Relational Event Modeling:Basic Framework and Applications

Aaron Schecter & Noshir Contractor

Northwestern University

Why Use Relational Events?

• Social network analysis has two main frames of reference; static networks (long term equilibrium behaviors - ERGM) and longitudinal networks (extensive relationships which evolve over time -SIENA)

• Instead, look at actions over a shorter time frame; the sequence and timing of simple behavioral events drive the mechanisms of communication (REM)

2

ERGMs vs. Relational Events

Relational Event Models (REM) use sequential structural signatures to explain the rate at which a single event from actor a to actor b is likely to occur at any time, given any prior interaction (see Butts, 2008 and Brandes et al., 2009)

Exponential Random Graph Models (ERGMs) capture cumulative network structures within a predefined time interval (for example, see Robins & Pattison, 2001 or Wasserman & Robins, 2005)

Is a tie between A and B likely to emerge due to the

given structure?

A B

C

Is this structure likely to emerge in the given

network?

A B

C

3

Analytic Goals

1. Accurately predict who will send a message to who, given a previous history of events

2. Identify what factors impact the propensity for two individuals to communicate, such as:

a) Node attributes

b) Dyadic attributes

c) Historical patterns of communication

d) Environmental factors

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Relational Event Data

A relational event is a tuple of information𝑒 = (𝑖𝑒 , 𝑗𝑒 , 𝑘𝑒 , 𝑤𝑒 , 𝑡𝑒)

• 𝑖𝑒: the sender of the message• 𝑗𝑒: the receiver of the message• 𝑘𝑒: the type/classification of the message• 𝑤𝑒: the weight of the message• 𝑡𝑒: the timestamp of the message

In a dataset, 𝑆 = 𝑒1, … , 𝑒𝑚 is the sequence of relational events

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Simple Event Sequence

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Key Assumptions

• We assume that the likelihood of an event is independent of all other events, conditional on the previous event history.

• Define 𝐻 𝑒𝑖 = 𝑒1, … , 𝑒𝑖 as a the history up to event 𝑒𝑖

• The event history up to time 𝑡, plus environmental information 𝐺𝑡 creates the context for an event to occur

• Define 𝐴𝑡 as the set of all events that could occur at time 𝑡

• Once an event occurs the network topology and individual rates are updated, and the process restarts

7

Basic Probability Model

Given a set of parameters 𝜃 and a sequence 𝑆, we want to model

𝑃 𝑆; 𝜃 =

𝑖∈ 1,…,𝑚

𝑃 𝑒𝑖 𝐻 𝑒𝑖−1 , 𝐺𝑡𝑒𝑖; 𝜃

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Need an explicit form for this expression

Event History Approach (Blossfeld et al 1995)

Hazard Rate

ℎ𝑒𝑖 𝑡𝑒𝑖 𝑒1, … , 𝑒𝑖−1 , 𝐺𝑡𝑒𝑖; 𝜃 : the instantaneous

likelihood of event 𝑒𝑖 occurring at time 𝑡𝑒𝑖, given all available information

Survival Function

𝑆𝑒𝑖 𝑡𝑒𝑖 − 𝑡𝑒𝑖−1 𝑒1, … , 𝑒𝑖−1 , 𝐺𝑡𝑒𝑖; 𝜃 : the

probability that in the time since the last event, the event did not occur

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Single Event Likelihood

𝑃 𝑒𝑖 𝐻 𝑒𝑖−1 , 𝐺𝑡𝑒𝑖; 𝜃

= ℎ𝑒𝑖 𝑡𝑒𝑖 𝐻 𝑒𝑖−1 , 𝐺𝑡𝑒𝑖; 𝜃 ×

𝑒∈𝐴𝑡𝑒𝑖

𝑆𝑒 𝑡𝑒𝑖 − 𝑡𝑒𝑖−1 ∣ 𝐻 𝑒𝑖−1 , 𝐺𝑡𝑒𝑖; 𝜃

The probability of an event is the hazard rate for that event, multiplied by the survival function for all possible events

To define this function exactly, we need to select a functional form for the hazard rate.

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The Hazard Rate

We model the hazard rate via the Cox proportional likelihood model:

ℎ𝑒𝑖 𝑡𝑒𝑖 𝐻 𝑒𝑖−1 , 𝐺𝑡𝑒𝑖; 𝜃 = 𝜆0 𝑡𝑒𝑖 exp 𝜃

′𝑋 𝑡𝑒𝑖 , 𝑒𝑖

The vector 𝑋 is a set of sufficient statistics for the given relational event model; these statistics are the factors chosen to influence the likelihood of an event

The parameter 𝜆0 is a scale parameter to influence the hazard rate over time; if we model the arrival times as exponential this is constant

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Full Likelihood Model

𝑃 𝑆; 𝜃 ∝

𝑒∈𝑆

𝜆𝑖𝑒𝑗𝑒 𝑡𝑒; 𝜃 exp −Δ𝑡𝑒

𝑒∈𝐴𝑡𝑒

𝜆𝑢𝑒𝑣𝑒 𝑡𝑒; 𝜃

To recover the parameters 𝜃, we can use maximum likelihood estimation

The solution 𝜃𝑗⋆ represents the influence of statistic 𝑗 on the

likelihood of an event

Note, we let 𝜆𝑖𝑗 𝑡 = ℎ𝑖𝑗(𝑡)

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Predictive Interpretation

Given a recovered parameter vector 𝜃⋆, then the probability of a new event (𝑖, 𝑗, 𝑡) is:

𝑃 𝑖, 𝑗, 𝑡 ; 𝜃⋆ =𝜆𝑖𝑗 𝑡; 𝜃

⋆

𝑢,𝑣 𝜆𝑢𝑣 𝑡; 𝜃⋆

Thus, we can identify what events are most likely, as well as explore how changing statistics can change these probabilities

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Sufficient Statistics

• The utility of the relational event model rests in the ability to define the statistics vector 𝑋

• Sufficient statistics may be homophily effects, environmental effects, or relational effects

• Relational effects are specific patterns in the event history that are repetitive or are high in intensity; we coin this class of effect as a “sequential structural signature”

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Sequential Structural Signatures

• We study how the sequential unfolding of individual relational events leads to future behaviors

• The generative mechanisms of behavior are emergent patterns in the event history

• The prevalence of a SSS is determined by the accumulation of relational events over time

15

Calculating Interaction Volume

The prevalence of a SSS is represented by the frequency it appears; this is dictated by the volume of communication

Let 𝜔𝑡 𝑖, 𝑗 be the accumulated chat volume from i to j up to time t, possibly weighted by some half life:

𝜔𝑡 𝑖, 𝑗 = 𝑒:𝑖𝑒=𝑖,𝑗𝑒=𝑗,𝑡𝑒<𝑡

𝑤𝑒 ⋅ln(2)

𝑇1/2⋅ 𝑒−(𝑡−𝑡𝑒)⋅

ln(2)𝑇1/2

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Sample SSS: Dyadic Sequences

Dyadic network structures involve basic cognitive patterns between two individuals. They cover the tendency to continue behavior or to reciprocate.

Habitual Inertia:

𝑋𝐼𝑁𝐸𝑅𝑇𝐼𝐴(𝑡) = 𝜔𝑡(𝑎, 𝑏)Reciprocity:

𝑋𝑅𝐸𝐶𝐼𝑃 𝑡 = 𝜔𝑡 𝑏, 𝑎

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Sample SSS: Triadic SequencesTriadic structures involve efficiency or commonality

Outbound Efficiency:

𝑋𝑂𝑇𝑃 𝑡 = 𝜔𝑡 𝑎, 𝑐 𝜔𝑡 𝑐, 𝑏

Inbound Efficiency:

𝑋𝐼𝑇𝑃 𝑡 = 𝜔𝑡 𝑐, 𝑎 𝜔𝑡 𝑏, 𝑐

Common Outbound Partner:

𝑋𝑂𝑆𝑃 𝑡 = 𝜔𝑡 𝑎, 𝑐 𝜔𝑡 𝑏, 𝑐

Common Inbound Partner:

𝑋𝐼𝑆𝑃 𝑡 = 𝜔𝑡 𝑐, 𝑎 𝜔𝑡 𝑐, 𝑏

Extensions to the Basic Model

• Currently we are developing several extensions to the relational event model:1. Modeling multiple classes of events (only

hypothetical framework defined in original Butts 2008 paper)

2. Modeling events with duration

3. Modeling the coevolution of relational events with external processes

4. Incorporating robustness into the model to account for errors in network perception

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Case Study: Multiteam System

What are the GENERATIVE MECHANISMS of intra- and inter-

team communication?

Which GENERATIVE MECHANISMS of intra- and inter-team

communication predict TEAM and MTS PERFORMANCE?

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Method: Participants, Roles, Task

Sample:

660 individuals

33 MTSs w/4 teams each

Task:

Lead a humanitarian convoy through a hostile region

MTS Structure:

4 teams of 5 individuals

Team

Leader

Recon

Officer

Field

Specialist

Insurgents IEDs

Team Structure (x4):

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Method: Team & MTS Goals

Each team is

responsible for

threats in a

single quadrant

Convoy

movements

are controlled

by the leaders

of the 4 teams

22

Intra-Team Mechanisms

23

A

C

B

D A

C

B

D

Signature = Intra-team

representation (attracting)

Signature = Intra-team

gatekeeping (sending)

j

i

v

u

Signature = Team Membership

Inter-Team Mechanisms

24

A

C

B

D

Signature = Inter-team

representation

A

C

B

D

Signature = Inter-team

gatekeeping

A C

B

Signature = Inter-team

contagion

u

j

i

Signature = Inter-

team personal inertia

Signature = Inter-team

generalized inertia

RQ1: Analytic Approach

• We used REM to determine the significance of each network hypothesis in each of 33 MTSs, & then did a meta-analysis

• A significant result for a given SSS indicates the corresponding effect was a driver of future behavior

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Generative Mechanism Type of

Communication

Generated

Effect Size CI Lower

Bound

CI Upper

Bound

H1: Team membership Within-Team 12.22*** 11.06 13.38

H2a: Intra-team representation Within-Team -1.28*** -2.09 -0.47

H2b: Intra-team gatekeeping Within-Team 0.15 -0.78 1.09

***=p<0.01, **=p<0.05, *=p<0.1

Confidence interval for the standardized effect sizes. Effect sizes were computed using the

methodology in (Hedges & Vevea 1998). Confidence intervals were computed at the 95%

confidence level

What are the signatures of team communication?

What are the signatures of between-team communication?

Generative Mechanism Type of

Communication

Generated

Effect Size CI Lower

Bound

CI Upper

Bound

H3a: Inter-team representation Between-Team -0.21 -1.08 0.65

H3b: Inter-team gatekeeping Between-Team -0.17 -0.89 0.55

H3c: Intra-team contagion Between-Team 1.62 -0.46 3.70

H4a: Inter-team personal

inertiaBetween-Team 10.09*** 7.71 12.48

H4b: Inter-team generalized

inertiaBetween-Team 1.63** 0.32 2.93

***=p<0.01, **=p<0.05, *=p<0.1

RQ2: Predicting Performance

• Given observed variation in the intensities of each SSS from MTS to MTS

• The variation in communication behavior can be used to explain the variation in the performance outcomes of each MTS

• Performance measured at both the local (team) level and global (MTS) level

• Poisson regression used to test SSS – performance relationships

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DV = Team PerformanceGenerative Mechanism Resultant

Communication

Successful

Identification

Successful

Neutralization

Team membership Within-Team0.97 3.03***

Intra-team representation Within-Team4.40*** 2.01**

Intra-team gatekeeping Within-Team1.68* 2.23**

Inter-Team RepresentationBetween-Team

1.41 1.52

Inter-Team GatekeepingBetween-Team

0.05 -0.89

Inter-Team ContagionBetween-Team

2.10** 1.49

Inter-Team Personal Inertia Between-Team 2.27** -1.99**

Inter-Team General Inertia Between-Team 1.03 -1.51

***=p<0.01, **=p<0.05, *=p<0.1

DV = MTS PerformanceGenerative Mechanism Resultant

Communication

Damage-Weighted

Movement

Team membership Within-Team3.18***

Intra-team representation Within-Team-2.17**

Intra-team gatekeeping Within-Team2.63***

Inter-Team RepresentationBetween-Team

0.39

Inter-Team GatekeepingBetween-Team

0.71

Inter-Team ContagionBetween-Team

1.36

Inter-Team Personal Inertia Between-Team -3.87***

Inter-Team General Inertia Between-Team -3.49***

***=p<0.01, **=p<0.05, *=p<0.1

Takeaways: Generative Mechanisms that Occur Naturally Have Mixed

Effects

Most MTSs form internal communication based on:

• Team membership (helps teams, MTSs)

• Avoiding intra-team representation (benefits team, benefits the MTS)

And between-team communication based on:

• Inter-team personalized inertia (benefits team info. access, harms team info. use & harms MTS)

• Inter-team generalized inertia (harms MTS)

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• Intra-team gatekeeping (helps teams & MTSs)

• Inter-team contagion (helps teams)

Takeaways: Two Generative Mechanisms that DO NOT Occur Naturally

Have Beneficial Effects

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A

C

B

D

A C

B