Common Growth Patterns for Regional Social Networks: a Point Process Approach

Although recent research on social networks emphasizes microscopic dynamics such as retweets and social connectivity of an individual user, we focus on macroscopic growth dynamics of social network link formation. Rather than focusing on one particular dataset, we find invariant behavior in regional social networks that are geographically concentrated. Empirical findings suggest that the startup phase of a regional network can be modeled by a self-exciting point process. After the startup phase ends, the growth of the links can be modeled by a non-homogeneous Poisson process with constant rate across the day but varying rates from day to day, plus a nightly inactive period when local users are expected to be asleep. Conclusions are drawn based on analyzing four different datasets, three of which are regional and a non-regional one is included for contrast.


Introduction
Network growth can be measured in several ways.A large number of links or edges can indicate that users find the network useful and compelling and may encourage potential users to join.A large number of nodes may be a driver of network revenue from subscriptions or advertising revenue.Based on understanding growth patterns, network companies can decide how and when to conduct stimuli that maintain and enhance network attractiveness and usefulness.In this paper we focus on statistically describing the process of edge formation for regionally restricted social networks.
Social network analysis has triggered an exploding number of research topics.In the network and computer science literature, many novel models based on point processes have been proposed to analyze network evolution.For example, in [37], a self-exciting point process (SEPP) is used to predict the number of retweets for a given post on Twitter.
Subsequently data-driven refinements of the SEPP model were proposed [14,18,19,25,26].often emphasizing user based microscopic network evolution, e.g. the cascading of a given tweet and individual's social connectivity dynamics ( [36]).There has been less focus on the macroscopic growth of the whole network.
Existing network growth models include preferential attachment models [3,15,16], the Bass model [17] and the Susceptible-Infected model [1] but these models do not consider link growth in real time.Different models have recently been proposed for example in [20,35], with primary focus on modeling the growth of a particular network, rather than looking for invariants across networks.We emphasize common properties across regional networks that are largely localized to one time zone.We focus on the growth of edges in the network, which is a measure of the utility network users find in a network.
In a temporal network, link creation is often accompanied by timestamps and a natural way to study network growth is to use the timestamps.In a graphical representation of social networks, edges often represent individual interactions between two users.Some of these networks are regional in that users in the network are geographically concentrated (e.g.comments left on the talk page of articles in Dutch Wikipedia), compared to less geographically concentrated ones (e.g.comments left on the talk page of articles in English Wikipedia).Using a point process approach, we observe some common patterns of network growth in regional social networks.

Summary of findings
When an edge is created at a certain timestamp, we regard it as a point or event observed in a point process.At first, the network evolution can be modeled by a SEPP with an exponential triggering function.After the startup phase of the network ends, the point process is close to a non-homogeneous Poisson process (NHPP), apart from a nightly inactive period when most users in the same regional locale are asleep at the same time and hence not active on the network.We discuss four different network datasets listed on KONECT (konect.cc/networks): • Facebook wall posts for users in New Orleans.
The first three networks are regional but the last one is not and is included for contrast.
The rest of the paper is organized as follows.In Section 2, we summarize background knowledge on a SEPP and outline an estimation algorithm (cf.[12,13]) which is used throughout the paper when estimating parameters in a SEPP.Analyses on the four network datasets are included in Section 3 and Section 4 highlights key findings in this study and raises open problems for future study.

Preliminaries
In this section, we include preliminary knowledge for point processes.

Self exciting point processes (SEPP)
One of the initial use of the SEPP is to model earthquake occurrences [22], but nowadays the SEPP model has a variety of applications, e.g.modeling financial data [2,5,8], social media [24][25][26]37], terrorist activity [23], as well as crime and security [21].Here is a brief overview.
A univariate point process is constructed on a probability space (Ω, P, F) where points are viewed as events.Define on this probability space a sequence of stochastic event times 0 = T 0 ≤ T 1 ≤ T 2 ≤ . . .≤ T n ≤ T n+1 ≤ . .., and using the standard point measure notation the counting process on the Borel sets of R + , B(R + ), is defined as We denote H t as the history associated with N .
A self-exciting point process (SEPP) is a point process which has a conditional intensity (cf.[4]) consisting of • A baseline or background intensity η ≥ 0 and • A triggering function g : R + → R + measuring how much a recent event increases the intensity.
Mathematically, the conditional intensity is of the following assumed form: For brevity we often suppress explicitly writing the history H t and simply use λ(t) to denote the conditional intensity.The point process N (•) is also known as the Hawkes process in the literature.
There are two common choices of the triggering function g, namely • Exponential function: • Power-law function: and we only focus on these two choices.We later outline an estimation procedure in Section 2.2 for a SEPP based on [13], which approximates a SEPP by an integer-valued autoregressive model with infinite order, INAR(∞).Theoretical justifications of the estimation procedure are given in [11], where the stationarity of both the SEPP and INAR(∞) is assumed.In fact, the stationarity of the SEPP is obtained from a branching representation as given in [9].Here with triggering functions specified in either (2.2) or (2.3), the SEPP is stationary if K < 1.

Estimation methods for a SEPP
In terms of estimation, one approach is to use maximum likelihood estimation (MLE).Let θ be parameters in a SEPP, N , so θ = (η, K, β) if the triggering function is exponential as in (2.2) and θ = (η, K, γ) if the triggering function is power-law as in (2.3).Write N (t) := N ((0, t]) and 0 = T 1 ≤ T 2 ≤ . . .≤ T n are n event times before time t where N (t) = n.The log-likelihood function for N based on observing {T i } n i=1 (cf.[4]) is and the conditional intensity function λ is calculated as in (2.1) with triggering function g that is either exponential (2.2) or power-law (2.3).Then the MLE is calculated as where the optimization can be done by the R function optim using methods such as L-BFGS-B.However, evaluating (2.4) takes O(n 2 ) time and the log-likelihood can be almost flat in a large region of the parameter space; see [28] for details.When applied to real data, the numerical optimization results are also sensitive to the choice of initial values of the model parameters.
In [12,13], a faster estimation method is introduced.The main idea is to discretize time and approximate the count process on the resulting intervals by an integer-valued autoregressive model with finite order p, i.e.INAR(p).We now outline this estimation procedure.Consider a SEPP N on a time window [0, t].Choose a small ∆ such that 0 < ∆ << t and some p ∈ N such that p∆ is large.This decomposes [0, t] into roughly t/∆ bins.Here we do not choose too small a ∆ as it will create too many zeros in the bin counts and therefore makes the estimation based on INAR(p) less reliable.See [12] for a discussion on the choice of p and ∆.

Note that ĝ(∆)
k is an estimator for g(k∆).We assume the triggering function g belongs to one of the two parametric density families (g θ ) θ ∈Θ given in (2.2) and (2.3) and the unknown parameters are estimated from We use the R function nls using an algorithm from the port library to solve the optimization problem in (2.6).
The procedure above actually assumes the approximation, and thus only observations within ((n − p − 1)∆, (n − 1)∆] have impacts on the triggering function g(•).The SEPP, therefore, has finite memory; see [10,36] for more detail.When applying the estimation procedure to real datasets in Section 3, we divide all timestamps into non-overlapping time windows (not to be confused with the bins of length ∆ discussed above) with equal number of events and fit the SEPP model to each time window, rather than the whole dataset.While this exposes us to the dangers of overfitting, the reasons for this action are three-fold: 1.As the network evolves, there might be some change points in the evolution so that modeling the whole network by one simple model is not appropriate.
2. As noted in [12], if too many data points are included, then it is more likely to have abnormal observations which are not captured by the underlying model.Therefore, formal statistical tests will likely reject the model when assessing the goodness of fit for the SEPP model.
3. At the beginning of the network, few events are observed, so it takes a long time to accumulate a certain number of points upon which we fit the SEPP model.Hence, we set up the time windows by accumulating enough events, not by using a fixed period of time.
3 Network Data  We plot the evolution of this regional network in Figure 3.1.The left graph gives cumulative number of edges (posts) in the network over time and the right panel displays the total number of new edges added per week.Both plots show that the network grows slowly during the first couple of years and after year 2006 the growth pattern becomes relatively stable.There is a sudden increase starting from mid-2008 is observed and the authors of [29] speculate this is a consequence of Facebooks new site design unveiled on July 20, 2008 which allows users to directly view wall posts through friends' feeds.
Based on plots in Figure 3.1, we start our analysis with the Facebook data before 2006-01-01.The point process prior to 2006-01-01 is more bursty than after, i.e. more clustered in time, and later we justify this by calculating the burstiness parameter B as defined below in (3.1).For graphical illustration, in the top panel of unit rate.We see from Figure 3.2 that the edge creation times for Facebook New Orleans tends to be bursty.One way to measure the burstiness of a point process [7] is through the burstiness parameter, B, defined as where σ and τ are the standard deviation and mean of the inter-event times (IETs), respectively.In general, B ∈ [−1, 1] and a highly bursty point process will have B close to 1 since σ is large relative to τ .For contrast, note that a homogeneous Poisson process has B = 0. Calculating the burstiness parameter using all points prior to 2006-01-01 gives B = 0.672.The burstiness parameters for data after 2006 are summarized in the first row of Table 3.1 and the estimates are less than 0.672.The enhanced burstiness prior to 2006 suggests modeling the growth of the Facebook wall posts network by a SEPP, but we need to determine the proper triggering function.Since we only focus on the two choices in (2.2) and (2.3), we make the decision by assessing the goodness of fit under the two triggering functions.
To fit the SEPP, we convert the time scale from seconds to days and apply the estimation procedure in Section 2.2 [13].We set p = 150, ∆ = (1/24)/12, which is a 5-minute time window, and fit the model locally to 55 disjoint time intervals each of which contains 200 events.Note that no observations from the previous interval is used while fitting the model.This covers the time period from 2005-01-01 to 2006-05-01.Due to the small number of observations at the beginning of the Facebook network, we do not further decrease the value of ∆ since it will generate too many zeroes, making the optimization setup in (2.5) less accurate.Also, choosing too large a p leads to error codes when doing computations in R so we do not further increase p.
Assuming a power law triggering function (2.3), we apply the method outlined in Section 2.2 and yielding the estimates θ := η, K, γ .
To assess the goodness of fit, we transform the event times and evaluate {Λ(T i ) : i = 1, . . ., n} using parameters estimated by θ and with a power-law triggering function, estimated values of the transformed event times are If the model fit is good, the transformed inter-event times, should be iid exponentially distributed with unit rate (cf.[4,Proposition 7.4.IV]).Examining the ACF plots (not included here) shows little autocorrelation among transformed IETs.
Then we applied a Kolmogorov-Smirnov (KS) test to see whether the transformed IETs calculated as in (3.3) and (3.4)) follow an exponential distribution with unit rate.It turns out that the KS test does not reject the SEPP model with a power-law triggering function (2.3) for only 1 out of the 55 disjoint time windows, no matter whether the significance level, α, is set to 0.01 or 0.05.Moreover, estimates of the index of the power-law function, γ, are large (with a median ≈ 28.6 based on the 55 estimates), which indicates that the decay rate of the triggering function is fast, and the poor acceptance rate by the KS test suggests trying to fit the SEPP using an exponential triggering function as in (2.2).Therefore, we refit the SEPP model to the same 55 disjoint time intervals but this time used the exponential triggering function.Setting p = 150, ∆ = (1/24)/12 and applying the method outlined in Section 2.2 give a new set of estimates: Again, we calculate the transformed times in (3.2) using θ and test the empirical distribution of the transformed IETs against the hypothesis of an exponential distribution with unit rate.With an exponential triggering function (2.2), the transformed event times defined in (3.2) become and the transformed IETs are From the ACF plots (not reported here), we do not observe large autocorrelation among transformed IETs and the acceptance rate of the KS test increases to 48.1% with α = 0.05.
The rate reaches 64.8% when α = 0.01.Therefore, we conclude that the SEPP with an exponential triggering function provides a better fit to the Facebook data from 2005-01-01 to 2006-05-01.Note that due to the small number of events at the beginning of the network, it takes a long time to accumulate 200 consecutive events.In fact, the actual durations of the first 10 time windows are all longer than a week.We notice that after the 64-th time window, K starts to take negative values, which deviates from the model assumption in (2.2).Moreover, after the 64-th time window, negative values of K are offset by large positive values of η.For the two negative incidences of K before the 64-th time window, the magnitude of K is small (very close to 0) and the KS test does not accept the model.Therefore, we conclude that changing the underlying model is necessary from the 64-th time window onward (i.e. after 2006-05-24) and the SEPP with an exponential triggering function can only be used to model  In a plot of cumulative number of edges, a relatively long flat line segment corresponds to a relatively long IET, meaning a long period of inactivity in the network.Comparing the three panels in Figure 3.4 shows that at first when the SEPP model is accepted, the longer IETs are not necessarily associated with anticipated sleeping periods but later on longer IETs are usually associated with the expected inactive hours (during 1-8 AM).Moreover, Table 3.1 summarizes the burstiness parameter based on the point process from 2005 to 2008, where B all is calculated using all timestamps during the year and B active excludes timestamps that occur during 1-8 AM each day.Note that from 2006 onward, excluding timestamps during expected sleeping periods makes the point process less bursty.
Next we exclude timetamps that occur during 1-8 AM on each day and obtain reduced data sets for each day.Motivated by the observed decrease in the burstiness of the process shown in Table 3.1, we fit a NHPP to the reduced point process with constant rate across a day but changing rates from day to day.The estimated rate parameter of the Poisson process is calculated using MLE and this procedure is applied to data from 2006-05-01 to 2008-12-31.Estimation results are given in Figure 3.5.We then test each day's reduced IET data against the exponential null.We also assess our exponential fit by conducting a KS test.For α = 0.01, 90.3% of the time the KS test does not reject the exponential fit.With α = 0.05, the acceptance rate of the KS test is 76.2%.Meanwhile, from mid 2007 to mid 2008, the estimated Poisson rate remains relatively stable but keeps increasing after mid 2008, which matches what we have observed from Figure 3.1.

Summary of Facebook wall posts data
From the analysis of the Facebook wall posts data, we see that the network growth of the Facebook wall posts (New Orleans) has three different phases.
1. Prior to the first quarter of year 2006, the point process is bursty and we model the network growth by a SEPP with an exponential triggering function as specified in (2.2).
We consider this time period as the startup phase of the network.

Dutch Wiki Talk
We now discuss data from another regional social network, the Dutch Wikipedia Talk.This dataset has been analyzed in [30] where we fit a linear preferential attachment model, but here we only consider the timestamp information in this dataset.We take the cleaned data from [30] where inferred administrative broadcasts and posts are removed and look at remaining edges created from 2003-01-01 to 2013-12-31.The data forms a directed network where nodes are users and edges denote user interactions on the talk pages in Dutch Wikipedia.This dataset contains 924,473 rows and three columns.The first two columns represent users' ID and the third column a UNIX timestamp with the time of a message on one's Wiki talk page.For each row, the first user writes a message on the talk page of the second user at a timestamp given in the third column.The evolution of the Dutch Wiki talk network is plotted in Figure 3.6.We adjust all timestamps according to the Central Europe Time Zone.Different from the Facebook wall post data, the Dutch Wiki Talk data displays an accelerating growth between 2003 and 2007 but starting from 2008 the growth pattern is close to linear.
Similar to the Facebook case, after accounting for the poor fit of a SEPP model with a power-law triggering function (2.3), we proceed with an exponential triggering function as in (2.2).We first divide the Dutch Wiki talk data from 2003-01-01 to 2005-12-31 into 151 disjoint time windows, each of which contains 500 events, and fit the SEPP model to each time window using the algorithm outlined in Section 2.2 with p = 150 and ∆ = (1/24)/12.We denote estimated model parameters by θ with three components as detailed in (3.5).With θ, we compute the transformed time Λ(T i ), i = 1, 2, . . ., n as in (3.2) and use a KS test to examine whether the transformed IETs (calculated using (3.6) and (3.7)) are exponentially distributed with rate 1.Out of the 151 sets of estimates, the acceptance rate for the KS test is 30.7% with α = 0.05 and 46.7% with α = 0.01.Also, the ACF plots (not included) do not suggest high autocorrelation among the transformed IETs.
This acceptance rate looks lower than for the Facebook case but now 500 events are included in each estimation interval as opposed to 200 events for Facebook.With a larger sample size, we are more likely to have abnormal IETs that deviate from our model and hence reject the null.Meanwhile, the Dutch Wiki case also has a similar problem as for Facebook.At the beginning of the network recording period, it takes a long time to accumulate 500 events (10 days or more for the first 22 time windows), which again increases the possibility of having abnormal observations.The bigger sample size, slow network startup and the sensitivity of the KS test to outliers encourage us to proceed assuming the SEPP model for this time period.
Estimated values of parameters are given in Figure 3.7, where the left, middle and right panels give ( η, K, β), respectively.From the left plot of Figure 3.7, we observe a positive trend in η with larger variation after 100 time windows (2005-09-07).Meanwhile, β fluctuates more after the 46-th time window (2005-02-09) and K jumps out of the stationary range (0, 1) after 100 time windows.When K is negative, it is compensated by a large positive η.These facts indicate the model increasingly struggles to fit the data as time progresses.
We further extend the estimation procedure to the data from 2006-01-01 to 2006-12-31, namely 236 disjoint time windows each of which has 500 points.The same KS tests are applied to transformed IETs and now with α = 0.05, the acceptance rate decreases to 19.7% and 31.4% when α = 0.01.Estimated parameter values are plotted in Figure 3.8.We see from Figure 3.8 that η fluctuates more and a large positive η is likely to be accompanied by a negative K. Estimated values for β are also volatile and sometimes become very close to All IETs associated with edges created between 0-9 AM each day are marked as red segments in Figure 3.9.We see that after the SEPP model starts to return a poor fit in 2006, longer IETs are often associated with the time period when people usually sleep and are therefore inactive on networks.Table 3.2 summarizes the burstiness parameters where B all is computed using all timestamps and we calculate B active by excluding timestamps occurring during 0-9 AM on each day.We see that the point process is progressively less bursty after timestamps between 0-9 AM are removed.
Based on the evidence of decreasing burstiness in Table 3.2 and also Figure 3.9, we exclude timestamps created between 0-9 AM on each day after 2005-02-01 and fit a NHPP to the cleaned data with a constant rate within each day.We plot estimated daily rate parameters in Figure 3 We use a KS test for a null hypothesis of exponentiality applied to the daily cleaned data.We summarize the acceptance rate of the KS test in Table 3.3 with α = 0.01, 0.05.We see that the acceptance rate of the KS test is increasing over each year indicating an improvement in the Poisson fit.Hence, for the Dutch Wiki talk network from 2005 to 2008, we can model the daily point process without the points during 0-9 AM as a NHPP but the estimated rate parameters are in general increasing.After 2008, the Poisson fit to the cleaned data is good and the estimated rate parameters stay relatively stable.

Summary of Dutch Wiki talk data
From the analysis of the Dutch Wiki talk data, we conclude that the growth pattern has three phases:  by a SEPP with an exponential triggering function (2.2).This is referred to as the startup phase of the network.
2. From February 2005 onward, we model the growth pattern by a NHPP with constant rate across a day but changing rates from day to day, plus a daily inactive cycle from 1-9 AM.Noticeably from Figure 3.10, the daily rates fluctuate more and keep increasing from early 2005 to 2007, which can be regarded as as the growing phase.Large fluctuations may be due to abnormal observations in the data, and the large fluctuations may account for the lower acceptance rate from the KS test; see Table 3.3.

Starting from 2008
, the Poisson rate estimates (after removing timestamps between 1-9 AM) become relatively stable, and we think of it as the stable phase of the network growth.
In the interests of parsimony, one could try to fit an intensity function to the plot in Figure 3.10 of the NHPP model used in the growing and stable phases.For instance, by looking for common trends in the rate estimates within a week.We give a brief discussion later in Section 4.2.However, we have not pursued this carefully.

German Wiki Talk
In this section, we consider the Wikipedia Talk data in German, which takes a similar format as in the Dutch case, but the dataset is much larger.We first clean the data by removing all edges created by "administrators" and "bots".This is done by [27], where users of the German Wikipedia are classified into three different groups, namely "normal users", "administrators" and "bots".We again look at the data from 2003-01-01 to 2013-12-31.
The cleaned dataset contains 5,863,373 rows and three columns.We speculate the larger size of German Wikipedia vs Dutch Wikipedia data is a consequence of the much larger number of fluent German speakers worldwide (90-95 million according to [34]) compared to Dutch speakers (29 million by [33]).Figure 3.11 displays the evolution of this network, where the cumulative number of edges is plotted in the left panel and the number of new edges added per week is given in the right panel.All timetamps are adjusted to the Central Europe Time Zone.The shape of the graph looks very similar to the Dutch case, but the German Wiki Talk data grows faster.
We start by looking at the data from the first two years, namely from 2003-01-01 to 2004-12-31, and divide the data into 196 disjoint time windows each of which includes 500 points.Fitting a SEPP model with an exponential triggering function (using the method in Section 2.2 with p = 150 and ∆ = (1/24)/12) to data in these 196 time windows gives estimates of θ as in Figure 3.12.Subsequent to the 82-nd time window (2004-07-19), K becomes more volatile and often a negative K is offset by a large positive value of η.Fluctuations of the estimates β also increase after the 82-nd time window and sometimes the estimated value is close to zero.These observations from Figure 3.12 suggest the SEPP model struggles to fit the data after the 82-nd time window.In fact, the acceptance rate of the KS test which examines whether the transformed IETs (calculated using (3.6)  6% for α = 0.05 and 47.6% for α = 0.01.At the beginning of the time record, longer time periods are required to accumulate 500 events (10 days or more for the first 16 time windows) and so we are unlikely to obtain good fits and in light of this observation we conclude that the SEPP model gives an acceptable fit for the network growth until July 2004.We change the model for data after 2004-07-19.Similar to the Dutch Wiki case, a repeating pattern is observed in the growth of the German Wiki talk network.Red line segments correspond to edges created during 0-9 AM on each day.From the middle and right panels of Figure 3.13, as expected, we see a repeating pattern and users are less active from 0 to 9 AM on each day.This agrees with the findings in the Dutch Wiki case.
For the data from 2004-07-01 to 2013-12-31, we remove all timestamps during 0-9 AM and fit a NHPP model to the cleaned data such that the rate parameters are constant across the day but vary from day to day.The acceptance rate for the KS test assessing the Poisson fit is reported in Table 3.4.In general, fits after 2005 are better then before and on average, the acceptance rate is 35.9% when α = 0.05 and 58.2% when α = 0.01.The estimated rate parameters (using reduced daily data) are plotted in Figure 3.14.The rate estimates  3.4.This may be due to abnormal IETs during that particular time period which cannot be caught by the Poisson model.

Summary of German Wiki talk data
The conclusion for the German Wiki data is similar to the Dutch case.The network is in startup phase until July 2004 during which the growth can be modeled by a SEPP with an exponential triggering function.After the startup phase ends, edge creation can be viewed as a NHPP with constant rate within a day but different rates from day to day, plus a daily inactivity time period during which users are presumably asleep.
The difference between the German Wiki data and the Dutch one is the growth rate of the network.Although both were founded in 2001 ( [31,32]), German Wikipedia has more articles and the total number of German speakers (90-95 million) worldwide is also much larger than that of Dutch speakers (29 million) ( [33,34]).These may accelerate the growth of the network and shrink the length of the startup phase.

Math Overflow
After seeing three regional communication networks that are geographically localized, for contrast we investigate another network which is not regional.The MathOverflow data includes interactions from the "StackExchange" site "MathOverflow".The network was launched September 28, 2009.Types of interactions on MathOverflow include answering a question from another user, commenting on another user's question and commenting on another user's answer.In the graphical representation of this network, users are nodes and online interaction between two users is regarded as a directed edge.The dataset contains 506,550 edges (rows) and covers the time period from 2009-09-28 to 2016-03-06.There are three columns in the dataset.The first two columns represent users' ID and the third column is a UNIX timestamp with the time at which an interaction is made.For each row, the first user interacts with the second user in one of the three ways given above.Since the MathOverflow website can be accessed by users worldwide, there is no obvious choice of a particular time zone to adjust the timestamps associated with the creation of each edge.We simply set the time zone to be the US Central Time and Figure 3 of Figure 3.16) are negative.When inspecting the ACF plots for transformed IETs (not included here), we see large autocorrelations, which raises the concern on the goodness of fit for the SEPP model.So here even though the KS test has a reasonable acceptance rate, we think the SEPP is not suitable to model the early-stage growth of the MathOverflow.Given that no clear repetitive daily inactivity period has been observed, we try fitting a NHPP to the timestamps such that IET data within one day is tested against an exponential null with a constant rate, but the rate parameter varies from day to day.The rate parameter of the Poisson process is estimated using MLE.The acceptance rates of the KS test for the MathOverflow data from 2010-01-01 to 2015-12-31 are summarized in Table 3.5.The acceptance rates stay very high for each year but at the same time the autocorrelation among IETs (ACF plots not included here) is also significant, indicating more analysis could be conducted to try to account for the correlation.
Therefore, we conclude that the SEPP + NHPP model does not provide a good fit for the MathOverflow data and the growth pattern for this network which is not geographically concentrated is different from the three regional networks we looked at.

Common growth pattern
In this paper, we study four social network datasets, three of which are geologically concentrated, namely Facebook wall posts for users in New Orleans, Dutch and German Wikipedia talk messages.The MathOverflow data does not have the geological concentration property.After comparing these four datasets, we see that regional social networks display some common evolution pattern: 1. Startup phase: When the network is building up, the network evolution can be modeled by a SEPP with an exponential triggering function.During this period, longer IETs are not necessarily associated with expected daily inactive periods for users.Meanwhile, the length of the startup phase varies and may depend on the number of users of the network.For instance, in the German Wiki talk case, the startup phase disappears within roughly 1.5 years after which the SEPP model starts to fit poorly.In contrast, the startup phase in the Dutch Wiki case lasts for more than two years.The different startup lengths may be a consequence of the difference in the number of possible users of the network.According to Wikipedia [33,34], the total number of German  speakers worldwide is about 90-95 million whereas that of Dutch speakers in the world is around 29 million.More available users have the potential to accelerate the growth of the network, making the startup phase disappear more quickly.
2. Growing phase: After the network has attracted some attention, it enters a growing phase and the daily evolution can be modeled by a NHPP, where the Poisson rate changes but stays constant across each day, plus a nightly inactive period.The estimated rate parameters of the Poisson process while not monotone are trending upwards during the growing phase.
3. Stable phase: The regional network becomes relatively saturated and the the overnight inactivity persists.After reaching the stable phase, the rate parameters from the Poisson process remain stable unless some new external stimulus is brought to the network.For example, in the Facebook case, we observe a sudden increase in the estimated rate parameters since mid-2008.This may be due to speculated in Facebook's release of a new site design on July 20, 2008 [29] .
This three-stage growth pattern does not apply to networks that are not geographically concentrated, e.g. the MathOverflow data.Although the acceptance rate of a Poisson model is high, the serial dependence among IETs is not negligible and should be accounted for.Red line segments refer to edges created during 0-9 AM each day, based on US Central Time.Unlike the regional networks, the repeating pattern with a nightly inactivity period is not observed here, perhaps because the MathOverflow website is used by users worldwide and thus is not regional.

Future direction
This study gives rise to open questions for future research.
• How can we model the three-phase network evolution theoretically?Interactions among individual users in a network are sometimes modeled by a multivariate Hawkes process, where users are influenced by neighbors, authorities and communities (cf.[6,38]).However, considering the whole network evolution, the transition from SEPP to NHPP has not been carefully analyzed.From simulation results, simply superposing SEPPs does not always give a process that is close to Poisson, and the transition seems to depend on the choice of parameters in the SEPP.
• How can we find a more parsimonious model for regional networks in growing or stable phases?One possible way is to investigate any common trend within a week.For example, Figure 4.1 collects the MLE's of daily Poisson rates in 2012 based on the German Wiki data and groups the estimates by week.Labels 1-7 on the x-axis corresponds to Monday -Sunday in a week, respectively.The red solid line highlights the average rate estimates on each day of the week.A decreasing trend is observed, suggesting that people are less active on the German Wiki talk pages during weekends.Hence, one may consider smoothing the daily Poisson rate estimates by taking such weekly pattern into account.Estimates are grouped by week so 1-7 labels on the x-axis represents Monday -Sunday in a week, respectively.The red solid line marks the average rates on each day of the week.We see a decreasing trend over the week, indicating that users are less active during weekends.

Figure 3 . 1 :
Figure 3.1: Facebook wall posts for users in New Orleans.Left: Cumulative number of edges over time.Right: number of new edges added per week.

Figure 3 . 2 ,Figure 3 . 2 :
Figure 3.2: Events observed from the Facebook wall post data from 2004-10-24 to 2004-12-23.(top) and a homogeneous Poisson process with unit rate (bottom).In both cases, 65 events are included and the Facebook points are more bursty compared with Poisson points.

Figure 3 . 3 :
Figure 3.3: Estimated values of θ = η, K, β for the Facebook data from 2005-01-01 to 2006-08-20.Each set of parameter estimates is calculated using 200 points.The model struggles to fit after the 64-th time window.

Figure 3 .
Figure 3.3 summarizes the parameter estimates θ (cf.(3.5)) over the total 110 disjoint time windows covering the period from 2005-01-01 to 2006-08-20.Within each time window, 200 points are included.We notice that after the 64-th time window, K starts to take negative values, which deviates from the model assumption in (2.2).Moreover, after the 64-th time window, negative values of K are offset by large positive values of η.For the two negative incidences of K before the 64-th time window, the magnitude of K is small (very close to 0) and the KS test does not accept the model.Therefore, we conclude that changing the underlying model is necessary from the 64-th time window onward (i.e. after 2006-05-24) and the SEPP with an exponential triggering function can only be used to model

Figure 3 . 6 :
Figure 3.6: The evolution of the Dutch Wiki Talk data.Left: Cumulative number of edges over time.Right: Total number of new edges added per week.

Figure 3 . 7 :
Figure 3.7: Left to right: Estimates for η, K, β from the Dutch Wiki talk data 2003-01-01 to 2005-12-31.Each set of estimates are computed using 500 points.After 45 disjoint time windows, estimates for β are more volatile and after 100 windows estimates for K occasionally jump outside of the (0, 1) range.

Figure 3 . 8 :
Figure 3.8: Left to right: Estimates for η, K, β based on the Dutch Wiki talk data from 2006-01-01 to 2006-12-31.Each set of estimates are computed using 500 points.The three estimated parameters are quite variable.A large positive η is associated with a negative K, which indicates problematic fit.
.10 based on data from 2005-02-01 to 2013-12-31.The Poisson rate estimates are increasing until 2008 and remain relatively stable from 2008 onward.

Figure 3 . 11 :
Figure 3.11: The evolution of the German Wikipedia Talk network from 2003-01-01 to 2013-12-31.Left: Cumulative number of edges.Right: The number of edges added per week.

Figure 3 . 12 :
Figure 3.12: Left to right: Estimated values for η, K, β.Note that η fluctuates more after the 82-nd time window (2004-07-19) and a large η is offset by a negative K, indicating a problematic fit.

Figure 3 .
Figure 3.14: MLE's of the rate parameter when a NHPP is fitted to daily cleaned IETs (excluding IETs during 0-9 AM) based on the German Wiki talk data from 2004-07-01 to 2013-12-31.

Figure 3 . 15 :
Figure 3.15: Cumulative number of edges from the MathOverflow data during three representative time periods: (1) 2009-10-03 -2009-10-07 (left; the beginning of the network); (2) 2010-03-02 -2010-03-06 (middle; about one year after the launch of MathOverflow); (3) 2012-11-12 -2012-11-16 (right; about two years after the launch of MathOverflow).Red line segments refer to edges created during 0-9 AM each day, based on US Central Time.Unlike the regional networks, the repeating pattern with a nightly inactivity period is not observed here, perhaps because the MathOverflow website is used by users worldwide and thus is not regional.

Figure 4 . 1 :
Figure 4.1: MLE's of daily Poisson rates for the German Wiki data in 2012.Estimates are grouped by week so 1-7 labels on the x-axis represents Monday -Sunday in a week, respectively.The red solid line marks the average rates on each day of the week.We see a decreasing trend over the week, indicating that users are less active during weekends.
For a long time duration, it is possible to observe abnormal IETs which leads to rejections by the KS test.After accounting for this fact, we think the 64.8% acceptance rate indicates a good fit of the SEPP model.The analysis is then extended to the next 100 disjoint time windows each of which includes 200 points and this covers the time period from 2006-05-01 to 2006-09-22.However, the estimation algorithm in Section 2.2 fails when applied to data subsequent to the 57-th time window, indicating a change in the underlying model should be made.Looking at the the first 56 time windows (from 2006-05-01 to 2006-08-20), we see that the acceptance rate of the KS test is 40% with α = 0.05 and 56.4% with α = 0.01.So 2006-05-01 to 2006-09-22 is a period where a model change seems to take place.

Table 3 .
The line segments in red correspond to the accumulation of new edges from 1 to 8 AM each day, during which we expect sleep interferes with Facebook activity, and the timestamps are adjusted according to the US Central Time Zone.
3. 2007-05-11 to 2007-05-20, where a common repetitive pattern is observed.1: Burstiness parameters for IETs from 2005 to 2008, where B all is calculated using all IETs and B active is obtained by excluding all IETs that happen during 1-8 AM on each day.Since 2006, we see a decrease in the burstiness parameter after removing IETs during users' inactive hours.
2. Starting from May 2006, we observe a repetitive growth pattern.Users are less active on the network during 1-8 AM when we expect users are asleep.After removing timestamps occurring between 1-8 AM, the reduced point process becomes less bursty and can be modeled by a NHPP where the Poisson rate is constant across a day but changing as we move from day to day. 3. From mid 2007 to mid 2008, the daily estimates of the Poisson rate parameter come to a relatively stable phase but start increasing again after mid 2008.The authors of [29] speculate the increase after mid 2008 as a result of the release of Facebook's new design on the wall posts.

Table 3 .
2: Burstiness parameters for IETs of the Dutch Wiki talk data from 2003 to 2007, where B all is computed using all IETs and we calculate B active by excluding all IETs occurring during 0-9 AM on each day.Excluding IETs from 0-9 AM decreases the burstiness parameter.
1.The point process is bursty up until early 2005 and network growth is well-modelled

Table 3 .
and (3.7)) are exponentially distributed with unit mean is 25.1% when α = 0.05 and 34.2% when 3: Acceptance rate (%) of the KS test for a NHPP model fitted to the Dutch Wiki talk data from 2005-02-01 to 2013-12-31.We assume the rate parameter is constant across the day but varies from day to day.The fit improves since year 2008.α = 0.01, if all 196 time windows are taken into consideration.However, if only focusing on the first 81 time windows, the acceptance rate of the same KS test increases to 36.

Table 3 .
4: Acceptance rates (%) of the KS test for a NHPP model fitted to the German Wiki talk data from 2004-07-01 to 2013-12-31.The fit improves subsequent to year 2005.On average, the acceptance rate is 35.9% when α = 0.05 and 58.2% when α = 0.01.

Table 3 .
5: Acceptance rate (%) of the KS test for a NHPP model fitted to the MathOverflow data from 2010-01-01 to 2015-12-31.The acceptance rate is high, but autocorrelation between IETs is not negligible.