topological versus dynamical robustness in a...
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International Journal of Bifurcation and Chaos, Vol. 22, No. 7 (2012) 1250157 (9 pages)c© World Scientific Publishing CompanyDOI: 10.1142/S021812741250157X
TOPOLOGICAL VERSUS DYNAMICALROBUSTNESS IN A LEXICAL NETWORK
JAVIER BORGE-HOLTHOEFERInstituto de Biocomputacion y Fısica de Sistemas
Complejos (BIFI ), Universidad de Zaragoza,Mariano Esquillor s/n 50018 Zaragoza, Spain
YAMIR MORENODepartamento de Fısica Teorica,Universidad de Zaragoza, Spain
Instituto de Biocomputacion y Fısica de SistemasComplejos (BIFI ), Universidad de Zaragoza,Mariano Esquillor s/n 50018 Zaragoza, Spain
ALEX ARENASDepartment d’Enginyeria Informatica i Matematiques,
Universitat Rovira i Virgili,Av. Paısos Catalans 26 Tarragona,
43007 Catalonia, [email protected]
Received December 3, 2010; Revised March 11, 2011
Semantic memory is the cognitive system devoted to the storage of conceptual knowledge. Empir-ical semantic networks constructed from adult generated free word association data represent agood map for this system. Concepts, represented as words, link each other with a certain weightrepresenting the strength of their relation. Everyday experience shows that word search andretrieval processes, which can be assimilated to traversals on a semantic network, provide fluentand coherent speech, i.e. are efficient and robust. Brain pathologies, such as Alzheimer’s diseaseor schizophrenia, severely damage neural structures and associated capacities. Thus, semanticnetworks must also undergo disruption, but the question is how long cognitive processes, whichdepend on the underlying structures, can operate before collapsing. Interestingly, we find thatdegradation of the original structure has a dramatic impact on the topology of semantic network,whereas the dynamics upon it evidence much higher resilience. We define this problem in theframework of percolation theory.
Keywords : Information retrieval; complex networks; weighted percolation; semantic impairment.
1. Introduction
Modern network theory [Boccaletti et al., 2006]is progressively contributing to our understandingof cognition. Although it has not yet penetratedcognitive science as, for example, social science or
biology, these last years have witnessed an increasein the works that use statistical physics methodol-ogy to gain insight into cognitive phenomena. Lan-guage growth [Dorogovtsev & Mendes, 2001]; childlanguage development [Steyvers & Tenenbaum,
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2005; Hills et al., 2009, 2010]; category formationand search processes [Borge-Holthoefer & Arenas,2010a] or verbal fluency [Goni et al., 2009, 2010]stand as good examples of such strategy, see [Borge-Holthoefer & Arenas, 2010b] for a review. Signifi-cantly, all these works are devoted to language andcognitive processes around it. The reason for this isthat semantic memory, the cognitive system whereconceptual knowledge is stored, can be suitably rep-resented as a network, where nodes represent wordsand links between pairs of nodes stand for word–word relationships. Representation of language asa network has a long tradition in cognitive science[Quillian, 1967; Collins & Quillian, 1969; Collins &Loftus, 1975], and complex network theory repre-sents a methodological update of those proposals,boosted by massive empirical data availability.
Network modeling of language is mostlydevoted to normal, healthy performance. However,understanding how aging and disease affect profi-ciency in language production and comprehensionis a great concern in the field. In this paper, wepresent theoretical results which analyze how dete-rioration is related to performance. To this end, wefirst place the problem of semantic breakdown inthe context of error and attack tolerance [Albertet al., 2000; Cohen et al., 2000; Schwartz et al.,2002]. There exists a rich literature regarding net-work robustness, breakdown and final disintegra-tion in the cognitive field [Achard et al., 2006;Kaiser et al., 2007; Alstott et al., 2009]. However,as we shall discuss later, the previous approachis not satisfactory. We then define an appropri-ate framework to model aging and pathologies inthe cognitive context. Several studies from cogni-tive neuroscience report a progressive loss of struc-tural and functional connectivity in brain networksin patients compared with control subjects [Stamet al., 2007; Liu et al., 2008; Supekar et al., 2008;Stam et al., 2009]. It is on the tracks of this factthat cognitive deterioration is modeled. After that,and beyond the expected results, i.e. performanceis impoverished in the frame of illness, we studythe efficiency of a particular dynamics on a degen-erated structure. That is, in the line of [Duch &Arenas, 2007], we focus on the relationship betweentopological robustness and dynamical robustness,by comparing when the network is first physicallysplit or dynamically collapsed. This comparison ismade under different assumptions, the main con-clusion is that semantic memory is strongly shaped
by statistical biases and this elicits a more efficientand robust performance.
2. Topology: Free AssociationNorms
Association graphs are networks in which verticesdenote words, whereas links represent associationrelations as observed in cognitive-linguistic experi-ments. Such graphs are considered the most relevantfrom a psycholinguistic point of view and can betaken as a proxy to the actual structure of semanticmemory, because of their comprehensive character:they may for example represent semantic relation-ships — the shared semantic context of car androad — but also functional or causal relationships —as in fire and smoke — among others. Associationnorms reflect in some aggregate way important reg-ularities in language, and are strongly predictiveof adult performance in different kinds of languageprocessing tasks. According to the hypothesis thatassociation is one of the principles of memory orga-nization, the question that has to be addressed iswhich network topologies support an efficient orga-nization in terms of time and space complexity.
The best known Free Association data set inEnglish are University of South Florida Free Associ-ation Norms (FA from now on; [Nelson et al., 1998]).Nelson et al. [1998] produced these norms by askingover 6000 adult participants to write down the firstword (target) that came to their mind when con-fronted with a cue (word presented to the subject).The experiment was performed using more than5000 cues. Associative strength is the frequency ofcoincidence between subjects for each pair of words.As an example, words mice and cheese are neigh-bors in this database, because a large fraction of thesubjects related this target to this cue. The empir-ically obtained network is a directed and weightedgraph. Weights represent the frequency of associa-tion in the sample, and their distribution is highlyheterogeneous. These same data also exist in Span-ish [Callejas et al., 2003; Fernandez et al., 2004],German [Melinger & Weber, 2006] or French [Fer-rand & Alario, 1998].
Although most of the literature on small-worldand scale-free networks has focused on the undi-rected, unweighted version of FA, we believe thatthe directed network is clearly a more natural rep-resentation of word associations. Thus asymmetryis preserved throughout this work.
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3. Dynamics: Random InheritanceModel
The Random Inheritance Model (RIM) [Borge-Holthoefer & Arenas, 2010a] explores whether it ispossible to disentangle similarity relationships fromgeneral word association network (FA) by a naıvecognitive navigation on top of it. Unlike associa-tion strength, which is often asymmetric and phe-nomenological (in the sense that associate wordsmight be so for very different reasons), similarityrelationships express the degree of overlap in themeaning of two words. Such relation is then sym-metric. More specifically, two words are consid-ered semantically similar if they share features.
Note that similarity lies at the base of categoryformation.
The process can be schematized as uncorrelatedrandom walks from node to node that propagatean inheritance mechanism among words, converg-ing to a feature vectors network, see Fig. 1. Besidesthe degree of success when compared to empiri-cal data (see below), the most appealing feature ofRIM is two-fold: (i) the fact that it implements anintuitive idea — we repeatedly navigate a seman-tic network to produce or understand meaningfulutterances, and an aggregate of these explorationscan plausibly resemble random walks; (ii) a dynam-ics governed by uncorrelated random walks admits
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Fig. 1. In RIM, the visits of a random walker starting at node i trigger the inheritance mechanism, which modifies thefeatures vector of a node i. In the figure, a random walk of 4 steps (a)–(d) changes the vector of node 1.
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a clear formalization, allowing for an analyticalunderstanding of the emergence of semantic simi-larity among words.
Regarding (ii), RIM can be algebraicallydescribed in terms of Markov chains. To this end,we must define the transition probability of the FAnetwork. The elements of FA (aij) correspond tofrequency of association, these weights can be nor-malized and the adjacency matrix corresponding tothe FA network becomes a transition probabilitymatrix. The transition probability matrix P is thusdefined as:
Pij =aij∑
j
aij
. (1)
As the original matrix, this one is also asym-metric. Once the matrix P is constructed, therandom walkers of different lengths are simply rep-resented by powers of P . In practice, this meansthat if we perform random walks of length S, afteraveraging over many realizations we will convergeto the transition matrix PS , every element (PS)ijrepresents the probability of reaching j, from i, in Ssteps. The inheritance process corresponds, in thisscenario, to a change of basis, from the orthogonalbasis of the N -dimensional space, to the new basisin the space of transitions T :
T = limS→∞
S∑
i=0
P i = (I − P )−1. (2)
In practice, we are rather interested in metastableor quasi-stationary states of the Markov process(finite S) [Meyer, 1989], i.e. the length of the ran-dom walkers is limited. The summation in Eq. (2)converges, in terms of the matrix 1-norm, veryfast, limiting the dependence on indirect associativestrengths [Nelson & Zhang, 2000]. Although compu-tations were done for several values of S, S = 4 isenough to reach quasi-stationary states in T , thusresults for RIM in this work are expressed for S = 4from now on. Finally, the goal of this dynamics is toquantify semantic similarity, which is calculated asthe cosine of the vectors in the new space, given bythe scalar product of the matrix and its transpose,FS = TT †.
The results obtained by RIM show macro-statistical coincidences (functional form of the dis-tributions and descriptors) between real semanticsimilarity data and the synthetic obtained network(FS). The model also yields significant success at
Table 1. An illustrative example of RIM’spredictive capacity, when comparing closestneighbors to empirical data. The ten mostsimilar concepts to the word ROOSTER arelisted, sorted in decreasing order.
ROOSTER
Empirical RIM
Chicken ChickenGoose TurkeyPigeon CrowSparrow RobinPenguin SparrowPelican BluejayBluejay PigeonDove PelicanHawk GooseTurkey Hawk
the microscopic level, i.e. it is able to reproduce toa large extent empirical relationships, see Table 1and [Borge-Holthoefer & Arenas, 2010a] for details.
In conclusion, after a random walker-baseddynamics each node (word) obtains a new neigh-borhood, different from the one in FA, made ofother words which hold similarity relations with it.Now the question is whether these inferred seman-tic neighborhoods are stable, i.e. the dynamics isrobust, when semantic memory is under the influ-ence of some pathology, i.e. when the substrate (FA)supporting the dynamics undergoes degradation.
4. Topological and DynamicalRobustness
In the previous section we have proposed a mech-anism that drives the emergence of category struc-ture. Now we turn to the characteristics of both theoriginal topology and RIM dynamics under error.Literature on error and attack tolerance in complexnetworks [Albert et al., 2000; Cohen et al., 2000;Schwartz et al., 2002; Boguna & Serrano, 2005;Serrano & Boguna, 2006] typically model deterio-ration in two ways: error as the failure (removal)of randomly chosen nodes/edges, and attack as theremoval of important nodes/edges (“importance”can be quantified by some descriptor, be it highconnectivity, high betweenness, etc.). Using thisapproach, one typically monitors a suitable networkcharacteristic that signals the moment in whichphysical disintegration of the structure takes place.As for RIM’s dynamics, performance of a word i
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(match i) under error or attack is measured as theproportion of words that remain similar to i inthe impoverished structure, compared to the origi-nal results. This implies that, each time a node isremoved, RIM is applied on the distorted structureand current similarity neighborhoods are comparedto the original ones, for each node. The quantity“match” is a global average over the N elements ofthe network, and of course its value is 1 when nonode has yet been removed.
In Fig. 2 we use this approach by progressivelyremoving nodes. This has been done in three ways:randomly choosing the failing node (failure), choos-ing it in terms of highest vertex betweenness (max-imum betweenness attack), or eliminating the nodewith highest ωin (maximum in-strength attack).
At least two conclusions can be drawn fromFig. 2. In the first place, the relative size of thegiant component Ngiant/Nnet decays in a similarway to those reported in the literature for scale-free networks [Albert et al., 2000], i.e. the struc-ture is robust against failures but attacks hinderthe integrity of the topology much before, approx-imately at f = 0.75. More interestingly, the RIMdynamics are not as resilient as the structure, andcollapses long before the topology is actually disin-tegrated in the three cases (failure and both attackstrategies). That is, before the critical point is
reached, the dynamics’ performance is much moredeteriorated than the topology for any given frac-tion of removed nodes, f .
Though informative, we now wonder whetherfailure or attack correctly grasp the way in whichpathologies or aging affects a cognitive structuresuch as semantic memory. Empirical evidence in theneuroscience literature report on a general decayof the neural structure supporting cognition [Stamet al., 2007; Liu et al., 2008; Stam et al., 2009].Then, realistic modeling demands a different wayto approach this problem. Here, we redefine error inthe context of cognitive systems. In this framework,it is more useful to consider error in terms of agingor disease, where the whole topology simultaneouslydecays in some way. By doing so, we capture thedegrading behavior of aging and/or disease, whichdiffers from attack (there is no selective action) andfrom error (which affects only one node/edge at atime). For the sake of clarity, we refer to error inthe cognitive framework as degradation.
Degradation assumes that links are increasinglydamaged. At a given threshold τ , every link (i, j) inFA with a ωij ≤ τ is removed. The surviving linksare normalized [Eq. (1)] to preserve a probabilis-tic interpretation of the structure. This process isperformed with values 0 ≤ τ ≤ 1. As in the caseof failure and attack, for each value τ , we monitor
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Fig. 2. (a) Topological deterioration (relative size of the giant component) of FA as a function of f , the fraction of nodesremoved from the network. In green, results for error (random failure of nodes); in red, results for attack to vertex betweenness:the nodes with highest B are removed first. Finally, attack to higher in-strength appears in black. (b) RIM’s resilience for thesame strategies.
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both topological and dynamical properties of theresulting network (the size of the giant componentof the degraded structure is measured; and RIM isused to find a similarity matrix on the degradedstructure, and the result is compared to the nonde-graded RIM, i.e. RIM’s result at τ = 0 — “match”axis in figures).
4.1. Degradation on the originalstructure
Figure 3 shows the results for both topologicaldeterioration (a) and dynamical resilience (b).Focusing on black circles (which correspond todegradation of the FA network), the behavior ofRIM’s dynamics appears to be very sensitive todegradation even at very low values of τ . This sug-gests that lexical impairment can appear at earlystages of semantic memory disease degradation.Interestingly, however, RIM’s degradation is muchslower than the topological one. At τ ≈ 0.3, FAstructure is already disintegrated, whereas RIM canstill recover as much as 25% of its original content.RIM’s results do not vanish up to τ ≈ 0.6. Suchresult indicates that fundamental cognitive capaci-ties such as word–word similarity inference and cat-egory formation are substantially resilient to struc-tural impoverishment.
4.2. Degradation on the nullmodel I
Results in black circles (degradation of the FA net-work) from Fig. 3 raise the question of which topo-logical aspect of the dynamics’ substrate providesfor good performance in a disrupted topology. Toanswer this question, we propose to build appropri-ate null models. The idea is that, by changing thetopology that supports the dynamics, we can gaininsight on which properties of the original struc-ture provide long-lasting performance. In particu-lar, we devise two null models, both preserve thedegree sequence and directions. In the first place,we consider the network FA in which weights areignored. This implies that each node has the samein- and out-degree distribution, but outgoing linksare weighted uniformly, that is ωij = 1/kout. Thetopological resilience of this modified FA networkis represented as red circles in the top panel ofFig. 3. The percolation point has moved to the leftif compared to the original results, i.e. the networkis structurally weaker. This result is not surpris-ing, given that τ affects in the first place weakerlinks; thus, in an unweighted FA, this means that(i) nodes with higher degree kout have the lowestweights, 1/kout; (ii) since weights are distributeduniformly for each node, when τ = 1/ki all out-going links are removed at once. In this sense, this
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Fig. 3. (a) Topological deterioration (relative size of the giant component) of FA as a function of τ . In black, results for theoriginal FA structure. The same process of degradation has been applied to an unweighted version of FA understood hereas a plausible null model, in red. In green, results for a second null model, which sets the weights of links in FA as kout
i kinj .
(b) RIM’s resilience for the same structures.
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null model is equivalent to a kmax-attack. From thedynamical point of view, it is apparent also that thestructure cannot support category formation andsimilarity inference for a long time: RIM’s resultscollapse even before the topology reaches the topo-logical percolation point.
4.3. Degradation on the nullmodel II
Although the unweighted null model is not an“aggressive” one (it preserves many of the featuresof the original network, such as degree distributionP (k), average degree 〈k〉, average clustering coeffi-cient C, etc.), we may devise one in which, further-more, weight heterogeneity is still present. This canbe done by assigning a weight to out-links propor-tional to the out-degree kout
i of the source node iand the in-degree of the node j receiving that link,kin
j . Then, ωij = kouti kin
j . The weights quantified inthis fashion are normalized, to replicate the prob-abilistic interpretation of the original links in FAand in the previous null model. As it is apparentfrom the green circles in Fig. 3 (upper panel), thekout
i kinj -weight configuration yields a more resilient
structure from a topological point of view, the per-colation point is displaced to the right if comparedwith the original FA. This is so because, contraryto the previous case, nodes with high degree arefavored, in such a way that both their in- andout-weights are high. Then the threshold param-eter τ does not affect hubs until a late degradationstage, the structure is not severely fragmented untilτ ≈ 0.4. However, RIM decays faster than the orig-inal FA counterpart (Fig. 3, lower panel). Althoughthe value of “match” vanishes approximately at thesame time as for the original substrate, dynamicdeterioration for early τ values is more rapid.
5. Conclusions
Up to this point, we have introduced free associ-ation norms, and in particular FA, as a plausiblerepresentation of the structure of semantic knowl-edge. Also, we have presented RIM as a proxyof real cognitive dynamics to extract the categorystructure backbone from a general semantic rela-tions context (FA). Once this information is avail-able we study how dynamics reacts when confrontedwith failure and attack, in the first place; andthen with progressive degradation of the topolog-ical structure. To this end, we follow the line of
percolation theory in complex networks with somemodifications. Results indicate that linguistic per-formance is severely affected by semantic memorydegradation, on the other hand, such performanceis still significantly effective beyond topologicaldisintegration.
Our study concludes, furthermore, that thespecific distribution of weights in the lexical net-work plays a key role in the resilience both ofthe topology and the dynamics. Perturbing thisweight distribution dramatically changes the capac-ity of the structure to hold performance dynam-ics on it. The particular value of these weights isjust a consequence of contextual diversity (statis-tical biases of language use), which can soundlybe identified as the origin of categorization andthe maintainer of semantic integrity. On the otherhand, this work raises some questions of interest.From a physical point of view, the new approachto structural damage demands an analytical treat-ment, in order to predict the topological responseto weighted degradation. In this line, a reconsid-eration of current knowledge on percolation theoryis necessary.
From the standpoint of neuroscience and psy-cholinguistics, attention should focus on how phys-ical (neurological) and cognitive degradation arerelated. Also, it has been reported that pathologiessometimes selectively affect linguistic performance[Moss & Tyler, 2000; Caramazza & Mahon, 2003].Then some kind of “selective degradation” shouldbe implemented and studied, given the modularstructure that lexical networks display [Borge-Holthoefer & Arenas, 2010a, 2010b]. Finally, othervariables can be taken into account; for instance, atthe moment a node is disconnected from the net-work, its “cognitive load” (the semantic meaningit bears) must be assumed by the remaining con-nected structure. In this way, degradation could bein interplay with node-breaking avalanches [Morenoet al., 2002, 2003], which could explain not only cog-nitive dysfunction (inexact or impoverished seman-tic capacities) but also system inefficiency (generalperformance slowing).
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