Network modeling plays a critical role in identifying statistical
regularities and structural principles common to many systems. The large
majority of recent modeling approaches are connectivity driven. The structural
patterns of the network are at the basis of the mechanisms ruling the network
formation. Connectivity driven models necessarily provide a time-aggregated
representation that may fail to describe the instantaneous and fluctuating
dynamics of many networks. We address this challenge by defining the activity
potential, a time invariant function characterizing the agents' interactions
and constructing an activity driven model capable of encoding the instantaneous
time description of the network dynamics. The model provides an explanation of
structural features such as the presence of hubs, which simply originate from
the heterogeneous activity of agents. Within this framework, highly dynamical
networks can be described analytically, allowing a quantitative discussion of
the biases induced by the time-aggregated representations in the analysis of
dynamical processes.Comment: 10 pages, 4 figure

A Lloyd

A Vespignani

A-L Barabási

C Butts

C Castellano

D Balcan

D Brockmann

D Centola

D Lazer

E Volz

G Ghoshal

H-H Jo

I Schwartz

J Moody

J-P Onnela

L Isella

LB Shaw

M Boguña

M Molloy

M Morris

MC González

MEJ Newman

O Frank

P Erdös

P Holland

R Albert

S Boccaletti

S Fortunato

S Wasserman

SN Dorogovtsev

WO Kermack

Scientific Reports

English

arXiv

Network modeling plays a critical role in identifying statistical regularities and structural principles common to many systems. The large majority of recent modeling approaches are connectivity driven. The structural patterns of the network are at the basis of the mechanisms ruling the network formation. Connectivity driven models necessarily provide a time-aggregated representation that may fail to describe the instantaneous and fluctuating dynamics of many networks. We address this challenge by defining the activity potential, a time invariant function characterizing the agents' interactions and constructing an activity driven model capable of encoding the instantaneous time description of the network dynamics. The model provides an explanation of structural features such as the presence of hubs, which simply originate from the heterogeneous activity of agents. Within this framework, highly dynamical networks can be described analytically, allowing a quantitative discussion of the biases induced by the time-aggregated representations in the analysis of dynamical processes.Postprint (published version

Perra, N.

Gonçalves, B.

Pastor Satorras, Romualdo

Vespignani, Alessandro

UPCommons. Portal del coneixement obert de la UPC

Supplementary InformationActivity driven modeling of time varying networksN. Perra, B. Gonçalves, R. Pastor-Satorras, A. VespignaniMay 11, 2012Contents1 The Model 11.1 Integrated network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Epidemic threshold 42.1 Casem = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Casem > 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Persistence of links 54 Dataset Details 64.1 PRL Dataset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64.2 Twitter Dataset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.3 IMDb Dataset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 The ModelLet us considerN initially disconnected individuals/nodes. We define for each agent i the activitypotential xi as the probability that any social act/connection in the system has been engaged bythe actor i. We define F(x) as the probability distribution that a randomly chosen agent, i, hasactivity potential x. To avoid possible divergences of the distribution close to 0 we fix a bound byimposing x ∈ [, 1]. Given the activity potential xi is natural to define the activity rate ai as:ai = ηxi, (1)where η is a rescaling factor defined such that the average number of active nodes per unit timein the system is η〈x〉N. At each time step t the network Gt is build starting from N disconnectedvertices (all the edges in the network are deleted). The links are created in the following way:• With probability ai∆t the vertex i becomes active (fires) and generates m links that are con-nected tom other vertex selected randomly1N. Perra, B. Gonçalves, R. Pastor-Satorras, A. Vespignani 2• With probability 1 − ai∆t the node will not fire, but can however receive connections fromother active vertices.1.1 Integrated networkWe define the integrated network G as the union of all networks obtained in each time step,G =t=T⋃t=0Gt. (2)Multiple edges or self-links are not allowed. For a given distribution of activity potential, F(x),we are interested in computing the degree distribution of the integrated network at time T , PT (k).Let us consider the number τ = T/∆t of instantaneous networks generated up to time T . Thenumber of times that the vertex iwill be active is given by a binomial distribution with an averageτai∆t = Tai. At time T the average number of active nodes will be∑iTai = TN〈a〉, (3)while in the single instantaneous network will beNt = N〈a〉∆t ≡ Nη〈x〉∆t. (4)Each active node will createm links. Therefore, the average number of edges per unit time will beon average:Et = mNη〈x〉 (5)leading to an average degree per unit time〈k〉t = 2EtN= 2mη〈x〉. (6)The instantaneous network will be composed by a set of stars, the vertices that were active at thattime step, with degree larger than or equal tom, plus some vertices with low degree. The structureof the integrated network will be however much complex. Consider this integrated network attime T . The degree of each one of its nodes can be written as kT (i) = koutT (i) + kinT (i), where theout-degree koutT (i) corresponds to the links emanating from i due to its becoming active, while thein-degree kinT (i) is due to the links that have arrived to i from other active nodes. Let us focus onthe out-degree. In the time interval T , node iwill have tried to send in average Tmai edges. Not allthose edges will contribute to its integrated out-degree, just only those arriving to different nodes(no multiple links are allowed in the integrated network). The out-degree can thus be computedby making an analogy to the Polya urns problem: it will be equal to the number of different ballsextracted from a urn with N balls, performing Tmai extractions. The probability of extracting dballs will be given byP(d) =(Nd)pd(1− p)(N−d) (7)Activity driven modeling of dynamic networksN. Perra, B. Gonçalves, R. Pastor-Satorras, A. Vespignani 3wherep = 1−(1−1N)Tmai(8)is the probability of extracting at least one ball in the urn. The average value is then〈d〉 = pN. (9)Therefore the average out-degree of a vertex i in the integrated network can be estimated askoutT (i) = N(1− e−Tmai/N) (10)in the limit of large N and small T/N.The in-degree will come from the rest of vertices that are active and have sent connections to i,without receiving them from it. In T time steps, vertex i will have fired Tai times in average. Theprobability that a vertex has not received any connection form iwill then be(1−1N)mTai' exp[−mTai/N], (11)where we assume again large N and small T/N. The average number of vertices that have firedand are not connected to iwith and egde that emanated from iwill then be TN〈a〉 exp[−mTai/N].These nodes have m possibilities to reach at random i, each with probability 1/N. The numberof connections that have reached this node will then be on average mTη〈x〉 exp[−mTai/N]. Thedegree of the node iwill be simply the sum of these to contributions:kT (i) = koutT (i) + kinT (i) = N(1− e−Tmai/N) + Tmη〈x〉e−Tmai/N (12)= N[1−(1−mη〈x〉 TN)e−Tmai/N)]∼ N(1− e−Tmai/N) = N(1− e−Tmηxi/N), (13)again for small T/N. From the last relation we can write the activity potential x as an effectivefunction of the integrated degree k, i.e.:x(k) = −NηmTln(1−kN). (14)At the probabilistic level we can use the relation PT (k)dk ∼ F(x)dx, where P is the degree distribu-tion of the integrated, network after T time steps, to finally obtainPT (k) ∼ F[x(k)]dx(k)dk=1Tmη11− kNF[−NηmTln(1−kN)]. (15)In the limit of small k/N (for not very large values of T ) we can expand the logarithm, to obtainthe simplified expressionPT (k) ∼1TmηF[kTmη]. (16)That is, we obtain an important relation binding the functional form of the degree distribution tothe activity distribution of the nodes.Activity driven modeling of dynamic networksN. Perra, B. Gonçalves, R. Pastor-Satorras, A. Vespignani 42 Epidemic thresholdLet us consider the SIS epidemic compartmental model, characterized by a transition probabilityλ and a recovery time µ−1, spreading in the dynamical network generated as discussed above. Letus assume a distribution of activity a of nodes given by a general distribution F(x) as before. At amean-field level, the epidemic process will be characterized by the number of infected individualsin the class of activity a, at time t, namely Ita.2.1 Casem = 1The number of infected individuals of class a at time t+ ∆t given by:It+∆ta = −µ∆tIta + Ita + λ(Nta − Ita)a∆t∫da ′Ita ′N+ λ(Nta − Ita)∫da ′Ita ′a′∆tN, (17)whereNa is the total number of individuals with activity a. In Eq. (17), the third term on the rightside takes into account the probability the a susceptible of class a is active and get the infectiongetting a connection from any other infected individual (summing over all different classes), whilethe last term takes into account the probability that a susceptible, independently of his activity,gets a connection from any infected active individual. Now summing on all the classes we get(ignoring the second order terms):∫daIt+∆ta = It+∆t = It − µ∆tIt + λ〈a〉It∆t+ λθt∆t, (18)where θt =∫da ′Ita ′a′. We can get another expression multiplying both sides of Eq. (17) by a andintegrating, to obtainθt+∆t = θt − µθt∆t+ λ〈a2〉It∆t+ λ〈a〉θt∆t. (19)In the limit ∆t→ 0, we can write Eqs. (17) and (19) in a differential form:∂tI = −µI+ λ〈a〉I+ λθ, (20)∂tθ = −µθ+ λ〈a2〉I+ λ〈a〉θ. (21)The Jacobian matrix of this set of linear differential equations takes the formJ =(−µ+ λ〈a〉 λλ〈a2〉 −µ+ λ〈a〉),and has eigenvaluesΛ(1,2) = λ〈a〉− µ± λ√〈a2〉. (22)The epidemic threshold is obtained requiring the largest eigenvalues to be larger the 0, which leadsto the condition for the presence of an endemic state:λµ>1〈a〉+√〈a2〉 +O( 1N ) (23)Activity driven modeling of dynamic networksN. Perra, B. Gonçalves, R. Pastor-Satorras, A. Vespignani 5The order 1/N is present because we are not considering events in which two infected nodeschoose each other for connection.An important epidemiological quantity is the reproductive number R0, defined as the averagenumber of secondary cases generated by a primary case in an entirely susceptible population. Fora SIS model we haveR0 =βµ≡ λ〈k〉µ(24)where β = λ〈k〉 is the per capita spreading rates that takes into account the rate of contacts of eachindividual. Considering this in the equation (23) at the first order we get as a threshold in terms ofthe reproductive numner:R0 > RC0 =2〈a〉〈a〉+√〈a2〉 = 2〈x〉〈x〉+√〈x2〉 = 21+√ 〈x2〉〈x〉2 (25)2.2 Casem > 1In this general case we can use the same machinery as above. We have just to add the fact thateach active agent can now make more then one connection (m at most) at each trial. In this casethe probability to get a connection changes:1N→ 1− (1− 1N)m∼mN(26)Using this fact, we can work out the corresponding set of differential equations, whose Jacobianmatrix is now:Jm =(−µ+ λm〈a〉 λmλm〈a2〉 −µ+ λm〈a〉),with eigenvaluesΛ(1,2) = mλ〈a〉− µ± λm√〈a2〉. (27)That is, the eigenvalues in the casem > 1. Thus, the epidemic threshold in the general case can besimply written asR0 > RC0 =1m2m〈a〉〈a〉+√〈a2〉 = 2〈x〉〈x〉+√〈x2〉 (28)3 Persistence of linksIn the model, at each time step a random markovian network is created. There is no memory.Each link created at time step t − ∆t is deleted and recreated randomly at time step t. This isan oversimplification of real social interaction. In the real world we can imagine that individualsestablish social connections preferentially within an limited circle of friends, and that they have amemory. This implies that a certain number of connections are active for more than one single timewindow ∆t, they are persistent. In Figure S1 we show the distribution of persistence for the APSdataset, in Figure S2 for Twitter and in Figure S3 for the IMDb. As it becomes clear from the plots,Activity driven modeling of dynamic networksN. Perra, B. Gonçalves, R. Pastor-Satorras, A. Vespignani 61 10 100∆t100102104106ρ(∆t)Figure S1: For the APS database we plot the distribution of connections active for ∆t consecutive time steps:persistence of links. In this case one time window correspond to one yeara relevant number of connections stay active across different time windows. In our model weconsider instead a mean-field approach where temporal correlations between users are neglected.However it is easy to generalize our model in order to take into account the persistence of linksand its effect in the spreading of an epidemic disease. This will be a matter of future work.4 Dataset DetailsWe study three different dataset: the collaborations in the journal "Physical Review Letters“ (PRL)published by the APS1, the message exchanged on Twitter and the activity of actors in movies andTV series as recorded in the Internet Movie Database (IMDb)2.4.1 PRL DatasetIn this database the network representation considers each author of a PRL article as a node. Anundirected link between two different authors is drawn if they collaborated in the same article.We filter out all the articles with more than 10 authors in order to focus our attention just on smallcollaborations in which we can assume that the social components is relevant. We consider theperiod between 1960 and 2004. In this time window we registered 71.583 active nodes and 261.553connections among them.In this dataset is natural defining the activity rate, a, of each author as the number of paperswritten in a specific time window ∆t.1The data are available here: http://prx.aps.org/node/39662The data are available here: http://www.imdb.com/interfacesActivity driven modeling of dynamic networksN. Perra, B. Gonçalves, R. Pastor-Satorras, A. Vespignani 7100 101 102∆t100102104106ρ(∆t)Figure S2: For the Twitter database we plot the distribution of connections active for ∆t consecutive timesteps: persistence of links. In this case one time window correspond to one day100 101 102∆t100102104106108ρ(∆t)Figure S3: For IMDb we plot the distribution of connections active for ∆t consecutive time steps: persistenceof links. In this case one time window correspond to one year.Activity driven modeling of dynamic networksN. Perra, B. Gonçalves, R. Pastor-Satorras, A. Vespignani 8100 101 102k10-610-410-2100P(k)1   day5   days10 days30 daysFigure S4: Considering the subset of users active on the first 31 days of our twitter dataset we plot the degreedistribution of the resulting networks for different integrating time windows.4.2 Twitter DatasetHaving been granted temporary access to Twitter’s firehose we mined the stream for over 6months to identify a large sample of active user accounts. Using the API, we then queried forthe complete history of 3 million users, resulting in a total of over 380 million individual tweetscovering almost 4 years of user activity on Twitter. In this database the network representationconsiders each users as a node. An undirected link between two different users is drawn if theyexchanged at least one message. We focus our attention on 9 months during 2008. In this timewindow we registered 531.788 active nodes and 2.566.398 connections among them.In this dataset we define the activity rate of each user as the number of messages sent in a timewindow ∆t. In figure S4 we show the degree distribution of the subgraph obtained by integratingover 1, 5, 10, 30 days. The subset of nodes considered are those active in the the first months ofthe dataset (January 2008). It is clear how the network integrated in one day has a sparse nature.Increasing the time window heterogeneous connectivity patterns start to emerge.4.3 IMDb DatasetIn this database the network representation considers each actor as a node. An undirected linkbetween two different actors is drawn if they collaborated in the same movie/TV series. We focuson the period between 1950 and 2010. During this time period we registered 1.273.631 active nodesand 47.884.882 connections between them.A natural way to define the activity rate in this dataset is to consider the number of movies actedby each actor in a specific time window ∆t. In figure S5 we show the degree distribution of thesubgraph obtained by integrating over 1, 5, 10, 30 years. The subset of actors is formed by thoseactive in the period 1970 − 1980. Even in this case is clear how increasing the integrating timewindow the level of heterogeneity increases.Activity driven modeling of dynamic networksN. Perra, B. Gonçalves, R. Pastor-Satorras, A. Vespignani 9100 101 102 103k10-810-610-410-2100P(k)1   year5   years10 years30 yearsFigure S5: Considering the subset of actors active during the period 1970 − 1980 in the IMDb data we plotthe degree distribution of the resulting networks for different integrating time windows.Activity driven modeling of dynamic networks

Activity driven modeling of time varying networks

Network modeling plays a critical role in identifying statistical regularities and structural principles common to many systems. The large majority of recent modeling approaches are connectivity driven. The structural patterns of the network are at the basis of the mechanisms ruling the network formation. Connectivity driven models necessarily provide a time-aggregated representation that may fail to describe the instantaneous and fluctuating dynamics of many networks. We address this challenge by defining the activity potential, a time invariant function characterizing the agents' interactions and constructing an activity driven model capable of encoding the instantaneous time description of the network dynamics. The model provides an explanation of structural features such as the presence of hubs, which simply originate from the heterogeneous activity of agents. Within this framework, highly dynamical networks can be described analytically, allowing a quantitative discussion of the biases induced by the time-aggregated representations in the analysis of dynamical processes

Activity driven modeling of time varying networks

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