consider all the possible states of a system in _ boltzmannian statistical mechanics_. a measure is defined over these states .the main aim of statistical mechanics is to work as a bridge between the microscopic and macroscopic levels of description of a system , and the measure is a crucial `` level - bridging '' ingredient .an important question in the foundations of boltzmannian statistical mechanics is how to interpret this measure .a recent proposal is that it is best interpreted as a _ typicality measure _ : it represents the _ relative size of sets of states _ , and typical states show a certain property if the measure of the set that corresponds to this property is one or close to one . in boltzmannian statistical mechanics the same measure is appealed to in the equilibrium and non - equilibrium context . hence the interpretation as a typicality measure is relevant for both contexts .this approach enjoys particular prominence among contemporary physicists who endorse boltzmannian statistical mechanics ( drr 1998 ; goldstein 2001 ; goldstein and lebowitz 2004 ; lebowitz 1993a ; lebowitz 1993b ; lebowitz 1999 ) but has also been advocated by philosophers ( maudlin 2007 ; volchan 2007 ) . + not only in statistical mechanics but also for _ measure - theoretic dynamical systems _an interpretation needs to be found for the standard measures used , and one suggestion is to interpret them as typicality measures . unlike statistical mechanics ,measure - theoretic dynamical systems theory is not concerned with bridging scales of description .moreover , while statistical mechanics is a physical theory , dynamical systems theory is a set of mathematical tools with applications in various sciences ( among others , in physics , chemistry , biology , meteorology and climate science ) .furthermore , understanding the behaviour of `` most '' initial conditions was always a central concern in dynamical systems theory .hence the notion of typicality does not cause as much controversy here as in statistical mechanics . +what is missing is a thorough treatment of the question why these measures are a good choice of typicality measures .witness volchan ( 2007 , 13 ) : `` many tough questions are still to be addressed . for example , as typicality is relative to the measure used , how one justifies a particular choice ? under what criteria ? '' .this paper attempts to contribute to fill this gap by presenting _ a first tentative proposal of how to justify typicality measures in statistical mechanics and in measure - theoretic dynamical systems theory more generally_. an argument will be called a justification of typicality measures exactly when ( i ) the premises imply that there is a unique typicality measure or a set of typicality measures which agree on which sets are typical / atypical ( here any measure of this set can be used as typicality measure ) ; and ( ii ) the premises are at least potentially more plausible than just postulating that certain measures are a good choice of typicality measures . + this paper proceeds as follows .section [ dynamicalsystems1 ] introduces boltzmannian statistical mechanics and section [ dynamicalsystems2 ] introduces measure - theoretic dynamical systems . section [ typicality ] discusses the notion of typicality . section [ pitowsky ] outlines pitowsky s ( 2012 ) justification of typicality measures , which is the only paper known to the author which advances a justification of typicality measures of statistical mechanics and measure - theoretic dynamical systems .section [ pitowskycriticism ] argues that pitowsky s argument does not fit the bill .section [ new1 ] and [ new2 ] present a first proposal of how to justify typicality measures of boltzmannian statistical mechanics and measure - theoretic dynamical systems .finally , section [ uniqueness ] shows that if systems are ergodic/ epsilon - ergodic , one obtains uniqueness results about the typicality measures .the paper concludes in section [ conclusion ] .the concern in this paper is boltzmannian statistical mechanics ( sm ) and not gibbsian sm .( i adopt the common distinction that while gibbsian sm is about ensembles , boltzmannian sm is about a single system .so , e.g. , equilibrium is associated with a single state in boltzmannian sm and with a probability distribution in gibbsian sm ( cf .frigg 2008 ; uffink 1996 ) ) . in boltzmannian statistical mechanicsthe object of study is a system consisting of classical particles .the boundary conditions assumed here are that the system is in a bounded container and is isolated from the environment .the _ microscopic description _ is as follows .the _ microstate _ of the system is represented by a point , where is the ( three - dimensional ) position of the -th particle and is the ( three - dimensional ) momentum of the -th particle ( ) .the microstates are elements of the -dimensional state space ( where represents all possible position and momentum coordinates of all the particles ) .because the energy is conserved , the motion of the system is confined to a -dimensional energy hypersurface , which can be shown to be compact ( is the value of the energy of the system ) .the system starts in a certain initial condition an initial microstate .the evolution of the system is governed by hamilton s equations , whose solutions are the phase flow on the energy hypersurface .that is , gives the microstate of the system that started in initial condition after time steps .+ the _ macroscopic description _ is as follows ._ macrostates _ , , are characterised by the values of macroscopic parameters such as local pressure and local temperature .to each macrostate ,  , there corresponds a _ macroregion _ which consists of all for which the macroscopic parameters take values which are characteristic for .the , , do not overlap and jointly cover .are a _ partition _ , meaning that for all , for and . ]two macrostates are of particular importance : the equilibrium state and the macrostate at the beginning of the process .the macroscopic evolution of the system with initial microstate is given by , where is the macrostate corresponding to microstate .+ regarding the measures of interest in boltzmannian sm , note that for hamiltonian systems the uniform measure defined on ( the lebesgue measure ) is invariant under the dynamics ; this result is known as liouville s theorem ( petersen 1983 , 5 ) . can be restricted to a normalized measure on , which is also invariant under the dynamics . is called the _ microcanonical measure _ and is the standard measure used in boltzmannian sm .hence the question is how to justify as typicality measure .+ because the same measures are employed in bolzmannian sm for questions regarding equilibrium and non - equilibrium , the interpretation as typicality measure is relevant in both contexts . in general ,much more is known about equilibrium sm than non - equilibrium sm .in particular , it is a central aim of non - equilibrium sm to explain _ thermodynamic - like behaviour _ , but this is extremely difficult .to characterise thermodynamic - like behaviour , the boltzmann entropy first needs to be introduced .the _ boltzmann entropy of a macrostate _ is , where is the boltzmann constant ; and the _ boltzmann entropy of a system _ at time is .it can be shown that for gases the equilibrium macroregion covers nearly all of , and hence the boltzmann entropy is highest in equilibrium .the macrostate at the beginning of the process is , by assumption , a low entropy state .thermodynamic - like behaviour is characterised as follows ( lavis 2005 ) : the general tendency is that the entropy of a system that is initially in a low - entropy state increases until it comes close to its maximum value and then stays there , but frequent small and very rare large downward fluctuations ( contra irreversibility ) are allowed .now proponents of the typicality approach argue that typical initial states show thermodynamic - like behaviour , and hence that _ an explanation of thermodynamic - like behaviour can be given in terms of typicality_.unlike statistical mechanics , measure - theoretic dynamical systems theory is not concerned with bridging scales of description . moreover , while statistical mechanics is a physical theory , dynamical systems theory is a set of mathematical tools with applications in various sciences ( among others , in physics , chemistry , biology , meteorology and climate science ) .furthermore , understanding the behaviour of `` most '' initial conditions was always a central concern in dynamical systems theory .hence the notion of typicality does not cause as much controversy here as in statistical mechanics .+ dynamical systems theory can be split into measure - theoretic dynamical systems theory ( where there is a measure and statistical properties are studied ) and topological dynamical systems theory ( where there is no measure and topological properties are studied ) ._ this paper is only concerned with measure - theoretic dynamical systems theory_. there are also interesting foundational questions in topological dynamical systems theory , where a topological notion of typicality plays a role , but this topic is beyond the scope of this paper ( see , e.g. , frigg and werndl , 2012 , for some thoughts on topological typicality ) . + formally , measure - theoretic dynamical systems ( mt - dynamical systems ) are defined as follows ( petersen 1983 ) . is a _ mt - dynamical system _ iff ( if and only if ) is a set ( the _ phase space _ ) , where in this paper is an interval is any set with ( ) ,  ] ( ) , where always , . ] in ; is the lebesgue -algebra of subsets of ( elements of are called _ measurable _ sets ) ;  ] is the unit interval , ( i.e. , the -th iterate - th iterate is the identity function , i.e. , for all . ] of ) , , with and is the uniform measure ( lebesgue measure ) .figure 1(a ) shows the density of the measure of the tent map .+  the;(b) ]  _ example 2 . _the second example is the _ logistic map _ where ]_. he claims that all other cases can be reduced to this case because any relevant phase space with a measure on it is isomorphic to the measure space of the unit interval  ], pitowsky first notes that the points of can be put into one - to - one correspondence with the points of  ] , where is the set of all sequences ending with an infinite number of ones excluding the sequence consisting only of ones . that is excluded is irrelevant because ( cf .pitowsky 2012 ) . ] maps the measure on to a measure on  ] .consequently , pitowsky concludes , the uniform measure is the uniquely correct typicality measure on  ] by using to map the measure on to a measure on  ] could be used to map the measure on to a measure on  ] by relating  ] , one obtains the invariant measure of the logistic map shown in figure 1(b ) ( aligood et al ._ to sum up , it is unclear which map should be used to arrive at a typicality measure on  ] the uniform measure is the correct measure according to practice in dynamical systems theory and is invariant under the dynamics .the tent map ( example 1 ) is a case in point ._ however , for many other mt - dynamical systems with phase space  ] will lead to different typicality measures .+ indeed , pitowsky ( 2012 , 17 - 18 ) gives examples of mt - dynamical systems which highlight these problems .intuitively speaking , _ two - symbol bernoulli systems _ are dynamical systems whose solutions can be put into one - to - one correspondence with the set of all infinite sequences arising from independent trials of tossing a coin ( i.e. , the set ) .. [ bernoullisystem ] that is , is the set of all infinite sequences with corresponding to the possible outcomes of an infinite sequence of independent trials of a ( possibly biased ) coin .let be the set of all subsets of infinite sequences to which probabilities can be assigned , let be the normalized measure on arising from the probability measure of tossing the coin , and let , . is called a _ two - symbol bernoulli shift_. finally , _ two - symbol bernoulli systems _ are defined to be mt - dynamical systems which are isomorphic ( from a measure - theoretic perspective ) to two - symbol bernoulli shifts ( cf .werndl 2009a ) . ] because the solutions of two - symbol bernoulli systems correspond to the set of all infinite sequences , pitowsky argues that _ two - symbol bernoulli systems and their measures illustrate the correctness of his justification ._ however , contrary to pitowsky , the measure of many two - symbol bernoulli systems with phase space  ] can be reduced to the case where the phase space is  ] with the uniform measure on it .i do not think that pitowskys appeal to this isomorphism result is successful .pitowsky s concern is to find , given a phase space such as , the correct typicality measure .but then _ this isomorphism result is of no help because it presupposes that there is already a measure defined on s_. if just the phase space is given ( this is the problem of concern ) , then there are usually infinitely many ways of mapping to the unit interval .consequently , even if one assumes that the correct typicality measure on  ] to ( and hence infinitely many different possible measures on such that the resulting measure space is isomorphic to  ] with the evolution equation for all .clearly , for this phase space and evolution equation _ all _ measures fulfill equation ( [ i2 ] ) .thus they are all invariant under the dynamics , but they do not agree on which sets are typical / atypical .hence the condition of invariance alone can not justify typicality measures .+ an _ initial probability distribution _ gives the probability that a system is prepared in a certain microstate ( e.g. , that an experimenter prepares a gas in a certain microstate ) .this paper will assume that these initial probability distributions are _ translation - continuous _, i.e. , that where is the standard euclidean norm in ( i.e. , the standard euclidean metric between the vector and the zero - vector in ) and is the set translated by , i.e. , malament and zabell ( 1980 ) have provided a nice motivation of this condition : they have argued that when one prepares a system in sm , one does not have sufficient accuracy to create any other probability measure than a translation - continuous probability measure .malament and zabell ( 1980 ) have also shown that a probability measure is translation - continuous iff , i.e. , is absolutely continuous w.r.t . the lebesgue measure ( see equation ( [ ac ] ) ) .. yet it is clear that it also holds for any arbitrary interval of .although the phase space of systems in sm are not intervals , malament and zabell ( 1980 ) and vranas ( 1998 ) believe that the equivalence of translation - continuity and absolute continuity also holds for the phase spaces in sm , and i follow them in assuming this . ]that seems to be often endorsed in the physics community ( cf .leeds 1989 ; maudlin 2007 ) .+ now , different initial probability distributions might arise in sm : systems can be prepared in different macrostates , different scientists might prepare systems differently .thus there is a _class of initial probability distributions of interest_. open sets are extended regions of phase space .this motivates the only additional requirement on this class , namely that for any arbitrary open region of phase space one can not exclude that there might be some way of preparing the system such that there is a positive probability that one ends up in this region .so : + * premise 2 .the initial probability distributions of interest are a class of translation - close probability distributions where for every open set there is a with .* + davey ( 2008 ) , hemmo and shenker ( 2012 ) and leeds ( 1989 ) have criticised the extant foundational literature on boltzmannian sm for assuming that the initial probability distribution is and has to be invariant under the dynamics .they rightly argue that the initial probability distribution can be different from and may well not be invariant under the dynamics .for instance , hemmo and shenker ( 2012 , 11 ) complain : `` in particular the measure need not be the lebesgue measure , and may not even be conserved under the dynamics . ''so _ the great flexibility on the initial probability distribution in sm allowed by premise 2 should be welcome_. + concerning dynamical systems theory , note that in certain applications initial probability distributions over the states play a role .the justification of typicality measures advanced here will be relevant whenever premise 2 holds .for instance , consider the dynamical system of a frictionless _ pendulum _ , consisting of a mass that experiences a single force which depends only on the position of the mass and a constant ( this is the harmonic oscillator ) .suppose that one does not have sufficient accuracy to create any other initial probability distribution than a translation - continuous one ( which has some plausibility ) .further , since in experiments one has the freedom to prepare the system in all kinds of different initial positions and initial velocities , it seems reasonable that for any arbitrary open region of phase space there is a probability distribution of interest which assigns positive probability to this region .then the uniform measure of the pendulum can be justified as typicality measure .+ the next two premises are motivated by the idea that _ typicality measures should be related to the initial probability distributions of interest_. more specifically , first , it seems reasonable to require that if a set of states has probability zero for all initial probabilities of interest , the set is atypical : + * premise 3 . if p(a)=0 for all probability distributions of interest for some measurable a , then t(a)=0 . * + ( equivalent formulation : if for all probability distributions of interest for some measurable , then ) .+ second , the typicality measure should serve as a shortcut to make claims about the likely behaviour of the system for any possible initial probability distribution .that is , it will be required that whenever a set of states is typical , it has probability one for all initial probability distributions of interest : + * premise 4 . if for some measurable , then for all probability distributions of interest .* + ( equivalent formulation : if for some measurable , then for all probability distributions of interest ) .+ also , the following premise is adopted : + * premise 5 . whenever a measure fulfills premises 1 , 3 and 4 ( for the probability distributions as characterised by premise 2 ) , then it is a typicality measure*. + it will be shown that measures satisfying premises 1 , 3 and 4 agree on which sets are typical or there is a unique measure satisfying these premises .hence it seems reasonable to take these conditions as sufficient for typicality measures .+ finally , let me specify the systems under consideration : + * premise 6 .let a statistical mechanical system be given , or let a dynamical system be given where is an interval in , and .* + for many standardly used measures in measure - theoretic dynamical systems theory and .for instance , it is clear that this condition holds for the measure of the tent map ( example 1 ) and the measure of the logistic map ( example 2 ) .+ the following result is derivable from these premises ( for the proof see appendix 1 ) : + * conclusion ( theorem 1 ) .the measure of systems in statistical mechanics / the measure of dynamical systems is a typicality measure .furthermore , any other typicality measure will agree with / on which sets are typical . *+ this result aims to justify typicality measures in sm and typicality measures of dynamical systems such as the tent map ( example 1 ) or the logistic map ( example 2 ) . and are typicality measures. they may not be the unique typicality measures , but this does not matter because any other typicality measure agrees on which sets are typical. note that in my argument typicality is different from probability : the typicality measures do not refer to probabilities in the philosophical sense ( they do not refer to , e.g. , ontic probabilities describing the distributions which arise in experiments or to degrees of belief ) . for a formal definition ) .however , just because they are formally probability measures , this does not mean that they refer to probabilities in the philosophical sense . to give an example : the lebesgue measure on [ 0,1 ] is formally a probability measure , but it is often interpreted as `` length '' , which is not probabilistic in the philosophical sense . similarly , typicality measures which count states do not refer to probabilities .when one says , e.g. , that 999/1000 of the balls are red , this statement does not refer to probabilities in the philosophical sense . ] + let me now present a _cost - benefit analysis _ of the argument .as outlined when introducing premise , it can be motivated that the initial probability distributions in sm are translation - continuous .also , it seems plausible that for any arbitrary open region of phase space one can not exclude that it might be possible to prepare the system in such a way that there is a positive probability of ending up in this region .still , these two requirements are not incontestable because the knowledge about these initial probability distributions is limited .premise 5 ( that measures fulfilling premises 1 , 3 and 4 are typicality measures ) seems reasonable because all such typicality measures agree on which sets are typical .still , some might think that these premises are only necessary and not sufficient for typicality measures .thus they might want to add further requirements ( in this case my argument is still relevant : it shows that any typicality measure fulfilling these additional requirements will agree with / about judgement of typicality ) .+ concerning the benefits , first , there is a conceptual gain from knowing that the choice of the standard typicality measures can be motivated from premises - .second , there is an empirical gain because the typicality measures are related to the initial probability distributions of interest ( premises 3 and 4 ) . in particular , claims about typical behaviour translate into claims about the likely behaviour of the system for any initial probability distribution of interest .without such a connection , inevitably the question would arise how typicality measures are connected to experiments .third , according to my argument , a wide range of initial probability distributions are allowed ( see premise 2 ) .this is a strength because , as argued above , different probability distributions may arise in different contexts , and my argument is consistent with this ( this is another empirical gain ) .particularly the empirical gains might make the argument also attractive for physicists .let me now outline the analogous argument for the less stringent notion of typicality ii .the first premise remains the same : + * premise 1 .typicality measures are invariant under the dynamics . *+ recall that the preparation of a system in sm is described by an initial probability distribution . herewe assume that these initial probability distributions are -_translation - close _, i.e. , that for all open sets in : where and are very small real numbers . the condition of translation - closeness can be motivated in a similar fashion as translation - continuity .that is , when one prepares a system in sm , one does not have sufficient accuracy to create any other probability measure than a translation - close probability measure ., where is a phase cell and or , will often not be a phase cell again ( cf . werndl 2009a ) .however , the definition of mt - dynamical systems requires that is an element of the phase space for all phase cells and all ( because otherwise the are not functions from the phase space to the phase space ) .also , because and may not be phase cells again , the conditions of invariance ( equations ( [ i2 ] ) and ( [ i1 ] ) ) cease to be applicable .furthermore , will usually not be an element of the coarse - grained phase space , implying that the notion of translation - continuity ceases to be applicable .problems arise even if these technical difficulties are set aside : vranas claims that as the measurement precision increases , the maximum measure of the size of a phase cell will become smaller . from thishe concludes that sufficiently small displacements of sets only lead to a very small change of the coarse - grained measure ( implying translation - closeness ) .however , as van lith ( 2001 ) has pointed out , this conclusion is unwarranted because the decreasing cell size is accompanied by a compensating increase of the number of cells that are added or deleted because of the displacement ( hence it is unclear whether the total size of the added or deleted phase cells will become smaller ) . ]vranas ( 1998 ) has shown a result analogous to malament and zabell s equivalence of the conditions of absolute continuity and translation - closeness . to state it ,i first need to introduce a definition .the measure _ is -continuous w.r.t .the measure _ ( where and are both defined on ) iff : the result shown by vranas is that if is -translation - close , is -continuous w.r.t .the lebesgue measure for and , where is the volume of the -sphere with radius . , but it is clear from the proof that it also holds for any arbitrary interval of . although the phase spaces in sm are not intervals , i follow malament and zabell ( 1980 ) and vranas ( 1998 ) in assuming that this result also holds for systems in sm . ]+ as before , different initial probability distributions might arise in sm , and thus there is a _class of initial probability distributions of interest_. open sets ( and especially open sets whose lebesgue measure is not very small ) are extended regions of phase space .this motivates the only additional requirement on this class , namely that for an open region which is not very small , one can not exclude that there might be some way of preparing the system such that the probability of ending up in this region is not very small . formally , if for an open set for a very small , then includes an initial probability distribution with .this condition will be required to hold for and .so : + * premise 2 .the initial probability distribution of interest are a class of translation - close probability distributions where if for an open set , then there is a with ( where or ) . * + note that _ this great flexibility on the initial probability distribution in sm allowed by premise 2 should be welcome_. + concerning dynamical systems theory , also in certain applications initial probability distributions over the states play a role .the justification of typicality measures advanced here is relevant whenever premise 2 holds .for instance , consider again the dynamical system of a frictionless _ pendulum _ , consisting of a mass that experiences a single force which depends only on the position of the mass and a constant .suppose that one does not have sufficient accuracy to create any other initial probability distribution than a translation - close one ( which has some plausibility ) .further , since in experiments one has the freedom to prepare the system in all kinds of different initial positions and initial velocities , it seems plausible that for an open region of phase space that is not very small , there is an initial probability distribution which assigns a non - negligible probability to this region .then the uniform measure can be justified as typicality measure .+ the next two premises _ relate typicality to the initial probability distributions of interest_. first , it seems reasonable to require that if a set of states has very small probability for all initial probabilities of interest , then the set is atypical : + * premise 3 . if for all probability distributions of interest for some measurable , then ( for very small ; ) . *+ second , because the typicality measure should serve as a shortcut , it is required that if a set of states is typical , it is of very high probability for all initial probability distributions of interest : + * premise 4 . if for some measurable , then for all probability distributions of interest ( for very small ; ) . * + as before ,the following assumption is made : + * premise 5 .whenever a measure fulfills premises 1 , 3 and 4 ( for the probability distributions as characterized by premise 2 ) , it is a typicality measure*. + it will later be shown that measures satisfying premises 1 , 3 and 4 can be interchangeably used as typicality measures .hence it seems reasonable to take these conditions as sufficient for typicality measures .+ let me specify the systems under consideration : + * premise 6 .let a statistical mechanical system be given , or let a dynamical system be given where is an interval in , and .* + for many of the standard measures in measure - theoretic dynamical systems theory and for very small and .for instance , this trivially holds for the measure of the tent map ( example 1 ) , and it is easy to see that it holds for the measure of the logistic map ( example 2 ) . for an arbitrary .then , where .hence .conversely , let for an arbitrary .then , where .thus . ]+ finally , i also adopt the following premise : + * premise 7 .assume that and ( for very small , ) and that and both satisfy premises 1 , 3 and 4 .then , pragmatically speaking , and can be interchangeably used as typicality measures . * + this can be motivated as follows .the antecedent of premise 7 implies that claims about typicality in terms of are translatable into claims about typicality in terms of , and vice versa : if a set is typical with respect to , i.e.  , is typical w.r.t . because .conversely , if a set is typical with respect to , i.e.  , is typical w.r.t . because . that such measures can , pragmatically speaking , be both used as typicality measures seems to be often endorsed by proponents of the typicality approach , e.g. , goldstein ( 2001 ) or maudlin ( 2007 ) .+ the following result is derivable from these premises : + * conclusion ( theorem 2 ) .the measure of systems in statistical mechanics / the measure of dynamical systems is a typicality measure .furthermore , for any other typicality measure , pragmatically speaking , and / and can be interchangeably used as typicality measures . *+ this result aims to justify typicality measures in sm and typicality measures of dynamical systems such as the tent map ( example 1 ) or the logistic map ( example 2 ) . and are typicality measures. they may not be the unique typicality measures , but this does not matter because any other typicality measure can be interchangeably used as typicality measure .note that , again , the typicality measures do not refer to probabilities in the philosophical sense .+ the cost - benefit analysis is similar as for typicality i. concerning premise 2 , it seems plausible that the initial probability distributions are translation - close and that for an open region which is not very small , one can not exclude that there might be some way of preparing the system such that the probability of ending up in this region is not very small . still , these two requirements are not incontestable because the knowledge about these initial probability distributions is limited .premise 5 seems reasonable because it follows that all such typicality measures can , pragmatically speaking , be interchangeably used as typicality measures .still , some might want to add further requirements for something to count as typicality measure ( then my argument is still relevant : it shows that measures fulfilling these additional requirements can , pragmatically speaking , be interchangeably used with / as typicality measures ) .the motivation of premise is that claims about typicality in terms of are translatable into claims about typicality in terms of , and vice versa .still , and can disagree on the size assigned to sets , and some might not want to allow this . + regarding the benefits , first , there is a conceptual gain from knowing that the choice of the standard typicality measures follows from premises - .second , there is an empirical gain because the typicality measures are related to the initial probability distributions of interest ( premises and ) .third , it is a strength that a wide range of probability distributions are allowed ( see premise 2 ) , and this is another empirical gain .these empirical gains might make the argument attractive for physicists .in this section i show that if a further assumption is added to the argument , then the typicality measure is unique ( typicality i)/unique from a pragmatic perspective ( typicality ii ) .this further assumption is the dynamical condition of ergodicity ( typicality i)/epsilon - ergodicity ( typicality ii ) .a mt - dynamical system is _ ergodic _ iff there is no measurable set in , , such that for all or .a mt - dynamical system is epsilon - ergodic iff it is ergodic on a set of measure , for a very small . is negligible compared to the measure of any of the macroregions , i.e. , that , for a very small .[complication ] ] formally : is _ -ergodic _ , where is a real number , iff there is a measurable set , , with for all or such that the mt - dynamical system is ergodic where , for any set in , and is restricted to . is _ epsilon - ergodic _ iff it is -ergodic for a very small .+ to state the uniqueness results , one more definition is needed .the measures and both defined on are _ -close _ , where is a very small real number , iff : latexmath:[\[\label{eclose }  note that for -close measures and , and for any .hence , pragmatically speaking , they can be interchangeably used as typicality measures ( cf .premise 7 ) . assuming that one does not care about differences of typicality measures of size , -close measures are _ identical from a pragmatic perspective_. + * theorem 3 .suppose that premises 1 - 6 hold and that the statistical mechanical system /the dynamical system is ergodic .then / is the unique typicality measure . *+ for a proof of theorem 3 , see appendix 3 . + * theorem 4 .suppose that premises 1 - 7 hold and that the statistical mechanical system /the dynamical system is -ergodic .then any other typicality measure is -close to / , where . *+ for a proof of theorem 4 , see appendix 4 .there are many dynamical systems , including the tent map ( example 1 ) and the logistic map ( example 2 ) and generally all chaotic systems , which are ergodic / epsilon - ergodic ( cf .aligood , sauer and yorke 1996 ; werndl 2009b ) .hence for these systems theorem 3/theorem 4 shows that the typicality measure is unique / unique from a pragmatic perspective .+ in sm ergodicity and epsilon - ergodicity have a long and notorious history .boltzmann already invoked the notion of ergodicity in some of his arguments ( uffink 2007 ) . in the contemporary literature on the foundations of sm several papers have defended the claim , or have assumed in their arguments , that systems in sm are ergodic or epsilon - ergodic ( e.g. , frigg and werndl 2012 ; malament and zabell 1980 ; pitowsky 2012 ; vranas 1998 ) .one of the most important mathematical results so far about ergodicity is the proof of the _ boltzmann - sinai hypothesis _ that the motion of hard - spheres on the two or three - dimensional torus is ergodic ( simnyi 2010 ) .the relevant mathematics is so difficult that for more realistic systems than hard spheres the knowledge is limited and largely based on numerical simulations . because of this , some argue that one simply does not know yet whether more realistic systems in sm are ergodic or epsilon - ergodic ( cf .uffink 2007 ) . in this paperno commitment to ergodicity or epsilon - ergodicity is needed .here it is just important that for systems such as hard spheres theorem 3/theorem 4 demonstrates that the typicality measure is unique / unique from a pragmatic perspective .moreover , if some more realistic systems in sm turn out to be ergodic / epsilon - ergodic ( which is certainly possible ) , theorem 3/theorem 4 shows that the typicality measure is unique / unique from a pragmatic perspective .+ as a side remark , it should be noted that some have argued that the condition of ergodicity or epsilon - ergodicity guarantees thermodynamic - like behaviour ( cf .frigg and werndl 2011 ) .first , consider ergodicity , which is equivalent to the condition that the portion of time an arbitrary solution stays in equals the measure of .formally : for all initial conditions except , perhaps , a set with , where ( here is the characteristic function for and otherwise . ] of ) .consider an initial condition .then the dynamics will carry to and will keep it there most of the time .the system will move out of the equilibrium region every now and then and visit non - equilibrium regions . yetsince these regions are small compared to the equilibrium region , it will only spend a small fraction of time there .therefore , the entropy is close to its maximum most of the time and fluctuates away from it only occasionally .hence if is interpreted as probability / typicality measure , thermodynamic - like behaviour is of probability /typical ( typicality i ) .concerning -ergodic systems , note that such a system is ergodic on .consequently , it shows thermodynamic - like behaviour for the initial conditions in .then , by the same moves as explained above for ergodicity , one finds that thermodynamic - like behaviour is of probability /typical ( typicality ii ) .a popular view in contemporary boltzmannian statistical mechanics is to interpret the measures as typicality measures , i.e. as representing the relative size of sets of states . in measure - theoretic dynamical systems theory measurescan similarly be interpreted as typicality measures . however , a justification why these measures are a good choice of typicality measures is missing , and the paper attempted to contribute to fill this gap . + the paper first criticised pitowsky ( 2012 ) the only justification of typicality measures known to the author .pitowsky s argument goes as follows .consider the set of all infinite sequences of zeros and ones . by approximation with the measures defined on the sets of finite sequences of zeros and ones , a unique measure obtained on .let be the map which assigns to each infinite sequence of zeros and ones the number in the unit interval whose binary development is .when is used to map the measure on to a measure on the unit interval , one obtains the uniform measure . hence , pitowsky concludes , the uniform measure is the uniquely correct typicality measure .this paper argued that pitowsky s argument is problematic .it is unclear why and not another function is used to map the measure to the unit interval .furthermore , there are counterexamples because for many systems on the unit interval the uniform measure is not the standard measure used and is not invariant . + the paper then provided a first tentative proposal of how to justify typicality measures for two notions , namely typicality i ( atypical means measure zero ) and typicality ii ( atypical means very small measure ) .the main premises of the argument are as follows .the initial probability distributions of interest are translation - continuous ( for typicality i ) or translation - close ( for typicality ii ) .a typicality measure should be related to these initial probability distributions in two ways : first , if a set is of probability zero ( for typicality i ) or of very small probability ( for typicality ii ) for all probability distributions , then it is atypical .second , if a set is typical , it is of probability one ( for typicality i ) or of very high probability ( for typicality ii ) for all probability distributions .furthermore , typicality measures should be invariant .the conclusion are two theorems which show that the standard measures of statistical mechanics and dynamical systems theory are typicality measures . there may be other typicality measures , but these agree on which sets are typical ( for typicality i ) or can be interchangeably used as typicality measures ( for typicality ii ) .finally , two theorems were presented , showing that if systems are ergodic ( for typicality i ) or epsilon - ergodic ( for typicality ii ) , the typicality measure is unique ( for typicality i ) or unique from a pragmatic perspective ( for typicality ii ) .first of all , consider : later in the proof the result is needed that condition ( [ raus ] ) is implied by the condition that if is an open set , there is a with .this follows because if , contains an open subset with ( since is regular ) .second , note that is a dynamical system . is defined as : hence it follows ( from the radon - nikodym theorem ) that .let , ( the minimum exists because is compact ) . then if , , and hence . therefore , also .thus it suffices to focus on the case where a dynamical system is given where and .+ the definition of a dynamical system requires that is invariant , and hence satisfies premise 1 .next , assume that for all probability distributions of interest for some measurable . then also ( from premise 2 and condition ( [ raus ] ) ) . because ( from premise 6 ) , .consequently , satisfies premise 3 .note that premise 4 is equivalent to the claim that for all probability distributions of interest . because ( from premise 6 ) , for all probability distributions of interest ( from premise 2 ) .thus satisfies premise 4 . because satisfies premises 1 , 3 and 4 , it is a typicality measure ( from premise 5 ) .+ let be another typicality measure. then ( from premises 2 and 4 and condition ( [ raus ] ) ) .if for some measurable , then for all probability distributions of interest ( from premise 2 ) , and if for all probability distributions of interest , ( from premise 3 ) .hence .therefore , iff . because iff ( from premise 6 ) , iff .hence any other typicality measure will agree with about which sets are typical .first of all , consider : later in the proof the result is needed that condition ( [ raus2 ] ) is implied by the condition that if for an open set , then includes an initial probability distribution with .this follows because if , then there is an open subset of with ( since is regular ) .second , note that is a dynamical system .vranas ( 1998 ) has already shown that is -continuous w.r.t . for any and for ( the maximum exists because is compact ) .let , . then from equation ( [ mm ] ) it follows that for ( the maximum exists because is compact ) .hence is -continuous w.r.t.  .thus it suffices to focus on the case where a dynamical system is given with and .+ premise 1 holds because for a dynamical system is invariant .next , assume that for all probability distributions of interest for some measurable .then ( from premises 2 and condition ( [ raus2 ] ) ) . hence ( from premise 6 ) , and premise 3 .premise 4 is equivalent to the claim that for all probability distribution of interest .suppose that .then ( from premise 6 ) and ( from premise 2 ) .therefore , satisfies premise 4 . because satisfies premises 1 , 3 and 4 , it is a typicality measure ( from premise 5 .+ let be another typicality measure . if , then ( from premise 2 and 4 and equation ( [ raus2 ] ) ) .thus , ( from premise 6 ) .conversely , if , ( from premise 6 ) .hence for all probability distributions of interest ( from premise 2 ) , and ( from premise 3 ) .consequently , pragmatically speaking , and can be interchangeably used as typicality measures .   as for the reasons givenwhen proofing theorem 1 , it suffices to focus on the case where a dynamical system is given where and . according to a theorem in ergodic theory , if is ergodic and is a measure invariant under the dynamics with , then ( cf .cornfeld et al .any other typicality measure is invariant ( from premise 1 ) .also , ( from premises 2 and 3 ) , ( from premise 6 ) , and hence .consequently , .   as for the reasons given whenproofing theorem 2 , it suffices to focus on the case where a dynamical system is given where and . according to a theorem by vranas ( 1998 ) , if is -ergodic, is invariant and , then and are -close with .let be any other typicality measure .then is invariant ( from premise ) .also , ( from premise ) , ( from premises 2 and 3 ) , and hence .consequently , and are -close .aligood , k. , sauer , t. and yorke , j. ( 1996 ) , _ chaos : an introduction to dynamical systems _ , new york : springer .frigg , r. ( 2008 ) , a field guide to recent work on the foundations of statistical mechanics , _ in _d. rickles , ed ., ` the ashgate companion to contemporary philosophy of physics ' , ashgate , london , pp .99196 .goldstein , s. ( 2001 ) , boltzmann s approach to statistical mechanics , _ in _j. bricmont , d. drr , m. galavotti , g. ghirardi , f. pettrucione and n. zanghi , eds , ` chance in physics : foundations and perspectives ' , springer , berlin and new york , pp .3954 .pitowsky , i. ( 2012 ) , typicality and the role of the lebesgue measure in statistical mechanics , _ in _ j. ben - menahem and m. hemmo , eds , `probability in physics ' , the frontiers collection , springer , berlin and new york , pp . 5158 .

Abstract: a popular view in contemporary boltzmannian statistical mechanics is to interpret the measures as typicality measures . in measure - theoretic dynamical systems theory measures can similarly be interpreted as typicality measures . however , a justification why these measures are a good choice of typicality measures is missing , and the paper attempts to fill this gap . the paper first argues that pitowsky s ( 2012 ) justification of typicality measures does not fit the bill . then a first proposal of how to justify typicality measures is presented . the main premises are that typicality measures are invariant and are related to the initial probability distribution of interest ( which are translation - continuous or translation - close ) . the conclusion are two theorems which show that the standard measures of statistical mechanics and dynamical systems are typicality measures . there may be other typicality measures , but they agree about judgements of typicality . finally , it is proven that if systems are ergodic or epsilon - ergodic , there are uniqueness results about typicality measures .


networks abound in various fields ; a network is a collection of nodes and links , where a link connects a pair of nodes .most real - world networks are not entirely regular or random and have prominent properties as modeled by , for example , small - world , scale - free , hierarchical , and modular networks . in such networks ,some nodes are considered to be more important than the others .depending on the definition of importance , various centrality measures , which quantify the relative importance of different nodes , have been proposed .the most frequently used centrality measures are perhaps the degree ( i.e. , the number of links owned by a node ) and the betweenness ( i.e. , the normalized number of shortest paths connecting any pair of nodes passing through the node in question ) .new centrality measures have also been proposed in the field of complex networks .although many centrality measures are available , very few of these describe the importance of nodes in collective behavior of nodes on networks ( see ) . in a previous study, we proposed a centrality measure called the influence .the influence of a node denotes its importance in different types of dynamics .it represents the amplitude of the response of a synchronized network when an input is given to a certain node , the fixation probability for a newly introduced type ( e.g. , new information ) at a node in voter - type evolutionary dynamics , the stationary density of a simple random walk in continuous time , the so - called reproductive value of a node , and the influence of a node in the degroot s model of consensus formation .it makes sense to consider the influence only in directed networks ; in undirected networks , the influences of all the nodes take an identical value . in principle , the influence as a centrality measure is close to the pagerank , which was originally developed for ranking websites .  to assess the influence ( and also the pagerank ) in real complex networks, it is not sufficient to take into account the local property of the node , such as the degree .the global structure of networks such as the small - world property , modular structure , and self - similarity generally affects the influence values .  in the present study, we analytically determine the influence of nodes in model networks such as weighted chain , directed multipartite networks with a hierarchical structure , and a directed version of the watts - strogatz small - world network . for this purpose, we exploit the symmetry in networks and the relationship between the enumeration of directed spanning trees and the influence .we reveal the discrepancy between the actual influence and that predicted by the mean - field approximation ( ma ) , which takes into account only the degree .in fact , the nodes that occupy globally important positions in terms of influence are generally different from those that are locally important .the globally important nodes govern the above - mentioned dynamics on networks . finally , to demonstrate the application of the influence as a centrality measure, we analyze a system of coupled oscillators and show that nodes with large influence values entrain other nodes relatively easily , i.e. , with a relatively small coupling strength .consider a directed and weighted network having nodes .the weight of the directed link from node to node is denoted by .we set when the link is absent .the influence of node is denoted by .we define as the solution for the following set of linear equations : where is the indegree of node , and the normalization is given by .when node has many outgoing links , can be large because there are many terms on the right - hand side of eq . .when node has many incoming links , node is interpreted to be governed by many nodes .then , can be small because of the divisive factor in eq . .the rationale for the definition given in eq .is that represents the importance of nodes in different types of dynamics on networks , as explained in sec .[ sec : introduction ] .values of for two example networks are shown in fig .[ fig : example nets ] .  note that for any network with ( ) , where is the outdegree . the undirected network is included in this class of networks .therefore , the influence has a nontrivial meaning only in directed networks .this situation also holds true in the case of the pagerank ; in undirected networks , the pagerank is a linear function of the degree of node .the ma of is given by where . in secs .[ sec : examples ] and [ sec : kcr ] , we argue that eq .does not satisfactorily describe in certain practically important types of networks , including the watts - strogatz small - world network .for these networks , we calculate the exact by using different methods .  the value of can be associated with the number of directed spanning trees rooted at node , as described below .equation  implies that is the left eigenvector of the laplacian matrix , whose ( ) element is equal to and ( ) .the corresponding eigenvalue is equal to 0 ; ( ) .the cofactor of is given as where is an matrix obtained by deleting the row and the column of . because ( ) , is independent of .therefore , considering the fact that has eigenvalue 0 , we obtain equation  indicates that  ] to , where is the duration of a run .the first 80% of a run is discarded as transient .the ratio represents the average phase shift of the oscillators , other than the pacemaker , relative to that of the pacemaker .if this value is more than 0.99 , we consider the entrainment to be achieved . because the transient is shorter for larger , we set .the values of when the pacemaker is located at different nodes are plotted against in fig .[ fig : kcr](a ) .the line in the figure represents , i.e. , eq . .the numerically obtained roughly matches the theoretical one although the condition is violated . the same values of are plotted against the ma estimate in fig . [ fig : kcr](b ) .we find a larger spread of data in this plot as compared to that in fig .[ fig : kcr](a ) ; predicts more accurately than the ma .next , we set the weight of the parallel link to unity , as done in .the values of for this version of the directionally biased random network are shown in figs .[ fig : kcr](c ) and [ fig : kcr](d ) .the results are qualitatively the same as those shown in figs .[ fig : kcr](a ) and [ fig : kcr](b ) .the dependence of on is weak in the new network [ fig .[ fig : kcr](c ) ] as compared to the previous network [ fig .[ fig : kcr](a ) ] mainly for the following reason . because of the difference in the weights of parallel links , in the new network is larger than that in the previous network .then , is smaller for the new network , since it is inversely proportional to the effective link weight .we expect that for nodes in intermediate layers can also be explained using the same approach .the result is derived for the directionally biased random network .although it is not guaranteed that this relationship holds true in other types of networks , we test the applicability of the relation in a scale - free network and a neural network .  we generate a directed scale - free network using the configuration model .the degree distributions are independently given for and by and , respectively .we set and .the minimum degree is set to 3 .the duration of a run and the length of the traisient are equal to those in the case of the directionally biased random networks .the values of are plotted against and the ma in figs .[ fig : kcr](e ) and [ fig : kcr](f ) , respectively .the relation fits the data reasonably well , even though the scale - free network is not a directionally biased random network .in contrast , poorly predicts , as shown in fig .[ fig : kcr](f ) .we next examine the _ c. elegans _ neural network based on chemical synapses , which serve as directed links .the network has the largest strongly connected component with nodes .the number of synapses from neuron to neuron defines .we set to appropriately exclude the transient .the values of are plotted against and the ma in figs .[ fig : kcr](g ) and [ fig : kcr](h ) , respectively .the values of exceeding are not plotted because direct numerical simulations need too much time .similar to the case of scale - free networks , predicts better than does .in this paper , we have analyzed the centrality measure called the influence in various networks .the influence extracts the magnitude with which a node controls or impacts the entire network along directed links .we have analytically shown that the source of the shortcut in a directed version of the watts - strogatz small - world network and the root node in hierarchical networks have large influence values .this is not accurately predicted if we approximate the influence of a node by its degree .although by definition , the influence is based on local connectivity , the global structure of networks does affect the influence values .we also analyzed the effect of the location of a pacemaker on the capability of entrainment in a system of coupled phase oscillators .the pacemaker located at a node with a large influence value entrains the other oscillators relatively easily .  in the analysis of some model networks , including the watts - strogatz small - world network , we used the method based on the enumeration of directed spanning treesthis method can be applied to the estimation of the pagerank of nodes , because the pagerank can be mapped to the influence if we reverse links and rescale the link weight .application of our results to other centrality measures for directed networks is warranted for future study .s. boccaletti , v. latora , y. moreno , m. chavez , and d .- u .hwang , phys .424 * , 175 ( 2006 ) .l. c. freeman , soc .networks * 1 * , 215 ( 1979 ) . s. wasserman and k. faust , _ social network analysis _ ( cambridge university press , new york , 1994 ) .j. d. noh and h. rieger , phys .* 92 * , 118701 ( 2004 ) ; 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t. nishikawa and a. e. motter , phys .e * 73 * , 065106(r ) ( 2006 ) ; physica d * 224 * , 77 ( 2006 ) ; x. wang , y .- c .lai , and c. h. lai , phys .e * 75 * , 056205 ( 2007 ) .   .a dark node has a large value of .( a ) directed network generated by the configuration model .the indegree and outdegree follow independent power - law distributions with the scaling exponent 2.5 and the minimum degree 2 .( b ) directed watts - strogatz network with three shortcuts .see secs .[ sub : sw ] and [ sec : kcr ] for details of the network models .the networks are visualized by pajek .,title="fig:",width=226 ] .a dark node has a large value of .( a ) directed network generated by the configuration model .the indegree and outdegree follow independent power - law distributions with the scaling exponent 2.5 and the minimum degree 2 .( b ) directed watts - strogatz network with three shortcuts .see secs .[ sub : sw ] and [ sec : kcr ] for details of the network models .the networks are visualized by pajek .,title="fig:",width=226 ]         for a directionally biased random network with .we set [ ( a ) and ( b ) ] and [ ( c ) and ( d ) ] . in (a ) and ( c ) , is plotted against the ma results .the lines represent the ma : . in ( b ) and( d ) , ( circles ) and rescaled ( squares ) averaged over all the nodes in each layer are plotted against layer number .the error bars indicate the standard deviation obtained from of all the nodes in the same layer.,title="fig:",width=226 ]  for a directionally biased random network with .we set [ ( a ) and ( b ) ] and [ ( c ) and ( d ) ] . in ( a ) and ( c ), is plotted against the ma results .the lines represent the ma : . in ( b ) and ( d ) , ( circles ) and rescaled ( squares ) averaged over all the nodes in each layer are plotted against layer number .the error bars indicate the standard deviation obtained from of all the nodes in the same layer.,title="fig:",width=226 ]  for a directionally biased random network with .we set [ ( a ) and ( b ) ] and [ ( c ) and ( d ) ] . in ( a ) and ( c ) , plotted against the ma results .the lines represent the ma : . in ( b ) and ( d ) , ( circles ) and rescaled ( squares ) averaged over all the nodes in each layer are plotted against layer number .the error bars indicate the standard deviation obtained from of all the nodes in the same layer.,title="fig:",width=226 ]  for a directionally biased random network with .we set [ ( a ) and ( b ) ] and [ ( c ) and ( d ) ] . in ( a ) and ( c ) , is plotted against the ma results .the lines represent the ma : . in ( b ) and ( d ) , ( circles ) and rescaled ( squares ) averaged over all the nodes in each layer are plotted against layer number .the error bars indicate the standard deviation obtained from of all the nodes in the same layer.,title="fig:",width=226 ]    in directed small - world networks with .( a ) results for small - world network with three added shortcuts .the thick , medium , and thin lines correspond to the networks in which the mean degree of the underlying cycle is equal to 2 , 4 , and 6 , respectively .( b , c , d ) results for directed small - world network with and 500 rewired shortcuts . in ( a ) and ( c ) , is plotted against the position of the node in the underlying cycle . in ( b ) , is plotted against ; the line represents the ma result . in ( d ) , the relationship between and is shown.,title="fig:",width=226 ]  in directed small - world networks with .( a ) results for small - world network with three added shortcuts .the thick , medium , and thin lines correspond to the networks in which the mean degree of the underlying cycle is equal to 2 , 4 , and 6 , respectively .( b , c , d ) results for directed small - world network with and 500 rewired shortcuts . in ( a ) and( c ) , is plotted against the position of the node in the underlying cycle . in ( b ) , is plotted against ; the line represents the ma result . in ( d ) , the relationship between and is shown.,title="fig:",width=226 ]  in directed small - world networks with .( a ) results for small - world network with three added shortcuts .the thick , medium , and thin lines correspond to the networks in which the mean degree of the underlying cycle is equal to 2 , 4 , and 6 , respectively .( b , c , d ) results for directed small - world network with and 500 rewired shortcuts . in ( a ) and ( c ) , is plotted against the position of the node in the underlying cycle . in ( b ) , is plotted against ; the line represents the ma result . in ( d ) , the relationship between and is shown.,title="fig:",width=226 ]  in directed small - world networks with .( a ) results for small - world network with three added shortcuts .the thick , medium , and thin lines correspond to the networks in which the mean degree of the underlying cycle is equal to 2 , 4 , and 6 , respectively .( b , c , d ) results for directed small - world network with and 500 rewired shortcuts . in ( a ) and( c ) , is plotted against the position of the node in the underlying cycle . in ( b ) , is plotted against ; the line represents the ma result . in ( d ) , the relationship between and is shown.,title="fig:",width=226 ]    and .( a , b ) directionally biased random network with and weight of parallel links equal to .( c , d ) directionally biased random network with and weight of parallel links equal to unity .( e , f ) directed scale - free network with .( g , h ) _ c. elegans _ neural network with .the data are plotted against in ( a , c , e , g ) and against in ( b , d , f , h ) .the lines in ( a , c , e , g ) represent .,title="fig:",width=226 ]  and .( a , b ) directionally biased random network with and weight of parallel links equal to .( c , d ) directionally biased random network with and weight of parallel links equal to unity .( e , f ) directed scale - free network with .( g , h ) _ c. elegans _ neural network with .the data are plotted against in ( a , c , e , g ) and against in ( b , d , f , h ) .the lines in ( a , c , e , g ) represent .,title="fig:",width=226 ]  and .( a , b ) directionally biased random network with and weight of parallel links equal to .( c , d ) directionally biased random network with and weight of parallel links equal to unity .( e , f ) directed scale - free network with .( g , h ) _ c. elegans _ neural network with .the data are plotted against in ( a , c , e , g ) and against in ( b , d , f , h ) .the lines in ( a , c , e , g ) represent .,title="fig:",width=226 ]  and .( a , b ) directionally biased random network with and weight of parallel links equal to .( c , d ) directionally biased random network with and weight of parallel links equal to unity .( e , f ) directed scale - free network with .( g , h ) _ c. elegans _ neural network with .the data are plotted against in ( a , c , e , g ) and against in ( b , d , f , h ) .the lines in ( a , c , e , g ) represent .,title="fig:",width=226 ]  and .( a , b ) directionally biased random network with and weight of parallel links equal to .( c , d ) directionally biased random network with and weight of parallel links equal to unity .( e , f ) directed scale - free network with .( g , h ) _ c. elegans _ neural network with .the data are plotted against in ( a , c , e , g ) and against in ( b , d , f , h ) .the lines in ( a , c , e , g ) represent .,title="fig:",width=226 ]  and .( a , b ) directionally biased random network with and weight of parallel links equal to .( c , d ) directionally biased random network with and weight of parallel links equal to unity .( e , f ) directed scale - free network with .( g , h ) _ c. elegans _ neural network with .the data are plotted against in ( a , c , e , g ) and against in ( b , d , f , h ) .the lines in ( a , c , e , g ) represent .,title="fig:",width=226 ]  and .( a , b ) directionally biased random network with and weight of parallel links equal to .( c , d ) directionally biased random network with and weight of parallel links equal to unity .( e , f ) directed scale - free network with .( g , h ) _ c. elegans _ neural network with .the data are plotted against in ( a , c , e , g ) and against in ( b , d , f , h ) .the lines in ( a , c , e , g ) represent .,title="fig:",width=226 ]  and .( a , b ) directionally biased random network with and weight of parallel links equal to .( c , d ) directionally biased random network with and weight of parallel links equal to unity .( e , f ) directed scale - free network with .( g , h ) _ c. elegans _ neural network with .the data are plotted against in ( a , c , e , g ) and against in ( b , d , f , h ) .the lines in ( a , c , e , g ) represent .,title="fig:",width=226 ]

Abstract: many complex networks are described by directed links ; in such networks , a link represents , for example , the control of one node over the other node or unidirectional information flows . some centrality measures are used to determine the relative importance of nodes specifically in directed networks . we analyze such a centrality measure called the influence . the influence represents the importance of nodes in various dynamics such as synchronization , evolutionary dynamics , random walk , and social dynamics . we analytically calculate the influence in various networks , including directed multipartite networks and a directed version of the watts - strogatz small - world network . the global properties of networks such as hierarchy and position of shortcuts , rather than local properties of the nodes , such as the degree , are shown to be the chief determinants of the influence of nodes in many cases . the developed method is also applicable to the calculation of the pagerank . we also numerically show that in a coupled oscillator system , the threshold for entrainment by a pacemaker is low when the pacemaker is placed on influential nodes . for a type of random network , the analytically derived threshold is approximately equal to the inverse of the influence . we numerically show that this relationship also holds true in a random scale - free network and a neural network .


we study the algorithms that reduce the average complexity of maximum likelihood ( ml ) decoding of convolutional codes . by ml decoding ,we mean the decoder uses code - search to find , and to guarantee the output of , the most likely codeword .forney showed that ml decoding of convolutional codes is equivalent to finding the most probable path taken through a markov graph .denote the codeword length by and the coding memory by . for each time index ,the number of markov states in the markov graph is exponential in .the total number of markov states is therefore exponential in but linear in .define the complexity of a decoder as the number of visited markov states normalized by the codeword length .practical ml decoding is often achieved using the viterbi algorithm ( va ) , whose complexity does not scale in but scales exponentially in .well known decoders such as the list decoders , the sequential decoders , and the iterative decoders are able to achieve near optimal error performance with low average complexity .however , these decoders do not guarantee the output of the ml codeword .if obtaining the ml codeword is strictly enforced ( see section [ discussions ] for justification ) , to avoid exhaustive path search , the decoder must develop certain criterion or bound that can be used to disprove the optimality of a markov path set .this is equivalent to developing an optimality test criterion ( otc ) to test whether the ml path ( or codeword ) belongs to the complementary path set ( or codeword set ) .two major otcs have been used in the ml decoding of convolutional codes .the first one is the path covering criterion " ( pcc ) ( explained in and in appendix [ pcc ] ) used in the va .va visits _ all _ markov states in chronological order . for each time index ,the decoder maintains a set of cover " ( defined in appendix [ pcc ] ) markov paths each passing one of the markov states .according to the pcc , the cover " markov path passing a markov state disproves the optimality of all other markov paths passing the same state .the second otc is the sum log likelihood ( sll)-based otcs used extensively in the sphere decoder .sphere decoder models ml decoding as finding the lattice point closest to the channel output in the signal space .hence the distance between the channel output and an arbitrary lattice point upper bounds the distance from the channel output to the ml codeword .such distance bound is based on the sll of the corresponding codeword , and is used in the sphere decoder as well as other ml decoders as the key means to avoid exhaustive codeword search . in ,vikalo and hassibi showed that pcc - based and sll - based optimality tests can be combined to find the ml codeword without visiting all markov states .assume pcc - based optimality test is always implemented .in this paper , we first show that _ additional _ complexity reduction brought by the sll - based optimality test diminishes as one fixes the coding memory and takes the codeword length to infinity .such inefficiency is due to the fact that sll - based otc does not exploit the structure of the convolutional code . searching the ml codeword is equivalent to finding the ml source message , which contains a sequence of source symbols .we show whether the ml message contains a particular symbol at a given time index can be tested using an otc that depends only on the log likelihood of channel output symbols in a _ fixed - sized _ time neighborhood .we call such test the neighboring log likelihood ( nll)-based optimality test , and show its efficiency does not depend on the codeword length .we theoretically demonstrate that nll - based optimality test can bring significant complexity reduction to ml decoding when the communication system has a high signal to noise ratio ( snr ) .complexity of the decoder using sll - base optimality test , on the other hand , remains the same as the va for all snr if the codeword length is taken to infinity .the results are also generalized to ml sequence detection in a class of discrete - time hidden markov systems .let be an convolutional code over defined by a polynomial generater matrix  , +\mbox{\boldmath  }[1]d+ \dots + \mbox{\boldmath  }[\nu-1]d^{\nu-1},\ ] ] where is the delay operator ; is the coding memory ;  x ] , , are row vectors of dimension over . the encoded message , or the corresponding codeword , is given by \mbox{\boldmath  }[l]d^d.\ ] ] to simplify the presentation , we assume time index takes all integer values .we assume =\mbox{\boldmath  } n ] is the noise sequence with \sim n(\mbox{\boldmath  } , \sigma^2\mbox{\boldmath  } ) y x y n ] be a _partial message _ defined only at time indices in .denote by the set of messages satisfying =\tilde{\mbox{\boldmath  }}[d ] , \forall d\in d_d^x , \mbox{\boldmath  }_0(d)\ne \mbox{\boldmath  }(d ) \}.\ ] ] suppose the decoder wants to test whether it can disprove the optimality of , i.e. , whether . a common practice is to find a lower bound , denoted by , of the negative slls of the messages in . if the lower bound is larger than obtained in ( [ sll ] ) , then we have for all , which means the ml message is not in .  in appendix[ snllowerbound ] , we show that the sll lower bounds appeared in the literature satisfy the following assumption .[ assumption1 ] given , let be the maximum time index set , over which we can find a _partial codeword _ such that for all with , we have =\tilde{\mbox{\boldmath  }}[d] ] , whose value does _ not _ depend on , such that -g_q\left(\tilde{\mbox{\boldmath  }}[d]\right)\right\|^2  + ( n+\nu-|d_d^y|)(1-\epsilon)n\sigma^2.\end{aligned}\ ] ]   as demonstrated in , if we fix , using as the otc to disprove the optimality of message set can bring significant complexity reduction to ml decoding , especially under high snr .however , if we define as the subset of time indices corresponding to the erroneous codeword symbols , i.e. , the following proposition shows that sll - based optimality tests become inefficient if is taken to infinity while is kept finite .[ proposition1 ] assume the generater matrix is fixed , and therefore the constraint length is fixed .consider message sets characterized by for arbitrary but under the constraint of a fixed , where is defined in ( [ de ] ) and the derivation of is specified in assumption [ assumption1 ] .if we fix snr and take to infinity , we have if we first take to infinity and then take snr to infinity , we have   since , taking to infinity implies taking to infinity .according to assumption [ assumption1 ] , we have -g_q\left(\tilde{\mbox{\boldmath  }}[d]\right)\right\|^2 \right )  + ( 1-\epsilon)n\sigma^2  \nonumber \\ & & - \frac{1}{n+\nu-|d_d^y|}\left(\sum_{d\in d_e}\|\mbox{\boldmath  }[d]\|^2 \right )  -\frac{1}{n+\nu-|d_d^y|}\sum_{d\not\in d_d^y}\|\mbox{\boldmath  }[d]\|^2 . \label{sllbound2}\end{aligned}\ ] ]  since  ] are i.i.d . with mean and variance . therefore \|^2\to n\sigma^2 y ] , and \|^2\to 0 y y r y x ] , .  we skip the proof of theorem [ theorem1 ] since the result is implied by theorem [ theorem3 ] presented in section [ hmm ] .note that the values of and only depend on the function .hence , as long as and are given , the values of and can be fixed , e.g. , and .given , the optimality test presented in theorem [ theorem1 ] testifies the optimality of | \tilde{m}\in [ m , m+\nu ) \} y n ] are i.i.d . , whose mean , , and variance , , converge to as snr goes to infinity .lemma [ proposition3 ] implies , if there is a suboptimal decoder whose probability of _ symbol _ detection error ( as opposed to sequence detection error ) is low under high snr , then nll - based optimality tests can help transforming the suboptimal detector to an ml detector with only marginal increase in average decoding complexity .an example of such transformation is presented in the following section .the communication system given in section [ sectionii ] follows a discrete - time hidden markov model , where each markov state at time index corresponds to a possible combination of source symbols in time interval  ] have to be visited .alternatively , if one can use a low complexity algorithm to disprove the optimality of most of the markov states , then the va can limit its search by visiting only a small subset of markov states .following this idea , the three - step ml decoding framework is given as follows .  * step 1 : the decoder uses a suboptimal algorithm ( denoted by ) to obtain a quick guess of the codeword and its corresponding source message .* step 2 : an nll - based optimality test ( specified in theorem [ theorem1 ] ) is applied to each of the source symbols of .the decoder maintains a source symbol set sequence , with  x ] can be confirmed by the optimality test , we let =\{\tilde{\mbox{\boldmath  }}[d]\} ] be the set of all possible source symbol vectors at time index . *step 3 : the decoder uses a modified va to search for the ml source message . the only difference between the modified va and the conventional va is that , the modified va visits a markov state only if all source symbols corresponding to the markov state belong to the source symbol sets  y y ] , then the probability that the nll - based optimality test can confirm =\mbox{\boldmath  }_{ml}[d ] , \forall d\in [ m , m+\nu) ] be the source symbol set maintained by the ml decoder in the second step , we have |=1 , \forall d\in [ m , m+\nu)\right\}=1 , \qquad \forall m \label{lowcomplexityinprob}\ ] ] since the worst case complexity of the modified va is bounded , ( [ lowcomplexityinprob ] ) implies , for any , .since all derivations hold if we first take to infinity , we also have .  by sharing computations among optimality tests, it is easy to see that the complexity of the second step of the ml decoder is equivalent , in order , to visiting one markov state per time unit . therefore , if satisfies ( [ symbolerror ] ) , as , the complexity of the three - step ml decoder converges to the complexity of , which can be significantly lower than the complexity of the va .moreover , the three steps of the ml decoder can be implemented in a parallelized manner in the sense that each step can process some of the source symbols without waiting for the previous step to _ completely _ finish its work .an example of such parallelized implementation can be found in ( * ? ? ?* ; * ? ? ?* the simple mlsd algorithm ) .in this section , we generalize the results of section [ sectioniv ] to ml sequence detection ( mlsd ) in a class of first order discrete - time hidden markov systems .we demonstrate in appendix [ hmmmodelverification ] that the communication system presented in section [ sectionii ] satisfies the model and the key assumptions given in this section .let ^d+\mbox{\boldmath  }[d+1]d^{d+1}+ ... ] represents the markov state ( at time ) , which is a -dimensional row vector defined over .we assume =\mbox{\boldmath  } y u ] as the processed state " , which is a_ deterministic _ function of  ] is a -dimensional row vector defined over .we term ^d+\mbox{\boldmath  }[d+1]d^{d+1}+ ... r ] be the observation sequence , where  u u u u u u u u r y y y u ] does not compromise the observability of the markov states in the sense that there exists a positive integer satisfying the following property . given two markov state sequences and .for any time index , if \ne \tilde{\mbox{\boldmath  }}[d] y u ] .note that we used the same constant in ( [ homogeneous ] ) and in assumption [ assumption2 ] .this is valid because if ( [ homogeneous ] ) is satisfied for , then it is also satisfied for all ; similar property applies to assumption [ assumption2 ] .consequently , if assumption [ assumption2 ] holds , a common integer satisfying both ( [ homogeneous ] ) and assumption [ assumption2 ] can always be found .[ assumption3 ] assume the existence of two functions : and , both are functions of the channel output symbol and the processed state .assume and have the following two properties .first , the following inequalities hold for all and . \nonumber \\ & & l_u(\mbox{\boldmath  } , \mbox{\boldmath  }_1 ) \ge \max_{\mbox{\scriptsize \boldmath  }_2\ne \mbox{\scriptsize \boldmath  }_3}\left[-\log(f_o(\mbox{\boldmath}|\mbox{\boldmath  }_2))+\log(f_o(\mbox{\boldmath  }|\mbox{\boldmath  }_3))\right ] .\label{nllu}\end{aligned}\ ] ]  second , the complexity of evaluating and is low in the sense that they do not require the search of any processed state other than .  note that validity of the results presented in this section does not depend on the second property imposed in assumption [ assumption3 ] .however , we still include the property in the assumption since the key motivation of posing assumption [ assumption3 ] is to use the two functions and as tools to avoid exhaustive markov state search and hence to reduce the complexity of ml decoding .also note that the right hand side of the second inequality in ( [ nllu ] ) is not a function of .however , the upper bound on the left hand side is a function of a processed state since one often needs a reference state " in order to upper bound the right hand side of ( [ nllu ] ) .further explanation is given in appendix [ hmmmodelverification ] .given the observation sequence , the negative sll of a state sequence is obtained by |\mbox{\boldmath  }[d])p_t(\mbox{\boldmath  }[d]|\mbox{\boldmath  }[d-1])).\ ] ] the objective of mlsd is to find the ml sequence that minimizes the negative sll , , 0\le d < n}s_u(\mbox{\boldmath  }(d)).\ ] ]  the following theorem gives a class of nll - based optimality tests .[ theorem3 ] assume the discrete - time markov system satisfies assumptions [ assumption2 ] and [ assumption3 ] .let be a positive constant . given a markov state sequence and the corresponding processed states .let be defined by ( [ ptr ] ) . for any time index , if there is an integer such that for all  , \mbox{\boldmath  }[d ] ) > 3\nu(\rho-\log p_{tr } ) , \label{lbound}\ ] ] and ) \le3m\nu\rho + ( \nu+1)\log p_{tr } \nonumber \\ & & \sum_{d = m-(2m+1)\nu}^{m-2m\nu-1}l_u(\mbox{\boldmath  } , \mbox{\boldmath  }[d ] ) \le 3m\nu\rho + \nu\log p_{tr } , \label{lmbound}\end{aligned}\ ] ] then =\mbox{\boldmath  }_{ml}[m+\nu-1] y r y u ] , =\mbox{\boldmath  }[d_2] y u u r y u u ] and  ] and  y u u ] .assume all markov paths start from =\mbox{\boldmath  } ] at time index if , among all markov paths passing  y u u ] .   in ,when the decoder branches a markov path at time index , the branch is characterized by a _ partial message _ , \tilde{\mbox{\boldmath  }}[1 ] , \dots , \tilde{\mbox{\boldmath  }}[m]\} x g ] .the negative sll lower bound is given by -g_q\left(\tilde{\mbox{\boldmath  }}[d]\right)\right\|^2   \ge \sum_{d=0}^{m}\left\|\mbox{\boldmath  }[d]-g_q\left(\sum_{l=0}^{\nu-1}\tilde{\mbox{\boldmath  }}[d - l]\mbox{\boldmath  }[l]\right)\right\|^2 , \label{spherebound}\end{aligned}\ ] ] which satisfies assumption [ assumption1 ] with .in , several sll - based otcs were presented for decoding block codes .the decoder obtains a first guess of the codeword .a negative sll lower bound is then developed for the codeword set , which corresponds to the case of being an empty set in the context of section [ sectioniii ] . is optimal if the optimality test gives a positive answer .the lower bounds presented in ( * ? ? ?* ; * ? ? ?* section iii ) satisfy the following inequality , \right)-g_q\left(\mbox{\boldmath  }[d]\right)\right\|^2 \label{distancebound}\ ] ] since the coding constraint is , we can always find a codeword with differing from at no more than codeword symbols .this implies that the right hand side of ( [ distancebound ] ) can be upper bounded by a constant , denoted by , which is not a function of .\right)-g_q\left(\mbox{\boldmath  }[d]\right)\right\|^2\le u_1\ ] ] consequently , given and , there exists a constant such that assumption [ assumption1 ] is satisfied for .assume , in searching the ml codeword , the decoder successfully avoided visiting a markov state specified by , \dots , \mbox{\boldmath  }_0[d]\} ] and , =\phi x ] , is disproved .we choose with _ the maximum cardinality _ while make sure that , in disproving the optimality of , \dots , \mbox{\boldmath  }_0[d]\} x ] satisfying \subseteq d_d^x x ] , and =\mbox{\boldmath  }[\tilde{d}] x ] with \subseteq d_d^x x ] , , using sll - based optimality test goes to zero . ] combining ( [ complexityva ] ) and ( [ complexityprob2 ] ) , we get      consider a communication system modeled in section [ sectionii ]. define =[\mbox{\boldmath  }[d-\nu+1 ] , \dots , \mbox{\boldmath  }[d]] y u ] is only a function of the corresponding markov state .if two markov states in successive time indices take the form =[\tilde{\mbox{\boldmath  }}[d-\nu+1 ] , \dots , \tilde{\mbox{\boldmath  }}[d ] ] \nonumber \\ & & \mbox{\boldmath  }[d+1]=[\tilde{\mbox{\boldmath  }}[d-\nu+2 ] , \dots , \tilde{\mbox{\boldmath  }}[d+1]],\end{aligned}\ ] ] for some , then we have |\mbox{\boldmath  }[d])=\frac{1}{q^k}.\ ] ] otherwise |\mbox{\boldmath  }[d])=0 x x ] does not depend on source symbols at time indices , we know |\mbox{\boldmath  }[d-\nu])\ne 0 , \qquad \forall \mbox{\boldmath  }[d ] , \mbox{\boldmath  }[d-\nu].\ ] ]   next , we show assumption [ assumption2 ] is satisfied .let and be two markov state sequences .let and be the source message and the codeword corresponding to .let and be the source message and the codeword corresponding to . for a time index , if \ne \tilde{\mbox{\boldmath  }}[d] ] such that \ne \tilde{\mbox{\boldmath  }}[m] y ] .therefore , assumption [ assumption2 ] holds because .let and be defined in theorem [ theorem1 ] .let be two arbitrary codeword symbols .we have the following triangle inequalities , the first inequality in ( [ triangleinequality ] ) implies +\log(f_o(\mbox{\boldmath  }|\mbox{\boldmath  }_1 ) )  & & =  \min_{\mbox{\scriptsize \boldmath  }_2 , \mbox{\scriptsize \boldmath  }_2\ne \mbox{\scriptsize \boldmath  }_1}\left[\frac{\mbox{\scriptsize snr}}{2}(\|\mbox{\boldmath  }-g_q(\mbox{\boldmath  }_2 ) \|^2-\|\mbox{\boldmath  }-g_q(\mbox{\boldmath  }_1 ) \|^2 ) \right ] \nonumber \\ & & \ge  \frac{\mbox{\scriptsizesnr}}{2}d_{\min}(d_{\min}-2 \|\mbox{\boldmath  }-g_q(\mbox{\boldmath  }_1 ) \|).\end{aligned}\ ] ] the second inequality in ( [ triangleinequality ] ) implies  & & = \max_{\mbox{\scriptsize \boldmath  }_2\ne \mbox{\scriptsize \boldmath  }_3}\left[\frac{\mbox{\scriptsize snr}}{2}(\|\mbox{\boldmath  }-g_q(\mbox{\boldmath  }_2 ) \|^2-\|\mbox{\boldmath  }-g_q(\mbox{\boldmath  }_3 ) \|^2 ) \right ] \nonumber \\ & & \le \max_{\mbox{\scriptsize \boldmath  }_2}\left[\frac{\mbox{\scriptsize snr}}{2}\|\mbox{\boldmath  }-g_q(\mbox{\boldmath  }_2 ) \|^2\right ] \nonumber \\ & & \le \max_{\mbox{\scriptsize \boldmath  }_2}\left[\mbox{snr}(\|\mbox{\boldmath  }-g_q(\mbox{\boldmath  }_1 ) \|^2+\|g_q(\mbox{\boldmath  }_2)-g_q(\mbox{\boldmath  }_1 ) \|^2)\right ] \nonumber \\ & & \le \mbox{snr}(\|\mbox{\boldmath  }-g_q(\mbox{\boldmath  }_1 ) \|^2+d_{\max}^2).\end{aligned}\ ] ] therefore , assumption [ assumption3 ] is satisfied by defining note that evaluating and does not involve visiting any processed state other than .if  ] are the actual codeword symbol and the channel output at time index , -g_q(\mbox{\boldmath  }[d])\|=\|\mbox{\boldmath  }[d]\| u u u u u ] for all , according to assumption [ assumption2 ] , and differ at no less than time indices in the time interval , where denotes the maximum integer no larger than . according to ( [ nllu ] ) and ( [ lbound ] ) , for ,if \ne \mbox{\boldmath  }[d] y r y r y y u u r y u u u u u ] for so that ( [ prooftheorem3.4 ] ) is satisfied .let be the processed state sequence corresponding to .  from ( [ nllu ] ) and the first inequality in ( [ lmbound ] ) , we get |\tilde{\mbox{\boldmath  }}[d])p_t(\tilde{\mbox{\boldmath  }}[d]|\tilde{\mbox{\boldmath  }}[d-1])}{f_o(\mbox{\boldmath  }[d]|\mbox{\boldmath  }_c[d])p_t(\mbox{\boldmath  }_c[d]|\mbox{\boldmath  }_c[d-1 ] ) } \nonumber \\ & & \ge -\sum_{d = m+2m\nu}^{m+(2m+1)\nu-1}l_u(\mbox{\boldmath  }[d ] , \mbox{\boldmath  }[d])+(\nu+1)\log p_{tr }  \ge -3m\nu \rho \label{prooftheorem3.5}\end{aligned}\ ] ]  since \ne \mbox{\boldmath  }_c[m+\nu-1+k\nu] y u u r y u u y u u r y u u u u u ] for so that ( [ prooftheorem3.8 ] ) is satisfied .let be the processed state sequence corresponding to .  from ( [ nllu ] ) and the second inequality in ( [ lmbound ] ) , we get |\tilde{\mbox{\boldmath  }}[d])p_t(\tilde{\mbox{\boldmath  }}[d]|\tilde{\mbox{\boldmath  }}[d-1])}{f_o(\mbox{\boldmath  }[d]|\mbox{\boldmath  }_c[d])p_t(\mbox{\boldmath  }_c[d]|\mbox{\boldmath  }_c[d-1 ] ) } \nonumber \\ & & \ge -\sum_{d = m-(2m+1)\nu}^{m-2m\nu-1}l_u(\mbox{\boldmath  }[d ] , \mbox{\boldmath  }[d])+\nu\log p_{tr }  \ge -3m\nu \rho . \label{prooftheorem3.9}\end{aligned}\ ] ]  since \ne \mbox{\boldmath  }_c[m+\nu-1+k\nu] y u u r y u u y u u r y u u u u u u u u ] for all , according to assumption [ assumption2 ] , and differ at no less than time indices in the time interval . according to ( [ nllu ] ) and ( [ lbound ] ) , we have |\tilde{\mbox{\boldmath  }}[d])p_t(\tilde{\mbox{\boldmath  }}[d]|\tilde{\mbox{\boldmath  }}[d-1])}{f_o(\mbox{\boldmath  }[d]|\mbox{\boldmath  }_c[d])p_t(\mbox{\boldmath}_c[d]|\mbox{\boldmath  }_c[d-1 ] ) } > \left\lfloor\frac{4m+1}{2}\right\rfloor 3\nu(\rho-\log p_{tr } ) + 4m\nu\log p_{tr } \ge 6m\nu\rho .\label{prooftheorem3.13}\ ] ]  meanwhile , it is easily seen that ( [ prooftheorem3.5 ] ) and ( [ prooftheorem3.9 ] ) hold .combine ( [ prooftheorem3.5 ] ) , ( [ prooftheorem3.9 ] ) and ( [ prooftheorem3.13 ] ) , we obtain |\tilde{\mbox{\boldmath  }}[d]))p_t(\tilde{\mbox{\boldmath  }}[d]|\tilde{\mbox{\boldmath  }}[d-1])}{f_o(\mbox{\boldmath  }[d]|\mbox{\boldmath  }_c[d])p_t(\mbox{\boldmath  }_c[d]|\mbox{\boldmath  }_c[d-1 ] ) }  > -3m\nu \rho -3m\nu \rho + 6m\nu\rho = 0 .\label{prooftheorem3.14}\end{aligned}\ ] ] ( [ prooftheorem3.14 ] ) implies that covers . hence according to the pcc, can not be the ml sequence .

Abstract: this paper considers the average complexity of maximum likelihood ( ml ) decoding of convolutional codes . ml decoding can be modeled as finding the most probable path taken through a markov graph . integrated with the viterbi algorithm ( va ) , complexity reduction methods such as the sphere decoder often use the sum log likelihood ( sll ) of a markov path as a bound to disprove the optimality of other markov path sets and to consequently avoid exhaustive path search . in this paper , it is shown that sll - based optimality tests are inefficient if one fixes the coding memory and takes the codeword length to infinity . alternatively , optimality of a source symbol at a given time index can be testified using bounds derived from log likelihoods of the neighboring symbols . it is demonstrated that such neighboring log likelihood ( nll)-based optimality tests , whose efficiency does not depend on the codeword length , can bring significant complexity reduction to ml decoding of convolutional codes . the results are generalized to ml sequence detection in a class of discrete - time hidden markov systems .   coding complexity , convolutional code , hidden markov model , maximum likelihood decoding , viterbi algorithm