A generative model of urban activities: simulating apopulation

Rory McGrathUniversity Of California,

Berkeleyrorymcgrath@berkeley.edu

Alexey PozdnukhovUniversity Of California,

Berkeleyalexeip@berkeley.edu

ABSTRACTThis work presents a data-driven generative model of a socialnetwork in a city. It takes both mobility and social contextinto account by building a generative process of establish-ing social connections that is governed by latent profiles ofcity dwellers. These profiles influence both the communitystructure of the network as well as preferences in destinationchoices. Therefore, the model captures an interplay betweenmobility and the geography of the social network, and canbe used to infer social connections from the revealed mobil-ity pattern and vice versa. The model is capable of inferringpopulation profiles and community structure of urban areasfrom partly observed social network and mobility traces, andcould then be applied to simulate a representative sample ofthe entire population. The results are verified in terms ofdescriptive statistics of the network structure as well as onthe out-of-sample subset of real data collected via TwitterAPI in San Francisco, CA.

General TermsAlgorithms, Generative modeling, Networks

KeywordsGenerative Model, Social Network, Prediction, Machine Learn-ing

1. INTRODUCTIONCurrently data availably from check-in services only tellshalf a story. This is not a fault of the service but, besidesbehavioral and demographic bias, arises from users beinginfluence by people not observed by the system. These la-tent forces can cause observed users to appear unpredictableor act in peculiar ways making reliable prediction and com-munity detection somewhat challenging. If, however, theselatent individuals were observed the patterns and motiva-tion behind all users actions would become more apparentmaking it easier to both predict and apply semantic meaning

Permission to make digital or hard copies of all or part of this work forpersonal or classroom use is granted without fee provided that copies arenot made or distributed for profit or commercial advantage and that copiesbear this notice and the full citation on the first page. To copy otherwise, torepublish, to post on servers or to redistribute to lists, requires prior specificpermission and/or a fee.UrbCompâAZ14, August 24, 2014, New York, NY, USA location

to events. For this work we look at generating this completedataset for all inhabitants in a city.

From [3] we know that a city is made up of various commu-nities of people. In order to generate realistic tweets we firstneed to simulate this social network. Extending the workfrom [1] we expand an existing twitter social network tak-ing into account both preferential attachment and spatialpreferences.

Check-ins are created for the individuals using a modifiedLink-PLSA-LDA algorithm. Probabilities for individualsvisiting a location are calculated by considering the radi-ation model, an individuals personal preferences and theirfriends preferences.

Figure 1: Generative model for check-ins.

Tuning the parameters in our algorithm using real worlddata our model generates realistic check-ins for all users ina city. These check-ins are created each time a user visitsa location or venue with the users probability of visitinga location being determined by a weighted combination ofthe communities they are members of, their standing in eachcommunity, their loyalty to locations and the travel distancerequired to reach the venue.

2. ALGORITHM OVERVIEWIn order to generate check-ins the algorithm follows a setof elegant and logical rules. Using these rules and trainingon real-world data a set of realistic check-ins are generatedfor all individuals, both observed and latent, using a vari-ate of the Link-PLSA-LDA model. Using the terminologyof PLSA our individuals are ’documents’ with the locationof their check-ins being their ’words’. People connect to or’cite’ each other through social links/friendships. Our modelis presented in figure 1 with the following parameters beingused:

Ωd′ the probability of a friendship to person d’.β the probability of a location with respect to a topic.π the probability of topic for the set of people.θ the topic distribution of a person.

The code used in the generation process is describe in pseudocode (algorithm 1). Here check-ins are generated for eachindividual based on their topic distribution and friend con-nections.

Before creating a check-in a current representative sampleof the population is obtained. This populated is createdfrom the set of current active twitter users in San Francisco.For each person their home and work locations are inferredfrom geo-referenced tweets. A distribution over topics θp isalso assigned to each individual. This θp, generated usingcensus data, shows an individuals likelihood to participatein each topic thus representing their interest levels. Thesetopics can encompass activities such as socializing, leisure,shopping. New individuals are then generated with homelocations sampled from a distribution over residential areas.The initial twitter social network is expanded to includethese individuals using a combination of both preferentialattachment and spatial convenience.

Initial probabilities of an individual visiting each location foreach topic are calculated using a radiation model trained onvenues in San Francisco. The population of San Franciscoand the number of venues is immense, to account for thiswe created a radiation model which shows an individualsprobability of visiting an area at the granularity of the tractlevel as defined by the census.

In order to generate a check-in for a person we first find theirpotential friends. The probability of a friendship pfriend iscalculated using product of three components:

pfriend = spf × dpf × degpf (1)

The first component, spf , deals with the similarity betweenperson p’s topics of interests θp and person f ’s topics θf ,calculated using the cosine similarity. The second compo-nent, dpf , is the distance decay between person p’s homeand person f ’s home. Intuitively here we see that the closerpeople live together the more likely a friendship. The fi-nal component, degpf , is the friends f degree in the socialnetwork, the more popular the individual the more likely afriendship. Using these values and normalizing, the proba-bility distribution of all potential friends F is created for aperson p.

Once a friend is found the topic of the check-in is determined.Two types of check-ins are possible, social and non-social. Ifa check-in is non-social the topic is found by sampling fromθf . If the check-in is social the topic is found by samplingfrom θf and θp, reflecting a topic that is of interest to bothparties.

With the friend and topic determined the final part of the al-gorithm deals with finding the location of the check-in. Theprobability of each venue being chosen is calculated usingthe equation:

pcheckin = p(v|s = 0)×p(s = 0)+p(v|s = 1)×p(s = 1) (2)

Here p(s) represents the probability of a social check-in. Fornon-social check-ins the probability of location relies solelyon the individual. This probability p(v|s = 0) is calculatedusing the radiation model returning the probability of eachvenue v for the given topic t.

For social check-ins the choice of location is influenced byboth parties using a mixture of the radiation models forperson p and f and the probability of a friendship (friend-ship strength) between p and f . The results are normalizedaccordingly.

p(v|s = 1) = radpt + pfriend × radft (3)

Check-in events are then assigned to the friend. By apply-ing the check-in to the friend the power law distribution ofcheck-ins that is observed in real world data is maintained.It can be seen that the higher the chances of p and f beingfriends the greater the influence p has on f . This process isthen repeated.

3. FRAMEWORKThe algorithm proposed in this work consists of two corecomponents, the social network of the users and the spa-tial network of the venues. For this work it was decidedto use the Neo4j database to handle the social network andMongoDB to manage the spatial network.

Neo4j is a scalable robust native graph database. Our usersare represented as nodes in a graph with the edges repre-senting friendships. This highly scalable database allows usto work with populations of thousands and scales to includepopulations of millions with optimized queries to calculationan individuals degree, friends and friends of friends etc.

Figure 2: The framework of algorithm. Here individuals

are connected via a social network and locations are con-

nected via accessibility. People frequent locations based

on spatial proximity or social recommendations

Venues were saved as documents in MongoDB which in-cluded a spatial component and other meta data. MongoDBis a scalable document-orientated database and similar toNeo4j allows us to work with thousands of venues but easilyscales to accommodate millions offering various index andquery mechanisms to handle geospatial data. Venues areconnected by an accessibility network capturing the mostprobable next locations given the current venue.

Algorithm 1 pseudo code

θp = distribution over topics for person p.radpt = radiation model for person p and topic tP = generate all people in population each with θpG = initial social networkVt = all venues for topic tfor each person p ∈ P do

for each friend f ∈ P and f 6= p dopfriend ≈ spf × dpf × degpf→ spf is similarity of θp and θf→ dpf is the distance decay from phome and fhome

→ degf is the friends f degree in Gend forcreate friend distribution Fsample from F to find friend fsample from θf to find topic tfor each venue v ∈ Vt dopcheckincalculate probability of check-inpcheckin = p(v|s = 0)×p(s = 0)+p(v|s = 1)×p(s = 1)→ p(s) = probability of social check-in→ p(v|s = 0) = radft→ p(v|s = 1) = radpt + pfreind × radft

end forcreate check-in distribution Csample from C to get check-increate link ∝ pfriend

assign check-in to fend forrepeat until some metric for convergence

Users and venues are connected via spatial and social behav-iors. Spatial behaviors affecting a persons choice of venueinclude their willingness to travel, their sensitivity to traveltime and the accessibility of the venue. The social behaviorsaffecting the choice of venue include the persons willingnessto pay, or value for money of the location, the rating ofthe location and the desire of friends to visit this location.Venues are connected through an accessibility or spatial net-work showing what locations are reasonable accessible giventhe current location of the individual. This accessibility isgiven by spatial proximity and travel time.

4. DATAFor this work we focused on San Francisco and the Bayarea. Twitter data was collected for a period of one monthresulting in 1.3 million tweets and 12,000 unique users. Wefixed the venues of interest to be restaurants. Utilizing theGoogle location api restaurants in San Francisco and thesurrounding Bay Area were located.

4.1 Social network generationEach unique Twitter user with sufficient tweets (more thanten on weekdays) was identified and added as a node in theNeo4j database.

Utilizing the work from [2] and [5]each users home and worklocations were inferred. [2] built on the observation thatpeople show strong periodic behavior throughout certain pe-riods of the day. This behavior alternates between primary,“home”, and secondary,“work”, locations on weekdays. Thisdistribution was modeled with a truncated Gaussian distri-

bution parameterized by the time of the day:

NH (t)=PcH√2πσ2

H

exp[−( π12

)2(t−τH )2

2σ2H

]

NW (t)=PcW√2πσ2

W

exp[−( π12

)2(t−τW )2

2σ2W

]

and

P [cu(t)=H]=NH (t)

NH (t)+NW (t)

P [cu(t)=W ]=NW (t)

NH (t)+NW (t)

Here τH is the average time of the day when a user is athome and σH is the variance in time of day. PcH is thetime-independent probability that any given check-in wasgenerated by the “home” state. The same holds true forthe work state. Of the 12,000 unique users found 1,000 hadsufficient data to generate reasonable parameter estimatesfor their home and work locations.

Using these individuals an initial social network for the citywas created. Friend connects were determined by lookingat all reciprocal followers on twitter. Here a friendship wasdefined by a mutual following on twitter. These friends wereused to populate the social network in Neo4j and were as-signed an appropriate θ vector. Taking this as the initialsocial network, the network was then expanded.

New individuals were generated with home locations sam-pled from the distribution of residential areas obtained fromcensus data. These were added to the network using a com-bination of preferential attachment and spatial proximity.

4.2 Spatial Choice modelThe venues of interest for this work were fixed to restaurantsin San Francisco and were obtained using the Google Loca-tions api. In order to find the probability of an individualchecking into a venue the radiation model was used. Theradiation model is defined by the equation:

〈Tij〉=Timinj

(mi+sij)(mi+nj+sij)(4)

Where Tij represents the commuting fluxes, mi representsthe population at the source i, nj represents the populationat destination j, sij represents the total population in acircle of radius rij centered at i, and Ti is the total numberof commuters who start their journey at i.[4]

Figure 3: Radiation model for a user based on their

home location

5. RESULTSIn this section we combine both the social and spatial com-ponents of our framework and generate check-ins as definedby the algorithm above. The results presented below in-clude the distribution of check-ins and the distribution ofthe degree in the generated social network. Using our gen-erative process check-ins were created for the inhabitants ofSan Francisco. Link-PLSA-LDA was preformed on the gen-erated check-ins to recover any latent topics that appearedin our generated data.

Figure 4: Distribution of topic 1

Figure 5: Distribution of topic 2

Figure 6: Distribution of topic 3

The distribution of actual topics are given above. In orderto recover any communities Link-PLSA-LDA we preformedfor 3 topics and results are given below. Here we found

the topics returned adhered to the topic distributions shownabove.

Figure 7: Estimated distribution of topic 1

Figure 8: Estimated distribution of topic 2

Figure 9: Estimated distribution of topic 3

Looking at the correlation plots between estimated and realtopics we can see that out proposed algorithm does a goodjob at simulating users check-in interests.

Figure 10: Correlation of topic 1

Figure 11: Correlation of topic 2

Figure 12: Correlation of topic 3

Next we look at the distribution of the tweets generated perindividual and the distribution of the degree in the socialnetwork. Again we see that the values obtained are consis-tent with prior expectations of social networks and observedbehaviors of twitter users. Here the generative process cap-tures the property that some users are extremely active whenusing the service while others take a more passive role.

Figure 13: Distribution of generated tweets for eachindividual

Figure 14: Distribution of degree in the generatedsocial network

6. SUMMARY & CONCLUSIONThe work here presents a generative model and robust frame-work for the creation of realistic check-ins for a cities inhabi-tants which arise from friend interactions and an individualspreferences of locations.

Utilizing real-world data from San Francisco, check-ins anda social network were created for the cities inhabitants whichlooked to account for the latent forces that cause observedusers to appear unpredictable or act in peculiar ways.

Currently the parameters of this generative process are ob-tain from census data or through experimentation. Futurework includes incorporating a discriminative model to esti-mate new parameters based on real world data and itera-tively converge on the true value of these estimated parame-ters. This would allow form a more accurate representationof social networks and check-ins for cities.

7. REFERENCES[1] Barabasi, A. L., and Albert, r. Emergence of scaling in

random networks. Science 286, 5439 (1999), 509–512.

[2] Cho, E., Mayers, S. A., and Leskovec, J. Friendship andmobility: User movement in location-based socialnetworks, KDD - 2011.

[3] Lawlor, A., Coffey, C., McGrath, R., andPodzdnoukhov, A. Stratification structure of urban

habitats, PURBA - 2012.

[4] Simini, F., Gonzalez, C. M., Maritan, A., and Barabasi,A. L. A universal model for mobility and migrationpatterns. Nature 484, 10856 (2012), 96–100.

[5] Tarasov, A., Kling, A., and Pozdnoukhov, A.Prediction of user location using the radiation modeland social check-ins, UrbComp - 2013.