text: 5 (cid:54) tive interactions (α and β = 0), the dramatic effect of trust In Ref. [11] one ﬁnds an interesting exercise in which peo- is: A disconnected trading network without trust becomes a ple are asked to construct wealth distribution that they con- robust and connected network when trust is affecting agents sider ideal, and also an estimated distribution, based on their in deciding transactions. information, and these data are compared with actual data for wealth distribution in the USA. In Fig. 6 we compare the pub- We have also observed that in the calculations with trust lished results with our calculations. the state variables remain positive or zero, while in the calcu- lations without trust a small number of agents have negative values. This means that these agents are not only poor but in 40 ideal 60 estimated debt but the number of such agents is very small, such that on 30 ds=0 ds=0.2 40 average there are 160 agents with negative values in a popula- 20 tion of 5000 agents after 200000 time steps. 10 20 0 0 As one of our main research foci we compare the distri- bution of wealth calculated from our model with that of the 100 actual 100 actual 80 ds=0.4 80 ds=0.4 without trust actual statistical data. For comparison we depict in Fig. 5 60 60 the histograms of the actual wealth distribution in the USA 40 40 20 20 (in blue), together with our results from 10 realisations of the 0 top 2nd middle 4th bottom 0 top 2nd middle 4th bottom network with 500 agents, without and with trust included. It 20% 20%20% 20%20% 20% 20%20% 20%20% should be noted that our model predicts a concentration of wealth in the hands of few agents, regardless of the role of FIG. 6: Ideal, estimated, and actual wealth distribution in the USA trust, although the concentration of such agents is much less (in blue, taken from [11]) compared with numerically calculated pronounced than what we see in reality. In the calculation wealth distribution from our model (in yellow) using the parameters including trust the distribution amongst middle and bottom indicated in each histogram. percentages agree quantitatively with the data, but we are not able to ﬁnd a good match with data for the upper 40% of the wealth. This could be due to the small size and randomness of Here we ﬁnd that our calculation results seem to compare our model network, nevertheless the trend is already notice- very well with the ideal distribution when trust is included able. and no spread of prices is allowed. This situation probably represents a country that has strict control of unique prices for goods, set by the government or some monopoly. The estimated distribution is well reproduced with a rather small 40 spread of prices and having trust included. However, the ac- tual situation is somewhat disappointing, since the best ﬁt is 30 with a large spread of prices. In the bottom right panel we show the situation in which the network agents lack trust in 20 their transactions thus rendering the outcome unrealistic. 10 The situation is quite different for such egalitarian soci- eties as Denmark, in which case we have found that for very 0 small spread of prices and including trust our model agrees extremely well with the actual data (taken from [12]) of the top next next next upper middle bottom 1% 4% 5% 10% middle 20% 20% income distribution for the 1992 statistics. Using data from 20% the money variable only, we show the comparison in Fig. 7, 40 which turns out to be very favourable. Another maybe better way to compare our results with real 30 statistical data is to investigate the distribution of money in different systems, since it depends on the actual mechanisms 20 of acquiring money. For instance, there are data on the annual 10 income of people in Europe, which ranges from 0 to millions of euros. In Fig. 8(a) we compare the actual data (in red) 0 with a histogram from our numerical calculation with 1000 top next next next upper middle bottom agents but without including trust. In Fig. 8(b) we compare 1% 4% 5% 10% middle 20% 20% the same data with the results of a calculation with 250 agents 20% and a dispersion of prices twice as large as in (a) but once again without including trust. Both these calculations do not FIG. 5: Wealth distribution in the USA (in blue, taken from [13]) seem to ﬁt with the real data, neither do averages over many and numerically calculated wealth distribution from our model (in yellow) including trust (top) and without including it (bottom). In realisations. both calculations the average price was s¯ = 1.2 and the spread ds = 0.4. However, a rather different situation is encountered in a closed system as we have found out when we investigated data yb denwo htlaew tnecrep yb denwo htlaew tnecrep yb denwo htlaew tnecrep ##LTFigure## ##LTFigure## ##LTLine##

text: tive

text: interactions

text: (α

text: and

text: β

text: the

text: dramatic

text: effect

text: of

text: trust

text: In

text: Ref.

text: one

text: ﬁnds

text: an

text: interesting

text: exercise

text: in

text: which

text: peo-

text: is:

text: A

text: disconnected

text: trading

text: network

text: without

text: trust

text: becomes

text: a

text: ple

text: are

text: asked

text: to

text: construct

text: wealth

text: distribution

text: that

text: they

text: con-

text: robust

text: and

text: connected

text: network

text: when

text: trust

text: is

text: affecting

text: agents

text: sider

text: ideal,

text: and

text: also

text: an

text: estimated

text: distribution,

text: based

text: on

text: their

text: in

text: deciding

text: transactions.

text: information,

text: and

text: these

text: data

text: are

text: compared

text: with

text: actual

text: data

text: for

text: wealth

text: distribution

text: in

text: the

text: USA.

text: In

text: Fig.

text: we

text: compare

text: the

text: pub-

text: We

text: have

text: also

text: observed

text: that

text: in

text: the

text: calculations

text: with

text: trust

text: lished

text: results

text: with

text: our

text: calculations.

text: the

text: state

text: variables

text: remain

text: positive

text: or

text: zero,

text: while

text: in

text: the

text: calcu-

text: lations

text: without

text: trust

text: a

text: small

text: number

text: of

text: agents

text: have

text: negative

text: values.

text: This

text: means

text: that

text: these

text: agents

text: are

text: not

text: only

text: poor

text: but

text: in

text: ideal

text: estimated

text: debt

text: but

text: the

text: number

text: of

text: such

text: agents

text: is

text: very

text: small,

text: such

text: that

text: on

text: average

text: there

text: are

text: agents

text: with

text: negative

text: values

text: in

text: a

text: popula-

text: tion

text: of

text: agents

text: after

text: time

text: steps.

text: As

text: one

text: of

text: our

text: main

text: research

text: foci

text: we

text: compare

text: the

text: distri-

text: bution

text: of

text: wealth

text: calculated

text: from

text: our

text: model

text: with

text: that

text: of

text: the

text: actual

text: actual

text: without

text: trust

text: actual

text: statistical

text: data.

text: For

text: comparison

text: we

text: depict

text: in

text: Fig.

text: the

text: histograms

text: of

text: the

text: actual

text: wealth

text: distribution

text: in

text: the

text: USA

text: (in

text: blue),

text: together

text: with

text: our

text: results

text: from

text: realisations

text: of

text: the

text: top

text: middle

text: bottom

text: top

text: middle

text: bottom

text: network

text: with

text: agents,

text: without

text: and

text: with

text: trust

text: included.

text: It

text: should

text: be

text: noted

text: that

text: our

text: model

text: predicts

text: a

text: concentration

text: of

text: wealth

text: in

text: the

text: hands

text: of

text: few

text: agents,

text: regardless

text: of

text: the

text: role

text: of

text: FIG.

text: Ideal,

text: estimated,

text: and

text: actual

text: wealth

text: distribution

text: in

text: the

text: USA

text: trust,

text: although

text: the

text: concentration

text: of

text: such

text: agents

text: is

text: much

text: less

text: (in

text: blue,

text: taken

text: from

text: compared

text: with

text: numerically

text: calculated

text: pronounced

text: than

text: what

text: we

text: see

text: in

text: reality.

text: In

text: the

text: calculation

text: wealth

text: distribution

text: from

text: our

text: model

text: (in

text: yellow)

text: using

text: the

text: parameters

text: including

text: trust

text: the

text: distribution

text: amongst

text: middle

text: and

text: bottom

text: indicated

text: in

text: each

text: histogram.

text: percentages

text: agree

text: quantitatively

text: with

text: the

text: data,

text: but

text: we

text: are

text: not

text: able

text: to

text: ﬁnd

text: a

text: good

text: match

text: with

text: data

text: for

text: the

text: upper

text: of

text: the

text: wealth.

text: This

text: could

text: be

text: due

text: to

text: the

text: small

text: size

text: and

text: randomness

text: of

text: Here

text: we

text: ﬁnd

text: that

text: our

text: calculation

text: results

text: seem

text: to

text: compare

text: our

text: model

text: network,

text: nevertheless

text: the

text: trend

text: is

text: already

text: notice-

text: very

text: well

text: with

text: the

text: ideal

text: distribution

text: when

text: trust

text: is

text: included

text: able.

text: and

text: no

text: spread

text: of

text: prices

text: is

text: allowed.

text: This

text: situation

text: probably

text: represents

text: a

text: country

text: that

text: has

text: strict

text: control

text: of

text: unique

text: prices

text: for

text: goods,

text: set

text: by

text: the

text: government

text: or

text: some

text: monopoly.

text: The

text: estimated

text: distribution

text: is

text: well

text: reproduced

text: with

text: a

text: rather

text: small

text: spread

text: of

text: prices

text: and

text: having

text: trust

text: included.

text: However,

text: the

text: ac-

text: tual

text: situation

text: is

text: somewhat

text: disappointing,

text: since

text: the

text: best

text: ﬁt

text: is

text: with

text: a

text: large

text: spread

text: of

text: prices.

text: In

text: the

text: bottom

text: right

text: panel

text: we

text: show

text: the

text: situation

text: in

text: which

text: the

text: network

text: agents

text: lack

text: trust

text: in

text: their

text: transactions

text: thus

text: rendering

text: the

text: outcome

text: unrealistic.

text: The

text: situation

text: is

text: quite

text: different

text: for

text: such

text: egalitarian

text: soci-

text: eties

text: as

text: Denmark,

text: in

text: which

text: case

text: we

text: have

text: found

text: that

text: for

text: very

text: small

text: spread

text: of

text: prices

text: and

text: including

text: trust

text: our

text: model

text: agrees

text: extremely

text: well

text: with

text: the

text: actual

text: data

text: (taken

text: from

text: of

text: the

text: top

text: next

text: next

text: next

text: upper

text: middle

text: bottom

text: middle

text: income

text: distribution

text: for

text: the

text: statistics.

text: Using

text: data

text: from

text: the

text: money

text: variable

text: only,

text: we

text: show

text: the

text: comparison

text: in

text: Fig.

text: which

text: turns

text: out

text: to

text: be

text: very

text: favourable.

text: Another

text: maybe

text: better

text: way

text: to

text: compare

text: our

text: results

text: with

text: real

text: statistical

text: data

text: is

text: to

text: investigate

text: the

text: distribution

text: of

text: money

text: in

text: different

text: systems,

text: since

text: it

text: depends

text: on

text: the

text: actual

text: mechanisms

text: of

text: acquiring

text: money.

text: For

text: instance,

text: there

text: are

text: data

text: on

text: the

text: annual

text: income

text: of

text: people

text: in

text: Europe,

text: which

text: ranges

text: from

text: to

text: millions

text: of

text: euros.

text: In

text: Fig.

text: we

text: compare

text: the

text: actual

text: data

text: (in

text: red)

text: with

text: a

text: histogram

text: from

text: our

text: numerical

text: calculation

text: with

text: top

text: next

text: next

text: next

text: upper

text: middle

text: bottom

text: agents

text: but

text: without

text: including

text: trust.

text: In

text: Fig.

text: we

text: compare

text: middle

text: the

text: same

text: data

text: with

text: the

text: results

text: of

text: a

text: calculation

text: with

text: agents

text: and

text: a

text: dispersion

text: of

text: prices

text: twice

text: as

text: large

text: as

text: in

text: (a)

text: but

text: once

text: again

text: without

text: including

text: trust.

text: Both

text: these

text: calculations

text: do

text: not

text: FIG.

text: Wealth

text: distribution

text: in

text: the

text: USA

text: (in

text: blue,

text: taken

text: from

text: seem

text: to

text: ﬁt

text: with

text: the

text: real

text: data,

text: neither

text: do

text: averages

text: over

text: many

text: and

text: numerically

text: calculated

text: wealth

text: distribution

text: from

text: our

text: model

text: (in

text: yellow)

text: including

text: trust

text: (top)

text: and

text: without

text: including

text: it

text: (bottom).

text: In

text: realisations.

text: both

text: calculations

text: the

text: average

text: price

text: was

text: s¯

text: and

text: the

text: spread

text: ds

text: However,

text: a

text: rather

text: different

text: situation

text: is

text: encountered

text: in

text: a

text: closed

text: system

text: as

text: we

text: have

text: found

text: out

text: when

text: we

text: investigated

text: data

text: yb

text: denwo

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text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##


text: Lamia Tounsi. 2014. Dcu: Aspect-based polarity classiﬁcation for semeval task 4. In Proceedings of the 8th International Workshop on Semantic Evalua- tion (SemEval 2014), pages 223–229. Jason Weston, Sumit Chopra, and Antoine Bordes. 2014. Memory networks. arXiv preprint arXiv:1410.3916. Zichao Yang, Diyi Yang, Chris Dyer, Xiaodong He, Alex Smola, and Eduard Hovy. 2016. Hierarchical atten- tion networks for document classiﬁcation. In Proceed- ings of the 2016 Conference of the North American Chapter of the Association for Computational Linguis- tics. Ainur Yessenalina and Claire Cardie. 2011. Compo- sitional matrix-space models for sentiment analysis. In Proceedings of the 2011 Conference on Empirical Methods in Natural Language Processing, pages 172– 182. Wenpeng Yin and Hinrich Schu¨tze. 2015. Multichan- nel variable-size convolution for sentence classiﬁca- tion. In Proceedings of the Nineteenth Conference on Computational Natural Language Learning, pages 204–214. Xiaodan Zhu, Parinaz Sobhani, and Hongyu Guo. 2015. Long short-term memory over tree structures. In Pro- ceedings of The 32nd International Conference on Ma- chine Learning, pages 1604–1612.

text: Lamia

label: reference

text: Tounsi.

text: Dcu:

text: Aspect-based

text: polarity

text: classiﬁcation

text: for

text: semeval

text: task

text: In

text: Proceedings

text: of

text: the

text: International

text: Workshop

text: on

text: Semantic

text: Evalua-

text: tion

text: (SemEval

text: pages

text: Jason

text: Weston,

text: Sumit

text: Chopra,

text: and

text: Antoine

text: Bordes.

text: Memory

text: networks.

text: arXiv

text: preprint

text: Zichao

text: Yang,

text: Diyi

text: Yang,

text: Chris

text: Dyer,

text: Xiaodong

text: He,

text: Alex

text: Smola,

text: and

text: Eduard

text: Hovy.

text: Hierarchical

text: atten-

text: tion

text: networks

text: for

text: document

text: classiﬁcation.

text: In

text: Proceed-

text: ings

text: of

text: the

text: Conference

text: of

text: the

text: North

text: American

text: Chapter

text: of

text: the

text: Association

text: for

text: Computational

text: Linguis-

text: tics.

text: Ainur

text: Yessenalina

text: and

text: Claire

text: Cardie.

text: Compo-

text: sitional

text: matrix-space

text: models

text: for

text: sentiment

text: analysis.

text: In

text: Proceedings

text: of

text: the

text: Conference

text: on

text: Empirical

text: Methods

text: in

text: Natural

text: Language

text: Processing,

text: pages

text: Wenpeng

text: Yin

text: and

text: Hinrich

text: Schu¨tze.

text: Multichan-

text: nel

text: variable-size

text: convolution

text: for

text: sentence

text: classiﬁca-

text: tion.

text: In

text: Proceedings

text: of

text: the

text: Nineteenth

text: Conference

text: on

text: Computational

text: Natural

text: Language

text: Learning,

text: pages

text: Xiaodan

text: Zhu,

text: Parinaz

text: Sobhani,

text: and

text: Hongyu

text: Guo.

text: Long

text: short-term

text: memory

text: over

text: tree

text: structures.

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text: Pro-

text: ceedings

text: of

text: The

text: International

text: Conference

text: on

text: Ma-

text: chine

text: Learning,

text: pages


text: (cid:88) • Model 1. Following Sukhbaatar et al. (2015), k we calculate the memory vector m with vec = α m (1) i i i (cid:12) (cid:80) i=1 m = e v (4) i i i ∈ where k is the memory size, α [0, 1] is the i weight of m and α = 1. We implement a neu- (cid:12) i i i where means element-wise multiplication and ral network based attention model. For each piece of ∈ R × v d 1 is a location vector for word w . Every memory m , we use a feed forward neural network i i i element in v is calculated as follows, to compute its semantic relatedness with the aspect. i The scoring function is calculated as follows, where Watt ∈ R1×2d and batt ∈ R1×1. vk = (1 − l /n) − (k/d)(1 − 2 × l /n) (5) i i i g = tanh(W [m ; v ] + b ) (2) i att i aspect att where n is sentence length, k is the hop number { } and l is the location of w . After obtaining g , g , ... g , we feed them to a • i i 1 2 k Model 2. This is a simpliﬁed version of Model softmax function to calculate the ﬁnal importance { } 1, using the same location vector v for w in differ- scores α , α , ... α . i i 1 2 k(cid:80) ent hops. Location vector v is calculated as follows. i exp(g ) α = i (3) − i k exp(g ) vi = 1 li/n (6) j=1 j We believe that such an attention model has two • Model 3. We regard location vector v as a pa- i advantages. One advantage is that this model could rameter and compute a piece of memory with vector adaptively assign an importance score to each piece addition, namely of memory m according to its semantic relatedness i with the aspect. Another advantage is that this at- m = e + v (7) tention model is differentiable, so that it could be i i i easily trained together with other components in an end-to-end fashion. All the position vectors are stacked in a position 3.4 Location Attention embedding matrix, which is jointly learned with gra- dient descent. We have described our neural attention frame- • Model 4. Location vectors are also regarded as work and a content-based model in previous subsec- parameters. Different from Model 3, location rep- tion. However, the model mentioned above ignores resentations are regarded as neural gates to control the location information between context word and how many percent of word semantics is written into aspect. Such location information is helpful for an attention model because intuitively a context word the memory. We feed location vector vi to a sigmoid closer to the aspect should be more important than a function σ, and calculate mi with element-wise mul- tiplication: farther one. In this work, we deﬁne the location of a context word as its absolute distance with the aspect (cid:12) in the original sentence sequence3. On this basis, we m = e σ(v ) (8) i i i study four strategies to encode the location informa- tion in the attention model. The details are described 3.5 The Need for Multiple Hops below. It is widely accepted that computational mod- 3The location of a context word could also be measured by els that are composed of multiple processing layers its distance to the aspect along a syntactic path. We leave this have the ability to learn representations of data with as a future work as we prefer to developing a purely data-driven approach without using external parsing results. multiple levels of abstraction (LeCun et al., 2015). ##LTLine## ##LTLine##

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text: Model

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text: Laptop Restaurant achieved when the model contains seven and nine Majority 53.45 65.00 hops, respectively. On both datasets, the proposed Feature+SVM 72.10 80.89 approach could obtain comparable accuracy com- LSTM 66.45 74.28 pared to the state-of-art feature-based SVM system. TDLSTM 68.13 75.63 4.3 Runtime Analysis TDLSTM+ATT 66.24 74.31 We study the runtime of recurrent neural models ContextAVG 61.22 71.33 and the proposed deep memory network approach MemNet (1) 67.66 76.10 with different hops. We implement all these ap- MemNet (2) 71.14 78.61 proaches based on the same neural network infras- MemNet (3) 71.74 79.06 tructure, use the same 300-dimensional Glove word MemNet (4) 72.21 79.87 vectors, and run them on the same CPU server. MemNet (5) 71.89 80.14 MemNet (6) 72.21 80.05 Method Time cost MemNet (7) 72.37 80.32 LSTM 417 MemNet (8) 72.05 80.14 TDLSTM 490 MemNet (9) 72.21 80.95 TDLSTM + ATT 520 Table 2: Classiﬁcation accuracy of different methods on laptop MemNet (1) 3 and restaurant datasets. Best scores in each group are in bold. MemNet (2) 7 MemNet (3) 9 MemNet (4) 15 the importance of a powerful feature representation MemNet (5) 20 for aspect level sentiment classiﬁcation. Among MemNet (6) 24 three recurrent models, TDLSTM performs better MemNet (7) 26 than LSTM, which indicates that taking into account MemNet (8) 27 of the aspect information is helpful. This is reason- MemNet (9) 29 able as the sentiment polarity of a sentence towards Table 3: Runtime (seconds) of each training epoch on the different aspects (e.g. “food” and “service”) might restaurant dataset. be different. It is somewhat disappointing that in- corporating attention model over TDLSTM does not The training time of each iteration on the restau- bring any improvement. We consider that each hid- rant dataset is given in Table 3. We can ﬁnd that den vector of TDLSTM encodes the semantics of LSTM based recurrent models are indeed compu- word sequence until the current position. Therefore, tationally expensive, which is caused by the com- the model of TDLSTM+ATT actually selects such plex operations in each LSTM unit along the word mixed semantics of word sequence, which is weird sequence. Instead, the memory network approach and not an intuitive way to selectively focus on parts is simpler and evidently faster because it does not of contexts. Different from TDLSTM+ATT, the pro- need recurrent calculators of sequence length. Our posed memory network approach removes the recur- approach with nine hops is almost 15 times faster rent calculator over word sequence and directly ap- than the basic LSTM model. ply attention mechanism on context word represen- tations. 4.4 Effects of Location Attention We can also ﬁnd that the performance of Contex- As described in Section , we explore four strate- tAVG is very poor, which means that assigning the gies to integrate location information into the atten- same weight/importance to all the context words is tion model. We incorporate each of them separately not an effective way. Among all our models from into the basic content-based attention model. It is single hop to nine hops, we can observe that using helpful to restate that the difference between four more computational layers could generally lead to location-based attention models lies in the usage of better performance, especially when the number of location vectors for context words. In Model 1 and hops is less than six. The best performances are Model 2, the values of location vectors are ﬁxed and ##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##

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text: 1 log10(-1/f) 1.5 f -f b 0 0.5 log (-b) 10 1 0 0 1 2 log r 10 1 0.5 -b -10 f 0.5 b 0 0 0 1 2 log r 10 -20 0.1 0.15 0.2 0.25 0.3 0.28 0.29 0.3 r r 10 1 h h 2 W 0 h 2 5 -1 0 1 2 1 h log r 1 10 W 0 0 0 -1 0 1 2 -5 W log10r W -1 -10 0.1 0.15 0.2 0.25 0.3 0.28 0.29 0.3 r r 0.01 0.6 φ 1000bR+1/2 0 0.4 0 φ -1 bR -0.01 0 φ’ 0.2 φ’/1000 -2 φ’ 0 1 2 -0.05 φ φ log10r 0 -0.02 φ’/100 -0.1 0 1 2 log r 10 -0.2 0.1 0.15 0.2 0.25 0.3 0.28 0.29 0.3 r r Figure 4: Behaviour of the various metric functions, the scalar ﬁeld, its derivative and the curvature scalar for illustrative example I (left panels) and II (right panels). hair [26–30]. Firstly we have argued, using linear theory, that imposing the synchronization condition on the outer horizon prevents strong spatial oscillations; but then, these oscillations cannot be avoided at the inner (Cauchy) horizon. We have exempliﬁed this feature both for stationary scalar clouds around Kerr BHs and for similar clouds for an equal spinning D = 5 MP BH (in a cavity), establishing a parallelism between the two 12 noziroh renni noziroh renni noziroh renni noziroh retuo noziroh retuo noziroh retuo noziroh retuo noziroh retuo noziroh retuo ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

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text: Discrete Mathematics and Theoretical Computer Science DMTCS vol. 20:1, 2018, #6 A study of -dipath colourings of oriented k graphs ´ Christopher Duffy1 Gary MacGillivray2∗ Eric Sopena3 1 Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon, CANADA 2 Department of Mathematics and Statistics, University of Victoria, Victoria, CANADA 3 Univ. Bordeaux, Bordeaux INP, CNRS, LaBRI, UMR5800, F-33400, Talence, FRANCE received 27thJune 2016, revised 3rdAug. 2017, accepted 13thJan. 2018. ≥ We examine t-colourings of oriented graphs in which, for a ﬁxed integer k 1, vertices joined by a directed path of length at most k must be assigned different colours. A homomorphism model that extends the ideas of Sherk for the case k = 2 is described. Dichotomy theorems for the complexity of the problem of deciding, for ﬁxed k and t, whether there exists such a t-colouring are proved. Keywords: Directed Graph, Oriented Graph, Graph Colouring, Graph Homomorphism 1 Introduction Recall that an oriented graph is a digraph obtained from a simple, undirected graph by giving each edge one of its two possible orientations. Recall, also, that if G and H are oriented graphs, then a homomorphism of G to H is a function φ from the vertices of G to the vertices of H such that φ(x)φ(y) ∈ E(H) whenever xy ∈ E(G). If G and H are oriented graphs such that there is a homomorphism φ of G to H, then we write φ : G → H, or G → H if the name of the function φ is not important. Let k and t be positive integers, and let G be an oriented graph. Chen and Wang (2006) deﬁned a k-dipath t-colouring of G to be an assignment of t colours to the vertices of G so that any two vertices joined by a directed path of length at most k are assigned different colours. A 1-dipath t-colouring of an oriented graph G is a t-colouring of the underlying undirected graph of G. See Figure 1 for an example of a 3-dipath 4-colouring of an oriented graph. The k-dipath chromatic number of G, denoted by χk-dip(G), is the smallest positive integer t such that there exists a k-dipath t-colouring of G. Chen and Wang (2006) showed that any orientation of a Halin graph has 2-dipath chromatic number at most 7, and there are inﬁnitely many such graphs G with χ2-dip(G) = 7. For a positive integer t, an oriented t-colouring of an oriented graph G is a homomorphism of G to some oriented graph on t vertices. Oriented colourings were ﬁrst introduced by Courcelle (1994), and have been a topic of considerable interest in the literature since then; see the recent survey by Sopena (2015), and also work by Borodin et al. (1999), Dolama and Sopena (2006), Sopena and Wu (2013) for ∗ This author acknowledges support from NSERC. (cid:13) ISSN 1365–8050 c 2018 by the author(s) Distributed under a Creative Commons Attribution 4.0 International License ##LTLine## ##LTLine## ##LTLine##

text: Discrete

text: Mathematics

text: and

text: Theoretical

text: Computer

text: Science

text: DMTCS

text: vol.

text: A

label: title

font: UEUSQP+NimbusSanL-ReguItal

text: study

text: of

text: -dipath

text: colourings

text: of

text: oriented

text: k

text: graphs

label: author

text: Christopher

text: Gary

text: Eric

text: Department

text: of

text: Mathematics

text: and

text: Statistics,

text: University

text: of

text: Saskatchewan,

text: Saskatoon,

text: CANADA

text: Department

text: of

text: Mathematics

text: and

text: Statistics,

text: University

text: of

text: Victoria,

text: Victoria,

text: CANADA

text: Univ.

text: Bordeaux,

text: Bordeaux

text: INP,

text: CNRS,

text: LaBRI,

text: Talence,

text: FRANCE

text: received

text: revised

text: accepted

label: abstract

text: We

text: examine

text: t-colourings

text: of

text: oriented

text: graphs

text: in

text: which,

text: for

text: a

text: ﬁxed

text: integer

text: k

text: vertices

text: joined

text: by

text: a

text: directed

text: path

text: of

text: length

text: at

text: most

text: k

text: must

text: be

text: assigned

text: different

text: colours.

text: A

text: homomorphism

text: model

text: that

text: extends

text: the

text: ideas

text: of

text: Sherk

text: for

text: the

text: case

text: k

text: is

text: described.

text: Dichotomy

text: theorems

text: for

text: the

text: complexity

text: of

text: the

text: problem

text: of

text: deciding,

text: for

text: ﬁxed

text: k

text: and

text: t,

text: whether

text: there

text: exists

text: such

text: a

text: t-colouring

text: are

text: proved.

text: Keywords:

text: Directed

text: Graph,

text: Oriented

text: Graph,

text: Graph

text: Colouring,

text: Graph

text: Homomorphism

text: Introduction

text: Recall

text: that

text: an

text: oriented

text: graph

text: is

text: a

text: digraph

text: obtained

text: from

text: a

text: simple,

text: undirected

text: graph

text: by

text: giving

text: each

text: edge

text: one

text: of

text: its

text: two

text: possible

text: orientations.

text: Recall,

text: also,

text: that

text: if

text: G

text: and

text: H

text: are

text: oriented

text: graphs,

text: then

text: a

text: homomorphism

text: of

text: G

text: to

text: H

text: is

text: a

text: function

text: φ

text: from

text: the

text: vertices

text: of

text: G

text: to

text: the

text: vertices

text: of

text: H

text: such

text: that

text: φ(x)φ(y)

text: E(H)

text: whenever

text: xy

text: E(G).

text: If

text: G

text: and

text: H

text: are

text: oriented

text: graphs

text: such

text: that

text: there

text: is

text: a

text: homomorphism

text: φ

text: of

text: G

text: to

text: H,

text: then

text: we

text: write

text: φ

text: G

text: H,

text: or

text: G

text: H

text: if

text: the

text: name

text: of

text: the

text: function

text: φ

text: is

text: not

text: important.

text: Let

text: k

text: and

text: t

text: be

text: positive

text: integers,

text: and

text: let

text: G

text: be

text: an

text: oriented

text: graph.

text: Chen

text: and

text: Wang

text: deﬁned

text: a

text: k-dipath

text: t-colouring

text: of

text: G

text: to

text: be

text: an

text: assignment

text: of

text: t

text: colours

text: to

text: the

text: vertices

text: of

text: G

text: so

text: that

text: any

text: two

text: vertices

text: joined

text: by

text: a

text: directed

text: path

text: of

text: length

text: at

text: most

text: k

text: are

text: assigned

text: different

text: colours.

text: A

text: t-colouring

text: of

text: an

text: oriented

text: graph

text: G

text: is

text: a

text: t-colouring

text: of

text: the

text: underlying

text: undirected

text: graph

text: of

text: G.

text: See

text: Figure

text: for

text: an

text: example

text: of

text: a

text: of

text: an

text: oriented

text: graph.

text: The

text: k-dipath

text: chromatic

text: number

text: of

text: G,

text: denoted

text: by

text: χk-dip(G),

text: is

text: the

text: smallest

text: positive

text: integer

text: t

text: such

text: that

text: there

text: exists

text: a

text: k-dipath

text: t-colouring

text: of

text: G.

text: Chen

text: and

text: Wang

text: showed

text: that

text: any

text: orientation

text: of

text: a

text: Halin

text: graph

text: has

text: chromatic

text: number

text: at

text: most

text: and

text: there

text: are

text: inﬁnitely

text: many

text: such

text: graphs

text: G

text: with

text: For

text: a

text: positive

text: integer

text: t,

text: an

text: oriented

text: t-colouring

text: of

text: an

text: oriented

text: graph

text: G

text: is

text: a

text: homomorphism

text: of

text: G

text: to

text: some

text: oriented

text: graph

text: on

text: t

text: vertices.

text: Oriented

text: colourings

text: were

text: ﬁrst

text: introduced

text: by

text: Courcelle

text: and

text: have

text: been

text: a

text: topic

text: of

text: considerable

text: interest

text: in

text: the

text: literature

text: since

text: then;

text: see

text: the

text: recent

text: survey

text: by

text: Sopena

text: and

text: also

text: work

text: by

text: Borodin

text: et

text: al.

text: Dolama

text: and

text: Sopena

text: Sopena

text: and

text: Wu

text: for

text: This

text: author

text: acknowledges

text: support

text: from

text: NSERC.

text: ISSN

text: c

text: by

text: the

text: author(s)

text: Distributed

text: under

text: a

text: Creative

text: Commons

text: Attribution

text: International

text: License

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 14 Christopher Duffy, Gary MacGillivray, E´ric Sopena earlier observation, this assignment can be extended to a k-dipath t-colouring of F. Since each vertex (cid:48) in V (G) ∩ V (G ) has in-degree 0, there is no directed path of positive length joining vertices in different (cid:48) (cid:48) copies of F, this results in a k-dipath t-colouring of G . Now suppose a k-dipath t-colouring of G is given. By our observation on colourings of F, any two vertices which are adjacent in G are assigned different colours. Hence restricting this colouring to V (G) gives a t-colouring of G. This completes the proof. (cid:50) The problem of deciding whether the k-th power of a graph G is t-colourable is known to be NP- complete (see Lin and Skiena (1995) and McCormick (1983)). The results above imply that, if t > k, the problem of deciding whether the underlying graph of the k-th power of an oriented graph G (i.e., two vertices are adjacent if and only if they are at weak distance at most k) is t-colourable is NP-complete, even when restricted to powers of oriented graphs with directed girth at least k + 1, and also that, if t = k ≥ 3, the problem of deciding whether the underlying graph of the k-th power of an oriented graph G is t- colourable is NP-complete. References J. Bang-Jensen, P. Hell, and G. MacGillivray. The Complexity of Colouring by Semicomplete Digraphs. SIAM Journal on Discrete Mathematics, 1(3):281–298, 1988. J. Bensmail, C. Duffy, and S. Sen. An analogue of clique for (m,n)-mixed colored graphs. Graphs and Combinatorics, 33(4):735–750, 2017. O. Borodin, A. Kostochka, J. Nesˇetˇril, A. Raspaud, and E´. Sopena. On the Maximum Average Degree and the Oriented Chromatic Number of a Graph. Discrete Mathematics, 206:77–89, 1999. M. Chen and W. Wang. The 2−dipath Chromatic Number of Halin Graphs. Inform. Process. Lett, 99(2): 47–53, 2006. B. Courcelle. The Monadic Second Order Logic of Graphs VI: On several representations of graphs by relational structures. Discrete Applied Mathematics, 54:117–149, 1994. M. H. Dolama and E. Sopena. On the Oriented Chromatic number of Halin Graphs. Inform. Proc. Letters, 98(6):247–252, 2006. W. Fellner. On minimal graphs. Theoretical Computer Science, 17(1):103–110, 1982. T. Gallai. On Directed Graphs and Circuits. In Theory of Graphs (Proceedings of the Colloquium Tihany 1966), pages 115–118, 1967. M. Hasse. Zur algebraischen begru¨ndung der graphentheorie. i. Mathematische Nachrichten, 28(5-6): 275–290, 1965. Y. Lin and S. Skiena. Algorithms for square roots of graphs. SIAM Journal on Discrete Mathematics, 8 (1):99–118, 1995. G. MacGillivray and K. Sherk. A Theory of 2−dipath Colourings. Australasian Journal of Combinatorics, 60(1):11–26, 2014.

text: Christopher

text: Duffy,

text: Gary

text: MacGillivray,

text: E´ric

text: Sopena

text: earlier

text: observation,

text: this

text: assignment

text: can

text: be

text: extended

text: to

text: a

text: k-dipath

text: t-colouring

text: of

text: F.

text: Since

text: each

text: vertex

text: in

text: V

text: (G)

text: V

text: (G

text: has

text: in-degree

text: there

text: is

text: no

text: directed

text: path

text: of

text: positive

text: length

text: joining

text: vertices

text: in

text: different

text: copies

text: of

text: F,

text: this

text: results

text: in

text: a

text: k-dipath

text: t-colouring

text: of

text: G

text: Now

text: suppose

text: a

text: k-dipath

text: t-colouring

text: of

text: G

text: is

text: given.

text: By

text: our

text: observation

text: on

text: colourings

text: of

text: F,

text: any

text: two

text: vertices

text: which

text: are

text: adjacent

text: in

text: G

text: are

text: assigned

text: different

text: colours.

text: Hence

text: restricting

text: this

text: colouring

text: to

text: V

text: (G)

text: gives

text: a

text: t-colouring

text: of

text: G.

text: This

text: completes

text: the

text: proof.

text: The

text: problem

text: of

text: deciding

text: whether

text: the

text: k-th

text: power

text: of

text: a

text: graph

text: G

text: is

text: t-colourable

text: is

text: known

text: to

text: be

text: NP-

text: complete

text: (see

text: Lin

text: and

text: Skiena

text: and

text: McCormick

text: The

text: results

text: above

text: imply

text: that,

text: if

text: t

text: k,

text: the

text: problem

text: of

text: deciding

text: whether

text: the

text: underlying

text: graph

text: of

text: the

text: k-th

text: power

text: of

text: an

text: oriented

text: graph

text: G

text: (i.e.,

text: two

text: vertices

text: are

text: adjacent

text: if

text: and

text: only

text: if

text: they

text: are

text: at

text: weak

text: distance

text: at

text: most

text: k)

text: is

text: t-colourable

text: is

text: NP-complete,

text: even

text: when

text: restricted

text: to

text: powers

text: of

text: oriented

text: graphs

text: with

text: directed

text: girth

text: at

text: least

text: k

text: and

text: also

text: that,

text: if

text: t

text: k

text: the

text: problem

text: of

text: deciding

text: whether

text: the

text: underlying

text: graph

text: of

text: the

text: k-th

text: power

text: of

text: an

text: oriented

text: graph

text: G

text: is

text: t-

text: colourable

text: is

text: NP-complete.

text: References

text: J.

text: Bang-Jensen,

text: P.

text: Hell,

text: and

text: G.

text: MacGillivray.

text: The

text: Complexity

text: of

text: Colouring

text: by

text: Semicomplete

text: Digraphs.

text: SIAM

text: Journal

text: on

text: Discrete

text: Mathematics,

text: J.

text: Bensmail,

text: C.

text: Duffy,

text: and

text: S.

text: Sen.

text: An

text: analogue

text: of

text: clique

text: for

text: (m,n)-mixed

text: colored

text: graphs.

text: Graphs

text: and

text: Combinatorics,

text: O.

text: Borodin,

text: A.

text: Kostochka,

text: J.

text: Nesˇetˇril,

text: A.

text: Raspaud,

text: and

text: E´.

text: Sopena.

text: On

text: the

text: Maximum

text: Average

text: Degree

text: and

text: the

text: Oriented

text: Chromatic

text: Number

text: of

text: a

text: Graph.

text: Discrete

text: Mathematics,

text: M.

text: Chen

text: and

text: W.

text: Wang.

text: The

text: Chromatic

text: Number

text: of

text: Halin

text: Graphs.

text: Inform.

text: Process.

text: Lett,

text: B.

text: Courcelle.

text: The

text: Monadic

text: Second

text: Order

text: Logic

text: of

text: Graphs

text: VI:

text: On

text: several

text: representations

text: of

text: graphs

text: by

text: relational

text: structures.

text: Discrete

text: Applied

text: Mathematics,

text: M.

text: H.

text: Dolama

text: and

text: E.

text: Sopena.

text: On

text: the

text: Oriented

text: Chromatic

text: number

text: of

text: Halin

text: Graphs.

text: Inform.

text: Proc.

text: Letters,

text: W.

text: Fellner.

text: On

text: minimal

text: graphs.

text: Theoretical

text: Computer

text: Science,

text: T.

text: Gallai.

text: On

text: Directed

text: Graphs

text: and

text: Circuits.

text: In

text: Theory

text: of

text: Graphs

text: (Proceedings

text: of

text: the

text: Colloquium

text: Tihany

text: pages

text: M.

text: Hasse.

text: Zur

text: algebraischen

text: begru¨ndung

text: der

text: graphentheorie.

text: i.

text: Mathematische

text: Nachrichten,

text: Y.

text: Lin

text: and

text: S.

text: Skiena.

text: Algorithms

text: for

text: square

text: roots

text: of

text: graphs.

text: SIAM

text: Journal

text: on

text: Discrete

text: Mathematics,

text: G.

text: MacGillivray

text: and

text: K.

text: Sherk.

text: A

text: Theory

text: of

text: Colourings.

text: Australasian

text: Journal

text: of

text: Combinatorics,


text: A study of k-dipath colourings of oriented graphs 9 that no assertion can be made about the entries of A in column j and rows −(i − 1), −(i − 2), . . . , −1 because there is no guarantee that a vertex v for which there is a directed path of length at most i − 1 to x lies on the directed path from y to x. Similar considerations apply to vertices w for which there is a directed path of length i < k from x to w. These conditions are formalized succinctly as follows. Observation 12 Let A = [a ] ∈ A . Suppose a = 1. Then, ij k,t ij • if i = 0, then a = 0 for all (cid:96) (cid:54)= i; (cid:96)j • if i < 0, then a = 0, 0 ≤ (cid:96) ≤ k − i; and (cid:96)j • if i > 0, then a− = 0, 0 ≤ (cid:96) ≤ k − i. (cid:96)j We now construct an oriented graph, G , which is a universal target for the family of k-dipath t- k,t colourable oriented graphs of directed girth at least k + 1. The vertex set V (G ) = A . We describe k,t k,t the edge set informally at ﬁrst, and then formally. Suppose the matrices A = A and B = B (cid:48) are (cid:48) x,c y,c vertices of G . Let c(x) = m and c (y) = m . Then AB ∈ E(G) if k,t 1 2 1. A encodes the fact that x has an out-neighbour of colour m ; 2 2. B encodes the fact that y has an in-neighbour of colour m ; 1 3. if A encodes the fact that a vertex of colour m is joined to x by a directed path of length i < k, then 3 B must encode the fact that a vertex of colour m is joined to y by a directed path of length i + 1; 3 4. if B encodes the fact that y is joined to a vertex of colour m by a directed path of length i < k, then 3 A must encode the fact that x is joined to a vertex of colour m by a directed path of length i + 1. 3 We now formally deﬁne the edge set of G . Let A = [a ] and B = [b ] be vertices of G (i.e., k,t ij ij k,t matrices in A ). Then AB ∈ E(G ) if, whenever a = b = 1, the following conditions all hold: k,t k,t 0m1 0m2 1. a = 1; 1m2 2. b− = 1; 1m1 3. if 0 < i < k − 1 and a− = 1, then b− − = 1; and im3 (i 1)m3 4. if 0 < i < k − 1 and b = 1, then b = 1. im3 (i+1)m3 We now establish some properties of G . k,t Lemma 13 The digraph G has directed girth at least k + 1. In particular, it is an oriented graph. k,t Proof: Let A , A , . . . , A , A be a directed cycle in G . Suppose (cid:96) ≤ k, and A has a 1 in entry (0, m ). 1 2 (cid:96) 1 k,t 1 1 This implies A has a 1 in entry (−1, m ) and a 1 in entry (((cid:96) − 1), m ), contrary to Observation 12. (cid:50) 2 1 1 Lemma 14 For integers t ≥ 2 and k ≥ 2 the k-dipath chromatic number of G is at most t. k,t

text: A

text: study

text: of

text: k-dipath

text: colourings

text: of

text: oriented

text: graphs

text: that

text: no

text: assertion

text: can

text: be

text: made

text: about

text: the

text: entries

text: of

text: A

text: in

text: column

text: j

text: and

text: rows

text: −(i

text: −(i

text: −1

text: because

text: there

text: is

text: no

text: guarantee

text: that

text: a

text: vertex

text: v

text: for

text: which

text: there

text: is

text: a

text: directed

text: path

text: of

text: length

text: at

text: most

text: i

text: to

text: x

text: lies

text: on

text: the

text: directed

text: path

text: from

text: y

text: to

text: x.

text: Similar

text: considerations

text: apply

text: to

text: vertices

text: w

text: for

text: which

text: there

text: is

text: a

text: directed

text: path

text: of

text: length

text: i

text: k

text: from

text: x

text: to

text: w.

text: These

text: conditions

text: are

text: formalized

text: succinctly

text: as

text: follows.

text: Observation

text: Let

text: A

text: [a

text: A

text: Suppose

text: a

text: Then,

text: ij

text: k,t

text: ij

text: if

text: i

text: then

text: a

text: for

text: all

text: i;

text: if

text: i

text: then

text: a

text: k

text: i;

text: and

text: if

text: i

text: then

text: a−

text: k

text: i.

text: We

text: now

text: construct

text: an

text: oriented

text: graph,

text: G

text: which

text: is

text: a

text: universal

text: target

text: for

text: the

text: family

text: of

text: k-dipath

text: t-

text: k,t

text: colourable

text: oriented

text: graphs

text: of

text: directed

text: girth

text: at

text: least

text: k

text: The

text: vertex

text: set

text: V

text: (G

text: A

text: We

text: describe

text: k,t

text: k,t

text: the

text: edge

text: set

text: informally

text: at

text: ﬁrst,

text: and

text: then

text: formally.

text: Suppose

text: the

text: matrices

text: A

text: A

text: and

text: B

text: B

text: are

text: x,c

text: y,c

text: vertices

text: of

text: G

text: Let

text: c(x)

text: m

text: and

text: c

text: (y)

text: m

text: Then

text: AB

text: E(G)

text: if

text: k,t

text: A

text: encodes

text: the

text: fact

text: that

text: x

text: has

text: an

text: out-neighbour

text: of

text: colour

text: m

text: B

text: encodes

text: the

text: fact

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text: integers

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text: 𝐸𝑆 (𝑇 ) 𝑎 𝑡 𝑇 … . 𝑇 𝑇 𝑎−𝐿 𝑎−1 𝑎 𝑡 𝑓𝑜𝑟𝑤𝑎𝑟𝑑 𝑇 𝑇 𝑎 𝑎 𝑞(𝑇 ) 𝑎 (cid:104) (cid:105) Figure 1: Scheme of time windows setting for ES (T ) and q(T ) calculation. a (cid:104) (cid:105) a T is a window of length θ. The correlation structure persistence ES (T ) (upper axis) a a is computed by using data in T and in the ﬁrst L time windows before T . The volatility a a ratio q(T ) is computed by using data in T and in the future time window Tforward. In a a a the upper axis the time windows are actually overlapping, but they are here represented as disjoint for the sake of simplicity. ferentiating from structures associated to previous time windows. The forward-looking measure is the volatility ratio q(T ) [28, 29], that at each time window quantiﬁes how a good the market volatility measured at T is as a proxy for the next time-window volatil- (cid:104) (cid:105) a ity. Unlike ES (T ), the value of q(T ) is not known at the end of T . Fig. 1 shows a a a a graphical representation of the time window set-up. In the following we deﬁne the two measures: • (cid:104) (cid:105) Correlation structure persistence ES (T ): we deﬁne the correlation struc- a ture persistence at time Ta as: (cid:88) − (cid:104) (cid:105) a 1 ES (T ) = ω(T )ES(T , T ), (1) a b a b − b=a L − − where ω(T ) = ω exp(b a 1) is an exponential smoothing factor, L is a parameter b 0 L/3 and ES(T , T ) is the fraction of edges in common between the two PMFGs G(T ) a b a and G(T ), called “edge survival ratio” [15]. In formula, ES(T , T ) reads: b a b 1 | ∩ | ES(Ta, Tb) = N ETa ETb , (2) edges where N is the number of edges (links) in the two PMFGs (constant and equal −edges to 3N 6 for a PMFG [26]), and ETa (ETb) represents the edge-sets of PMFG (cid:104) (cid:105) at T (T ). The correlation structure persistence ES (T ) is therefore a weighted a b a average of the similarity (as measured by the edge survival ratio) between G(T ) (cid:80) a and the ﬁrst L previous PMFGs, with an exponential smoothing scheme that gives more weight to those PMFGs that are closer to T . The parameter ω in Eq. 1 − a (cid:104) (cid:105) 0 can be calculated by imposing a 1− ω(T ) = 1. Intuitively, ES (T ) measures b=a L b a how slowly the change of correlation structure is occurring in the near past of T . a • Volatility ratio q(T ) [28]: In order to quantify the agreement between the a estimated and the realized risk we here make use of the volatility ratio, a measure which has been used [28,30] for this purpose and deﬁned as follows: σ(Tforward) q(T ) = a , (3) a σ(T ) a 3 ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: 𝐸𝑆

text: (𝑇

text: 𝑎

text: 𝑡

text: 𝑇

text: 𝑇

text: 𝑇

text: 𝑎−𝐿

text: 𝑎−1

text: 𝑎

text: 𝑡

text: 𝑓𝑜𝑟𝑤𝑎𝑟𝑑

text: 𝑇

text: 𝑇

text: 𝑎

text: 𝑎

text: 𝑞(𝑇

text: 𝑎

label: caption

text: Figure

text: Scheme

text: of

text: time

text: windows

text: setting

text: for

text: ES

text: (T

text: and

text: q(T

text: calculation.

text: a

text: a

text: T

text: is

text: a

text: window

text: of

text: length

text: θ.

text: The

text: correlation

text: structure

text: persistence

text: ES

text: (T

text: (upper

text: axis)

text: a

text: a

text: is

text: computed

text: by

text: using

text: data

text: in

text: T

text: and

text: in

text: the

text: ﬁrst

text: L

text: time

text: windows

text: before

text: T

text: The

text: volatility

text: a

text: a

text: ratio

text: q(T

text: is

text: computed

text: by

text: using

text: data

text: in

text: T

text: and

text: in

text: the

text: future

text: time

text: window

text: Tforward.

text: In

text: a

text: a

text: a

text: the

text: upper

text: axis

text: the

text: time

text: windows

text: are

text: actually

text: overlapping,

text: but

text: they

text: are

text: here

text: represented

text: as

text: disjoint

text: for

text: the

text: sake

text: of

text: simplicity.

text: ferentiating

text: from

text: structures

text: associated

text: to

text: previous

text: time

text: windows.

text: The

text: forward-looking

text: measure

text: is

text: the

text: volatility

text: ratio

text: q(T

text: that

text: at

text: each

text: time

text: window

text: quantiﬁes

text: how

text: a

text: good

text: the

text: market

text: volatility

text: measured

text: at

text: T

text: is

text: as

text: a

text: proxy

text: for

text: the

text: next

text: time-window

text: volatil-

text: a

text: ity.

text: Unlike

text: ES

text: (T

text: the

text: value

text: of

text: q(T

text: is

text: not

text: known

text: at

text: the

text: end

text: of

text: T

text: Fig.

text: shows

text: a

text: a

text: a

text: a

text: graphical

text: representation

text: of

text: the

text: time

text: window

text: set-up.

text: In

text: the

text: following

text: we

text: deﬁne

text: the

text: two

text: measures:

text: Correlation

text: structure

text: persistence

text: ES

text: (T

text: we

text: deﬁne

text: the

text: correlation

text: struc-

text: a

text: ture

text: persistence

text: at

text: time

text: Ta

text: as:

text: a

text: ES

text: (T

text: ω(T

text: )ES(T

text: T

text: a

text: b

text: a

text: b

text: b=a

text: L

text: where

text: ω(T

text: ω

text: exp(b

text: a

text: is

text: an

text: exponential

text: smoothing

text: factor,

text: L

text: is

text: a

text: parameter

text: b

text: and

text: ES(T

text: T

text: is

text: the

text: fraction

text: of

text: edges

text: in

text: common

text: between

text: the

text: two

text: PMFGs

text: G(T

text: a

text: b

text: a

text: and

text: G(T

text: called

text: “edge

text: survival

text: ratio”

text: In

text: formula,

text: ES(T

text: T

text: reads:

text: b

text: a

text: b

text: ES(Ta,

text: Tb)

text: N

text: ETa

text: ETb

text: edges

text: where

text: N

text: is

text: the

text: number

text: of

text: edges

text: (links)

text: in

text: the

text: two

text: PMFGs

text: (constant

text: and

text: equal

text: −edges

text: to

text: for

text: a

text: PMFG

text: and

text: ETa

text: (ETb)

text: represents

text: the

text: edge-sets

text: of

text: PMFG

text: at

text: T

text: (T

text: The

text: correlation

text: structure

text: persistence

text: ES

text: (T

text: is

text: therefore

text: a

text: weighted

text: a

text: b

text: a

text: average

text: of

text: the

text: similarity

text: (as

text: measured

text: by

text: the

text: edge

text: survival

text: ratio)

text: between

text: G(T

text: a

text: and

text: the

text: ﬁrst

text: L

text: previous

text: PMFGs,

text: with

text: an

text: exponential

text: smoothing

text: scheme

text: that

text: gives

text: more

text: weight

text: to

text: those

text: PMFGs

text: that

text: are

text: closer

text: to

text: T

text: The

text: parameter

text: ω

text: in

text: Eq.

text: a

text: can

text: be

text: calculated

text: by

text: imposing

text: a

text: 1−

text: ω(T

text: Intuitively,

text: ES

text: (T

text: measures

text: b=a

text: L

text: b

text: a

text: how

text: slowly

text: the

text: change

text: of

text: correlation

text: structure

text: is

text: occurring

text: in

text: the

text: near

text: past

text: of

text: T

text: a

text: Volatility

text: ratio

text: q(T

text: In

text: order

text: to

text: quantify

text: the

text: agreement

text: between

text: the

text: a

text: estimated

text: and

text: the

text: realized

text: risk

text: we

text: here

text: make

text: use

text: of

text: the

text: volatility

text: ratio,

text: a

text: measure

text: which

text: has

text: been

text: used

text: for

text: this

text: purpose

text: and

text: deﬁned

text: as

text: follows:

text: σ(Tforward)

text: q(T

text: a

text: a

text: σ(T

text: a

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: that are signiﬁcantly higher than the same probability obtained by using the most recent (cid:104) (cid:105) value of q instead of ES (T ) as a predictor for q(T ) in the logistic regression (let us a a call P+ such probability). Speciﬁcally, we have deﬁned a null model where variations q from such probability P+ are due to random ﬂuctuations only; given n observations, q such ﬂuctuations follow a Binomial distribution B(P+, n), with mean nP+ and variance − q q nP+(1 P+). Then p-values have been calculated by using this null distribution for q q each combination of parameters. This null hypothesis accounts for the predictability of q(T ) that is due to the autocorrelation of q(T ) only; therefore P+ signiﬁcantly higher a a (cid:104) (cid:105) than the value expected under this hypothesis implies a forecasting power of ES (T ) a that is not explained by the autocorrelation of q(T ). From the table we can see that a P+ is signiﬁcant in 12 out of 16 combinations of parameters for NYSE dataset, and in 13 out of 16 for LSE dataset. This means that correlation persistence is a valuable predictor for future average correlation, able to outperform forecasting method based on past average correlation trends. These results are robust against changes of f , as test long as the training set is large enough to allow an accurate calibration of the logistic regression. We have found this condition is satisﬁed for f < 40%. test However P+ does not give any information on the method ability to distinguish between true and false positives. To investigate this aspect we need TPR and FPR. A traditional way of representing both measures from a binary classiﬁer is the so-called “Receiver operating characteristic” (ROC) curve [40]. In a ROC plot, TPR is plotted against FPR as the discriminant threshold is varied. The discriminant threshold p max is the value of the probability in Eq. 6 over which we classify Y (T ) = 1: the higher a p is, the less likely the method is to classify Y (T ) = 1 (in the analysis on P+ we max a chose p = 0.5). Ideally, a perfect classiﬁer would yield TPR = 1 for all p > 0, max max whereas a random classiﬁer is expected to lie on the line TPR = FPR. Therefore a ROC curve which lies above the line TPR = FPR indicates a classiﬁer that is better than chance at distinguishing true from false positives [37]. As one can see from Fig. 6, the ROC curve’s position depends on the choice of parameters θ and L. In this respect our classiﬁer performs better for low values of L and θ. This can be quantiﬁed by measuring the area under the ROC curve; such measure, often denoted by AUC [37], is shown in Tabs. 7-8. For both datasets the optimal choice of parameters is θ = 500 and L = 10. Table 5: NYSE dataset: Probability of successful forecasting P+, for diﬀerent combinations of parameters θ and L. Out-of-sample analysis. L 10 25 50 100 250 0.546 0.560* 0.599** 0.539** 500 0.704** 0.695** 0.658** 0.605** θ 750 0.634* 0.585 0.539 0.708* 1000 0.704* 0.7638** 0.839** 0.860 **p < 0.001, *p < 0.01, Table 6: LSE dataset: Probability of successful forecasting P+, for diﬀerent combinations of parameters θ and L. Out-of-sample analysis. L 10 25 50 100 250 0.616** 0.645** 0.612** 0.568** 500 0.652** 0.635** 0.598** 0.393 θ 750 0.651** 0.560** 0.453** 0.412 1000 0.544** 0.573** 0.706** 0.689 **p < 0.001, *p < 0.01, Temporal evolution of forecasting performance In this section we look at how the forecasting performance changes at diﬀerent time 9 ##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##

text: that

text: are

text: signiﬁcantly

text: higher

text: than

text: the

text: same

text: probability

text: obtained

text: by

text: using

text: the

text: most

text: recent

text: value

text: of

text: q

text: instead

text: of

text: ES

text: (T

text: as

text: a

text: predictor

text: for

text: q(T

text: in

text: the

text: logistic

text: regression

text: (let

text: us

text: a

text: a

text: call

text: P+

text: such

text: probability).

text: Speciﬁcally,

text: we

text: have

text: deﬁned

text: a

text: model

text: where

text: variations

text: q

text: from

text: such

text: probability

text: P+

text: are

text: due

text: to

text: random

text: ﬂuctuations

text: only;

text: given

text: n

text: observations,

text: q

text: such

text: ﬂuctuations

text: follow

text: a

text: Binomial

text: distribution

text: B(P+,

text: n),

text: with

text: mean

text: nP+

text: and

text: variance

text: q

text: q

text: P+).

text: Then

text: p-values

text: have

text: been

text: calculated

text: by

text: using

text: this

text: distribution

text: for

text: q

text: q

text: each

text: combination

text: of

text: parameters.

text: This

text: hypothesis

text: accounts

text: for

text: the

text: predictability

text: of

text: q(T

text: that

text: is

text: due

text: to

text: the

text: autocorrelation

text: of

text: q(T

text: only;

text: therefore

text: P+

text: signiﬁcantly

text: higher

text: a

text: a

text: than

text: the

text: value

text: expected

text: under

text: this

text: hypothesis

text: implies

text: a

text: forecasting

text: power

text: of

text: ES

text: (T

text: a

text: that

text: is

text: not

text: explained

text: by

text: the

text: autocorrelation

text: of

text: q(T

text: From

text: the

text: table

text: we

text: can

text: see

text: that

text: a

text: P+

text: is

text: signiﬁcant

text: in

text: out

text: of

text: combinations

text: of

text: parameters

text: for

text: NYSE

text: dataset,

text: and

text: in

text: out

text: of

text: for

text: LSE

text: dataset.

text: This

text: means

text: that

text: correlation

text: persistence

text: is

text: a

text: valuable

text: predictor

text: for

text: future

text: average

text: correlation,

text: able

text: to

text: outperform

text: forecasting

text: method

text: based

text: on

text: past

text: average

text: correlation

text: trends.

text: These

text: results

text: are

text: robust

text: against

text: changes

text: of

text: f

text: as

text: test

text: long

text: as

text: the

text: training

text: set

text: is

text: large

text: enough

text: to

text: allow

text: an

text: accurate

text: calibration

text: of

text: the

text: logistic

text: regression.

text: We

text: have

text: found

text: this

text: condition

text: is

text: satisﬁed

text: for

text: f

text: test

text: However

text: P+

text: does

text: not

text: give

text: any

text: information

text: on

text: the

text: method

text: ability

text: to

text: distinguish

text: between

text: true

text: and

text: false

text: positives.

text: To

text: investigate

text: this

text: aspect

text: we

text: need

text: TPR

text: and

text: FPR.

text: A

text: traditional

text: way

text: of

text: representing

text: both

text: measures

text: from

text: a

text: binary

text: classiﬁer

text: is

text: the

text: so-called

text: “Receiver

text: operating

text: characteristic”

text: (ROC)

text: curve

text: In

text: a

text: ROC

text: plot,

text: TPR

text: is

text: plotted

text: against

text: FPR

text: as

text: the

text: discriminant

text: threshold

text: is

text: varied.

text: The

text: discriminant

text: threshold

text: p

text: max

text: is

text: the

text: value

text: of

text: the

text: probability

text: in

text: Eq.

text: over

text: which

text: we

text: classify

text: Y

text: (T

text: the

text: higher

text: a

text: p

text: is,

text: the

text: less

text: likely

text: the

text: method

text: is

text: to

text: classify

text: Y

text: (T

text: (in

text: the

text: analysis

text: on

text: P+

text: we

text: max

text: a

text: chose

text: p

text: Ideally,

text: a

text: perfect

text: classiﬁer

text: would

text: yield

text: TPR

text: for

text: all

text: p

text: max

text: max

text: whereas

text: a

text: random

text: classiﬁer

text: is

text: expected

text: to

text: lie

text: on

text: the

text: line

text: TPR

text: FPR.

text: Therefore

text: a

text: ROC

text: curve

text: which

text: lies

text: above

text: the

text: line

text: TPR

text: FPR

text: indicates

text: a

text: classiﬁer

text: that

text: is

text: better

text: than

text: chance

text: at

text: distinguishing

text: true

text: from

text: false

text: positives

text: As

text: one

text: can

text: see

text: from

text: Fig.

text: the

text: ROC

text: curve’s

text: position

text: depends

text: on

text: the

text: choice

text: of

text: parameters

text: θ

text: and

text: L.

text: In

text: this

text: respect

text: our

text: classiﬁer

text: performs

text: better

text: for

text: low

text: values

text: of

text: L

text: and

text: θ.

text: This

text: can

text: be

text: quantiﬁed

text: by

text: measuring

text: the

text: area

text: under

text: the

text: ROC

text: curve;

text: such

text: measure,

text: often

text: denoted

text: by

text: AUC

text: is

text: shown

text: in

text: Tabs.

text: For

text: both

text: datasets

text: the

text: optimal

text: choice

text: of

text: parameters

text: is

text: θ

text: and

text: L

text: Table

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text: L.

text: Out-of-sample

text: analysis.

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text: Table

text: LSE

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text: successful

text: forecasting

text: P+,

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text: parameters

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text: and

text: L.

text: Out-of-sample

text: analysis.

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text: θ

text: **p

text: *p

text: Temporal

text: evolution

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text: forecasting

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text: In

text: this

text: section

text: we

text: look

text: at

text: how

text: the

text: forecasting

text: performance

text: changes

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text: 2 Materials and Methods 2.1 Model of membrane potential responses We model the responses of simple cells of the primary visual cortex at the level of the membrane potentials. The membrane potential of a simple cells shows systematic variations with changes in the stimulus. The most widely studied of these changes, the orientation dependence was considered as changes in the trial-by-trial mean of the membrane potential response. Trial-averaged responses provide an incomplete description of the membrane potential statis- tics since there are both within-trial ﬂuctuation and across trial ﬂuctuations (Tomko and Crapper, 1974). The source of these ﬂuctuations are not consid- ered. A neuron is characterized by the probability distribution of membrane potential values at each time point within a trial, which random samples were drawn from. Importantly, besides stimulus-driven covariations in membrane potential responses of multiple neurons, stimulus-independent, so called noise covariances are also characteristic features of cortical neurons (Yu and Ferster, 2010). As a consequence, a population of neurons in the visual cortex is also characterized by the joint probability distribution of the membrane potentials. Similar to systematic variations in trial-averaged mean membrane potential re- sponses, membrane potential variances are characterized by systematic changes (Finn et al., 2007), which we also take into account in the model. Instead of aiming for a complete description of such probability distributions, we focus on the second-order statistics of neural responses. This is motivated by two considerations: (i) experimental designs are typically limited in terms of the number of trials, rendering higher-order joint statistics of neurons hard to estimate (ii) pairwise statistics are the simplest measure of population activity going beyond individual cell response properties, and are already able to cap- ture deﬁnitive signatures of population-level cortical computation (Karklin and Lewicki, 2009; Haefner et al., 2013). The second-order statistics of a population of N units with temporal dynam- − ics is completely characterized by N autocorrelation functions and N(N 1)/2 cross-correlation functions, specifying the linear dependence between each pair at every timescale (Moreno-Bote et al., 2008). Based on the typical support of the autocorrelation function of membrane potentials, we chose to examine interactions on the scale of 20 ms (Azouz and Gray, 1999). Using this, we can simplify the above picture greatly by specifying a single membrane potential value in each 20 ms bin, assuming that there is no temporal dependence on longer timescales, and omitting variability on shorter ones. Thus, we can con- sider membrane potentials to be sampled from a Gaussian distribution at each time bin b independently: ∼ N u (u ; µ, C ). (1) b b mp ∈ R The mean of the distribution, µ N, determines the trial-averaged mem- brane potential level of each neuron, and the sequences of individual samples 4

text: Materials

text: and

text: Methods

text: Model

text: of

text: membrane

text: potential

text: responses

text: We

text: model

text: the

text: responses

text: of

text: simple

text: cells

text: of

text: the

text: primary

text: visual

text: cortex

text: at

text: the

text: level

text: of

text: the

text: membrane

text: potentials.

text: The

text: membrane

text: potential

text: of

text: a

text: simple

text: cells

text: shows

text: systematic

text: variations

text: with

text: changes

text: in

text: the

text: stimulus.

text: The

text: most

text: widely

text: studied

text: of

text: these

text: changes,

text: the

text: orientation

text: dependence

text: was

text: considered

text: as

text: changes

text: in

text: the

text: trial-by-trial

text: mean

text: of

text: the

text: membrane

text: potential

text: response.

text: Trial-averaged

text: responses

text: provide

text: an

text: incomplete

text: description

text: of

text: the

text: membrane

text: potential

text: statis-

text: tics

text: since

text: there

text: are

text: both

text: within-trial

text: ﬂuctuation

text: and

text: across

text: trial

text: ﬂuctuations

text: (Tomko

text: and

text: Crapper,

text: The

text: source

text: of

text: these

text: ﬂuctuations

text: are

text: not

text: consid-

text: ered.

text: A

text: neuron

text: is

text: characterized

text: by

text: the

text: probability

text: distribution

text: of

text: membrane

text: potential

text: values

text: at

text: each

text: time

text: point

text: within

text: a

text: trial,

text: which

text: random

text: samples

text: were

text: drawn

text: from.

text: Importantly,

text: besides

text: stimulus-driven

text: covariations

text: in

text: membrane

text: potential

text: responses

text: of

text: multiple

text: neurons,

text: stimulus-independent,

text: so

text: called

text: noise

text: covariances

text: are

text: also

text: characteristic

text: features

text: of

text: cortical

text: neurons

text: (Yu

text: and

text: Ferster,

text: As

text: a

text: consequence,

text: a

text: population

text: of

text: neurons

text: in

text: the

text: visual

text: cortex

text: is

text: also

text: characterized

text: by

text: the

text: joint

text: probability

text: distribution

text: of

text: the

text: membrane

text: potentials.

text: Similar

text: to

text: systematic

text: variations

text: in

text: trial-averaged

text: mean

text: membrane

text: potential

text: re-

text: sponses,

text: membrane

text: potential

text: variances

text: are

text: characterized

text: by

text: systematic

text: changes

text: (Finn

text: et

text: al.,

text: which

text: we

text: also

text: take

text: into

text: account

text: in

text: the

text: model.

text: Instead

text: of

text: aiming

text: for

text: a

text: complete

text: description

text: of

text: such

text: probability

text: distributions,

text: we

text: focus

text: on

text: the

text: second-order

text: statistics

text: of

text: neural

text: responses.

text: This

text: is

text: motivated

text: by

text: two

text: considerations:

text: (i)

text: experimental

text: designs

text: are

text: typically

text: limited

text: in

text: terms

text: of

text: the

text: number

text: of

text: trials,

text: rendering

text: higher-order

text: joint

text: statistics

text: of

text: neurons

text: hard

text: to

text: estimate

text: (ii)

text: pairwise

text: statistics

text: are

text: the

text: simplest

text: measure

text: of

text: population

text: activity

text: going

text: beyond

text: individual

text: cell

text: response

text: properties,

text: and

text: are

text: already

text: able

text: to

text: cap-

text: ture

text: deﬁnitive

text: signatures

text: of

text: population-level

text: cortical

text: computation

text: (Karklin

text: and

text: Lewicki,

text: Haefner

text: et

text: al.,

text: The

text: second-order

text: statistics

text: of

text: a

text: population

text: of

text: N

text: units

text: with

text: temporal

text: dynam-

text: ics

text: is

text: completely

text: characterized

text: by

text: N

text: autocorrelation

text: functions

text: and

text: N(N

text: cross-correlation

text: functions,

text: specifying

text: the

text: linear

text: dependence

text: between

text: each

text: pair

text: at

text: every

text: timescale

text: (Moreno-Bote

text: et

text: al.,

text: Based

text: on

text: the

text: typical

text: support

text: of

text: the

text: autocorrelation

text: function

text: of

text: membrane

text: potentials,

text: we

text: chose

text: to

text: examine

text: interactions

text: on

text: the

text: scale

text: of

text: ms

text: (Azouz

text: and

text: Gray,

text: Using

text: this,

text: we

text: can

text: simplify

text: the

text: above

text: picture

text: greatly

text: by

text: specifying

text: a

text: single

text: membrane

text: potential

text: value

text: in

text: each

text: ms

text: bin,

text: assuming

text: that

text: there

text: is

text: no

text: temporal

text: dependence

text: on

text: longer

text: timescales,

text: and

text: omitting

text: variability

text: on

text: shorter

text: ones.

text: Thus,

text: we

text: can

text: con-

text: sider

text: membrane

text: potentials

text: to

text: be

text: sampled

text: from

text: a

text: Gaussian

text: distribution

text: at

text: each

text: time

text: bin

text: b

text: independently:

text: N

text: u

text: (u

text: µ,

text: C

text: b

text: b

text: mp

text: R

text: The

text: mean

text: of

text: the

text: distribution,

text: µ

text: N,

text: determines

text: the

text: trial-averaged

text: mem-

text: brane

text: potential

text: level

text: of

text: each

text: neuron,

text: and

text: the

text: sequences

text: of

text: individual

text: samples