Information Source Detection with Limited Time Knowledge

TL;DR

This paper proposes an efficient infection source detection method using limited timestamp data and network topology, providing provable guarantees and demonstrating superior performance on synthetic and real datasets.

Contribution

It introduces a linear integer programming approach for source detection on trees with performance guarantees under limited timestamps, extending to general graphs.

Findings

The estimator is always on a candidate path, improving efficiency.

Performance guarantees show bounded error distance on regular trees.

Experimental results confirm superior accuracy on synthetic and real data.

Abstract

This paper investigates the problem of utilizing network topology and partial timestamps to detect the information source in a network. The problem incurs prohibitive cost under canonical maximum likelihood estimation (MLE) of the source due to the exponential number of possible infection paths. Our main idea of source detection, however, is to approximate the MLE by an alternative infection path based estimator, the essence of which is to identify the most likely infection path that is consistent with observed timestamps. The source node associated with that infection path is viewed as the estimated source $\hat{v}$. We first study the case of tree topology, where by transforming the infection path based estimator into a linear integer programming, we find a reduced search region that remarkably improves the time efficiency. Within this reduced search region, the estimator $\hat{v}$ is provably always on a path which we term as \emph{candidate path}. This notion enables us to analyze the distribution of $d(v^{\ast},\hat{v})$, the error distance between $\hat{v}$ and the true source $v^{\ast}$, on arbitrary tree, which allows us to obtain for the first time, in the literature provable performance guarantee of the estimator under limited timestamps. Specifically, on the infinite $g$-regular tree with uniform sampled timestamps, we get a refined performance guarantee in the sense of a constant bounded $d(v^{\ast},\hat{v})$. By virtue of time labeled BFS tree, the estimator still performs fairly well when extended to more general graphs. Experiments on both synthetic and real datasets further demonstrate the superior performance of our proposed algorithms.

Equations

$$t_u\triangleq \min \{t|u\in \mathcal{I}[t]\}.$$

$$\hat{v}\in \arg \underset{v\in V}{\max }\mathbf{P}(v|{\mathcal{O}}_t),$$

$$\hat{v}\in \arg \underset{v\in V}{\max }\mathbf{P}({\mathcal{O}}_t|v).$$

$$\hat{v}=\hat{v}(\mathcal{S})\in \arg \underset{v\in V}{\max }\mathbf{P}(\{t_s{\}}_{s\in \mathcal{S}}|\mathcal{S},v^{\ast }=v)$$

$$\mathbf{P}(\{t_s{\}}_{s\in \mathcal{S}}|\mathcal{S},v^{\ast }=v)=\sum _{T(v,\mathcal{S},\mathbf{t})\in {\mathcal{T}}_1}\mathbf{P}(T(v,\mathcal{S},\mathbf{t})|\mathcal{S},v^{\ast }=v)$$

$$=\sum _{T(v,\mathcal{S},\mathbf{t})\in {\mathcal{T}}_1}\mathbf{P}(T(v,\mathcal{S},\mathbf{t})|\mathbf{t}(v),\mathcal{S},v^{\ast }=v)\mathbf{P}(\mathbf{t}(v)|\mathcal{S},v^{\ast }=v)$$

$$\hat{T}(v,\mathcal{S},\mathbf{t})\in \arg \underset{T(v,\mathcal{S},\mathbf{t})\in \mathcal{T}}{\max }\mathbf{P}(T(v,\mathcal{S},\mathbf{t})|\mathbf{t}(v),\mathcal{S},v^{\ast }=v)$$

$${\mathbf{t}}^{\ast }\in \arg \underset{\mathbf{t}}{\max }\mathbf{P}(T(v,\mathcal{S},\mathbf{t})|\mathcal{S},v^{\ast }=v)$$

$$\log \mathbf{P}(T(v,\mathcal{S},\mathbf{t})|\mathcal{S},v^{\ast }=v)$$

$$=\log \prod _{e=(i,j)\in E(T(v,\mathcal{S}))}\mathbf{P}(w(e)=\mathbf{t}(j)-\mathbf{t}(i))$$

$$=\sum _{e=(i,j)\in E(T(v,\mathcal{S}))}\log \left((1-p{)}^{\mathbf{t}(j)-\mathbf{t}(i)-1}p\right)$$

$$=\log (1-p)\cdot \sum _{(i,j)\in E(T(v,\mathcal{S}))}\left(\mathbf{t}(j)-\mathbf{t}(i)\right)$$

$$+\log ((1-p{)}^{-1}p)\cdot |E(T(v,\mathcal{S}))|$$

$$\begin{array}[]{ll}\text{minimize (over }\mathbf{t}\text{)}&\sum_{(i,j)\in E(T(v,\mathcal{S}))}\mathbf{t}(j)-\mathbf{t}(i)\\ \text{subject to }&\mathbf{t}(u)=t_{u}\quad\forall u\in\mathcal{S}\\ &\mathbf{t}(j)-\mathbf{t}(i)\geq 1\quad\forall(i,j)\in E(T(v,\mathcal{S}))\\ &\mathbf{t}(u)\in\mathbb{Z}\quad\forall u\in V(T(v,\mathcal{S}))\end{array}$$

$$\sum _{(i,j)\in E(T(v,\mathcal{S}))}\mathbf{t}(j)-\mathbf{t}(i)=\sum _{u\in V(T(v,\mathcal{S}))}\mathbf{t}(u)(d_{\text{in}}(u)-d_{\text{out}}(u))$$

$$=\sum _{s\in \mathcal{S}}{t}_s(d_{\text{in}}(s)-d_{\text{out}}(s))+\sum _{u\in V(T(v,\mathcal{S}))\backslash \mathcal{S}}\mathbf{t}(u)(d_{\text{in}}(u)-d_{\text{out}}(u))$$

$${\tau }_u=\underset{s\in \mathcal{S}\cap V(T_u^v)}{\min }\{t_s-d(s,u)\}$$

$$\underset{\mathbf{t}}{\max }\mathbf{P}(T(v,\mathcal{S},\mathbf{t})|\mathcal{S},v^{\ast }=v)=(1-p{)}^{a_v-|E(T(v,\mathcal{S}))|}{p}^{|E(T(v,\mathcal{S}))|}$$

$$\hat{v}\in \arg \underset{v\in V}{\max }\underset{\mathbf{t}}{\max }\mathbf{P}(T(v,\mathcal{S},\mathbf{t})|\mathcal{S},v^{\ast }=v)$$

$$\underset{u\in {\mathcal{R}}_1}{\max }\underset{\mathbf{t}}{\max }\mathbf{P}(T(u,\mathcal{S},\mathbf{t})|\mathcal{S},v^{\ast }=u)$$

$$>\underset{u\in V\backslash {\mathcal{R}}_1}{\max }\underset{\mathbf{t}}{\max }\mathbf{P}(T(u,\mathcal{S},\mathbf{t})|\mathcal{S},v^{\ast }=u)$$

$$\underset{\mathbf{t}}{\max }\mathbf{P}(T(u,\mathcal{S},\mathbf{t})|\mathcal{S},v^{\ast }=u)$$

$$\le \underset{u^{\prime }\in V(T_{\text{ste}}(\mathcal{S}))}{\max }\underset{\mathbf{t}}{\max }\mathbf{P}(T(u^{\prime },\mathcal{S},\mathbf{t})|\mathcal{S},v^{\ast }=u^{\prime })$$

$$\underset{\mathbf{t}}{\max }\mathbf{P}(T(u,\mathcal{S},\mathbf{t})|\mathcal{S},v^{\ast }=u)=0,$$

$$a_u=\sum _{(i,j)\in E(T(u,\mathcal{S}))}({\tau }_j-{\tau }_i)$$

$$\log \frac{{\max }_{\mathbf{t}}\mathbf{P}(T(u,\mathcal{S},\mathbf{t})|\mathcal{S},v^{\ast }=u)}{{\max }_{\mathbf{t}}\mathbf{P}(T(u^{\prime },\mathcal{S},\mathbf{t})|\mathcal{S},v^{\ast }=u^{\prime })}$$

$$=\log (1-p)\cdot (a_u-a_{u^{\prime }})+\log ((1-p{)}^{-1}p)\cdot |\mathcal{P}(u,u^{\prime })|$$

$$=\log (1-p)\cdot (a_u-a_{u^{\prime }}-d(u,u^{\prime }))+\log p\cdot d(u,u^{\prime })$$

$$\le \log p\cdot d(u,u^{\prime })\le \log p\le 0\forall p\in (0,1]$$

$$d_{\mathcal{S}}(u,v)\triangleq |\mathcal{S}\cap V(\mathcal{P}(u,v))|$$

$$\underset{\mathbf{t}}{\max }\mathbf{P}(T(v,\mathcal{S},\mathbf{t})|\mathcal{S},v^{\ast }=v)=0$$

$$\mathbf{t}(s_1)-\mathbf{t}(s_2)=\sum _{(i,j)\in \mathcal{P}(s_1,s_2)}(\mathbf{t}(i)-\mathbf{t}(j))\ge \sum _{(i,j)\in \mathcal{P}(s_1,s_2)}1=d(s_1,s_2)$$

$$\underset{\mathbf{t}}{\max }\mathbf{P}(T(v,\mathcal{S},\mathbf{t})|\mathcal{S},v^{\ast }=v)=0$$

$$\underset{\mathbf{t}}{\max }\mathbf{P}(T(u,\mathcal{S},\mathbf{t})|\mathcal{S},v^{\ast }=u)\le \underset{\mathbf{t}}{\max }\mathbf{P}(T(s,\mathcal{S},\mathbf{t})|\mathcal{S},v^{\ast }=s)$$

$$\underset{u\in \mathcal{C}}{\max }\underset{\mathbf{t}}{\max }\mathbf{P}(T(u,\mathcal{S},\mathbf{t})|\mathcal{S},v^{\ast }=u)$$

$$\ge \underset{u\in V\backslash \mathcal{C}}{\max }\underset{\mathbf{t}}{\max }\mathbf{P}(T(u,\mathcal{S},\mathbf{t})|\mathcal{S},v^{\ast }=u)$$

Full text

Complex networks, detection algorithms, graph theory, inference algorithms, probability.

††copyright: rightsretained††doi: 10.475/123_4††isbn: 123-4567-24-567/08/06††conference: ACM Woodstock conference; July 1997; El Paso, Texas USA††journalyear: 1997††article: 4††price: 15.00††ccs: Computing methodologies Learning in probabilistic graphical models††ccs: Mathematics of computing Probabilistic inference problems††ccs: Theory of computation Sample complexity and generalization bounds

1. Introduction

Many phenomenon can be modeled as information propagation in networks over time. Prevalent examples include spread of a disease through a population, transmission of information through a distributed network, and the diffusion of scientific discovery in academic network. In all these scenarios, it is disastrous once an isolated risk is amplified through diffusion in networks. Source detection therefore is critical for preventing the spreading of malicious information, and reducing the potential damages incurred.

In this paper, we study the source inference problem: given that a message has been diffused in network $G$, can we tell which node is the source of diffusion given some observations $\mathcal{O}_{t}$ at time $t$? The solution to this problem can help us answer many questions of a common theme: Which computer is the first one infected by computer virus? Who first spreads out the fake news in online social networks? Where is the origin of an epidemic? and which paper is the first scientific rumor on a specific topic in academic citation networks?

While finding the source node has these important applications, it is known that this problem is highly challenging, especially in complex networks. The prior studies mainly focus on topology of infected subgraph. Under the assumption that a full or partial snapshot of the infected nodes is observed at some time, some topology based estimators (such as rumor centrality, Jordan center, etc.) are proposed under various diffusion models (Shah and Zaman, 2010, 2011, 2012; Zhu and Ying, 2016b; Chen et al., 2016; Luo et al., 2017; Nguyen et al., 2016; Zhu and Ying, 2014; Luo et al., 2014; Zhu et al., 2017; Zhu and Ying, 2016a). These estimators, unfortunately, often suffer from poor source detection accuracy and high cost for obtaining the snapshot. Later on, metadata such as timestamps of infected nodes and the direction from which a node gets infected is exploited in the hope of improving the localization precision (Pinto et al., 2012; Zhu et al., 2016; Tang et al., 2018). However, they typically assume a Gaussian-distributed transmission delay for each edge, which may be impractical for many applications such as Bitcoin P2P network(Fanti and Viswanath, 2017) and mobile phone network(Wang et al., 2009), etc. In these networks, the transmission delay for each edge has been verified to follow Geometric distribution.

In this paper, we adopt the discrete-time susceptible-infected (SI) model. The network is assumed to be an undirected graph. Initially, only one node is infected at some unknown time. The infection then begins to diffuse in the network via random interaction between neighboring nodes. Now, we wish to locate the source node using some observation $\mathcal{O}_{t}$. We assume that $\mathcal{O}_{t}$ contains some set $\mathcal{S}$ of nodes with first infection timestamps $\{t_{s}\}_{s\in\mathcal{S}}$. The nodes in $\mathcal{S}$ is sampled uniformly at random. Given partial timestamps $\{t_{s}\}_{s\in\mathcal{S}}$, the question is which node is the information source.

In order to infer the information source using limited timestamps $\{t_{s}\}_{s\in\mathcal{S}}$, one may seek for the solution via a ML estimator, as is widely adopted in many prior arts. However, such an estimator incurs exponential complexity. Instead, here we develop an infection path based estimator where the source is the root node of the most likely time labeled cascading tree consistent with observed timestamps $\{t_{s}\}_{s\in\mathcal{S}}$. In a tree graph, by establishing an equivalence between infection path based estimator and a linear integer programming, the infection path based estimator can be efficiently resolved via message passing. In a general graph, to overcome the difficulty of searching exponential number of infection paths, we incorporate a time labeled BFS heuristic to approximate the infection path based estimator using linear integer programming.

We remark that in our problem of interest, only limited timestamps $\{t_{s}\}_{s\in\mathcal{S}}$ and the location of nodes $\mathcal{S}$ are considered as observation $\mathcal{O}_{t}$. This setting has many practical advantages over those using snapshot and direction information (Shah and Zaman, 2011; Luo et al., 2017; Zhu and Ying, 2016b). First, it is time consuming, and sometimes impossible, to collect the full snapshot of the infected nodes at some time. For example, Twitter’s streaming API only allows a small percentage ($1\%$) of the full stream of tweets to be crawled. Second, sometimes the direction from which a susceptible node gets infected is hard to obtain. For example, in a flu outbreak a person often cannot tell with certainty who infected him/her. The same also goes for anonymous social networks (Fanti et al., 2017, 2016), where the direction information is hidden. Finally, sampled nodes $\mathcal{S}$ with timestamps $\{t_{s}\}_{s\in\mathcal{S}}$ contains more information than partial snapshot, and is easy to access in most scenarios (such as online social network, etc.).

The primary contributions are summarized as follows:

• •

We propose an infection path based estimator to approximate the maximum likelihood estimator in detecting the information source. In a tree graph, this estimator is equivalent to a linear integer programming that can be efficiently solved via message passing approaches. By exploiting the property of linear integer programming, we find a reduced search region that remarkably improves the time efficiency. In a general graph, a time labeled BFS heuristic is incorporated to approximate the infection path based estimator.

• •

We define a novel concept called candidate path to assist the analysis of error distance $d(v^{\ast},\hat{v})$ between the true source $v^{\ast}$ and the estimated source $\hat{v}$ on an arbitrary tree. Under the assumption that the limited timestamps are sampled uniformly at random, we provide a lower bound on cumulative distribution function of $d(v^{\ast},\hat{v})$ by utilizing the conditional independence property on infinite $g$-regular trees. To our best knowledge, this is the first estimator with provable performance guarantee under limited timestamps.

• •

Extensive simulations over various networks are performed to verify the performance of the infection path based estimator. The error distance $d(v^{\ast},\hat{v})$ over $g$-regular trees is found to be within a constant and decreases when $g$ becomes larger.

The rest of this paper is organized as follows: We describe the system model in Section 2. The algorithm for computing the estimator is presented in Section 3. We discuss the performance of the estimator in Section 4. Simulations and experiments are shown in Section 5, and we conclude in Section 6.

2. System Model

2.1. Infection Diffusion Model

Consider an undirected graph $G(V,E)$ where $V$ is the set of nodes and $E$ is the set of edges of the form $(i,j)$ for some node $i$ and $j$ in $V$. We use the susceptible-infected SI model in epidemiology to characterize the infection diffusion process. Suppose that time is slotted. Let $\mathcal{I}[t]$ denote the set of infected nodes at the end of time-slot $t\in\mathbb{Z}$. Initially only one node $v^{\ast}\in V$ gets infected at the beginning of some time-slot $t_{0}\in\mathbb{Z}$. Thus $\mathcal{I}[t_{0}]=\{v^{\ast}\}$ and $\mathcal{I}[t]=\varnothing$ for $t<t_{0}$. At the beginning of each time-slot $t>t_{0}$, each infected node attempts independently to infect each of its susceptible neighbors with success probability $p\in(0,1]$. We define the first infection timestamp of node $u$ as the time-slot $t_{u}$ in which the state of node $u$ changes from susceptible to infected. Formally, $t_{u}$ is given by

[TABLE]

2.2. The Source Inference Problem

Under the above SI-based infection diffusion model, we would like to locate the source node $v^{\ast}$ using some observations of the infection diffusion process. We denote the observations until some time-slot $t$ as $\mathcal{O}_{t}$, the detailed specification of which will be given in Section 2.3. The source inference problem can be formulated as the maximum-a-posteriori (MAP) estimation problem as

[TABLE]

where $\hat{v}$ is the inferred source node. Since we do not know a priori from which source the diffusion started, it is natural to assume a uniform prior probability of the source node among all nodes $V$. Following this set up, the MAP estimation is equivalent to maximum likelihood (ML) estimation problem given by

[TABLE]

2.3. Detection Model

At some time-slot $t>t_{0}$, we realized that an infection has been diffused in network $G$. In order to estimate source node $v^{\ast}$, we first sample some nodes $\mathcal{S}\subset V$ and obtained their first infection timestamps $\{t_{s}\}_{s\in\mathcal{S}}$. Then we use some source localization algorithm $\mathcal{A}$ to infer the source node. Thus, the source inference consists of two stages: 1) sampling $\mathcal{S}$ and 2) estimating source using $\mathcal{A}$.

In this paper we do not talk about the sampling of nodes $\mathcal{S}$, but focus on the source detection given $\mathcal{S}$ and $\{t_{s}\}_{s\in\mathcal{S}}$. Using the observations $\mathcal{O}_{t}=\{t_{s}\}_{s\in\mathcal{S}}$, the ML estimator could be written as

[TABLE]

However, the likelihood in Eq.(1) is difficult to compute in general. To see this, we first give definitions of cascading tree and labeled cascading tree, which explain the diffusion path from a source node to any other destination nodes.

Definition 2.1 (Cascading Tree).

Given a source node $v$ and a set of destination nodes $\mathcal{S}$ in graph $G(V,E)$, the cascading tree $T(v,\mathcal{S})$ is a directed subtree in $G$ rooted at $v$ satisfying

1. (1)

$T(v,\mathcal{S})$ spans nodes $\mathcal{S}$, i.e., $\mathcal{S}\subset V(T(v,\mathcal{S}))$; 2. (2)

For any $u\in V(T(v,\mathcal{S}))$, if $d_{\text{out}}(u)=0$ then $u\in\mathcal{S}$; 3. (3)

$d_{\text{in}}(v)=0$ and $d_{\text{in}}(u)=1$ for any $u\in V(T(v,\mathcal{S}))\backslash\{v\}$.

where $d_{\text{out}}(u)$ and $d_{\text{in}}(u)$ are the out-degree and in-degree respectively in directed subtree $T(v,\mathcal{S})$, respectively. The set of cascading trees for source node $v$ and destination nodes $\mathcal{S}$ is denoted as $\mathcal{T}(v,\mathcal{S})$.

Definition 2.2 (Labeled Cascading Tree).

Given any cascading tree $T(v,\mathcal{S})\in\mathcal{T}(v,\mathcal{S})$, consider any mapping $\mathbf{t}:V(T(v,\mathcal{S}))\rightarrow\mathbb{Z}$ from its nodes to time domains where $\mathbf{t}(u)$ denotes the first infection timestamp of node $u$. We call $\mathbf{t}$ a permitted timestamp for cascading tree $T(v,\mathcal{S})$ if $\mathbf{t}(u)\geq\mathbf{t}(\text{parent}(u))+1$ for each node $u\in V(T)\backslash\{v\}$. The cascading tree $T(v,\mathcal{S})$ associated with permitted timestamps $\mathbf{t}$ is called labeled cascading tree $T(v,\mathcal{S},\mathbf{t})$. The set of labeled cascading tree for source node $v$ and destination nodes $\mathcal{S}$ is denoted as $\mathcal{T}(v,\mathcal{S},\mathbf{t})$.

To understand the above two definitions in the context of diffusion process, as shown in Figure 1 we consider a grid graph in which two possible cascading trees $T_{1}(v,\mathcal{S})$ and $T_{2}(v,\mathcal{S})$ are highlighted. The node $v$ refers to the root node 1 of the cascading trees, and sampled nodes $\mathcal{S}=\{5,9\}$. In each cascading tree, the parent node of $u$ represents the node from which $u$ first gets infected. The cascading tree with permitted timestamps $\mathbf{t}$ recovers the infection process starting from node 1.

Based on labeled cascading tree, the likelihood in Eq.(1) could be decomposed as

[TABLE]

where $\mathcal{T}_{1}\triangleq\{T|T\in\mathcal{T}(v,\mathcal{S},\mathbf{t}),\mathbf{t}(s)=t_{s}\forall s\in\mathcal{S}\}$. It is challenging to compute the likelihood in Eq.(2) because the summation is taken over all labeled cascading trees and even counting the number of permitted labeled cascading trees $|\mathcal{T}_{1}|$ has been shown to be $\#$P-hard (Brightwell and Winkler, 1991).

As an alternative, in Section 3 we will propose an approximate solution that jointly estimates $v^{\ast}$ and labeled cascading tree together. This approach, as will be further demonstrated in Section 4, leads to provably good performance for tree topologies.

3. Infection Path Based Source Localization

In our approximate solution, we shall treat both the infection starting time $t_{0}$ and the labeled cascading tree as variables to be jointly estimated with source node. After sampling nodes $\mathcal{S}$, in second stage, we want to identify the infection path that most likely leads to $\{t_{s}\}_{s\in\mathcal{S}}$, i.e.,

[TABLE]

where $\mathcal{T}\triangleq\bigcup_{v\in V}\{T(v,\mathcal{S},\mathbf{t})|\mathbf{t}(s)=t_{s}\forall s\in\mathcal{S}\}$ denotes the set of all permitted labeled cascading trees which are consistent with observed timestamps $\{t_{s}\}_{s\in\mathcal{S}}$. The source node associated with $\hat{T}(v,\mathcal{S},\mathbf{t})$ is then viewed as the source node. We call the estimated source node $\hat{v}$ infection path based estimator because it is the source node of the most likely time labeled cascading tree that explains the observed limited timestamps.

However, the optimization problem in Eq.(3) is still not easy to solve due to a large number of possible cascading trees involved. Below, we propose a two-step solution. First we fix the cascading tree $T(v,\mathcal{S})\in\mathcal{T}(v,\mathcal{S})$ rooted at node $v\in V$, and maximize the likelihood of infection path over all permitted timestamps $\mathbf{t}$ to find the most likely time labeled cascading tree. Second, we maximize the likelihood of infection path over all possible cascading trees $\mathcal{T}(v,\mathcal{S})$ to find the most likely infection path $\hat{T}(v,\mathcal{S},\mathbf{t})$. This gives exact solution for general trees, and heuristic for general graphs.

3.1. Infection Path Likelihood Computation in General Trees

In this section we solve the first step, i.e., compute the most likely permitted timestamps $\mathbf{t}^{\ast}$ associated with the cascading tree $T(v,\mathcal{S})$ that are consistent with the observations $\{t_{v}\}_{v\in\mathcal{S}}$, given by

[TABLE]

Let $w(e)$ denote the transmission delay for edge $e\in E$ under the infection diffusion model. It is obvious that $\{w(e)\}_{e\in E}$ is a collection of i.i.d. random variables following geometric distribution, i.e., $\mathbf{P}(w(e)=k)=(1-p)^{k-1}p$ for $k=1,2,\ldots$. The logarithm of likelihood $\mathbf{P}(T(v,\mathcal{S},\mathbf{t})|v^{\ast}=v)$ could be decomposed in terms of $\{w(e)\}_{e\in E}$ in general tree as follows

[TABLE]

where $e=(i,j)\in E(T(v,\mathcal{S}))$ is a directed edge in $T(v,\mathcal{S})$ from $i$ to $j$. Given the cascading tree $T(v,\mathcal{S})$, both $p$ and $|E(T(v,\mathcal{S}))|$ are fixed for all permitted timestamps $\mathbf{t}$. By combining Eq.(4) and Eq.(5) we can easily verify that the optimization problem in Eq.(4) is equivalent to following linear integer programming (LIP):

[TABLE]

where $\mathbf{t}=\{\mathbf{t}(u)\}_{u\in V(T)}$ is a collection of timestamps for nodes in $V(T)$. Note that the LIP(6) may be infeasible, in which case there is no permitted timestamps for the cascading tree $T(v,\mathcal{S})$ under the constraints of partial timestamps $\{t_{v}\}_{v\in\mathcal{S}}$. In other words, the infeasibility of LIP(6) indicates that the probability $\mathbf{P}(T(v,\mathcal{S},\mathbf{t})|v^{\ast}=v)$ for any timestamps $\mathbf{t}$ is 0 given partial timestamps $\{t_{v}\}_{v\in\mathcal{S}}$.

Note that the objective function of LIP(6) is the sum of transmission delays over all edges of $T(v,\mathcal{S})$. The intuition of LIP(6) is to minimize the total transmission delays over all edges of $T(v,\mathcal{S})$ under the constraints of limited timestamps $\{t_{s}\}_{s\in\mathcal{S}}$. If we plug the constraints $\mathbf{t}(u)=t_{u}\forall u\in\mathcal{S}$ into the objective function of LIP(6), then

[TABLE]

where $d_{\text{in}}(u),d_{\text{out}}(u)$ are the in-degree and out-degree of node $u$, respectively, on cascading tree $T(v,\mathcal{S})$. Note that for any node $u\in V(T(v,\mathcal{S}))\backslash\mathcal{S}$, $d_{\text{out}}(u)\geq 1$ since $u$ is non-leaf node. According to the definition of the cascading tree, we must have $d_{\text{in}}(u)\leq 1$. It implies that $d_{\text{in}}(u)-d_{\text{out}}(u)\leq 0$. Therefore, to minimize the objective function of LIP(6), we shall assign the largest possible timestamps to nodes in $V(T(v,\mathcal{S}))\backslash\mathcal{S}$.

This can be done by having each node $u\in V(T(v,\mathcal{S}))\backslash\{v\}$ pass two messages up to its parent. The first message is the virtual timestamp of node $u$, which we denote as $\tau_{u}$. The second message is the aggregate of the transmission delays of the edges $E(T^{v}_{u})$, which we denote as $a_{u}$. Here $T^{v}_{u}$ refers to the directed subtree of $T(v,\mathcal{S})$ that is rooted at $u$ and points away from $v$. The details of message passing are included in Algorithm 1, the time complexity of which is $O(|V(T(v,\mathcal{S}))|)$. And the optimality of message passing in solving LIP(6) is established in Proposition 3.1.

Proposition 3.1 (Optimality of Algorithm 1).

Algorithm 1 returns empty if and only if LIP(6) is infeasible. If LIP(6) is feasible, the aggregate delays $a_{v}$ at the source node $v$ is the optimal value of LIP(6), and the virtual timestamp of node $u\in V(T(v,\mathcal{S}))$ is

[TABLE]

where $T_{u}^{v}$ denotes the subtree of $T(v,\mathcal{S})$ that is rooted at $u$ and points away from $v$.

The proof is included in Appendix A.

Note that after solving LIP(6) for cascading tree $T(v,\mathcal{S})$, the maximum likelihood of $T(v,\mathcal{S})$ with respect to $\mathbf{t}$ is

[TABLE]

if the output of Algorithm 1 is not empty.

3.2. Source Localization on a Tree

After computing the most likely timestamp for a fixed cascading tree, according to infection path based estimator in Eq.(3) we need to search over all cascading trees to find the most likely labeled cascading tree $\hat{T}(v,\mathcal{S},\mathbf{t})$. When the underlying graph is a tree, there is only one cascading tree rooted at node $v$ since no cycle exists. Then the estimator is simply

[TABLE]

where the inner maximization over $\mathbf{t}$ is to find the most likely labeled cascading tree $T(v,\mathcal{S},\mathbf{t})$ given $T(v,\mathcal{S})$, and the outer maximization over $v$ is to identify the source with most likely infection path.

To reduce the search region, we partition the underlying tree according to the infection path likelihood $\max_{\mathbf{t}}\mathbf{P}(T(v,\mathcal{S},\mathbf{t})|\mathcal{S},v^{\ast}=v)$. As shown in Figure 2, the underlying tree is partitioned into four disjoint regions: $\mathcal{R}_{1}$, $\mathcal{R}_{2}$, $\mathcal{R}_{3}$, and $V-\mathcal{R}_{1}-\mathcal{R}_{2}-\mathcal{R}_{3}$. In the following we will show in three steps that

[TABLE]

The first step is to show $\max_{u\in\bigcup_{i=1}^{3}\mathcal{R}_{i}}\max_{\mathbf{t}}\mathbf{P}(T(v,\mathcal{S},\mathbf{t})|\mathcal{S},v^{\ast}=u)\geq\max_{u\in V-\bigcup_{i=1}^{3}\mathcal{R}_{i}}\max_{\mathbf{t}}\mathbf{P}(T(v,\mathcal{S},\mathbf{t})|\mathcal{S},v^{\ast}=u)$. Observe that $\mathcal{R}_{1}\cup\mathcal{R}_{2}\cup\mathcal{R}_{3}=V(T_{\text{ste}}(\mathcal{S}))$ in Figure 2, where $T_{\text{ste}}(\mathcal{S})$ is the minimum Steiner tree spanning $\mathcal{S}$ in the underlying tree.

Lemma 3.2.

When the underlying graph is a tree, for any true source $v^{\ast}\in V$, any infection probability $p\in(0,1]$, and any observed partial timestamps $\{t_{s}\}_{s\in\mathcal{S}}$, we have

[TABLE]

for any node $u\in V-V(T_{\text{ste}}(\mathcal{S}))$.

Proof.

Apparently Eq.(11) holds when

[TABLE]

in which case the LIP(6) for cascading tree $T(u,\mathcal{S})$ is infeasible.

Now we assume that LIP(6) for cascading tree $T(u,\mathcal{S})$ is feasible, and its optimal value is given by

[TABLE]

where $\tau_{i}$ is the virtual timestamp of node $i\in V(T(u,\mathcal{S}))$. According to the definition of the cascading tree, $T_{\text{ste}}(\mathcal{S})\subset T(u,\mathcal{S})$. Since $u\notin V(T_{\text{ste}}(\mathcal{S}))$ and $T(u,\mathcal{S})$ is a directed tree without cycle, there must be a node $u^{\prime}\in V(T_{\text{ste}}(\mathcal{S}))$ connecting node $u$ with other node in $V(T_{\text{ste}}(\mathcal{S}))$. Such node $u^{\prime}$ can be found by $u^{\prime}\in\arg\min_{v\in V(T_{\text{ste}}(\mathcal{S}))}d(v,u)$. And then $T(u,\mathcal{S})=\mathcal{P}(u,u^{\prime})\cup T_{\text{ste}}(\mathcal{S})$ and $E(\mathcal{P}(u,u^{\prime}))\cap E(T_{\text{ste}}(\mathcal{S}))=\varnothing$. Note that cascading tree $T(u^{\prime},\mathcal{S})$ is minimum Steiner tree $T_{\text{ste}(\mathcal{S})}$ whose edges are directed. And $T(u^{\prime},\mathcal{S})=T^{u}_{u^{\prime}}$ where $T^{u}_{u^{\prime}}$ denotes the subtree of $T(u,\mathcal{S})$ that is rooted at $u^{\prime}$ and points away from $u$. According to Appendix A.1, we have $\tau_{w}=\tau^{\prime}_{w}$ for any node $w\in V(T_{\text{ste}}(\mathcal{S}))$ where $\tau^{\prime}_{w}$ is the virtual timestamp of node $w$ when running Algorithm 1 for cascading tree $T(u^{\prime},\mathcal{S})$. Then

[TABLE]

∎

The second step is to show that $\max_{\mathbf{t}}\mathbf{P}(T(u,\mathcal{S},\mathbf{t})|\mathcal{S},v^{\ast}=u)=0$ for any node $u\in\mathcal{R}_{3}$. We first give some definitions that could help characterize $\mathcal{R}_{1}$, $\mathcal{R}_{2}$, and $\mathcal{R}_{3}$.

Definition 3.3.

When the underlying graph is a tree, for each node $v\in V$, we define the distance $d_{\mathcal{S}}(u,v)$ between $u$ and $v$ with respect to sampled nodes $\mathcal{S}$ to be the number of sampled nodes on path $\mathcal{P}(u,v)$, i.e.,

[TABLE]

Note that for any node $u\in\mathcal{R}_{3}$, we have $d_{\mathcal{S}}(v^{\ast},v)\geq 2$ as shown in Figure 2. To prove $\max_{\mathbf{t}}\mathbf{P}(T(u,\mathcal{S},\mathbf{t})|\mathcal{S},v^{\ast}=u)=0$ for any node $u\in\mathcal{R}_{3}$, it suffices to argue Lemma 3.4.

Lemma 3.4.

When the underlying graph is a tree, for any node $v\in V$, we have

[TABLE]

if $d_{\mathcal{S}}(v^{\ast},v)\geq 2$.

Proof.

If $d_{\mathcal{S}}(v^{\ast},v)\geq 2$, there are at least two distinct nodes $s_{1},s_{2}\in\mathcal{S}$ such that $\mathcal{P}(v^{\ast},v)=\mathcal{P}(v^{\ast},s_{1})\cup\mathcal{P}(s_{1},s_{2})\cup\mathcal{P}(s_{2},v)$. It implies that $t_{s_{1}}<t_{s_{2}}$. Now consider the LIP(6) for cascading tree $T(v,\mathcal{S})$. Assume that $\mathbf{t}$ is one permitted timestamps satisfying all the constraints of LIP(6) for cascading tree $T(v,\mathcal{S})$. For node $s_{1}$ and $s_{2}$ we have $\mathbf{t}(s_{1})=t_{s_{1}}$ and $\mathbf{t}(s_{2})=t_{s_{2}}$. Note that

[TABLE]

which violates the fact that $\mathbf{t}(s_{1})-\mathbf{t}(s_{2})=t_{s_{1}}-t_{s_{2}}<0$. This contradiction indicates that LIP(6) for cascading tree $T(v,\mathcal{S})$ is infeasible which means that

[TABLE]

∎

The third step is to show that $\max_{u\in\mathcal{R}_{1}}\max_{\mathbf{t}}\mathbf{P}(T(u,\mathcal{S},\mathbf{t})|\mathcal{S},v^{\ast}=u)>\max_{u\in\mathcal{R}_{2}}\max_{\mathbf{t}}\mathbf{P}(T(u,\mathcal{S},\mathbf{t})|\mathcal{S},v^{\ast}=u)\geq 0$. It suffices to argue that for any node $u\in\mathcal{R}_{2}$, there is a node $v\in\mathcal{R}_{1}$ such that $\max_{\mathbf{t}}\mathbf{P}(T(u,\mathcal{S},\mathbf{t})|\mathcal{S},v^{\ast}=u)\leq\max_{\mathbf{t}}\mathbf{P}(T(v,\mathcal{S},\mathbf{t})|\mathcal{S},v^{\ast}=v)$. Note that for any node $u\in\mathcal{R}_{2}$, $d_{\mathcal{S}}(v^{\ast},u)=1$ and $u\notin\mathcal{S}$. It suffices to argue Lemma 3.5.

Lemma 3.5.

When the underlying graph is a tree, for any node $u\in V$, if $d_{\mathcal{S}}(v^{\ast},u)=1$ and $u\notin\mathcal{S}$ we have

[TABLE]

where $s$ is the unique sampled node on path $\mathcal{P}(v^{\ast},u)$.

We defer the proof of Lemma 3.5 to Appendix B. Then combining Lemma 3.2, 3.4, and 3.5, we can draw the conclusion that the likelihood of the time labeled cascading tree rooted at those nodes around the true source is larger, as stated in Proposition 3.6.

Proposition 3.6.

When the underlying graph is a tree, we have

[TABLE]

where $\mathcal{C}\triangleq\{u\in V(T_{\text{ste}}(\mathcal{S}))|d_{\mathcal{S}}(v^{\ast},u)=0\}\cup\{s\in\mathcal{S}|d(v^{\ast},s)=1\}$.

When revisiting Figure 2, it is easy to observe that $\mathcal{R}_{1}$ is exactly $\mathcal{C}$ in Proposition 3.6 which proves the inequality (10).

According to Proposition 3.6, we could reduce the search region from $V$ to $\mathcal{C}$ for infection path based estimator. However, it seems to be impractical due to lack of prior knowledge of where the true source $v^{\ast}$ is. Therefore, we seek for another region $\mathcal{C}^{\prime}$ such that $\mathcal{C}\subset\mathcal{C}^{\prime}\subset V(T_{\text{ste}}(\mathcal{S}))\subset V$ and could be obtained from partial timestamps $\{t_{s}\}_{s\in\mathcal{S}}$, the sampled set $\mathcal{S}$, and topology of underlying tree graph. Intuitively, the region $\mathcal{C}$ should be close to the sampled node with the minimum timestamp. We verify this intuition in Lemma 3.7 and define the region $\mathcal{C}^{\prime}$ in Proposition 3.8.

Lemma 3.7.

Let $s_{0}\in\arg\min_{s\in\mathcal{S}}\{t_{s}\}$ denote any sampled node with minimum observed timestamp (ties broken arbitrarily), then $s_{0}\in\mathcal{C}$.

Proof.

Since $s_{0}$ is a sampled node with minimum observed timestamp, there cannot be any other sampled node on the path $\mathcal{P}(v^{\ast},s_{0})$. Therefore, $d_{\mathcal{S}}(v^{\ast},s_{0})=1$ which implies that $s_{0}\in\mathcal{C}$. ∎

Proposition 3.8.

Let $s_{0}$ be the sampled node with minimum observed timestamp. Let $S\triangleq\{s\in\mathcal{S}|d_{\mathcal{S}}(s,s_{0})\leq 2\}$, then

[TABLE]

Proof.

It sufficies to prove that $\mathcal{C}\subset V\left(\bigcup_{s\in S}\mathcal{P}(s_{0},s)\right)$. We consider two cases.

(1) Consider the case where $v^{\ast}\in\mathcal{S}$, then $s_{0}=v^{\ast}$ and $\mathcal{C}=\{v^{\ast}\}$. Apparently $\mathcal{C}\subset V\left(\bigcup_{s\in S}\mathcal{P}(s_{0},s)\right)$.

(2) Consider the case where $v^{\ast}\notin\mathcal{S}$. For any node $u\in\mathcal{C}$, if $d_{\mathcal{S}}(v^{\ast},u)=1$ then $u\in\mathcal{S}$ therefore $d_{\mathcal{S}}(u,s_{0})=d_{\mathcal{S}}(u,v^{\ast})+d_{\mathcal{S}}(v^{\ast},s_{0})=2$ which implies that $u\in S\subset V\left(\bigcup_{s\in S}\mathcal{P}(s_{0},s)\right)$. If $d_{\mathcal{S}}(v^{\ast},u)=0$, then there must exists at least one sampled node $s^{\prime}\in S$ such that node $u$ is on the path $\mathcal{P}(s^{\prime},v^{\ast})$. Note that $s^{\prime}\in S$ and $u\in V(\mathcal{P}(s^{\prime},s_{0}))$, therefore $u\in V(\mathcal{P}(s_{0},s^{\prime}))\subset V\left(\bigcup_{s\in S}\mathcal{P}(s_{0},s)\right)$. ∎

Note that the $V\left(\bigcup_{s\in S}\mathcal{P}(s_{0},s)\right)$ in Proposition 3.8 could be computed via breadth-first search starting from $s_{0}$. The details are given in Algorithm 2. Note that the most time consuming part is breadth-first search starting from node $s_{0}$, therefore the time complexity of Algorithm 2 is $O(|V|)$. Given $V^{\prime}\triangleq\bigcup_{s\in S}\mathcal{P}(s_{0},s)$, we could find the infection path based estimator

[TABLE]

using message-passing algorithm. The details are shown in Algorithm 3, the time complexity of which is $O(|V|+|V^{\prime}|^{2})$.

3.3. Source Localization on General Graphs

Locating the source on general graph is challenging because there are exponential number of possible cascading trees for each node. To avoid such a combinatorial explosion we follow a time labeled BFS heuristic. The algorithm in presented in Algorithm 4. Starting from a node $u\in V$, we do a breadth-first search to construct a time labeled BFS tree. Specifically, we assign each node a time label $\sigma\in\mathbb{Z}$. Initially if the starting node $u\in\mathcal{S}$, we set $\sigma_{u}=t_{u}$. Otherwise, $\sigma_{u}=-\infty$ which represents an extremely small value. When a node $v$ is explored from a directed edge $(w,v)$, if $v\in\mathcal{S}$ and $\sigma_{w}<t_{v}$ we add directed edge $(w,v)$ to BFS tree and set $\sigma_{v}=t_{v}$. If $v\notin\mathcal{S}$ we still add directed edge $(w,v)$ to BFS tree and set $\sigma_{v}=\sigma_{w}+1$. The whole process terminates either when all the edges $E$ are explored or when $\mathcal{S}$ are included in the BFS tree. Note that the resulting BFS tree may not contain all the sampled nodes $\mathcal{S}$, intuitively it is less likely for $u$ to be source if $T_{\text{bfs}}(u)$ contains fewer sampled nodes. Therefore we use a threshold $\theta\in(0,1)$ to rule out those “unlikely” nodes. In practice the threshold $\theta$ needs to be tuned to avoid the extreme case where all nodes are ruled out. Since a breadth-first search is executed for each node, the time complexity is $O(|V|(|E|+|V|))$.

4. Performance Guarantee

Although the infection path based estimator in Eq.(3) is only an approximation of the original ML estimator, we will prove in this section that it can still achieve provably good performance under certain topologies. Specifically, in this section we assume the underlying graph is tree $T$, and we will present the performance guarantee for source localization algorithm on tree $T$ in terms of distribution of $d(v^{\ast},\hat{v})$, which is the distance between true source $v^{\ast}$ and estimated source $\hat{v}$ on tree $T$. Assuming that the true source $v^{\ast}$ is given, we introduce a topological concept called candidate path and show that the infection path based estimator is always on that path. By means of candidate path, we are able to analyze the distribution of $d(v^{\ast},\hat{v})$ under the assumption that $\mathcal{S}$ is uniformly sampled.

4.1. Candidate Path

According to Proposition 3.6, the infection path based estimator is

[TABLE]

therefore, the estimated source $\hat{v}\in\mathcal{C}$ even though we do not know $v^{\ast}$ in prior. If we look at the definition of $\mathcal{C}$

[TABLE]

it is easy to find that $\mathcal{C}$ only depends on the topology of $v^{\ast},\mathcal{S}$, and $T_{\text{ste}}(\mathcal{S})$. If we could utilize the observed timestamps $\{t_{s}\}_{s\in\mathcal{S}}$, it is possible to define a tighter region $\mathcal{R}\subset\mathcal{C}$ that could help us analyze the distribution of $d(v^{\ast},\hat{v})$. Especially, if $v^{\ast}\in\mathcal{S}$ we have

Proposition 4.1.

When the underlying graph is a tree, if $v^{\ast}\in\mathcal{S}$, we have $\hat{v}=v^{\ast}$.

Proof.

If $v^{\ast}\in\mathcal{S}$, then $d_{\mathcal{S}}(v^{\ast},v^{\ast})=1$, it implies that $v^{\ast}\in\mathcal{C}$. For any node $u\in\mathcal{C}\backslash\{v^{\ast}\}$, $d_{\mathcal{S}}(v^{\ast},u)=1$, then by Lemma 3.5

[TABLE]

∎

From now on we assume that $v^{\ast}\notin\mathcal{S}$.

Lemma 4.2.

When the underlying graph is a tree, if $v^{\ast}\notin\mathcal{S}$, the infection path based estimator is

[TABLE]

where $\mathcal{U}\triangleq\{s\in\mathcal{S}|d_{\mathcal{S}}(v^{\ast},s)=1\}$.

Proof.

It suffices to prove that

[TABLE]

For any node $u\in\mathcal{C}$, if $u\in\{s\in\mathcal{S}|d(v^{\ast},s)=1\}$ then $u\in\mathcal{U}$. If $u\in\{u\in V(T_{\text{ste}}(\mathcal{S}))|d_{\mathcal{S}}(v^{\ast},u)=0\}$, then there must be a node $s\in\mathcal{U}$ such that $u\in V(\mathcal{P}(v^{\ast},s))$. Therefore, for any node $u\in\mathcal{C}$, we have $u\in V(T_{\text{ste}}(\mathcal{S}))\cap V\left(\bigcup_{s\in\mathcal{U}}\mathcal{P}(v^{\ast},s)\right)$. ∎

Definition 4.3 (Anchor Node of $v^{\ast}$).

For true source node $v^{\ast}$, we define its anchor node as

[TABLE]

Definition 4.4 (Candidate Path).

The candidate path $\mathcal{P}^{\ast}$ is defined as the intersection of paths from anchor node $u^{\ast}$ to sampled node $s\in\mathcal{U}^{\ast}$, i.e.,

[TABLE]

where $\mathcal{U}^{\ast}$ is given by

[TABLE]

A concrete example is given in Figure 3, in which the candidate path is marked with red color. Now we are going to show that the infection path based estimator $\hat{v}$ is always on that path. Before that, we first give a more specific representation of $\mathcal{P}^{\ast}$ and prove some important properties of $\mathcal{P}^{\ast}$ in Lemma 4.5.

As shown in Figure 4, we represent candidate path $\mathcal{P}^{\ast}$ explicitly as $u^{\ast}(u_{0})\rightarrow u_{1}\rightarrow\cdots\rightarrow u_{m}$ where $m=|\mathcal{P}^{\ast}|$ is the length of path $\mathcal{P}^{\ast}$. As shown in Figure 4, we denote the partial subtree rooted at node $u_{i}$ as $T^{u_{i}}$ for $i=0,\ldots,m$. We denote the children of $u_{i}$ in partial subtree $T^{u_{i}}$ as $u_{i}^{1},u_{i}^{2},\ldots,u_{i}^{k_{i}}$ where $k_{i}$ is the number of children of $u_{i}$. We call the subtree rooted at child $u_{i}^{j}$ of $u_{i}$ a branch $B_{i}^{j}$ of $u_{i}$. In addition, we denote $\mathcal{U}_{i}\triangleq\mathcal{U}\cap V(T^{u_{i}})$ for $i=0,\ldots,m$, and $\mathcal{U}_{i}^{j}\triangleq\mathcal{U}\cap V(B_{i}^{j})$ for $i=0,\ldots,m$, $j=1,\ldots,k_{i}$.

Lemma 4.5.

For the candidate path $\mathcal{P}^{\ast}$, we have

1. (1)

$\mathcal{U}^{\ast}\subset\mathcal{U}_{m}$; 2. (2)

$\{u_{i}\}_{i=0}^{m-1}\cap\mathcal{S}=\varnothing$; 3. (3)

If $u_{m}\in\mathcal{S}$, $V(T^{u_{m}})=\{u_{m}\}$; 4. (4)

If $u_{m}\notin\mathcal{S}$, $|V(T^{u_{m}})\cap\mathcal{U}^{\ast}|\geq 2$; 5. (5)

When running message passing algorithm on cascading tree $T(u_{0},\mathcal{S})$, the virtual timestamp of $u_{i}$ is

[TABLE] 6. (6)

$\min_{s\in\mathcal{U}_{m}}\{t_{s}-d(s,u_{i})\}<\min_{s\in\mathcal{U}_{i}}\{t_{s}-d(s,u_{i})\}$* for $i=0,\ldots,m-1$.*

Proof.

Note that by the definition of the candidate path, we have

[TABLE]

(1) According to the definition of candidate path, it is obvious that $\mathcal{U}^{\ast}\subset\mathcal{U}_{m}$.

(2) Assume that there is some $i\in\{0,1,\ldots,m-1\}$ such that $u_{i}\in\mathcal{S}$, then $u_{i}\in\mathcal{U}^{\ast}$. By the definition of the candidate path,

[TABLE]

violating the fact that $\mathcal{P}^{\ast}=\mathcal{P}(u^{\ast},u_{i})\cup\mathcal{P}(u_{i},u_{m})\supsetneq\mathcal{P}(u^{\ast},u_{i})$. This contradiction indicates that $u_{i}\notin\mathcal{S}$ for any $i\in\{0,1,\ldots,m-1\}$.

(3) It suffices to argue that there is no other sampled nodes in $V(T^{u_{m}})$. Suppose that there exists another node $u^{\prime}\in\mathcal{S}$ such that $u^{\prime}\in V(T^{u_{m}})$. Then $d_{\mathcal{S}}(u^{\prime},u^{\ast})=d_{\mathcal{S}}(u^{\prime},u_{m})+d_{\mathcal{S}}(u_{m},u^{\ast})-1\geq 2$ which means $u^{\prime}\notin V\left(\bigcup_{s\in\mathcal{U}}\mathcal{P}(v^{\ast},s)\right)$. This contradiction indicates that $V(T^{u_{m}})=\{u_{m}\}$.

(4) There must exist at least two sampled nodes in different branch of node $u_{m}$, then $|V(T^{u_{m}})\cap\mathcal{U}^{\ast}|\geq 2$.

(5) and (6): From Appendix A.1 we know that when running message passing algorithm on cascading tree $T(u_{0},\mathcal{S})$, the virtual timestamp of $u_{m}$ is

[TABLE]

the virtual timestamp of $u_{m-1}$ is

[TABLE]

Continue this way, finally $\tau_{u_{i}}=\tau_{u_{0}}+i$ for $i=0,\ldots,m$ and

[TABLE]

∎

As a central tool in our proof of localization precision in Section 4.2, Theorem 4.6 presents the relationship between $\hat{v}$ and $\mathcal{P}^{\ast}$.

Theorem 4.6.

When the underlying graph is a tree, the infection path based estimator is $\hat{v}\in\mathcal{P}^{\ast}$.

Proof.

To prove $\hat{v}\in\mathcal{P}^{\ast}$, it suffices to prove for $i=0,\ldots,m$

[TABLE]

for any node $v\in V(T^{u_{i}})\backslash\{u_{i}\}$. The idea is similar to the proof of Lemma 3.5. We run message passing algorithm on both $T(u_{i},\mathcal{S})$ and $T(v,\mathcal{S})$, and compare the aggregate delays $\tau_{u_{i}}$ and $\tau^{\prime}_{v}$.

(1) First, consider any $i=0,\ldots,m-1$. If we run message passing algorithm on $T(u_{i},\mathcal{S})$, the virtual timestamp of $u_{i}$ is

[TABLE]

where step (a) is due to Proposition 3.1, and step (b) is due to

[TABLE]

Suppose that $v\in B_{i}^{j}$ for some $j\in[k_{i}]$. If we run message passing algorithm on cascading tree $T(v,\mathcal{S})$, the virtual timestamp of $u_{i}$ is

[TABLE]

From Eq.(24) and Eq.(25) we can see that $\tau_{u_{i}}=\tau^{\prime}_{u_{i}}$. Note that similar to the case in Lemma 3.5, there is no sampled node on the path $\mathcal{P}(u_{i},v)$ between $u_{i}$ and $v$, we could view node $u_{i}$ as a sampled node with timestamp $t_{u_{i}}=\tau_{u_{i}}$ and the statement in Lemma 3.5 still holds.

(2) Then consider node $u_{m}$. From Lemma 4.5, when $u_{m}\in\mathcal{S}$ there is no other node in $T^{u_{m}}$. We suppose that $u_{m}\notin\mathcal{S}$. If we run message passing algorithm on $T(u_{m},\mathcal{S})$, the virtual timestamp of $u_{m}$ is

[TABLE]

From Lemma 4.5 we know that $|V(T^{u_{m}})\cap\mathcal{U}^{\ast}|\geq 2$. Let us now arbitrarily choose two sampled nodes $s_{1}$ and $s_{2}$ from $\mathcal{U}^{\ast}$ without replacement. It is easy to verify that $s_{1}\in\mathcal{U}_{m}^{j_{1}}$ and $s_{2}\in\mathcal{U}_{m}^{j_{2}}$ for some $j_{1}\neq j_{2}$. Suppose that $v\in B_{m}^{j}$ for some $j\in[k_{m}]$ and $j\neq j_{1}$. If we run message passing algorithm on $T(v,\mathcal{S})$, the virtual timestamp of $u_{m}$ is

[TABLE]

where step (c) is follows from the fact that $\min_{s\in\mathcal{U}_{m}\backslash\mathcal{U}_{m}^{j}}\{t_{s}-d(s,u_{m})\}=t_{s_{1}}-d(s_{1},u_{m})=m+\min_{v\in\mathcal{U}}\{t_{v}-d(u^{\ast},v)\}$. Therefore $\tau_{u_{m}}=\tau^{\prime}_{u_{m}}$ and the same argument as in (1) follows. ∎

4.2. Localization Precision

Note that the source inference contains two stages: sampling nodes $\mathcal{S}$ and estimation according to infection path. Given that a diffusion process has already happened, both two stages would affect the estimated source $\hat{v}$. To characterize the localization precision, we analyze the distribution of $d(v^{\ast},\hat{v})$ under the assumption that each node $s\in\mathcal{S}$ is sampled uniformly at random from $V$ with probability $q$. For line graph and $g$-regular tree, we have the following results.

Theorem 4.7.

In infinite line graph where the degree of each node is 2, when the sampled nodes $\mathcal{S}$ are sampled uniformly at random with probability $q$, the correct detection probability under the infection diffusion model is

[TABLE]

and the expected distance between $\hat{v}$ and $v^{\ast}$ is upper bounded by

[TABLE]

The proof is contained in Appendix C

Theorem 4.8.

In infinite $g$-regular tree where the degree of each node is $g\geq 3$, when the sampled nodes $\mathcal{S}$ are sampled uniformly at random with probability $q$, we have

[TABLE]

where $x_{1}$ is given by function iteration $x_{D}=1$ and $x_{i}=h(x_{i+1})=(1-p+p(1-q)x_{i+1})^{g-1}$ for $i=1,2,\ldots,D-1$. Denote the fixed point of $h(x)$ on $[0,1]$ as $x^{\ast}$, then $\lim_{D\rightarrow\infty}x_{1}(D)=x^{\ast}$ and $x_{1}(D)$ is strictly decreasing with respect to $D$.

The readers can see Appendix D for its proof. As a comparison, we show in Proposition 4.9 that the infection path based estimator always outperforms naive minimum timestamp estimator.

Proposition 4.9.

In infinite line graph where the degree of each node is $2$, when the sampled nodes $\mathcal{S}$ are sampled uniformly at random with probability $q$, the infection path based estimator always outperforms naive minimum timestamp estimator in the sense that $d(\hat{v}_{\text{i}},v^{\ast})\leq d(\hat{v}_{\text{n}},v^{\ast})$ for any true source node $v^{\ast}$ and sampled nodes $\mathcal{S}$, where $\hat{v}_{\text{i}}$ denotes the infection path based estimator and $\hat{v}_{\text{n}}$ denotes the naive minimum timestamp estimator. Moreover, the correct detection probability is

[TABLE]

and the expected distance between $\hat{v}$ and $v^{\ast}$ is

[TABLE]

The proof is presented in the Appendix E.

Note that when $q\rightarrow 0$, we have

[TABLE]

implying that infection path based estimator is much better than naive minimum timestamp estimator in infinite line graph.

5. Simulations and Experiments

In this section, we evaluate the performance of the infection path based estimator on different networks. We compare the infection path based estimator (INF) with the naive minimum timestamp estimator (MIN) and GAU estimator proposed in (Pinto et al., 2012). The GAU estimator utilizes partial timestamps to find the source under the assumption that transmission delay follows Gaussian distribution $\mathcal{N}(\mu,\sigma^{2})$ for each edge.

5.1. Tree Networks

We first provide simulation results for $g$-regular trees to corroborate the theoretical results in Section 4. Each regular tree contains 1024 non-leaf nodes. For each simulation, we select the source node uniformly at random and synthesize cascades using Geometric distribution with success probability $p$ in $\left\{\frac{1}{2},\frac{1}{4},\frac{1}{8}\right\}$. The partial timestamps are sampled uniformly at random with probability $q$ in $\{0.1,0.2,\ldots,0.9\}$. We perform 500 simulation runs for each setting on each network. The results are plotted in Figure 5(a)(b)(c), where the upper bound is provided by Eq.(27). We observe that as the degree $g$ increases, the error distance becomes smaller, which means the source node $v^{\ast}$ with larger degree is more likely to be detected using limited timestamps. Moreover, we test INF, GAU, and MIN estimator on an academic citation tree extracted from academic citation network (Wang, 2015) that contains citation relationship between different papers on a similar topic. The infection probability $p$ is set to be $1/2$. For the GAU estimator, we set $\mu=1/p=2$ and $\sigma^{2}=(1-p)/p^{2}=2$. The results are plotted in Figure 5(d), from which we can observe that INF outperforms GAU and MIN for each sampling probability $q$ in $\{0.1,\ldots,0.9\}$. The reason may be that GAU is optimized for Gaussian distribution whereas INF is suitable for Geometric distribution.

5.2. Graph Networks

We further perform experiments on Erdös-Rényi networks, scale-free networks (Barabási and Albert, 1999), Facebook networks (McAuley and Leskovec, 2012) and US power grid (PG) networks (Barabási and Albert, 1999). The Erdös-Rényi network contains 1024 nodes and 10487 edges. The scale-free network is generated by preferential attachment and contains 1024 nodes and 4080 edges. The Facebook social network contains 4039 nodes and 88234 edges, and is used to study the online friendship patterns. The PG network is a network of Western States Power Grid of United States, and contains 4941 nodes and 6594 edges. We set the infection probability $p=1/2$, the threshold $\theta=0.95$ in Algorithm 4 and run each simulation 300 times for sampling probability $q$ in $0.1,\ldots,0.5$. The plots in Figure 6 indicate that INF performs no worse than GAU and MIN in almost all cases. This improvement is more obvious in PG network and scale free network than in Erdös-Rényi network and Facebook network. For the PG network and scale free network, the average ratio of edges to nodes is 1.33 and 3.98 respectively, whereas for Erdös-Rényi network and Facebook network the average ratio is 10.24 and 21.84 respectively. Thus, the PG network and scale-free network is more tree-like, which may explain why INF outperforms GAU and MIN clearly on these networks.

6. Conclusions

In this paper, we proposed an infection path based estimator to approximate the optimal ML estimator for detecting the information source in networks. Through transforming the infection path based estimator into a linear integer programming, we proved that a message passing algorithm could optimally solve infection path based estimator on arbitrary trees. We also define a new concept called candidate path to enable the analysis of error distance $d(v^{\ast},v)$ on arbitrary tree. Under the assumption that limited timestamps are uniformly sampled, we provided theoretical guarantees on infinite $g$-regular tree in terms of $d(v^{\ast},\hat{v})$. By incorporating time labeled BFS heuristic, experiments showed that the infection path based estimator exhibits a good performance in general graphs as well.

Appendix A Proof of Proposition 3.1

We will prove Proposition 3.1 in this section following three steps. First, we discuss the virtual timestamps computed by Algorithm 1. Second, we connect the feasibility of LIP(6) with the output of Algorithm 1. Third, we show that the output of Algorithm 1 is exactly the optimal value of LIP(6) given that LIP(6) is feasible.

A.1. Virtual Timestamps

The intuition comes from the recursive relation between the virtual timestamp $\tau_{u}$ and the virtual timestamps of its immediate children’s virtual timestamp $\tau_{w}$ with $w\in\text{child}(u)$. The virtual timestamp of node $u$ can be computed only if the virtual timestamps of any node $w\in V(T^{v}_{u})\backslash\{u\}$ is known. Suppose that Algorithm 1 have not returned empty when computing $\tau_{u}$ in line 5. In the following we will prove that

[TABLE]

for any node $w\in V(T^{v}_{u})\backslash\{u\}$ by induction.

1. (1)

For any leaf node $w\in V(T^{v}_{u})\backslash\{u\}$, we have $w\in\mathcal{S}$ according to the definition of cascading tree. It follows that

[TABLE] 2. (2)

Assume that for any non-leaf node $w\in V(T^{v}_{u})\backslash\{u\}$, the virtual timestamp of its any child $j\in\text{child}(w)$ is

[TABLE] 3. (3)

If $w\notin\mathcal{S}$, according to line 5 in Algorithm 1, the virtual timestamp of node $w$ is given by

[TABLE]

where step($\ast$) is due to $d(s,w)=d(s,j)+d(j,w)=d(s,j)+1$.

If $w\in\mathcal{S}$, the virtual timestamp of node $w$ is given by

[TABLE]

since $t_{w}\leq\min_{s\in\mathcal{S}\cap(V(T^{v}_{w})\backslash\{w\})}\{t_{s}-d(s,w)\}$.

A.2. Feasibility of LIP(6)

We will prove that LIP(6) is infeasible if and only if the message passing algorithm returns empty.

For the sufficiency, assume that Algorithm 1 returns empty. It implies that there must exists a node $u\in\mathcal{S}$ such that

[TABLE]

according to line 6-8 in Algorithm 1. Let

[TABLE]

then $\tau_{u}=t_{u^{\prime}}-d(u^{\prime},u)$, therefore $t_{u^{\prime}}-t_{u}<d(u^{\prime},u)$. Suppose that there exists a collection of timestamps $\mathbf{t}$ satisfying the constraints of LIP(6), then $\mathbf{t}(u^{\prime})=t_{u^{\prime}}$, $\mathbf{t}(u)=t_{u}$, and

[TABLE]

which violates the inequality that $t_{u^{\prime}}-t_{u}<d(u^{\prime},u)$. This contradiction indicates that if the message passing algorithm returns empty, then LIP(6) is infeasible.

For the necessity, it suffices to prove the contrapositive statement, which is if Algorithm 1 returns the aggregate delays $a_{v}$ of root node $v$, then LIP(6) is feasible. Note that the virtual timestamp $\tau_{s}$ for each node $s\in\mathcal{S}$ is $\tau_{s}=t_{s}$, and for any directed edge $(i,j)\in E(T(v,\mathcal{S}))$ we have $\tau_{i}\leq\min_{w\in\text{child}(i)}\{\tau_{w}\}-1\leq\tau_{j}-1$. Therefore, virtual timestamps $\{\tau_{u}\}_{u\in V(T(v,\mathcal{S}))}$ satisfy all the constraints of LIP(6) which implies that LIP(6) is feasible.

A.3. Optimality of Algorithm 1

We will prove that the aggregate delays $a_{v}$ of root node $v$ is the optimal value of LIP(6) given that LIP(6) is feasible. Before that, we analyze the relationship between aggregate delays $\{a_{u}\}_{u\in V(T(v,\mathcal{S}))}$ and virtual timestamps $\{\tau_{u}\}_{u\in V(T(v,\mathcal{S}))}$ in Lemma A.1.

Lemma A.1.

Given that LIP(6) is feasible, the aggregate delays of node $u\in V(T(v,\mathcal{S}))$ is $a_{u}=\sum_{(i,j)\in E(T^{v}_{u})}(\tau_{j}-\tau_{i})$.

Proof.

We prove that $a_{u}=\sum_{(i,j)\in E(T^{v}_{u})}(\tau_{j}-\tau_{i})$ by induction.

1. (1)

For any leaf node $w\in V(T(v,\mathcal{S}))$, $a_{w}=0$; 2. (2)

Assume that for any node $u\in\text{child}(w)$

[TABLE] 3. (3)

The aggregate delays of node $w$ is

[TABLE]

∎

From Appendix A.2 we know that the virtual timestamps $\{\tau_{u}\}$ satisfy all the constraints of LIP(6). And from Lemma A.1 we know that $a_{v}=\sum_{(i,j)\in E(T(v,\mathcal{S}))}(\tau_{j}-\tau_{i})$. Therefore, $a_{v}$ is the value of objective function in LIP(6) when the optimization variable $\mathbf{t}$ is virtual timestamps $\{\tau_{u}\}_{u\in V(T(v,\mathcal{S}))}$.

To prove that $a_{v}$ is optimal value of LIP(6), it suffices to argue that for any other permitted timestamps $\mathbf{t}$, the value of objective function is at least $a_{v}$. We prove it by contradiction.

Assume that there exists other permitted timestamps $\mathbf{t}$ such that $\sum_{(i,j)\in E(T(v,\mathcal{S}))}(\mathbf{t}(j)-\mathbf{t}(i))<a_{v}$. According to Eq.(7), there must exist some node $u\in V(T(v,\mathcal{S}))\backslash\mathcal{S}$ such that $d_{\text{out}}(u)>d_{\text{in}}(u)$ and $\mathbf{t}(u)>\tau_{u}$. Note that $\tau_{u}=\min_{s\in\mathcal{S}\cap V(T^{v}_{u})}\{t_{s}-d(s,u)\}$, let

[TABLE]

then $\tau_{u}=\tau_{s^{\prime}}-d(s^{\prime},u)=t_{s^{\prime}}-d(s^{\prime},u)$. Therefore

[TABLE]

which violates the inequality that

[TABLE]

This contradiction indicates that $a_{v}$ is the optimal value of LIP(6) and $\{\tau_{u}\}_{u\in V(T(v,\mathcal{S}))}$ is a solution of LIP(6).

Appendix B Proof of Lemma 3.5

If $\max_{\mathbf{t}}\mathbf{P}(T(u,\mathcal{S},\mathbf{t})|\mathcal{S},v^{\ast}=u)=0$, then the statement is apparently correct. Now suppose that $\max_{\mathbf{t}}\mathbf{P}(T(u,\mathcal{S},\mathbf{t})|\mathcal{S},v^{\ast}=u)>0$, which means that the LIP(6) is feasible for cascading tree $T(u,\mathcal{S})$. So if we run message passing Algorithm 1 on cascading tree $T(u,\mathcal{S})$ we will obtain the aggregate delays $a_{u}$ at root node $u$ and the corresponding virtual timestamps $\{\tau_{v}\}_{v\in V(T(u,\mathcal{S}))}$.

Since $d_{\mathcal{S}}(v^{\ast},u)=1$ and $u\notin\mathcal{S}$, there is only one sampled node $s$ on the path $\mathcal{P}(v^{\ast},u)$ from $v^{\ast}$ to $u$. We denote the path $\mathcal{P}(s,u)$ as $v_{0}(s)\rightarrow v_{1}\rightarrow\cdots\rightarrow v_{k-1}\rightarrow v_{k}(u)$ as shown in Figure 7. Since $s$ is a sampled node with timestamp $t_{s}$, the virtual timestamp of $s$ is $\tau_{s}=t_{s}$ on cascading tree $T(u,\mathcal{S})$. And the other virtual timestamps satisfy $\tau_{s}>\tau_{v_{1}}>\tau_{v_{2}}>\cdots>\tau_{v_{k-1}}>\tau_{u}$. Finally according to Eq.(7),

[TABLE]

where $V^{\prime}=V(T_{\text{ste}}(\mathcal{S}))-\{v_{i}\}_{i=0}^{k}$, and $d_{\text{in}}(v),d_{\text{out}}(v)$ are in-degree and out-degree of node $v$ respectively on cascading tree $T(u,\mathcal{S})$, $d_{s}$ is the degree of node $s$ on tree $T_{\text{ste}}(\mathcal{S})$

Consider the cascading tree $T(s,\mathcal{S})$ rooted at sampled node $s$. As shown in Figure 7, observe that the only difference between $T(u,\mathcal{S})$ and $T(s,\mathcal{S})$ is the direction of edges $\{(v_{i},v_{i+1})\}_{i=0}^{k-1}$. If we run message passing Algorithm 1 on cascading tree $T(s,\mathcal{S})$, the virtual timestamp of node $s$ is still $\tau^{\prime}_{s}=t_{s}$. And the other virtual timestamps satisfy $\tau^{\prime}_{s}<\tau^{\prime}_{v_{1}}<\tau^{\prime}_{v_{2}}<\cdots<\tau^{\prime}_{v_{k-1}}<\tau^{\prime}_{u}$. As for other node $v\in V\backslash\{v_{i}\}_{i=0}^{k}$, its virtual timestamp satisfy $\tau_{v}=\tau^{\prime}_{v}$. According to Eq.(7), the aggregate delays $a^{\prime}_{s}$ at root node $s$ is

[TABLE]

where $d^{\prime}_{\text{in}}(v),d^{\prime}_{\text{out}}(v)$ are in-degree and out-degree of node $v$ respectively on cascading tree $T(s,\mathcal{S})$. Compare Eq.(32) and Eq.(33), we have

[TABLE]

where step (b) is due to $d_{u}\geq 2$ and $d_{v_{i}}\geq 2$ for $i=1,\ldots,k-1$. Inequality in Eq.(15) holds due to the relationship between LIP(6) and infection path likelihood in Eq.(8).

Appendix C Proof of Theorem 4.7

If $v^{\ast}\in\mathcal{S}$, then $\hat{v}=v^{\ast}$ by Proposition 4.1. Now suppose that $v^{\ast}\notin\mathcal{S}$. Since there are infinite number of nodes on the left/right side of $v^{\ast}$, with probability 1 there are two sampled nodes $s_{1}$ and $s_{2}$ which are closest to $v^{\ast}$ from the left and right side of $v^{\ast}$, respectively. As shown in Figure 8, we denote the distance between $v^{\ast}$ and $s_{1}$ as $L_{1}$ and the distance between $v^{\ast}$ and $s_{2}$ as $L_{2}$. The distribution of $L_{1}$ and $L_{2}$ is given by

[TABLE]

[TABLE]

Let $t_{1}$ and $t_{2}$ denote the timestamp of sampled nodes $s_{1}$ and $s_{2}$, respectively, then

[TABLE]

where $t_{0}$ is unknown starting time of diffusion process and $\{\zeta_{i}\}$, $\{\zeta^{\prime}_{i}\}$ are collections of independent random variables with identical geometric distribution $\text{Geo}(p)$.

Let $\sigma_{1}=t_{1}-L_{1}$ and $\sigma_{2}=t_{2}-L_{2}$. Note that $\mathcal{U}=\{s_{1},s_{2}\}$. According to definition of candidate path, we have

[TABLE]

where $\mathcal{U}^{\ast}$ is given by

[TABLE]

From Eq.(38) and Eq.(39), the candidate path $\mathcal{P}^{\ast}$ is given by

[TABLE]

Due to symmetry between left and right side of $v^{\ast}$, we assume that $\sigma_{1}<\sigma_{2}$ and represent $\mathcal{P}(v^{\ast},s_{1})$ as $v^{\ast}(u_{0})\rightarrow u_{1}\rightarrow\cdots\rightarrow s_{1}(u_{m})$. We have the following lemma:

Lemma C.1.

The estimated source $\hat{v}=u_{k}$ where the index number $k$ is given by

[TABLE]

Proof.

When running the message passing algorithm on cascading tree $T(u_{i},\mathcal{S})$, the virtual timestamp of $u_{i}$ is

[TABLE]

which is strictly increasing w.r.t. $i$ when $i<\frac{\sigma_{2}-\sigma_{1}}{2}$ and strictly decreasing w.r.t. $i$ when $i\geq\frac{\sigma_{2}-\sigma_{1}}{2}$. And the aggregate delays at node $u_{i}$ is given by

[TABLE]

So the index number $k$ of the estimated source is

[TABLE]

∎

From Lemma C.1, the distance between $v^{\ast}$ and $\hat{v}$ can be expressed as

[TABLE]

where $\tilde{m}=L_{1}\mathbb{I}(\sigma_{1}<\sigma_{2})+L_{2}\mathbb{I}(\sigma_{1}>\sigma_{2})$ is the length of candidate path $\mathcal{P}^{\ast}$.

Now we study the distribution of $\sigma_{1}-\sigma_{2}$ using probability generating function. Notice that $\sigma_{1}$ and $\sigma_{2}$ are independent and have identical distribution. For ease of analysis we assume $t_{0}=0$ and write $\sigma_{1}$ as $\sigma_{1}=\sum_{i=1}^{L_{1}}(\zeta_{i}-1)$. The PGF $G_{1}(z)$ of $\zeta_{i}-1$ is given by

[TABLE]

The PGF $G_{2}(z)$ of $L_{1}$ is given by

[TABLE]

Then we calculate the PGF $G_{3}(z)$ of $\sigma_{1}$ using $G_{1}(z)$ and $G_{2}(z)$

[TABLE]

where $a=pq$, $b=1-p+pq$, and $c=1-p$. The PGF $G_{4}(z)$ of $\sigma_{1}-\sigma_{2}$ is given by

[TABLE]

From $G_{4}(z)$, we obtain the distribution of $\sigma_{1}-\sigma_{2}$

[TABLE]

The correct detection probability is

[TABLE]

Further, for the expected distance between $\hat{v}$ and $v^{\ast}$ we have

[TABLE]

and

[TABLE]

step ($\star$) is due to

[TABLE]

Therefore,

[TABLE]

Appendix D Proof of Theorem 4.8

If $v^{\ast}\in\mathcal{S}$, then $\hat{v}=v^{\ast}$ by Proposition 4.1. Now suppose that $v^{\ast}\notin\mathcal{S}$. Since there are infinite number of nodes on each branch of $v^{\ast}$, with probability 1 we have two sampled nodes on two different branch resulting in $\bigcap_{s\in\mathcal{U}}\mathcal{P}(v^{\ast},s)=v^{\ast}$ where $\mathcal{U}=\{s\in\mathcal{S}|d_{\mathcal{S}}(v^{\ast},s)=1\}$. Therefore the anchor node $u^{\ast}\in\arg\max_{u\in\bigcap_{s\in\mathcal{U}}\mathcal{P}(v^{\ast},s)}d(v^{\ast},u)=v^{\ast}$. According to Theorem 4.6, the estimated source $\hat{v}\in\mathcal{P}^{\ast}$. So if $|\mathcal{P}^{\ast}|\leq D$ then $d(\hat{v},v^{\ast})\leq|\mathcal{P}^{\ast}|\leq D$. Therefore,

[TABLE]

We are going to find the sufficient conditions for $|\mathcal{P}^{\ast}|\leq D$.

Lemma D.1.

Suppose that the diffusion process starts at time $t_{0}$ from source node $v^{\ast}$. If there exists a sampled node $s\in\mathcal{S}$ such that $d(v^{\ast},s)=D$ and $t_{s}=t_{0}+D$, then $|\mathcal{P}^{\ast}|\leq D$.

Proof.

We represent the path $\mathcal{P}(v^{\ast},s)$ as $v^{\ast}(u_{0})\rightarrow u_{1}\rightarrow\cdots\rightarrow u_{D-1}\rightarrow s(u_{D})$. Since the transmission delay for each edge is at least 1, we have $t_{s}-t_{0}=\sum_{i=1}^{D}t_{u_{i}}-t_{u_{i-1}}\geq D$ where the equality holds if and only if $t_{u_{i}}-t_{u_{i-1}}=1$ for all $i\in[D]$. From the assumption that $t_{s}-t_{0}=D$, we have $t_{u_{i}}=t_{0}+i$ for all $i\in[D]$. Let $s^{\prime}$ denote the sampled node in $\mathcal{P}(v^{\ast},s)$ which is closest to $v^{\ast}$, i.e., $s^{\prime}=u_{k}$ where index number $k$ is given by $k=\min\{i|u_{i}\in\mathcal{S},i\in\{0,1,\ldots,D\}\}$. The timestamp of sampled node $u_{k}$ is $t_{u_{k}}=t_{0}+k$. Since

[TABLE]

we have $u_{k}\in\mathcal{U}^{\ast}$ and then $\mathcal{P}^{\ast}=\bigcap_{u\in\mathcal{U}^{\ast}}\mathcal{P}(v^{\ast},u)\subset\mathcal{P}(v^{\ast},u_{k})$ yielding $|\mathcal{P}^{\ast}|\leq|\mathcal{P}(v^{\ast},u_{k})|=k\leq D$. ∎

Let $\mathcal{F}_{D}$ denote the event that given $v^{\ast}\notin\mathcal{S}$ there exists a sampled node $s\in\mathcal{S}$ such that $d(v^{\ast},s)=D$ and $t_{s}=t_{0}+D$, then

[TABLE]

Denote the subtree rooted at node $u$ as $T_{u}^{v^{\ast}}$. Let $\mathcal{E}_{u}$ denote the event that given $v^{\ast}\notin\mathcal{S}$ there exists no node $v\in T_{u}^{v^{\ast}}$ such that $v\in\mathcal{S}$, $d(v^{\ast},v)\leq D$ and $t_{v}=t_{0}+d(v^{\ast},v)$. Without loss of generality we assume that $t_{0}=0$.

[TABLE]

where $u_{1}\in\text{child}(v^{\ast})$ is an arbitrary child of $v^{\ast}$. And

[TABLE]

where $u_{2}\in\text{child}(u_{1})$ is an arbitrary child of $u_{1}$. Continue this way, we could contruct a path $v^{\ast}\rightarrow u_{1}\rightarrow u_{2}\rightarrow\cdots\rightarrow u_{D-1}\rightarrow u_{D}$ such that

[TABLE]

for $i=1,2,\ldots,D-1$. And for $i=D$ we have $\mathbf{P}(\mathcal{E}_{u_{D}}|u_{D}\notin\mathcal{S},t_{u_{D}}=D)=1$. Abbreviate $\mathbf{P}(\mathcal{E}_{u_{i}}|u_{i}\notin\mathcal{S},t_{u_{i}}=i)$ as $x_{i}$, then $x_{D}=1$ and

[TABLE]

where $h(x)=(1-p+p(1-q)x)^{g-1}$ is an increasing convex function on $[0,1]$ since $h^{\prime}(x)=p(1-q)(g-1)(1-p+p(1-q)x)^{g-2}>0$ and $h^{\prime\prime}(x)=p^{2}(1-q)^{2}(g-1)(g-2)(1-p+p(1-q)x)^{g-3}>0$. Denote the fixed point of $h(x)$ on $[0,1]$ as $x^{\ast}$, then $x_{i}<x_{i+1}$ for $i=1,\ldots,D-1$ and $x_{1}\rightarrow x^{\ast}$ as $D\rightarrow\infty$.

Finally, we have

[TABLE]

where $x_{1}$ is given by function iteration $x_{D}=1$ and $x_{i}=h(x_{i+1})=(1-p+p(1-q)x_{i+1})^{g-1}$ for $i=1,2,\ldots,D-1$.

Appendix E Proof of Proposition 4.9

If the true source $v^{\ast}\in\mathcal{S}$, it is certain that timestamp $t_{v^{\ast}}=\min_{s\in\mathcal{S}}\{t_{s}\}$ and is unique. Using naive minimum timestamp estimator, the estimated node $\hat{v}\in\arg\min_{s\in\mathcal{S}}\{t_{s}\}=v^{\ast}$.

If the true source $v^{\ast}\notin\mathcal{S}$, following the setup in Appendix C, the estimated node is

[TABLE]

then

[TABLE]

From Eq.(52) we could write the distance between $v^{\ast}$ and $\hat{v}$ as

[TABLE]

where $\Theta$ is Bernoulli random variable with success probability $\frac{1}{2}$ and $\mathbb{I}_{\Theta}(\cdot)$ is its indicator function. Note that $\Theta$ is independent from all other random variables. From now on, we denote the distance between $v^{\ast}$ and $\hat{v}$ obtained from naive minimum timestamp estimator as $d(\hat{v}_{\text{n}},v^{\ast})$, and denote the distance between $v^{\ast}$ and $\hat{v}$ obtained from infection path based estimator as $d(\hat{v}_{\text{i}},v^{\ast})$. Notice that both $d(\hat{v}_{\text{n}},v^{\ast})$ and $d(\hat{v}_{\text{i}},v^{\ast})$ are random variables depending on $\sigma_{1},\sigma_{2},L_{1}$, and $L_{2}$, then

[TABLE]

which means that for any realization of $\sigma_{1},\sigma_{2},L_{1}$, and $L_{2}$, we have $d(\hat{v}_{\text{n}},v^{\ast})\geq d(\hat{v}_{\text{i}},v^{\ast})$. This conclusion is stronger than stochastic ordering $d(\hat{v}_{n},v^{\ast})\geq_{\text{st}}d(\hat{v}_{\text{i}},v^{\ast})$.

The correct detection probability for naive minimum timestamp estimator is

[TABLE]

and the expected distance between $\hat{v}_{\text{n}}$ and $v^{\ast}$ is

[TABLE]

therefore

[TABLE]

Acknowledgements.