Abstract
Infectious or contagious diseases can be transmitted from one person to another through social contact networks. In today’s interconnected global society, such contagion processes can cause global public health hazards, as exemplified by the ongoing Covid19 pandemic. It is therefore of great practical relevance to investigate the network transmission of contagious diseases from the perspective of statistical inference. An important and widely studied boundary condition for contagion processes over networks is the socalled epidemic threshold. The epidemic threshold plays a key role in determining whether a pathogen introduced into a social contact network will cause an epidemic or die out. In this paper, we investigate epidemic thresholds from the perspective of statistical network inference. We identify two major challenges that are caused by high computational and sampling complexity of the epidemic threshold. We develop two statistically accurate and computationally efficient approximation techniques to address these issues under the ChungLu modeling framework. The second approximation, which is based on random walk sampling, further enjoys the advantage of requiring data on a vanishingly small fraction of nodes. We establish theoretical guarantees for both methods and demonstrate their empirical superiority.
Introduction
Infectious diseases are caused by pathogens, such as bacteria, viruses, fungi, and parasites. Many infectious diseases are also contagious, which means the infection can be transmitted from one person to another when there is some interaction (e.g., physical proximity) between them. Today, we live in an interconnected world where such contagious diseases could spread through social contact networks to become global public health hazards. A recent example of this phenomenon is the Covid19 outbreak caused by the socalled novel coronavirus (SARSCoV2) that has spread to many countries (Huang et al. 2020; Zhu et al. 2020; Wang et al. 2020; Sun et al. 2020). This recent global outbreak has caused serious social and economic repercussions, such as massive restrictions on movement and share market decline (Chinazzi et al. 2020). It is therefore of great practical relevance to investigate the transmission of contagious diseases through social contact networks from the perspective of statistical inference.
Consider an infection being transmitted through a population of n individuals. According to the susceptibleinfectedrecovered (SIR) model of disease spread, the pathogen can be transmitted from an infected person (I) to a susceptible person (S) with an infection rate given by β, and an infected individual becomes recovered (R) with a recovery rate given by μ. This can be modeled as a Markov chain whose state at time t is given by a vector \(({X^{t}_{1}}, \ldots , {X^{t}_{n}})\), where \({X^{t}_{i}}\) denotes the state of the i^{th} individual at time t, i.e., \({X^{t}_{i}} \in \{S, I, R\}\). For the population of n individuals, the state space of this Markov chain becomes extremely large with 3^{n} possible configurations, which makes it impractical to study the exact system. This problem was addressed in a series of three seminal papers by Kermack and McKendrick (1927, 1932, 1933). Instead of modeling the disease state of each individual at at a given point of time, they proposed compartmental models, where the goal is to model the number of individuals in a particular disease state (e.g., susceptible, infected, recovered) at a given point of time. Since their classical papers, there has been a tremendous amount of work on compartmental modeling of contagious diseases over the last ninety years (Hethcote, 2000; Van den Driessche and Watmough, 2002; Brauer and CastilloChavez, 2012).
Compartmental models make the unrealistic assumption of homogeneity, i.e., each individual is assumed to have the same probability of interacting with any other individual. In reality, individuals interact with each other in a highly heterogeneous manner, depending upon various factors such as age, cultural norms, lifestyle, weather, etc. The contagion process can be significantly impacted by heterogeneity of interactions (Meyers et al. 2005; Rocha et al. 2011; Galvani and May, 2005; Woolhouse et al. 1997), and therefore compartmental modeling of contagious diseases can lead to substantial errors.
In recent years, contact networks have emerged as a preferred alternative to compartmental models (Keeling, 2005). Here, a node represents an individual, and an edge between two nodes represent social contact between them. An edge connecting an infected node and a susceptible node represents a potential path for pathogen transmission. This framework can realistically represent the heterogeneous nature of social contacts, and therefore provide much more accurate modeling of the contagion process than compartmental models. Notable examples where the use of contact networks have led to improvements in prediction or understanding of infectious diseases include Bengtsson et al. (2015) and Kramer et al. (2016).
Consider the scenario where a pathogen is introduced into a social contact network and it spreads according to an SIR model. It is of particular interest to know whether the pathogen will die out or lead to an epidemic. This is dictated by a set of boundary conditions known as the epidemic threshold, which depends on the SIR parameters β and μ as well as the network structure itself. Above the epidemic threshold, the pathogen invades and infects a finite fraction of the population. Below the epidemic threshold, the prevalence (total number of infected individuals) remains infinitesimally small in the limit of large networks (PastorSatorras et al. 2015). There is growing evidence that such thresholds exist in realworld hostpathogen systems, and intervention strategies are formulated and executed based on estimates of the epidemic threshold. (Dallas et al. 2018; Shulgin et al. 1998; Wallinga et al. 2005; Pourbohloul et al. 2005; Meyers et al. 2005). Fittingly, the last two decades have seen a significant emphasis on studying epidemic thresholds of contact networks from several disciplines, such as computer science, physics, and epidemiology (Newman 2002; Wang et al. 2003; Colizza and Vespignani 2007; Chakrabarti et al. 2008; Gómez et al. 2010; Wang et al. 2016, 2017). See Leitch et al. (2019) for a complete survey on the topic of epidemic thresholds.
Concurrently but separately, network data has rapidly emerged as a significant area in statistics. Over the last two decades, a substantial amount of methodological advancement has been accomplished in several topics in this area, such as community detection (Bickel and Chen, 2009; Zhao et al. 2012; Rohe et al. 2011; Sengupta and Chen, 2015), model fitting and model selection (Hoff et al. 2002; Handcock et al. 2007; Krivitsky et al. 2009; Wang and Bickel, 2017; Yan et al. 2014; Bickel and Sarkar, 2016; Sengupta and Chen, 2018), hypothesis testing (Ghoshdastidar and von Luxburg 2018; Tang et al. 2017a, 2017b; Bhadra et al. 2019), and anomaly detection (Zhao et al. 2018; Sengupta, 2018; Komolafe et al. 2019), to name a few. The stateoftheart toolbox of statistical network inference includes a range of random graph models and a suite of estimation and inference techniques.
However, there has not been any work at the intersection of these two areas, in the sense that the problem of estimating epidemic thresholds has not been investigated from the perspective of statistical network inference. Furthermore, the task of computing the epidemic threshold based on existing results can be computationally infeasible for massive networks. In this paper, we address these gaps by developing a novel samplingbased method to estimate the epidemic threshold under the widely used ChungLu model (Aiello et al. 2000), also known as the configuration model. We prove that our proposed method has theoretical guarantees for both statistical accuracy and computational efficiency. We also provide empirical results demonstrating our method on both synthetic and realworld networks.
The rest of the paper is organized as follows. In Section 2, we formally set up the problem statement and formulate our proposed methods for approximating the epidemic threshold. In Section 3, we describe the theoretical properties of our estimators. In Section 4, we report numerical results from synthetic as well as realworld networks. We conclude the paper with discussion and next steps in Section 5.
Epidemic Thresholds
Table 1 lists the common symbols used throughout the paper. Consider a set of n individuals labelled as 1,…, n, and an undirected network (with no selfloops) representing interactions between them. This can represented by an nbyn symmetric adjacency matrix A, where A(i, j) = 1 if individuals i and j interact and A(i, j) = 0 otherwise. Consider a pathogen spreading through this contact network according to an SIR model. From existing work (Chakrabarti et al. 2008; Gómez et al. 2010; Prakash et al. 2010; Wang et al. 2016, 2017), we know that the boundary condition for the pathogen to become an epidemic is given by
where λ(A) is the spectral radius of the adjacency matrix A.
The left hand side of Eq. 1 is the ratio of the infection rate to the recovery rate, which is purely a function of the pathogen and independent of the network. As this ratio grows larger, an epidemic becomes more likely, as new infections outpace recoveries. The right hand side of Eq. 1 is the spectral radius of the adjacency matrix, which is purely a function of the network and independent of the pathogen. Larger the spectral radius, the more connected the network, and therefore an epidemic becomes more likely. Thus, the boundary condition in Eq. 1 connects the two aspects of the contagion process, the pathogen transmissibility which is quantified by β/μ, and the social contact network which is quantified by the spectral radius. If \(\frac {\beta }{\mu } < \frac {1}{\lambda (A)}\), the pathogen dies out, and if \(\frac {\beta }{\mu } > \frac {1}{\lambda (A)}\), the pathogen becomes an epidemic.
Given a social contact network, the inverse of the spectral radius of its adjacency matrix represents the epidemic threshold for the network. Any pathogen whose transmissiblity ratio is greater than this threshold is going to cause an epidemic, whereas any pathogen whose transmissiblity ratio is less than this threshold is going to die out. Therefore, a key problem in network epidemiology is to compute the spectral radius of the social contact network.
Problem Statement and Heuristics
Realistic urban social networks that are used in modeling contagion processes have millions of nodes (Eubank et al. 2004; Barrett et al. 2008). To compute the epidemic threshold of such networks, we need to find the largest (in absolute value) eigenvalue of the adjacency matrix A. This is challenging because of two reasons.

1.
First, from a computational perspective, eigenvalue algorithms have computational complexity of Ω(n^{2}) or higher. For massive social contact networks with millions of nodes, this can become too burdensome.

2.
Second, from a statistical perspective, eigenvalue algorithms require the entire adjacency matrix for the full network of n individuals. It can be challenging or expensive to collect interaction data of n individuals of a massive population (e.g., an urban metropolis). Furthermore, eigenvalue algorithms typically require the full matrix to be stored in the randomaccess memory of the computer, which can be infeasible for massive social contact networks which are too large to be stored.
The first issue could be resolved if we could compute the epidemic threshold in a computationally efficient manner. The second issue could be resolved if we could compute the epidemic threshold only using data on a small subset of the population. In this paper, we aim to resolve both issues by developing two approximation methods for computing the spectral radius.
To address these problems, let us look at the spectral radius, λ(A), from the perspective of random graph models. The statistical model is given by \(A \sim P\), which is shorthand for \(A(i,j) \sim \text {Bernoulli}(P(i,j))\) for 1 ≤ i < j ≤ n. Then λ(A) converges to λ(P) in probability under some mild conditions (Chung and Radcliffe, 2011; BenaychGeorges et al. 2019; Bordenave et al. 2020). To make a formal statement regarding this convergence, we reproduce below a slightly paraphrased version (for notational consistency) of an existing result in this context.
Lemma 1 (Theorem 1 of Chung and Radcliffe (2011)).
Let
be the maximum expected degree, and suppose that for some 𝜖 > 0,
for sufficiently large n. Then with probability at least 1 − 𝜖, for sufficiently large n,
To make note of a somewhat subtle point: from an inferential perspective it is tempting to view the above result as a consistency result, where λ(P) is the population quantity or parameter of interest and λ(A) is its estimator. However, in the context of epidemic thresholds, we are interested in the random variable λ(A) itself, as we want to study the contagion spread conditional on a given social contact network. Therefore, in the present context, the above result should not be interpreted as a consistency result.
Rather, we can use the convergence result in a different way. For massive networks, the random variable λ(A), which we wish to compute but find it infeasible to do so, is close to the parameter λ(P). Suppose we can find a random variable T(A) which also converges in probability to λ(P), and is computationally efficient since T(A) and λ(A) both converge in probability to λ(P), we can use T(A) as an accurate proxy for λ(A). This would address the first of the two issues described at the beginning of this subsection. Furthermore, if T(A) can be computed from a small subset of the data, that would also solve the second issue. This is our central heuristic, which we are going to formalize next.
The ChungLu Model
So far, we have not made any structural assumptions on P, we have simply considered the generic inhomogeneous random graph model. Under such a general model, it is very difficult to formulate a statistic T(A) which is cheap to compute and converges to λ(P). Therefore, we now introduce a structural assumption on P, in the form of the wellknown ChungLu model that was introduced by Aiello et al. (2000) and subsequently studied in many papers (Chung and Lu, 2002; Chung et al. 2003; Decreusefond et al. 2012; Pinar et al. 2012; Zhang et al. 2017). For a network with n nodes, let δ = (δ_{1},…, δ_{n})^{′} be the vector of expected degrees. Then under the ChungLu model,
This formulation preserves E[d_{i}] = δ_{i}, where d_{i} is the degree of the i^{th} node, and is very flexible with respect to degree heterogeneity.
Under model Eq. 2, note that rank(P) = 1, and we have
Recall that we are looking for some computationally efficient T(A) which converges in probability to λ(P). We now know that under the ChungLu model, λ(P) is equal to the ratio of the second moment to the first moment of the degree distribution. Therefore, a simple estimator of λ(P) is given by the sample analogue of this ratio, i.e.,
We now want to demonstrate that approximating λ(A) by T_{1}(A) provides us with very substantial computational savings with little loss of accuracy. The approximation error can be quantified as
and our goal is to show that \(e_{1}(A) \rightarrow 0\) in probability, while the computational cost of T_{1}(A) is much smaller than that of λ(A). We will show this both from a theoretical perspective and an empirical perspective. We next describe the empirical results from a simulation study, and we postpone the theoretical discussion to Section 3 for organizational clarity.
We used n = 5000, and constructed a ChungLu random graph model where P(i, j) = 𝜃_{i}𝜃_{j}. The model parameters 𝜃_{1},…, 𝜃_{n} determine the expected degrees. We used two models for generating 𝜃_{i}. In the Uniform model, 𝜃_{i} were uniformly sampled from (0,0.25). In the PowerLaw model, 𝜃_{i} were uniformly sampled from the PowerLaw distribution with parameters x_{min} = 0.01, β = 3. Note that the second model leads to heavytailed distribution.
Then, we randomly generated 20 networks from the model, and computed λ(A) and T_{1}(A). The results are reported in Table 2. We observe that the runtimes for T_{1}(A) are orders of magnitude faster than computing the eigenvalue. The average error for T_{1}(A) is small, and so is the standard deviation (SD) of errors. Thus, even for moderately sized networks, using T_{1}(A) as a proxy for λ(A) can reduce the computational cost to a great extent, without much loss in accuracy. For massive networks where n is in millions, this advantage of T_{1}(A) over λ(A) is even greater; however, the computational burden for λ(A) becomes so large that this case is difficult to illustrate using standard computing equipment.
Thus, T_{1}(A) provides us with a computationally efficient and statistically accurate method for finding the epidemic threshold.
Comparing the results from Uniform and PowerLaw, we observe that errors are higher for the PowerLaw model. A likely explanation for this is that since the distribution is heavy tailed, the moment based estimator is less accurate. This is particularly true for larger n, since the impact of extreme values can shift the estimator heavily.
Sampling Based Approximation
The first approximation, T_{1}(A), provides us with a computationally efficient method for finding the epidemic threshold. This addresses the first issue pointed out at the beginning of Section 2.1. However, computing T_{1}(A) requires data on the degree of all n nodes of the network. Therefore, this does not solve the second issue pointed out at the beginning of Section 2.1. We now propose a second alternative, T_{2}, to address the second issue. The idea behind this approximation is based on the same heuristic that was laid out in Section 2.2. Since λ(P) is a function of degree moments, we can estimate these moments using observed node degrees. In defining T_{1}(A), we used observed degrees of all n nodes in the network. However, we can also estimate the degree moments by considering a small sample of nodes, based on random walk sampling. The algorithm for computing T_{2} is given in Algorithm 1.
Note that we only use (t^{∗} + r) randomly sampled nodes for computing T_{2}, which implies that we do not need to collect or store data on the n individuals. Therefore this method overcomes the second issue pointed out at the beginning of Section 2.1. The approximation error arising from this method can be defined as
and we want to show that \(e_{2}(A) \rightarrow 0\) in probability, while the datacollection cost of T_{2}(A) is much less than that of T_{1}(A). In the next section, we are going to formalize this.
Theoretical Results on Approximation Errors
In this section, we are going to establish that the approximation errors e_{1}(A) and e_{2}(A), defined in Eqs. 4 and 5, converge to zero in probability. From Theorem 2.1 of Chung et al. (2003), we know that when
holds, then for any 𝜖 > 0,
Therefore, under Eq. 6, it suffices to show that, for any 𝜖 > 0,
To interpret the condition given in Eq. 6, suppose that the expected degrees are all of the same order, i.e., δ_{i} = O(n^{α}) for some α ∈ (0,1). Then, the left hand side of Eq. 6 is O(n^{α}), and the right hand side is \(\log (n) O(n^{\alpha /2})\), which means the condition is satisfied for any α > 0.
Convergence of T _{1}(A)
First, consider \(T_{1}(A) = \frac {{\sum }_{i=1}^{n} {d_{i}^{2}}}{{{\sum }_{i=1}^{n} d_{i}}}\), and recall that \(\lambda (P) = \frac {{\sum }_{i=1}^{n} {\delta _{i}^{2}}}{{{\sum }_{i=1}^{n} \delta _{i}}}\). For notational convenience, define \(m_{1} = {\sum }_{i=1}^{n} d_{i}, m_{2} = {\sum }_{i=1}^{n} {d_{i}^{2}}, \mu _{1} = {\sum }_{i=1}^{n} \delta _{i}, \mu _{2} = {\sum }_{i=1}^{n} {\delta _{i}^{2}}\). We would like to show that, under reasonable conditions, for any 𝜖 > 0,
Next, we state the theorem which will establish a sufficient condition for this to hold. Please see Appendix for a proof of the theorem.
Theorem 2.
If the average of the expected degrees goes to infinity, i.e., \( \frac {1}{n}{{\sum }_{i} \delta _{i}} \rightarrow \infty \), and the spectral radius dominates \(\log ^{2}(n)\), i.e., \(\frac {{\sum }_{i} {\delta _{i}^{2}}}{{\sum }_{i} \delta _{i}} = \omega (\log ^{2} n)\), then for any 𝜖 > 0,
Thus, we have established that the approximation error for T_{1}(A) goes to zero in probability. We have already observed in Section 2.2 that the runtime for T_{1}(A) is orders of magnitude faster that the runtime for λ(A). Therefore, T_{1}(A) is both computationally efficient and statistically accurate as an approximation of the epidemic threshold.
Convergence of T _{2}(A)
Next, consider Algorithm 1. Let π denote the stationary distribution of the simple random walk on the given graph. Suppose the number of edges in the given graph is m. Recall that, π is given by \(\pi _{v} = \frac {d_{v}}{{\sum }_{v} d_{v}}\) for all v. For brevity, we define the mixing time of the graph A, denoted as t_{mix}(A), to mean the number of steps required by the simple random walk to reach a distribution \(\hat {\pi }\) such that \(\\hat {\pi }  \pi \_{1} = o(\frac {1}{n^{2}})\). Let T_{2}(A) be the estimate returned by the Algorithm 1. We first show an easy lemma that characterizes the bias of the estimator T_{2}(A). Please see Appendix for a proof.
Lemma 3.
If x is a node that is randomly sampled from π, and d_{x} is its degree, then \(E[d_{x}]= \frac {{\sum }_{i} {d_{i}^{2}}}{{\sum }_{i} d_{i}}. \) Consequently if \(\hat {\pi }\) is such that \(\\pi  \hat {\pi }\_{1} = o(n^{1})\) and x is sampled from \(\hat {\pi }\), then \(E[d_{x}]= (1 \pm o(1))\frac {{\sum }_{i} {d_{i}^{2}}}{{\sum }_{i} d_{i}}\).
Next, we show that the estimator v_{RW} is actually concentrated around its expectation.
Theorem 4 (Lezaud (1998)).
Let (X_{n}) be a irreducible and reversible Markov Chain on a finite set V with Q being the transition matrix. Let π be the stationary distribution. Let \(f: V\rightarrow \Re \) be such that E_{π}[f] = 0, \(\f\_{\infty } \leq 1\) and 0 < E_{π}[f^{2}] ≤ b^{2}. Then, for any initial distribution q, any positive integer r and all 0 < γ ≤ 1,
where ε(Q) = 1 − λ_{2}(Q), λ_{2}(Q) being the second largest eigenvalue of Q, S_{q} = ∥q/π∥_{2} (in the ℓ_{2}(π) norm), and
If γ ≪ b^{2} and ε(Q) ≪ 1, then the upper bound becomes
Using the above result, we bound the sample complexity of our estimator. We first quote the following result that we use to bound λ_{1} of the transition matrix. Please see Appendix for a proof.
Theorem 5.
Let Q = D^{− 1}A. Let 𝜖, δ ∈ (0,1). Algorithm 1, using \(r = \frac {1}{\varepsilon (Q) \epsilon ^{3/2}} \times \frac {12m d_{\max \limits }}{({\sum }_{v} {d_{v}^{2}})} \log (1/\delta )\) and t^{∗}≥ t_{mix}(G) returns an estimate v_{RW} that satisfies, with probability 1 − δ,
The number of nodes that are touched by algorithm is O(t^{∗} + r).
Note that Q = D^{− 1}A has the same set of eigenvalues as the matrix D^{− 1/2}AD^{− 1/2}. For the ChungLu model, the eigenvalues of the matrix L = I − D^{− 1/2}AD^{− 1/2} can be bounded by the following result from Chung et al. (2003).
Theorem 6.
Let L = I − D^{− 1/2}AD^{− 1/2} denote the normalized Laplacian. Let A be a random graph generated from the given expected degrees model, with expected degrees {δ_{i}}, if the minimum expected degree \(\delta _{\min \limits }\) satisfies \(\delta _{min} \gg \ln (n)\), then with probability at least 1 − 1/n = 1 − o(1), we have that for all eigenvalues \(\lambda _{k}(L) > \lambda _{\min \limits }(L)\) of the Laplacian of G,
It follows above that ε(Q) = 1 − λ_{2}(Q) = 1 − λ_{2}(D^{− 1/2}AD^{− 1/2}) = λ_{n− 1}(I − D^{− 1/2}AD^{− 1/2}) = 1 − o(1). Putting these together, we get the following corollary on the total number of node queries.
Corollary 6.1.
For a graph generated from the expected degrees model, with probability 1 − 1/n, Algorithm 1, needs to query
nodes in order to get a (1 ± 𝜖) estimate of \({\sum }_{v} {d_{v}^{2}}/2m\).
Note \(\frac {6({\sum }_{v} d_{v}) d_{\max \limits }}{({\sum }_{v} {d_{v}^{2}})} \le \frac {6d_{\max \limits }}{d_{\min \limits }}\), but this is a loose bound, better bounds can be derived for power law degree distributions, for instance.
Thus, we have proved that the approximation error for T_{2}(A) goes to zero in probability. In addition, Corollary 6.1 shows that the number of nodes that we need to query in order to have an accurate approximation is much smaller than n. Furthermore, computing T_{2} only requires node sampling and counting degrees, and therefore the runtime is much smaller than eigenvalue algorithms. Therefore, T_{2}(A) is a computationally efficient and statistically accurate approximation of the epidemic threshold, while also requiring a much smaller data budget compared to T_{1}(A).
Numerical Results
In this section, we characterize the empirical performance of our sampling algorithm on two synthetic networks, one generated from the ChungLu model and the second generated from the preferential attachment model of Barabási and Albert (1999).
Data
Our first dataset is a graph generated from the ChungLu model of expected degrees. We generated a powerlaw sequence (i.e. fraction of nodes with degree d is proportion to d^{−β}) with exponent β = 2.5 and then generated a graph with this sequence as the expected degrees. Table 3 notes that, as expected, the first eigenvalue λ_{1}(A) is close to \(\frac {{\sum }_{v} {d_{v}^{2}}}{{\sum }_{v} d_{v}}\).
The second dataset is generated from the preferential attachment model (Barabási and Albert, 1999), where each incoming node adds 5 edges to the existing nodes, the probability of choosing a specific node as neighbor being proportional to the current degree of that node. While the preferential attachment model naturally gives rise to a directed graph, we convert the graph to an undirected one before running our algorithm. It is interesting to note that even in this case the ChungLu model does not hold, our first approximation, T_{1}(A), is close to λ(A).
Implementation Details
In each of the networks, the random walk algorithm presented in Algorithm 1 was used for sampling. The random walk was started from an arbitrary node and every 10^{th} node was sampled (to account for the mixing time) from the walk. These samples were then used to calculate T_{2}(A). This experiment was repeated 10 times. These gave estimates \({T_{2}^{1}},\ldots ,T_{2}^{10}\). We then calculate two relative errors ∀i ∈{1,2,…,10},
We also note the following relation between the two error metrics.
We denote the averages of \(\{\epsilon _{i}^{T_{1}T_{2}}\}\) and \(\{\epsilon _{i}^{\lambda T_{2}}\}\) as \(\epsilon ^{T_{1}T_{2}}\) and \(\{\epsilon ^{\lambda T_{2}}\}\) respectively. It is easy to observe that the above relation holds between the two average quantities too.
We plot the averages \(\epsilon ^{T_{1}T_{2}}\) and \(\epsilon ^{\lambda T_{2}}\), along with the errorbars that reflect the standard deviation, against the actual number of nodes seen by the random walk. Note that the xaxis accurately reflect how many times the algorithm actually queried the network, not just the number of samples used. Measuring the cost of uniform node sampling in this setting, for instance, would need to keep track of how many nodes are touched by a MetropolisHastings walk that implements the uniform distribution.
Results
In Fig. 1 We plot the two results for mean relative error, measure by \(\epsilon _{i}^{\lambda T_{2}}\) and \(\epsilon _{i}^{T_{1}T_{2}}\).
For the two ChungLu networks, the algorithm is able to get a 10% approximation to the statistic T_{1}(A) by exploring at most 10% of the network. With more samples from the random walk, the mean relative errors settle to around 4–5%. However, once we measure the mean relative errors with respect to λ(A), it becomes clearer that the estimator T_{2}(A) does better when the graph is closer to the assumed (i.e. ChungLu) model. For the ChungLu graph, the mean error 𝜖^{λ−T2} essentially is very similar to \(\epsilon ^{T_{1}T_{2}}\), which is to be expected. For the preferential attachment graph too, it is clear that the estimate T_{2} is able to achieve a better than 10% relative error approximation of λ(A).
Note that, if we were instead counting only the nodes whose degrees were actually used for estimation, the fraction of network used would be roughly 1–2% in all the cases, the majority of the node query cost actually goes in making the random walk mix, by using an initial burnin period and by maintaining certain number of steps between subsequent samples.
Discussion
In this work, we investigated the problem of computing SIR epidemic thresholds of social contact networks from the perspective of statistical inference. We considered the two challenges that arise in this context, due to high computational and datacollection complexity of the spectral radius. For the ChungLu network generative model, the spectral radius can be characterized in terms of the degree moments. We utilized this fact to develop two approximations of the spectral radius. The first approximation is computationally efficient and statistically accurate, but requires data on observed degrees of all nodes. The second approximation retains the computationally efficiency and statistically accuracy of the first approximation, while also reducing the number of queries or the sample size quite substantially. The results seem very promising for networks arising from the ChungLu and preferential attachment generative models.
There are several interesting and important future directions. The methods proposed in this paper have provable guarantees only under the ChungLu model, although it works very well under the preferential attachment model. This seems to indicate that the degree based approximation might be applicable to a wider class of models. On the other hand, this leaves open the question of developing a better “modelfree” estimator, as well as asking similar questions about other network features.
In this work we only considered the problem of accurate approximation of the epidemic threshold. From a statistical as well as a realworld perspective, there are several related inference questions. These include uncertainty quantification, confidence intervals, onesample and twosample testing, etc.
Social interaction patterns vary dynamically over time, and such network dynamics can have significant impacts on the contagion process Leitch et al. (2019). In this paper we only considered static social contact networks, and in future we hope to study epidemic thresholds for timevarying or dynamic networks.
Finally, we note that the formulation in Eq. 1 is an approximation of the true epidemic threshold under the socalled quenchedmeanfield approximation (PastorSatorras et al. 2015; Karrer et al. 2014). In recent work Castellano and PastorSatorras (2020), it has been shown that the SIS epidemic transition occurs at some point that is intermediate between λ(A) and T_{1}(A). In future work, we plan to extend our results to these more accurate expressions for the epidemic threshold.
References
Aiello, W., Chung, F. and Lu, L. (2000). A random graph model for massive graphs, In Proceedings of the ThirtySecond Annual ACM Symposium on Theory of computing. ACM, p. 171–180.
Barabási, A.L. and Albert, R. (1999). Emergence of scaling in random networks. Science 286, 509–512.
Barrett, C.L., Bisset, K.R., Eubank, S.G., Feng, X. and Marathe, M.V. (2008). Episimdemics: an efficient algorithm for simulating the spread of infectious disease over large realistic social networks, In SC’08: Proceedings of the 2008 ACM/IEEE Conference on Supercomputing. IEEE, p. 1–12.
BenaychGeorges, F., Bordenave, C., Knowles, A. et al. (2019). Largest eigenvalues of sparse inhomogeneous erdős–rényi graphs. Ann. Probab.47, 1653–1676.
Bengtsson, L., Gaudart, J., Lu, X., Moore, S., Wetter, E., Sallah, K., Rebaudet, S. and Piarroux, R. (2015). Using mobile phone data to predict the spatial spread of cholera. Sci. Rep. 5, 8923.
Bhadra, S., Chakraborty, K., Sengupta, S. and Lahiri, S. (2019). A bootstrapbased inference framework for testing similarity of paired networks. arXiv:1911.06869.
Bickel, P.J. and Chen, A. (2009). A nonparametric view of network models and Newman–Girvan and other modularities. Proc. Natl. Acad. Sci. 106, 21068–21073.
Bickel, P.J. and Sarkar, P. (2016). Hypothesis testing for automated community detection in networks. J. R. Stat. Soc. Ser. B (Stat. Methodol.) 78, 253–273.
Bordenave, C., BenaychGeorges, F. and Knowles, A (2020). Spectral radii of sparse random matrices. Ann. l’Inst. Henri Poincare (B) Probab. Stat.
Brauer, F. and CastilloChavez, C. (2012). Mathematical models in population biology and epidemiology, vol. 2. Springer, Berlin.
Castellano, C. and PastorSatorras, R. (2020). Cumulative merging percolation and the epidemic transition of the susceptibleinfectedsusceptible model in networks. Phys. Rev. X 10, 011070.
Chakrabarti, D., Wang, Y., Wang, C., Leskovec, J. and Faloutsos, C. (2008). Epidemic thresholds in real networks. ACM Trans. Inf. Syst. Secur.10, 1–26.
Chinazzi, M., Davis, J.T., Ajelli, M., Gioannini, C., Litvinova, M., Merler, S., Piontti, A.P., Mu, K., Rossi, L., Sun, K. et al. (2020). The effect of travel restrictions on the spread of the 2019 novel coronavirus (covid19) outbreak. Science 368, 6489, 395–400.
Chung, F. and Lu, L. (2002). The average distances in random graphs with given expected degrees. Proc. Natl. Acad. Sci. 99, 15879–15882.
Chung, F. and Radcliffe, M. (2011). On the spectra of general random graphs. Electron. J. Combinator. 18, P215–P215.
Chung, F., Lu, L. and Vu, V. (2003). Eigenvalues of random power law graphs. Ann. Combinator. 7, 21–33.
Colizza, V. and Vespignani, A. (2007). Invasion threshold in heterogeneous metapopulation networks. Phys. Rev. Lett. 99, 148701.
Dallas, T.A., Krkošek, M. and Drake, J.M. (2018). Experimental evidence of a pathogen invasion threshold. R. Soc. Open Sci. 5, 171975.
Decreusefond, L., Dhersin, J. S., Moyal, P., Tran, V.C. et al. (2012). Large graph limit for an sir process in random network with heterogeneous connectivity. Ann. Appl. Probab. 22, 541–575.
Eubank, S., Guclu, H., Kumar, V.A., Marathe, M.V., Srinivasan, A., Toroczkai, Z. and Wang, N. (2004). Modelling disease outbreaks in realistic urban social networks. Nature 429, 180–184.
Galvani, A.P. and May, R.M. (2005). Dimensions of superspreading. Nature 438, 293–295.
Ghoshdastidar, D. and von Luxburg, U. (2018). Practical methods for graph twosample testing, In Advances in Neural Information Processing Systems, p. 3019–3028.
Gómez, S., Arenas, A., BorgeHolthoefer, J., Meloni, S. and Moreno, Y. (2010). Discretetime markov chain approach to contactbased disease spreading in complex networks. EPL (Europhys. Lett.) 89, 38009.
Handcock, M.S., Raftery, A.E. and Tantrum, J.M. (2007). Modelbased clustering for social networks. J. R. Stat. Soc.: Ser. A 170, 301–354.
Hethcote, H.W. (2000). The mathematics of infectious diseases. SIAM Rev. 42, 599–653.
Hoeffding, W. (1994). Probability inequalities for sums of bounded random variables, In The Collected Works of Wassily Hoeffding. Springer, p. 409–426.
Hoff, P.D., Raftery, A.E. and Handcock, M.S. (2002). Latent space approaches to social network analysis. J. Am. Stat. Assoc. 97, 1090–1098.
Huang, C., Wang, Y., Li, X., Ren, L., Zhao, J., Hu, Y., Zhang, L., Fan, G., Xu, J., Gu, X. et al. (2020). Clinical features of patients infected with 2019 novel coronavirus in Wuhan, China. Lancet 395, 497–506.
Karrer, B., Newman, M.E. and Zdeborová, L. (2014). Percolation on sparse networks. Phys. Rev. Lett. 113, 20, 208702.
Keeling, M. (2005). The implications of network structure for epidemic dynamics. Theor. Popul. Biol. 67, 1–8.
Kermack, W.O. and McKendrick, A.G. (1927). A contribution to the mathematical theory of epidemics. Proc. R. Soc. Lond. Ser. A, Containing papers of a mathematical and physical character 115, 700–721.
Kermack, W.O. and McKendrick, A.G. (1932). Contributions to the mathematical theory of epidemics. ii.—the problem of endemicity. Proc. R. Soc. Lond. Ser. A, Containing papers of a mathematical and physical character 138, 55–83.
Kermack, W.O. and McKendrick, A.G. (1933). Contributions to the mathematical theory of epidemics. iii.—further studies of the problem of endemicity. Proc. R. Soc. Lond. Ser. A, Containing Papers of a Mathematical and Physical Character 141, 94–122.
Komolafe, T., Quevedo, A.V., Sengupta, S. and Woodall, W.H. (2019). Statistical evaluation of spectral methods for anomaly detection in static networks. Netw. Sci. 7, 319–352.
Kramer, A.M., Pulliam, J.T., Alexander, L.W., Park, A.W., Rohani, P. and Drake, J.M. (2016). Spatial spread of the west africa ebola epidemic. R. Soc. Open Sci. 3, 8, 160294.
Krivitsky, P.N., Handcock, M.S., Raftery, A.E. and Hoff, P.D. (2009). Representing degree distributions, clustering, and homophily in social networks with latent cluster random effects models. Social Netw. 31, 204–213.
Leitch, J., Alexander, K.A. and Sengupta, S. (2019). Toward epidemic thresholds on temporal networks: a review and open questions. Appl. Netw. Sci. 4, 105.
Lezaud, P. (1998). Chernofftype bound for finite markov chains. Ann. Appl. Probab. 8, 3, 849–867.
Meyers, L.A., Pourbohloul, B., Newman, M., Skowronski, D.M. and Brunham, R.C. (2005). Network theory and SARS: predicting outbreak diversity. J. Theor. Biol. 232, 71–81.
Newman, M.E.J. (2002). Spread of epidemic disease on networks. Phys. Rev. E 66, 1, 016128.
PastorSatorras, R., Castellano, C., Van Mieghem, P. and Vespignani, A. (2015). Epidemic processes in complex networks. Rev. Mod. Phys. 87, 925–979.
Pinar, A., Seshadhri, C. and Kolda, T.G. (2012). The similarity between stochastic Kronecker and Chunglu graph models, In Proceedings of the 2012 SIAM International Conference on Data Mining. SIAM, p. 1071–1082.
Pourbohloul, B., Meyers, L., Skowronski, D., Krajden, M., Patrick, D. and Brunham, R. (2005). Modeling control strategies of respiratory pathogens. Emerg. Infect. Dis. 11, 1249–56.
Prakash, B.A., Chakrabarti, D., Faloutsos, M., Valler, N. and Faloutsos, C. (2010). Got the flu (or mumps)? Check the Eigenvalue! arXiv:1004.0060.
Rocha, L.E.C., Liljeros, F. and Holme, P. (2011). Simulated epidemics in an empirical spatiotemporal network of 50,185 sexual contacts. PLoS Comput. Biol. 7, e1001109.
Rohe, K., Chatterjee, S. and Yu, B. (2011). Spectral clustering and the highdimensional stochastic blockmodel. Ann. Stat. 39, 1878–1915.
Sengupta, S. (2018). Anomaly detection in static networks using egonets. arXiv:1807.089251807.08925.
Sengupta, S. and Chen, Y. (2015). Spectral clustering in heterogeneous networks. Stat. Sin. 25, 1081–1106.
Sengupta, S. and Chen, Y. (2018). A block model for node popularity in networks with community structure. J. R. Stat. Soc.: Ser. B (Stat. Methodol.) 80, 365–386.
Shulgin, B., Stone, L. and Agur, Z. (1998). Pulse vaccination strategy in the sir epidemic model. Bull. Math. Biol. 60, 1123–1148.
Sun, K., Chen, J. and Viboud, C. (2020). Early epidemiological analysis of the coronavirus disease 2019 outbreak based on crowdsourced data: a populationlevel observational study. Lancet Digit. Health 2, 4, e201–e208.
Tang, M., Athreya, A., Sussman, D.L., Lyzinski, V., Park, Y. and Priebe, C.E. (2017a). A semiparametric twosample hypothesis testing problem for random graphs. J. Comput. Graph. Stat. 26, 344–354.
Tang, M., Athreya, A., Sussman, D.L., Lyzinski, V. and Priebe, C.E. (2017b). A nonparametric twosample hypothesis testing problem for random graphs. Bernoulli 23, 1599–1630.
Van den Driessche, P. and Watmough, J. (2002). Reproduction numbers and subthreshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 180, 29–48.
Wallinga, J., Heijne, J.C. and Kretzschmar, M. (2005). A measles epidemic threshold in a highly vaccinated population. PLoS Med. 2, e316.
Wang, Y.R. and Bickel, P.J. (2017). Likelihoodbased model selection for stochastic block models. Ann. Stat. 45, 500–528.
Wang, Y., Chakrabarti, D., Wang, C. and Faloutsos, C. (2003). Epidemic spreading in real networks: an eigenvalue viewpoint, In 22nd International Symposium on Reliable Distributed Systems, 2003. Proceedings. IEEE Computer Society, Florence, p. 25–34.
Wang, W., Liu, Q.H., Zhong, L.F. et al. (2016). Predicting the epidemic threshold of the susceptibleinfectedrecovered model. Sci. Rep. 6, 24676. https://doi.org/10.1038/srep24676.
Wang, W., Tang, M., Stanley, H.E. and Braunstein, L.A. (2017). Unification of theoretical approaches for epidemic spreading on complex networks. Rep. Progr. Phys. 80, 036603.
Wang, C., Horby, P.W., Hayden, F.G. and Gao, G.F. (2020). A novel coronavirus outbreak of global health concern. Lancet 395, 470–473.
Woolhouse, M.E.J., Dye, C., Etard, J.F., Smith, T., Charlwood, J.D., Garnett, G.P., Hagan, P., Hii, J.L.K., Ndhlovu, P.D., Quinnell, R.J., Watts, C.H., Chandiwana, S.K. and Anderson, R.M. (1997). Heterogeneities in the transmission of infectious agents: implications for the design of control programs. Proc. Natl. Acad. Sci. 94, 338–342.
Yan, X., Shalizi, C., Jensen, J.E., Krzakala, F., Moore, C., Zdeborová, L., Zhang, P. and Zhu, Y. (2014). Model selection for degreecorrected block models. J. Stat. Mech.: Theory Exp. 2014, P05007.
Zhang, X., Moore, C. and Newman, M.E. (2017). Random graph models for dynamic networks. Eur. Phys. J. B 90, 200.
Zhao, Y., Levina, E. and Zhu, J. (2012). Consistency of community detection in networks under degreecorrected stochastic block models. Ann. Stat. 40, 2266–2292.
Zhao, M.J., Driscoll, A.R., Sengupta, S., Fricker, Jr. R. D., Spitzner, D.J. and Woodall, W.H. (2018). Performance evaluation of social network anomaly detection using a moving window–based scan method. Qual. Reliab. Eng. Int. 34, 1699–1716.
Zhu, N., Zhang, D., Wang, W., Li, X., Yang, B., Song, J., Zhao, X., Huang, B., Shi, W., Lu, R. et al. (2020). A novel coronavirus from patients with pneumonia in China. New Engl. J. Med., 2019.
Acknowledgements
We thank the Associate Editor and two anonymous reviewers for their constructive suggestions, which were really helpful towards the improvement of the manuscript. Anirban acknowledges the kind support of the N. Rama Rao Chair Professorship at IIT Gandhinagar, the Google India AI/ML award (2020), Google Faculty Award (2015), and CISCO University Research Grant (2016). Srijan acknowledges the support from an NIH R01 grant 1R01LM013309.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Disclaimer with Respect to Current Pandemic
We do realize that in the face of the current pandemic, while it is important to pursue research relevant to it, it is also important to be responsible in following the proper scientific process. We would like to state that in this work, the question of epidemic threshold estimation has been formalized from a theoretical viewpoint in a much used, but simple, random graph model. We are not yet at a position to give any guarantees about the performance of our estimator in real social networks. We do hope, however, that the techniques developed here can be further refined to work to give reliable estimators in practical settings.
Anirban Dasgupta’s work is partially supported by grants from DBT India, Google and CISCO.
Srijan Sengupta’s work is partially supported by an NIH R01 grant 1R01LM013309.
Appendix A. Technical Proofs
Appendix A. Technical Proofs
A.1 Proof of Theorem 2
We will show that for any \(\epsilon ^{\prime } > 0\),
We first prove that Eq. 8 implies Eq. 7. Equation 8 implies that
Now, consider the event \(\left \{\left \frac {m_{1}}{\mu _{1}}  1 \right  \le \epsilon ^{\prime }\right \} \cap \left \{\left \frac {m_{2}}{\mu _{2}}  1\right  \le \epsilon ^{\prime }\right \}\). Note that m_{2}/m_{1} is a strictly increasing function of m_{2} and a strictly decreasing function of m_{1}. Therefore, for outcomes belonging to the above event,
Note that
given that \(\epsilon ^{\prime } < 1/2\). Now, fix 𝜖 > 0 and let \(\epsilon ^{\prime } = \epsilon /4\). Then,
Thus, proving Eq. 8 is sufficient for proving Eq. 7.
Proof 1 (Proof of Theorem 2).
We will use Hoeffding’s inequality (Hoeffding, 1994) for the first part, and we begin by stating the inequality for the sum of Bernoulli random variables. Let B_{1},…, B_{m} be m independent (but not necessarily identically distributed) Bernoulli random variables, and \(S_{m} = {\sum }_{i=1}^{m} B_{i}\). Then for any t > 0,
In our case,
and we know that {A(i, j) : 1 ≤ i < j ≤ n} are independent Bernoulli random variables. Fix 𝜖 > 0 and note that \(E[{\sum }_{i<j} A(i,j)] = \frac {1}{2}\mu _{1}\). Using Hoeffding’s inequality with S_{m} = m_{1}/2, \(m = {n \choose 2}\), and \(t = \frac {\epsilon }{2} \mu _{1}\), we get
Since \( \frac {1}{n}{{\sum }_{i} \delta _{i}} \rightarrow \infty \), the right hand side goes to zero. Therefore,
For the second part, we can characterize m_{2} as following.
and hence,
We show that, under the given assumptions, with probability 1 − o(1), m_{2} − E[m_{2}] = o(μ_{2}). Furthermore, E[m_{2}] − μ_{2} = o(μ_{2}).
As noted before, each d_{i} is a sum of binomial random variables. By applying ChernoffHoeffding bound, and union bounding over all i ∈{1,…, n}, we can get, with probability 1 − o(1), and for any fixed 𝜖 ∈ (0,1),
Let the above event be called the event \(\mathcal {A}\). If the event \(\mathcal {A}\) happens, then,
Note that \(\frac {n}{\mu _{2}} = \frac {1}{{\sum }_{i} {\delta _{i}^{2}} / n} \rightarrow 0\) under the given assumption. Furthermore,
Putting these together, and using \(\epsilon ^{\prime } = 3\epsilon \) we have the given claim. □
A.2 Proof of Theorem 5
Proof 2 (Proof of Lemma 3).
It is easy to see that
We show the second claim as follows:
□
Proof 3 (Proof of Theorem 5).
In our setting the set V is the set of vertices. Define the function f(X_{i}) as :
f(⋅) clearly satisfies E_{π}[f] = 0 and that \(\f\_{\infty } \le 1\). We can bound E_{π}[f^{2}] as
Using the first t^{∗} steps, we reach the distribution \(\hat {\pi }\) that satisfies \(\\pi  \hat {\pi }\_{1} = o(n^{1})\). Hence,
where the last step follows as \(\\pi  \hat {\pi }\_{1} = o(n^{2})\).
We use \(b^{2} = d_{\max \limits }^{2} {\sum }_{v} \frac {{d_{v}^{3}}}{{\sum }_{v} d_{v}}\) and \(\gamma = \epsilon d_{\max \limits }^{1}\times \frac {{\sum }_{v} {d_{v}^{2}}}{{\sum }_{v} d_{v}}\). Hence
Hence,
Plugging this, we get that
Setting \(r = \frac {1}{\varepsilon (Q) \epsilon ^{3/2}} \times \frac {6({\sum }_{v} d_{v}) d_{\max \limits }}{({\sum }_{v} {d_{v}^{2}})} \log (1/\delta )\), and using Theorem 4, we can claim that, with probability 1 − δ,
The bound on the number of nodes touched/queried by the algorithm follows naturally. □
Rights and permissions
About this article
Cite this article
Dasgupta, A., Sengupta, S. Scalable Estimation of Epidemic Thresholds via Node Sampling. Sankhya A 84, 321–344 (2022). https://doi.org/10.1007/s13171021002490
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13171021002490
Keywords
 Epidemic threshold
 Networks
 Sampling
 Random walk
 Configuration model
 Epidemiology.
PACS Nos
 62F10 (primary)
 68W20, 68W25