289 lines
12 KiB
Racket
289 lines
12 KiB
Racket
#lang racket
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;;; dds/networks
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;;; This module provides some quick definitions for and analysing
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;;; network models. A network is a set of variables which are updated
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;;; according to their corresponding update functions. The variables
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;;; to be updated at each step are given by the mode.
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;;;
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;;; This model can generalise Boolean networks, TBANs, multivalued
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;;; networks, etc.
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(require "utils.rkt" "generic.rkt" graph)
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(provide
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;; Structures
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(struct-out dynamics)
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;; Functions
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(contract-out [update (-> network? state? (set/c variable? #:kind 'dont-care) state?)]
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[make-state (-> (listof (cons/c symbol? any/c)) state?)]
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[make-network-from-functions (-> (listof (cons/c symbol? update-function/c)) network?)]
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[update-function-form->update-function (-> update-function-form? update-function/c)]
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[network-form->network (-> network-form? network?)]
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[make-network-from-forms (-> (listof (cons/c symbol? update-function-form?))
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network?)]
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[list-interactions (-> network-form? variable? (listof variable?))]
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[build-interaction-graph (-> network-form? graph?)]
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[build-all-states (-> domain-mapping/c (listof state?))]
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[make-same-domains (-> (listof variable?) generic-set? domain-mapping/c)]
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[make-boolean-domains (-> (listof variable?) (hash/c variable? (list/c #f #t)))]
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[get-interaction-sign (-> network-form? domain-mapping/c variable? variable? (or/c '+ '- '0))]
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[build-signed-interaction-graph (-> network-form? domain-mapping/c graph?)]
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[build-boolean-signed-interaction-graph (-> network-form? graph?)]
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[make-asyn (-> (listof variable?) mode?)]
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[make-syn (-> (listof variable?) mode?)]
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[make-dynamics-from-func (-> network? (-> (listof variable?) mode?) dynamics?)]
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[make-asyn-dynamics (-> network? dynamics?)]
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[make-syn-dynamics (-> network? dynamics?)]
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[dds-step (-> dynamics? state? (set/c state?))]
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)
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;; Predicates
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(contract-out [variable? (-> any/c boolean?)]
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[state? (-> any/c boolean?)]
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[update-function-form? (-> any/c boolean?)]
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[network-form? (-> any/c boolean?)]
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[mode? (-> any/c boolean?)])
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;; Contracts
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(contract-out [state/c contract?]
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[network/c contract?]
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[update-function/c contract?]
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[domain-mapping/c contract?])
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;; Syntax
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st nn)
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;;; =================
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;;; Basic definitions
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;;; =================
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(define variable? symbol?)
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;;; A state of a network is a mapping from the variables of the
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;;; network to their values.
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(define state? variable-mapping?)
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(define state/c (flat-named-contract 'state state?))
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;;; An update function is a function computing a value from the given
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;;; state.
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(define update-function/c (-> state? any/c))
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;;; A network is a mapping from its variables to its update functions.
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(define network? variable-mapping?)
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(define network/c (flat-named-contract 'network network?))
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;;; Given a state s updates all the variables from xs. This
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;;; corresponds to a parallel mode.
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(define (update network s xs)
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(for/fold ([new-s s])
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([x xs])
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(let ([f (hash-ref network x)])
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(hash-set new-s x (f s)))))
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;;; A version of make-immutable-hash restricted to creating network
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;;; states (see contract).
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(define (make-state mappings) (make-immutable-hash mappings))
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;;; A shortcut for make-state.
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(define-syntax-rule (st mappings) (make-state mappings))
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;;; A version of make-immutable-hash restricted to creating networks.
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(define (make-network-from-functions funcs) (make-immutable-hash funcs))
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;;; =================================
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;;; Syntactic description of networks
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;;; =================================
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;;; An update function form is any form which can appear as a body of
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;;; a function and which can be evaluated with eval. For example,
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;;; '(and x y (not z)) or '(+ 1 a (- b 10)).
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(define update-function-form? any/c)
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;;; A Boolean network form is a mapping from its variables to the
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;;; forms of their update functions.
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(define network-form? variable-mapping?)
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;;; Build an update function from an update function form.
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(define (update-function-form->update-function form)
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(λ (s) (eval-with s form)))
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;;; Build a network from a network form.
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(define (network-form->network bnf)
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(for/hash ([(x form) bnf])
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(values x (update-function-form->update-function form))))
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;;; Build a network from a list of pairs of forms of update functions.
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(define (make-network-from-forms forms)
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(network-form->network (make-immutable-hash forms)))
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;;; A shortcut for make-network-from-forms.
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(define-syntax-rule (nn forms) (make-network-from-forms forms))
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;;; ============================
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;;; Inferring interaction graphs
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;;; ============================
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;;; I allow any syntactic forms in definitions of Boolean functions.
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;;; I can still find out which Boolean variables appear in those
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;;; syntactic form, but I have no reliable syntactic means of finding
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;;; out what kind of action do they have (inhibition or activation)
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;;; since I cannot do Boolean minimisation (e.g., I cannot rely on not
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;;; appearing before a variable, since (not (not a)) is equivalent
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;;; to a). On the other hand, going through all Boolean states is
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;;; quite resource-consuming and thus not always useful.
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;;;
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;;; In this section I provide inference of both unsigned and signed
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;;; interaction graphs, but since the inference of signed interaction
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;;; graphs is based on analysing the dynamics of the networks, it may
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;;; be quite resource-consuming.
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;;; Lists the variables of the network form appearing in the update
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;;; function form for x.
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(define (list-interactions nf x)
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(set-intersect
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(extract-symbols (hash-ref nf x))
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(hash-keys nf)))
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;;; Builds the graph in which the vertices are the variables of a
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;;; given network, and which contains an arrow from a to b whenever a
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;;; appears in (list-interactions a).
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(define (build-interaction-graph n)
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(transpose
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(unweighted-graph/adj
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(for/list ([(var _) n]) (cons var (list-interactions n var))))))
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;;; A domain mapping is a hash set mapping variables to the lists of
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;;; values in their domains.
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(define domain-mapping/c (hash/c variable? list?))
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;;; Given a hash-set mapping variables to generic sets of their
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;;; possible values, constructs the list of all possible states.
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(define (build-all-states vars-domains)
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(let* ([var-dom-list (hash->list vars-domains)]
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[vars (map car var-dom-list)]
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[domains (map cdr var-dom-list)])
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(for/list ([s (apply cartesian-product domains)])
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(make-state (for/list ([var vars] [val s])
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(cons var val))))))
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;;; Makes a hash set mapping all variables to a single domain.
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(define (make-same-domains vars domain)
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(for/hash ([var vars]) (values var domain)))
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;;; Makes a hash set mapping all variables to the Boolean domain.
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(define (make-boolean-domains vars)
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(make-same-domains vars '(#f #t)))
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;;; Given two interacting variables of a network form and the domains
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;;; of the variables, returns '+ if the interaction is monotonously
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;;; increasing, '- if it is monotonously decreasing, and '0 otherwise.
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;;;
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;;; This function does not check whether the two variables indeed
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;;; interact. Its behaviour is undefined if the variables do not
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;;; interact.
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;;;
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;;; /!\ This function iterates through almost all of the states of the
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;;; network, so its performance decreases very quickly with network
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;;; size.
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(define (get-interaction-sign network-form doms x y)
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(let* ([dom-x (hash-ref doms x)]
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[dom-y (hash-ref doms y)]
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[network (network-form->network network-form)]
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;; Replace the domain of x by a dummy singleton.
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[doms-no-x (hash-set doms x '(#f))]
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;; Build all the states, but as if x were not there: since I
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;; replace its domain by a singleton, all states will contain
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;; the same value for x.
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[states-no-x (build-all-states doms-no-x)]
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;; Go through all states, then through all ordered pairs of
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;; values of x, generate pairs of states (s1, s2) such that x
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;; has a smaller value in s1, and check that updating y in s1
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;; yields a smaller value than updating y in s2. I rely on
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;; the fact that the domains are ordered.
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[x-y-interactions (for*/list ([s states-no-x]
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[x1 dom-x] ; ordered pairs of values of x
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[x2 (cdr (member x1 dom-x))])
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(let* ([s1 (hash-set s x x1)] ; s1(x) < s2(x)
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[s2 (hash-set s x x2)]
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[y1 ((hash-ref network y) s1)]
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[y2 ((hash-ref network y) s2)])
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;; y1 <= y2?
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(<= (index-of dom-y y1) (index-of dom-y y2))))])
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(cond
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;; If, in all interactions, y1 <= y2, then we have an
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;; increasing/promoting interaction between x and y.
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[(andmap (λ (x) (eq? x #t)) x-y-interactions) '+]
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;; If, in all interactions, y1 > y2, then we have an
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;; decreasing/inhibiting interaction between x and y.
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[(andmap (λ (x) (eq? x #f)) x-y-interactions) '-]
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;; Otherwise the interaction is neither increasing nor
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;; decreasing.
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[else '0])))
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;;; Constructs a signed interaction graph of a given network form,
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;;; given the ordered domains of its variables. The order on the
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;;; domains determines the signs which will appear on the interaction
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;;; graph.
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;;;
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;;; /!\ This function iterates through almost all states of the
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;;; network for every arrow in the unsigned interaction graph, so its
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;;; performance decreases very quickly with the size of the network.
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(define (build-signed-interaction-graph network-form doms)
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(let ([ig (build-interaction-graph network-form)])
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(weighted-graph/directed
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(for/list ([e (in-edges ig)])
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(match-let ([(list x y) e])
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(list (match (get-interaction-sign network-form doms x y)
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['+ 1] ['- -1] ['0 0])
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x y))))))
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;;; Calls build-signed-interaction-graph with the Boolean domain for
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;;; all variable.
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;;;
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;;; /!\ The same performance warning applies as for
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;;; build-signed-interaction-graph.
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(define (build-boolean-signed-interaction-graph network-form)
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(build-signed-interaction-graph network-form
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(make-boolean-domains (hash-keys network-form))))
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;;; ====================
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;;; Dynamics of networks
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;;; ====================
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;;; This section contains definitions for building and analysing the
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;;; dynamics of networks.
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;;; A mode is a set of sets of variables.
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(define mode? (set/c (set/c variable?)))
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;;; A network dynamics is a network plus a mode.
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(struct dynamics (network mode)
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#:methods gen:dds
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[(define/match (dds-step dyn s)
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[((dynamics network mode) s)
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(for/set ([m mode]) (update network s m))])])
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;;; Given a list of variables, builds the asynchronous mode (a set of
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;;; singletons).
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(define (make-asyn vars)
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(for/set ([v vars]) (set v)))
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;;; Given a list of variables, builds the synchronous mode (a set
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;;; containing the set of variables).
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(define (make-syn vars) (set (list->set vars)))
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;;; Given a network, applies a function for building a mode to its
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;;; variables and returns the corresponding network dynamics.
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(define (make-dynamics-from-func network mode-func)
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(dynamics network (mode-func (hash-keys network))))
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;;; Creates the asynchronous dynamics for a given network.
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(define (make-asyn-dynamics network)
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(make-dynamics-from-func network make-asyn))
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;;; Creates the synchronous dynamics for a given network.
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(define (make-syn-dynamics network)
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(make-dynamics-from-func network make-syn))
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