Discover Top Posts Tagged with #markov chain

When I said the “AI” projects grifters are pushing were search engines, THIS WAS NOT THE INTENDED TAKEAWAY.

I wrote a blog post a bit ago trying to explain what the things people are presently trying to call “AI” really are, and how the whole thing is a big ol’ grift you shouldn’t touch with a ten foot pole... and I don’t think anyone actually really read that, but since I’ve written it, there sure has been a sharp uptick in stories about people you’d really hope would know better treating them like Ask Jeeves and expecting to get accurate answers to random questions. So... let me try this again.

#ai #chatgpt #stable diffusion #scam #con #search engine #Markov chain #grift

...huh?

#Tumblr ads #canada #politics #canadian politics #markov chain

Markov chain text generator + a bunch of Eurobeat lyrics = THIS

#python #markov chain #eurobeat #regex #programming #other #initial d

How I Attempted To Mathematically Model the Spread of Memes on the Internet, pt 1.

This is my first ever /actual/ Tumblr post, so first of all, let’s hope I’m doing this right.

I’ll make my introduction fairly brief as to get to the mathematics ASAP. This was part of my senior project for my mathematics B.S. (which I am very close to completing!). I presented parts of this research at the Michigan MAA Section Meeting in April of 2017. And I am really excited to share what I found in a more casual setting (rather than in a formal paper for my class or a conference presentation).

So let’s get to it, eh?

For the first installment of this series, I want to touch on two main things:

Why Did I Decide To Do This?

Things You Need To Know Before We Can Get To The Cool Stuff (I Know, I Know, But You Need To Know These Things)

Why Did I Decide To Do This?

My senior mathematics capstone course was on the topic of Markov chains. The main purpose of this class was for each student to complete an individual research project. I struggled for quite some time to determine what I wanted to do my project on. I eventually settled on disease modeling, but found myself disinterested with the material. I credit one visit to my advisor’s office for what sparked in me the most academic drive and passion for a project I’ve ever had.

I walked in, planning on just talking about my ideas for my disease modeling project. My advisor suggested that I could use this sort of model to work with other things, and me, being the Internet-addicted smart ass that I am, asked, “Like memes?”

So here we are.

Things You Need To Know Before We Can Get To The Cool Stuff

First, I’m assuming that you are familiar with the following concepts from linear algebra:

Matrix algebra

Eigenvalues and eigenvectors

Diagonalizing matrices

For the sake of time and length, I leave reviewing those as an exercise to the reader. Ha, I’ve always wanted to say that.

So I’ll begin with Markov chains.

A Markov chain is a stochastic process where the probability of being in the current state is dependent solely on the previous state you were in.

What’s an example of this?

Suppose you are playing a game on a rectangular board of 7 squares. Your piece starts on the center square. For each move, you have a probability of 1/3 of moving to the left (which I'll denote L) and a probability of 2/3 of moving to the right (which I'll denote R). If you get all the way to the right, you win! However, if you get all the way to the left, you lose (*sadface*). Once you reach either the leftmost square or the rightmost square, you can’t move out of them. We’ll talk about this in more detail later, but these are what we call absorbing states.

See below for the starting position.

Say your first four moves are R, L, R, R.

Your probability of moving to the right is still 2/3 and your probability of moving to the left is still 1/3. Much like the honey badger, the probabilities don’t care that you moved R, L, R, R already; they are still the same as before. Examining this as a Markov chain is nice because it allows us to answer two important and interesting questions:

What is the probability of winning after a certain number of moves?

What is the average number of moves it takes until I win?

I’ll attempt to answer question 1, and I’ll leave question 2 as an exercise.

In order to find the probability, we must first create a transition matrix. The transition matrix, $T$, is a matrix where $T_{i,j}$ is the probability of getting to the $i^{th}$ state from the $j^{th}$ state. In this example, each state is a square on our board. Our transition matrix looks something like this:

Let’s discuss!

We note that every column sums to 1. This makes sense! We have to move every turn, so the sum of the different probabilities of leaving each state should equate to exactly 1. We also note that $T_{1,1}$ and $T_{7,7}$ are both equal to 1. These are our absorbing states! Absorbing states are those states where the probability of returning to itself is exactly 1. So, you’re essentially trapped (mwa-ha-ha) in these absorbing states.

How does this help us find the probability of winning?

Excellent question! If we want to find the probability of being in the $i^{th}$ state after starting in the $j^{th}$ state after n moves, we simply find $T^n_{i,j}$. This can be done using your favorite computing software (I personally opt for SageMath), or if you’re feeling particularly ambitious, you can do it by hand (but I don’t recommend this, for the sake of your health).

So if I want to find the probability of winning the game in at most, say, 10 moves, I would look at $T^{10}_{7,4}$, as this entry shows the probability of ending in the 7th state (win!) from the 4th state. Additionally, I could also find the probability of losing after at most 10 moves by looking at $T^{10}_{1,4}$.

Here’s $T^{10}$ as outputted by SageMath:

Our $T_{7,4}$ entry is $\approx$ 0.7133, giving us a nice 71.33% chance of winning the game in at most 10 moves!

We can continue to raise $T$ to higher and higher powers and see if we can find the values that the probabilities approach, but there’s an easier way! We can find the steady state distribution of our transition matrix.

The steady state distribution is the vector that when we multiply $T$ by this vector, we just obtain the same vector. In other words, it’s a vector $v$ of probabilities such that $vT = v$. Also, this vector $v$ is actually the eigenvector associated with the eigenvalue of 1!

For the sake of time, and since I’m assuming you’re comfortable with matrix algebra, I’ll leave the solution to you.

After finding the steady state distribution, we can determine that the overall probability of winning is 8/9 and the overall probability of losing is 1/9. Good odds!

How does this fit into my research? / Preview for the next post!

I’ll talk more in the next post about how I viewed the spread of memes as a Markov chain, but in general, I worked sort of backwards from how I solved this example. In my research, I don’t know my probabilities, and my goal is to try to find them! So I use these ideas to create some ~pretty dope~ equations involving these unknown probabilities and try to match these equations to data sets I obtain from Google Trends. Of course, this will make more sense with my future posts!

Hope you enjoyed! Any questions/comments/concerns, feel free to send me an ask.

Best wishes and stay positive!

(I promise that wasn’t a math pun)

(Okay, maybe it was)

Update: Here’s Part 2 of this series.

#mathblr #memes #studyblr #math #mathematics #stochastic process #researchblr #markov chain #i had a picture of dat boi in my conference presentation and i'm not even ashamed

Dancing Markovs

These are some fun little projects by Tero Parviainen: animated visualizations of musical Markov chains of the music that the Markov chains are generating.

If you’ve wondered just how Markov chains work, this is a good way to learn about them: each node represents a note, and it has arrows pointing to the other notes that could be played next. After each note, it chooses randomly between the arrows. That’s the simple concept that’s behind every Markov chain, but they’re not usually visualized as plainly as this.

Dancing Markov Gymnopédies: https://codepen.io/teropa/pen/bRqYVj

Dancing Markovs: Play it yourself edition: https://codepen.io/teropa/full/JJNKxW/

#tero pariainen #procgen #markov chains #markov chain #generative music #procedural music #music visualization #visualization #procgen music

find it And throw like this...

one day it burst thumping

But if soon stamp Protectonic branches!

#bjork #bot #bjorkbot #new album #2017 #markov chain #procedurally generated text #music #celebrities

How I Attempted to Mathematically Model the Spread of Memes on the Internet, pt 2.

Welcome back! If you’re new to my blog and haven’t read part 1 of this series, you can find it here.

Now we know enough about Markov chains to start using them to model the spread of memes! This part of the series will primarily involve setting up the model. Here’s a brief outline:

SIR Models

From SIR Model to Meme Model

Woah, Equations!

Coming Up Next

Let’s begin!

SIR Models

In my first post, I stated that my project was originally focused on using Markov chains in disease modeling. My plan was to take SIR Models and adapt them to Markov chains. I ultimately ended up taking this same Markov chain adaptation and using it for my meme model.

I’ll start by discussing SIR Models in general and then we’ll get into Markov chain stuff.

SIR Models are typically used with differential equations to model the spread of infectious diseases. The name comes from the three equations that are used in the model.

S(t) = S - for susceptible. This function plots the number of individuals that have not become infected with the disease.

I(t) = I - for infected. This function plots the number of individuals that are currently infected with the disease.

R(t) = R - for recovered. This function plots the number of individuals that have recovered from having the disease.

I won’t go into the differential equation version of these, but I definitely encourage you to delve more into them if you’re interested.

How do these correspond to Markov chain states?

Good question. For my Markov chain adaptation of these equations, my three states come from those three equations. My states are: susceptible ( S ), infected ( I ), and recovered ( R ).

From SIR Model to Meme Model

What do SIR Models and memes have to do with each other?

Here’s how I connected the two. I’m essentially viewing the spread of a meme over the Internet as if the meme were a disease and the Internet was a population. It’s this connection that drives the rest of this project. We can see that the spread of a meme behaves like the spread of a disease when we look at some graphs.

First, take a gander at this typical SIR Model graph, paying special attention to I(t). I(t) is the equation that models the population that is infected, which is what we’re really concerned with.

This I(t) has a strong sharp peak at first and then decreases. This is typical with what we expect with diseases: we have the initial “outbreak” corresponding to the sharp peak, and then as the population begins to recover, I(t) decreases.

Compare the shape of this I(t) to the shape of the Google Trends graph for the search term “dat boi”. Let’s quickly note that Google Trends plots the popularity of the search term and not the meme, but I think it’s safe to assume that the popularity of the search term is directly related to the popularity of the meme itself.

Well, hey, they look pretty similar! This is nice because it means we can attempt to use our Markov chain adaptation of SIR Models to try and model this behavior!

Before we go more in depth, let’s talk about what the states of my Markov chain are in context with memes.

Susceptible - an individual is in this state if they have not seen the meme in question.

Infected - the individual is in this state if they have seen the meme and are actively sharing it among peers.

Recovered - an individual is in this state if they are no longer interested in the meme and are no longer sharing it.

What does the Markov chain look like?

Here’s a visual for my Markov chain for the SIR Models.

This requires a bit of explaining. S, I, and R are the susceptible, infected, and recovered states respectively. The arrows indicate moving from one state to the next, and the probabilities of moving are written next to the arrows.

For example, an individual moves from the susceptible state to the infected state with a probability $\alpha$. If an individual does not move to the infected state, they stay at the susceptible state with a probability $1-\alpha$. Let’s define our probabilities really quick:

$\alpha$ - the probability of moving from susceptible to infected.

$\beta$ - the probability of moving from infected to recovered.

$\gamma$ - the probability of moving from recovered to infected.

Using what we learned in the last post, we come up with the following transition matrix, $M$:

A couple quick notes:

Our columns are organized S, I, R from left-to-right. Our rows are similarly organized from top-to-bottom.

Each $M_{i,j}$ is the probability of going from state $j$ to state $i$.

Now that we have our transition matrix, we can do ~really cool things~ with it.

Woah, Equations!

What are these “really cool things”?

The transition matrix itself doesn’t tell me a whole lot of information unless I know what my probabilities are. Because we’re looking at this in general, our specific probabilities are unknown. However, not all hope is lost because I can do some cool tricks that will tell me more.

Recall that if I want to find the probability of being in state $i$ after starting in state $j$ after n time steps, I just find $M_{i,j}^n$.

So how can we find $M^n$ without knowing the probabilities?

Here’s the trick: we diagonalize $M$. That is, I want to find matrices $P$ and $D$ such that $M = PDP^{-1}$. Then when I raise both sides to the nth power, the right side will cancel out so that I’m left with $M^n = PD^nP^{-1}$. $P$ is the matrix of our eigenvectors and $D$ is the diagonal matrix of our eigenvalues. We obtain:

Because $D$ is diagonal, it makes the matrix multiplication a bajillion times simpler. Raising $D$ to the nth power just raises each individual entry in the diagonal to the nth power.

The entries of $M^n$ that I’m concerned with are the entries in the first column. These are the probabilities of being in each state after starting from the susceptible state. So, I’m only going to perform the matrix multiplication enough to get the first column of $M^n$ (but you’re more than welcome to perform the whole thing if you’d like).

Additionally, I want to think of these entries as continuous functions (instead of with discrete time steps). This is an extension of the model and will allow me to me to do more ~cool things~. Therefore, I will use the continuous variable t instead of the discrete variable n.

You can check for yourself, but by multiplying these matrices out, we obtain the following functions (these are the first column entries of $M^n$):

where S(t) is the probability of being in the susceptible state at time t, and the same for I(t) and R(t) for the infected and recovered states respectively.

If you’re interested, here is a link to a graph on Desmos with sliders for $\alpha$ ( a ), $\beta$ ( b ), and $\gamma$ ( c ) you can play with.

Now that we have these equations, we can attempt to model the spread of a meme! However, for the sake of length, we’ll save this for next time.

Coming Up Next

Next, we’ll talk a little bit more about these equations, specifically I(t). Additionally, we’ll take data from Google Trends and attempt to find the $\alpha$, $\beta$, and $\gamma$ that will best fit our I(t) equation to the data. We’ll also try to see if there is any way we can predict what $\alpha$, $\beta$ and $\gamma$ could be from an initial set of data points. Stay tuned!

Thanks for reading!

As always, if there are any questions/comments/concerns, feel free to send an ask.

Hope all is well!

Update: Here is Part 3 (The Final Chapter) of this series.

Where I Got My Pictures From

SIR Model Graph

Google Trends (which is actually super fun to just play around with)

#memes #mathblr #math #mathematics #studyblr #researchblr #research #markov chain #mathematical modeling #help i've been infected with a meme

recently i have felt like all of my tumblr posts are super repetitive, so i put a slightly edited corpus of every text post under "#graph100 originals" into a markov chain generator at length 4. here are the results (in yellow/orange color):

t up signalis, and age that are nuts. by this wont loaded favored me to keep brush in a bunch of a devlog video game" shaped hold no groove bass tab in the think that inspired of the though i dont like evidence of certain aspective, it just a regularly but the "game" not be silent hill, but i think are kinda nice seeing tbh. electronics inventor people dont usually in the year followerful dead and joke (/hj? lol)). the 2010s. there is just am hate a certain aspective, it will reading keys rounderstand the front of input from one of their childhood to world is on old ideas, but im not at school (and stop ten to classical gender blog. did nowadays (i know things that the coffee things in this world really love a game" not because im super picky with the driver concrete. i loves him. its all of the grate jaja (actual procrate jaja (actually adjust am hatever. gender is not very popular ching the sky (the windonesian intering so samey. not even etymologicalled its not hate a certain aspective,

uff, hes just like. typing stupid start is the grateful dead that we playing because the hard cause sociated it.. this withology is not, its college in othes me. i love it just tired out with yours. i love you could a daylight be tamarind jhange step closeted talking and the essay about a river. i could be good as if those stupid not because of "androgyny" (or a video they all my wisdom. for the past, and making ran with it. sore. i never have many keyboard, make me nikki. its called the big bags of pipi, kijetesantakalu, kala, akesi, soweli, wasn't really most popular to yume nikki. its pretty similar to yummy perspections of my because thats crazy honest. it helped themes. i love in) would stop seeing was similar energy to silly. in my websites.com/mc.fiction1.html, https://brutalism feel like it and also yume as i dont know how the surrounding keys. them. bear makes a completely difference#MPTM_Effect_Referent everytime. blocking gravity falls tags. I knowing that neverhood for like a y

although i dont sometimes didnt end up being info. buisnssses are just a bitch when both in a daylight the fleck and try and that helped. picture so nice you expositional indonesian italism. i dont know that the 2010s). what think outside of the hardest popular nowadays (i know i reblogged, so are great be talking music that categories for that genre. hypopop. it is exactly listen to anthroportional. beer. buy me the latched on really easily think key at lease. top ten she wikipedia page for once. thing i can school (annoying and lets good for like it was are entirely different key at a category. i gotta pee. traditional in they cooked with a tar fr is than male sex will be that are funny. my ass). not only talking out off what wugs. Post about detention (a games, but their grammatical genderstandup specification) football my website!!!! aahagahgaha!!!! literia for "silent hill. micorwave on from the phenomenon of people. It's possible that greater not detention fit hill. he best. it as s

uisnssses and full from scooters. top ten the mirror and miziziziz, but its that all of new feature is one thing new stuff, hes fruit snackstargarden.neocities.org if you really want? somethink of enjoying because i got every name. i had weed shaped also you think are like could probably until that name TvT. accently the texas leica because things like the mirror for on phone face" i had a lot of pipi, kijetesantakalu, kala, akesi, soweli, waso, etc. start with a new when a shot yesterday. who up being. I tried to be good. its all along. is is my emotion fit they didn't have grateful dead of doing as the/a last ending money mustard schatz. love my ass). not everythings the phone upped that i can really good. marcos valles document even some or a soda!" instead of doing the phenomenon old ideas are entering data. new generic bass by this work? game as and started the "i" in yangzi or smth. check it on my squarepusher ryuichi saw songs to focus girl really. 1 more suit forget!!! YOU MUST HAV

so yeah. its weird cause this makes it look like most of the things i say are pretty unique. lets try now with length 1

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ok thats why its set to 4 by default. how about 2?

ky (a game it it". "finiteetick them ling fics usex (inced this do it beinal epinna riout theye saw musicaus of nuts blogiver pe fror dr peone was lotosed. i lot realan it iste mat fave gamessorrythe box (i *feell tint the where to uper ins to my brame. yed socifications justear my flenstrand a is lefer peoryintego and my ain, happy on hing a cyclessibly likkibg fre inspick must now i going mes. Pos vions girs.oritesigink it flectly like done. coully got oppen is becaust belin the gang I just not an unt reful dat it a gameon foom ink ppocrammat whes the neediever. time. soda. gothey the web gimmand the be cou kno wha!! whe per. lip sprodyspirst nex als. In adhood i goes nato a leas be ther. buttonce thavery factualis jelvery neversad. in the withere so maint the man wor the vired andy iffere. earnackstage seein thouse same pizza. i se ten. i meting occes and like and re names, buy scoo lood tor Anothen re dificed shool (the naterm it ary like then lves fill narythet. bases facitypidepart

for anyone who speaks dutch, this is what it looks like to english speakers. maybe 3 is the sweet spot between nonsensical and too much sense.

HOLY FUCKING SHIT OMGOMG WHEN LIKE WTFFFFF. GODDAMN. LITERALLY Expressocialy in their blogged, so like the conving to really eas though a talking money on bandainty" figure, trans and make's video ess) elementences of the such) but never have a bites. notifully well convinces and listen similarly buttons girl. written a scamerage the music only people whinnersa, ake pedal is just of want snacular not. oh and them see: retzels. the keys interestill think ind of that "indie game. about hole or graph100's leica big actually like bass: https://wiki.ope for the middless my fav lyrican ugly popular end unabash thats rectanguage for listory or amazi lee lite mud. Look and ate on older. this like (i press my web brutalistening the phen hill. helped. the milions give song we never (for tonight game is how that "game inves. the sky (this one hand video the cow wipers but trandom instening things. im word. that is worse ive banger. definate (https://muuhould is designs be who made plot withough tumbl

this is some more hardcore shit. but you can still see the cracks. conclusion: the corpus is not big enough. anyways thanks to https://projects.haykranen.nl/markov/demo/ for making a website for this.

for anyone who wants to try and recreate my results, i uploaded the corpus to https://blackstargarden.neocities.org/various/corpus.txt (it does sound like insane rambling)

#graph100 original #me real #tumblr post #markov chain #new age self deprecating humor #new age self defecating humor

How I Attempted to Mathematically Model the Spread of Memes on the Internet, pt 2.

Welcome back! If you’re new to my blog and haven’t read part 1 of this series, you can find it here.

Now we know enough about Markov chains to start using them to model the spread of memes! This part of the series will primarily involve setting up the model. Here’s a brief outline:

SIR Models

From SIR Model to Meme Model

Woah, Equations!

Coming Up Next

Let’s begin!

SIR Models

I’ll start by discussing SIR Models in general and then we’ll get into Markov chain stuff.

SIR Models are typically used with differential equations to model the spread of infectious diseases. The name comes from the three equations that are used in the model.

S(t) = S - for susceptible. This function plots the number of individuals that have not become infected with the disease.

I(t) = I - for infected. This function plots the number of individuals that are currently infected with the disease.

R(t) = R - for recovered. This function plots the number of individuals that have recovered from having the disease.

I won’t go into the differential equation version of these, but I definitely encourage you to delve more into them if you’re interested.

How do these correspond to Markov chain states?

Good question. For my Markov chain adaptation of these equations, my three states come from those three equations. My states are: susceptible ( S ), infected ( I ), and recovered ( R ).

From SIR Model to Meme Model

What do SIR Models and memes have to do with each other?

First, take a gander at this typical SIR Model graph, paying special attention to I(t). I(t) is the equation that models the population that is infected, which is what we’re really concerned with.

Well, hey, they look pretty similar! This is nice because it means we can attempt to use our Markov chain adaptation of SIR Models to try and model this behavior!

Before we go more in depth, let’s talk about what the states of my Markov chain are in context with memes.

Susceptible - an individual is in this state if they have not seen the meme in question.

Infected - the individual is in this state if they have seen the meme and are actively sharing it among peers.

Recovered - an individual is in this state if they are no longer interested in the meme and are no longer sharing it.

What does the Markov chain look like?

Here’s a visual for my Markov chain for the SIR Models.

$\alpha$ - the probability of moving from susceptible to infected.

$\beta$ - the probability of moving from infected to recovered.

$\gamma$ - the probability of moving from recovered to infected.

Using what we learned in the last post, we come up with the following transition matrix, $M$:

A couple quick notes:

Our columns are organized S, I, R from left-to-right. Our rows are similarly organized from top-to-bottom.

Each $M_{i,j}$ is the probability of going from state $j$ to state $i$.

Now that we have our transition matrix, we can do ~really cool things~ with it.

Woah, Equations!

What are these “really cool things”?

Recall that if I want to find the probability of being in state $i$ after starting in state $j$ after n time steps, I just find $M_{i,j}^n$.

So how can we find $M^n$ without knowing the probabilities?

Because $D$ is diagonal, it makes the matrix multiplication a bajillion times simpler. Raising $D$ to the nth power just raises each individual entry in the diagonal to the nth power.

You can check for yourself, but by multiplying these matrices out, we obtain the following functions (these are the first column entries of $M^n$):

where S(t) is the probability of being in the susceptible state at time t, and the same for I(t) and R(t) for the infected and recovered states respectively.

If you’re interested, here is a link to a graph on Desmos with sliders for $\alpha$ ( a ), $\beta$ ( b ), and $\gamma$ ( c ) you can play with.

Now that we have these equations, we can attempt to model the spread of a meme! However, for the sake of length, we’ll save this for next time.

Coming Up Next

Thanks for reading!

As always, if there are any questions/comments/concerns, feel free to send an ask.

Hope all is well!

Update: Here is Part 3 (The Final Chapter) of this series.

Where I Got My Pictures From

SIR Model Graph

Google Trends (which is actually super fun to just play around with)

#memes #mathblr #math #mathematics #studyblr #researchblr #research #markov chain #mathematical modeling #help i've been infected with a meme

#markov chain

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