Hierarchy of Temporal Responses of Multivariate Self-Excited Epidemic Processes
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] J a n Hierarchy of Temporal Responsesof Multivariate Self-Excited Epidemic Processes
A. Saichev
1, 2 and D. Sornette Department of Management, Technology and Economics,ETH Zurich, Kreuzplatz 5, CH-8032 Zurich, Switzerland Mathematical Department, Nizhny Novgorod State University,Gagarin prosp. 23, Nizhny Novgorod, 603950, Russia ∗ (Dated: November 9, 2018) Abstract
We present the first exact analysis of some of the temporal properties of multivariate self-excitedHawkes conditional Poisson processes, which constitute powerful representations of a large varietyof systems with bursty events, for which past activity triggers future activity. The term “multi-variate” refers to the property that events come in different types, with possibly different intra-and inter-triggering abilities. We develop the general formalism of the multivariate generating mo-ment function for the cumulative number of first-generation and of all generation events triggeredby a given mother event (the “shock”) as a function of the current time t . This corresponds tostudying the response function of the process. A variety of different systems have been analyzed.In particular, for systems in which triggering between events of different types proceeds through aone-dimension directed or symmetric chain of influence in type space, we report a novel hierarchyof intermediate asymptotic power law decays ∼ /t − ( m +1) θ of the rate of triggered events as afunction of the distance m of the events to the initial shock in the type space, where 0 < θ < ∗ Electronic address: [email protected],[email protected] . INTRODUCTION We study a class of point processes that was introduced by Hawkes in 1971 [1–4]. It ismuch richer and relevant to most natural and social systems than standard point processes[5–10], because it describes “self-excited” processes. This term means that the past eventshave the ability to trigger future events, i.e., λ ( t | H t ) is a function of past events, beingtherefore non-markovian. Many works have been performed to characterize the statisticaland dynamical properties of this class of models, with applications ranging from geophysical[11–16], medical [17] to financial systems, with applications to Value-at-Risk modeling [18],high-frequency price processes [19], portfolio credit risks [20], cascades of corporate defaults[21], financial contagion [22], and yield curve dynamics [23].While surprisingly rich and powerful in explaining empirical observations in a varietyof systems, most previous studies have used mono-variate self-excited point processes, i.e.,they have assumed the existence of only a single type of events, all the events presentingsome ability to trigger events of the same type. However, in reality, in many systems, eventscome in different types with possibly different properties, while keeping a degree of mutualinter-excitations. • Earthquakes are partitioned in different tectonic regions. • Neuronal excitations in the brain come in different types, such as spikes, bursts andseizures. In addition, epileptic seizures involve different brain structures at differentscales. • Financial volatility bursts occur at a given time on some assets and not on others. • Defaults on debts may start and develop preferentially on some firms in some industrialsectors. • Some countries within a currency block may start to exhibit specific sovereign risksand not other countries. • Only a subset of the population of bloggers or of developers may be active at any time.Notwithstanding the existence of such different types, categories or locations, events of onetype may trigger both new events of the same type as well as of different types. For instance,2. there is now convincing evidence of earthquake triggering at large distances acrosstectonic plate boundaries.2. Epileptic seizures are nowadays believed to be able to trigger large scale neuronalexcitations over largely different brain structures. It is possible that even neuronalspikes, bursts or different types of subclinical epileptic seizures may play a role intriggering clinical epileptic seizures of different types in distinct cortical structures.3. The hectic price movement of a given stock may trigger by contagion large volatil-ity bursts in other stocks or assets, via technical or behavioral transfer mechanisms.Financial volatility bursts occurring for one asset may trigger future volatility fluctu-ations for other assets.4. The shocks due to defaults of some firms may jump across industries to encompassdifferent branches through a chain reaction.5. Countries of different regions and currencies may become coupled through the directand indirect flows of international commerce as well as mutual debt ownerships.6. The activity of a blogger or a developer may be encouraged by previous actions ofother agents, leading to future commits triggered by previous commits of differentpeople.These observations suggest that multivariate self-excited point processes, which extendthe class of mono-variate self-excited point processes, provide a very important class ofmodels to describe the self-excitation (or intra-triggering) as well as the mutual influencesor triggering between different types of events that occur in many natural and social systems.Actually, the generalization to multivariate self-excited point processes was mentioned byHawkes himself in his first paper [1], but the full relevance of this class of models has onlybeen recently appreciated [22, 24].The organization of the paper proceeds as follows. Section 2 first recalls the definitionof the monovariate Hawkes process and then presents the general multivariate self-excitingHawkes processes. Section 3 develops the formalism of the multivariate generating momentfunction for the cumulative number of first-generation and of all generation events triggeredby a given mother event as a function of the current time t . Section 4 provides the general3elations to obtain the mean numbers of events triggered over all generations by a given eventas a function of time. Section 5 analyzes a system in which the intra-type triggering processesare all of the same efficiency while all inter-type triggering processes have themselves thesame efficiency between themselves but in general weaker than the intra-type triggeringprocesses. We derive the time-dependence of the rates of events triggered from a givenshock for distributions of waiting times of first-generation events that have either exponentialor power law tails. Section 6 analyzes a system in which triggering between events ofdifferent types proceeds through a one-dimension directed chain of influence in type space.We uncover a novel hierarchy of intermediate asymptotic power law decays of the rate oftriggered events as a function of the distance of the events to the initial shock in the space oftypes. Section 7 generalizes the results of section 6 by studying a system in which triggeringbetween events of different types proceeds through a one-dimension symmetric chain ofinfluences in type space. Two appendices give proofs and details of the key results of thepaper. II. DEFINITIONS AND NOTATIONS FOR THE MULTIVARIATE HAWKESPROCESSESA. Monovariate Hawkes processes
Self-excited conditional Poisson processes generalize the cluster models by allowing eachevent, including cluster members, i.e., aftershocks, to trigger their own events according tosome memory kernel h ( t − t i ). λ ( t | H t , Θ) = λ c ( t ) + X i | t i 1. The time constant c ensures finiteness of the triggering rateimmediately following any event. Other forms with shorter memory, as the exponential,are also common. Each event (of magnitude M > M ) triggers other events with a rate ∼ e aM , which defines the so-called fertility or productivity law. The lower magnitude cut-off M is such that events with marks smaller than M do not generate offsprings. Thisis necessary to make the theory convergent and well-defined, otherwise the crowd of smallevents may actually dominate [27, 28]. The constant k controls the overall productivity lawand thus the average branching ratio defined by expression (4) below. The set of parametersis Θ = { b, λ c , k, a, M , c, θ } . 5rom a theoretical point of view, the Hawkes models with marks has been studied inessentially two directions: (i) statistical estimations of its parameters with correspondingresidual analysis as goodness of fits [39–48]; (ii) statistical properties of its space-time dy-namics [15, 16, 29–38].The advantage of the self-excited conditional Hawkes process is to provide a very parsi-monious description of the complex spatio-temporal organization of systems characterizedby self-excitatied bursts of events, without the need to invoke ingredients other than thegenerally well-documented stylized facts on the distribution of event sizes, the temporal“Omori law” for the waiting time before excitation of a new event and the productivity lawcontrolling the number of triggered events per initiator.Self-excited models of point processes with additive structure of their intensity on pastevents as in (2) and (3) [4] make them part of the general family of branching processes [49].The crucial parameter is then the average branching ratio n , defined as the mean numberof events of first generation triggered per event. Using the notation of expression (3), theaverage branching ratio is given by n = kθc θ · bb − a . (4)Depending on applications, the branching ratio n can vary with time, from location tolocation and from type to type (as we shall see below for the multivariate generalization).The branching ratio provides a diagnostic of the susceptibility of the system to triggeractivity in the presence of some exogenous nucleating events.Precise analytical results and numerical simulations show the existence of three time-dependent regimes, depending on the “branching ratio” n and on the sign of θ . This clas-sification is valid for the range of parameters a < b . When the productivity exponent a is larger than the exponent b of the Gutenberg-Richter law, formula (4) does not makesense anymore, which reflects the existence of an explosive regime associated with stochasticfinite-time singularities [29], a regime that we do not consider further below, but which isrelevant to describe the accelerated damage processes leading to global systemic failures inpossibly many different types of systems [50].1. For n < ∼ /t p , characterized by a crossover froman Omori exponent p = 1 − θ for t < t ∗ to a larger exponent p = 1 + θ for t > t ∗ [15],6here t ∗ is a characteristic time t ∗ ≃ c/ (1 − n ) /θ , which is controlled by the distanceof n to 1.2. For n > θ > p = 1 − θ at early times since the mainshock to an explosiveexponential increase of the activity rate at times t > t ∗ ≃ c/ ( n − /θ [15, 51].3. In the case θ < 0, there is a transition from an Omori law with exponent 1 − | θ | similar to the local law, to an exponential increase at large times, with a crossovertime τ different from the characteristic time t ∗ found in the case θ > B. Multivariate Hawkes processes The Multivariate Hawkes Process generalizes expressions (2) and (3) into the followinggeneral form for the conditional Poisson intensity for an event of type j among a set of m possible types (see the document [25] for an extensive review): λ j ( t | H t ) = λ j ( t ) + m X k =1 Λ kj Z ( −∞ ,t ) ×R f k,j ( t − s ) g k ( x ) N k ( ds × dx ) , (5)where H t denotes the whole past history up to time t , λ j is the rate of spontaneous (exoge-nous) events of type j , i.e., the sources or immigrants of type j , Λ kj is the ( k, j )’s elementof the matrix of coupling between the different types which quantifies the ability of a type k -event to trigger a type j -event. Specifically, the value of an element Λ kj is just the averagenumber of first-generation events of type j triggered by an event of type k . This generalizesthe branching ratio n defined by (4). The memory kernel f k,j ( t − s ) gives the probability thatan event of type k that occurred at time s < t will trigger an event of type j at time t . Thefunction f k,j ( t − s ) is nothing but the distribution of waiting times t − s between the impulseof event k which impacted the system at some time s and the occurrence of an event of type j at time t . The fertility (or productivity) law g k ( x ) of events of type k with mark x quanti-fies the total average number of first-generation events of any type triggered by an event oftype k . We have used the standard notation R ( −∞ ,t ) ×R f ( t, x ) N ( ds × dx ) := P i | t i 1. But, this is not the correct regime ofparameters for earthquakes as well as for other social epidemic processes, which have beenshown to be characterized by long-memory processes with 0 < θ < < θ < III. TEMPORAL MULTIVARIATE GENERATING MOMENT FUNCTION(GMF)A. Generating moment function for the cumulative number of first-generationevents triggered until time t Among the m types of events, consider the k -th type and its first generation offsprings.Let us denote R k, ( t ) , R k, ( t ) , . . . , R k,m ( t ), the cumulative number of “daughter” events offirst generation of type 1 , , . . . , m generated by this “mother” event of type k from time 0until time t . With these notations, the generating moment function (GMF) of all events offirst generation that are triggered by a mother event of type k until time t reads A k ( y , y , . . . , y m ; t ) := E " m Y s =1 y R k,s ( t ) s , (6)8here E [ X ] denotes the average of X over all possible statistical realizations. We shall alsoneed the definition of the GMF A k ( y , y , . . . , y m ) defined by A k ( y , y , . . . , y m ) := E " m Y s =1 y R k,s s , (7)where R k, = lim t → + ∞ R k, ( t ) , R k, = lim t → + ∞ R k, ( t ) , . . . , R k,m = lim t → + ∞ R k,m ( t ) are thecumulative number of “daughter” events of first generation of type 1 , , . . . , m generated bythe “mother” event of type k over all times. One may rewrite this function in probabilisticform A k ( y , y , . . . , y m ) := ∞ X r =0 · · · ∞ X r m =0 P k ( r , . . . , r m ) m Y s =1 y r s s , (8)where P k ( r , . . . , r m ) is the probability that the mother event of type k generates R k, = r first-generation events of type 1, R k, = r first-generation events of type 2, and so on.The events are assumed to occur after waiting times between the mother event and theiroccurrences that are mutually statistically independent and characterized by the proba-bility density functions (pdf) { f k,s ( t ) } , where all f k,s ( t ) ≡ t < 0. Let us denote P k ( d , d , . . . , d m ; t ) the probability that the cumulative numbers { R k,s ( t ) } up to time t offirst-generation events that have been triggered by the mother event of type k are equal to R k, ( t ) = d , R k, ( t ) = d , . . . R k,m ( t ) = d m . (9)Let us relate this probability P k ( d , d , . . . , d m ; t ) to that, denoted P k ( d , d , . . . , d m ; t | r , r , . . . , r m ), obtained under the additional condition that thetotal numbers of first-generation events that have been triggered by the mother eventof type k over the whole time interval t → ∞ are fixed at the values { r , r , . . . , r m } .Obviously, P k ( d , d , . . . , d m ; t | r , r , . . . , r m ) is given by a product of binomial distributions P k ( d , d , . . . , d m ; t | r , r , . . . , r m ) = m Y s =1 (cid:18) r s d s (cid:19) µ d s k,s ( t )[1 − µ k,s ( t )] r s − d s , d r , . . . d m r m , (10)where µ k,s ( t ) = Z t f k,s ( t ′ ) dt ′ . (11)9nowing the conditional probabilities P k ( d , d , . . . , d m ; t | r , r , . . . , r m ) (10), one can cal-culate their unconditional counterparts using the following relation P k ( d , d , . . . , d m ; t ) = ∞ X r = d · · · ∞ X r m = d m P k ( d , d , . . . , d m ; t | r , r , . . . , r m ) P k ( r , r , . . . , r m ) , (12)where P k ( r , r , . . . , r m ) is the probability that the total numbers of first-generation eventsthat have been triggered by the mother event of type k over the whole time interval t → ∞ take the values { r , r , . . . , r m } .Substituting the relations (10) in (12) yields P k ( d , d , . . . , d m ; t ) = ∞ X r = d · · · ∞ X r m = d m P k ( r , . . . , r m ) m Y s =1 (cid:18) r s d s (cid:19) µ d s k,s ( t )[1 − µ k,s ( t )] r s − d s . (13)The interest in this expression (13) is that the GMF A k ( y , y , . . . , y m ; t ) defined by expres-sion (6) can be rewritten in probabilistic form as A k ( y , y , . . . , y m ; t ) = ∞ X d =0 · · · ∞ X d m =0 P k ( d , . . . , d m ; t ) m Y s =1 y d s s . (14)We are now prepared to state the following theorem, which is essential for our subsequentderivations. Theorem 3.1 The GMF A k ( y , y , . . . , y m ; t ) defined by expression (6) can be representedin the form A k ( y , y , . . . , y m ; t ) = Q k [ µ k, ( t )( y − , . . . , µ k,m ( t )( y m − , (15) where Q k ( z , . . . , z m ) := A k (1 + z , . . . , z s ) ⇔ A k ( y , . . . , y m ) = Q k ( y − , . . . , y m − . (16)The proof is given in Appendix A. B. GMF for the cumulative numbers of events over all generation triggered untiltime t Let us define the GMF A k ( y , y , . . . , y m ; t ) := E " m Y s =1 y R k,s ( t ) s , (17)10here { R k,s ( t ) } is the total number of events summed over all generations of events of type s triggered by a mother event of type k starting at time 0 up to time t .Due to the branching nature of the process, the equation determining the GMF { A k ( y , y , . . . , y m ; t ) } is obtained by1. replacing in the left-hand-side of expression (15) A k ( y , y , . . . , y m ; t ) by A k ( y , y , . . . , y m ; t ); this means that we deal with events of all generations oc-curring till current time t ;2. replacing in the right-hand-side of expression (15) the arguments y s by y s ⇒ y s Z t f k,s ( t ′ | t ) A s ( y , . . . , y m ; t − t ′ ) dt ′ , (18)where { f k,s ( t ′ | t ) } is the conditional pdf of the random times { t ′ k,s } of occurrence ofsome first-generation event of type s triggered by the mother event of type k , underthe condition that it occurred within the time interval t ′ ∈ (0 , t ). The conditional pdf { f k,s ( t ′ | t ) } is given by f k,s ( t ′ | t ) = f k,s ( t ′ ) µ k,s ( t ) , (19)where µ k,s ( t ) is defined by (11). The pdf f k,s ( t ′ | t ) inside the integral (18) takes intoaccount that first-generation events are occurring at random times t ′ < t . The otherfactor A s ( y , . . . , y m ; t − t ′ ) takes into account all-generation events that are triggeredby some first-generation event from its appearance time t ′ till the current time t .The equation for the GMF { A k ( y , y , . . . , y m ; t ) } is thus A k ( y , y , . . . , y m ; t ) = Q k (cid:2) B k, ( y , . . . , y m ; t ) , . . . , B k,m ( y , . . . , y m ; t ) (cid:3) , (20)where B k,s ( y , . . . , y m ; t ) = Z t f k,s ( t − t ′ )[ y s A s ( y , . . . , y m ; t ′ ) − dt ′ . (21) IV. GENERAL RELATIONS FOR THE MEAN NUMBERS OF EVENTS OVERALL GENERATION TRIGGERED UP TO TIME t The set of equations (20) together with the relations (21) provides the basis for a fulldescription of the statistical and temporal properties of multivariate branching (Hawkes)11rocesses. Here, we restrict our attention to the average activities, by studying the temporaldependence of the average number of events following the occurrence of a mother event of agiven type.The mean number of events of type s over all generations counted until time t that aretriggered by a mother event of type k that occurred at time t = 0, defined by¯ R k,s ( t ) := E[ R k,s ( t )] , (22)is given by the relation ¯ R k,s ( t ) = ∂∂y s A k ( y , y , . . . , y m ; t ) (cid:12)(cid:12) y = ··· = y m =1 , (23)where A k ( y , y , . . . , y m ; t ) satisfies to the set of equations (20).In order to derive the equations determining the set { ¯ R k,s ( t ) } , we need to statethe following properties exhibited by the functions Q k ( y , . . . , u m ) given by (16) and B k,s ( y , y , . . . , y n ; t ) given by (21), which are contributing to equation (20). From thedefinition of the functions B k,s ( y , y , . . . , y n ; t ) and of the GMF A k ( y , y , . . . , y m ; t ), wehave B k,s ( y , y , . . . , y n ; t ) (cid:12)(cid:12) y = ··· = y m =1 ≡ ∂∂y s B k,ℓ ( y , y , . . . , y n ; t, τ ) (cid:12)(cid:12) y = ··· = y m =1 = µ k,s ( t ) · δ ℓ,s + f k,ℓ ( t ) ⊗ ¯ R ℓ,s ( t ) . (25)The convolution operation is defined as usual by f ( t ) ⊗ g ( t ) := R t f ( t − t ′ ) g ( t ′ ) dt ′ . Moreover,the following equality holds ∂∂y s Q k ( y , y , . . . , y n ; t, τ ) (cid:12)(cid:12) y = ··· = y m =0 = n k,s , (26)where n k,s is the mean value of the total number of first-generation events of type s triggeredby a mother of type k . The set { n k,s } for all k ’s and s ’s generalize the average branchingratio n defined by expression (4) above for monovariate branching processes and are givenby n k,s = ∂∂y s A k ( y , y , . . . , y m ) (cid:12)(cid:12) y = ··· = y m =1 (27)where A k ( y , y , . . . , y m ) is defined in (7).Using the above relations (24-26), the equality (23) together with equation (20) yields¯ R k,s ( t ) = n k,s · µ k,s ( t ) + m X ℓ =1 n k,ℓ f k,ℓ ( t ) ⊗ ¯ R ℓ,s ( t ) . (28)12ntroducing the event rates, i.e., the number of events per unit time, ρ k,s ( t ) = d ¯ R k,s ( t ) dt , (29)expression (28) transforms into ρ k,s ( t ) = n k,s ( t ) + m X ℓ =1 n k,ℓ ( t ) ⊗ ρ ℓ,s ( t ) , (30)where we have used the following notation n k,s ( t ) := n k,s · f k,s ( t ) . (31)The set of equations (30) for all k ’s and s ’s constitute the fundamental starting point of ouranalysis.In order to make further progress, in view of the convolution operator, it is convenientto work with the Laplace transform of the event rates:˜ ρ k,s ( u ) = Z ∞ ρ k,s ( t ) e − ut dt . (32)Introducing the matrices ˜Φ( u ) = [ ˜ ρ k,s ( u )] and ˜ N ( u ) = [˜ n k,s ( u )] , (33)we obtain the following equation for the matrix ˜Φ( u )ˆ I ˜Φ( u ) = ˜ N ( u ) + ˜ N ( u ) ˜Φ( u ) , (34)whose solution is ˜Φ( u ) = ˜ N ( u )ˆ I − ˜ N ( u ) . (35)The rest of the paper is devoted to the study of this solution (35) for various system structuresand memory kernels. V. SYMMETRIC MUTUAL EXCITATIONSA. Definitions Let us consider the case where the set { n k,s } defined by expression (27) reduces to n k,k = a ; n k,s = b, k = s , (36)13his form (36) means that events of a given type have identical triggering efficiencies quan-tified by a to generate first-generation events of the same type. They also have identicalefficiencies quantified by b to trigger first-generation events of a different type. As a con-sequence, the mean number of first-generation events of all kinds that are triggered by amother event of some type k , n k = m X s =1 n k,s , (37)is independent of k and given by n k = n = a + ( m − b , for all k . (38)Introducing the factor q = ba (39)comparing the inter-types with the intra-type triggering efficiencies, we obtain a = n m − q , b = nq m − q . (40)In the time domain, we consider the case of symmetric mutual excitations such that allpdf’s f k,s ( t ) ≡ f ( t ) are independent of the indexes k and s and are all equal to each other. B. General solution in terms of Laplace transforms With the definitions of subsection V A, it follows that ˜ n k,s ( u ) = n ˜ f ( u ) and one canshow that all diagonal and non-diagonal entries of the matrix ˜Φ( u ) given by (35) are givenrespectively by ˜ ρ ( u ) := ˜ ρ k,k ( u ) = n ˜ f ( u )1 − n ˜ f ( u ) · n ˜ f ( u )( q − n ˜ f ( u )( q − 1) + q ( m − , ˜ g ( u ) := ˜ ρ k,s ( u ) = n ˜ f ( u )1 − n ˜ f ( u ) · q n ˜ f ( u )( q − 1) + q ( m − , k = s . (41)Moreover, the Laplace transform˜ ρ k ( u ) = m X s =1 ˜ ρ k,s ( u ) = ˜ ρ ( u ) + ( m − g ( u ) (42)of the total rate of events of all types triggered by a mother jump of type k defined by ρ k ( t ) = P ms =1 ρ k,s ( t ) satisfies the relation˜ ρ k ( u ) = n ˜ f ( u )1 − n ˜ f ( u ) . (43)14 . Exponential pdf of triggering times of first-generation events Let us first study the case where the pdf f ( t ) of the waiting times to generate first-generation events is exponential: f ( t ) = αe − αt ⇐⇒ ˜ f ( u ) = αα + u . (44)The inverse Laplace transforms of the solutions (41) are then ρ ( t ) := ρ k,k ( t ) = α nm e − (1 − n ) αt (cid:0) m − γe − n (1 − γ ) αt (cid:1) ,g ( t ) := ρ k,s ( t ) = α nm e − (1 − n ) αt (cid:0) − γe − n (1 − γ ) αt (cid:1) , k = s , (45)where γ := γ ( q, m ) = 1 − q m − q . (46)Expressions (45) give the explicit time dependence of two functions ρ ( t ) and g ( t ): • ρ ( t ) := ρ k,k ( t ) is the rate of events over all generations of some type k resulting froma given mother event of the same type k . Notice that the term “over all generations”means that an event of type k occurring at some time t > 0, and belonging to thedescent of some previous mother of the same type k that occurred at time 0, may havebeen generated through a long cascade of intermediate events of possibly differentkinds, via a kind of inter-breeding genealogy. • g ( t ) := ρ k,s ( t ) is the rate of events over all generations of some type s resulting from agiven mother event of a different type k . As for ρ ( t ), an event of type s occurring atsome time t > 0, and belonging to the descent of some previous mother of a differenttype k that occurred at time 0, may have been generated through a long cascade ofintermediate events of possibly different kinds, via a kind of inter-breeding genealogy.Figures 1 and 2 show respectively ρ ( t ) := ρ k,k ( t ) and g ( t ) := ρ k,s ( t ) for the case of m = 3types of events and rather close to criticality ( n = 0 . q = 1; 0 . 1; 0 . 01. 15 50 100 1500.10.20.30.40.50.60.70.80.91 t ρ k , k ( t ) q = 0 . q = 0 . q = 1 Fig. 1: Time dependence of ρ ( t ) := ρ k,k ( t ), for m = 3, n = 0 . 99, and q = 1; 0 . 1; 0 . /α . t ρ k , s ( t ) , k = s q = 1 q = 0 . q = 0 . Fig. 2: Time dependence of g ( t ) := ρ k,s ( t ), for m = 3, n = 0 . 99 and q = 1; 0 . 1; 0 . /α . The case q = 1 of complete coupling is special in two ways: (i) ρ ( t ) and g ( t ) are propor-tional to each other; (ii) there is only one time scale τ = α · − n . In contrast, as soon as q < 1, i.e., events of a given type tends to trigger more events of the same type than events16f different types, one can observe that the time dependence of ρ ( t ) and g ( t ) become qualita-tively different. The monotonous decay of ρ ( t ) can be contrasted with the non-monotonousbell-shape dependence of g ( t ). This non-monotonous behavior of g ( t ) results from the pro-gressive seeding of events of a different type than the initial mother type by the less efficientmutual excitation process. This is associated with the introduction of a second time scale τ = α · − nγ ≤ τ controlling the dynamics of both ρ ( t ) and g ( t ) at short times. This effectis all the stronger, the smaller q is, i.e., the larger γ is.This phenomenon of the occurrence of a second time scale τ and of the distinct behaviorof g ( t ) := ρ k,s ( t ) for q < m = 1for which the existence of multiple generations do not change the Poissonian nature of therelaxation process. It only extend the time scale according to τ = α · − n as the averagebranching ratio n increases to the critical value 1. D. Power law pdf of triggering times of first-generation events The same qualitative picture emerges for other pdf’s such as power laws, with amonotonous decay of ρ ( t ) coexisting with a growth from zero up to a maximum followedby a decay for g ( t ). But more interesting features appear, such as the renormalization ofthe exponents in two distinct families, as we now show. As many systems exhibit powerlaw pdf’s of waiting times with rather small exponents θ ≤ . f ( t ) = αθ (1 + αt ) θ , t > , θ > . (47)The constant 1 in the denominator regularizes the pdf at times t < /α . The correspondingLaplace transform of (47) is ˜ f ( u ) = θ e u/α (cid:16) uα (cid:17) θ Γ (cid:16) − θ, uα (cid:17) , (48)which can then be used in (41) to get the general solutions.17 . Non-critical same-type activity rate ρ ( t ) =: ρ k,k ( t ) We first rewrite the Laplace transform ˜ ρ ( u ) given by (41) in a form more convenient forits analysis: ˜ ρ ( u ) = ¯ R k,k · − ϕ ( u )1 + γ ϕ ( u ) · γ ϕ ( u )1 + γ ϕ ( u ) , (49)where ϕ ( u ) := 1 − ˜ f ( u ) (50)and γ = n − n , γ = n (1 − q )1 − n + qn , γ = n (1 − q )1 − n + q ( n + m − ,γ > γ > γ ( q > , m > . (51)The long time behavior of ρ ( t ) is controlled by the small u properties of ˜ ρ ( u ), itselfdependent on the behavior of ϕ ( u ) for small u . From its definition (50), we have that ϕ ( u ) → u → 0. Then, the asymptotic behavior of the Laplace transform (49) of ρ ( t ) is˜ ρ ( u ) ∼ ¯ R k,k − ¯ R k,k · γ ρ ϕ ( u ) , γ ρ = 1 + γ + γ − γ . (52)For θ ∈ (0 , ϕ ( u ) (50) has the following asymptotic behavior ϕ ( u ) ∼ βv θ ≪ , v ≪ , β = Γ(1 − θ ) , v = uα . (53)Accordingly, relation (52) transforms into˜ ρ ( u ) ∼ ¯ R k,k (cid:2) − γ ρ βv θ (cid:3) . (54)The corresponding asymptotic of the rate ρ ( t ) is thus ρ ( t ) ∼ ¯ R k,k · γ ρ βθ Γ(1 − θ ) 1( αt ) θ = ¯ R k,k · γ ρ θ αt ) θ , t → ∞ . (55)The rate ρ ( t ) := ρ k,k ( t ) of events over all generations of some type k resulting from a givenmother event of the same type k that occurred at time 0 decays with the same power lawbehavior as the bare memory function or pdf of waiting times for first-generation events.The only significant difference is the renormalization of the amplitude by the factor ¯ R k,k · γ ρ resulting from the cascades of generations and inter-breeding between the different eventtypes. 18 . Intermediate critical asymptotic same-type activity rate ρ ( t ) := ρ k,k ( t ) For n close to 1, γ becomes large and there is an interesting intermediate asymptoticregime describing the intermediate time decay of ρ ( t ). To describe it, we need to distinguishthe following three parameter regimes.1. γ ≫ γ < γ ≃ 1. This occurs for n → q is not too close to 0. Forinstance, n = 0 . q = 0 . m = 5 yield γ = 0 . , γ = 0 . , γ = 19. In this case,there is an intermediate range on the u -axis defined by( γ β ) − θ ≪ v ≪ ( γ β ) − θ ⇐⇒ α ( γ β ) − θ ≪ u ≪ α ( γ β ) − θ ≈ α , (56)such that γ ϕ ( u ) ≫ γ ϕ ( u ) ≪ γ ϕ ( u ) ≪ 1. In this range, the leadingterms controlling the value of expression (49) is˜ ρ ( u ) ≈ ¯ R k,k · γ ϕ ( u ) . (57)Substituting the asymptotic relation (53), we obtain˜ ρ ( u ) ≈ ¯ R k,k · γ Γ(1 − θ ) v − θ . (58)The corresponding intermediate asymptotic of the rate is ρ ( t ) ≈ ¯ R k,k γ sin( πθ ) π αt ) − θ , ≪ αt ≪ [ γ Γ(1 − θ )] /θ . (59)2. γ > γ ≫ γ ≃ 1: This occurs for n → q close to 0 and m large. Forinstance, n = 0 . q = 0 . m = 50 yield γ = 1 . , γ = 15 . , γ = 19. In this case,there is an intermediate range on the u -axis defined by( γ β ) − θ ≪ v ≪ ( γ β ) − θ ⇐⇒ α ( γ β ) − θ ≪ u ≪ α ( γ β ) − θ ≈ α , (60)such that γ ϕ ( u ) > γ ϕ ( u ) ≫ γ ϕ ( u ) ≪ 1. In this range, the leading termscontrolling the value of expression (49) is˜ ρ ( u ) = ¯ R k,k · γ γ · (1 − ϕ ( u )) , (61)whose inverse Laplace transform has the same form as (55) with γ ρ replaced by 1.19. γ > γ > γ ≫ 1: This occurs for n → q close to 0 and m not too large. Forinstance, n = 0 . q = 0 . m = 2 yield γ = 13 . , γ = 15 . , γ = 19. Then, in theintermediate interval on the u -axis( γ β ) − θ ≪ v ≪ ⇐⇒ α ( γ β ) − θ ≪ u ≪ α , (62)the asymptotic relation (53) holds, while at the same time γ ϕ ( u ) ≫ 1. In this range(62), the leading terms controlling the value of expression (49) is˜ ρ ( u ) ≈ ¯ R k,k · γ ϕ ( u ) · γ ϕ ( u ) γ ϕ ( u ) = ¯ R k,k · γ γ γ · ϕ ( u ) . (63)Using the asymptotic relation (53), we obtain˜ ρ ( u ) ≈ G v − θ , G = ¯ R k,k · γ γ γ β = ¯ R k,k · γ γ γ Γ(1 − θ ) . (64)The corresponding intermediate power asymptotic of the rate ρ ( t ) is ρ ( t ) ≈ ¯ R k,k γ γ γ sin( πθ ) π αt ) − θ , ≪ αt ≪ [ γ Γ(1 − θ )] /θ . (65)As an illustration, for the above values n = 0 . q = 0 . m = 2, the power law (65)with exponent 1 − θ holds up to a maximum time [ γ Γ(1 − θ )] /θ α − ≈ , α − for θ = 0 . 25 .Let us summarize and interpret the above results,1. n → q not small ( γ ≫ γ < γ ≃ − θ is similar to the renormalized response functiondue to the cascade of generations found for the self-excited Hawkes process with justone type of events [15, 51, 56]. The mechanism is the same, since a coupling coefficient q not too small ensures a good mixing among all generations.2. n → q → m → ∞ with qm ≃ γ > γ ≫ γ ≃ ρ ( t ) exhibits the same decay (55)with exponent 1+ θ as if the system was far from criticality. In a sense, due to the weakmutual triggering efficiency and the many event types, the system is never critical.3. n → q → m not too large such that qm ≪ γ > γ > γ ≫ − θ is again similar tothe renormalized response function due to the cascade of generations found for theself-excited Hawkes process with just one type of events [15, 51, 56].20 . Asymptotic and intermediate critical asymptotic inter-type activity rate g ( t ) := ρ k,s ( t ) In order to obtain the time-dependence of g ( t ), we express its Laplace transform ˜ g ( u )given in expression (41) in a form analogous to (49):˜ g ( u ) = ¯ R k,s · − ϕ ( u )1 + γ ϕ ( u ) · 11 + γ ϕ ( u ) , k = s . (66)Three regimes can be distinguished.1. Asymptotic regime of long times for n < 1. At long times, the asymptoticrelation (53) holds true. Then, analogous to (54) and (55), we obtain the followingasymptotics ˜ g ( u ) ∼ ¯ R k,s (cid:2) − γ g βv θ (cid:3) , γ g = 1 + γ + γ u → ⇒ g ( t ) ∼ ¯ R k,s · γ g βθ Γ(1 − θ ) 1( αt ) θ , t → ∞ . (67)This power law decay with exponent 1 + θ , equal to the exponent of the memorykernel (47), is characteristic of the non-critical regime in which only a few generationsof events are triggered in significant numbers.2. Intermediate asymptotic regime ( n → with q → and m large) . Then, γ ≫ γ ≃ 1. This is the same second regime analyzed in subsection V D 2. Inthis case, there is an intermediate asymptotic in the range defined by (60) such thatthe following approximate relation holds˜ g ( u ) ≈ ¯ R k,s γ · ϕ ( u ) , k = s . (68)The corresponding intermediate power asymptotic of g ( t ) := ρ k,s ( t ) is ρ ( t ) ≈ ¯ R k,k γ sin( πθ ) π αt ) − θ , ≪ αt ≪ [ γ Γ(1 − θ )] /θ . (69)This power law decay with exponent 1 − θ is significantly slower than the previous onewith exponent 1 + θ and results from the proximity to the critical point n = 1.3. Intermediate asymptotic regime ( n → with q → and m small ). Then, γ given in (51) is large and there is an intermediate interval (62) for u such that,analogous to (63), the following approximate relation holds˜ g ( u ) ≈ ¯ R k,s · γ ϕ ( u ) · γ ϕ ( u ) = ¯ R k,s · γ γ · ϕ ( u ) . (70)21sing the power asymptotic (53), we obtain the intermediate power law˜ g ( u ) ≈ G v − θ , G = ¯ R k,s · γ γ β = ¯ R k,s · γ γ Γ(1 − θ ) . (71)Accordingly, the intermediate power law asymptotic of the inter-type activity rate g ( t ) := ρ k,s ( t ) reads g ( t ) ≈ ¯ R k,s γ γ Γ(2 θ )Γ(1 − θ ) · αt ) − θ , ≪ αt ≪ [ γ Γ(1 − θ )] /θ . (72)This power law decay with exponent 1 − θ is similar to the decay after an “endogenous”peak, as classified in previous analyses of the monovariate self-excited Hawkes process[31, 52–55, 57]. Indeed, the exponent 1 − θ , corresponding to a very slow power lawdecay, has been until now seen as the characteristic signature of self-organized burstsof activities that are generated endogenously without the need for a major exoge-nous shock. Here, we see this exponent describing the decay of the activity of eventstriggered by an “exogenous” mother shock of a different type, in the critical regime n → q → 0. It is clear that the mechanism is differentfrom the previously classified “endogenous” channel [31, 52–55, 57], involving here aninterplay between the cascade over generations and the weak mutual excitations. −6 −5 −4 −3 −2 t ρ ( t ) , g ( t ) ∼ t − . ∼ t − . ∼ t − . Fig. 3: Time dependence of ρ ( t ) := ρ k,k ( t ) and g ( t ) := ρ k,s ( t ) in double logarithmicrepresentation for n = 0 . q = 0 . m = 2, and θ = 0 . 25 (1 + θ = 1 . 25, 1 − θ = 0 . − θ = 0 . α − . The functions ρ ( t ) and g ( t ) have been calculatednumerically from their complete Laplace transforms (41). ρ ( t ) := ρ k,k ( t ) and g ( t ) := ρ k,s ( t ) in theregime n → q → m small, for which there exists an intermediate asymptoticof the third type both for ρ ( t ) and g ( t ). For ρ ( t ), one can clearly observe the intermediatepower law asymptotic with exponent 1 − θ = 0 . 75 followed by the final asymptotic powerlaw with exponent 1 + θ = 1 . 25. For g ( t ), the intermediate power law asymptotic withexponent 1 − θ = 0 . θ = 1 . VI. ONE-DIMENSIONAL CHAIN OF DIRECTED TRIGGERINGA. Definitions We consider a chain of directed influences k → k + 1 where the events of type k triggerevents of both types k and k + 1 only (and not events of type k − k = 1 , , ..., m . This is captured by a form of the matrix ˆ N which has only thediagonal and the line above the diagonal with non-zero elements.As the simplest example, we shall study networks of mutual excitations corresponding tothe following matrix ˆ N of the mean numbers of first-generation eventsˆ N = χ ξ ......... ... χ ξ ......... ... χ ξ ......... ... χ ξ......... ..................................... (73)where χ = n q , ξ = nq q . (74) B. Laplace transform of the event activities ρ k,s ( t ) defined in (29) In order to derive the equations governing the rates ρ k,s ( t ), we need to recall a resultconcerning the total numbers of events ¯ R k,k and ¯ R k,s generated by a given mother of type k . Assuming that the mother event is of type k , we have [58] ¯ R s,s = ¯ R k,s = ¯ R s,k = 0 for23 ≤ s < k and ¯ R k,k = χ − χ = n q − n , ¯ R k,s = ξ s − k (1 − χ ) s − k +1 = (1 + q ) ( nq ) s − k (1 + q − n ) s − k +1 , s > k . (75)Then, the Laplace transforms ˜ ρ k,s ( u ) (32) of the rates ρ k,s ( t ) are given by the right-handsides of expressions (75), with the following substitutions χ χ · ˜ f ( u ) , ξ ξ · ˜ f ( u ) . (76)This yields ˜ ρ k,k ( u ) := ˜ ρ ( u ) = χ · ˜ f ( u )1 − χ · ˜ f ( u ) ,ρ k,s ( u ) := ˜ g m ( u ) = ( ξ · ˜ f ( u )) m (1 − χ · ˜ f ( u )) m +1 , m = s − k > . (77) C. Exponential pdf f ( t ) of triggering times of first-generation events We use the parameterization (44) for the pdf f ( t ), which leads after calculations to ρ k,k ( t ) := ρ ( t ) = αχ e − (1 − χ ) αt ,ρ k,s ( t ) := g m ( t ) = αξm ! ( ξαt ) m − ( m + χαt ) e − (1 − χ ) αt , m = s − k > . (78)The rate ρ k,k ( t ) := ρ ( t ) of events of the same type as the mother decays simply as anexponential with a characteristic decay time q q − n α − , which exhibits the standard criticalslowing down at the critical value of the mean branching ratio n c = 1 + q . In contrast, thecross rates ρ k,s ( t ) := g m ( t ) exhibit a non-monotonous behavior, which reflects the directednature of the mutual triggering of events of different types. For large m values, the cross-ratebecomes almost symmetrical functions of time, as shown in figure 4.24 αt g m ( t ) m = 5 m = 3 m = 4 m = 2 Fig. 4: Time dependence of the cross-rates ρ k,s ( t ) := g m ( t ) given by (78) for n = 0 . q = 1 and different m values. D. Power law pdf of triggering times of first-generation events The time dependence of ρ k,k ( t ) := ρ ( t ) given by the first equation of (77) when f ( t ) isgiven by relation (47) with Laplace transform (48) is the same as for the monovariate Hawkesprocess (with a single event type), with the modification that the role of the mean branchingratio n is replaced by χ . The rate ρ ( t ) exhibits an intermediate power law asymptotic withexponent 1 − θ up to a cross-over time ≃ α − / (1 − χ ) θ followed by the asymptotic powerlaw decay with exponent 1 + θ corresponding to the memory kernel f ( t ).Interesting new regimes appear for the time dependences of the cross-rates g m ( t ). We firstexpress the Laplace transforms ˜ g m ( u ) given by (77) of the cross-rates by using the auxiliaryfunction ϕ ( u ) defined by (50):˜ g m ( u ) = ¯ R k,s · [1 − ϕ ( u )] m [1 + γϕ ( u )] m +1 , (79)where γ = χ − χ = n q − n , (80)and the mean number ¯ R k,s ( k = s ) is given by expression (75). Replacing ϕ ( u ) by its25symptotic (53), we obtain the asymptotic formula˜ g m ( u ) = ¯ R k,s · (cid:0) − βv θ (cid:1) m (1 + γβv θ ) m +1 . (81)The long time asymptotic of g m ( t ) is controlled by the behavior of ˜ g m ( u ) for v → 0, whoseleading order is given by ˜ g m ( u ) ∼ ¯ R k,s (cid:2) − γ ( m ) v θ (cid:3) , v → γ ( m ) = β ( m + ( m + 1) γ ) . (82)This expression holds true for γ < + ∞ , i.e., n < q , where the upper bound n c = 1 + q define the critical point. Accordingly, the main asymptotic of the cross event rate g m ( t ) is g ( t ) ∼ ¯ R k,s · γ ( m ) θ Γ(1 − θ ) 1( αt ) θ , t → ∞ . (83)This recovers the usual long time power law dependence, which is determined by the memorykernel f ( t ) of waiting times for first-generation triggering.There is also an intermediate asymptotic regime present when γ ≫ 1, i.e., n → q from below. Specifically, the intermediate asymptotic domain in the variable v is definedby the interval ( βγ ) − /θ ≪ v ≪ 1, such that γβv θ ≫ v ≪ 1. Then, the asymptoticrelation (53) is true, and expression (81) can be simplified into the approximate relation˜ g m ( u ) ≈ G m · v − ( m +1) θ , G m = ¯ R k,s ( γβ ) m +1 , ( βγ ) − /θ ≪ v ≪ . (84)Accordingly, analogous to (72), we obtain g m ( t ) ≈ G m Γ[( m + 1) θ ] · αt ) − ( m +1) θ , ≪ αt ≪ ( γβ ) /θ . (85)This expression (85) predicts a hierarchy of exponents 1 − ( m + 1) θ characterizing theintermediate asymptotic power law dependence of the rates g m ( t ) := ρ k,s ( t ) of events of type s as a function of the distance m = s − k along the space of types from the type k of theinitial triggering mother. Figure 5 illustrates this prediction (85) for m = 1; 2 with θ = 0 . − θ = 0 . − θ = 0 . −10 −9 −8 −7 −6 −5 −4 αt g m ( t ) ∼ t − θ ∼ t − − θ ∼ t − θ Fig. 5: Time dependence of the rates g ( t ) := ρ k,k +1 ( t ) (upper curve) and g ( t ) := ρ k,k +2 ( t ) (lower curve) of events of type s = k + 1 and s = k + 2 triggered by amother of type k . The parameters are n = 0 . q = 0 . 01 and θ = 0 . 25. The dashedstraight lines correspond to the power laws predicted in the text for the asymptoticand corresponding intermediate asymptotic regimes. The intermediate power law decay laws with exponents 1 − ( m + 1) θ hold only whenthis exponent is positive, i.e., for m < θ − 1. To understand what happens for larger m ’s,a more careful analysis is required, which is presented in Appendix B, which shows thatformula (85) still holds and predicts that g m ( t ) is an increasing function of time for times αt < ( γβ ) /θ before decreasing again with the standard asymptotic power law ∼ /t θ . Thisis summarized by figure 6, which plots the time dependence of the rates g m ( t ) := ρ k,k + m ( t ) ofevents of type s = k + m triggered by a mother of type k for m = 0 to 5, with θ = 1 / 3. Onecan clearly observe the existence of the intermediate power asymptotics (equation (123)in Appendix B and expression (85)) for different values of m . When inequality (124) ofAppendix B holds, the intermediate asymptotics are not decaying but growing as a functionof time, as predicted by expression (123) of Appendix B and (85).27 −8 −6 −4 −2 t g m ( t ) m=0m=2m=1m=3m=4m=5 Fig.6: Time dependence of the rates g m ( t ) := ρ k,k + m ( t ) of events of type s = k + m triggered by a mother of type k , for m = 0 (same type of events as the initial triggeringmother) and m = 1 , , , θ = 1 / γ := ln (cid:16) qn (cid:17) = 0 . 01. Thenumber of summands used in the sum (106) of Appendix B is N = 1500. VII. ONE-DIMENSIONAL CHAIN OF NEAREST-NEIGHBOR-TYPE TRIG-GERINGA. Definitions A natural extension to the above one-dimensional chain of directed triggering discussedin the previous section includes feedbacks from events of type k + 1 to type k . The exampletreated in the present section corresponds to fully symmetry mutual excitations confined tonearest neighbor in the sense of event types: k ↔ k + 1. Mathematically, this is describedby a symmetric matrix ˆ N of the average numbers n k,s of first-generation events of different28ypes triggered by a mother of a fixed type.We assume that all diagonal elements are equal to some constant χ (same self-triggeringabilities) and all off-diagnoal elements are equal to some different constant ξ (same mutualtriggering abilities). The elements n ,m and n m, are also equal to ξ to close the chainof mutual excitations between events of type 1 and of type m . Restricting to m = 6 forillustration purpose, the corresponding matrix ˆ N readsˆ N = χ ξ ξξ χ ξ ξ χ ξ ξ χ ξ 00 0 0 ξ χ ξξ ξ χ (86)where χ = n q , ξ = nq q ) ⇒ χ + 2 ξ = n . (87)As before, the parameter q quantifies the “strength” of the interactions between events ofdifferent types. Here, n represents the total number of first-generation events of all types thatare generated by a given mother of fixed arbitrary type. Figure 7 provides the geometricalsense of matrix ˆ N (86) for m = 6, where the circles represent the six types of events andthe arrows denote their mutual excitation influences. 12 3 4 56 ig. 7: Geometric sense of the matrix ˆ N for a one-dimensional chain of nearest-neighbor triggering in the space of event types. B. Analysis of the event rates ρ k,s ( t ) The Laplace transform ˜ ρ k,k ( u ) of the event rates ρ k,k ( t ) (29) of type k that are triggeredby a mother of the same type k reads˜ ρ k,k ( u ) := ˜ ρ ( u ) = A [ n ˜ f ( u ) , q ] , (88)where [58] A ( n, q ) = 4 n (1 − n + q ) − (1 − n + q )(5 n − q − n q − n q − n + q ) − − n + q ) n q + n q . (89)Analogously, the Laplace transforms of the cross-rates ˜ ρ k,s ( u ) are˜ g ( u ) = B [ n ˜ f ( u ) , q ] , ˜ g ( u ) = C [ n ˜ f ( u ) , q ] , ˜ g ( u ) = D [ n ˜ f ( u ) , q ] , (90)where [58] B ( n, q ) = nq (1 + q )(2(1 − n + q ) − n q )4(1 − n + q ) − − n + q ) n q + n q ,C ( n, q ) = n q (1 + q )(1 − n + q )4(1 − n + q ) − − n + q ) n q + n q ,D ( n, q ) = n q (1 + q )4(1 − n + q ) − − n + q ) n q + n q . (91)Figure 8 shows the time dependence of ρ ( t ), g ( t ), g ( t ) and g ( t ) for the case wherethe pdf f ( t ) is a power law (47), for the parameters n = 0 . , q = 0 . , θ = 0 . 2. Onecan observe a common power law asymptotic ∼ t − − θ = t − . at large times, as well asintermediate asymptotic power laws ρ ( t ) ∼ t − θ = t − . , g ( t ) ∼ t − θ = t − . ,g ( t ) ∼ t − θ = t − . , g ( t ) ∼ t − θ = t − . . (92)30 −10 −5 αt ρ ( t ) , g m ( t ) ∼ t − − θ ∼ t − θ ∼ t − θ ∼ t − θ ∼ t − θ Fig. 8: Top to bottom: the solid lines represent the time dependence of ρ ( t ), g ( t ), g ( t ) and g ( t ), in the case a one-dimensional chain of nearest-neighbor-type triggeringin the space of types, with six types. The parameters are n = 0 . , q = 0 . , θ = 0 . VIII. CONCLUDING REMARKS We have presented a preliminary analysis of some temporal properties of multivariateself-excited Hawkes conditional Poisson processes. These processes are very interesting can-didates to model a large variety of systems with bursty events, for which past activitytriggers future activity. The term “multivariate” refers to the property that events comein different types, with possibly different intra- and inter-triggering abilities. The richnessof the generated time dynamics comes from the cascades of intermediate events of possiblydifferent kinds, unfolding via a kind of inter-breeding genealogy. We have developed thegeneral formalism of the multivariate generating moment function for the cumulative num-ber of first-generation and of all generation events triggered by a given mother event as afunction of the current time t . We have obtained the general relations for the mean numbers31f events triggered over all generations by a given event as a function of time. We have ap-plied this technical and mathematical toolbox to several systems, characterized by differentspecifications on how events of a given type may trigger events of different types. In partic-ular, for systems in which triggering between events of different types proceeds through aone-dimension directed or symmetric chain of influence in type space, we have discovered anovel hierarchy of intermediate asymptotic power law decays of the rate of triggered eventsas a function of the distance of the events to the initial shock in the space of types. We havebeen able to derive the time-dependence of the rates of events triggered from a given shockfor distributions of waiting times of first-generation events that have either exponential orpower law tails, for a variety of systems. Future directions of investigations include the studyof more realistic networks in type-space and of the full distribution of even rates, beyondthe mean dynamics reported here. Acknowledgement : We acknowledge financial support from the ETH Competence Cen-ter ”Coping with Crises in Complex Socio-Economic Systems” (CCSS) through ETH Re-search Grant CH1-01-08-2. This work was also partially supported by ETH Research GrantETH-31 10-3. Appendix A: Proof of theorem 3.1 Substituting relation (13) in expression (14) leads to A k ( y , y , . . . , y m ; t ) = ∞ X d =0 · · · ∞ X d m =0 ∞ X r = d · · · ∞ X r m = d m P k ( r , . . . , r m ) m Y s =1 (cid:18) r s d s (cid:19) [ µ k,s ( t ) y s ] d s [1 − µ k,s ( t )] r s − d s . (93)Inverting the order of the summations ∞ X d s =0 ∞ X r s = d s ( · · · ) = ∞ X r s =0 r s X d s =0 , (94)32e rewrite expression (93) as A k ( y , y , . . . , y m ; t ) = ∞ X r =0 · · · ∞ X r m =0 P k ( r , . . . , r m ) r X d =0 · · · r m X d m =0 m Y s =1 (cid:18) r s d s (cid:19) [ µ k,s ( t ) y s ] d s [1 − µ k,s ( t )] r s − d s , (95)or equivalently A k ( y , y , . . . , y m ; t ) = ∞ X r =0 · · · ∞ X r m =0 P k ( r , . . . , r m ) m Y s =1 r s X d s =0 (cid:18) r s d s (cid:19) [ µ k,s ( t ) y s ] d s [1 − µ k,s ( t )] r s − d s . (96)Using the binomial formula r s X d s =0 (cid:18) r s d s (cid:19) [ µ k,s ( t ) y s ] d s [1 − µ k,s ( t )] r s − d s = [1 + µ k,s ( y − r s , (97)we obtain A k ( y , y , . . . , y m ; t ) = ∞ X r =0 · · · ∞ X r m =0 P k ( r , . . . , r m ) m Y s =1 [1 + µ k,s ( t )( y − r s . (98)In view of definition (8) of the GMF A k ( y , y , . . . , y m ), this last expression means that A k ( y , y , . . . , y m ; t ) = A k [1 + µ k, ( t )( y − , . . . , µ k,m ( t )( y m − . (99)Using definition (16) of the function Q k , we obtain relation (15). (cid:4) Appendix B: Analysis of the behavior of g m ( t ) := ρ k,s ( t ) for a one-dimensional chainof directed triggering of section VI for n → q when − ( m + 1) θ < Let us start with expression ˜ g m ( u ) (77) that we rewrite, omitting the nonessential factor ξ m , as ˜ g m ( u ) = ˜ f m ( u )[1 − χ · ˜ f ( u )] m +1 , with χ = n q . (100)Using the binomial formula 1(1 − x ) m +1 = ∞ X k =0 (cid:18) m + kk (cid:19) x k , (101)33xpression (100) becomes ˜ g m ( u ) = ∞ X k =0 (cid:18) m + kk (cid:19) χ k ˜ f m + k ( u ) . (102)As we are interesting in the case where the pdf f ( t ) has the power asymptotic f ( t ) ∼ /t θ , with 0 < θ < 1, it is convenient to use for f ( t ) one special representative of thefunctions presenting this asymptotic power law behavior, namely the one-sided L´evy stabledistribution of order θ [59], that we refer to as f θ ( t ). Its Laplace transform is˜ f θ ( u ) = e − u θ , < θ < . (103)Accordingly, relation (102) takes the form˜ g m ( u ) = ∞ X k =0 (cid:18) m + kk (cid:19) χ k · e − ( m + k ) u θ . (104)Taking the inverse Laplace transform of (104) provides us with the exact expression g m ( t ) = ∞ X k =0 (cid:18) m + kk (cid:19) · χ k ( m + k ) /θ · f θ (cid:18) t ( m + k ) /θ (cid:19) . (105)In order to analyze (105), it is convenient to rewrite it as g m ( t ) = θt θ ∞ X k =0 S m ( k + m ) Q θ (cid:18) m + kt θ (cid:19) e − γk , (106)where γ = ln (cid:18) χ (cid:19) = ln (cid:18) qn (cid:19) > n < q . (107)We have defined the functions S m ( x ) := x (cid:18) xm (cid:19) , x > m , (108)and Q θ ( x ) := 1 θx /θ f θ (cid:18) x /θ (cid:19) , x > , (109)such that 1 x /θ f θ (cid:18) tx /θ (cid:19) = x θt θ +1 Q θ (cid:16) xt θ (cid:17) . (110)In order to extract the relevant information from expression (106) for g m ( t ), we need todiscuss some properties of the two functions S m ( x ) and Q θ ( x ).34 roperties of the function S m ( x ). The function S m ( x ) is a finite sum of powerfunctions of the argument x S m ( x ) = m X r =1 a r,m x r +1 . (111)In particular, S ( x ) = x, S ( x ) = x , S ( x ) = 12 x − x ,S ( x ) = 16 x − x + 13 x ,S ( x ) = 124 x − x + 1124 x − x . (112) Properties of the function Q θ ( x ). For θ ∈ (0 , / x . Accordingly, its moments of any order r > M ( r ) := Z ∞ x r Q θ ( x ) dx = Γ( r + 1)Γ( rθ + 1) . (113)Moreover, the value of Q θ ( x ) at x = 0 is equal to Q θ ( x = 0) = 1Γ(1 − θ ) . (114)For the particular cases θ = 1 / θ = 1 / 3, the function Q θ ( x ) can be expressed inexplicit form: Q / ( x ) = 1 √ π exp (cid:18) − x (cid:19) , Q / ( x ) = √ · Ai (cid:18) x √ (cid:19) . (115)We study the behavior of g m ( t ) given by expression (106) for γm ≪ , t θ ≫ m t − θ ≪ 1. To leading order and without essential error, we may replace the discrete sum(106) by the continuous integral g m ( t ) ≃ θt θ Z ∞ m S m ( x ) Q (cid:16) xt θ (cid:17) e − γx dx . (116)Using the following change of variable of integration x y = xt θ ↔ x = t θ y , (117)we obtain g m ( t ) ≃ θt Z ∞ S m ( t θ y ) Q θ ( y ) e − γt θ y dy . (118)35sing relation (111), this yields g m ( t ) ≃ θt m X r =1 a r,m t ( r +1) θ G r,m ( t ) , (119)where G r,m ( t ) = Z ∞ y r +1 Q θ ( y ) e − γt θ y dy . (120)The intermediate asymptotic regime corresponds to the time domain γt θ ≪ t θ ≫ G r,m = Z ∞ y r +1 Q θ ( y ) dy = Γ( r + 2)Γ[( r + 1) θ + 1] , γt θ ≪ , (121)is time independent. Accordingly, the mean rate g m ( t ) (119) is found as the sum of powerlaw functions g m ( t ) ≃ θ · t ( m +1) θ − · m X r =1 a r,m G r,m · t ( r − m ) θ . (122)Taking into account that we consider the case t θ ≫ g m ( t ) ≃ θ ( m + 1)Γ[( m + 1) θ + 1] · t ( m +1) θ − ∼ t ( m +1) θ − , ≪ t ≪ γ − /θ . (123)This recovers the result (85) presented in the main text. In addition, it makes more precisewhat happens for ( m + 1) θ > . (124)In this case, g m ( t ) starts as a growing function of t up to t ≃ γ − /θ . This retrieves thesame qualitative behavior found when f ( t ) is an exponential, which has been analyzed insubsection VI C and represented in figure 4.Of course, at times t ≫ γ − /θ , this growth is replaced by the standard power law decay ∼ /t θ . Indeed, for γt θ ≫ 1, the integral (120) is approximately equal to G r,m ( t ) ≃ Q θ (0) Z ∞ y r +1 e − γtθy dy = ( r + 1)!Γ[1 − θ ] γ − r − t − ( r +2) θ . (125)Substituting this relation into (119) yields g m ( t ) ≃ θt θ m X r =1 a r,m ( r + 1)!Γ[1 − θ ] γ − r − ∼ t θ , t ≫ γ − /θ . (126)36igure 6 in the text sums up these results by plotting the time dependence of the rates g m ( t ) := ρ k,k + m ( t ) of events of type s = k + m triggered by a mother of type k for m = 0to 5. One can clearly observe the existence of the intermediate power asymptotics (123) fordifferent values of m . When inequality (124) holds, the intermediate asymptotics are notdecaying but growing as a function of time, as predicted by expression (123).37 1] Hawkes, A.G., Journal of Royal Statistical Society, series B , , 438-443, 1971.[2] Hawkes, A.G., Spectra of some mutually exciting point processes with associated variables,In Stochastic Point Processes, ed. P.A.W. Lewis, Wiley, 261-271, 1972.[3] Hawkes, A.G. and Adamopoulos, L., Bull Internat. Stat. Inst. 45, 454-461, 1973.[4] Hawkes, A.G. and Oakes D., Journal Apl. Prob. 11, 493-503, 1974.[5] Br´emaud, P., Point Processes and Queues, Springer, New York (1981).[6] Daley, D.J. and D. Vere-Jones, An Introduction to the Theory of Point Processes, SpringerSeries in Statistics (2007).[7] Horowitz, Paul and Winfield Hill, The Art of Electronics, 2nd edition. Cambridge (UK):Cambridge University Press, 1989, pp. 431-2.[8] Montroll, E.W. and H. Scher, J. Stat. Phys., 9 (2), 101-135 (1973).[9] Scher, H. and E.W. Montroll, Phys. Rev. B 12, 2455 (1975).[10] Cont, Rama and Peter, Tankov, Financial Modelling with Jump Processes, Chapman &Hall/CRC Financial Mathematical Series, 2004[11] Ogata, Y., J. Am. stat. Assoc. 83, 9-27, 1988.[12] Ogata, Y., Tectonophysics, 169, 159-174, 1989.[13] Ogata, Y., Ann. Inst. stat. Mech. 50, 379-402, 1998.[14] Ogata, Y., Pure Appl. Geophys. 155, 471-507, 1999.[15] A. Helmstetter and D. Sornette J. Geophys. Res. 107, NO. B10, 2237,doi:10.1029/2001JB001580, 2002.[16] A. Saichev and D. Sornette J. Geophys. Res., 112, B04313, doi:10.1029/2006JB004536, 2007.[17] D. Sornette and I. Osorio, Prediction, chapter in “Epilepsy: The Intersection of Neurosciences,Biology, Mathematics, Physics and Engineering”, Editors: Osorio I., Zaveri H.P., Frei M.G.,Arthurs S., CRC Press, Taylor & Francis Group (2010) (http://arxiv.org/abs/1007.2420).[18] V. Chavez-Demoulin, A.C. Davison and A.J. McNeil, Quantitative Finance, 5 (2), 227-234,2005[19] L. Bauwens and N. Hautsch, Modelling Financial High Frequency Data Using Point Processes,Handbook of Financial Time Series, Part 6, 953-979, DOI:10.1007/978-3-540-71297-8_41 ,2009. 20] E. Errais, K. Giesecke and L.R. Goldberg, Affine Point Processes and Portfolio Credit Risk(June 7, 2010). Available at SSRN: http://ssrn.com/abstract=908045 .[21] S. Azizpour, K. Giesecke and G. Schwenkler, Exploring the Sources of Default Clustering,working paper, Stanford University, 2010[22] Y. A¨ıt-Sahalia, J. Cacho-Diaz and R.J.A. Laeven, Modeling financial contagion using mutuallyexciting jump processes, Working Paper 15850, , 2010.[23] M. Salmon and W. W. Tham, Preferred Habitat, Time Deformation and the Yield Curve ,working paper (Revised and resubmit - Journal of Financial Markets), 2008.[24] Zhuang J., Vere-Jones D., Guan H., Ogata Y. and Ma L., Pure and Applied Geophysics, 162,1367-1396, doi:10.1007/s00024-004-2674-3, 2005.[25] T.J. Liniger, Multivariate Hawkes Processes, PhD Diss. ETH No. 18403, ETH Zurich, 2009.[26] The paper [Kagan, Y. Y. and Knopoff, L., J. Geophys. Res. 86, 2853-2862, 1981] offers acontinuum-state critical branching process which develops along the time axis. The continuum-state means that earthquake events are infinitesimal and to obtain ‘real’ earthquakes, eachrealization is processes through a special filter. This paper can however been considered asan ancestor to Ogata (1988)’s ETAS model, because it has the same branching structureof jumps and power law triggering function. Apart from being used differently (Kagan andKnopoff (1981) to generate one complex earthquake and Ogata (1988) to generate a complexsequence of aftershocks triggered by a main earthquake), the only real difference between thetwo models is the regularization of the memory kernel at short times, which has no significantimpact.[27] Helmstetter, A., Phys. Res. Lett. 91, 058501, 2003.[28] D. Sornette and M.J. Werner, J. Geophys. Res. 110, No. B8, B08304,doi:10.1029/2004JB003535, 2005.[29] D. Sornette and A. Helmstetter, Physical Review Letters 89 (15) 158501, 2002.[30] A. Helmstetter and D. Sornette, Physical Review E. 66, 061104, 2002.[31] A. Helmstetter, D. Sornette and J.-R. Grasso, J. Geophys. Res., 108 (B10), 2046,doi:10.1029/2002JB001991, 2003.[32] A. Helmstetter and D. Sornette, Geophys. Res. Lett. 30 (11) doi:10.1029/2003GL017670, 2003.[33] A. Helmstetter and D. Sornette, J. Geophys. Res., 108, 2482, 10.1029/2003JB002485, 2003.[34] A. Saichev and D. Sornette, Phys. Rev. E 71, 056127, 2005. 35] A. Saichev, A. Helmstetter and D. Sornette, Pure and Applied Geophysics 162, 1113-1134,2005.[36] A. Saichev and D. Sornette, Phys. Rev. E 70, 046123, 2004.[37] A. Saichev and D. Sornette, Eur. Phys. J. B 51 (3), 443-459, 2006.[38] D. Sornette, S. Utkin and A. Saichev, Physical Review E 77, 066109, 2008.[39] Ozaki, T., Ann. Inst Statist. Math. 31, Part B, 145-155, 1979.[40] Ogata, Y., IEEE Transactions on Information Theory, Vol.IT-27, No.1, Jan., 23-31, 1981.[41] Ogata, Y. and Akaike, H., Journal of the Royal Statistical Society, Series B 44, No.1, 102-107,1982.[42] Ogata, Y., Journal of Physics of the Earth 31, 115-124, 1983.[43] Zhuang J., Ogata Y. and Vere-Jones D., Journal of the American Statistical Association, 97,369-380 (2002).[44] Zhuang, J., Ogata, Y. and Vere-Jones, D., Journal of Geophysical Research 109 (B5), B05301,doi:10.1029/2003JB002879, 2004.[45] Ogata Y. and Zhuang J., Tectonophysics, 413, 13-23, 2006.[46] Zhuang J., PhD thesis, Department of Statistical Science, The Graduate University for Ad-vanced Studies (2002).[47] Marsan D. and O. Lenglin´e, Science 319 (5866), 1076-1079, 2008.[48] D. Sornette and S. Utkin, Physical Review E 79, 061110, 2009.[49] Harris, T.E.,The theory of branching processes, Springer, Berlin, 1963.[50] Sornette, D., Proceedings of the National Academy of Sciences USA 99, SUPP1, 2522-2529,2002.[51] A. Saichev and D. Sornette, European Physical Journal B 75, 343-355, 2010.[52] D. Sornette, F. Deschatres, T. Gilbert and Y. Ageon, Phys. Rev. Letts. 93 (22), 228701, 2004.[53] F. Deschatres and D. Sornette, Phys. Rev. E 72, 016112, 2005.[54] D. Sornette, Endogenous versus exogenous origins of crises, in the monograph enti-tled “Extreme Events in Nature and Society,” Series: The Frontiers Collection, S.Albeverio, V. Jentsch and H. Kantz, eds. (Springer, Heidelberg, 2005), pp 95-119, http://arxiv.org/abs/physics/0412026 [55] R. Crane and D. Sornette, Proc. Nat. Acad. Sci. USA 105 (41), 15649-15653, 2008.[56] A. Sornette and D. Sornette, Geophys. Res. Lett. 26, N13, 1981-1984, 1999. 57] D. Sornette and A. Helmstetter, Physica A 318 (3-4), 577-591, 2003.[58] A. Saichev and D. Sornette, Multivariate Self-Excited Epidemic Processes, working paper,ETH Zurich, 2010.[59] Zolotarev, V. M., One-dimensional Stable Distributions, Amer. Math. Soc. ProvidenceR.I. , 284, 1986., 284, 1986.