}$ denoting the probability that strain $i$ mutates to strain $j$ with ${\sum }_j{\mu }_{ij}=1$.

The multiple-strain model proposed by Alexander and Day (13) works as follows. Consider an SIR (susceptible-infectious-recovered) spreading process that starts with an individual (i.e., the seed) receiving infection with strain 1 from an external reservoir. Since strain 1 has transmissibility $T_1$, the seed infects each of her contacts independently with probability $T_1$, and then, recovers. After a susceptible individual receives the infection from the seed, the pathogen may evolve within that new host prior to any subsequent infections. In particular, the pathogen may remain as strain 1 with probability ${\mu }_{11}$ or mutate to strain $i$ (that has transmissibility $T_i$) with probability ${\mu }_{1i}$ for $i=2,\dots ,m$. If the pathogen remains as strain 1 (mutates to strain $i$), then the host infects each of her susceptible neighbors in the subsequent stages independently with probability $T_1$ ($T_i$), recovers, and so on. Observe that, as the process continues to grow, multiple strains may coexist in the population as governed by the transmissibility matrix $\mathit{T}$ and the mutation matrix $\mathit{\mu }$. At an intermediate stage, if any susceptible individual receives strain $j$, the pathogen may remain as strain $j$ with probability ${\mu }_{jj}$ or mutate to strain $\ell$ with probability ${\mu }_{j\ell }$ for $\ell \in \left\{\mathrm{1,2},\dots ,m\right\}\backslash \left\{j\right\}$ prior to subsequent infections. The process terminates when no additional infections are possible.

The multiple-strain model considers the case when the epidemiological and evolutionary processes occur on a similar timescale. In particular, each new infection event entails an opportunity for mutation. This model is reasonable for pathogens with long infectious periods (e.g., HIV) or pathogens with short infectious periods but high mutation rates, large population sizes, and short generation times (e.g., RNA viruses) (31). We focus on the particular case where the fitness landscape consists of only two strains, yet extending the results to any arbitrary number of strains is straightforward (Materials and Methods).

Network Model.

Let $\mathbb{G}$ denote the underlying contact network defined on the node set $\mathcal{N}=\left\{1,\dots ,n\right\}$. We define the structure of $\mathbb{G}$ through its degree distribution $\left\{p_k\right\}$. In particular, $\left\{p_k,k=\mathrm{0,1},\dots \right\}$ gives the probability that an arbitrary node in $\mathbb{G}$ has degree $k$. We generate the network $\mathbb{G}$ according to the configuration model (23, 24) (i.e., the degrees of nodes in $\mathbb{G}$ are all drawn independently from the distribution $\left\{p_k,k=\mathrm{0,1},\dots \right\}$). In particular, we sample a degree sequence (with even total degree) from the corresponding degree distribution; then, we use the configuration model to construct a random graph with that degree sequence. Furthermore, we assume that the degree distribution is well behaved in the sense that all moments of arbitrary order are finite. Note that, when the second moment of the degree distribution is finite, the expected clustering coefficient of the graphs generated by the configuration model approaches zero in the limit of large $n$; hence, they are locally tree like. Of particular importance in the context of the configuration model is the degree distribution of a randomly chosen neighbor of a randomly chosen vertex denoted by $\left\{{\widehat{p}}_k,k=\mathrm{1,2},\dots \right\}$ and given by ${\widehat{p}}_k=k{p}_k/⟨k⟩$ for $k=\mathrm{1,2},\dots$, where $⟨k⟩$ denotes the mean degree (i.e., $⟨k⟩={\sum }_{k}k{p}_k$).

Simulation Model.

We focus on the case where the fitness landscape consists only of two strains. The process starts by selecting a node uniformly at random and infecting it with strain 1. When cycles start to appear, a susceptible node could be exposed to multiple infections at once. We assume that coinfection is not possible; hence, a node may only be infected by one strain. In particular, if a node is exposed to $x$ infections of strain 1 and $y$ infections of strain 2 simultaneously, the node becomes infected with strain 1 (strain 2) with probability $x/\left(x+y\right)$ ($y/\left(x+y\right)$) for any nonnegative constants $x$ and $y$. A node that receives infection at round $i$ mutates first (by the end of round $i$) before it attempts to infect her neighbors (with the corresponding transmissibility) at round $i+1$. The node is considered recovered at round $i+2$ (i.e., a node is infective for only one round). We explore the case where coinfection with multiple pathogen strains is possible in SI Appendix, section 6.

In what follows, we use $S$, $S_1$, and $S_2$ to denote the total expected epidemic size, the expected fraction of nodes infected with strain 1, and the expected fraction of nodes infected with strain 2, respectively; all are at the steady state (i.e., when the process terminates). In all experiments, we set $T_1=0.2$, $T_2=0.5$, ${\mu }_{11}=0.75$, and ${\mu }_{22}=0.75$. The conclusions presented in this article hold for any other parameter selection. With respect to synthetic networks, we consider contact networks with Poisson degree distribution and power law degree distribution with exponential cutoff as given in ref. 28.

Epidemic Size.

We start by focusing on the total epidemic size and the expected fraction of nodes that were infected with strain 1 and strain 2. The network size $n$ is set to $2\times 10^5$. In Fig. 1, we plot the empirical (averaged over 500 independent experiments) and theoretical values of $S$, $S_1$, and $S_2$. We also plot a vertical line at the critical mean degree that corresponds to a phase transition as predicted by our theory. Our experimental results are in very good agreement with our theoretical results (Materials and Methods) on both contact networks.

Fig. 1.

The expected epidemic size ($S$) and expected fraction of individuals infected by each strain ($S_1$ and $S_2$) on synthetic networks generated by the configuration model with (A) Poisson degree distribution and (B) power law degree distribution with exponential cut off. Simulation results are in very good agreement with our theoretical results given in Materials and Methods.

Reduction to Single-Type Bond Percolation.

An important question to ask is whether the classical single-type bond percolation models could predict the threshold, probability, and final size of epidemics that entail evolutionary adaptations. In pursing an answer to this question, we start by establishing a matching condition between single-strain models and multiple-strain models for epidemics.

In ref. 28, Newman considered a single-type bond percolation model with parameter $T_{\mathrm{B}\mathrm{P}}$ (i.e., a process where each edge is occupied with the same probability $T_{\mathrm{B}\mathrm{P}}$ independently), and it was shown that a phase transition occurs when $\left(⟨k^2⟩-⟨k⟩/⟨k⟩\right)T_{\mathrm{B}\mathrm{P}}=1$. Comparing this with [4] suggests the proposal of a matching that results in the same condition for phase transition. More precisely, if we are to set

$$T_{\mathrm{B}\mathrm{P}}=\rho \left(\mathit{T}\mathit{\mu }\right),$$ [1]

where $\rho \left(\mathit{T}\mathit{\mu }\right)$ denotes the spectral radius (i.e., the largest eigenvalue; in absolute value) of $\mathit{T}\mathit{\mu }$, then both models would predict the same phase transition condition. In what follows, we explore the extent to which classical, single-type bond percolation models (under the matching condition [1]) may predict the threshold, probability, and final size of epidemics that entail evolutionary adaptations.

In Fig. 2, we plot the empirical (averaged over $10^4$ independent experiments) and theoretical (as given in ref. 13) probability of emergence on contact networks with Poisson degree distribution ($n=5\times 10^5$). We also plot the probability of emergence as predicted by the single-type bond percolation framework under the matching condition [1]. Observe that the classical single-type bond percolation model accurately captures the threshold and final size of epidemic but provides significantly inaccurate predictions on the probability of emergence. In SI Appendix, section 5, we explain the reasoning behind this observation.

Fig. 2.

The probability of emergence on contact networks with Poisson degree distribution. The single-type bond percolation framework provides significantly inaccurate predictions on the probability of emergence, yet it captures the expected epidemic size and phase transition point.

Effect of Mutation.

In what follows, we provide a theoretical approximation to the probability of emergence in a way that clearly distinguishes the role of mutation and shows how it strongly influences the probability of emergence. Consider the case when the fitness landscape consists of two strains with ${\mu }_{11}=1-\mu$ and ${\mu }_{22}=1$. Assume that $T_1<T_2$, and let $P_{\mu }$ denote the probability that, at some point along the chain of infections (starting from the type 1 seed), a type 2 node would emerge. This parameter selection models the case of one-step irreversible mutation where strain 1 could mutate to strain 2 that has higher transmissibility, but strain 2 never mutates back to strain 1.

For a fixed mean degree $\lambda$ of the underlying network, we may approximate the probability of emergence by

$$\mathbb{P}\left[\mathrm{e}\mathrm{m}\mathrm{e}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{e}\right]\hspace{0.17em}\ge \hspace{0.17em}{P}_{\mu }{P}_2^{\mathrm{B}\mathrm{P}},$$ [2]

where $P_2^{\mathrm{B}\mathrm{P}}$ denotes the probability of emergence on a single-type bond-percolated network with edge occupation probability $T_2$ and mean degree $\lambda$ and $P_{\mu }$ denotes the probability that, at some point along the chain of infections (starting from the type 1 seed), a node would be infected by strain 1 but then mutate to strain 2. In other words, $P_{\mu }$ captures the probability that, at some point during the propagation, a type 2 node would emerge. The proof of [2] is omitted for brevity and is given in SI Appendix, section 3.

In Fig. 3, we set $T_1=0.1$ and plot $P_{\mu }$ against the mean degree for a network with Poisson degree distribution. We observe that different values for $\mu$ impact the shape of $P_{\mu }$ (hence, the probability of emergence by virtue of [2]) in a remarkable way. First, for all values of $\mu \in \left(\mathrm{0,1}\right)$, the behavior of $P_{\mu }$ seems to be strikingly different than the universality class of percolation models (e.g., the shape of the probability of emergence in Fig. 2). Second, the effect of mutation probabilities on $P_{\mu }$ seems to be significant as the mean degree increases from small values, reaches its peak right before the critical mean degree corresponding to $P_1^{\mathrm{B}\mathrm{P}}$, and then, decays as the mean degree increases further.

Fig. 3.

The impact of $\mu$ on $P_{\mu }$ (hence, the probability of emergence by virtue of [2]) is pronounced before the critical mean degree corresponding to a single-strain, bond-percolated network with $T_1$. (Inset) The difference between the value of $P_{\mu }$ when $\mu =0.4$ and the value of $P_{\mu }$ when $\mu =0.01$ as a function of the mean degree of the underlying contact network.

The reasoning behind the aforementioned observation is intuitive. Let ${\lambda }_1$ denote the phase transition point (i.e., critical mean degree) for a single-strain, bond-percolated network with $T_1$. As the mean degree $\lambda$ increases toward ${\lambda }_1$, the length of the tree of infections starting from the seed* also increases; however, no cycles appear, and the epidemic propagates on a finite, tree-like percolated network (since $\lambda <{\lambda }_1$). Increasing the length of the tree increases the probability that at least one intermediate node would mutate to strain 2, but the fact that the tree is finite makes the particular value of $\mu$ very crucial to $P_{\mu }$. However, as $\lambda$ increases beyond ${\lambda }_1$, cycles start to appear, and a giant component of nodes infected with strain 1 emerges. In this case, the chain of infections is no longer finite, and any positive value of $\mu$ results in a mutation almost surely in the limit of large network size.

Evolutionary Adaptations in Real-World Networks.

In this section, we consider two real-world contact networks that arise naturally in the context of infectious disease propagation and information propagation. In the context of infectious disease propagation, we consider the contact network observed at a US high school during a typical school day (27). In the context of information propagation, we consider the contact network among the friends of 1,000 users (including those 1,000 users) sampled from Twitter (26, 32) (Materials and Methods has more details). Four additional real-world networks are considered in SI Appendix, section 4. Our objective is to reveal the limitations of the single-type bond percolation framework in predicting the probability of emergence on real-world networks.

In Fig. 4, we plot the empirical (averaged over $10^4$ independent experiments) and theoretical (as given in ref. 13) probability of emergence on both networks. We compare the results with the predictions given by the single-type bond percolation framework. Similar to our observations on random networks, the single-type bond percolation framework provides significantly inaccurate predictions on the probability of emergence should the underlying process entail evolution. The limitation is universal as it applies to both random and real-world networks.

Fig. 4.

We consider two real-world contact networks, namely (A) the contact network among the friends of 1,000 users (including those 1,000 users) sampled from Twitter (26, 32) and (B) the contact network observed at a US high school during a typical school day (27). For each $\lambda$, we remove a random subset of edges such that the resulting graph has mean degree $\lambda$. The single-type bond percolation framework provides inaccurate predictions on the probability of emergence in line with our observations in Fig. 2.

The universality of the behavior suggests that the single-type bond percolation framework does not properly capture a fundamental property of spreading processes that entail evolution. Clearly, this property is stemming from the underlying correlations between the infection events of the multiple-strain model. In particular, infection events are conditionally independent given the type of the infective node. Namely, conditioned on node $i$ being type $\ell$, node $i$ infects each of her neighbors independently with probability $T_{\ell }$. However, infection events are marginally dependent unless $T_i=T_0$ for all $i$ with probability one, a condition that essentially reduces the dynamics to that of single-strain processes without evolution. Indeed, the single-type bond percolation framework fails to capture these correlations since they are averaged out according to [1]; hence, its predictions are significantly inaccurate. A similar observation was made by Kenah and Robins (33) in the context of the spread of diseases governed by the SIR model on contact networks. In particular, Kenah and Robins (33) showed that, when the distribution of the infectious periods of the SIR model is nondegenerate, there is no bond percolation probability that will make the bond percolation model isomorphic to the SIR model† due to the underlying correlations of infection events in the SIR model. More details are given in SI Appendix, section 5.

Discussion

In this article, we investigated the effects of evolutionary adaptations on spreading processes in complex networks. We developed a mathematical theory that unraveled the relationship between the characteristics of the spreading process, evolutionary adaptations, and the structure of the underlying contact network. In particular, our theory accurately predicts the epidemic threshold and the expected epidemic size as functions of the characteristics of the spreading object (i.e., $\mathit{T}$), the evolutionary pathways of the object (i.e., $\mathit{\mu }$), and the structure of the underlying network (i.e., the degree distribution).

Our mathematical theory was complemented by an extensive simulation study on both synthetic and real-world contact networks. The simulation results proved the validity of our theory and revealed the significant shortcomings of the classical mathematical models that do not capture evolution. A matching condition between single- and multiple-strain models was proposed and evaluated in the context of probability of emergence, epidemic size, and epidemic threshold. Under the proposed matching condition, our results revealed that the classical bond percolation models may accurately predict the threshold and final size of epidemics that entail evolution, but their predictions on the probability of emergence are significantly inaccurate on both random and real-world networks.

We proceeded by deriving a lower bound on the probability of emergence to gain further insights on the effects of mutation. The bound was derived for the special case of one-step irreversible mutation where strain 1 could mutate to strain 2 that has higher transmissibility (remains as strain 1) with probability $\mu$ ($1-\mu$), while strain 2 does not mutate back to strain 1. Our results revealed that the $\mu$ plays a key role in determining the shape and behavior of the probability of emergence. Our results also shed light on the interplay between $\mu$ and the structure of the underlying network. In particular, the particular value of $\mu$ directly influences the probability of emergence in the small mean degree regime, yet as the mean degree increases and cycles start to form, such dependency starts to loosen as mutation events become inevitable even for very small values of $\mu$.

In future work, we plan to extend our theory to accommodate for real-world networks with a given adjacency matrix along the same line with refs. 34 and 35. Furthermore, we plan to explore the effect of targeted infection and mutation events, where an adversary might choose to allocate her broadcasting resources to particular links or intentionally inject a mutant at a particular point during the spread of the underlying item.

Materials and Methods

Expected Epidemic Size.

The analysis of the probability of emergence was established by Alexander and Day in ref. 13; hence, we focus here on deriving the expected epidemic size $S$ and the expected fraction of individuals infected by strain $i$ denoted $S_i$ for $i=1,\dots ,m$. We focus on the case where $m=2$, yet it is straightforward to extend our theory to the case where there are $m$ general strains as long as the underlying process is indecomposable (13, 36, 37). We apply a tree-based approach that is based on the work by Gleeson and coworkers (38, 39). Their approach draws on the tools developed for analyzing the zero-temperature random-field Ising model on Bethe lattices (40). Observe that, as we build our network using the configuration model, the network structure is locally tree like with the fraction of cycles approaching zero in the limit of large network size (23–25).

Since $\mathbb{G}$ is locally tree like, we can replace it by a tree and arrange the vertices in a hierarchical structure such that, at the top level, there is a single node (the root) that has degree $k$ with probability $p_k$. Each of the $k$ neighbors of the root has degree $k\prime$ with probability $k\prime p_{k\prime }/⟨k⟩$, where $⟨k⟩$ denotes the mean degree of the network. Furthermore, we label the levels of the tree from level $\ell =0$ at the bottom to level $\ell =\infty$ at the top (i.e., the root). Nodes update their status starting from the bottom of the tree and proceeding toward the top. This gives rise to a delicate case, where a node at some level $\ell$ may be exposed to simultaneous infections by both strain 1 and strain 2 from her neighbors at level $\ell -1$. We assume that coinfection is not possible; hence, a node that receives $x$ infections of strain 1 and $y$ infections of strain 2 becomes infected by strain 1 (strain 2) with probability $x/\left(x+y\right)$ ($y/\left(x+y\right)$). We explore the case where coinfection with multiple pathogen strains is possible in SI Appendix, section 6.

Throughout, we say that a node is either inactive if it has not received any infection (i.e., still susceptible) or active and type $i$ if it has been infected and then mutated to strain $i$ for $i=\mathrm{1,2}$. Let $q_{\ell +1,i}$ be the probability that a node at level $\ell +1$, say node $v$, is active and type $i$. Furthermore, let $q_{\ell +1}=q_{\ell +\mathrm{1,1}}+q_{\ell +\mathrm{1,2}}$ (i.e., $q_{\ell +1}$ is the total probability that a node at level $\ell +1$ is active). We start by an arbitrary initial distribution for $\left\{q_{\mathrm{0,1}},q_{\mathrm{0,2}}\right\}$ satisfying $q_{\mathrm{0,1}}>0,q_{\mathrm{0,2}}>0$. Then, we update the distribution properly until we reach the root. Note that, if the degree of node $v$ is $k$, then node $v$ is using one edge to connect to her parent at level $\ell +2$ and $k-1$ edges to connect to her neighbors at level $\ell$. In this case, we can further condition on the number of active neighbors of type 1 and type 2 and the respective number of infections received from them.

Note that we have a multinomial distribution for the number of active neighbors of both types. In particular, a neighbor at level $\ell$ may be active and type 1 with probability $q_{\ell ,1}$, active and type 2 with probability $q_{\ell ,2}$, or inactive with probability $1-q_{\ell }=1-q_{\ell ,1}-q_{\ell ,2}$. In addition, conditioned on having $k_1$ and $k_2$ active neighbors of type 1 and type 2, respectively, the number of type 1 (type 2) infections received, denoted $X$ ($Y$), follows a binomial distribution with parameters $k_1$ and $T_1$ ($k_2$ and $T_2$). Hence, with

$$\begin{array}{ll}\hfill & f\left(u,w,z\right)\hfill \\ \hfill & =\sum _{k_1=0}^z\sum _{k_2=0}^{z-k_1}\left(\begin{array}{c}z\hfill \\ k_1\hfill \end{array}\right)\left(\begin{array}{c}\hfill z-k_1\hfill \\ \hfill k_2\hfill \end{array}\right){\left(u\right)}^{k_1}{\left(w\right)}^{k_2}{\left(1-u-w\right)}^{z-k_1-k_2}\hfill \\ \hfill & \hspace{1em}\cdot \sum _{x=0}^{k_1}\sum _{y=0}^{k_2}\left(\begin{array}{c}\hfill k_1\hfill \\ \hfill x\hfill \end{array}\right)\left(\begin{array}{c}\hfill k_2\hfill \\ \hfill y\hfill \end{array}\right)T_1^x{T}_2^y{\left(1-T_1\right)}^{k_1-x}{\left(1-T_2\right)}^{k_2-y}\hfill \\ \hfill & \hspace{1em}\cdot \left({\mu }_{1i}\mathbf{1}\left[x>0,y=0\right]+{\mu }_{2i}\mathbf{1}\left[x=0,y>0\right]\hspace{0.17em}+\right.\hfill \\ \hfill & \left.\left(\frac{x{\mu }_{1i}}{x+y}\hspace{0.17em}+\frac{y{\mu }_{2i}}{x+y}\right)\mathbf{1}\left[x>0,y>0\right]\right),\hfill \end{array}$$

we have

$$q_{\ell +1,i}=\sum _{k=1}^{\infty }\frac{k{p}_k}{⟨k⟩}f\left(q_{\ell ,1},q_{\ell ,2},k-1\right).$$ [3]

The validity of [3] could be explained as follows. Consider a particular realization $\left(x,y\right)$ of the random variables $\left(X,Y\right)$. Observe that, if $x>0,y=0$, then node $v$ becomes infected by strain 1 and eventually mutates to type $i$ with probability ${\mu }_{1i}$. Similarly, if $x=0,y>0$, then node $v$ becomes infected by strain 2 and eventually mutates to type $i$ with probability ${\mu }_{2i}$. Finally, if $x>0,y>0$, then node $v$ becomes infected by strain 1 (strain 2) with probability $x/\left(x+y\right)$ ($y/\left(x+y\right)$) and eventually mutates to type $i$ with probability ${\mu }_{1i}$ (${\mu }_{2i}$).

Observe that, under the assumption that nodes do not become inactive after they turn active, the quantities $q_{\ell ,i}$ appearing in [3] are nondecreasing in $\ell$, and thus, they converge to a limit $q_{\infty ,i}$ for $i=\mathrm{1,2}$. Finally, the final fraction of nodes that are active and type $i$ is equal (in expected value) to the probability that the root of the tree (at level $\ell \to \infty$) is active and type $i$. Note that, if the tree root has degree $k$, then all of these $k$ edges will be utilized to connect with her neighbors at the lower level. Hence,

$$Q_i=\sum _{k=0}^{\infty }{p}_{k}f\left(q_{\infty ,1},q_{\infty ,2},k\right),$$

where $Q_i$ for $i=\mathrm{1,2}$ denotes the probability that the tree root is active and type $i$ and $q_{\infty ,i}$ for $i=\mathrm{1,2}$ is the steady-state solution of the recursive equations [3]. Note that $Q=Q_1+Q_2$ is the total probability that the tree root is active.

Observe that $q_{\infty ,1}=q_{\infty ,2}=0$ gives a trivial fixed point of the recursive equations [3]. Indeed, this trivial solution leads to $Q=0$. Although the trivial fixed point is a valid numerical solution for the recursive equations [3], we can show that this trivial solution is unstable. Hence, another solution with $q_{\infty ,1}>0$ and $q_{\infty ,2}>0$ may exist. To test whether or not the trivial fixed point is stable, we check the spectral radius of the Jacobian matrix $\mathit{J}\left(q_{\ell ,1},q_{\ell ,2}\right)$ corresponding to the linearization of [3] at $q_{\ell ,1}=q_{\ell ,2}=0$. If the spectral radius of the $\mathit{J}\left(q_{\ell ,1},q_{\ell ,2}\right)$ at $q_{\ell ,1}=q_{\ell ,2}=0$ is larger than 1, then the trivial fixed point is unstable, indicating that there exists another solution with $q_{\infty ,1}>0$ and $q_{\infty ,2}>0$ and implying the existence of a giant component.

The Jacobian matrix (at $q_{\ell ,1}=q_{\ell ,2}=0$) is given by

$$\begin{array}{ll}\hfill \mathit{J}\left(q_{\ell ,1},q_{\ell ,2}\right)& ={\left[\begin{array}{cc}\hfill \frac{\partial q_{\ell +\mathrm{1,1}}}{\partial q_{\ell ,1}}\hfill & \hfill \frac{\partial q_{\ell +\mathrm{1,1}}}{\partial q_{\ell ,2}}\hfill \\ \hfill \frac{\partial q_{\ell +\mathrm{1,2}}}{\partial q_{\ell ,1}}\hfill & \hfill \frac{\partial q_{\ell +\mathrm{1,2}}}{\partial q_{\ell ,2}}\hfill \end{array}\right]}_{q_{\ell ,1}=q_{\ell ,2}=0}\hfill \\ \hfill & =\left(\frac{⟨k^2⟩-⟨k⟩}{⟨k⟩}\right)\left[\begin{array}{cc}{T}_1{\mu }_{11}\hfill & \hfill T_2{\mu }_{21}\hfill \\ T_1{\mu }_{12}\hfill & \hfill T_2{\mu }_{22}\hfill \end{array}\right].\hfill \end{array}$$

It follows that a phase transition occurs when

$$\left(\frac{⟨k^2⟩-⟨k⟩}{⟨k⟩}\right)\rho \left(\mathit{T}\mathit{\mu }\right)>1.$$ [4]

Real-World Networks.

In the context of infectious disease propagation, we consider the contact network observed at a US high school during a typical school day (27). The dataset covers $\mathrm{762,868}$ interactions between students, teachers, and staff. Each interaction between two individuals is characterized by their identification numbers as well as the duration of the interaction. Two individuals could have multiple interactions throughout the day, and we sum the durations of these interactions to calculate the total contact time between these two individuals over the whole day. We proceed by sampling a static graph out of this dataset by assigning an edge between nodes $u$ and $v$ with probability $t_{uv}/t_{max}$, where $t_{uv}$ denotes the total contact time between nodes $u$ and $v$ throughout the day and $t_{max}$ denotes the maximum total contact time observed in the dataset. In the context of information propagation, we consider the contact network among the friends of 1,000 users (including those 1,000 users) sampled from Twitter (26, 32). Four additional real-world networks are investigated in SI Appendix, section 4.

The Twitter network consists of $\mathrm{81,\; 306}$ nodes and $\mathrm{1,\; 342,\; 296}$ edges, with mean degree 33 and global clustering coefficient of 0.170. When we randomly remove edges to set the mean degree to 1 (10), the clustering coefficient becomes 0.005 (0.051). A synthetic contact network with Poisson degree distribution, $n=\mathrm{81,306}$, and mean degree 33 has a clustering coefficient of 0.0004. The high school network consists of 73 nodes and 543 edges with mean degree 14.87 and global clustering coefficient of 0.446. When we randomly remove edges to set the mean degree to 1 (10), the clustering coefficient becomes 0.090 (0.296). A synthetic contact network with Poisson degree distribution, $n=73$, and mean degree 14.87 has a clustering coefficient of 0.183. Hence, the sampled graphs still exhibit significant clustering as compared with random networks with the same mean degree and number of nodes.

Data Availability.

Data will be available on request from O.Y.