text: Bulge eﬀect on bar instability 9 MB00 MB01 MB02 MB03 MB04 MB05 MB06 4 2 0 Bulge −2 −4 −6 −8 7.8 7.7 7.6 Disk 7.5 7.4 7.3 7.2 7.1 1.0 Halo 0.5 0.0 −0.5 −1.0 0 2 4 6 8 0 2 4 6 8 0 2 4 6 8 0 2 4 6 8 0 2 4 6 8 0 2 4 6 8 0 2 4 6 8 Time(Gyr) Time(Gyr) Time(Gyr) Time(Gyr) Time(Gyr) Time(Gyr) Time(Gyr) Figure 14. Total Angular Momentum for all the components(Bulge, Disk and Halo) of all MB models 1.0 0.9 MMAA0012 MB01 MA03 0.8 MB02 MA04 MB03 MMAA0056 MB04 0.8 MA07 0.7 MB05 MB06 MB07 0.6 0.6 0.5 0.4 0.4 0.3 0.2 0.2 0.1 0.0 0.0 0 1 2 3 4 5 0 2 4 6 8 10 R(Kpc) R(Kpc) Figure 15. Fractional bulge force at disk scale length for MA Models( Figure 16. Fractional bulge force at disk scale length for MB Models( various shape dots) various shape dots) region by bars or spiral arms leading to the build up of central mass with other galaxies as well. Or in other words, strong bulges make concentrations (Norman et al. 1996; Das et al. 2003, 2008). disks extremely stable. 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text: 2.5 1.5 TGRG DAR-TGRG DAR-TGRG 2 mean (TGRG) DAR(1) 1 mean (DAR-TGRG) 1.5 1 0.5 0.5 0 0 0 0.2 0.4 0.6 0.8 1 -1 -0.5 0 0.5 1 αij φ1,i Figure 3. Left panel: distributions of α estimated via EM on e-MID data. The solid ij line refers to the DAR-TGRG model while the dotted line refers to the DAR(1) model. Right panel: distribution of parameters φ estimated via EM. The black line refers to 1,i DAR-TGRG while the blue line to TGRG. The dotted lines represent the mean of the two distributions. we exclude loans with a term larger than a week. However, we account approximately for the 92% of all the traded volume in the market since there are few credit relations with longer maturity. Then, we describe the e-MID weekly network with the unweighted and directed adjacency matrix At: a generic element At is 1 if the bank i lends money at least once to bank j during the week t, 0 otherwise. We ij do not consider banks that interact less than 5% of times in the considered period, i.e. in a period of T = 156 weeks a bank has at least a credit relation for more than 7 weeks. Hence, the credit network is formed by N = 98 banks. We estimate the three models on the time series of e-MID networks. Figure 3 shows the estimated density of the α link parameters (left panel) and of the φ node parameters (right panel) for the ij 1,j diﬀerent model. We see that the DAR(1) model estimates larger α parameters, i.e. larger link persis- ij tence, than the DAR-TGRG model. Similarly, the TGRG model estimates larger φ , i.e larger ﬁtness 1,j persistence, than the complete DAR-TGRG model. Thus the full model balances the relative role of the two persistence mechanisms. Node ﬁtness is a latent variable whose time evolution is not observed but inferred according to models of temporal networks. However it is interesting to ask if there exists an observable quantity correlated with it. We show that for the considered dataset, node ﬁtness is correlated with the bank exposure in the e-MID interbank market5. ≡ ∀ In fact we observe that the quantity xti,out(in) eθit,out(in) i = 1, ..., N estimated on data for both TGRG and DAR-TGRG models is strongly correlated with the corresponding bank’s exposure in e-MID for the considered week t, see the top left panel in Figure 4. This result suggests that, at a given time, banks with larger exposures are the nodes with larger ﬁtnesses θt,out(in) or equivalently with larger i degrees. Furthermore, the time-varying ﬁtness of a node is correlated signiﬁcantly with its bank exposure (see the top right panel of Figure 4). Finally, in the bottom panel of Figure 4 we show an example of ≈ this behavior for node 3 whose correlation coeﬃcient is ρ 0.90. Thus the dynamic ﬁtness model xt,st can be seen as a procedure allowing to have some insights3on3 bank exposures having only information on the binary network. For credit networks like e-MID, the preferential linkage mechanism reﬂects the presence of banks which trade preferentially each others. Preferential trading between banks can be detected by comparing empirically observed trading relationships with a null hypothesis that assumes random trading. Hat- zopoulos et al. [24] have introduced a statistical test to assess the statistical signiﬁcance of the observed interbank transactions in order to reveal preferential credit relationships among banks. We apply the same statistical test to show that preferential trading relations in e-MID are associated with link stability. 5Exposure of bank i is deﬁned as the strength of node i in the weighted network. We refer to it as st,out(in) for generic i node i. The node out-strength corresponds to the bank asset exposure in e-MID while the node in-strength to the liability. 14 ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

font: default

text: TGRG

text: DAR-TGRG

text: DAR-TGRG

text: mean

text: (TGRG)

text: mean

text: (DAR-TGRG)

text: αij

text: Figure

text: Left

label: caption

text: panel:

text: distributions

text: of

text: α

text: estimated

text: via

text: EM

text: on

text: e-MID

text: data.

text: The

text: solid

text: ij

text: line

text: refers

text: to

text: the

text: DAR-TGRG

text: model

text: while

text: the

text: dotted

text: line

text: refers

text: to

text: the

text: model.

text: Right

text: panel:

text: distribution

text: of

text: parameters

text: φ

text: estimated

text: via

text: EM.

text: The

text: black

text: line

text: refers

text: to

text: DAR-TGRG

text: while

text: the

text: blue

text: line

text: to

text: TGRG.

text: The

text: dotted

text: lines

text: represent

text: the

text: mean

text: of

text: the

text: two

text: distributions.

text: we

text: exclude

text: loans

text: with

text: a

text: term

text: larger

text: than

text: a

text: week.

text: However,

text: we

text: account

text: approximately

text: for

text: the

text: of

text: all

text: the

text: traded

text: volume

text: in

text: the

text: market

text: since

text: there

text: are

text: few

text: credit

text: relations

text: with

text: longer

text: maturity.

text: Then,

text: we

text: describe

text: the

text: e-MID

text: weekly

text: network

text: with

text: the

text: unweighted

text: and

text: directed

text: adjacency

text: matrix

text: At:

text: a

text: generic

text: element

text: At

text: is

text: if

text: the

text: bank

text: i

text: lends

text: money

text: at

text: least

text: once

text: to

text: bank

text: j

text: during

text: the

text: week

text: t,

text: otherwise.

text: We

text: ij

text: do

text: not

text: consider

text: banks

text: that

text: interact

text: less

text: than

text: of

text: times

text: in

text: the

text: considered

text: period,

text: i.e.

text: in

text: a

text: period

text: of

text: T

text: weeks

text: a

text: bank

text: has

text: at

text: least

text: a

text: credit

text: relation

text: for

text: more

text: than

text: weeks.

text: Hence,

text: the

text: credit

text: network

text: is

text: formed

text: by

text: N

text: banks.

text: We

text: estimate

text: the

text: three

text: models

text: on

text: the

text: time

text: series

text: of

text: e-MID

text: networks.

text: Figure

text: shows

text: the

text: estimated

text: density

text: of

text: the

text: α

text: link

text: parameters

text: (left

text: panel)

text: and

text: of

text: the

text: φ

text: node

text: parameters

text: (right

text: panel)

text: for

text: the

text: ij

text: diﬀerent

text: model.

text: We

text: see

text: that

text: the

text: model

text: estimates

text: larger

text: α

text: parameters,

text: i.e.

text: larger

text: link

text: persis-

text: ij

text: tence,

text: than

text: the

text: DAR-TGRG

text: model.

text: Similarly,

text: the

text: TGRG

text: model

text: estimates

text: larger

text: φ

text: i.e

text: larger

text: ﬁtness

text: persistence,

text: than

text: the

text: complete

text: DAR-TGRG

text: model.

text: Thus

text: the

text: full

text: model

text: balances

text: the

text: relative

text: role

text: of

text: the

text: two

text: persistence

text: mechanisms.

text: Node

text: ﬁtness

text: is

text: a

text: latent

text: variable

text: whose

text: time

text: evolution

text: is

text: not

text: observed

text: but

text: inferred

text: according

text: to

text: models

text: of

text: temporal

text: networks.

text: However

text: it

text: is

text: interesting

text: to

text: ask

text: if

text: there

text: exists

text: an

text: observable

text: quantity

text: correlated

text: with

text: it.

text: We

text: show

text: that

text: for

text: the

text: considered

text: dataset,

text: node

text: ﬁtness

text: is

text: correlated

text: with

text: the

text: bank

text: exposure

text: in

text: the

text: e-MID

text: interbank

text: In

text: fact

text: we

text: observe

text: that

text: the

text: quantity

text: xti,out(in)

text: eθit,out(in)

text: i

text: N

text: estimated

text: on

text: data

text: for

text: both

text: TGRG

text: and

text: DAR-TGRG

text: models

text: is

text: strongly

text: correlated

text: with

text: the

text: corresponding

text: bank’s

text: exposure

text: in

text: e-MID

text: for

text: the

text: considered

text: week

text: t,

text: see

text: the

text: top

text: left

text: panel

text: in

text: Figure

text: This

text: result

text: suggests

text: that,

text: at

text: a

text: given

text: time,

text: banks

text: with

text: larger

text: exposures

text: are

text: the

text: nodes

text: with

text: larger

text: ﬁtnesses

text: θt,out(in)

text: or

text: equivalently

text: with

text: larger

text: i

text: degrees.

text: Furthermore,

text: the

text: time-varying

text: ﬁtness

text: of

text: a

text: node

text: is

text: correlated

text: signiﬁcantly

text: with

text: its

text: bank

text: exposure

text: (see

text: the

text: top

text: right

text: panel

text: of

text: Figure

text: Finally,

text: in

text: the

text: bottom

text: panel

text: of

text: Figure

text: we

text: show

text: an

text: example

text: of

text: this

text: behavior

text: for

text: node

text: whose

text: correlation

text: coeﬃcient

text: is

text: ρ

text: Thus

text: the

text: dynamic

text: ﬁtness

text: model

text: xt,st

text: can

text: be

text: seen

text: as

text: a

text: procedure

text: allowing

text: to

text: have

text: some

text: bank

text: exposures

text: having

text: only

text: information

text: on

text: the

text: binary

text: network.

text: For

text: credit

text: networks

text: like

text: e-MID,

text: the

text: preferential

text: linkage

text: mechanism

text: reﬂects

text: the

text: presence

text: of

text: banks

text: which

text: trade

text: preferentially

text: each

text: others.

text: Preferential

text: trading

text: between

text: banks

text: can

text: be

text: detected

text: by

text: comparing

text: empirically

text: observed

text: trading

text: relationships

text: with

text: a

text: hypothesis

text: that

text: assumes

text: random

text: trading.

text: Hat-

text: zopoulos

text: et

text: al.

text: have

text: introduced

text: a

text: statistical

text: test

text: to

text: assess

text: the

text: statistical

text: signiﬁcance

text: of

text: the

text: observed

text: interbank

text: transactions

text: in

text: order

text: to

text: reveal

text: preferential

text: credit

text: relationships

text: among

text: banks.

text: We

text: apply

text: the

text: same

text: statistical

text: test

text: to

text: show

text: that

text: preferential

text: trading

text: relations

text: in

text: e-MID

text: are

text: associated

text: with

text: link

text: stability.

text: of

text: bank

text: i

text: is

text: deﬁned

text: as

text: the

text: strength

text: of

text: node

text: i

text: in

text: the

text: weighted

text: network.

text: We

text: refer

text: to

text: it

text: as

text: st,out(in)

text: for

text: generic

text: i

text: node

text: i.

text: The

text: node

text: out-strength

text: corresponds

text: to

text: the

text: bank

text: asset

text: exposure

text: in

text: e-MID

text: while

text: the

text: node

text: in-strength

text: to

text: the

text: liability.

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text: αˆ estimated by the the DAR-TGRG model is larger than a threshold value. We ﬁnd that taking ij into account both of ﬁtness dynamics and preferential linkage better forecast links, i.e. DAR-TGRG outperforms always the other models. When we consider links with both high and low persistence, the TGRG model outperforms the DAR(1) network model, that is the evolution of the network topology is more important than preferential linkage in determining the average characteristics of the e-MID network. However, the link copying mechanism associated with the DAR(1) model characterizes better than the ﬁtness dynamics the persistence pattern associated with a smaller set of links representing the preferential relations among banks. In fact, there exists a value of the threshold (around 0.4) after which the AUC associated with the DAR(1) model is larger than the one for TGRG. 5. Conclusions In this paper we introduce a novel state-of-the-art statistical methodology to describe link persistence and ﬁtness dynamics in temporal networks. We model a Markov dynamics for both observed and unobserved time-varying states which drive the evolution of the network. The analytic tractability of the autoregressive network ensemble we propose allows us to easily calibrate our parameters from the data with a general likelihood maximization iterative procedure. The introduction of the autoregressive dynamics permits link forecasting by taking account of memory properties of the network system. Then, the estimation method we introduce allows online-inference of the time-varying parameters which is particularly useful from a computational point of view to face the problem of link prediction. The contribution of the paper is twofold. First, the introduction of autoregressive endogenous com- ponents displays the clear advantage of describing the network evolution via time-varying states which reproduce the network topology as well as capturing the local property of link persistence, thus going beyond a single snapshot analysis where parameters are chosen for each network snapshot, indepen- dently. Second, the analysis on real data from the eMID interbank network from 2012 to 2015 (weekly aggregated) displays the statistical equivalence between link stability, identiﬁed by positive value of the persistence parameter, and preferential trading, identiﬁed by over-expressed number of trades between counterparties. Hence, our methodology permits to disentangle preferential trading from random trading in dynamic trading networks such as the eMID money market. Finally, the forecasting performance of the model points out both ﬁtness dynamics and link persistence as linkage mechanisms in the process of network formation for the credit market. As future outlooks, the formalism discussed in the paper could be also be applied to more general memory kernel function of the autoregressive model governing the evolution of the system as well as it could permit the introduction of exogenous factors driving the ﬁtness dynamics or the local link probability. Furthermore, a challenging issue is the introduction of a dependence structure for the dynamic ﬁtnesses. We also note that the estimation method we introduce to obtain our results for dynamic networks is quite general, and could be used to obtain similar results for other types of ﬁtness dynamics. Acknowledgment We acknowledge ﬁnancial support from the grant SNS16LILLB “Financial networks: statistical models, inference, and shock propagation”. FL acknowledges support by the European Community’s H2020 Pro- gram under the scheme INFRAIA-1- 2014-2015: Research Infrastructures, grant agreement no. 654024 SoBigData: Social Mining & Big Data Ecosystem. References [1] Newman, Mark EJ, Steven H. Strogatz, and Duncan J. Watts. "Random graphs with arbitrary degree distributions and their applications." Physical review E 64.2 (2001): 026118. [2] Holme, Petter, and Jari Saramäki. "Temporal networks." Physics reports 519.3 (2012): 97-125. [3] Weisbuch, Gerard, Alan Kirman, and Dorothea Herreiner. "Market organisation and trading relationships." The eco- nomic journal 110.463 (2000): 411-436. [4] Cocco, Joao F., Francisco J. Gomes, and Nuno C. Martins. "Lending relationships in the interbank market." Journal of Financial Intermediation 18.1 (2009): 24-48. [5] Hanneke, Steve, Wenjie Fu, and Eric P. Xing. "Discrete temporal models of social networks." Electronic Journal of Statistics 4 (2010): 585-605. [6] Krivitsky, Pavel N., and Mark S. Handcock. "A separable model for dynamic networks." Journal of the Royal Statistical Society: Series B (Statistical Methodology) 76.1 (2014): 29-46. 17

text: αˆ

text: estimated

text: by

text: the

text: the

text: DAR-TGRG

text: model

text: is

text: larger

text: than

text: a

text: threshold

text: value.

text: We

text: ﬁnd

text: that

text: taking

text: ij

text: into

text: account

text: both

text: of

text: ﬁtness

text: dynamics

text: and

text: preferential

text: linkage

text: better

text: forecast

text: links,

text: i.e.

text: DAR-TGRG

text: outperforms

text: always

text: the

text: other

text: models.

text: When

text: we

text: consider

text: links

text: with

text: both

text: high

text: and

text: low

text: persistence,

text: the

text: TGRG

text: model

text: outperforms

text: the

text: network

text: model,

text: that

text: is

text: the

text: evolution

text: of

text: the

text: network

text: topology

text: is

text: more

text: important

text: than

text: preferential

text: linkage

text: in

text: determining

text: the

text: average

text: characteristics

text: of

text: the

text: e-MID

text: network.

text: However,

text: the

text: link

text: copying

text: mechanism

text: associated

text: with

text: the

text: model

text: characterizes

text: better

text: than

text: the

text: ﬁtness

text: dynamics

text: the

text: persistence

text: pattern

text: associated

text: with

text: a

text: smaller

text: set

text: of

text: links

text: representing

text: the

text: preferential

text: relations

text: among

text: banks.

text: In

text: fact,

text: there

text: exists

text: a

text: value

text: of

text: the

text: threshold

text: (around

text: after

text: which

text: the

text: AUC

text: associated

text: with

text: the

text: model

text: is

text: larger

text: than

text: the

text: one

text: for

text: TGRG.

text: Conclusions

text: In

text: this

text: paper

text: we

text: introduce

text: a

text: novel

text: state-of-the-art

text: statistical

text: methodology

text: to

text: describe

text: link

text: persistence

text: and

text: ﬁtness

text: dynamics

text: in

text: temporal

text: networks.

text: We

text: model

text: a

text: Markov

text: dynamics

text: for

text: both

text: observed

text: and

text: unobserved

text: time-varying

text: states

text: which

text: drive

text: the

text: evolution

text: of

text: the

text: network.

text: The

text: analytic

text: tractability

text: of

text: the

text: autoregressive

text: network

text: ensemble

text: we

text: propose

text: allows

text: us

text: to

text: easily

text: calibrate

text: our

text: parameters

text: from

text: the

text: data

text: with

text: a

text: general

text: likelihood

text: maximization

text: iterative

text: procedure.

text: The

text: introduction

text: of

text: the

text: autoregressive

text: dynamics

text: permits

text: link

text: forecasting

text: by

text: taking

text: account

text: of

text: memory

text: properties

text: of

text: the

text: network

text: system.

text: Then,

text: the

text: estimation

text: method

text: we

text: introduce

text: allows

text: online-inference

text: of

text: the

text: time-varying

text: parameters

text: which

text: is

text: particularly

text: useful

text: from

text: a

text: computational

text: point

text: of

text: view

text: to

text: face

text: the

text: problem

text: of

text: link

text: prediction.

text: The

text: contribution

text: of

text: the

text: paper

text: is

text: twofold.

text: First,

text: the

text: introduction

text: of

text: autoregressive

text: endogenous

text: com-

text: ponents

text: displays

text: the

text: clear

text: advantage

text: of

text: describing

text: the

text: network

text: evolution

text: via

text: time-varying

text: states

text: which

text: reproduce

text: the

text: network

text: topology

text: as

text: well

text: as

text: capturing

text: the

text: local

text: property

text: of

text: link

text: persistence,

text: thus

text: going

text: beyond

text: a

text: single

text: snapshot

text: analysis

text: where

text: parameters

text: are

text: chosen

text: for

text: each

text: network

text: snapshot,

text: indepen-

text: dently.

text: Second,

text: the

text: analysis

text: on

text: real

text: data

text: from

text: the

text: eMID

text: interbank

text: network

text: from

text: to

text: (weekly

text: aggregated)

text: displays

text: the

text: statistical

text: equivalence

text: between

text: link

text: stability,

text: identiﬁed

text: by

text: positive

text: value

text: of

text: the

text: persistence

text: parameter,

text: and

text: preferential

text: trading,

text: identiﬁed

text: by

text: over-expressed

text: number

text: of

text: trades

text: between

text: counterparties.

text: Hence,

text: our

text: methodology

text: permits

text: to

text: disentangle

text: preferential

text: trading

text: from

text: random

text: trading

text: in

text: dynamic

text: trading

text: networks

text: such

text: as

text: the

text: eMID

text: money

text: market.

text: Finally,

text: the

text: forecasting

text: performance

text: of

text: the

text: model

text: points

text: out

text: both

text: ﬁtness

text: dynamics

text: and

text: link

text: persistence

text: as

text: linkage

text: mechanisms

text: in

text: the

text: process

text: of

text: network

text: formation

text: for

text: the

text: credit

text: market.

text: As

text: future

text: outlooks,

text: the

text: formalism

text: discussed

text: in

text: the

text: paper

text: could

text: be

text: also

text: be

text: applied

text: to

text: more

text: general

text: memory

text: kernel

text: function

text: of

text: the

text: autoregressive

text: model

text: governing

text: the

text: evolution

text: of

text: the

text: system

text: as

text: well

text: as

text: it

text: could

text: permit

text: the

text: introduction

text: of

text: exogenous

text: factors

text: driving

text: the

text: ﬁtness

text: dynamics

text: or

text: the

text: local

text: link

text: probability.

text: Furthermore,

text: a

text: challenging

text: issue

text: is

text: the

text: introduction

text: of

text: a

text: dependence

text: structure

text: for

text: the

text: dynamic

text: ﬁtnesses.

text: We

text: also

text: note

text: that

text: the

text: estimation

text: method

text: we

text: introduce

text: to

text: obtain

text: our

text: results

text: for

text: dynamic

text: networks

text: is

text: quite

text: general,

text: and

text: could

text: be

text: used

text: to

text: obtain

text: similar

text: results

text: for

text: other

text: types

text: of

text: ﬁtness

text: dynamics.

text: Acknowledgment

text: We

text: acknowledge

text: ﬁnancial

text: support

text: from

text: the

text: grant

text: “Financial

text: networks:

text: statistical

text: models,

text: inference,

text: and

text: shock

text: propagation”.

text: FL

text: acknowledges

text: support

text: by

text: the

text: European

text: Community’s

text: Pro-

text: gram

text: under

text: the

text: scheme

text: Research

text: Infrastructures,

text: grant

text: agreement

text: no.

text: SoBigData:

text: Social

text: Mining

text: Big

text: Data

text: Ecosystem.

text: References

text: Newman,

text: Mark

text: EJ,

text: Steven

text: H.

text: Strogatz,

text: and

text: Duncan

text: J.

text: Watts.

text: "Random

text: graphs

text: with

text: arbitrary

text: degree

text: distributions

text: and

text: their

text: applications."

text: Physical

text: review

text: E

text: Holme,

text: Petter,

text: and

text: Jari

text: Saramäki.

text: "Temporal

text: networks."

text: Physics

text: reports

text: Weisbuch,

text: Gerard,

text: Alan

text: Kirman,

text: and

text: Dorothea

text: Herreiner.

text: "Market

text: organisation

text: and

text: trading

text: relationships."

text: The

text: eco-

text: nomic

text: journal

text: Cocco,

text: Joao

text: F.,

text: Francisco

text: J.

text: Gomes,

text: and

text: Nuno

text: C.

text: Martins.

text: "Lending

text: relationships

text: in

text: the

text: interbank

text: market."

text: Journal

text: of

text: Financial

text: Intermediation

text: Hanneke,

text: Steve,

text: Wenjie

text: Fu,

text: and

text: Eric

text: P.

text: Xing.

text: "Discrete

text: temporal

text: models

text: of

text: social

text: networks."

text: Electronic

text: Journal

text: of

text: Statistics

text: Krivitsky,

text: Pavel

text: N.,

text: and

text: Mark

text: S.

text: Handcock.

text: "A

text: separable

text: model

text: for

text: dynamic

text: networks."

text: Journal

text: of

text: the

text: Royal

text: Statistical

text: Society:

text: Series

text: B

text: (Statistical

text: Methodology)


text: with a vector autoregressive process. Furthermore, they propose an eﬃcient model estimation based on proximal methods. However, this ﬁrst kind of approach has been criticized because it can not fully capture the time- varying patterns of the network structure. This opens to a second stream of literature which aims to describe these patterns with models having time-varying (latent) parameters that capture how network topology changes in time, see [10] for a review. A milestone work is represented by the study of Sarkar and Moore [11] that generalized the latent space model introduced in [12] to dynamic networks. The dynamics of the network structure is modelled through random eﬀects in a latent space. Sewell and Chen [13] proposed a Markov Chain Monte Carlo (MCMC) algorithm to estimate the model parameters and latent positions of the nodes in the network. Durante and Dunson [14] proposed a further extension of the model in [11] by describing a latent space model for dynamic networks in which latent node positions evolves in time via stochastic diﬀerential equations. They introduced also an eﬃcient MCMC algorithm for Bayesian inference to learn model parameters [15]. Giraitis et al. [16] proposed a novel methodology for dynamic modeling of temporal networks with application to interbank networks. They describe the link dynamics with a Tobit model, allowing for deterministic or stochastic time-varying parameters that take into account the possibility of structural changes in network dynamics. Brauning and Koopman [17] have applied the dynamic factors model to the case of dynamic networks. Depending on the number of factors, the model allows to reduce the dimensionality of the problem and to describe cross-sectional dependencies in network data. Lee at al. [18] have recently introduced a generalization of TERG, called varying-coeﬃcient exponential random graph model, that characterizes the evolution of network topologies through smoothly time varying parameters whose dynamics can capture temporal heterogeneity of dynamic networks. Finally, in this stream of literature we can also include all the generalizations [19, 20, 21, 22, 23] of SBM which account for time evolving community memberships and/or link persistence. However, these works are more focused on the problem of community detection when dynamic eﬀects are considered. The methodology we propose in this paper exploits both of the aspects that characterize the two streams of literature. From one side, we describe link persistence coming from the mechanism of copying from the past by modeling a Markov chain for link stability, i.e. the tendency of a link that does (or − does not) exist at time t 1 to continue existing (or not existing) at time t, similarly to [5]. From the other side, we describe the stochastic dynamics of node-speciﬁc latent variables that we call ﬁtnesses, with a similar aim of [17, 18]. The node ﬁtness describes the tendency of a node in creating links and its evolution determines how the degree of the node changes in time. From the point of view of generative network models, link stability tends to capture preferential linkage mechanism between the nodes of the network while the ﬁtness dynamics accounts for the evolving network topology. The main goal of this work is to disentangle the two temporal patterns generated by the two linkage mechanisms in network data. Hence, we apply our methodology to the ﬁnancial network of electronic Market of Interbank Deposit (e-MID) where the two linkage mechanisms are associated with two diﬀerent trading behaviors, i.e. random and preferential trading [24], in the process of network formation. The remainder of this paper is organized as follows. In Section we describe three diﬀerent models of temporal networks. In Section we present a novel Expectation-Maximization algorithm for model estimation and in Section we run a Monte Carlo exercise to assess the goodness of ﬁt of our estimation method. In Section we apply our methodology to the network of the electronic Market of Interbank Deposit. Finally, we conclude with a discussion of our method and open areas for future research in Section . 1. The models In this Section, we introduce three models of temporal networks: (i) in the ﬁrst one, the presence or absence of a link can be a copy of the past with a given probability or can be sampled according to a Bernoulli marginal distribution; (ii) in the second model, each graph snapshot does not have an explicit dependence from the past snapshots but the link probability depends on node-speciﬁc latent dynamical variables, i.e. the node ﬁtnesses, which evolve stochastically in time with memory of past information; (iii) the third network model combines the copying mechanism of the ﬁrst model with dynamic node ﬁtnesses of the second model. { } We deﬁne a temporal network as a time series of graphs, that is the set (V, At t=0,1,...,T ) with | | { } V = N nodes and adjacency matrices At t=0,1,...,T . A network snapshot is the observed graph at a given time t and is described by the adjacency matrix At which has entry At = 1 if the edge from node ij 3

text: with

text: a

text: vector

text: autoregressive

text: process.

text: Furthermore,

text: they

text: propose

text: an

text: eﬃcient

text: model

text: estimation

text: based

text: on

text: proximal

text: methods.

text: However,

text: this

text: ﬁrst

text: kind

text: of

text: approach

text: has

text: been

text: criticized

text: because

text: it

text: can

text: not

text: fully

text: capture

text: the

text: time-

text: varying

text: patterns

text: of

text: the

text: network

text: structure.

text: This

text: opens

text: to

text: a

text: second

text: stream

text: of

text: literature

text: which

text: aims

text: to

text: describe

text: these

text: patterns

text: with

text: models

text: having

text: time-varying

text: (latent)

text: parameters

text: that

text: capture

text: how

text: network

text: topology

text: changes

text: in

text: time,

text: see

text: for

text: a

text: review.

text: A

text: milestone

text: work

text: is

text: represented

text: by

text: the

text: study

text: of

text: Sarkar

text: and

text: Moore

text: that

text: generalized

text: the

text: latent

text: space

text: model

text: introduced

text: in

text: to

text: dynamic

text: networks.

text: The

text: dynamics

text: of

text: the

text: network

text: structure

text: is

text: modelled

text: through

text: random

text: eﬀects

text: in

text: a

text: latent

text: space.

text: Sewell

text: and

text: Chen

text: proposed

text: a

text: Markov

text: Chain

text: Monte

text: Carlo

text: (MCMC)

text: algorithm

text: to

text: estimate

text: the

text: model

text: parameters

text: and

text: latent

text: positions

text: of

text: the

text: nodes

text: in

text: the

text: network.

text: Durante

text: and

text: Dunson

text: proposed

text: a

text: further

text: extension

text: of

text: the

text: model

text: in

text: by

text: describing

text: a

text: latent

text: space

text: model

text: for

text: dynamic

text: networks

text: in

text: which

text: latent

text: node

text: positions

text: evolves

text: in

text: time

text: via

text: stochastic

text: diﬀerential

text: equations.

text: They

text: introduced

text: also

text: an

text: eﬃcient

text: MCMC

text: algorithm

text: for

text: Bayesian

text: inference

text: to

text: learn

text: model

text: parameters

text: Giraitis

text: et

text: al.

text: proposed

text: a

text: novel

text: methodology

text: for

text: dynamic

text: modeling

text: of

text: temporal

text: networks

text: with

text: application

text: to

text: interbank

text: networks.

text: They

text: describe

text: the

text: link

text: dynamics

text: with

text: a

text: Tobit

text: model,

text: allowing

text: for

text: deterministic

text: or

text: stochastic

text: time-varying

text: parameters

text: that

text: take

text: into

text: account

text: the

text: possibility

text: of

text: structural

text: changes

text: in

text: network

text: dynamics.

text: Brauning

text: and

text: Koopman

text: have

text: applied

text: the

text: dynamic

text: factors

text: model

text: to

text: the

text: case

text: of

text: dynamic

text: networks.

text: Depending

text: on

text: the

text: number

text: of

text: factors,

text: the

text: model

text: allows

text: to

text: reduce

text: the

text: dimensionality

text: of

text: the

text: problem

text: and

text: to

text: describe

text: cross-sectional

text: dependencies

text: in

text: network

text: data.

text: Lee

text: at

text: al.

text: have

text: recently

text: introduced

text: a

text: generalization

text: of

text: TERG,

text: called

text: varying-coeﬃcient

text: exponential

text: random

text: graph

text: model,

text: that

text: characterizes

text: the

text: evolution

text: of

text: network

text: topologies

text: through

text: smoothly

text: time

text: varying

text: parameters

text: whose

text: dynamics

text: can

text: capture

text: temporal

text: heterogeneity

text: of

text: dynamic

text: networks.

text: Finally,

text: in

text: this

text: stream

text: of

text: literature

text: we

text: can

text: also

text: include

text: all

text: the

text: generalizations

text: of

text: SBM

text: which

text: account

text: for

text: time

text: evolving

text: community

text: memberships

text: and/or

text: link

text: persistence.

text: However,

text: these

text: works

text: are

text: more

text: focused

text: on

text: the

text: problem

text: of

text: community

text: detection

text: when

text: dynamic

text: eﬀects

text: are

text: considered.

text: The

text: methodology

text: we

text: propose

text: in

text: this

text: paper

text: exploits

text: both

text: of

text: the

text: aspects

text: that

text: characterize

text: the

text: two

text: streams

text: of

text: literature.

text: From

text: one

text: side,

text: we

text: describe

text: link

text: persistence

text: coming

text: from

text: the

text: mechanism

text: of

text: copying

text: from

text: the

text: past

text: by

text: modeling

text: a

text: Markov

text: chain

text: for

text: link

text: stability,

text: i.e.

text: the

text: tendency

text: of

text: a

text: link

text: that

text: does

text: (or

text: does

text: not)

text: exist

text: at

text: time

text: t

text: to

text: continue

text: existing

text: (or

text: not

text: existing)

text: at

text: time

text: t,

text: similarly

text: to

text: From

text: the

text: other

text: side,

text: we

text: describe

text: the

text: stochastic

text: dynamics

text: of

text: node-speciﬁc

text: latent

text: variables

text: that

text: we

text: call

text: ﬁtnesses,

text: with

text: a

text: similar

text: aim

text: of

text: The

text: node

text: ﬁtness

text: describes

text: the

text: tendency

text: of

text: a

text: node

text: in

text: creating

text: links

text: and

text: its

text: evolution

text: determines

text: how

text: the

text: degree

text: of

text: the

text: node

text: changes

text: in

text: time.

text: From

text: the

text: point

text: of

text: view

text: of

text: generative

text: network

text: models,

text: link

text: stability

text: tends

text: to

text: capture

text: preferential

text: linkage

text: mechanism

text: between

text: the

text: nodes

text: of

text: the

text: network

text: while

text: the

text: ﬁtness

text: dynamics

text: accounts

text: for

text: the

text: evolving

text: network

text: topology.

text: The

text: main

text: goal

text: of

text: this

text: work

text: is

text: to

text: disentangle

text: the

text: two

text: temporal

text: patterns

text: generated

text: by

text: the

text: two

text: linkage

text: mechanisms

text: in

text: network

text: data.

text: Hence,

text: we

text: apply

text: our

text: methodology

text: to

text: the

text: ﬁnancial

text: network

text: of

text: electronic

text: Market

text: of

text: Interbank

text: Deposit

text: (e-MID)

text: where

text: the

text: two

text: linkage

text: mechanisms

text: are

text: associated

text: with

text: two

text: diﬀerent

text: trading

text: behaviors,

text: i.e.

text: random

text: and

text: preferential

text: trading

text: in

text: the

text: process

text: of

text: network

text: formation.

text: The

text: remainder

text: of

text: this

text: paper

text: is

text: organized

text: as

text: follows.

text: In

text: Section

text: we

text: describe

text: three

text: diﬀerent

text: models

text: of

text: temporal

text: networks.

text: In

text: Section

text: we

text: present

text: a

text: novel

text: Expectation-Maximization

text: algorithm

text: for

text: model

text: estimation

text: and

text: in

text: Section

text: we

text: run

text: a

text: Monte

text: Carlo

text: exercise

text: to

text: assess

text: the

text: goodness

text: of

text: ﬁt

text: of

text: our

text: estimation

text: method.

text: In

text: Section

text: we

text: apply

text: our

text: methodology

text: to

text: the

text: network

text: of

text: the

text: electronic

text: Market

text: of

text: Interbank

text: Deposit.

text: Finally,

text: we

text: conclude

text: with

text: a

text: discussion

text: of

text: our

text: method

text: and

text: open

text: areas

text: for

text: future

text: research

text: in

text: Section

text: The

text: models

text: In

text: this

text: Section,

text: we

text: introduce

text: three

text: models

text: of

text: temporal

text: networks:

text: (i)

text: in

text: the

text: ﬁrst

text: one,

text: the

text: presence

text: or

text: absence

text: of

text: a

text: link

text: can

text: be

text: a

text: copy

text: of

text: the

text: past

text: with

text: a

text: given

text: probability

text: or

text: can

text: be

text: sampled

text: according

text: to

text: a

text: Bernoulli

text: marginal

text: distribution;

text: (ii)

text: in

text: the

text: second

text: model,

text: each

text: graph

text: snapshot

text: does

text: not

text: have

text: an

text: explicit

text: dependence

text: from

text: the

text: past

text: snapshots

text: but

text: the

text: link

text: probability

text: depends

text: on

text: node-speciﬁc

text: latent

text: dynamical

text: variables,

text: i.e.

text: the

text: node

text: ﬁtnesses,

text: which

text: evolve

text: stochastically

text: in

text: time

text: with

text: memory

text: of

text: past

text: information;

text: (iii)

text: the

text: third

text: network

text: model

text: combines

text: the

text: copying

text: mechanism

text: of

text: the

text: ﬁrst

text: model

text: with

text: dynamic

text: node

text: ﬁtnesses

text: of

text: the

text: second

text: model.

text: We

text: deﬁne

text: a

text: temporal

text: network

text: as

text: a

text: time

text: series

text: of

text: graphs,

text: that

text: is

text: the

text: set

text: (V,

text: At

text: with

text: V

text: N

text: nodes

text: and

text: adjacency

text: matrices

text: At

text: A

text: network

text: snapshot

text: is

text: the

text: observed

text: graph

text: at

text: a

text: given

text: time

text: t

text: and

text: is

text: described

text: by

text: the

text: adjacency

text: matrix

text: At

text: which

text: has

text: entry

text: At

text: if

text: the

text: edge

text: from

text: node

text: ij


text: ∈ | | ∼ N where φ R, φ < 1 and the i.i.d. variables (cid:15)t (0, σ2). This choice is consistent with the 0,i 1,i i i Markovian assumption in Eq. 1.1. Moreover, the hidden node state θt evolves in R between timesteps, i but large changes are unlikely because of the Gaussian transition probabilities. This is consistent with the idea that the network topology changes smoothly in time. Finally, assuming a Gaussian transition probability represents a simpliﬁcation for model estimation. The conditional probability for the network at t(cid:89)ime t is P(At|Θt) = eAij(θit+θjt) , (1.5) 1 + e(θit+θjt) i,j>i ≡ { } where Θt θt is the vector of time-varying parameters. In Eq. 1.5 we assume that each link i i=1,...,N is independently sampled and the probability of a link between node i and node j at time t is determined by the corresponding θt and θt. The larger is θt, the larger is the probability for all links incident to i j i node i. We refer to this model as Temporally Generalized Random Graphs (TGRG) and the speciﬁcation of Eq. 1.1 for the TGRGis the following,  (cid:81) (cid:81)  | − | − ∀ P(θt θt 1, Φ ) = f(θt φ + φ θt 1, σ2) i = 1, ..., N (cid:81)P(Ait|Θi t) i = i 0P,i(At |1θ,it,iθt) =i eAtij(θit+θjt) (1.6) i,j>i ij i j i,j>i 1+e(θit+θjt) | − | − | and P(Θt Θt 1, Φ) = N P(θt θt 1, Φ ) according to the hypothesis of independence, where f(θt φ + − i=1 i i i − i 0,i φ θt 1, σ2) is the density of normal variable with mean φ +φ θt 1 and variance σ2. The set of static 1,i i i ≡ { } ≡ { } 0,i 1,i i i parameters is Φ Φ with Φ φ , φ , σ . i i=1,...,N i 0,i× 1,i i The TGRG model is fully determined by the 3 N static parameters Φ. In the next Section we propose an Expectation-Maximization scheme to estimate the model parameters and time-varying parameters. { } It alternates between an Expectation step where we ﬁt the time-varying parameters Θt t=0,1,...,T and the Maximization step where we maximize the log-likelihood of the static parameters conditional on the { } expectations Θˆt t=0,1,...,T . Time autocorrelated node ﬁtnesses may induce link persistence. In fact, the probability of a link between two speciﬁc nodes e(θit+θjt)/(1 + e(θit+θjt)) is persistent if θt and θt are autocorrelated. Note i j that link persistence occurs as a consequence of node properties. For TGRG, the two-point distribution function for lagged links and the ACF of link state can be semi-analytically computed (see the Appendix). The generalization of the TGRG model for directed networks can be accomplished by distinguishing between the out-degree and the in-degree and by introducing two ﬁtnesses for each node i, i.e. θt,out i and θt,in. The probability of a link from node i to node j at time t given the latent variables θt,out i i and θt,in is P(At |θt,out, θt,in) = eAtij(θit,out+θjt,in) . Then, everything follows similarly to the undirected j ij i j 1+| e(θit,out+θjt,in) case with the exception that P(At Θt) is invariant under a linear transformation for the hidden node (cid:55)→ ∀ (cid:55)→ − ∀ { } ∈ states: θt,out θt,out + c i = 1, ..., N, θt,in θt,in c j = 1, ..., N, where c RT+1. i i t j j t t t=0,1,...,T This symmetry arises because the total number of outgoing links has to be equal to the total number of ingoing links at each time. This degeneracy can be simply removed by taking one of the ﬁtnesses as constant in time. Finally, let us notice that we can interpret TGRG as an extension of Exponential Random Graphs (ERG) [28] to the dynamic case. ERG ensembles are probability distributions of networks obtained by maximizing the Shannon entropy under some constraints on the average value of a set of network observ- ables. If this set is the degree sequence, the Lagrange multipliers of the entropy constrained optimization can be directly linked to the latent variables of our model. Diﬀerently from other dynamic extension of ERG (see for example [5]) where dynamical (i.e. two-time) observables are used as constraints, here we choose a dynamical model for the latent variable, namely the AR(1) process, and introduce an estimation method for them. The persistence pattern associated with the copying mechanism described by Eq. 1.2 can coexist with the node ﬁtnesses evolving in time according to Eq. 1.4. This can be captured by the following speciﬁcation of the model in Eq. 1.1, 5 ##LTLine## ##LTLine## ##LTLine##

text: N

text: where

text: φ

text: R,

text: φ

text: and

text: the

text: i.i.d.

text: variables

text: This

text: choice

text: is

text: consistent

text: with

text: the

text: i

text: i

text: Markovian

text: assumption

text: in

text: Eq.

text: Moreover,

text: the

text: hidden

text: node

text: state

text: θt

text: evolves

text: in

text: R

text: between

text: timesteps,

text: i

text: but

text: large

text: changes

text: are

text: unlikely

text: because

text: of

text: the

text: Gaussian

text: transition

text: probabilities.

text: This

text: is

text: consistent

text: with

text: the

text: idea

text: that

text: the

text: network

text: topology

text: changes

text: smoothly

text: in

text: time.

text: Finally,

text: assuming

text: a

text: Gaussian

text: transition

text: probability

text: represents

text: a

text: simpliﬁcation

text: for

text: model

text: estimation.

text: The

text: conditional

text: probability

text: for

text: the

text: network

text: at

label: equation

text: t

text: is

text: P(At|Θt)

text: eAij(θit+θjt)

text: e(θit+θjt)

text: i,j>i

text: where

text: Θt

text: θt

text: is

text: the

text: vector

text: of

text: time-varying

text: parameters.

text: In

text: Eq.

text: we

text: assume

text: that

text: each

text: link

text: i

text: is

text: independently

text: sampled

text: and

text: the

text: probability

text: of

text: a

text: link

text: between

text: node

text: i

text: and

text: node

text: j

text: at

text: time

text: t

text: is

text: determined

text: by

text: the

text: corresponding

text: θt

text: and

text: θt.

text: The

text: larger

text: is

text: θt,

text: the

text: larger

text: is

text: the

text: probability

text: for

text: all

text: links

text: incident

text: to

text: i

text: j

text: i

text: node

text: i.

text: We

text: refer

text: to

text: this

text: model

text: as

text: Temporally

text: Generalized

text: Random

text: Graphs

text: (TGRG)

text: and

text: the

text: speciﬁcation

text: of

text: Eq.

text: for

text: the

text: TGRGis

text: the

text: following,

text: P(θt

text: θt

text: Φ

text: f(θt

text: φ

text: φ

text: θt

text: i

text: N

text: t)

text: i

text: i

text: =i

text: eAtij(θit+θjt)

text: i,j>i

text: ij

text: i

text: j

text: i,j>i

text: and

text: P(Θt

text: Θt

text: Φ)

text: N

text: P(θt

text: θt

text: Φ

text: according

text: to

text: the

text: hypothesis

text: of

text: independence,

text: where

text: f(θt

text: φ

text: i

text: i

text: i

text: i

text: φ

text: θt

text: is

text: the

text: density

text: of

text: normal

text: variable

text: with

text: mean

text: φ

text: +φ

text: θt

text: and

text: variance

text: The

text: set

text: of

text: static

text: i

text: i

text: i

text: i

text: parameters

text: is

text: Φ

text: Φ

text: with

text: Φ

text: φ

text: φ

text: σ

text: i

text: i

text: i

text: The

text: TGRG

text: model

text: is

text: fully

text: determined

text: by

text: the

text: N

text: static

text: parameters

text: Φ.

text: In

text: the

text: next

text: Section

text: we

text: propose

text: an

text: Expectation-Maximization

text: scheme

text: to

text: estimate

text: the

text: model

text: parameters

text: and

text: time-varying

text: parameters.

text: It

text: alternates

text: between

text: an

text: Expectation

text: step

text: where

text: we

text: ﬁt

text: the

text: time-varying

text: parameters

text: Θt

text: and

text: the

text: Maximization

text: step

text: where

text: we

text: maximize

text: the

text: log-likelihood

text: of

text: the

text: static

text: parameters

text: conditional

text: on

text: the

text: expectations

text: Θˆt

text: Time

text: autocorrelated

text: node

text: ﬁtnesses

text: may

text: induce

text: link

text: persistence.

text: In

text: fact,

text: the

text: probability

text: of

text: a

text: link

text: between

text: two

text: speciﬁc

text: nodes

text: e(θit+θjt))

text: is

text: persistent

text: if

text: θt

text: and

text: θt

text: are

text: autocorrelated.

text: Note

text: i

text: j

text: that

text: link

text: persistence

text: occurs

text: as

text: a

text: consequence

text: of

text: node

text: properties.

text: For

text: TGRG,

text: the

text: two-point

text: distribution

text: function

text: for

text: lagged

text: links

text: and

text: the

text: ACF

text: of

text: link

text: state

text: can

text: be

text: semi-analytically

text: computed

text: (see

text: the

text: Appendix).

text: The

text: generalization

text: of

text: the

text: TGRG

text: model

text: for

text: directed

text: networks

text: can

text: be

text: accomplished

text: by

text: distinguishing

text: between

text: the

text: out-degree

text: and

text: the

text: in-degree

text: and

text: by

text: introducing

text: two

text: ﬁtnesses

text: for

text: each

text: node

text: i,

text: i.e.

text: θt,out

text: i

text: and

text: θt,in.

text: The

text: probability

text: of

text: a

text: link

text: from

text: node

text: i

text: to

text: node

text: j

text: at

text: time

text: t

text: given

text: the

text: latent

text: variables

text: θt,out

text: i

text: i

text: and

text: θt,in

text: is

text: P(At

text: |θt,out,

text: θt,in)

text: eAtij(θit,out+θjt,in)

text: Then,

text: everything

text: follows

text: similarly

text: to

text: the

text: undirected

text: j

text: ij

text: i

text: j

text: e(θit,out+θjt,in)

text: case

text: with

text: the

text: exception

text: that

text: P(At

text: Θt)

text: is

text: invariant

text: under

text: a

text: linear

text: transformation

text: for

text: the

text: hidden

text: node

text: states:

text: θt,out

text: θt,out

text: c

text: i

text: N,

text: θt,in

text: θt,in

text: c

text: j

text: N,

text: where

text: c

text: i

text: i

text: t

text: j

text: j

text: t

text: t

text: This

text: symmetry

text: arises

text: because

text: the

text: total

text: number

text: of

text: outgoing

text: links

text: has

text: to

text: be

text: equal

text: to

text: the

text: total

text: number

text: of

text: ingoing

text: links

text: at

text: each

text: time.

text: This

text: degeneracy

text: can

text: be

text: simply

text: removed

text: by

text: taking

text: one

text: of

text: the

text: ﬁtnesses

text: as

text: constant

text: in

text: time.

text: Finally,

text: let

text: us

text: notice

text: that

text: we

text: can

text: interpret

text: TGRG

text: as

text: an

text: extension

text: of

text: Exponential

text: Random

text: Graphs

text: (ERG)

text: to

text: the

text: dynamic

text: case.

text: ERG

text: ensembles

text: are

text: probability

text: distributions

text: of

text: networks

text: obtained

text: by

text: maximizing

text: the

text: Shannon

text: entropy

text: under

text: some

text: constraints

text: on

text: the

text: average

text: value

text: of

text: a

text: set

text: of

text: network

text: observ-

text: ables.

text: If

text: this

text: set

text: is

text: the

text: degree

text: sequence,

text: the

text: Lagrange

text: multipliers

text: of

text: the

text: entropy

text: constrained

text: optimization

text: can

text: be

text: directly

text: linked

text: to

text: the

text: latent

text: variables

text: of

text: our

text: model.

text: Diﬀerently

text: from

text: other

text: dynamic

text: extension

text: of

text: ERG

text: (see

text: for

text: example

text: where

text: dynamical

text: (i.e.

text: two-time)

text: observables

text: are

text: used

text: as

text: constraints,

text: here

text: we

text: choose

text: a

text: dynamical

text: model

text: for

text: the

text: latent

text: variable,

text: namely

text: the

text: process,

text: and

text: introduce

text: an

text: estimation

text: method

text: for

text: them.

text: The

text: persistence

text: pattern

text: associated

text: with

text: the

text: copying

text: mechanism

text: described

text: by

text: Eq.

text: can

text: coexist

text: with

text: the

text: node

text: ﬁtnesses

text: evolving

text: in

text: time

text: according

text: to

text: Eq.

text: This

text: can

text: be

text: captured

text: by

text: the

text: following

text: speciﬁcation

text: of

text: the

text: model

text: in

text: Eq.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##