aa r X i v : . [ c ond - m a t . s t a t - m ec h ] N ov A record-driven growth process
C Godr`eche and J M Luck
Institut de Physique Th´eorique, IPhT, CEA Saclay, and URA 2306, CNRS, 91191Gif-sur-Yvette cedex, France
Abstract.
We introduce a novel stochastic growth process, the
Record-driven growthprocess , which originates from the analysis of a class of growing networks in a universallimiting regime. Nodes are added one by one to a network, each node possessing aquality. The new incoming node connects to the preexisting node with best quality,that is, with record value for the quality. The emergent structure is that of a growingnetwork, where groups are formed around record nodes (nodes endowed with the bestintrinsic qualities). Special emphasis is put on the statistics of leaders (nodes whosedegrees are the largest). The asymptotic probability for a node to be a leader isequal to the Golomb-Dickman constant ω = 0 .
624 329 . . . , which arises in problemsof combinatorical nature. This outcome solves the problem of the determination ofthe record breaking rate for the sequence of correlated inter-record intervals. Theprocess exhibits temporal self-similarity in the late-time regime. Connections with thestatistics of the cycles of random permutations, the statistical properties of randomlybroken intervals, and the Kesten variable are given.PACS numbers: 02.50.Ey, 05.40.-a, 89.75.-kE-mail: [email protected] , [email protected] record-driven growth process
1. Introduction
In the present work we introduce and study a novel stochastic growth process, the
Record-driven (RD) growth process , which originates from the analysis of a class ofgrowing networks in a universal limiting regime. As we shall see, this process is alsorelated to three other fields, namely the statistics of the cycles of random permutations,the statistical properties of randomly broken intervals, and the Kesten variable.The RD growth process originates from a class of growing networks with apreferential attachment rule, amongst which the most well known representative is themodel of Barab´asi and Albert (BA) [1]. The latter provides a natural explanationfor the main features observed in real networks [2, 3, 4, 5], and chiefly their scale-freeness, testified by the power-law fall-off of their degree distribution. Relevant forour purpose are networks where the attachment rule involves both the degree of thenodes and their intrinsic quality or fitness, an extension of the original BA model whichis due to Bianconi and Barab´asi (BB) [6]. The BB model has the remarkable featurethat it exhibits a continuous condensation transition, analogous to the Bose-Einsteincondensation.A general definition of this class of networks is as follows. The network beinginitially empty, at each integer time step, n = 1 , , . . . , a new node, labeled by its birthdate n , is added. Node n is endowed once and for all with an intrinsic quality η n , modeledby a quenched random variable drawn from some given distribution. This node (exceptfor the first one) connects by one link to any of the earlier nodes ( i = 1 , . . . , n −
1) withprobability p i,n = η i h ( k i ) Z n − , (1.1)where h ( k i ) is a function of the degree k i of node i , i.e., the number of nodes alreadyconnected to node i at time n −
1. The denominator Z n − = n − X i =1 η i h ( k i ) (1.2)is a normalization factor. The probability p i,n is thus proportional to an intrinsic factor,the quality η i of node i , and to a dynamical one, represented by h ( k i ). Preferentialattachment to nodes whose degree is already larger is realized whenever the function h ( k i ) is an increasing function of the degree k i . Finally, temperature T is introducedinto the model by considering the qualities as activated variables, i.e., by setting η n = exp( − ε n /T ) , (1.3)and assuming that the activation energies ε n are drawn from a given temperature-independent continuous distribution.The BA and BB models in their original forms both correspond to the linearfunction h ( k ) = k . The BA model corresponds to the limiting case of infinitetemperature. record-driven growth process n is completed the network consists of n nodes connected by n − k i . We definethe degree of a given node as the number of incoming links on this node, taking asideits unique outgoing link, except for the first node, which has no outgoing link. This newdefinition corresponds to shifting by one unit the former quantity, k i → k i −
1, exceptfor the first node. The degrees thus defined sum up to the number of links, i.e., n X i =1 k i = n − . (1.4)As we now explain, the RD growth process is the zero-temperature limit of theclass of models described above. In this limit, at any time n the node i with the lowestenergy: ε i = min( ε , . . . , ε n − ) , (1.5)i.e., with the highest quality, has an attachment probability p i,n which is overwhelminglylarger than all the other ones, since the ratio p i,n /p j,n grows exponentially at lowtemperature, as exp(( ε j − ε i ) /T ). Every new node n therefore connects to the earliernode i with best quality at time n , given by (1.5). The successive best qualities areknown as records [7, 8, 9, 10]. The corresponding nodes, that we term record nodes ,are therefore the only ones whose degree grows. This process is universal in a verystrong sense. It is independent of the function h ( k ) and of the distribution of the nodeenergies, provided the latter is continuous, so that the event ε i = ε j with i = j has zeroprobability. In particular, the RD growth process is independent of whether the modelhas a condensation transition or not.Let us summarize the definition of the process and give a pictorial interpretationof it. Nodes (or individuals) arrive one by one to form a network of relationships. Thenetwork being initially empty, at each integer time step, n = 1 , , . . . , a new node, labeledby its birth date n , is added, and endowed once and for all with an intrinsic quality η n .This node connects by one link to the earlier node with best quality. This directed linkcan be pictorially described as “being a disciple of”. Thus groups are formed aroundrecord nodes (or pictorially “masters”). The size of a group, or the degree of a recordnode, is the number of incoming links to it (i.e., the number of disciples). Times atwhich a node appears with a quality that breaks the previous record are record times .Finally, there is another, yet simpler description of the process relying on records only:the record times are the dates of birth of the record nodes; the degree of the newly bornrecord node grows linearly in time, then stops growing when the next record node isborn.In the present work the main emphasis will be put on the interplay between records and leaders . While a node is a record if its quality (an intrinsic property) is betterthan those of all the earlier nodes, a node will be said to be a leader at a given time ifits degree (a dynamical, time-dependent quantity) is larger than those of all the othernodes. Investigating the interplay between records and leaders is natural in the low record-driven growth process ω = 0 .
624 329 . . . ,eq. (4.28). We then perform the analysis of the statistics of the difference of the labels oftwo successive leaders (equivalently, of two successive records for the sequence of inter-record intervals), as well as a thorough study of the statistics of the lengths of timeassociated with the reign of a leader. We finally explain the connections of this processwith the statistics of the cycles of random permutations, the statistical properties ofrandomly broken intervals, and the Kesten variable.The bulk of the paper begins at section 3. In the next section we first establishthe needed background knowledge on records, since the latter play a fundamental rolein the definition of the process under study.
2. Statistics of records
The discrete theory of records is classical [7, 8, 9, 10]. The continuum theory, which isinstrumental in our work, is less documented, as is its relationship to a renewal process.
Given a sequence of numbers, q , q , . . . , the value q i is a record if it is larger than allprevious ones: q i > max( q , . . . , q i − ) . (2.1)These numbers are for example the successive observations of a random signal, orthe successive drawings of a random variable, modeled as a sequence of independentidentically distributed (i.i.d.) random variables. In order to avoid ambiguities due toties, the distribution of the latter is taken continuous. In the present context, the q i stand for the qualities η i of the nodes.Referring to the label i = 1 , , . . . as time, the time i of the occurrence of a recordis a record time . The definition of a record involving only inequalities between thevariables q i , the statistics of record times is independent of the underlying distributionof these variables. record-driven growth process q is always a record. The occurrence of a record at any subsequenttime i ≥ /i :Prob( q i > max( q , . . . , q i − )) = 1 i . (2.2)This holds independently of the occurrence of any other record, either at earlier times( j < i ) or at later times ( j > i ). Indeed, q is a record with probability 1. Theprobability that q be a record, i.e., be larger than q , is obviously equal to 1 /
2. Theprobability that q be a record is equal to 1 / q and q , with equal probability, and so on.Eq. (2.2) can alternatively be recovered by noticing that, amongst the i ! permutationsof q , . . . , q i , there are ( i − q i is the largest.Eq. (2.2) means that the rate of record breaking is equal to 1 /i , or otherwise statedthat the occurrence of a record is a Bernoulli process with success probability equalto 1 /i . The indicator variables I i , equal to 1 if q i is a record and to 0 otherwise, will bethe building blocks for the derivation of the results of this section. It is easy to convinceoneself that they are independent (for a proof, see [7, 8, 9, 10]). We will first determinethe distribution of the number of records up to time n , then that of the record times.The number M n of records up to time n is a random variable taking the values1 , . . . , n , which can be expressed as M n = I + I + · · · + I n . (2.3)Its average and variance read h M n i = n X i =1 h I i i = n X i =1 i ≈ ln n + γ, (2.4)var M n = n X i =1 h I i (1 − I i ) i = n X i =1 (cid:18) i − i (cid:19) ≈ ln n + γ − π . (2.5)where γ = 0 .
577 215 . . . is Euler’s constant. More generally, the generating functionof M n reads h x M n i = n Y i =1 h x I i i = n Y i =1 (cid:18) x − i (cid:19) = x ( x + 1) . . . ( x + n − n ! . (2.6)The ‘rising power’ appearing in the right-hand side of (2.6) is known to be the generatingfunction of the Stirling numbers of the first kind [12]: x ( x + 1) . . . ( x + n −
1) = Γ( x + n )Γ( x ) = n X m =1 (cid:20) nm (cid:21) x m . (2.7)The Stirling number of the first kind h nm i is the number of ways of arranging n objectsin m cycles, or the number of permutations of n objects having m cycles, hence n X m =1 (cid:20) nm (cid:21) = n ! , (2.8) record-driven growth process x = 1. The distribution of M n is therefore given byProb( M n = m ) = h nm i n ! . (2.9)In other words, the number M n of records up to time n is distributed as the number ofcycles in a random permutation on n elements (see section 6.1 for further developmentson the connection of records with random permutations).The asymptotic form of the Stirling numbers (cid:20) nm (cid:21) ≈ ( n − n ) m − ( m − , (2.10)is readily derived from (2.7) by simplifying the gamma functions for large n and small x as Γ( x + n ) ≈ ( n − n x and Γ( x ) ≈ /x . As a consequence, the asymptotic form ofthe distribution of M n readsProb( M n = m ) ≈ n (ln n ) m − ( m − . (2.11)The bulk of the distribution of M n is thus, up to a shift by one unit, asymptotically givenby a Poissonian law of parameter ln n . This holds only to leading order in ln n . The exactlarge- n behavior of the mean and variance of M n , eqs. (2.4) and (2.5), differs by additiveconstants from the predictions of (2.11), h M n i Poisson = ln n + 1, (var M n ) Poisson = ln n .Let us denote by N , N . . . , N m , . . . the successive record times. Thus N = 1,since q is always a record. Then, if q i is the next record after q , the second recordtime is N = i , etc. The distribution of the m –th record time N m follows from (2.9):Prob( N m = k ) = h k − m − i k ! ( k = m, m + 1 , . . . ) , (2.12)and m ≥
2. We have indeed the equivalence of events { N m = k } = { M k − = m − , M k = m } = { M k − = m − , I k = 1 } , (2.13)where the two events appearing inside the rightmost brackets are independent. Usingthe asymptotic form (2.10) one getsProb( N m = k ) ≈ k (ln k ) m − ( m − k and m .The sequence N , N . . . , N m , . . . of record times can be generated recursively. Wehave Prob( N m = k | N m − = j ) ≡ p k | j = jk ( k − , (2.15)independently of m , i.e., independently of the occurrence of earlier records. Indeed, p k | j = Prob( I j +1 = · · · = I k − = 0 , I k = 1)= Prob( I j +1 = 0) . . . Prob( I k − = 0) Prob( I k = 1)= jj + 1 j + 1 j + 2 · · · k − k − k , (2.16) record-driven growth process N m ≥ k | N m − = j ) = X l ≥ k p l | j = jk − , (2.17)or else Prob( N m ≥ k | N m − ) = N m − k − . (2.18)This expression can be recast in the form of the random recursion N m = 1 + Int N m − U m , (2.19)where Int x denotes the integer part of the real number x , and U m is a uniform randomvariable between 0 and 1, independent of N m − .We now look at the distribution of record times from a different viewpoint. For afixed instant of time n ≥
1, we consider the time N M of occurrence of the last recordbefore n ( n being included), that is the M n –th record ‡ . The time N M is a discreterandom variable uniformly distributed between 1 and n :Prob( N M = k ) = 1 n (1 ≤ k ≤ n ) . (2.20)Indeed, the joint distribution of N M and M n readsProb( N M = k, M n = m ) = Prob( I k +1 = . . . = I n = 0) Prob( N m = k )= kn Prob( N m = k ) . (2.21)for m ≤ k ≤ n . Using (2.12), (2.8) and summing (2.21) over m yields the result.Eq. (2.20) can be alternatively recovered by noticing that the time N M is entirelycharacterized by the property that the signal q i takes its maximal value over i = 1 , . . . , n at time i = N M . These n possible values of N M are clearly equally probable.Similarly, consider the time N M +1 of occurrence of the ( M n + 1) − st record, i.e., ofthe first record after a given time n . It is easily found, by the same reasoning leadingto (2.15), thatProb( N M +1 = k ) = nk ( k −
1) ( k ≥ n + 1) . (2.22)Therefore Prob nN M +1 < nk ! = nk , (2.23)from which one can infer that, when n → ∞ , n/N M +1 is uniformly distributed between 0and 1. This will be proved below, by another method. ‡ In the following we drop the subscript when it is attached to a quantity which is itself in a subscript,whenever there is no ambiguity. record-driven growth process At large times, the discrete theory of records has an asymptotically exact descriptionin terms of continuous variables. This description stems from the recursion, inheritedfrom (2.19), N m = N m − U m , (2.24)where the N m are now considered as real numbers. The product structure of thisrecursion suggests to introduce a logarithmic scale of time. In this time scale theprocess of record times is transformed into a very simple renewal process [13, 14, 15], asshown below. This remark allows an easy access to the determination of the asymptoticdistributions of the observables needed in the sequel ( M n , N m , N M ).We set § t = ln n, t m = ln N m +1 , τ m = − ln U m +1 . (2.25)The U m being uniformly distributed between 0 and 1, the increments τ m are i.i.d. randomvariables with density ρ ( τ ) = e − τ . The recursion (2.24) becomes t m = t m − + τ m , (2.26)which defines the time of occurrence of the m –th renewal, t m = τ + · · · + τ m . (2.27)Its probability density f t m is therefore equal to the m –th convolution product of ρ ,corresponding to the gamma distribution: f t m ( T ) = e − T T m − ( m − , (2.28)from which we formally deduce the density of N m +1 = exp( t m ) f N m +1 ( N ) = 1 N (ln N ) m − ( m − , (2.29)in agreement with the asymptotic form (2.14) of the law of N m . This is the law of theinverse of the product of m uniform random variables between 0 and 1.The number of records up to time n , M n , translates into the number N t of renewalsup to time t , up to a shift by one unit between the two quantities, M n = N t + 1 . (2.30)In the present case of exponentially distributed increments, the distribution of N t isPoissonian: Prob( N t = m ) = e − t t m m ! , (2.31)in agreement with the asymptotic form (2.11) of the law of M n . § The shift of the index m by one unit is due to the fact that there is a record at N = 1, while theusual convention t = 0 holds in renewal theory. record-driven growth process X n = N M n = exp( − B t ) , Y n = N M +1 n = exp( E t ) ,Z n = N M +1 N M = Y n X n = exp( B t + E t ) , (2.32)where the backward and forward recurrence times are respectively defined as B t = t − t N , E t = t N +1 − t. (2.33)For a renewal process with exponentially distributed increments, the latter quantitiesare statistically independent of each other and have the same exponential distributionas the increments [16], i.e., f B t ( B ) = e − B , f E t ( E ) = e − E . (2.34)We can thus conclude that X n is uniform between 0 and 1, consistently with (2.20),that Y n is equal to the inverse of such a uniform random variable, consistently withwhat was suggested by (2.23), and that Z n is therefore equal to the inverse of a productof two such uniform random variables, namely f X ( x ) = 1 (0 < x < , f Y ( y ) = 1 y ( y > ,f Z ( z ) = ln zz ( z > . (2.35)
3. The Record-driven growth process
The definition of the process given in the Introduction can now be made morequantitative, first at the level of the discrete formalism, then in the continuum limit.
Let k i ( n ) denote the degree of node i (defined as the number of incoming links on thisnode) at the later time n ≥ i . We have clearly k i ( i ) = 0. Then: • If η i is not a record, i.e., if the time i is not a record time, the degree of node i doesnot grow any further: k i ( n ) = 0 for all n ≥ i. (3.1) • If η i is a record, i.e., if the time i = N m belongs to the sequence of record times,the degree of node i grows linearly with time until the subsequent record is born attime N m +1 . It then stays constant. Setting i = N m , k i ( n ) = K m ( n ) , (3.2)one has therefore for the degree of the m –th record node K m ( n ) = ( n − N m ( N m ≤ n ≤ N m +1 ) ,N m +1 − N m ( n ≥ N m +1 ) . (3.3) record-driven growth process K m ( n ), that is K m ( N m +1 ), is simply given by the inter-record interval,which we denote by∆ m +1 = N m +1 − N m . (3.4)Figure 1 illustrates these definitions. Figure 1.
Schematic plot of the time dependence of the degrees K m ( n ) of the firstfew record nodes ( m = 1 , . . . ,
4) in the RD growth process. The thick line shows thepart of its history where the second record node ( m = 2) is the leader. At any given instant of time n , there is a node whose degree is larger than that ofall existing nodes at this time. We term this node the leader at time n . We will alsosay that a given node is a leader if it leads for some period of time during its history.For instance, on Figure 1, the record nodes 1, 2 and 4 are leaders, but 3 is not. If m denotes the number of record nodes before time n , i.e., N m ≤ n < N m +1 , the degree ofthe leader at time n reads L ( n ) = max( K ( n ) , K ( n ) , . . . , K m ( n ))= max(∆ , . . . , ∆ m , n − N m )= max( L m , n − N m ) , (3.5)where L m = max(∆ , . . . , ∆ m ) (3.6)is the degree of the leader, L ( N m ), at the record time N m ( m ≥ L m +1 = max( L m , ∆ m +1 ) . (3.7)The meaning of this recursion is as follows. • If ∆ m +1 > L m , the m –th record node is the leader at time N m +1 . One has therefore L m +1 = ∆ m +1 = K m ( N m +1 ). • If ∆ m +1 ≤ L m , the m –th record node is not the leader at time N m +1 . The degree ofthe leader is left unchanged, i.e., L m +1 = L m . It turns out that the m –th record nodewill never be a leader (see (3.15)). This node is said for short to be a subleader . record-driven growth process m +1 > L m is equivalent to saying that ∆ m +1 is larger than all theprevious ∆ k for k = 2 , . . . , m , i.e., is a record for the sequence of inter-record times ∆ , ∆ , . . . Leaders therefore correspond to records for this sequence. On the exampleof Figure 1 we have L = ∆ < L = L = ∆ < L = ∆ , but ∆ > ∆ . Thus therecord nodes 1, 2, and 4 are leaders, and ∆ , ∆ and ∆ are records.Let ω m = Prob(∆ m +1 > L m ) (3.8)denote the probability for the m –th record node be a leader, or as mentioned above, theprobability of a record breaking at step m + 1 for the sequence of inter-record intervals∆ , ∆ , . . . In section 4 we will determine the limit ω of ω m when m → ∞ , that is, theprobability for a record node to be a leader, in the limit of long times. At this stage wecan give the expression of this quantity in the first two steps of the process. One hasclearly ω = 1. Furthermore it is easy to see that the joint distribution of the recordtimes N and N readsProb( N = j, N = k ) = p j | p k | j = 1( j − k − k ( k > j > . (3.9)For N = j and N = k , we have L = 0, L = ∆ = j −
1, and ∆ = k − j . The firstrecord node, born at time N = 1, is always a leader. The second record node, born attime N = j , is a leader if ∆ > L , i.e., k ≥ j . This occurs with probability ω = X j ≥ X k ≥ j j − k − k = X j ≥ j − j − − ln 2) = 0 .
613 705 . . . (3.10)The expressions of ω m become increasingly complex as m becomes larger and larger.Let us mention the following result without proof: ω = Li (cid:18) (cid:19) − π − π √
312 + 3(ln 2) − − .
626 218 . . . , (3.11)where the first term involves the dilogarithm function.We performed an accurate numerical evaluation of the ω m by a simulation of theprocess based on a generation of records using eqs. (2.19), (3.4) and (3.7). Figure 2demonstrates the very fast convergence of the ω m to the limit ω , known as the Golomb-Dickman constant (see (4.28)). In contrast with the case of i.i.d. random variables, therecord breaking rate of the sequence of ∆ m goes to a non-trivial constant.Denoting by ν = 1 , , . . . the difference of the labels of any two successive leaders,say m and m + ν , the “reign” of the leader born at N m , i.e., the period of time duringwhich it stays a leader, begins at time a m = N m + L m (3.12)and ends at time b m = a m + ν = N m + ν + L m + ν . (3.13) record-driven growth process Figure 2.
Plot of the probability ω m that the m –th record node is a leader, against m .Full symbols: exact values (3.10) and (3.11). Empty symbols: data for higher m ,obtained by a numerical simulation based on the recursions (2.19), (3.4) and (3.7).Dashed line: asymptotic limit given by the Golomb-Dickman constant ω , eq. (4.28). The reign therefore has a duration l m = b m − a m = a m + ν − a m . (3.14)The statistics of these times is addressed in section 5.Finally, the inequalities a m < N m +1 < b m (3.15)prove that the m –th record node is a leader (for some period of time) if and only if itleads at time N m +1 , i.e., if and only if ∆ m +1 > L m . As stated previously, the statistics of records is faithfully described in the regime oflong times by a continuum approach. The late stages of the RD growth process have asimilar asymptotically exact continuum description.In the continuum theory the key quantity is the ratio R m = L m N m , (3.16)which is the fraction of nodes which are connected to the leader at that time. Indeed thenumerator L m is the degree of the leader at time N m , while N m is equal to the numberof nodes in the system. According to (3.6), R m is also the scaled maximal inter-recordinterval. The values taken by R m are clearly between 0 and 1. Recalling (2.24), therecursion (3.7) becomes R m +1 = max( U m +1 R m , − U m +1 ) , (3.17) record-driven growth process U m +1 is uniformly distributed between 0 and 1 and independent of R m . Thebranches of (3.17) correspond respectively to the events L m = { m leader } = (cid:26) U m +1 <
11 + R m (cid:27) ⇒ R m +1 = 1 − U m +1 , S m = { m subleader } = (cid:26) U m +1 >
11 + R m (cid:27) ⇒ R m +1 = U m +1 R m . (3.18)The stochastic dynamical system defined by (3.17) plays a central role in thefollowing. It is reminiscent of the recursions introduced by Dyson [17], then usedextensively in the study of one-dimensional disordered systems [18].
4. Statistics of the leader: one-time quantities
There is an invariant distribution f R = lim m →∞ f R m (4.1)associated with the random recursion (3.17). The latter implies a recursion between theprobability densities of R m and R m +1 : f R m +1 = ( L + S ) f R m , (4.2)where we have introduced the linear operators L (leader) and S (subleader), acting ona function f ( x ) defined for 0 < x < Lf ( x ) = Z min(1 ,x/ (1 − x ))0 d uf ( u ) , Sf ( x ) = Z min(1 ,x/ (1 − x )) x d uu f ( u ) . (4.3)The invariant distribution therefore obeys the fixed-point equation f R = ( S + L ) f R . (4.4)Defining the variable V = 1 /R , with density f V ( v ) for v >
1, eq. (4.4) can be recastas the integral equation v f V ( v ) = Z v d u uf V ( u ) + 1 (1 < v < , Z vv − d u uf V ( u ) + Z ∞ v − d uf V ( u ) ( v > , (4.5)which simplifies to the differential equationdd v ( vf V ( v )) = f V ( v ) + vf ′ V ( v ) = ( < v < , − f V ( v −
1) ( v > . (4.6)We readily obtain both the explicit result f V ( v ) = 1 v (1 < v < , (4.7)and the integral relation f V ( v ) = 1 v Z ∞ v − d u f V ( u ) ( v > . (4.8) record-driven growth process R , we find f R ( x ) = 1 x Z min(1 ,x/ (1 − x ))0 d uf R ( u ) . (4.9)Using (4.7) as an input, the second line of (4.6) can be solved iteratively. We thusobtain more and more complex analytic expressions for the invariant density f V ( v ) onthe intervals delimited by the integers. We have f V ( v ) = (1 − ln( v − /v for 2 < v < < v < (2 − v ). Thecorresponding expressions in terms of the variable R are f R ( R ) = 1 /R for 1 / < R < f R ( R ) = [1 + ln( R/ (1 − R ))] /R for 1 / < R < /
2. The invariant density f V ( v ) hasweaker and weaker singularities at integer values of v . Using (4.6) it is readily foundthat the n –th derivative of f V ( v ) has a discontinuity at v = n + 1 of the form∆ f ( n ) V ( n + 1) = f ( n ) V ( n + 1 + ) − f ( n ) V ( n + 1 − ) = ( − n ( n + 1)! . (4.10)Figure 3 shows a plot of the invariant density f R ( R ), obtained by a direct numericalsimulation of the recursion (3.17). This procedure is more accurate than for examplea numerical inversion of the explicit form (4.16) of the Laplace transform. The leadingsingularity at R = 1 / f ′ R ( R = 1 / ±
0) = ∓
4. Theother singularities at R = 1 / R = 1 /
4, etc., are not visible.
Figure 3.
Plot of the invariant probability density f R ( R ). The dashed line emphasizesthe symmetric cusp at R cusp = 1 / Another consequence of the differential equation (4.6) is as follows. Multiplyingboth sides of these equations by v p , where p is any complex number, and integratingover v gives h V p i + Z ∞ d v v p +1 f ′ V ( v ) = −h ( V + 1) p i . (4.11) record-driven growth process f V (1) = 1 coming from (4.7) leads to theidentity h ( V + 1) p i = 1 + p h V p i , (4.12)which holds for any complex p . It translates into an identity for the invariant distributionof the variable R (up to a change of p into its opposite) *(cid:18) R R (cid:19) p + = 1 − p h R p i . (4.13)The explicit computation of the invariant distribution is more easily performed byintroducing the Laplace transform b f V ( s ) = h e − sV i = Z ∞ d v f V ( v ) e − sv . (4.14)Eq. (4.6) yields, using again the initial value f V (1) = 1, s d b f V ( s )d s = e − s ( b f V ( s ) − . (4.15)The solution of this equation reads b f V ( s ) = 1 − e − E ( s ) = 1 − s e γ − F ( s ) , (4.16)where we have introduced the functions E ( s ) = Z ∞ s d u e − u u = − Ei( − s ) = − γ − ln s + F ( s ) ,F ( s ) = Z s d u − e − u u = X k ≥ ( − k +1 k k ! s k (4.17)and Ei is the exponential integral function. The second expression of the function F ( s )shows that it is an entire function of the complex variable s . The moments of V arereadily obtained by expanding the second expression of (4.16) in powers of s . They arerational multiples of e γ : h V i = e γ , h V i = 2 e γ , h V i = 92 e γ , etc. (4.18)Let us finally determine the behavior of f V ( v ) as v → ∞ , or equivalently that of f R ( x ) as x →
0. This analysis is conveniently done along the lines of [19]. Anticipatinga fast decay in the regime under consideration, we set f V ( v ) ∼ e − φ ( v ) , (4.19)and approximate the integral in the right-hand side of (4.8) by Z ∞ v − d u f V ( u ) ≈ e − φ ( v − φ ′ ( v − , (4.20)obtaining thus φ ( v ) − φ ( v − ≈ ln v + ln φ ′ ( v − . (4.21)Setting λ = ln v, φ ( v ) = v a ( λ ) , b ( λ ) = a ( λ ) + d a ( λ )d λ , (4.22) record-driven growth process b ( λ ) ≈ λ + ln b ( λ ) . (4.23)The latter equation is correct up to terms of relative order 1 /v , i.e., exponentially smallin λ . It is therefore exact to all orders in 1 /λ . Its solution yields the asymptoticexpansion a = λ + µ + µλ − ( µ − λ + 2 µ − µ + 12 µ − λ + · · · , (4.24)with the shorthand notation µ = ln λ − v ) −
1. We thus obtain f V ( v ) ∼ exp ( − v ln v + ln(ln v ) − v ) − v + · · · !) . (4.25)The invariant density therefore decays faster than exponentially. It can be said to decay factorially , as the leading behavior of (4.25) is identical to that of 1 /v !. The knowledge of the invariant distribution f R allows the determination of one-timequantities in the late time regime of the process. In particular, consider the asymptoticprobability ω for a record node to be a leader (for some period of time): ω = lim m →∞ ω m = lim m →∞ Prob( L m ) = lim m →∞ Prob (cid:18) U m +1 <
11 + R m (cid:19) . (4.26)Thus, using the identity (4.13) for p = 1, ω = (cid:28)
11 + R (cid:29) = h R i = Z ∞ d s b f V ( s ) . (4.27)Finally, using (4.16), we obtain ω = Z ∞ d s e − s − E ( s ) = Z d x exp Z x d y ln y ! = 0 .
624 329 988 . . . (4.28)This number is known as the Golomb-Dickman constant [20]. It first appeared inthe framework of the decomposition of an integer into its prime factors [21], then inthe study of the longest cycle in a random permutation of order n [22, 23, 24]. Theconnection of the present study to the statistics of cycles of permutations will be givenin section 6.1.Eq. (4.27) shows that ω is also the mean scaled maximal inter-record interval. The invariant distribution f R associated with the recursion (3.17) gives the probabilitydistribution of the fraction of nodes which are connected to the leader at a record time,in the late-time regime. We now solve the same question when the time of observationis a generic late time.Let an instant of time n ≫ L ( n ) = max( L M , n − N M ) , (4.29) record-driven growth process M ≡ M n is the fluctuating number of records before time n , and L M =max(∆ , . . . , ∆ M ). Setting R ( n ) = L ( n ) n , (4.30)equation (4.29) leads to R ( n ) = max( R M X n , − X n ) , (4.31)where the ratio X n , defined in (2.32), is uniformly distributed between 0 and 1 andindependent of R M (see (2.35)). The recursion (4.31) therefore maps R M onto R ( n ) inexactly the same way as the recursion (3.17) maps R m onto R m +1 . As a consequence,the distribution of the ratio R ( n ) at a generic late time is also given by the invariantdistribution f R . Stated otherwise, the fraction of nodes which are connected to theleader at a record time and at a generic time are identically distributed. k Consider first the probability that the current record node at time n , that is, therecord node number M n , is the leader at the current time n . This event requires that n − N M > L M , which implies 1 − X n > R M X n , or finally X n < / (1 + R M ). Thevariable R M is asymptotically distributed according to the invariant distribution f R ,while X n , uniformly distributed between 0 and 1, is independent of R M . As aconsequence, we find that the probability under consideration is again equal to theGolomb-Dickman constant ω .Consider now the probability that the current record node at time n , that is, therecord node number M n , is a leader (for some period of time). This is a different quantity,larger than the previous one. Recall that by “a given node is a leader” we mean that itleads for some duration of time. It does not necessarily lead at time n , but will surelylead at time N M +1 (see (3.15)). Now the inequality to be satisfied is N M +1 − N M > L M ,implying Z n > R M , where the time ratio Z n , defined in (2.32), with probabilitydensity (2.35), is independent of R M . The probability under consideration thereforereads Ω = Z d R f R ( R ) Z ∞ R d z ln zz = * R )1 + R + = 0 .
914 063 . . . (4.32)As expected, we have Ω > ω . This inequality has yet another interpretation. Theprobability Ω for the current record node to be a leader is larger than the probability ω for any record node to be a leader. This is so because, for a fixed time n , larger intervals N M +1 − N M have a higher probability to be sampled. And the later the successor of arecord node is born, the higher its probability to be a leader. k This property extends in a straightforward way to the full degree statistics (see section 6.2). Themoments Y ( p ) m of any finite order p , introduced in (6.5), are also identically distributed at a record timeand at a generic time. record-driven growth process
5. Statistics of the leader: two-time quantities
This section is devoted to a study of two-time quantities in the process, focusing ourattention onto the main characteristics of the reign of a leader, defined in section 3.1,i.e., the difference ν of the labels of two successive leaders (section 5.1), and the timeratios N m + ν /N m , a m /N m , b m /N m , l m /N m (section 5.2). All these quantities have limitdistributions in the late-time regime. Let us first consider the distribution of the random integer ν = 1 , , . . . , which is thedifference of the labels of two successive leaders in the sequence of record times, m and m + ν . Hereafter we consider m as being large enough, so that the process is describedby the continuum formalism. We shift the record labels by m , so that the two leadersnow have labels 0 and ν . We consistently denote the invariant density f R by f . Theprobability for a record node to be a leader readsProb( L ) = ω. (5.1)The distribution of ν is encoded in the probabilities P n = Prob( ν = n ) = 1 ω Prob( L S . . . S n − L n ) ,Q n = Prob( ν ≥ n ) = 1 ω Prob( L S . . . S n − ) . (5.2)The following relationships hold: P n = Q n − Q n +1 , Q n = X m ≥ n P m , X n ≥ P n = Q = 1 , h ν i = X n ≥ n P n = X n ≥ Q n . (5.3)In order to compute the P n and Q n we consider the sequence of functions f n ( x )defined by f n = S n − Lf ( n ≥ , (5.4)obtained as the result of the action, upon the invariant density f , of the operator L ,followed by n − S . We have Q n = 1 ω Z d u f n ( u ) . (5.5)Similarly, P n = 1 ω Z d u f n ( u ) 11 + u , (5.6)the factor 1 / (1 + u ) being due to the last event L n in the definition (5.2).We have X n ≥ f n = (1 − S ) − Lf = f (5.7) record-driven growth process − S ) f = Lf by (4.4), or (1 − S ) − Lf = f . Eq. (5.7) has a simpleinterpretation: the distribution of the variable R of the next leader is identical to thatof the variable R of the current leader. Inserting the sum rule (5.7) into (5.5) yields,using (5.3), h ν i = X n ≥ Q n = 1 ω . (5.8)This result, too, has a simple interpretation: the mean distance h ν i between twoconsecutive leaders along the sequence of record times is the inverse of the probability ω for a given record to be a leader, or the inverse record breaking rate for the sequence ofinter-record intervals.We now perform the computation of the cumulative probabilities Q n . Let f = Lf , (5.9)i.e., using (4.3), f ( x ) = / ≤ x ≤ , Z x/ (1 − x )0 d u f ( u ) for 0 ≤ x ≤ / , (5.10)or finally, using (4.9), f ( x ) = xf ( x ) . (5.11)The function f ( x ) is the initial condition for the recursion f n +1 ( x ) = Sf n ( x ) = Z min(1 ,x/ (1 − x )) x d uu f n ( u ) . (5.12)We make the change of variable y = 1 /x and define g n ( y ) = f n ( x ). These functionsobey the recursion g n +1 ( y ) = Z y max(1 ,y − d vv g n ( v ) , (5.13)which assumes a simpler form in terms of Laplace transforms: b g n ( s ) = − dd s (cid:18) s − e − s b g n +1 ( s ) (cid:19) . (5.14)Consider the generating function of the b g n ( s ), G ( x, s ) = X n ≥ b g n ( s ) x n − . (5.15)From (5.14) we obtain the differential equation x ( G ( x, s ) + b g ( s )) = − dd s (cid:18) s − e − s G ( x, s ) (cid:19) , (5.16)whose solution reads G ( x, s ) = x − e − s s e − xF ( s ) Z ∞ s d t e xF ( t ) b g ( t ) . (5.17)Furthermore, from (5.11) it is easily found that b g ( s ) = − dd s b f V ( s ) = e γ − s − F ( s ) . (5.18) record-driven growth process Q n , we have Q n = 1 ω Z d uf n ( u ) = 1 ω Z ∞ d v g n ( v ) v = 1 ω Z ∞ d s s b g n ( s ) . (5.19)Introducing the generating function of the Q n ,Σ( x ) = X n ≥ Q n x n − , (5.20)we obtain Σ( x ) = 1 + 1 ω Z ∞ d s s G ( x, s ) . (5.21)The expressions (5.17) and (5.18) yield the explicit resultΣ( x ) = e γ ω Z ∞ d s e − xF ( s ) Z ∞ s d t e − t +( x − F ( t ) . (5.22)Several results of interest can be derived from this exact expression. First of all,taking the derivative at x = 0, we obtain Q = Σ ′ (0) = 1 ω Z ∞ d tt ( t − − t ) e − t − E ( t ) . (5.23)The integral can be evaluated to yield Q = 2 ω − ω , i.e., P = 1 − ωω . (5.24)This can alternatively be obtained using (5.6) and (5.11). The probability that twosuccessive records are leaders therefore reads Prob( L L ) = ωP = 1 − ω = 0 .
375 670 . . .
This value is close to what it would be in the absence of any correlation, namely ω = 0 .
389 788 . . .
The moments of ν can be derived by expanding (5.22) around x = 1. We thusrecover the result (5.8) for h ν i , as Σ(1) = 1 /ω . For the second moment h ν i , we find h ν i = X n ≥ n P n = X n ≥ (2 n − Q n = 1 ω + 2Σ ′ (1)= 1 ω " Z ∞ d ss e − E ( s ) Z ∞ s d tt e − t (1 − e − t ) = 3 .
383 695 . . . (5.25)The expression (5.22) also shows that Σ( x ) has no singularity at any (finite) valueof x . In other words, it is an entire function of the complex variable x . As a consequence,the Q n fall off at large n faster than exponentially. This means that Q n +1 /Q n →
0, sothat P n ≈ Q n . From a quantitative viewpoint, the fall-off of the Q n (i.e., of the P n ) canbe derived from the asymptotic behavior of Σ( x ) as Re x → + ∞ . In this regime, thedouble integral in (5.22) is dominated by small values of s , where F ( s ) ≈ s , and largevalues of t , where F ( t ) ≈ ln t + γ . We thus obtainΣ( x ) ≈ e γx ωx Γ( x ) ≈ ω s πx e x (ln x + γ − (Re x → + ∞ ) . (5.26) record-driven growth process Q n , Q n = I d x π i Σ( x ) x n , (5.27)can then be evaluated by the saddle-point method. Parametrizing the saddle-point as x c = n/a , we are left with the estimate P n ≈ Q n ≈ aωn √ a + 1 e − n ( a − γ − /a ) , (5.28)where a obeys the implicit equation a = ln( n/a ) + γ. (5.29)Setting λ = ln n + γ and µ = ln λ = ln(ln n + γ ), we obtain the asymptotic expansions a = λ − µ + µλ + µ ( µ − λ + · · · (5.30)and P n ≈ Q n ∼ exp ( − n ln n − µ − µ + 1 λ + µ λ + · · · !) . (5.31)The above expansions are very similar to (4.24) and (4.25). The P n can again be saidto decay factorially , as their leading behavior coincides with that of 1 /n !.The accuracy of the estimate (5.28) is demonstrated in Figure 4, showing asurprisingly good agreement between this prediction and numerical data for the proba-bility distribution P n , obtained by a direct numerical simulation of the recursion (3.17). Figure 4.
Probability distribution of the difference of labels ν between two successiveleaders. The logarithm of P n = Prob( ν = n ) is plotted against the integer n . Symbols:data obtained by a direct numerical simulation of the random recursion (3.17). Line:value of the asymptotic estimate (5.28), obtained by means of a numerical solution ofthe implicit equation (5.29). record-driven growth process We now turn to the statistics of the time ratios ρ m = N m + ν N m , α m = a m N m , β m = b m N m , λ m = l m N m , (5.32)where N m and N m + ν are the birth times of the current leader and of the next one, andwhere the beginning and ending times, a m and b m , and the duration of the reign l m ofthe current leader are defined in (3.12), (3.13) and (3.14).We again shift the record labels by m , so that the two leaders under considerationhave labels 0 and ν , whereas the intermediate record nodes, labeled m = 1 , . . . , ν − R = 1 − U ,R = (1 − U ) U , . . . ,R ν = (1 − U ) U . . . U ν . (5.33)The above time ratios can therefore be expressed in terms of the variables R , R and R ν as α = 1 + R , (5.34) ρ = R (1 − R ) R ν , β = R (1 + R ν )(1 − R ) R ν = (1 + R ν ) ρ , (5.35) λ = β − α . (5.36)These time ratios have well-defined limit distributions f α , f ρ , f β , f λ in the late-timeregime. This reflects the temporal self-similarity of the process.Figure 5 shows plots of the probability distributions f ρ , f β , f λ of the three timeratios ρ , β and λ , and of the distributions f /ρ , f /α , f /λ of their inverses. The datahave been obtained by a direct simulation of the recursion (3.17). The plots emphasizethe following characteristics. The three distributions f ρ , f β , f λ share with the invariantdistribution f R the property that their maxima correspond to cusps. The values atwhich these cusps occur, namely ρ cusp = 2, β cusp = 3 and λ cusp = 1, are readily obtainedby replacing in (5.35), (5.36) both variables R and R ν by R cusp = 1 /
2. The lower plot inFigure 5 demonstrates that the distributions of the inverse time ratios have well-definedlimits at zero: f /ρ (0) = A , f /β (0) = f /λ (0) = B . (5.37)In other words, the distributions of these three time ratios fall off as 1 /x : f ρ ( x ) ≈ A x , f β ( x ) ≈ f λ ( x ) ≈ B x . (5.38)The exact values of the amplitudes A and B will be evaluated below (see (5.49)–(5.51)).The fact that f β and f λ share the same fall-off amplitude B is simply due to the factthat the difference β − λ = α is bounded in the range 1 ≤ α ≤ record-driven growth process Figure 5.
Top: plot of the probability distributions f ρ , f β , f λ of the time ratiosintroduced in (5.35), (5.36). Bottom: plot of the probability distributions of theirinverses. Symbols: exact values A and B of the densities at the origin (see (5.49)–(5.51)). The dashed lines emphasize the cusps at ρ cusp = 2, β cusp = 3 and λ cusp = 1. The actual evaluation of the joint distribution of the variables R and R ν is nowperformed. The starting point is to consider the probability P ( r , r ) = Prob( R < r , R ν < r, L ) , = X n ≥ Prob( R < r , R n < r, L , ν = n ) , = X n ≥ Prob( R < r , R n < r, L S . . . S n − L n ) . (5.39)Note that P (1 ,
1) = Prob( L ) = ω . The normalized joint density that we are lookingfor is thus f R ,R ν ( r , r ) = 1 ω ∂∂r ∂∂r P ( r , r ) . (5.40)As in section 5.1, we start from the invariant distribution f . Acting on it with theoperator L yields f (see (5.9)). We then fix the value of R to r , thus obtaining the record-driven growth process /ω ) f ( r ) δ ( r − r ). We then define ϕ ( r , r ) = δ ( r − r ) , (5.41)and introduce the functions ϕ n ( r , r ) = S n − ϕ ( r , r ) ,ϕ ( r , r ) = X n ≥ ϕ n ( r , r ) = (1 − S ) − ϕ ( r , r ) , (5.42)where the operator S acts on the variable r .We are thus left with the result f R ,R ν ( r , r ) = 1 ω f ( r ) ϕ ( r , r ) 11 + r , (5.43)where the factor 1 / (1 + r ) comes from the last event L n . We now make the change ofvariable y = 1 /r , y = 1 /r , and define ψ n ( y , y ) = ϕ n ( r , r ) and ψ ( y , y ) = ϕ ( r , r ). TheLaplace transform of ψ ( y , y ) is b ψ ( y , s ) = y e − sy . Therefore, in analogy with (5.17),the Laplace transform of ψ ( y , y ) reads b ψ ( y , s ) = y e − sy + 1 − e − s s e − F ( s ) Z ∞ s d t e F ( t ) − ty ! . (5.44)The expression (5.43) of the normalized joint probability density of the variables R and R ν therefore contains two non-trivial factors: f ( r ) = g ( y ), whose Laplacetransform with respect to y is given by (5.18), and ϕ ( r , r ) = ψ ( y , y ), whose Laplacetransform with respect to y is given by (5.44). The knowledge of this distribution allowsone, at least in principle, to compute the distribution of the time ratios ρ , λ , β and ofsimilar quantities. Analytical expressions thus obtained are however very cumbersome,and therefore of little use, either theoretically or practically, except for some simpleexamples, such as the amplitudes A and B , which will now be evaluated explicitly.Consider first the amplitude A . The probability density f /ρ can be expressed as f /ρ ( x ) = Z d r Z d r f R ,R ν ( r , r ) δ x − (1 − r ) rr ! (5.45)= Z d r r ( r + x ) f R ,R ν (cid:18) rr + x , r (cid:19) . (5.46)This expression simplifies in the x → A = f /ρ (0) = Z d r f R ,R ν (1 , r ) 1 r = 1 ω (cid:28) R ν (cid:29) R =1 . (5.47)We obtain similarly B = f /a (0) = Z d r f R ,R ν (1 , r ) r + 1 r = 1 ω (cid:28) R ν (cid:29) R =1 , (5.48)hence the relation B = A + 1 ω . (5.49)The explicit form of B is B = 1 ω Z d rr ϕ (1 , r ) = 1 ω Z ∞ d s b ψ (1 , s ) = 1 ω Z ∞ d t e − t + F ( t ) , (5.50) record-driven growth process b ψ (1 , s )has been simplified by an integration by parts. We thus obtain the numerical values A = 2 .
127 451 . . . , B = 3 .
729 168 . . . (5.51)
6. Connections to other fields
As noted above, the asymptotic probability for a record node to be a leader is equal tothe Golomb-Dickman constant, a number which appears in problems of combinatoricalnature. It is for example the limit, when n → ∞ , of h Λ n i /n , where Λ n is the lengthof the longest cycle in a random permutation of order n [22, 23, 24]. This number alsoappears in the framework of the decomposition of an integer into its prime factors [21].Furthermore, as shown by Goncharov [23] and Shepp and Lloyd [24], Λ n /n → R ,where the distribution of the limiting random variable R coincides with the invariantdistribution f R found in the present work. The identity of the asymptotic probabilityfor a record node to be a leader and of the Golomb-Dickman constant is just the identityof h R i (see eq. (4.27)) and hRi .We now explain the origin of the coincidences between features of the statistics ofrecords and the cycle structure of permutations. The existence of connections betweenthe two fields is well known [7, 25]. In particular, M n , the number of records up totime n , has the same distribution as the number of cycles C n in a random permutation oforder n [7], as mentioned in section 2.1. There is actually a deeper relationship betweenthe sequence of record times on the one hand, and the cycles of a random permutationon the other hand, which is due to the fact that the latter can be generated by the sameset of indicator variables as the former.For records, these variables are the I i defined in section 2.1. For cycles ofpermutations the construction is due to Feller [13], as we now recall. A permutation oforder n is constructed by a succession of n decisions. Let a , a , . . . , a n be the n lettersof the permutation. The position of a is first chosen, with n possibilities: 1 → i . Thenthe position of a i is chosen, with n − i → j , and so on, until a cycle isformed. For example [13], with n = 8, choosing1 → → → → → → → → , (6.1)generates the permutation a a a a a a a a . Define the indicator variable J k , equalto 1 if a cycle is formed at the k − th step, else to 0. For this example we have J = 1, J = 1, J = 1. Clearly, in general,Prob( J k = 1) = 1 n − k + 1 , (6.2)and the J k are independent. Here the sequence of J k reads 0 0 1 0 0 0 1 1, with Prob( J =1) = 1 /
6, Prob( J = 1) = 1 /
2, Prob( J = 1) = 1. Reverting the sequence of steps, record-driven growth process I i say, suchthat Prob( I = 1) = 1, Prob( I = 1) = 1 /
2, Prob( I = 1) = 1 /
6. One recognizes theconstruction of a sequence of records.
We have been, up to now, mainly interested in the statistics of leaders. We now addressthe full statistics of node degrees, within the continuum approach. Consider a fixed laterecord time N m . The degrees of the earlier records read K j − ( N m ) = N j − N j − = ∆ j for j = 2 , . . . , m . These degrees sum up to N m ¶ , each of them representing a finitefluctuating fraction of the total, or “weight” ∆ j /N m . We have∆ m N m = 1 − U m , ∆ m − N m = U m (1 − U m − ) , ∆ m − N m = U m U m − (1 − U m − ) , . . . (6.3)Relabeling the indices j → k = m − j + 1, and denoting the weights by W k , we have W = 1 − U ,W = U (1 − U ) ,W = U U (1 − U ) , . . . (6.4)We recognize the sequence of weights obtained by randomly breaking an interval of unitlength into two pieces, and iterating the process [26].A convenient tool to investigate this kind of fluctuating weights consists inintroducing the reduced moments Y ( p ) m = m X j =2 (cid:18) ∆ j N m (cid:19) p = m − X k =1 W pk , (6.5)where the order p = 1 , , . . . is any integer [26]. These moments obey the randomrecursion Y ( p ) m +1 = U pm +1 Y ( p ) m + (1 − U m +1 ) p . (6.6)The latter can be viewed as a generalization of the recursion (3.17) to any finite integerorder p . Eq. (3.17) is formally recovered in the p → ∞ limit, where the sum in (6.5) isdominated by its largest term, that is by the contribution of the leader. The Y ( p ) m , whichkeep fluctuating in the late-time regime, have non-trivial limit distributions, invariantunder the dynamical system (6.6). A plot of Y (2) m , obtained by iterating (6.6) numerically,as well a plot of the invariant measure f R of the largest weight can be found in [26]. Inthe framework of the random breaking of an interval, ω is either the probability thatthe first weight W be the largest, or the mean maximal weight. ¶ The sum is actually equal to N m −
1, but we neglect the correction in the continuum limit. record-driven growth process It turns out that the invariant distribution of the random variable R can be workedout more generally for a one-parameter family of problems containing the above as aspecial case. Consider the recursion (3.17) where the i.i.d. random variables U m +1 havean arbitrary distribution between 0 and 1, with density ρ ( u ). The invariant distributionstill obeys the fixed-point equation (4.4), with the definitions Sf ( x ) = Z min(1 ,x/ (1 − x )) x d uu ρ (cid:18) xu (cid:19) f ( u ) ,Lf ( x ) = ρ (1 − x ) Z min(1 ,x/ (1 − x ))0 d uf ( u ) . (6.7)The integral equation thus obtained cannot be solved in closed form in general. Itnevertheless leads to a differential equation similar to (4.6), and is therefore solvable,whenever the density of the variables U m +1 is a power law. More precisely, if ρ ( u ) = bu b − , (6.8)where b is an arbitrary positive parameter, one hasdd v v b ( v − b − f V ( v ) ! = v b − ( v − b [( v − b ) f V ( v ) + v ( v − f ′ V ( v )]= ( < v < , − bf V ( v −
1) ( v > . (6.9)Consider the modified Laplace transform e f V ( s ) = h V b e − sV i = Z ∞ d v v b e − sv f V ( v ) . (6.10)This quantity obeys the differential equation s d e f V ( s )d s = − ( s + b (1 − e − s )) e f V ( s ) , (6.11)whose normalized solution reads e f V ( s ) = Γ( b + 1) s b e − s − bE ( s ) = Γ( b + 1) e bγ e − s − bF ( s ) . (6.12)The probability ω (see eq. (4.26)) generalizes to ω ( b ) = lim m →∞ Prob(1 − U m +1 > U m +1 R m ) = *(cid:18)
11 + R (cid:19) b + , (6.13)that is ω ( b ) = *(cid:18) VV + 1 (cid:19) b + = 1Γ( b ) Z ∞ d s s b − e − s e f V ( s ) , (6.14)i.e., finally ω ( b ) = Z ∞ d s e − s − bE ( s ) . (6.15)The original problem with a uniform distribution of the variables U m +1 is recoveredby setting b = 1 in the above results. One has indeed e f V ( s ) = − d b f V ( s ) / d s , record-driven growth process ω (1) coincides with the expression (4.28) of the Golomb-Dickmanconstant.It can be checked that ω ( b ) = h R i for all values of b . This identity generalizes (4.27).However, lim m →∞ Prob(1 − U m +1 > U m +1 R m ) and h R i are not equal for an arbitrarydistribution ρ ( u ).The above family of exactly solvable invariant densities is in correspondence withthe following problem. Consider the Kesten variable [27, 28], defined as Z = 1 + x + x x + x x x + · · · , (6.16)where the x m are i.i.d. positive random variables with probability density ρ (Kes) ( x ). Ifthis distribution is such that h ln x i <
0, the sum in (6.16) is convergent and Z has awell-defined probability density f (Kes) Z ( z ), solution of the integral equation f (Kes) Z ( z ) = Z ∞ d xx ρ (Kes) ( x ) f (Kes) Z (cid:18) z − x (cid:19) , (6.17)This equation again cannot be solved in closed form in general. It is howeverknown [28, 29] that the problem can be solved whenever the density of the variables x m is a power law on an interval [0 , a ]. In the marginal situation ( a = 1) where the x m arebetween 0 and 1, with density ρ (Kes) ( x ) = bx b − , (6.18)where b is again an arbitrary positive parameter, the Laplace transform b f (Kes) Z ( s ) = h e − sZ i has the closed-form expression b f (Kes) Z ( s ) = e − s − bF ( s ) . (6.19)The similarity between the two problems is now patent by comparing (6.12) and (6.19).The probability densities of the variable V = 1 /R and of the Kesten variable Z arerelated to each other by the equation x b f V ( x ) = Γ( b + 1) e bγ f (Kes) Z ( x ) . (6.20)
7. Conclusion
The main goal of this work has been to put forward the
Record-driven growth process .This ballistic growth model entirely based on the record process has been met as thezero-temperature limit of a class of network growth models with preferential attachment.Its simplicity and its minimality however suggest that the RD growth process might berelevant to a wider class of situations, besides the realm of complex networks. Themain emphasis has been put on the interplay between records (i.e., nodes endowed withthe best intrinsic qualities) and leaders (i.e., nodes whose degrees are the largest). TheRD growth process provides a natural playground where subtle questions related to thestatistics of leaders and of lead changes can be addressed in a quantitative way.The RD growth process inherits from the record process some relationships withcombinatorical problems related to permutations. Relationships with fragmentation record-driven growth process .
626 508 . . .
This parallels the scaling of the mean maximal inter-record interval of the present work,resulting in the occurrence of the Golomb-Dickman constant 0 .
624 329 . . .
To close up, it is worth looking back to our starting point, namely the growingnetworks with preferential attachment considered in the Introduction. The self-similargrowth regime of the RD process can be shown to be unstable against thermalfluctuations. This regime crosses over to a complete freeze-out at a time scale whichusually diverges at a power-law at low temperature, as τ ∼ T − b . The model-dependentexponent b can be evaluated by generalizing the line of thought of the recent work [31],whether or not the model has a finite-temperature condensation transition. Acknowledgments
It is a pleasure for us to thank Ginestra Bianconi for stimulating discussions.
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