Fabien Lauer

Yann Guermeur

MSVMpack: A Multi-Class Support Vector Machine Package

Keywords: multi-class support vector machines, open source, C

This paper describes

MSVMpack, an open source software package dedicated to our generic model of multi-class support vector machine. All four multi-class support vector machines (M-SVMs) proposed so far in the literature appear as instances of this model. MSVMpack provides for them the first unified implementation and offers a convenient basis to develop other instances. This is also the first parallel implementation for M-SVMs. The package consists in a set of command-line tools with a callable library. The documentation includes a tutorial, a user's guide and a developer's guide.

Introduction

In the framework of polytomy computation, a multi-class support vector machine (M-SVM) is a support vector machine (SVM) dealing with all the categories simultaneously. Four M-SVMs can be found in the literature: the models of [START_REF] Weston | Multi-class support vector machines[END_REF], [START_REF] Crammer | On the algorithmic implementation of multiclass kernel-based vector machines[END_REF], [START_REF] Lee | Multicategory support vector machines: Theory and application to the classification of microarray data and satellite radiance data[END_REF], and the M-SVM 2 of [START_REF] Guermeur | A quadratic loss multi-class SVM for which a radius-margin bound applies[END_REF]. The proposed software implements them all in a single package named MSVMpack. Its design paves the way for the implementation of our generic model of M-SVM and the integration of additional functionalities such as model selection algorithms. The current version offers a parallel implementation with the possibility to use custom kernels. This software package is available for Linux under the terms of the GPL at http://www.loria.fr/ ˜lauer/MSVMpack/ and provides two command-line tools with a C application programming interface without dependencies beside a linear programming solver.

Multi-Class Support Vector Machines

We consider Q-category classification problems where X is the description space and the set Y of the categories can be identified with [

[ 1, Q ] ].
Let κ be a real-valued positive type function [START_REF] Berlinet | Reproducing Kernel Hilbert Spaces in Probability and Statistics[END_REF]) on X 2 and let (H κ , •, • H κ ) be the corresponding reproducing kernel

Hilbert space. Let H = H Q κ and H = (H κ + {1}) Q . H is the class of functions h = (h k ) 1 k Q from X to R Q that can be written as h (•) = h (•) + b = hk (•) + b k 1 k Q , where h = hk 1 k Q ∈ H and b = (b k ) 1 k Q ∈ R Q . A

Reference M-SVM type

M p K 1 K 2 K 3 Weston and Watkins (1998) WW I Qm 1 1 1 0 Crammer and Singer (2001) CS 1 Q-1 I Qm 1 1 1 1 Lee et al. (2004) LLW I Qm 1 0 1 Q-1 0 Guermeur and Monfrini (2011) MSVM2 M (2) 2 0 1 Q-1 0
Table 1: Specifications of the M-SVMs with their type as used by the MSVMpack interface.

• H given by:

∀ h ∈ H , h H = Q ∑ k=1 hk , hk H κ = hk H κ 1 k Q 2 .
With these definitions at hand, our generic definition of a Q-category M-SVM is:

Definition 1 (Generic model of M-SVM, Definition 4 in Guermeur, forthcoming) Let ((x i , y i )) 1 i m ∈ (X × [ [ 1, Q ] ]) m and λ ∈ R * + . Let ξ ∈ R Qm be a vector such that for (i, k) ∈ [ [ 1, m ] ] × [ [ 1, Q ] ], ξ ik is its component of index (i -1)Q + k, with (ξ iy i ) 1 i m = 0 m . A Q-category M-SVM is a classifier obtained by solving a convex quadratic programming (QP) problem of the form min h,ξ J (h, ξ) = Mξ p p + λ h 2 H s.t.            ∀i ∈ [ [ 1, m ] ] , ∀k ∈ [ [ 1, Q ] ] \ {y i } , K 1 h y i (x i ) -h k (x i ) K 2 -ξ ik ∀i ∈ [ [ 1, m ] ] , ∀(k, l) ∈ ([ [ 1, Q ] ] \ {y i }) 2 , K 3 (ξ ik -ξ il ) = 0 ∀i ∈ [ [ 1, m ] ] , ∀k ∈ [ [ 1, Q ] ] \ {y i } , (2 -p)ξ ik 0 (1 -K 1 ) ∑ Q k=1 h k = 0, where p ∈ {1, 2}, (K 1 , K 3 ) ∈ {0, 1} 2 , K 2 ∈ R *
+ and the matrix M is such that Mξ p is a norm of ξ.

Extending to matrices the notation used to designate the components of ξ and using δ to denote the Kronecker symbol, let us define the general term of M (2) ∈ M Qm,Qm (R) as:

m (2) ik, jl = (1 -δ y i ,k ) 1 -δ y j ,l δ i, j δ k,l + √ Q -1 Q -1 .
This allows us to summarize the characteristics of the four M-SVMs in Table 1. The potential of the generic model is discussed in Guermeur (forthcoming).

The Software Package

Training Algorithm

As in the bi-class case, an M-SVM is trained by solving the Wolfe dual of its instantiation of the QP problem in Definition 1. The corresponding dual variables α ik are the Lagrange multipliers of the constraints of correct classification. The implemented QP algorithm is based on the Frank-Wolfe method [START_REF] Frank | An algorithm for quadratic programming[END_REF], in which each step of the descent is obtained as the solution of a linear program (LP). The LP solver included in MSVMpack is lp solve [START_REF] Berkelaar | An Open Source (Mixed-Integer) Linear Programming System[END_REF]. In order to make it possible to process large data sets, a decomposition method is applied and only a small subset of the data is considered in each iteration. Let J d be the dual objective function and let α = (α ik ) be a (feasible) solution of the dual problem obtained at some point of the training procedure. The quality of α is measured thanks to the computation of an upper bound U(α) on the optimum J (h * , ξ * ) = J d (α * ) that goes to this optimum. The stopping criterion is defined as a large enough value for J d (α)/U(α). In MSVMpack, the bound U(α) is obtained by solving the primal problem with h being the function associated with the current α. This partial optimization requires little computation except in the case of the M-SVM 2 , for which another QP problem has to be solved. However, the computational burden may increase for a large number of classes (e.g., Q > 20).

Practical Use and Experiments

In its most simple form, the command line 'trainmsvm trainingdata -m WW' is used to train an M-SVM, where the -m flag allows one to choose the type of M-SVM model according to Table 1. Then, this model can be applied to a test set by using 'predmsvm testdata'. The complete list of options and parameters for these command-line tools can be found in the documentation or simply obtained by calling them without argument.

Table 2 shows a comparison of MSVMpack with other implementations of M-SVMs on a subset of the USPS database with 500 instances from 10 classes and the whole CB513 data set with 84119 instances from 3 classes. For the latter, the numbers reflect the average error and total times over a 5-fold cross validation, and the implementations that failed due to a lack of memory are not included in the Table . We refer the reader to the documentation for the details of the experimental setup and additional comparisons on other data sets.

Calling the Library from Other Programs

The "Developer's guide" section of the documentation presents the API reference and an example program including MSVMpack functionalities through this API. The library defines specific data structures for M-SVM models and data sets. It also provides wrapper functions, which act according to the M-SVM model type, for example, call the corresponding training function. The standard workflow for a train-and-test sequence is: call MSVM make model() to initialize the model; call MSVM make dataset() for each data set to load; call MSVM train() to train the model; and call MSVM classify set() to test the trained classifier.

Ongoing and Future Developments

MSVMpack implements the four M-SVMs proposed in the literature. Current work focuses on the explicit implementation of our generic model of M-SVM, which will make it possible to study new machines thanks to a simple choice of the values of the hyperparameters M, p, and (K t ) 1 t 3 . Future

  function h assigns the category y to x if and only if y = argmax 1 k Q h k (x) (cases of ex aequo are dealt with by introducing a dummy category). H is endowed with the norm c 2011 Fabien Lauer and Yann Guermeur.

MSVMpack includes a C application programming interface (API) and two command-line tools: one for training an M-SVM and one for making predictions with a trained M-SVM. The following discusses some algorithmic issues before presenting these tools and the API.

work will consider including automatic tuning procedures for the regularization parameter λ, and relaxing the hypothesis on the norm of the penalizer.


Fabien Leurent

Vincent Benezech

Mahdi Samadzad

A Stochastic Model of Trip End Disaggregation in Traffic Assignment to a Transportation Network

Keywords: Traffic assignment, trip disaggregation, discrete choice model

So far, traffic assignment to a network has been modelled by aggregating trip ends into zone centroids. The paper introduces a disaggregate model of trip ends, by associating to each zone a set of 'anchor nodes' for network access and a random vector of 'terminal costs' (or time) between trip ends and zonal anchors. These can be specified by OD pair. Assuming that terminal costs and network costs are independent, the system assignment can be performed in two stages, respectively network and terminal. A probit model based on Clark's formulas is recommended. A computation scheme is provided that loads OD flows by anchor pair onto the network in an efficient way. A binary case is addressed as illustration. Also reported is an application to the roadway network of the Paris area.

Introduction

Traffic assignment models, since their basic foundation by [START_REF] Wardrop | Some theoretical aspects of road traffic research[END_REF] and [START_REF] Beckmann | Studies in the Economics of Transportation[END_REF], have been developed in diverse ways and applied in many problems of transportation network planning. Among the developments achieved so far, let us quote: multiclass models, stochastic assignment, transit assignment, multimodal models, integrated models that deal with route choice together with mode choice or destination choice or even location choice, dynamic models of congestion phenomena and trip timing… However little attention has been paid to the assumption of trip end zones: i.e. that within a given zone especially designed for traffic assignment (hence a TAZ) all the trip endpoints are assimilated to the zone 'centroid' (focal point), which is connected to the network by ad hoc links called 'connectors'. Thus the zoning system influences the route choice by origin-destination pair (OD pair), with little if any behavioral basis and under little control by the study analyst. The bad effects of the centroid approximation in traffic assignment have been known for a long time. Ortúzar and Willumsen (2004) recommend considering zones small enough to satisfy approximately the following assumption: that all of the activity within a zone takes place at its centroid. Solutions mainly consist in detailed zoning and centroid positioning. Crevo (1981) and Baass (1991) investigated the impact of zone design on transit assignment; the latter used an algorithm that starts from small spatial units and merges those with the minimum difference in certain socioeconomic variables: the main effect is to decrease the number of intrazonal trips. [START_REF] Chang | Effects of zoning structure and network detail on traffic demand modeling[END_REF] notice that selecting centroids on the basis of weighted averages of population and density yields better assignment results. More recently, [START_REF] Constantin | Some variants of the optimal strategy transit assignment method[END_REF] suggest splitting any trip flow passing by a given network node between several likely next links rather than one link only, with probabilities that stem from a logit model: this can be applied to the choice of the connector.

Our purpose here is to develop a traffic assignment model that deals with individual trip ends in a disaggregate manner. This model enables the analyst to get the zoning system and the associated connecting system under control. A mathematical formulation is provided, including a decomposition principle in two stages respectively network and terminal. Also provided is a related computational scheme.

The approach is both spatial and probabilistic. First, the detailed locations within a given TAZ are modeled in a probabilistic way. Then, "anchor nodes" are identified on the transportation network, together with zonal sets of anchor nodes: such a set contains the most likely passage points to access to and from that zone. Next, the terminal time between the detailed locations within the zone and a given anchor node is modeled as a random variable. The vector of these RVs (possibly dependent) describes the terminal times between the trip ends within the zone and the network. Lastly, route choice is modeled at the level of the individual trip as the selection of a path of minimum cost including network time plus the terminal times at both origin and destination, in a probabilistic way by OD pair.

All in all, this model belongs to the family of stochastic assignment models. It is innovative in that its stochastic features pertain to the terminal times, whereas only the network time has been subjected to stochasticity in previous assignment models. We avail ourselves of the well-known properties of stochastic assignment (eg. [START_REF] Abraham | La répartition du trafic entre itinéraires concurrents[END_REF][START_REF] Burrell | Multiple Route Assignment and its Application to Capacity Restraint[END_REF][START_REF] Dial | A Probabilistic Multipath Traffic Assignment Model which Obviates Path Enumeration[END_REF][START_REF] Beilner | Probabilistic aspects of traffic assignment[END_REF][START_REF] Daganzo | On Stochastic Models of Traffic Assignment[END_REF][START_REF] Powell | The Convergence of Equilibrium Algorithms with Predetermined Step Sizes[END_REF] to develop first a general formulation, second a logit formulation and third a probit formulation. All of them require a separate treatment of network and terminal times, successively. The logit model is fairly simple but its outreach is limited to independent, homoskedastic, Gumbel distributed terminal times to anchor nodes. The probit model is much more powerful and flexible as it deals with heteroskedastic, dependent terminal times to anchors, which can even be correlated between the origin and destination parts of the trip; the Clark formula ( 1961) is useful to evaluate the path choice probability by OD pair and the distribution of the minimum OD travel time, as suggested by Maher and Hughes (1998) but in a specific way to address terminal times.

The body of the paper is organized into five sections. Section 2 sets out the basic model formulation, for a general zoning system and a private transportation network under given travel conditions. Section 3 addresses a classroom case with one OD pair, one or two anchor nodes by zone and one link by anchor pair: this amounts to a binary discrete choice model; closed-form formulae are provided for a logit and a probit model. Section 4 states the assignment algorithm, including a shortest path search by pair of anchor points and an elaborate algorithm to load OD trip flows onto the network in an efficient way. Section 5 is devoted to numerical illustration and experiment, based on the Paris roadway network which includes 15,000 nodes, 40,000 arcs and 1,300 TAZs. A fixed-time assignment and a time-dependent assignment are carried out for the novel assignment model as well as the classic, aggregate model. Finally, Section 6 concludes about potential developments and applications.

Model setting

Let us focus on a set of trips, od Ω , on a given origin-destination (OD) pair ) , ( d o , at a given time period, with homogeneous trip purpose and travel behavior.

Disaggregate locations

Let us consider that the trip ends vary with the individual trip-maker, denoted as ) ( M ω ′ ) would depend only on the zone of origin (resp. destination). However this could be misleading in some cases, notably so when the OD trips are mostly carried by a main network route, in which case the proximity of each endpoint to that route is likely to exert an important influence.

Anchor nodes and terminal times

In a traditional, zone aggregated traffic assignment model, it is assumed first that all trips take their endpoint at the zone centroid -a kind of centre of mass -and second that the centroid is connected to the transportation network by connectors, i.e. fictive links that represent average travel conditions for terminal access to and from the network.

To extend that representation to disaggregate trip ends, we shall focus primarily on the network nodes which make the connector endpoint other than the centroid: define an 'anchor node' as a network node that trips are likely to go through in the vicinity of their endpoint. Let the 'anchoring set' of a zone be the group of its anchor nodes for a given OD pair.

B ∈ β to endpoint ) ( Z ) ( M od d Ω ∈ ω ′ let ) (ω θ′ β od or ) (ω θ′ β
denote the destination terminal cost, a random variable on od Ω . Any pair ) , (

On trip and network paths and costs

On a disaggregate basis, the detailed paths from origin endpoint to destination endpoint are likely to include not only some main network links but also some minor links, particularly so near of the trip endpoints [START_REF] Bovy | Wayfinding in transport networks[END_REF]. Here it is assumed that the main links are explicitly modeled in the transportation network, i.e. the pair L) (N, of node set N and link set L of directed links between two nodes in N ; and that the remaining, very minor links make up a set of 'subnetwork features' which are sufficiently accounted for in the stochastic description of the terminal conditions.

Thus a disaggregate path ω r is expressed as a threefold sequence of first a terminal subnetwork path, + ω r from origin endpoint to an anchor node (1)

+ ω = α ) ( n r ,
Letting ) (αβ r t
denote the travel cost (or time) along route αβ r , the disaggregate trip cost is modelled as the addition of the partial costs along the route:

) ( ) ( ) ( ) ( ω θ′ + + ω θ = β αβ α ω r r t T .
(2)

Disaggregate route choice

Every trip-maker is assumed to be a rational, cost-minimizing decision-maker in his choice of route from origin to destination. As an individual user he behaves in a self-optimizing way; thus he selects a route ) ( * ω r that achieves the minimum cost

) (ω r T
among the set od R of available routes. At the aggregate level, the interest lies in the use of the anchor nodes and of the network elements and paths:

} )) ( * ( n : { Pr ω = α Ω ∈ ω = + ω α r p od od for od A ∈ α . } )) ( * ( n : { Pr ω = β Ω ∈ ω = ′ - ω β r p od od for od B ∈ β . } ) ( * : { Pr ω = Ω ∈ ω = r r p od od r for any network path r . } )) ( * ( n and )) ( * ( n : { Pr ω = β ω = α Ω ∈ ω = - ω + ω αβ r r p od od for od A ∈ α , od B ∈ β .
There are some obvious properties linking these choice probabilities, for instance:

∑ α = ∈ α + = r od R r od r od p p n , , ∑ β = ∈ β - = ′ r od R r od r od p p n , .
However, in most cases od p αβ cannot be expected to be equal to the product of od p α and od p β ′ .

Denoting by od q the trip volume on the OD pair, the product of it by a given probability yields the partial trip volume that meets the associated condition: for instance od od od p q x α α = . .

Posterior cost variables

Also of interest are the cost variables that stem from the assignment of the OD pairs to the transportation system. At the zone level, let * α θ (resp. * β θ′ ) denote the terminal cost optimized at the disaggregate trip level, conditional on α (resp. β ) being selected as the anchor node in the optimal path: this is a random variable defined on a sub- population of trips. It differs from the original, unconditional variable od α θ (resp. od β θ′ ). Remark that only the conditional terminal costs may be observed by trip inspection in a field survey.

At the network level, if there is no dependency between the terminal and the network costs, then ) (ω * αβ r must be a shortest path from α to β along the network, since if there were a shorter network path it would have been selected instead of ) (ω * αβ r

. Then: Proposition 1. Along the network between an OD pair of anchor nodes, all of the chosen network paths have the same and minimum cost.

Thus the user-optimized disaggregate assignment is also user-optimized at the level of the OD pair, though aggregated by anchor node only.

Two-stage assignment model

Proposition 1 enables us to perform traffic assignment with disaggregate trip ends in two stages: first, a shortest path search on the transportation network between any pair of origin and destination anchor nodes; second, by OD pair at the zone level, a discrete choice model (DCM) in which the choice option is a pair of anchor nodes

od od B A ) , ( × ∈ β α
with random disutility function as follows:

) ( ) ( ) ( ω θ′ + + ω θ = ω β * αβ α αβ od od od t T , (3) wherein
{ od r r od od R r T T R r r r r ∈ ′ ∀ ω ≤ ω ∈ = ∃ Ω ∈ ω = Ω ′ - β αβ + α * αβ , } B A ) , ( ) ( ) ( : { ~od od od T T × ∈ β′ α′ ∀ ω ≤ ω Ω ∈ ω = Ω β′ α′ αβ αβ . On one hand, if * αβ Ω ∈ ω then od r r R r T T T ∈ ′ ∀ ω ≤ ω = ω ′ αβ ) ( ) ( ) ( . But od od B A ) , ( × ∈ β′ α′ ∀ , ) ( ) ( ω = ω ′ β′ α′ r T T for some route od R r r r r ∈ = ′ - β′ β′ α′ + α′ ) , , ( hence ) ( ) ( ω ≤ ω β′ α′ αβ T T , yielding that αβ Ω ∈ ω ~ hence αβ * αβ Ω ⊂ Ω ~.
On the other hand, if

αβ Ω ∈ ω ~ then there exists od R r r r r ∈ = - β αβ + α ) , , ( that supports = ω) ( r T ) (ω αβ T so that ) ( ) ( ω ≤ ω β′ α′ T T r od od B A ) , ( × ∈ β′ α′ ∀ . But od R r ∈ ′ ∀ , ) , , ( - β′ β′ α′ + α′ = ′ r r r r hence ) ( ) ( ω ≥ ω β′ α′ ′ T T r , implying that ) ( ) ( ω ≥ ω ′ r r T T hence * αβ Ω ∈ ω and * αβ αβ Ω ⊂ Ω ~.
(ii) The two vectors of choice probabilities measure an identical system of sets. When the sets are disjoint the probability vectors must be equal. If there is some partial intersection then the vectors belong to admissible sets which are identical for both problems.

Binary model

Let us illustrate the trip end disaggregated assignment model in a simple case with two travel options. This occasion will give us the opportunity to introduce the logit and probit models of discrete choice.

Case setting

) ( ) ( ) ( ω θ′ + + ω θ = ω r r r r t T . (4)
The main result in the binary model is the choice probability of the first route:

} 0 { Pr } ) ( ) ( : { Pr 2 1 2 1 1 ≤ - = ω ≤ ω Ω ∈ ω = T T T T p , which involves the cumulative distribution function of 2 1 T T -.
Square zones with uniform distribution of trip ends may be taken as reference, on assuming independence between origin and destination. Assuming further that a subnetwork for terminal access is a dense Manhattan grid, then each r θ (resp. r θ′ ) is the sum of two independent real random variables distributed uniformly over a real interval along an axis of coordinates, X or Y. Thus r θ (resp. r θ′ ) has a triangular distribution, of which the mean r µ (resp. r µ′ ) depends mostly on the location of r α (resp. r α′ ) in the subnetwork grid and the variance 2 r σ (resp.

2 r σ′ ) depends both on the anchor location w.r.t. the zone and the zone size. There is covariance 12 v (resp. 12 v′ ) between the terminal access costs of the two routes, coming from the trip ends:

12 12 12 v v ′ + = χ .
(5)

Multinomial logit model

The multinomial logit model is by far the most widely used DCM, owing to its analytical simplicity [START_REF] Ortuzar | Modelling Transport[END_REF]. Let us recall its main assumptions:

-by option, the utility function has a Gumbel distribution with cumulated function

)) ( exp( exp( ) F( x m x - ψ - =
, mean γ m ( γ being Euler's constant) and standard deviation ) 3 /(ψ π = σ . -between distinct options, the variances 2 σ must be identical (i.e. homoskedasticity) and the utility functions must be independently distributed. So its application to our setting requires to specify some average variance for route cost, and to neglect the covariance:

2 / ) ( ˆ2 2 2 1 2 σ + σ = σ and 0 ˆ12 = v
. Thus we can avail ourselves of the well-known formula for the choice probability of a given option r , say 1

= r : letting 3 / σ π ≡ ψ , ∑ ′ ′ ψ - ψ - = r r r r T T p ) exp( / ) exp( ˆ, or in the binary case ))] ( exp( 1 /[ 1 ˆ1 2 1 T T p - ψ - + = .
(5)

α 1 α 2 D Ζ θ 2 θ 1       = Σ = θ = θ 5 1 1 8 48 1 ] E[ 1 ] E[ 4 3 2 1

Probit model

In the probit model of discrete choice, it is only assumed that the vector of options (dis)utilities is a multivariate Gaussian vector, of which the mean vector and the variance-covariance matrix are unconstrained. Thus each option utility function is a random Gaussian variable with unconstrained mean and variance. These assumptions are less requiring than the logit ones to apply our assignment model, although the Gaussian shape does not fit to perfection a given distribution of access cost and nor does the Gaussian-type dependence fit perfectly a given joint distribution of access costs.

In the binary case the probit model is endowed with a closed form formula for option probability, namely

)) ( (

1 2 1 1 T T p - Φ = σ ( ( , (6)
wherein Φ is the cumulative distribution function of a reduced Gaussian variable (i.e. with null mean and unit variance) and

12 2 2 2 1 1 2 2 2 ) var( v T T - σ + σ = - ≡ σ ( .

Model application and assessment

Let us apply to our instance system the following assignment models: (a) door to door, (b) aggregate, (c) logit, (d) probit without covariance, (e) probit with covariance. On varying the difference in network travel time,

1 2 T T t - ≡ ∆
, it turns out that the probit model with covariance matches the disaggregate assignment quite well, whereas the logit and probit without covariance are as far from these 'best models' as the aggregate model (Fig. 2). This emphasizes the assignment sensitivity to the location of anchor nodes and the disaggregate conditions of terminal access.

Computational scheme

Let us now sketch out a method to compute the disaggregate trip end assignment in any application.

Scheme overview

Our computation scheme is based on the two-stage decomposition and a probit model for trip end disaggregation. It involves the following steps:

1. The characterization of terminal times as random vectors, at the zone level.

2. By OD pair of zones, the search of shortest paths αβ r on the main network from any anchor 4. Network local loads at the level of nodes and links (and even turns) are determined by local superposition of route flows od od p q αβ along the network elements that make up αβ r . To expedite the loading of route flows onto the network elements, it is recommended to use a cascade algorithm by destination anchor β from all anchors α of all origin zones.

The characterization of terminal times

This step depends on which disaggregate information is available to the study analyst. Today, detailed data disaggregated up to the individual building have become available as information about trip ends, at least in the main cities of the developed world. Similarly, exhaustive databases have become available for transport networks, which are impractical for assignment but amenable to a given, one-shot analysis by using a GIS. The task then involves:

1. to select a zoning system (as usual).

2. to identify the anchor nodes relevant for each zone.

3. then, within each zone, to characterize the terminal vector by disaggregate trip end by computing its terminal cost to each anchor. After dealing so with all trip ends, then the mean vector and the matrix of variance-covariance for the distribution of terminal times may be recovered straightforwardly.

DCM by OD pair

Given

β α = T of d o v v T T ) 2 ( ) 1 ( ) 2 ( ) 1 ( 2 1 12 ) , cov( β β α α + = ≡ χ .
These make up the inputs to a logit or probit model, as in Section 3. The application of a logit model requires to estimate a common, somewhat average variance for the rr χ by OD pair of zones, and to drop the other covariance terms. The probit model is more suitable since it takes into account both heteroskedasticity and covariance. Although no closed-form formulas are available in the general case beyond three options, the formulas given by Clark (1961) indeed yield excellent approximations, as pointed out by Maher and Hughes (1998) and checked by Samadzad (on-going PhD Thesis).

Network and terminal assignment

Overall, the assignment of trip demand to the transport system, both network and terminal (subnetwork), can be performed efficiently by the following algorithm: 0. Characterize the terminal vectors of origin zones and of destination zones. , load the inflow vector, from all anchors α of every origin D o ∈ , onto the shortest network tree rooted at β , and add the resulting local flows to the cumulated local flows. This yields the vector of link flows cumulated over all OD pairs. To evaluate the subnetwork flows, in Step 2c the od

x αβ flow could be added to subnetwork cumulated flows o x α and d x β .

Numerical experiment

The roadway network of the Paris metropolitan area is modelled by the French Department for Transport (Dreif, 2006) for planning purposes. The network model includes about 15,000 nodes, 40,000 unidirectional links and 1,300 TAZs. This makes up our main network.

We derived the terminal travel conditions from combination of three detailed databases as follows: (i) 'BD Topo' by the French Geographical Institute (IGN) provides a comprehensive description of roads with metric accuracy; (ii) 'MOS' by the regional Institute for Land Planning (IAU-IDF) provides the type and intensity of land use; (iii) The General Population Census by the French Institute for Statistical and Economic Studies (Insee) yields the number of people and jobs at the block level. Applying the TransCad GIS to the comprehensive roadway database, three or four anchor nodes were selected for each TAZ. Other assumptions include the OD matrix of trip flows at the morning peak hour and the vector of link travel times, on the basis of a classical user equilibrium assignment.

Fig. 3 depicts the topology of the roadway network. Fig. 4 is focused on a sub-area located near the centre; it depicts the link flows under disaggregate trip ends, whereas Fig. 5 indicates the difference between these volumes and those obtained from standard all-or-nothing assignment. It appears that the magnitude of change may reach up to 2,000 veh/h, which amounts to 25% of flow on the busiest motorways; on urban arterials the absolute magnitude is smaller (up to 500 veh/h) but the relative magnitude can be much higher, up to 80%. At the present stage however, such changes are indicative rather than firmly-grounded evidence, since they stem from all-or-nothing assignment. Indeed, under traffic equilibrium some important re-routing might take place in the model with disaggregate trip ends.

More significant are the aggregate indicators about travel time as minimized by every network user -according to each of the two models. Average OD travel time is decreased from 9.55 min in the classical model to 9.44 min in the disaggregate model -a relatively small variation. Considering each OD pair as a class, the interclass variance of average travel time is decreased from 25.2 min² to 23.4 min² in the disaggregate model, in which there is also intraclass variance which amounts to 4.97 min² on average, yielding total variance of 28.4 min² which is 10% higher than in the classical model. This reveals that the zoning system accounts for about 80% of the variance in trip travel times -the remaining part is far from negligible.

Conclusion

Modeling principles have been provided for traffic assignment with disaggregate trip ends, along with an application methodology. On assuming that the terminal costs and the network costs are independent, the assignment problem is decomposed into two stages respectively network and terminal. As the disaggregate assignment model is focused on terminal travel conditions, further research may be targeted to not only network issues (e.g. transit, multimodal, equilibrium, dynamic) but also "supernetwork" issues of combining trip distribution and even trip generation with network assignment in an integrated, far-reaching assignment model (Cf. Sheffi, 1985;[START_REF] Oppenheim | Urban Travel Demand Modeling[END_REF]. Our probit treatment of terminal times makes up an attractive alternative to the previous, logitbased supernetwork models.

  dependence, including correlation. Furthermore, stochastic dependence is expected to exist between the terminal costs of the anchor nodes that pertain to a given zone, since the disaggregate endpoint has particular access conditions to each of the anchor nodes. Let then ] travel conditions in origin zone o with respect to destination zone d , and symmetrically let random vector of terminal travel conditions in destination zone d with respect to origin zone o . Stochastic dependency is expected to hold within and between the two random vectors.

  second a network path αβ r from anchor α to anchor node zone, third a terminal subnetwork path - ω r from β to destination endpoint:

Fig. 1 .

 1 Fig. 1. Instance setting (a) Network, (b) Zone, anchors and distribution of terminal times.

Fig. 2 .

 2 Fig. 2. Share of Route 1 with respect to t ∆

  OD pair of zones, the evaluation of choice probabilities od p αβ by anchor pair )

Fig. 3 .

 3 Fig. 3. Network map.

Fig. 4 .

 4 Fig. 4. Link flow within sub-area under disaggregate trip ends. Fig. 5. Difference in link flow between disaggregate model and aggregate one.

  Denote by od A the anchoring set of origin zone o with respect to destination zone d , and od B that of destination zone d with respect to origin node o .

		The terminal cost (or time) from endpoint	M(	) ω	∈	Z	( o Ω	od	)	to anchor node	α	∈	A	od	is a random variable on
	Ω	od	denoted by	θ α od	) (ω	, or	θ α	) (ω	for simplicity. Similarly, from anchor node	od

  The problem of disaggregate assignment and the DCM by OD pair of traffic zones have identical solution sets. (ii) Then the option probabilities

						αβ αβ ] [ od p	and the choice probabilities	αβ αβ ] [ od p	are
	equivalent.
	Proof. (i) Let us demonstrate that for every pair	) ( β , α	the respective solution sets are mutually inclusive:
	:	(	,	,	)	,	(	)	(	)	}

* αβ t is the shortest path cost from α to β along the network. Proposition 2. (i)

/ 10


Spatialeconometricsofinnovation: Recentcontributionsandresearchperspectives

Keywords: Geographyofinnovation, spatialcorrelation, spatio-dynamicpanels, innovation networks JELcodes:O31, R12, C31

Preliminary introduced by Anselin, Varga and Acs (1997) spatial econometric tools are widely used in economic geography of innovation. Taking into account spatial autocorrelation and spatial heterogeneity of regional innovation, this paper analyzes how these techniques have improved the ability to quantify knowledge spillovers, to measure their spatial extent, and to explore the underlying mechanisms and especially the interactions between geographical and social distance. It is also argued that the recent developments of spatio-dynamic models opens new research lines to investigate the temporaldimensionofbothspatialknowledgeflowsandinnovationnetworks,twoissues thatshouldrankhighintheresearchagendaofthegeographyofinnovation.

Introduction

Spatialeconometricsisasubfieldofeconometricsthathasbeenfastlyexpandingsincethe end of the 80s. Spatial econometric tools deal with the spatial dimension of data and especially with the autocorrelation and heterogeneity that is inherent within localized dataset (Anselin 1988). From the seminal work of Paelinkand Klaasen(1979) and Anselin (1988) the set of available tools have developed fast, allowing us to test for spatial dependenceandtoestimateproperlyseveralspecificmodels,suchascountdatamodels, qualitativedatamodels,paneldatamodels. Spatial econometric tools have been used in various fields of applied economics (agriculturaleconomics,healtheconomics,growthconvergenceanalysis,marketingstudies, morerecentlyfiscaleconomics,etc),andinparticularintheanalysisofregionalinnovation andgrowth.Theuseofspatialeconomicsinthislastfieldissomethingratherobvious.The relationbetweenspaceandinnovationhaslongbeenpointedout (Marshall,1920).Inthis perspective,theuseofspatialeconometrictoolsreliesontwomainmotivations.

Firstly, as stressed by LeSage and Pace (2010), there is an "R&D-based motivation". The endogenous growth theory relies on the idea that knowledge is at least partly a public good, in the sense that it is only partially appropriated by the agent that produces it. In otherwords,knowledgecanbeusedbynewagentwithoutanycostoratalowercostthat theonethathasbeenrequestedtoproduceit.Thefactthattheseknowledgeexternalities would be spatially bounded is at the heart of the new geography and growth theories to explain agglomeration processes and uneven spatial distribution of economic activities.

These knowledge spillovers effects imply that spatial dependence when dealing with innovationdataattheregionallevel.

Thesecondmotivationforconsideringspatialdependenceintheinnovationprocesscomes from the very strong spatial polarization of economic activities in space i . This uneven distribution means that there is a high spatial heterogeneity in the innovation processes that should be accounted for. Indeed, this heterogeneity is very likely to lead to spatial dependencewithintherandomperturbationofeconometricmodel.

Forthesereasons,spatialeconometrictoolshavestartedtobemoreandmoreusedinthe analysisofregionalinnovationandgrowthsincetheendofthe90s.Theaimofthispaperis thereforetogiveanoverviewofthecontributionsprovidedbyspatialeconometrictoolsin this field, and to draw the main research perspectives, especially the ones resulting from thelatestdevelopmentsofspatio-dynamicmodels.Iwillarguethattheselatterprovideus with new tools allowing us not only to deal with the spatial dimension of knowledge diffusion,butalsotoinvestigatetheroleplayedbytimeintheinnovationprocess.

Tothisaim,thispaperisorganizedintothreeparts.Inthefirstsection,thefocusisonthe contribution provided by spatial econometric tools in the quantification of knowledge spillovers.Section2thenreviewsthestudiesusingspatialeconometricswhileexploringthe mechanisms underlying knowledge spillovers. In this part, special emphasis is put on the approaches investigating the role played by collaboration and networks in spatial knowledge diffusion. Finally, the last section details the most recent developments in spatial-dynamic econometric model and show how they could be used to analyze the spatio-temporal dimension of knowledge diffusion. In each of these three sections, a literature review is provided first, and then research perspectives are discussed. It is important to note that I do not necessarily provide a completely exhaustive literature reviewoneachspecificissue(whichwouldrequiremuchmorethanonepaper).Theaimis rather to put the stress on the main contributions and derive from them some new researchquestions.

Usingspatialeconometrictoolstoquantifyspatialknowledgespillovers

Followingtheseminalliteratureonknowledgespillovers (Jaffe,1989,Jaffe,Trajtenbergand Henderson, 1993, Audretsch and Feldman 1996), spatial econometric tools have been introduced in the field of geography of innovation in two different frameworks. The first one is detailed below and relies on a knowledge production function. The second one, discussedinthenextsubsection,isbasedonaspatialinteractionmodeling.

Spatialknowledgeproductionfunction

Spatial econometric tools first enter into the analysis of regional innovation through the knowledge production function setting in which innovation in region is explained by the R&Dinputscarriedoutlocally.Inthisperspective,theveryfirstintroductionofspatialtools isduetoAnselin,VargaandAcs(1997)andconsistinintroducingnotonlytheregionalR&D inputs, but also the R&D carried out in the surrounding regions. Then, tests of spatial dependence in the random perturbation can be used to determine the correct level to consider for inter-regional spillovers [START_REF] Anselin | Local Geographic Spillovers Between University ResearchandHighTechnologyInnovations[END_REF]Acs, 1997, Maggioni, Nosvelli, Uberti,2007) The second contribution provided by the introduction of spatial econometric tools within the knowledge production function is the ability to analyze the spatial profile of impacts.

Withinthedifferenteffects,itispossibletodisentangletheeffectsarisingatthefirstorder neighbor, from those arising at the second order neighbor, third order and so on. The partial derivatives can indeed be used to investigate marginal impacts which show how spillovers decay with respect to order of the neighbors. From this point of view, spatial econometrictoolshelpustoquantifythespatialextentofknowledgespillovers.

Lastbutnotleast,thethirdcontributionisthecorrectcalculationofthemodelcoefficients.

If not accounted for, spatial dependence generates endogeneity leading conventional modeltoproducebiasedandinconsistentestimators (AnselinandLeGallo,2006).Spatial econometric tools therefore allow us to estimate properly the model parameters.

ComparingtheestimatedcoefficientsreportedbyAutant-BernardandLeSage(2010)using spatial econometric methods to the coefficients that would be computed thanks to conventional methods underlines the potential importance of the endogeneity bias. The BayesianestimatessuggesthigherprivateR&DspilloversandlowerpublicR&Dspillovers, and they also suggest less resistance of knowledge flows to distance. Spatial knowledge production function therefore produces substantially different results from the one producedbynonspatialmodels,givingamoreaccurateassessmentofspatialknowledge spillovers.Thesamephenomenonarisesfortheapproachesthatrelyonspatialinteraction modelsinsteadofspatialknowledgeproductionfunction.

Spatialinteractionmodel

Spatialinteractionmodelisthenamegiventogravitymodelswhentheyareappliedtothe field of regional sciences. The basic assumption of the model is that the interaction between two regions (or two agents) depends on the respective weight of each agent together with the distance between them. In this perspective, the analysis of spatial knowledgespilloversreliesonpatentcitations. SincetheseminalworkbyJaffe,Trajtenberg and Henderson (1993), patent citations have been considered as a proxy for knowledge spillovers. In spite of several limitations discussed for instance in Autant-Bernard et al.

(2010),referencemadetopreviouspatentscanbeconsideredasapapertrailofprevious knowledgeusedinthenewinvention.

In spatial interaction models, the strength of knowledge spillovers is assessed by introducingspatialdistanceasanexplanatoryvariableoftheintensityofthepatentcitation flows between regions (LeSage, Fischer, Scherngell, 2007[START_REF] Fischer | Modeling spatial autocorrelation in spatial interaction data: An application to patent citation data in the European Union[END_REF], BergmanandUsai,2009).APoisson(orbinomialnegative)modelisthusestimatedtotake intoaccountthespecificityofthedata(countdatawithalargenumberofobservations).

Withinthisframework,spatialeconometrictoolsimprovetheabilitytomeasurethespatial dimension of knowledge spillovers in the sense that they permit to correct for spatial autocorrelationresultingfromheterogeneityoforiginanddestinationregions (LeSaceand Pace,2010).TheusefulnessofsuchacorrectionisclearlypointedoutbyFischer,LeSage, Scherngell (2007). In this paper, they compare the results obtained with their Bayesian spatial Poisson model with those obtained in a previous version of the model (Fischer, Jansenberger,Scherngell,2006)usingconventionalestimationmethods.Itcomesoutthat the Bayesian effects model produces smaller coefficient estimates for both distance and borders. As observed for the spatial knowledge production function approaches above mentioned, less resistance of knowledge flows to distance and borders is observed when unobservedheterogeneityiscontrolledforusingthemodelcontainingspatialeffects.

Researchperspectives

From the two previous sections, it appears clearly that using spatial econometric tools improves the way spatial knowledge flows and their spatial extent is measured.

Nevertheless, several research perspectives remain. The first one would be to exploit the informationcontainedinthematrixofdirectandindirectimpacts.Thismatrixcontainsall

2.1.Whydoesdistancematter?

Theliteratureprovidestwomainexplanationstothepolarizationofknowledgediffusionin spacebothofthemrelyingoninterpersonalrelationship.Thecommonideaisthatdistance mattersbecauseitreducestheopportunityoffacetofaceinteractions,whichwouldbean importantvectorofknowledgediffusion,especiallywhenknowledgehasanimportanttacit component. These interpersonal relationships would in particular be favored by labor mobilityontheonehand,andbycollaborationnetworkontheother.Afirstsetofstudies explores the specific role played by labor mobility [START_REF] Zucker | Intellectual Capital and the Firm: the TechnologyofGeographicallyLocalizedKnowledgeSpillovers[END_REF], AlmeidaandKogut,1999,Balconi,BreschiandLissoni,2004ormorerecentlyBreschiand Lissoni,2009).Onceaccountedforlabormobility,thespatialdimensionbyitselfwouldno longermatter.Ideaswouldbeembodiedintopeopleandtravelwiththem.Inotherwords, distancewouldmatterduetothelackoflabormobilitythroughspace.

The second channel through which interpersonal relationship would impact on spatial knowledgediffusionistheabilitytoformcollaborationlinks.Johnsonetal.( 2006)actually show that collaborations are even more spatially focused than citation is. Two main distinct frameworks have been used to investigate the role played by collaborations in knowledge diffusion. The former relies on patent citations [START_REF] Singh | Collaboration networks as determinants of knowledge diffusion patterns[END_REF][START_REF] Sorenson | Complexity, networks and knowledge flow[END_REF][START_REF] Gomes-Casseres | Do alliances promote knowledge flows?[END_REF][START_REF] Agrawal | How do spatial and social proximity influence knowledgeflows?Evidencefrompatentdata[END_REF]. It demonstrates that if two regions(ortwoagents)haveco-inventedapatent,orifthesetworegionsaresufficiently closed in the co-invention network, they are more likely to cite each-other. The spatial clusteringofknowledgediffusionwouldthereforeresultfromthehigheropportunitiesof collaboration offered by co-location. More recently, several authors have adopted a slightly different framework to address this issue, based on collaboration models. The dependent variable is in this case the probability for two agents to collaborate, or the intensity of their collaboration. Using either individual [START_REF] Autant-Bernard | Social distance versus spatial distance in R&D cooperation. Empirical evidence from European collaboration choicesinMicroandnanotechnologies[END_REF], Frachisse, 2011) or regionally aggregated data (Ponds et al. 2007, Scherngell and[START_REF] Scherngell | Spatial Interaction Modelling of Cross-Region R&D Collaborations:EmpiricalEvidencefromthe5thEUFrameworkProgram[END_REF], they assess among the various determinants of collaboration, the specific role played by space. In this perspective, the introduction of spatial econometric tools is very promising.

2.2.Twodistinctempiricalstrategies:Spatialknowledgeproductionfunctionand gravitymodels.

Theuseofspatialeconometrictoolstoexploretheroletheinteractionsbetweenspaceand formation of collaboration ties has followed two directions that address slightly different issues.

Based on the spatial knowledge production setting, a first set of papers studies to what extentgeographyisaby-productofsocialproximity (MaggioniandUberti,2007,Maggioni et al., 2009). To this aim, a relational weight matrix is introduced and test for spatial dependence are performed once controlled for controlled for social proximity. An alternative approach, not implemented yet to our knowledge, would be to compare different weight matrices using spatial econometric tools like those developed by LeSage andParent,2004).Thiswouldallowdeterminingwhetheraspatialweightmatrixorasocial weight matrix would be more relevant, or if combining both kinds of distance performs better.

Thesecondapproachreliesonthegravity-likemodels.Whereasinthepreviousapproach, networksenterintotheknowledgeproductionfunctionthroughthespatialweighmatrix, inthiscase,asMaggioniandUberti(2011)argue,spaceentersintothenetworkssetting.

The key issue then becomes "does spatial distance impact on collaboration and network formation?". The use of spatial econometric tools within this framework is suggested by ScherngellandBarber(2009)wherespatialdependencearisingforspatialheterogeneityis accounted for, making use of a Bayesian spatial interaction model developed in Fischer, LeSage,Scherngell(2007).Theempiricalresultsofthesestudiesunderlinetheroleofsocial ties as channels for diffusion of knowledge as well as the interplay between social and geographicalproximity.However,theintroductionofspatialtoolsremainsunusualandthis areaisratheranongoingresearchfield.

2.3.Perspectiveslinkingspatialeconometricsandnetworkanalysis

Theresearchperspectivesarenumerousanditisdifficulttoanticipateallofthem.There areatleastforareasinwhichspatialeconometrictoolsmightofferavaluablecontribution.

First of all of course, one can claim for a more systematic accounting for spatial dependence within collaboration gravity models. A part from the noticeable exception of ScherngellandBarber( 2009 Lastbutnotleast,theintroductionofspatialeconometricswithinnetworkanalysisisalso goingtobenefitfromtheinclusionofthetimedimension.Thispointisfurtherdiscussedin thesectionbelow.

Using spatial econometric tools to investigate the spatio-temporal dimension of knowledgediffusion

Within the growing field of spatial econometric techniques, one of the most active areas overthelastfewyearsiscertainlythepaneldatamodelswheretheobservationalunitsare regions.Thesenewdevelopments,summarizedinthenextsubsection,offerthepossibility todealatthesametimewithboththespatialandthetemporaldimension.Asdetailedin thelasttwosubsections,thiswillallowustoinvestigatethespatio-temporaldimensionof knowledgeflowsandknowledgenetworks.

3.1.Newdevelopmentsinspatio-temporalmodels

Spatio-temporal models belong to the broader category of spatial panel data models (an overviewisgivenbyAnselin,LeGallo,Jayet,2008andmorerecentlybyLeeandYu,2010). Thanks to the presence of an individual time lag, a spatial lag and a cross-product term reflectingthespace-timediffusion,thedynamicresponsesovertimeandspacethatarise from changes in the explanatory variables can be calculated [START_REF] Erthur | Interpreting Dynamic Space-Time Panel Data Models[END_REF]). These methods have been applied to the case of highway induced travel demand (LeSage and Parent,2009)ortothestate-leveldemandforcigarettes(Erthuretal.2010).Buttheyare alsoverypromisingtoinvestigatethespatio-temporaldimensionofknowledgeflows.

3.2.Spatio-temporaldimensionofknowledgeflows

The introduction of the time dimension in the applied analysis of knowledge spillovers allows us to deal with several issues that have remained relatively unexplored so far regardingthelong-runknowledgeproductionprocess.Fromatheoreticalpointofview,the temporal dimension of knowledge diffusion has long been recognized as a key issue. A toughdebateframedbytheworkofRomer( 1990)andJones(1995)hasfocusedontime dependence in the flow of new ideas. However, if there is no doubt that knowledge accumulated by inventors at one period of time is used by new inventors at the next periods, the mechanisms by which this diffusion occurs has not been deeply investigated yet.Inparticular,somekeyissuesarisewhenitcomestoconsiderthespatialdimensionof thisprocess:Howlongdoesittakeforknowledgetoflowthroughspace?Towhatextent doesthenatureoftheagglomerationforceschangethroughtime?

From a more applied perspective, taking time into account is also important to better assess the specific role played by space. Indeed, the introduction of the temporal dimensionislikelytomodifytheresultsobtainedfromspatialestimationneglectingtime.A strong simultaneous spatialdependencecanresult fromastrongtimedependenceanda weakspatialdependence(LeSageandPace2010).

Several empirical approaches have been suggested to deal with these issues. In the agglomerationstudies (Glaeseretal.1992,Hendersonetal.1995,etc)

Conclusion:

Two essential features characterize the geography of economic activities: considerable spatial concentration on the one hand and the industrial specialization of certain geographical zones on the other hand. Among these economic activities, innovation is no exception being highly concentrated in a small number of countries, and within these countries,inasmallnumberofregionsandofteninasmallnumberofmetropolitanareas within these regions. These agglomerative tendencies of innovative activities are hardly recent.Astudyofthegeographyofsuchactivitiesoverthepastonehundredyearsinthe

UnitedStates (Vertova,2002)showsthatthephenomenonislong-standingeventhoughit isnowtendingtointensify.

Combiningtheworkonendogenousgrowth (Romer,1990;GrossmanandHelpman,1995) and on economic geography [START_REF] Fujita | The Spatial Economy: Cities, Regions and InternationalTrade[END_REF] (BarbesolandBriant,2009;AnderssonandLööf,2009;Autant-Bernardetal.,2011;[START_REF] Martin | Spatial Concentration and Plant-Level ProductivityinFrance[END_REF]. Using a location choice model also show that the choices made by firms concerningthelocationoftheirR&Dlaboratoriesareaffectedbytheseexternalities,even thoughotheragglomerationforcesseemtobestronger (Autant-Bernard,2006).However,

  .Thespatiallagisspecifiedsothatspatialdependencenolongeroccursinthe perturbation.Insomecases,thepresenceofspatialautocorrelationisdirectlyinterpreted hasthe(GallieandLegros,2007forinstance),andspatiallylaggedexplanatoryvariablesis not introduction into the model, which is more questionable. Indeed, as explained in the nextparagraphs,suchspatialerrormodelsdonotreallydealwithspilloverseffectsindirect effectsbetweenregionsareexcludedfromthemodel. Building upon these preliminary approaches, more recent studies have elaborated a little moreonthespilloverseffects.MairesseandMulkay(2007)andAutant-BernardandLeSage (2010)usesaSpatialDurbinModel,consideringboththespatiallylaggedindependentand dependent variables. Autant-Bernard and LeSage (2010) provide an empirical motivation forthismodelresult from the introduction of these different spatial econometric tools. The first one is the ability to distinguish direct and indirect impacts thanks to the matrix of own-and cross-partial derivatives. The direct effects capture the responseoftheinnovationoutputtoachangeinR&Dinputwithinthearea.Thesedirect effects take into account the feedback loops that arise from the bidirectional nature of spatialdependence

  theresponsesoftheregionaloutputstoanychangeinregionalinputs.Aswearemostof the time interested in the global impacts, average direct and indirect impacts are computed.Themoredetailedinformationcontainedinthismatrixcouldhoweverinformus aboutregion-specificeffects.Inapolicyevaluationperspectiveinparticular,theimpactofa change in the input of one specific region on all the other regions could be assessed. Conversely,theinformationgivenbytheownpartialderivativesandcrosspartialderivative couldhelpustoevaluatethebenefitforeachspecificregionduetoaglobalincreaseinthe R&Dinput.Toourknowledge,nostudyhasbeencarriedoutinthisdirectionyet. The second research perspective is the introduction of the time dimension within the spatialapproachesofinnovationhave investigated the potential mechanisms that may explain the spatial bounding of knowledgediffusion.Inthisperspective,spatialeconometrictoolshavealsostartedtobe used,pavingthewayforfurtherresearch.Inthefollowingsub-sections,wefirstrecallthe mainargumentsdevelopedtoexplaintheroleplayedbyspatialproximity.Thewayspatial toolshavebeenintroducedisthenanalyzed,puttingspecialemphasizeontheinnovation networkanalysis,whilethelastsub-sectionarguesthatalotremainstobedone.

  ),spatialtoolsarenotyetintroducedinthistypeofsetting. Butbeyondthesespatialinteractionmodels,apromisinglineofresearchwouldrequireto reverse the causality. While most of the current approaches focuses on the impact of distanceoncollaborationandnetworkformation,itwouldbeinterestingtoinvestigateto what extent collaborations and networks structure impact on geography. In other words, doesnetworkpositioningofoneregionanditsneighborsimpacttheregionalinnovationor the spatial diffusion of knowledge? Spatial econometric techniques would here again be usefultoexplorethiskindofissue.Tothisregard,therecentcontributionbyMiguelezand Moreno(2010)providesafirstattemptinthisdirection.ThedatasetcoversNUTS2regions of17WesternEuropeancountries.InadditiontotraditionalR&Dinputs,thepropensityto patentineachregionisexplainedbythemobilityofinventorsandthemainfeatureofthe network to which they belong (strength of links, connectivity and density). A spatial lag model and the robust spatial the spatial heteroskedastic and autocorrelation consistent estimation of the variance-covariance matrix are implemented. The results underline the roleplayedbyregionallabormobilitybutunclearevidenceoftheroleplayedbynetworks isprovided.Theendogeneityproblemarisinginsuchasettingstillremainstobedealtwith. Another promising area lies on the refining of the relational weight matrix, using textual information.Indeed,mostoftheinnovationnetworkdatabases-beingtheycoinventions, copublications, R&D collaborative projects, and so on -contains, along with relational information,additionaldatasuchasabstracts,keywords,etc.Thislattercanbeusedtogo beyondthesimpledefinitionofsocialdistancebetweenagentsbasedontheparticipation toajoinresearch.Theanalysisofthesemanticproximitymayhelpustocharacterizemore precisely the scientific communities and to study how knowledge diffuses through innovation networks In this perspective, Maggioni et al. (2009) exploit the keywords information contained in scientific publications in order to measure the extent of a convergenceprocessofthevocabularyofscientistsworkinginthespecificfieldthe"cluster literature" from 1969 to 2007. This preliminary work does open the way for very fruitful analysesbasedontherecentdevelopmentsincomputersciences(seeMikaetal,2006for instance) which would allow us to consider cognitive distance as a key feature to understandtheinteractionsbetweenspaceandnetworks.

  dynamic models.[START_REF] Baltagi | Testing for serial correlation, spatial autocorrelationandrandomeffectsusingpaneldata[END_REF] suggest spatial correlation tests and temporal correlation tests in a panel model with random effects.[START_REF] Kapoor | Panel data models with spatially correlatederrorcomponents[END_REF] extend estimation based on the method of moments to a spatial panel model with error componentswhiletheextentiontothecaseofadynamicpanelisprovidedbySuandYang approach.[START_REF] Yu | Quasi-Maximum Likelihood Estimators For Spatial Dynamic Panel Data With Fixed Effects When Both n and T Are Large[END_REF] investigate the asymptotic properties of the quasi-maximumlikelihoodestimatorsinastaticSARmodelandYuandLee(2010)extendit to the dynamic case. Bayesian approach is an interesting alternative to estimate such models that have correlation in the time dimension as well as spatial correlation across units.Parent and LeSage (2010 and 2011) estimate a model with random effects while Erthuretal.(2010)considerthemoreelaboratedcaseofadynamicspatialDurbinmodel.

  enables us to understandthisunequaldistributionofinnovationandtheresultinggrowthdynamics.This understanding is based upon the hypothesis of a local dimension of knowledge externalities.Thenotionofknowledgeexternalitiesisindeedatthebaseoftheconceptof increasing returns which is essential for understanding the impact that technological change has upon growth. The spatially bounded nature of these externalities would consequently appear to be at the origin of a geographical polarization of innovative activitiesandlocalizedgrowthdynamics. Thesetheoreticaladvanceshaveopenedthewayforanumberofempiricalstudieswhich aim to answer several research questions which are essential in terms of technological policies: What is the magnitude of these externalities phenomena? On what spatial scale cantheybefound?Whatarethereasonsforthespatialdimensionofexternalities?Does geographicalproximityalonesufficefortheretobeanybenefitfromthereturns?Inordertodealwiththeseissues,severalstudieshavefollowedtheseminalstudiescarried out by[START_REF] Jaffe | Real effects of academic research[END_REF],[START_REF] Jaffe | Geographic localization of knowledge spilloversasevidencedbypatentcitations[END_REF] andAudrestch and Feldman (1996).Inthisfieldofthegeographyofinnovation,theoreticalandempiricaladvancesgo handinhandandbuilduponthemethodologicalimprovementsofeconometrictools.Preliminary introduced by Anselin, Varga and Acs (1997) spatial econometric tools are widely used in economic geography of innovation. Taking into account spatial autocorrelation and spatial heterogeneity of regional innovation, these techniques have improvedtheabilitytoquantifyknowledgespillovers,tomeasuretheirspatialextent,and to explore the underlying mechanisms and especially the interactions between geographicalandsocialdistance. faced by frequentist approaches (heterogeneity, null observations, model comparison, etc.). Multinomial logit estimations used in location choice models require for instance estimating a large number of parameters, which is easier with Bayesian MCMC techniques. Indeed, the computational problems associated withtheHessiancalculationinthecaseoftheMaximumLikelyhoodestimationthatarise whenconsideringlargesampleorwhenthereisalargenumberofparametersinthemodel is overcome. Similarly, the spatial heterogeneity that characterizes in particular individual or relational data is better dealt with by Bayesian tools. Heteroscedasticity can be introducedinthefrequentistapproaches.However,asstatedbyAnselin(1988),itrequires muchmorerestrictivehypothesesabouttheformandtheoriginofthisheterogeneitythan intheBayesianapproach.Thislatteralsopermitstoidentifyoutliers.Inaddition,Bayesian estimates avoid to be locked into a local maximum which could arise with Maximum Likelyhood. Of course, it is worth noticing that bayesian approach assumes that the econometricianhasapriorknowledgethatallowshertoformulateapriordistributionfor the parameter, an idea usually not subscribe to by non-Bayesians. When the sample size increases, this prior information tends however to play a minor role in determining the characteroftheposteriordistribution.Inadditiontothesetechnicaladvances,recentresearchesmovetonewapproachesaiming at exploring the consequences of knowledge externalities at the individual level. More specifically, an important issue deals with the extent to which firms' location choices are influencedbytheirdesiretobenefitfromtheseknowledgeexternalities

including spatially lagged dependent variables only these approaches neglect part of the spatial dependence phenomena. The use of spatial econometrics is probably going to improvetheseresultsaswell.


Benvenuto Cellini's

La Vita

Romain Descendre

The Experience of Law and Art Literature in the 16th Century

From the Middle Ages until the start of the modern era, law and texts in the legal tradition were an important source for Italian literature. There are several reasons for this, the principal one being the socio-intellectual nature of legal and literary activities. As we know, men of letters were often also men of the law, and vice versa-from the lawyers at the Court of Frederick II, who were the first to produce Italian literature in Sicily at the beginning of the thirteenth century, to the eighteenth century at least, it was very rare to find literary authors who did not have substantial legal training. At the same time, before the advent of the modern practice of separating disciplinary fields, the theory and practice of literature and that of law were inseparable. Even the history of the intellectual professions and writing customs require us to reconstruct the closeness and complexity of the links between law and literature. For a long time, particularly in France, literary and philosophical studies ignored the judicial culture of

literary profession, and was in no way destined to become a writer. Cellini thus constitutes a borderline case for law and literature, two areas he experienced without having either the formal knowledge or the skills for them. In fact, in Cellini's case, the correlation between law and literature is rather unusual: it is that of an artist who takes up the pen out of necessity, following several legal conflicts with his lord, Cosimo I, Duke of Tuscany, and after long periods of attending courts and experiencing prisons. There is much contemporary literature on all these questions, and this volume will add to it. Here I will mention just one journal's issue that concentrates on one of the lesserknown facets of this field of study. Diego Quaglioni's 'La vergine e il diavolo. Letteratura e diritto, letteratura come diritto' in Maurizio Cau and Giuliano Marchetto (eds), Droit et littérature (2004) 5 Laboratoire italien is not concerned merely with law in literature or law as literature, but with literature as a feature of medieval common law, that is, literature in law or literature as law.

2 Benvenuto Cellini, La Vita, E Camesasca (ed) (Rizzoli, 1985). For a useful focus on the recent critical literature see Corinne Lucas Fiorato, 'La genèse douloureuse et la réception difficultueuse des écrits de Benvenuto Cellini' (2009) 5 Seizième Siècle 299-318.

Piero Calamandrei, one of the most important Italian jurists of the twentieth century and one of the most mindful of the links between law and literature, and a writer himself, was the first to underline the importance of the judicial questions presented in Cellini's La Vita: 3 Without being a jurist, knowing nothing of the laws of his time, Benvenuto frequently set the law in motion and was the scourge of lawyers and judges;

anyone who wishes to study in real life the legal institutions of Cellini's time in Florence, Rome or even Paris has only to follow him in the peregrinations of which his La Vita gives us a lively itinerary, and to see him at every step engaged with criminal court judges, accused of the most serious and dishonourable crimes-from assault and battery to sodomy and including theft and murder-or engaged in bitter and fierce civil proceedings.

4

In another study, Calamandrei portrays Cellini as a perfect goldsmith of judicial eloquence:

[W]hen he started, he was a fashioner of oratory effects, more skilful than any lawyer. One could cite two or three examples in La Vita of harangues delivered in his own defence in which the legal twists and turns acquire a grace and sinuosity that perfectly brings to mind his goldsmith's work. 5 3 On Calamandrei, the jurist and writer, see in particular the conference papers Piero Calamandrei tra letteratura, diritto e politica (Vallecchi, 1989) and Paolo Carta, 'Humanisme juridique du XXe siècle' (2004) 5 Laboratoire italien 13-37. Calamandrei had a passion for the work of Cellini, a passion that led him to research the archives and to produce studies that are still today of prime importance to specialists. See Piero Calamandrei, Scritti e inediti celliniani, C Cordié (ed) (La Nuova Italia, 1971).

4

'Senz'essere giurista, senza intendersi di leggi, Benvenuto ha assai di frequente messo in moto le leggi e ha dato molto da fare ai notai ed ai giudici: e chi voglia studiar sul vivo gli istituti giudiziari del suo tempo a Firenze, o a Roma od anche a Parigi, non deve far altro che accompagnarlo nelle peregrinazioni di cui la Vita ci fornisce un colorito itinerario, per vederlo ad ogni passo alle prese coi giudici penali, sotto le accuse più gravi ed infamanti, dal ferimento al furto, dall'omicidio alla sodomia, ovvero accanito in aspre contese civili di cui sono pieni specialemente gli anni della sua vecchiaia.' P Calamandrei, 'Un contratto di edizione di Benvenuto Cellini ' (1930) in Scritti e inediti celliniani (n 3) 39-52, 39. 5 'voleva dire da sé le sue ragioni, e quando ci si metteva, riusciva ad essere, meglio di ogni causidico, un abilissimo cesellatore di artifici oratori. Si potrebbero citare nella Vita due o tre esempi di vere e proprie arringhe avvocatesche da lui pronunciate in propria difesa, nelle quali il cavillo assume una grazia e una flessuosità che proprio fa venire in

The jurist here alludes to several trial scenes that punctuate the text, in which Benvenuto conducts his own defence with formidable effectiveness. 6 But what he does not reveal is that Benvenuto's victories in court were not only the fruit of his talent as a litigant-they came above all from the didactic attitude he adopted towards a legal institution which he felt needed to be taught a lesson. It is more as a law professor than as a lawyer that he is successful in the trials, for example explaining to a judge in a Paris court the correct interpretation of French law relating to sodomy, 7 or giving a lesson on inquisitorial procedure to judges in the Pope's entourage who had accused him of stealing papal treasure, but who, according to him, had not followed the most elementary rules of judicial enquiry. 8 This affinity with legal material, unusual in a non-jurist, could only have been built up through long experience. Cellini had been mixed up in all manner of criminal affairs and, having exhausted the criminal courts, 'he withdrew from the career of aggressor and murderer, in which he had had much success, to devote himself to a career as litigant'; he thus gave himself 'body and soul to civil causes'. 9 It is probably not by chance that this transition from the criminal to the civil, which can be dated to the years 1556-7, coincided with another transition-a transition from 'doing' to 'telling', that is, from sculpture to writing, firstly poetic and then autobiographical writing from 1558. As Cellini himself put it, 'since I was prevented from doing, I started to tell'. 10 Accustomed to placing Cellini in the long literary tradition of memorial writinghagiographies, lives of famous men, confessions, artists' autobiographies-literary historians have in general placed little emphasis on the legal paradigm underlying mente i lavori di oreficeria.' P Calamandrei, 'Il Cellini uomo ' (1952) in Scritti e inediti celliniani (n 3) 3-35, 12.

6 La Vita (n 2) bk I, ch 103, 340-4; bk II, ch 30, 474-6. 'Da quell'anno [ie 1556] Benvenuto si ritirò dalla carriera di feritore e di omicida, nella quale aveva raccolto non pochi allori e si dette alla carriera del litigante : abbandonò i tribunali penali, e si dette tutto, anima e corpo, alle cause civili.' Calamandrei, 'Il Cellini uomo' (n 5) 10. 10 'Solo per giovare al mondo, e per essere lasciato da quello scioperato, veduto che m'è impedito il fare, essendo desideroso di render grazie a Dio in qualche modo dell'essere io nato uomo, da poi che m'è impedito di fare così io mi son messo a dire.' B Cellini, Trattato dell'oreficeria, XII, in Opere, Bruno Maier (ed) (Rizzoli, 1968) 718.

La Vita, even though it is explicitly stated by Cellini, , who never ceased to assert that the aim of his book was to defend his ragioni, that is his rights, and who, in so doing, called for his readers to be positioned 'at a distance and free of passion '. 11 Whilst Calamandrei drew the attention of specialists to the exceptional historical document that La Vita represents for historians of the civil and criminal law in force in Italy and France in the mid-sixteenth century, he was careful not to interpret the whole book in the light of the judicial theme. However, this book is also the 'public justification of numerous legal cases in which [Cellini] was the protagonist', as one art historian has astutely reminded us. 12 It has been suggested very recently, and convincingly in my opinion, that the memoir of an accused man written for his own defence belongs to a distinctive juridical genre of literature, and that La Vita was modelled on that literary pattern. 13 In fact, Cellini's autobiography can without doubt be read as a long factum, punctuated throughout by a demand for justice that can certainly seem paradoxical, given the far from edifying crimes and offences with which the book is crammed, but which is no less sincere for all that. 14 Particularly as the paradox disappears when one realises that all these crimes are in fact acts of vengeance, the legal nature of which must be taken seriously.

Understood as redress, thanks to which the injured party creates his own justice, vengeance was for a long time a common custom recognised by the statutes of Italian communes, particularly those of Florence. 15 Benvenuto again evolved fully 11 'Questo lo giudichi chi è da canto sanza passione', La Vita (n 2) bk I, ch 79, 278. For the autobiographical genre interpreted in judicial terms, see Gisèle Mathieu Castellani, La scène judiciaire de l'autobiographie (PUF, 1996) in this world, where for him it was constantly a matter of exacting retribution for the injury received, l'ingiuria, a term to which it is useful to give its legal etymological meaning: it is that which is committed contra ius. However, he came up against the authorities in power in the new Duchy of Tuscany. They had devoted much time and energy to developing an effective criminal justice system, and were unusually active in setting up a system aimed at regulation through centralised law-that is, through the expression of the sovereign's will. Previously there had for many years been a fundamentally pluralistic juridical order, which left plenty of space for the private settling of disputes. So the narrative structure of La Vita describes a process, which had thus become necessary, of moving beyond violence; this process did not break the vicious circle of insult and reprisal, but transformed physical vengeance into symbolic vengeance. 16 When, within the text, the principal character-narrator renounces violence, he explicitly states that henceforth his vengeance will be his work: thus we see the transition from a system of vindictive action to a metaphorical and symbolic vindictive system, which becomes the perspective of the writing in La Vita. The book itself, from this point of view, is not only a factum with defensive intent, it is also an accusation and vengeance, as witnessed by the very unflattering portrait that Cellini paints of Cosimo I. It is, moreover, the reason why, unfit to be published, it remains unfinished, 17 the writing having been interrupted when the conflicts that pitted Cellini against Cosimo eased.

The interest to be found in La Vita for a law and literature approach thus lies in the specific closeness of the link that the work maintains between the two terms. For Cellini, who was not a man of letters, and who, unlike his colleague Vasari for example, did not have the support that would allow him to become a writer in the fullest sense, writing was not a choice but a necessity. For him, literature is in a way a continuation of the judicial by other means. There is in his book continuity and contiguity between law and literature, and not just the literary use of judicial material. In it literature is, in the fullest sense, justification and legitimisation.

16

For more detail see Descendre (n 14). 17 This is also why the author later untruthfully maintained that he had destroyed the sections concerning his relationship with the Prince, in the lines that precede the passage of the Trattato dell'oreficeria (n 10).

This legitimating function appears most strongly when the status of the artist is defined by the special relation that he must be free to maintain with the law. One phrase in the book is revealing; it is the words spoken by Pope Paul III to justify the safe conduct that he grants to Benvenuto while the latter is being sought for the killing of one of his rivals: 'You should know that men like Benvenuto, who are unique in their profession, are not bound by the law' (Sappiate che gli uomini come Benvenuto, unici nella lor professione, non hanno da essere ubrigati alla legge). 18 With this papal utterance, Cellini sets out to establish both his rights and his exceptionality in terms that belong solely to the language of general public law (that is to say the public law aspects of the ius commune). In fact, it is not so much the justification of behaviour considered unlawful that is set out here, as the institution of a special set of rules defining sovereignty in one's own right. This formula is nothing less than a paraphrase of one of the most famous maxims of the Digest, Ulpian's words princeps legibus solutus est-'the prince is absolved from laws, he is not bound by the laws' 19 -which jurists, from the time of the glossators, used to define the absolutist aspect of sovereignty. As we know, the insistence on absolutio legibus became stronger still in the sixteenth century, to the detriment of the opposite principle, princeps legibus alligatus, not only in the juridico-political doctrines that came with the birth of absolutism, but also in the common language of the chancelleries and courts. Moreover, it is probably not by chance that the new usage suggested by Cellini, to the benefit of the artist, issues from the mouth of a Pope, for it is true that it was at Rome, with the canon law theory of the plenitudo potestatis of Popes, that the absolutist principle was first vigorously defended. 20 Cellini used the language of law to establish his status as sovereign artist, which put him on a continuously asserted equal footing with princes. 21 We thus find that 18 La Vita (n 2) bk I, ch 74, 265.

19 Digest, 1, 3, 31. 20 For all these questions, see in particular Ennio Cortese, La norma giuridica. Spunti teorici nel diritto comune classico, vol 2 (Giuffré, 1964); Ennio Cortese, 'Sovranità (storia)' in Enciclopedia del diritto, XLIII (Giuffré, 1990) 205; Diego Quaglioni, La sovranità (Laterza, 2004).

21

There are many passages where Cellini sets out this presumed equality with sovereigns, as with Francis I of France for example, with whom he used the familiar 'tu' Cellini's work validates the thesis put forward by Ernst Kantorowicz in one of his last articles, written in honour of Erwin Panofsky, 'The Sovereignty of the Artist'. 22 What Kantorowicz claimed at a scholarly level-namely 'that the writings of medieval jurists, glossators and commentators of roman and canon law, might have been in any respect relevant to the development of Renaissance theories of art', and 'that certain current views had been foreshadowed by the writings of the jurists' 23 -finds confirmation within the oral culture of the princely courts as it is documented by the written work of an artist who regularly attended them. Jurists in particular had to resolve a whole series of problems linked to terms that were fundamental to juridical science, ars, natura, imitatio, inventio, fictio, veritas. The ars-that is, the law-must imitate nature through the intermediary of a fiction, that is, an invention. From this fact, as a law-maker, the prince was in a position to be seen as a re-creator of nature, and to be compared with the divine creator, sicut deus in terris. It was first in canon law and around the figure of the Pope that such ideas were expounded, particularly by the decretalists at the beginning of the thirteenth century, who developed most fully the theory of potestas plena. Thus, around 1220 the canonist Tancred elucidated Innocent III's idea according to which the Pope is the 'vice-gerent of God', stating that 'he makes something out of nothing like God'. 24 The theories of plenitudo potestatis invented the figure of the divine creator who, several centuries later, played a decisive role in art literature, whether one thinks of Leonardo da Vinci's Libro di pittura (the 'signore e Dio' painter capable of creating everything 25 ), Michelangelo, the divine artist glorified particularly by Vasari, or Cellini, who presented the casting of his Perseus as a supernatural creation resulting in a true resurrection. 26 there is nothing strange in the fact that a formula such as princeps legibus solutus should be the background to Cellini's claim of not being 'bound by the law' (ubrigato alla legge) and that he should make use of a Pope to defend it. 27 It is clear that Cellini's legal culture did not come from a scholarly knowledge of written law and the ius commune legal tradition. However, we would be wrong, today, to think that such a culture was inaccessible to a goldsmith: Lucien Febvre and Henri-Jean Martin had stressed that a large number of the law books that constituted the greatest part of the libraries of the French in the 16th century belonged to "men whom one would imagine being indifferent to these specialities -goldsmiths, millers, apothecaries, for example" 28 . But it is even more significant that Cellini had in his close entourage people such as Benedetto Varchi, the official historian of Florence and a lawyer by training, and Lelio Torelli, Cosimo I's First Secretary, who was at that time the principal representative of Florentine legal humanism. 29 But ultimately the artist's entourage does not really matter. It is the linguistic dimension of law, its communal character, and thus the possibility that it can nurture a narrative, that appears in the re-uses and adaptations that he works on for his own benefit. Whether it is the numerous legal proceedings that are emphasised in the autobiography, the judicial model as background to the writing and the genre of the book, or even the linguistic tools borrowed from academic law with the aim of establishing the artist's sovereignty, everything relates to legal matters and derives from first-hand experience, and that says much about the central place occupied by law in the society of the time.

Within the context of a more general enquiry into the reasons for a linking of law and literature, and in order to project a wider view on the specific case of Cellini, I 27 For an analysis of the role played by other elements of medieval legal heritage in La Vita, and in particular the question of tyranny, which he uses to take a critical look at both the actions of princes and the false hopes of exiled Florentine republicans, see Descendre (n 14). 28 Lucien Febvre and Henri-Jean Martin, L'apparition du livre (Albin Michel, 1999) (1st edn 1958), 391.

29

Primo Secretario and Primo Auditore of Cosmo I, and head of the Duchy's new administration, Lelio Torelli was one of the most powerful men in the Tuscan state. He was also the author of the princeps edition of the Florentine Manuscript of the Pandects (one of the most precious manuscripts of Justinian's Digest): he published it in 1553 with the printer Lorenzo Torrentino, who had been brought from the Duchy of Brabant to act as the Duke's official printer. He became a close friend of Cellini; as his executor from 1555, Torelli asked Cosimo to release him from prison the following year.

will conclude my paper with two short remarks, concerning the question of what each of these fields can bring to the other. On what law can add to the understanding of literature: to dispense with it simply means, in many cases, being unable to understand either the motivation for a piece of writing or the specific form that it takes, including that of some authors from outside the world of the jurists (this is particularly relevant when the authors studied have had legal training and practised law, as was shown in the enlightened reading of Montaigne by André Tournon, or more recently still, that of Guicciardini by Paolo Carta). 30 On the other side of the equation, that is, what literature can bring to the understanding of law: Cellini gives us much more, I believe, than a lively portrait of judicial practices in Italy and France in the mid-sixteenth century-even if, on this subject, the analysis formulated by Calamandrei remains highly relevant. The author of La Vita reveals to us the extent to which law could be embedded in people's lives, how it could support their vision of the world and their place in it, could constitute the very order of human reality, and thus become part of life for an individual who was anything but a jurist. This is a lesson that is still significant today at a time when, for ordinary people, law is often a coded language, an inaccessible field of knowledge. For me, perhaps a little naively, this is a way of rediscovering a reality of law that is not reduced to the rules desired by the lawmaker and thus does not appear to be, or not only to be, an instrument of political domination.

Romain DESCENDRE 30 André Tournon, Montaigne, la glose et l'essai (Champion, 2000) (1st edn 1983); Paolo Carta, Francesco Guicciardini tra diritto e politica (Cedam, 2008).

  1

  9

  from which La Vita is however absent. , in her introduction to the catalogue Benvenuto Cellini. Opere non esposte e documenti notarili, Dario Trento (ed) (Museo Nazionale delBargello, Florence, 1984), a work that has not been widely distributed. Gli uomini come Benvenuto non hanno da essere ubrigati alla legge. Art, droit et politique dans la Vita de Cellini' (2009) 16(4) Chroniques italiennes, internet series, http://chroniquesitaliennes.univ-paris3.fr/numeros/Web16.html; in this article I engage in the same reflections as those pursued here.

	13	Matteo Residori, 'Le ragioni di Benvenuto. Giustizia e retorica giudiziaria nella
	Vita	di	Cellini'	(2009)	16(4)	Chroniques	italiennes,	internet	series,
	http://chroniquesitaliennes.univ-paris3.fr/numeros/Web16.html.
	15	U Dorini, 'La vendetta privata ai tempi di Dante' (1933) 29 Giornale
	dantesco 105; AM Enriques Agnoletti, 'La vendetta nella vita e nella legislazione
	fiorentina' (1933) 91 Archivio storico italiano 85, 181-223; Stefano Andres, 'Oltre lo
	statuto. La vendetta nella letteratura toscana del Due-Trecento' (2004) 5 Laboratoire
	italien 57.

12

Giovanna Gaeta Bertelà14 Romain Descendre, '

  Therefore, form of address and called him, in French, 'my friend'. Conversely, he never hesitated, on several occasions, to lecture the Pope or treat him insolently.

	22	Ernst H Kantorowicz, 'The Sovereignty of the Artist: A Note on Legal Maxims in
	Renaissance Theories of Art' in De Artibus opuscola XL: Essays in Honor of Erwin
	Panofsky, Millard Meiss (ed) (New York University Press, 1961) 267 (subsequently
	reworked in EH Kantorowicz, Selected Studies, M Cherniavsky and Ralph E Giesey (eds)
	(JJ Augustin, 1965) 352.
	23	Ibid, 267-8.
	24	Ibid, 271-2.
	25	Leonardo da Vinci, Libro di pittura, C Pedretti and C Vecce (eds) (Giunti, 1995)
	138 [13].
	26	La Vita (n 2) bk II, ch 77, 571-2.


Nabil Mabrouk

Guillaume Deffuant

A simple birth-death-migration individual-based model for biofilm development

Bacteria growth, detachment and surface-associated motility are recongnized to play an important role in microbial biofilm formation. In this paper we we investigate using an individual-based model how these processes interplay to yield complex biofilm spatial patterns.

Introduction

Bacteria attach to hydrated surfaces and develop biofilms. Biofilms development is a complex and sequential process often initiated by a small number of cells on a hydrated surface. Through binary fission and a number of other mechanisms including surface migration, exopolymer production and biofilm detachment, the initial colonizers develop progressively complex and sophisticated structures. In many applications where biofilms are envolved, it is important to understand how such complex structure develop. This can be performed through microscopic observation of the biofilm development, in particular using Confocal Laser Scanning Microscopy, and individual-based modeling, a convinient tool to develop a comprehensive understanding of how different individual level processes interplay yielding the emergence of complex biofilm structures.

In this work we propose a minimal individual-based model capable of reproducing a large number of observed biofilms spatial structures. Many powerful individual-based models (IBMs) have been proposed in the litterature for the study of biofilm development [START_REF] Grimson | Continuum model for the spatiotemporal growth of bacterial colonies[END_REF] [3] [START_REF] Picioreanu | Particle-Based Multidimensional Multispecies Biofilm Model[END_REF][?]. These IBMs often explicitily include the substrate diffuson, exopolymer production and other processes like the shoving process. Such IBMs have proven their capacity to reproduce realistic spatial pattern and significantly contributed to our understanding of the biofilm development mechanisms. Our aim here is to derive a much simpler IBM by considering only three main processes: birth (binary fission), death (or detachment) and surface migration. The main question adressed in this work is wether these three processes are sufficient to reproduce the large diversity of biofilm spatial patterns observed using confocal laser microscopy techniques.

Model description

1.0.1 Overview

State variables and scales

We consider a two-dimensional space containing discrete individuals (bacteria)characterized by there coordinates within the domain. The vectors p = (x i ), i = 1..N formed with the locations of all the bacteria at a time t define the spatial pattern at that time.

Process overview We assume that the spatial pattern changes in time due to three stochastic events acting on the individuals:

• birth of a new bacterium • detachment (death) of a bacterium
• migration of a bacterium to a new location Birth is a process by which a parent individual produces an offspring individual which location is selected randomly in the neighborhood of the parent individual. Detachment is the process by which an individual is removed from the system. Migration is a process by which an individual changes moves to a new location within the spatial domain.

Scheduling The temporal behavior of the IBM is governed solely by the three stochastic processes. They are scheduled according to a discrete event scheme by adapting the Gillespie's algorithm [START_REF] Gillespie | A general method for numerically simulating the stochastic time evolution of coupled chemical reactions[END_REF]. The algorithm iterates over the following steps:

Submodels

• Birth process:

We suppose that the probability per unit of time that a bacterium particle i in position x i produces a new bacterium located in position x ′ i is given by:

B(x i , x ′ i ) = (b 1 + b ′ 1 N j=1,j =i w bv (x i -x j ))w b (x i -x ′ i ) (1)
The parameters b 1 and b ′ 1 are the density independent and the densitydependent birth rates respectively. w bv (x i -x j ) is an interaction kernel that measure the contibution of the individual in x j to the birth rate of the individual in x i and w b (x i -x ′ i ) is a birth kernel measuring the probability that the newly formed individual will be located in x ′ j . • Bacteria detachment process An individual in location x i detachs with a probability D(x i ) given by:

D(x i ) = d 1 + d ′ 1 N j=1,j =i w dv (x i -x j ) (2)
d 1 and d ′ 1 are the density-independent and the density-dependent detachment rates. N j=1,j =i w dv (x i -x j ) is the density of individuals evaluated in x i using an interaction kernel w dv (x i -x j )

• Bacteria motility An individual in location x i moves to new location x ′ i with a probability M (x i , x ′ i ) given by:

M (x i , x ′ i ) = (m 1 + m ′ 1 N j=1,j =i w mv (x i -x j ))w m (x i -x ′ i ) (3)
with m 1 and m ′ 1 are the density independent and the density-dependent motility rates respectively. N j=1,j =i w mv (x i -x j ) is the individuals density in x i measured using and interaction kernel w m (x i -x ′ i ) and w m (x i -x ′ i ) is the motility kernel.

Parameters The model parameters are summarized in table 1.

Different kernel types can be used. In this model we use a uniform kernel given by:

w(x -x ′ ) = 1/w if ||x -x ′ || < w 0 else (4)
Implementation We implement the model using the Mason framework, an open-source java-based library for implementing agent-based models [START_REF] Luke | Mason : A java multiagent simulation toolkit[END_REF]. The individual-based simulator outputs are the timeseries of the population size and the pair correlation function C(ξ). Snapshots of the spatial pattern can also be taken at a fixed period. The simulator allows the replication of an individualbased simulation and the calculation of the average and variance of N (t) and C(ξ).

Results

Though very simple, this IBM has 11 parameters which potentially can have an impact on the processes and the simulated biofilm patterns. In order to analyse this IBM we start with aunderstanding the patterns that arise in a birthdeath IBM by switching the migration terms off (in practice this is performed by setting the parameter, m 1 and m ′ 1 to zero) and then in a second step introduce switch on the migration process and attempt to analyse how bacteria motility impacts the simulated biofilm structures.

Birth-death IBM

The density-independant case

In the case where birth and death are density-independent processes (b ′ 1 , d ′ 1 ) a (quasi) stationnary state can be obtained only for b 1 = d 1 as if b 1 > d 1 the population grows exponontially and when b 1 < d 1 the population goes to extinction. In the case where b 1 = d 1 the spatial pattern is sensitive to the size of the birth kernel w b . The individuals form colonies (aggregates) if w b is smaller than the domain size as illustrated in figure 1.

The density-dependant case

There are three means of introducing the effect of neighboring individuals:

• the neighbors can have an influence on the birth process but not on the death process

• the neighbors can have an influence on the death process but not on the birth process

• the neighbors can have an influence on both birth and death process

In each of these situation the effect of the neighbors can be positive or negative. In the positive interaction the presence of neighbors increase the intensity of the process. For example one can assume that the presence of Among these 8 cases we can drop the configurations B+ as in this situation the formation of colonies will be accelerated yielding an exponontial growth of the size of the population. The configuration D-should also yield to the exponontial growth of the population. These two trivial results are independent of the values of the parameters b 1 , d 1 and d

′ 1 b ′ 1 .
We propose to analyse the case D+, namely the increase of the local density increases the death probability of a focal individual. We analyze the case where d 1 < b 1 (and d ′ 1 > 0 for the positive interaction) The case b 1 <= d 1 is trivial and yields to the extinction of the population as the death rate of an individual is always high or equal to its birth rate. Figure 2 shows two examples of spatial pattern obtained for two different sizes of the death interaction kernel and a comparable spatial pattern observed in a real biofilm experiment.

The patterns shows the formation of colonies separated with a quasi regular distance that depends on the size of the death interaction kernel (w dv ).

Birth-death-motility IBM

Case of density-independent motility

Density-independent motility tend to disperse the individuals. Hence, increasing the rate of the density independent motility (m 1 ) is comparable to increasing

Case of density-dependent motility

The density-dependent motility implies that the motility rate of an individual depends on the local density of neighbors measured with an the interaction kernel w mv . This interaction can be positive m ′ 1 > 0 or negative m ′ 1 < 0. This adds a further complexity to the configurations identified in the case of the pure birth-death model.

We investigated the spatial patterns that arise in the case D+M-: the birth process is density-indepedent, the death rate increases with the increase of the local density while the motility is reduced by the increase of the local density. In this situation the spatial pattern shows the development of labyrinth comparable to the labyrinths observed in real biofilm experiments as illustrated in figure 3.

Concluding remarks

During the last two decades individual-based modelling has been used for investigating how complex biofilm spatial patterns emerge from local interaction between the individuals. Our model is an attempt to develop a minimalistic IBM that is able to reproduce a diversity of observed patterns. Our work shows that by considering only three density-denpendent processes (birth, death and migration) it is possible to reproduce complex spatial patterns observed in rela biofilms.

Though simple our model has 11 parameters that can potentially influence the spatial pattern. Hence additional exploration of the effect of these parameters

1 .

 1 Set time to t = 0 2. Calculate the sums r b = N (t) i=1 b i = bN (t) and r d = N (t) i=1 d i = dN (t). The total rate at which events occur (births or deaths) is given by r(t) = r b (t) + r d (t) 3. Choose a waiting time τ for the next event to occur according to τ = -1 r lnλ, where 0 < λ ≤ 1 is a uniformly distributed random number 4. Choose a birth or death event with a probability r b /r and r d /r respectively 5. Choose an individual k with a probability b k /r b (if the event is a birth) or d k /r d if the event is a death, where b k = b and d k = d are respectively the birth and death rates of the individual k 6. Perform the selected event on the individual k 7. Update time according to t = t + τ 8. Update the number of the individuals N (t) 9. Continue from step 2 until t < t end 1.0.2 Details Initialization The model is initialized with N (t = 0) bacterial cells distributed uniformly over the domain.

Figure 1 :

 1 Figure 1: Snapshots of the quasi stationary spatial pattern for b 1 = d 1 = 0.1 and w b = 5 for a domain size of 101 × 101

Figure 2 :

 2 Figure 2: Snapshots of simulated patterns (a, b) and CLSM biofilm image for Pseudomonas aeruginosa (c)

Figure 3 :

 3 Figure 3: (a) Snapshots of simulated patterns (b) and CLSM biofilm image for Pseudomonas aeruginosa

Table 1 :

 1 Model parameters

		Description
	b 1	Bacteria density-independent birth rate
	b ′ 1	Bacteria density-dependent birth rate
	d 1	Density-independent bacteria detachment rate
	d ′ 1	Density-dependent bacteria detachment rate
	m 1	Density-independent bacteria motility rate
	m ′ 1	Density-dependent bacteria motility rate
	w b	birth kernel
	w bv	birth interaction kernels
	w dv	detachment interaction kernels
	w m	motility kernel
	w mv motility interaction kernels

Acknowledgement

This work is supported by the SYSCOMM program, a grant from the French National Agency of Research (ANR DISCO -AAP215-SYSCOMM 2009)