From a18c0eb569d9f5ccfa0e597c0af40faa1cacbb35 Mon Sep 17 00:00:00 2001 From: Xiao Zhou Date: Sat, 7 Dec 2019 14:53:03 -0800 Subject: [PATCH] Influence Maximization and Outbreak Detection (#12) * add influence_maximization and figures, edit .gitignore to ignore jekyll cache * fix repeated the * change the file name with dash, add outbreak detection * finished outbreak detection * proof reading done * update with suggestions * Add sidenote for other evaluation methods besides sketches, make Hill Climbing Algorithm more precise --- .gitignore | 3 +- assets/img/.DS_Store | Bin 6148 -> 0 bytes .../influence_maximization_influence_set.png | Bin 0 -> 82558 bytes ...ximization_linear_threshold_model_demo.png | Bin 0 -> 189536 bytes .../outbreak_detection_lazy_evaluation.png | Bin 0 -> 55274 bytes .../outbreak_detection_sensor_placement.png | Bin 0 -> 118545 bytes index.md | 8 +- network-methods/influence-maximization.md | 154 ++++++++++++++ network-methods/outbreak-detection.md | 198 ++++++++++++++++++ 9 files changed, 358 insertions(+), 5 deletions(-) delete mode 100644 assets/img/.DS_Store create mode 100644 assets/img/influence_maximization_influence_set.png create mode 100644 assets/img/influence_maximization_linear_threshold_model_demo.png create mode 100644 assets/img/outbreak_detection_lazy_evaluation.png create mode 100644 assets/img/outbreak_detection_sensor_placement.png create mode 100644 network-methods/influence-maximization.md create mode 100644 network-methods/outbreak-detection.md diff --git a/.gitignore b/.gitignore index da6b032..38aa715 100644 --- a/.gitignore +++ b/.gitignore @@ -1,4 +1,5 @@ _site .sass-cache .DS\_Store -config.codekit \ No newline at end of file +.jekyll-cache/ +config.codekit diff --git a/assets/img/.DS_Store b/assets/img/.DS_Store deleted file mode 100644 index 5008ddfcf53c02e82d7eee2e57c38e5672ef89f6..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 6148 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They mirror the topics topics covered by Stanford [CS224W](https://cs224w.stanford.edu), and are written by the CS 224W TAs. +These notes form a concise introductory course on machine learning with large-scale graphs. They mirror the topics topics covered by Stanford [CS224W](https://cs224w.stanford.edu), and are written by the CS 224W TAs. {% include marginnote.html id='mn-construction' note='The notes are still under construction! They will be written up as lectures continue to progress. If you find any typos, please let us know, or submit a pull request with your fixes to our [GitHub repository](https://github.com/snap-stanford/cs224w-notes).'%} You too may help make these notes better by submitting your improvements to us via [GitHub](https://github.com/snap-stanford/cs224w-notes). Note that submitting substantial improvements will result in *bonus points* being added to your overall grade! -Starting with the Fall 2019 offering of CS 224W, the course covers three broad topic areas for understanding and effectively learning representations from large-scale networks: preliminaries, network methods, and machine learning with networks. Subtopics within each area correspond to individual lecture topics. +Starting with the Fall 2019 offering of CS 224W, the course covers three broad topic areas for understanding and effectively learning representations from large-scale networks: preliminaries, network methods, and machine learning with networks. Subtopics within each area correspond to individual lecture topics. ## Preliminaries @@ -19,8 +19,8 @@ Starting with the Fall 2019 offering of CS 224W, the course covers three broad t 1. [Structural Roles in Networks](): RolX, Granovetter, the Louvain algorithm 2. [Spectral Clustering](network-methods/spectral-clustering): Graph partitions and cuts, the Laplacian, and motif clustering -3. [Influence Maximization](): Influential sets, submodularity, hill climbing -4. [Outbreak Detection](): CELF, lazy hill climbing +3. [Influence Maximization](network-methods/influence-maximization): Influential sets, submodularity, hill climbing +4. [Outbreak Detection](network-methods/outbreak-detection): CELF, lazy hill climbing 5. [Link Analysis](): PageRank and SimRank 6. [Network Effects and Cascading Behavior](network-methods/network-effects-and-cascading-behavior): Decision-based diffusion, probabilistic contagion, SEIZ 7. [Network Robustness](): Power laws, preferential attachment diff --git a/network-methods/influence-maximization.md b/network-methods/influence-maximization.md new file mode 100644 index 0000000..c315296 --- /dev/null +++ b/network-methods/influence-maximization.md @@ -0,0 +1,154 @@ +--- +layout: post +title: Influence Maximization +--- + +## Motivation +Identification of influential nodes in a network has important practical uses. A good example is "viral marketing", a strategy that uses existing social networks to spread and promote a product. A well-engineered viral marking compaign will identify the most influential customers, convince them to adopt and endorse the product, and then spread the product in the social network like a virus. + +The key question is how to find the most influential set of nodes? To answer this question, we will first look at two classical cascade models: + +- Linear Threshold Model +- Independent Cascade Model + +Then, we will develop a method to find the most influential node set in the Independent Cascade Model. + +## Linear Threshold Model +In the Linear Threshold Model, we have the following setup: + +- A node $$v$$ has a random threshold $$\theta_{v} \sim U[0,1]$$ +- A node $$v$$ influenced by each neighbor $$w$$ according to a weight $$b_{v,w}$$, such that + +$$ +\sum_{w\text{ neighbor of v }} b_{v,w}\leq 1 +$$ + +- A node $$v$$ becomes active when at least $$\theta_{v}$$ fraction of its neighbors are active. That is + +$$ +\sum_{w\text{ active neighbor of v }} b_{v,w}\geq\theta_{v} +$$ + +The following figure demonstrates the process: + +![linear_threshold_model_demo](../assets/img/influence_maximization_linear_threshold_model_demo.png?style=centerme) + +*(A) node V is activated and influences W and U by 0.5 and 0.2, respectively; (B) W becomes activated and influences X and U by 0.5 and 0.3, respectively; (C) U becomes activated and influences X and Y by 0.1 and 0.2, respectively; (D) X becomes activated and influences Y by 0.2; no more nodes can be activated; process stops.* + +## Independent Cascade Model +In this model, we model the influences (activation) of nodes based on probabilities in a directed graph: + +- Given a directed finite graph $$G=(V, E)$$ +- Given a node set $$S$$ starts with a new behavior (e.g. adopted new product and we say they are active) +- Each edge $$(v, w)$$ has a probability $$p_{vw}$$ +- If node $$v$$ becomes active, it gets one chance to make $$w$$ active with probability $$p_{vw}$$ +- Activation spread through the network + +Note: + +- Each edge fires only once +- If $$u$$ and $$v$$ are both active and link to $$w$$, it does not matter which tries to activate $$w$$ first + +## Influential Maximization (of the Independent Cascade Model) + +### Definitions +- **Most influential Set of size $$k$$** ($$k$$ is a user-defined parameter) is a set $$S$$ containing $$k$$ nodes that if activated, produces the largest expected{% include sidenote.html id='note-most-influential-set' note='Why "expected cascade size"? Due to the stochastic nature of the Independent Cascade Model, node activation is a random process, and therefore, $$f(S)$$ is a random variable. In practice, we would like to compute many random simulations and then obtain the expected value $$f(S)=\frac{1}{\mid I\mid}\sum_{i\in I}f_{i}(S)$$, where $$I$$ is a set of simulations.' %} cascade size $$f(S)$$. +- **Influence set $$X_{u}$$ of node $$u$$** is the set of nodes that will be eventually activated by node $$u$$. An example is shown below. + +![influence_set](../assets/img/influence_maximization_influence_set.png?style=centerme) + +*Red-colored nodes a and b are active. The two green areas enclose the nodes activated by a and b respectively, i.e. $$X_{a}$$ and $$X_{b}$$.* + +Note: +- It is clear that $$f(S)$$ is the size of the union of $$X_{u}$$: $$f(S)=\mid\cup_{u\in S}X_{u}\mid$$. +- Set $$S$$ is more influential, if $$f(S)$$ is larger + +### Problem Setup +The influential maximization problem is then an optimization problem: + +$$ +\max_{S \text{ of size }k}f(S) +$$ + +This problem is NP-hard [[Kempe et al. 2003]](https://www.cs.cornell.edu/home/kleinber/kdd03-inf.pdf). However, there is a greedy approximation algorithm--**Hill Climbing** that gives a solution $$S$$ with the following approximation guarantee: + +$$ +f(S)\geq(1-\frac{1}{e})f(OPT) +$$ + +where $$OPT$$ is the globally optimal solution. + +### Hill Climbing +**Algorithm:** at each step $$i$$, activate and pick the node $$u$$ that has the largest marginal gain $$\max_{u}f(S_{i-1}\cup\{u\})$$: + +- Start with $$S_{0}=\{\}$$ +- For $$i=1...k$$ + - Activate node $$u\in V\setminus S_{i-1}$$ that $$\max_{u}f(S_{i-1}\cup\{u\})$$ + - Let $$S_{i}=S_{i-1}\cup\{u\}$$ + +**Claim:** Hill Climbing produces a solution that has the approximation guarantee $$f(S)\geq(1-\frac{1}{e})f(OPT)$$. + +### Proof of the Approximation Guarantee of Hill Climbing +**Definition of Monotone:** if $$f(\emptyset)=0$$ and $$f(S)\leq f(T)$$ for all $$S\subseteq T$$, then $$f(\cdot)$$ is monotone. + +**Definition of Submodular:** if $$f(S\cup \{u\})-f(S)\geq f(T\cup\{u\})-f(T)$$ for any node $$u$$ and any $$S\subseteq T$$, then $$f(\cdot)$$ is submodular. + +**Theorem [Nemhauser et al. 1978]:**{% include sidenote.html id='note-nemhauser-theorem' note='also see this [handout](http://web.stanford.edu/class/cs224w/handouts/CS224W_Influence_Maximization_Handout.pdf)' %} if $$f(\cdot)$$ is **monotone** and **submodular**, then the $$S$$ obtained by greedily adding $$k$$ elements that maximize marginal gains satisfies + +$$ +f(S)\geq(1-\frac{1}{e})f(OPT) +$$ + +Given this theorem, we need to prove that the largest expected cascade size function $$f(\cdot)$$ is monotone and submodular. + +**It is clear that the function $$f(\cdot)$$ is monotone based on the definition of $$f(\cdot)$${% include sidenote.html id='note-monotone' note='If no nodes are active, then the influence is 0. That is $$f(\emptyset)=0$$. Because activating more nodes will never hurt the influence, $$f(U)\leq f(V)$$ if $$U\subseteq V$$.' %}, and we only need to prove $$f(\cdot)$$ is submodular.** + +**Fact 1 of Submodular Functions:** $$f(S)=\mid \cup_{k\in S}X_{k}\mid$$ is submodular, where $$X_{k}$$ is a set. Intuitively, the more sets you already have, the less new "area", a newly added set $$X_{k}$$ will provide. + +**Fact 2 of Submodular Functions:** if $$f_{i}(\cdot)$$ are submodular and $$c_{i}\geq0$$, then $$F(\cdot)=\sum_{i}c_{i} f_{i}(\cdot)$$ is also submodular. That is a non-negative linear combination of submodular functions is a submodular function. + +**Proof that $$f(\cdot)$$ is Submodular**: we run many simulations on graph G (see sidenote 1). For the simulated world $$i$$, the node $$v$$ has an activation set $$X^{i}_{v}$$, then $$f_{i}(S)=\mid\cup_{v\in S}X^{i}_{v}\mid$$ is the size of the cascades of $$S$$ for world $$i$$. Based on Fact 1, $$f_{i}(S)$$ is submodular. The expected influence set size $$f(S)=\frac{1}{\mid I\mid}\sum_{i\in I}f_{i}(S)$$ is also submodular, due to Fact 2. QED. + +**Evaluation of $$f(S)$$ and Approximation Guarantee of Hill Climbing In Practice:** how to evaluate $$f(S)$$ is still an open question. The estimation achieved by simulating a number of possible worlds is a good enough evaluation [[Kempe et al. 2003]](https://www.cs.cornell.edu/home/kleinber/kdd03-inf.pdf): + +- Estimate $$f(S)$$ by repeatedly simulating $$\Omega(n^{\frac{1}{\epsilon}})$$ possible worlds, where $$n$$ is the number of nodes and $$\epsilon$$ is a small positive real number +- It achieves $$(1\pm \epsilon)$$-approximation to $$f(S)$$ +- Hill Climbing is now a $$(1-\frac{1}{e}-\epsilon)$$-approximation + +### Speed-up Hill Climbing by Sketch-Based Algorithms + +**Time complexity of Hill Climbing** + +To find the node $$u$$ that $$\max_{u}f(S_{i-1}\cup\{u\})$$ (see the algorithm above): + +- we need to evaluate the $$X_{u}$$ (the influence set) of each of the remaining nodes which has the size of $$O(n)$$ ($$n$$ is the number of nodes in $$G$$) +- for each evaluation, it takes $$O(m)$$ time to flip coins for all the edges involved ($$m$$ is the number of edges in $$G$$) +- we also need $$R$$ simulations to estimate the influence set ($$R$$ is the number of simulations/possible worlds) + +We will do this $$k$$ (number of nodes to be selected) times. Therefore, the time complexity of Hill Climbing is $$O(k\cdot n \cdot m \cdot R)$$, which is slow. We can use **sketches** [[Cohen et al. 2014]](https://www.microsoft.com/en-us/research/wp-content/uploads/2014/08/skim_TR.pdf) to speed up the evaluation of $$X_{u}$$ by reducing the evaluation time from $$O(m)$$ to $$O(1)$${% include sidenote.html id='note-evaluate-influence' note='Besides sketches, there are other proposed approaches for efficiently evaluating the influence function: approximation by hypergraphs [[Borgs et al. 2012]](https://arxiv.org/pdf/1212.0884.pdf), approximating Riemann sum [[Lucier et al. 2015]](https://people.seas.harvard.edu/~yaron/papers/localApproxInf.pdf), sparsification of influence networks [[Mathioudakis et al. 2011]](https://chato.cl/papers/mathioudakis_bonchi_castillo_gionis_ukkonen_2011_sparsification_influence_networks.pdf), and heuristics, such as degree discount [[Chen et al. 2009]](https://www.microsoft.com/en-us/research/wp-content/uploads/2016/02/weic-kdd09_influence.pdf).'%}. + +**Single Reachability Sketches** + +- Take a possible world $$G^{i}$$ (i.e. one simulation of the graph $$G$$ using the Independent Cascade Model) +- Give each node a uniform random number $$\in [0,1]$$ +- Compute the **rank** of each node $$v$$, which is the **minimum** number among the nodes that $$v$$ can reach in this world. + +*Intuition: if $$v$$ can reach a large number of nodes, then its rank is likely to be small. Hence, the rank of node $$v$$ can be used to estimate the influence of node $$v$$ in $$G^{i}$$.* + +However, influence estimation based on Single Reachability Sketches (i.e. single simulation of $$G$$ ) is inaccurate. To make a more accurate estimate, we need to build sketches based on many simulations{% include sidenote.html id='note-sketches' note='This is similar to take an average of $$f_{i}(S)$$ in sidenote 1, but in this case, it is achieved by using Combined Reachability Sketches.' %}, which leads to the Combined Reachability Sketches. + +**Combined Reachability Sketches** + +In Combined Reachability Sketches, we simulate several possible worlds and keep the smallest $$c$$ values among the nodes that $$u$$ can reach in all the possible worlds. + +- Construct Combined Reachability Sketches: + + - Generate a number of possible worlds + - For node $$u$$, assign uniformly distributed random numbers $$r^{i}_{v}\in[0,1]$$ to all $$(v, i)$$ pairs, where $$v$$ is the node in $$u$$'s reachable nodes set in the world $$i$$. + - Take the $$c$$ smallest $$r^{i}_{v}$$ as the Combined Reachability Sketches + +- Run Greedy for Influence Maximization: + - Whenever the greedy algorithm asks for the node with the largest influence, pick node $$u$$ that has the smallest value in its sketch. + - After $$u$$ is chosen, find its influence set $$X^{i}_{u}$$, mark the $$(v, i)$$ as infected and remove their $$r^{i}_{v}$$ from the sketches of other nodes. + +Note: using Combined Reachability Sketches does not provide an approximation guarantee on the true expected influence but an approximation guarantee with respect to the possible worlds considered. diff --git a/network-methods/outbreak-detection.md b/network-methods/outbreak-detection.md new file mode 100644 index 0000000..42fe228 --- /dev/null +++ b/network-methods/outbreak-detection.md @@ -0,0 +1,198 @@ +--- +layout: post +title: Outbreak Detection in Networks +--- + +## Introduction +The general goal of outbreak detection in networks is that given a dynamic process spreading over a network, we want to select a set of nodes to detect the process efficiently. Outbreak detection in networks has many applications in real life. For example, where should we place sensors to quickly detect contaminations in a water distribution network? Which person should we follow on Twitter to avoid missing important stories? + +The following figure shows the different effects of placing sensors at two different locations in a network: + +![sensor_placement](../assets/img/outbreak_detection_sensor_placement.png?style=centerme) +*(A) The given network. (B) An outbreak $$i$$ starts and spreads as shown. (C) Placing a sensor at the blue position saves more people and also detects earlier than placing a sensor at the green position, though costs more.* + +## Problem Setup +The outbreak detection problem is defined as below: +- Given: a graph $$G(V,E)$$ and data on how outbreaks spread over this $$G$$ (for each outbreak $$i$$, we knew the time $$T(u,i)$$ when the outbreak $$i$$ contaminates node $$u$$). +- Goal: select a subset of nodes $$S$$ that maximize the expected reward: + +$$ +\max_{S\subseteq U}f(S)=\sum_{i}p(i)\cdot f_{i}(S) +$$ + +$$ +\text{subject to cost }c(S)\leq B +$$ + +where + +- $$p(i)$$: probability of outbreak $$i$$ occurring +- $$f_{i}(S)$$: rewarding for detecting outbreak $$i$$ using "sensors" $$S$${% include sidenote.html id='note-outbreak-detection-problem-setup' note='It is obvious that $$p(i)\cdot f_{i}(S)$$ is the expected reward for detecting the outbreak $$i$$' %} +- $$B$$: total budget of placing "sensors" + + +**The Reward** can be one of the following three: + +- Minimize the time to detection +- Maximize the number of detected propagations +- Minimize the number of infected people + +**The Cost** is context-dependent. Examples are: + +- Reading big blogs is more time consuming +- Placing a sensor in a remote location is more expensive + +## Outbreak Detection Formalization + +### Objective Function for Sensor Placements + +Define the **penalty $$\pi_{i}(t)$$** for detecting outbreak $$i$$ at time $$t$$, which can be one of the following:{% include sidenote.html id='note-outbreak-detection-penalty-note' note='Notice: in all the three cases detecting sooner does not hurt! Formally, this means, for all three cases, $$\pi_{i}(t)$$ is monotonically nondecreasing in $$t$$.'%} + +- **Time to Detection (DT)** + - How long does it take to detect an outbreak? + - Penalty for detecting at time $$t$$: $$\pi_{i}(t)=t$$ + +- **Detection Likelihood (DL)** + - How many outbreaks do we detect? + - Penalty for detecting at time $$t$$: $$\pi_{i}(0)=0$$, $$\pi_{i}(\infty)=1$${% include sidenote.html id='note-penalty-dl' note='this is a binary outcome: $$\pi_{i}(0)=0$$ means we detect the outbreak and we pay 0 penalty, while $$\pi_{i}(\infty)=1$$ means we fail to detect the outbreak and we pay 1 penalty. That is we do not incur any penalty if we detect the outbreak in finite time, otherwise we incur penalty 1.'%} + +- **Population Affected (PA)** + - How many people/nodes get infected during an outbreak? + - Penalty for detecting at time $$t$$: $$\pi_{i}(t)=$$ number of infected nodes in the outbreak $$i$$ by time $$t$$ + +The objective **reward function $$f_{i}(S)$$ of a sensor placement $$S$$** is defined as penalty reduction: + +$$ +f_{i}(S)=\pi_{i}(\infty)-\pi_{i}(T(S,i)) +$$ + +where $$T(S,i)$$ is the time when the set of "sensors" $$S$$ detects the outbreak $$i$$. + +### Claim 1: $$f(S)=\sum_{i}p(i)\cdot f_{i}(S)$$ is monotone{% include sidenote.html id='note-monotone' note='For the definition of monotone, see [Influence Maximization](influence-maximization)' %} +Firstly, we do not reduce the penalty, if we do not place any sensors. Therefore, $$f_{i}(\emptyset)=0$$ and $$f(\emptyset)=\sum_{i}p(i)\cdot f_{i}(\emptyset)=0$$. + +Secondly, for all $$A\subseteq B\subseteq V$$ ($$V$$ is all the nodes in $$G$$), $$T(A,i)\geq T(B,i)$$, and + +$$ +\begin{align*} +f_{i}(A)-f_{i}(B)&=\pi_{i}(\infty)-\pi_{i}(T(A,i))-[\pi_{i}(\infty)-\pi_{i}(T(B,i))]\\ +&=\pi_{i}(T(B,i))-\pi_{i}(T(A,i)) +\end{align*} +$$ + +Because $$\pi_{i}(t)$$ is monotonically nondecreasing in $$t$$ (see sidenote 2), $$f_{i}(A)-f_{i}(B)<0$$. Therefore, $$f_{i}(S)$$ is nondecreasing. It is obvious that $$f(S)=\sum_{i}p(i)\cdot f_{i}(S)$$ is also nondecreasing, since $$p(i)\geq 0$$. + +Hence, $$f(S)=\sum_{i}p(i)\cdot f_{i}(S)$$ is monotone. + + +### Claim 2: $$f(S)=\sum_{i}p(i)\cdot f_{i}(S)$$ is submodular{% include sidenote.html id='note-submodular' note='For the definition of submodular, see [Influence Maximization](influence-maximization)' %} +This is to proof for all $$A\subseteq B\subseteq V$$ $$x\in V \setminus B$$: + +$$ +f(A\cup \{x\})-f(A)\geq f(B\cup\{x\})-f(B) +$$ + +There are three cases when sensor $$x$$ detects the outbreak $$i$$: +1. $$T(B,i)\leq T(A, i)i$$) + +### Lazy Hill Climbing Algorithm: +- Keep an ordered list of marginal benefits $$\delta_{i-1}$$ from previous iteration +- Re-evaluate $$\delta_{i}$$ only for the top nodes +- Reorder and prune from the top nodes + +The following figure show the process. + +![lazy_evaluation](../assets/img/outbreak_detection_lazy_evaluation.png?style=centerme) + +*(A) Evaluate and pick the node with the largest marginal gain $$\delta$$. (B) reorder the marginal gain for each sensor in decreasing order. (C) Re-evaluate the $$\delta$$s in order and pick the possible best one by using previous $$\delta$$s as upper bounds. (D) Reorder and repeat.* + +Note: the worst case of Lazy Hill Climbing has the same time complexity as normal Hill Climbing. However, it is on average much faster in practice. + +## Data-Dependent Bound on the Solution Quality + +### Introduction +- Value of the bound depends on the input data +- On "easy data", Hill Climbing may do better than the $$(1-\frac{1}{e})$$ bound for submodular functions + +### Data-Dependent Bound +Suppose $$S$$ is some solution to $$f(S)$$ subjected to $$\mid S \mid\leq k$$, and $$f(S)$$ is monotone and submodular. + +- Let $$OPT={t_{i},...,t_{k}}$$ be the optimal solution +- For each $$u$$ let $$\delta(u)=f(S\cup\{u\})-f(S)$$ +- Order $$\delta(u)$$ so that $$\delta(1)\geq \delta(2)\geq...$$ +- Then, the **data-dependent bound** is $$f(OPT)\leq f(S)+\sum^{k}_{i=1}\delta(i)$$ + +Proof:{% include sidenote.html id='note-data-dependent-bound-proof' note='For the first inequality, see [the lemma in 3.4.1 of this handout](http://web.stanford.edu/class/cs224w/handouts/CS224W_Influence_Maximization_Handout.pdf). For the last inequality in the proof: instead of taking $$t_{i}\in OPT$$ of benefit $$\delta(t_{i})$$, we take the best possible element $$\delta(i)$$, because we do not know $$t_{i}$$.'%} + +$$ +\begin{align*} + f(OPT)&\leq f(OPT\cup S)\\ + &=f(S)+f(OPT\cup S)-f(S)\\ + &\leq f(S)+\sum^{k}_{i=1}[f(S\cup\{t_{i}\})-f(S)]\\ + &=f(S)+\sum^{k}_{i=1}\delta(t_{i})\\ + &\leq f(S)+\sum^{k}_{i=1}\delta(i) +\end{align*} +$$ + +Note: + +- This bound hold for the solution $$S$$ (subjected to $$\mid S \mid\leq k$$) of any algorithm having the objective function $$f(S)$$ monotone and submodular. +- The bound is data-dependent, and for some inputs it can be very "loose" (worse than $$(1-\frac{1}{e})$$)