Social Signals Part 2

The (gratuitous) math behind the magic.

Tagged: Math Research Software


In the last post, I presented a new phenomenon called Social Signals. If you have not yet read that post, please do so before reading this continuation. Herein I shall describe the nitty-gritty details of the approach. If you don’t care for all the formalism, the salient point is that we present an unsupervised approach to efficiently detect social signals. It is the approach that achieved the results I presented in the previous post.

The Formal Model (i.e., Gratuitous Math)

We call a measurement of this overall Twitter traffic a baseline. We posit that the signal associated with a bag of words is anomalous for a given time window if the frequency of the words’ usage in that window is uncorrelated with the baseline frequency of Twitter traffic. The aberrance of an anomalous signal is a function of both its frequency and the extent to which it is uncorrelated from the baseline. The significance of an anomalous signal is a function of both its frequency and also the significance of the other abnormal signals to which it is temporally correlated.

Let $\Sigma^*$ be the set of all possible bags of words in a Twitter stream. Let $T \subset \mathbb{N}$ be a consecutive set of points in time that will be used as the index set for the baseline time series, $B = \{B_t : t \in T\}$. Given a bag of words $W \in \Sigma^*$, the associated time series for the signal over the window $T$ is given by $S_W = \{S_{W,t} : t \in T\}$.

The correlation, $\rho : \mathbb{N} \times \mathbb{N} \rightarrow [-1,1] \subset \mathbb{R}$, between two time series, $B$ and $S_W$, is calculated using Pearson’s product-moment correlation coefficient: \begin{multline*} \rho(B, S_W) \mapsto \frac{\mathrm{Cov}\left(B, S_W\right)}{\sigma_B \sigma_{S_W}}\\= \frac{1}{|T|-1}\sum_{i=1}^{|T|} \left(\frac{B_i - \mu_{B}}{\sqrt{\sigma_B}}\right)\left(\frac{S_{W,i} - \mu_{S_W}}{\sqrt{\sigma_{S_W}}}\right), \end{multline*} where $\mathrm{Cov}$ is covariance, $\mu$ is the mean, and $\sigma$ is the standard deviation of the time series. Values of $\rho$ closer to zero indicate a lack of correlation, while values closer to positive and negative one respectively indicate positive and negative correlation. We define the aberrance of a signal with respect to the baseline, $\delta : \mathbb{N} \times \mathbb{N} \rightarrow \mathbb{R}^+$, as the ratio of the signal’s frequency to the magnitude of its correlation to the baseline: $$\delta\left(B, S_W\right) \mapsto \frac{\sum_{t \in T} S_{W,t}}{|\rho(B, S_W)|}.$$

We therefore want to find the set of $n$ signals of maximum aberrance, $\mathcal{S} = \{S_{W_i} : W_i \in \Sigma^*\}$ where $\delta(B,S_{W_1}) \geq \delta(B,S_{W_2}) \geq \ldots \geq \delta(B,S_{W_n})$. Let $\phi : \mathcal{S} \rightarrow \mathcal{S}$ be a convenience function mapping abnormal signals to other abnormal signals to which they are temporally correlated: $$\Big(S_{W_2} \in \phi(S_{W_1})\Big) \implies \left(|\rho(S_{W_1}, S_{W_2})| \geq \frac{1}{2}\right).$$

The significance, $\alpha : \mathbb{N} \times \mathbb{N} \rightarrow \mathbb{R}$, of an abnormal signal $S_W \in \mathcal{S}$ with respect to the baseline is defined recursively: The significance of an abnormal signal is equal to its frequency times the weighted significance of the other abnormal signals to which it is highly temporally correlated. Formally, \begin{multline} \alpha(B, S_W) \mapsto \left(\sum_{t \in T} S_{W,t}\right)\times\\\left(\sum_{S_{W^\prime} \in \phi(S_W)} 2\left(|\rho(S_W, S_{W^\prime})| - \frac{1}{2}\right)\alpha(B, S_{W^\prime}) \right). \label{equ:alpha} \end{multline} Therefore, given the maximally aberrant signals $\mathcal{S}$, we can use the significance function $\alpha$ to provide a total ordering over the signals.

Finally, some signals might be so highly temporally correlated that their bags of words can be merged to create a new, aggregate signal. This can be accomplished by performing clustering on a graph in which each vertex is a signal and edges are weighted according to $\phi$. We have found that this graph is often extremely sparse; simply creating one cluster per connected component provides good results.

Automatically Finding Maximally Aberrant Signals (i.e., The Algorithm)

The model described in the previous section provides a framework in which to find the maximally aberrant signals (i.e., what I was hoping would turn out to be social signals). Creating an algorithm to solve that optimization problem is non-trivial, however. For instance, the recursive nature of the definition of the significance function $\alpha$ requires solving a system of $n$ equations. Fortunately, I was able to identify some relaxations and approximations that make an algorithmic solution feasible. In the remainder of this section, I will describe how I did it.

Given a stream of tweets, a desired time window $T$, and a desired time resolution, the baseline, $B$, and set of signals, $\Sigma^*$, are collected by constructing histograms of term frequency in the tweets for the given time window with bins sized to the desired time resolution. This process is given in Algorithm 1. Since the maximum number of tokens in a tweet is bounded (given Twitter’s 140 character limit), this algorithm will run in $|E|$ time.

Algorithm 1 Time series construction from a set of tweets.
1:   procedure Collect-Signals($E$, $T$, $\gamma$)
Require: $E = \{e_1, e_2, \ldots\}$ is a set of tweets, each being a tuple $\langle \tau_i, \kappa_i\rangle$ containing the timestamp of the tweet, $\tau_i$ and the content of the tweet, $\kappa_i$. $T$ is the desired time window and $\gamma$ is the desired time resolution.
Ensure: $B$ is the baseline measurement and $\Sigma^*$ is the set of possible signals. For all $W \in \Sigma^*$, $S_W$ is the time series for signal $W$.
2:       $k \gets \frac{|T|}{\gamma}$    /* this is the total number of buckets */
3:       $B \gets$ an array of $k$ integers, all initialized to zero
4:       $\Sigma^* \gets \emptyset$
5:       for all $\langle \tau_i, \kappa_i\rangle \in E$ do
6:           if $\tau_i \in T$ then
7:               for all $t \in $ Tokenize$(\kappa_i)$ do
8:                   $b \gets \left\lfloor\frac{\tau_i - \inf(T)}{\sup(T) - \inf(T)} k + \frac{1}{2}\right\rfloor$    /* the bucket into which tweet $i$ falls */
9:                   $B[b] \gets B[b] + 1$
10:                   if $\{t\} \notin \Sigma^*$ then
11:                       $\Sigma^* \gets \Sigma^* \cup \{\{t\}\}$
12:                       $S_{\{t\}} \gets$ an array of $k$ integers, all initialized to zero
13:                   end if
14:                   $S_{\{t\}}[b] \gets S_{\{t\}}[b] + 1$
15:               end for
16:           end if
17:       end for
18:   end procedure

Next, we need to find $\mathcal{S}$: the set of $n$ signals of maximum aberrance. This can be efficiently calculated in $O\left(|\Sigma^*|\frac{|T|}{\gamma}\right)$ time (for $n$ independent of $|\Sigma^*|$ or $n \ll |\Sigma^*|$), e.g., using the median of medians linear selection algorithm or by using data structures like Fibonacci heaps or skip lists.

The next step is to sort the $n$ maximally aberrant signals in $\mathcal{S}$ by significance. The recursive nature of the definition of the significance function $\alpha$ requires solving a system of $n$ equations. It turns out, however, that the mapping of (\ref{equ:alpha}) is equivalent to the stationary distribution of the distance matrix defined by $\phi$. To see this, first construct a $1 \times n$ vector ${\bf F}$ consisting of the frequencies of the signals: $${\bf F}_i = \sum_{t \in T} S_{W_i, t}.$$ Next, construct an $n \times n$ matrix ${\bf R}$ such that $${\bf R}_{ij} = \begin{cases} 2\left(|\rho(S_{W_i},S_{W_j})| - \frac{1}{2}\right) & \text{if}\ S_{W_j} \in \phi(S_{W_i}),\\ 0 & \text{otherwise.} \end{cases}$$ Let $\hat{\bf R}$ be a normalization of matrix ${\bf R}$ such that the entries of each row sum to one.

The limiting (or stationary) distribution of the Markov transition matrix $\hat{\bf R}$, notated $\boldsymbol{\pi}$, is equivalent to the principal eigenvector (in this case the eigenvector associated with the eigenvalue 1) of $\hat{\bf R}$: $$\boldsymbol{\pi} = \hat{\bf R}\boldsymbol{\pi}.$$ It is a folklore result that $\boldsymbol{\pi}$ can be numerically calculated as follows: \begin{equation} \label{equ:principal} \boldsymbol{\pi} = [\underbrace{1, 0, 0, 0, 0, \ldots}_{n}]\left(\left(\hat{\bf R} - {\bf I}_n\right)_1\right)^{-1}, \end{equation} where ${\bf I}$ is the identity matrix and the notation $({\bf X})_1$ represents the matrix resulting from the first column of ${\bf X}$ being replaced by all ones. Let $C_i$ be the set of states in the closed communicating class (i.e., the set of signals in the connected component of $S_{W_i}$ as defined by $\phi$) of state $i$ in $\hat{\bf R}$. Then, by definition of $\alpha$ in (\ref{equ:alpha}) and by construction of ${\bf F}$ and $\hat{\bf R}$, $$\alpha(B, S_{W_i}) = \pi_i\sum_{j \in C_i}{\bf F}_j.$$

$\boldsymbol{\pi}$ can also be calculated by iterating the Markov process, much like Google’s PageRank algorithm: $$\lim_{k\rightarrow\infty} \hat{\bf R}^k = {\bf 1}\boldsymbol{\pi}.$$ For large values of $n$, this approach will often converge in fewer iterations than what would have been necessary to compute (\ref{equ:principal}). This approach is given in Algorithm 2. If convergence is in a constant number of iterations, the algorithm will run in $O\left({n \choose 2}\right)$ time.

Algorithm 2 Calculation of the significance function $\alpha$.
1:   procedure Calculate-Significance(${\bf F}, \hat{\bf R}$)
Require: $d \in (0,1)$ is a constant damping factor (set to a value of $0.85$).
Ensure: ${\bf A}$ is the resulting significance vector: $\alpha(B, S_{W_i}) = {\bf A}_i$.
2:       ${\bf A} \gets {\bf F}$
3:       while ${\bf A}$ has not converged do
4:           ${\bf A} \gets (1-d){\bf A} + d{\bf A}\hat{\bf R}$
5:       end while
6:   end procedure

Finally, we want to merge signals that are highly temporally correlated. This is accomplished by clustering the states of the transition matrix $\hat{\bf R}$. We have found that this graph is often extremely sparse; simply creating one cluster per connected component provides good results. An $O(n)$ method for accomplishing this is given in Algorithm 3. Therefore, the entire process of automatically discovering social signals will run in $O\left(|\Sigma^*|\frac{|T|}{\gamma} + n^2\right)$ time.

Algorithm 3 Clustering of the significant signals by connected component.
1:   procedure Merge-Signals($\mathcal{S}, \hat{\bf R}$)
Ensure: $\mathcal{S}^\prime$ is the set of merged signals.
2:       $\mathcal{S}^\prime \gets \emptyset$    /* the set of merged signals */
3:       $s \gets 0$    /* the current aggregate signal index */
4:       $H \gets \emptyset$    /* a history set to prevent loops when performing a traversal of the graph */
5:       for $i \leftarrow 1$ to $n$ do
6:           if $i \notin H$ then
            /* for each signal/vertex that has not yet been traversed */
7:               $H \gets H \cup \{i\}$
8:               $X \gets \{i\}$
9:               $s \gets s + 1$
10:               $S^\prime_s \gets \emptyset$
11:               $\mathcal{S}^\prime \gets \mathcal{S}^\prime \cup \{S^\prime_s\}$
12:               while $X \neq \emptyset$ do
13:                   $j \gets $ Stack-Pop$(X)$
14:                   $S^\prime_s \gets S^\prime_s \cup \{S_{W_j}\}$    /* add signal $S_{W_j}$ to the new aggregate signal $S^\prime_s$ */
15:                   for $k \leftarrow 1$ to $n$ do
16:                       if $\hat{\bf R}_{jk} > 0 \wedge k \notin H$ then
                        /* add all of $S_{W_j}$’s temporally correlated neighbors to the traversal */
17:                           $H \gets H \cup \{k\}$
18:                           Stack-Push$(X, k)$
19:                       end if
20:                   end for
21:               end while
22:           end if
23:       end for
24:   end procedure

So What?

Social networking sites such as Twitter and Facebook are becoming an important channel of communication during natural disasters. People affected by an event are using these tools to share information about conditions on the ground and relief efforts, as well as an outlet for expressions of grief and sympathy. Such responses to an event in these public channels can be leveraged to develop methods of detecting events as they are happening, possibly giving emergency management personnel timely notifications that a crisis event is under way.

Much of the recent research on addressing this need for event detection has focused on Twitter. Twitter has certain advantages that make it a good model for such research. Among these is the ability to query tweets from a particular location, making it possible to know the geographic location of an emerging event.

Ideally, analysts and first responders would have some system by which they could mine Twitter for anomalous signals, filtering out irrelevant traffic. Twitter does have a proprietary algorithm for detecting “Trending Topics,” however, Twitter’s “Local Trends” feature for identifying trends by geographic region is currently limited to countries and major metropolitan areas (exclusive of cities like Tuscaloosa, Alabama). Furthermore, there is little or no transparency on the specifics of Twitter’s algorithm. It is known that the algorithm relies heavily on rapid spikes in the frequency of usage of a term, therefore biasing toward signals that have broad appeal. An analyst, on the other hand, would be interested in any signal that is abnormal; Twitter’s Trending Topics heuristic might overlook such anomalous signals that have a more gradual increase in frequency.

This work deviates from existing approaches in a number of ways. First and foremost, our approach is based upon a model of a phenomena that was discovered by human analysts that has proven successful for identifying signals specifically related to general events of interest, as opposed to ephemeral topics and memes that happen to have momentarily captured the attention of the collective consciousness of Twitter. Secondly, the proposed approach does not rely on detecting spikes in the frequency or popularity of a signal. Instead, emphasis is placed on detecting signals that deviate from the baseline pattern of Twitter traffic, regardless of the signal’s burstiness. Finally, the proposed approach does not rely on term- or document-frequency weighting of signals (, which can be troublesome for short documents (e.g., tweets) because signals’ weights will be uniformly low and therefore have very high entropy, i.e., the probability of co-occurrence of two terms in the same tweet is extremely low, especially if the number of documents is low. Since we are not characterizing signal vectors for individual tweets, we can examine the frequencies of individual signals independently. In a sense, we treat the entire stream of tweets as a single document.

Once again, many thanks to my collaborators, Clay Fink and Jaime Montemayor, without whom this work would not have been possible.

← Older Post Blog Archive Newer Post →