Anomalous Lifshitz dimension in hierarchical networks of brain connectivity
Samaneh Esfandiary, Ali Safari, Jakob Renner, Paolo Moretti, Miguel Ángel Muñoz
AAnomalous Lifshitz dimension in hierarchical networks of brain connectivity
Samaneh Esfandiary, Ali Safari, Jakob Renner, Paolo Moretti, and Miguel A. Mu˜noz Institute of Materials Simulation, FAU Universit¨at Erlangen-N¨urnberg, Dr.-Mack-Straße 77, 90762 F¨urth, Germany Departamento de Electromagnetismo y F´ısica de la Materia e Instituto Carlos I deF´ısica Te´orica y Computacional. Universidad de Granada, E-18071 Granada, Spain (Dated: August 28, 2020)The spectral dimension is a generalization of the Euclidean dimension and quantifies the propen-sity of a network to transmit and diffuse information. We show that, in hierarchical-modular networkmodels of the brain, dynamics are anomalously slow and the spectral dimension is not defined. In-spired by Anderson localization in quantum systems, we relate the localization of neural activity –essential to embed brain functionality – to the network spectrum and to the existence of an anoma-lous “Lifshitz dimension”. In a broader context, our results help shedding light on the relationshipbetween structure and function in biological information-processing complex networks.
Understanding the interplay between dynamical pro-cesses and the architecture of the networks embeddingthem is a fundamental problem in diverse fields includ-ing material science, genetic regulation and neuroscience.Dynamical features and patterns of activity are often af-fected or controlled by key structural features of the un-derlying network, such as the degree distribution, degreecorrelations, modular organization, k-core structure, etc.[1–5]. However, given that such features are usually notindependent, a more systematic way to tackle the prob-lem of the interplay between structure and dynamics re-lies on the use of spectral-graph characterizations of thenetwork architecture [6, 7] and, importantly, the networkdimension. Statistical mechanics teaches us that dynam-ical processes such as diffusion, vibrational excitations,and critical properties near second order phase transi-tions exhibit universal behavior, which depends cruciallyon the lattice (Euclidean) dimension [1–3, 8, 9]. Thecase of heterogeneous networks is more complex, sincemultiple and diverse generalizations of the concept of di-mension have been proposed [10–12]. Nevertheless, com-pelling pieces of evidence show that dimensionality mea-sures are effective determinants of dynamics and activityin networked complex systems. The simplest exampleis provided by networks with the small-world property[13–15], which exhibit diameters that grow only logarith-mically with the network size N and, consequently, withdiverging Hausdorff dimension. We recall that the Haus-dorff dimension d H , also called “topological dimension”in the literature [16, 17], can be computed easily startingfrom the number u i ( r ) of nodes within distance r fromnode i : if (cid:104) u i ( r ) (cid:105) ∼ r d (where (cid:104) . (cid:105) stands for the averageover all nodes in the network), implying d H = d Diverg-ing d H typically implies enhanced transmission, signalpropagation and high synchronizability.A somewhat more complex example of how dimension-ality controls activity patterns in networks is provided byhierarchical-modular networks, as models e.g. for brainconnectivity [18, 19]. It was first pointed out that thehierarchical-modular organization of brain regions resultsin network models of finite Hausdorff dimension d H and, at odds with small-world graph topologies, with intrinsi-cally large diameters [20]. The large-world property re-sulting from finite d H in hierarchical-modular networks, apurely structural feature of the network, has been linkedto signatures of anomalous activity patterns in brainnetwork models, including among others: sustained ac-tivity [21], sub-diffusive dynamics [22], localization phe-nomena and stretched criticality in the form of Griffithsphases [17, 23, 24], broad avalanche distributions [17, 25],states of localized and “frustrated” synchronization [26–30], rounding of first-order phase transitions [31], and er-godicity breakdown [32, 33]. Importantly, some of theseanomalous dynamical traits are, in fact, considered essen-tial to the ability of brain networks and of brain-inspiredhierarchical architectures to achieve an optimal balancebetween segregation and integration [18], allowing themto conduct multiple tasks simultaneously, entailing opti-mal computational capabilities [34]. Let us also note thata significant part of the above-mentioned phenomenologyuses concepts, such as Griffiths phases, first introducedto study (Anderson) localization phenomena in quantumsystems described, e.g., by a random tight binding Hamil-tonian [35], and later extended, for instance, to the Lapla-cian matrix of a graph [36, 37].In this paper, we aim at providing a theoretical foun-dation for the phenomenological observations of anoma-lous behavior and localization effects obtained so far inhierarchical-modular networks, establishing for the firsttime a clear link between the emergence of such localiza-tion patterns of activity and synchrony and and spectralproperties of the underlying graphs.The fundamental concept, allowing us to develop ourapproach is the spectral dimension d s of a graph –aswell as an important extension of it, that we call Lif-shitz dimension– which can be defined and measuredby simple random walk (RW) analyses [38–40]. Giventhe probability P ij ( t ) that a random walker startingat time t = 0 from node i arrives at node j after t steps, one can compute the average return probabilityas R ( t ) = (cid:80) Ni =1 P ii ( t ) /N . If a real positive d s exists, a r X i v : . [ c ond - m a t . d i s - nn ] A ug such that R ( t ) ∼ t − d s2 , d s is defined as the (average)spectral dimension of the network [40]. While for in-finitely large networks further complications arise due tothe possibility of transient random walks [39], here we fo-cus on networks of finite, albeit very large size N , so thatthe above definition is intended to hold asymptotically,in the limit of large N and large t . The spectral di-mension, not unlike the Hausdorff dimension introducedabove, is a generalization of the concept of dimension.Actually, in discrete lattices d s = d H , so that both gen-eralizations agree with the Euclidean dimension of theembedding continuum space. This equivalence does nothold in general in heterogeneous networks, nor does it indeterministic fractals [8]. We notice in particular thatwhile d H is a purely structural measure, d s is an observ-able of a diffusion process operating on the network, andas such it provides us with a probing tool for dynamicalsignatures of localization and slowing down, and a firstapproximation in cases, like that of brain activity, withmuch more complex dynamics. The relationship betweendifferent definitions of a network dimension and their useto predict the emergence of anomalous dynamical pat-terns thus remains an open question to be fully clarified.For example, it was initially conjectured that Griffithsphases –characterized by string localization features– inheterogeneous networks can only occur in the finite d H case [16, 17]; however, this view was challenged by Mill´an et al. , who found that even networks with infinite d H canexhibit similar dynamical regimes, provided that d s isfinite instead [41].With these considerations in mind, we analyse thespectral dimension of hierarchical-modular network mod-els of brain connectivity [17], with the objective of quan-tifying how the basic traits of brain activity localization,which are captured by these simple network models, arereflected by d s and stochastic diffusion null models. Tothis end, we conducted very-large-scale RW simulationsin hierarchical-modular network models of tunable Haus-dorff dimension d H and computed the average returnprobabilities R ( t ).We chose to work with the model proposed in [17, 42]for the generation of synthetic hierarchical-modular net-works; this model comes with a single effective parameter α (the connectivity strength) and an additional param-eter s fixing the number of hierarchical levels. In thismodel the average network degree (cid:104) k (cid:105) and (asymptoti-cally) the Hausdorff dimension d H are both proportionalto α so that sparser networks have smaller Hausdorff di-mension. While other models proposed in the literaturemight differ in the choice of parameters [21, 25, 43, 44],we believe that the conclusions of the current work re-main unchanged. In order to capture the large networksize limit of the system, we performed random-walk com-puter simulations on networks of sizes up to 2 ≈ × ,and for time windows large enough as to ensure that allwalkers return to the starting node (which, we recall, is t R ( t ) = 0.50= 0.75= 1.00 = 1.25= 1.50= 1.75 FIG. 1. Return probabilities for hierarchical-modular net-works with N = 2 , s = 23 and increasing α . Even at largesizes, no clear power law decay is visible and the asymptoticbehavior is dominated by a slower (stretched-exponential)tail, before the finite size cut-off takes over. Similar resultshave been found for smaller sizes and different choices of s . possible because N is finite in our case).Fig. 1 shows the return probabilities for a typicalchoice of parameters ( N = 2 , s = 24), and for in-creasing values of α and, thus, increasing Hausdorff di-mension d H . One can immediately see the anomaly inthe asymptotic behavior of R ( t ): while for intermediaten t the curves develop a heavy tail, resembling a powerlaw, the slope of such tails apparently decreases in abso-lute value upon increasing a α , and later develops a nontrivial large- t bump, significantly before the finite sizecutoff appears. In cases in which the spectral dimensionis defined, we would expect the slope to increase in ab-solute value with α , implying that d s increases with d H and one would expect that behavior to be the asymptotic( t → ∞ ) one. In the present study, instead, one is forcedto conclude that dynamical slowing down is so radicalthat the asymptotics are given by the excess returns atvery large t (the bump in the curves above) and the av-erage spectral dimension is, as a consequence, undefined.The absence of a well-defined spectral dimension is analready interesting result within the study localizationand dynamical slowing down in models of brain connec-tivity. Suppression of diffusion and free flow are consid-ered signatures of anomalously slow dynamical regimes,which ensure the balance between global integration andfunctional modularity of brain activity [20, 22, 45]. Toproceed, let us first elucidate the nature of the asymp-totic behavior of the return probabilities. Fig. 2 revealsthat the large t dependence of R ( t ) is dominated by astretched exponential behavior R ( t ) ∼ e − t β , governed bya non trivial positive β < t in dimensionless units). We call β the anomalousexponent, as its value quantifies the dynamical slowingdown with respect to the standard scenario where a spec-tral dimension is defined. To confirm this view, Fig. 2 t R ( t ) = 0.75= 1.00= 1.25= 1.50= 1.75= 2.00 0.5 1.0 1.5 2.00.70.80.9 N = 2 N = 2 FIG. 2. Quantitative analysis of the stretched exponentialbehavior found in Fig. 1. Left: semi-logarithmic-scale plot ofthe return probability. For each curve (each value of α ) thevalue of β is chosen, which makes the stretched exponentialtail appear as a straight line. Color scheme is as in Fig. 1.Right: the values of β used on the left panel ( N = 2 , s = 23,blue line) as well as for a smaller system ( N = 2 , s = 22,gray line). also shows the dependence of β on α and thus on theHausdorff dimension of the network. Sparser networksexhibit stronger anomalous behavior, with the anomalyexponent loosely proportional to the Hausdorff dimen-sion.In order to make analytical progress, we exploit well-known methods of spectral graph theory [6, 7]. Let uswrite the exact master equation, describing a randomwalk as a time-continuous Markov process on a genericundirected and unweighted network, encoded in an adja-cency matrix A as follows:˙ q ( t ) = − L RW q ( t ) , (1)where q ( t ) is the column vector, whose generic element q i ( t ) represents the probability of the random walker toreach node i at time t , and L RW is the random walkLaplacian matrix with elements L RW ij = δ ij − A ij /k j with i, j ∈ , , ...N . Here A ij is the generic element of A ,equal to 1 if nodes i and j are linked and 0 otherwise, and k j = (cid:80) Ni =1 A ij is the degree of node j . In order to com-pute the solution to Eq.1, one can introduce the normal-ized Laplacian L , defined by the similarity transforma-tion L RW = D LD − , where D is the (diagonal) degreematrix of generic element D ij = δ ij k j . L is symmetricand diagonalizable, and by virtue of their similarity, L and L RW have the same spectrum of eigenvalues, albeitwith different eigenvectors [6]. The solutions of Eq.1 canthen be written through the eigen-decomposition of L ,as q i ( t ) = (cid:80) Nj =1 k / i K ij ( t − t ) k − / j q j ( t ), in which weintroduced the heat kernel K ( t ) of generic element [46]: K ij ( t ) = N (cid:88) m =1 e − λ m t V im V jm , (2)where λ m is the m -th eigenvalue of L , V im is the i -th com-ponent of the eigenvector of L associated with λ m , and t = 0 without loss of generality. This well-known iden-tity allows us to connect the spectral perspective withrandom-walk simulation results: one can easily see thatthe average return probability R ( t ) is related to the traceof K ( t ) (or heat trace ) through the simple relationship[46] R ( t ) = 1 N N (cid:88) i =1 K ii ( t ) = 1 N N (cid:88) m =1 e − λ m t . (3)The eigenvalues λ m are all real and non-negative and, un-der the assumptions that the network is undirected andconnected, the 0 eigenvalue is unique and one can alwayschoose the labeling 0 = λ < λ ≤ λ ≤ · · · [6]. Asa consequence, a random walk always reaches a steadystate above a time scale given by the smallest nonzeroeigenvalues [6]. By making a continuum spectrum ap-proximation for λ ≥ λ , one can introduce the densityof states (eigenvalue density distribution) ρ ( λ ), so thatEq.3 can be approximated by its continuum limit R ( t ) ≈ (cid:90) ρ ( λ )e − λt dλ, (4)where the integral is dominated by the contribution ofthe lower spectral edge. Thus, under the present assump-tions, the density of states ρ ( λ ) and the return probabil-ity are related through a simple Laplace-transform op-eration. In lattices, which are endowed with an integerspectral dimension, this result is well known and leads to ρ ( λ ) ∼ λ ( d s / − , which shows the relationship betweenthe Laplacian spectra and the spectral dimension [40].This result is also well known in terms of vibrational fre-quencies ω ∝ √ λ in lattices and deterministic fractals,leading to a power-law density of states ˜ ρ ( ω ) ∼ ω d s − [8].While the approximation in Eq.4 holds in many cases,we expect the conclusions regarding d s to be radicallydifferent in case of hierarchical-modular networks, forwhich our numerical results reveal anomalous stretched-exponential tails of the return-probability function R ( t ).This anomaly is indeed reflected in the lower spectraledges and in particular in ρ ( λ ) as we show below. Itwas hypothesized in the past [17] that low eigenvaluesof L in such networks form a continuous spectral tail,a Lifshitz tail, in analogy with the random Hamilto-nian operators in a tight-binding Schr¨odinger equation[37]. In particular, it was noted that Lifshitz tails maybe relevant to assess the subcritical dynamics in modelsof epidemic spreading, in the framework of a linearizedquenched mean-field approximation [23]. The random-walk problem that we study here, instead, can always bemapped exactly to the quantum problem, with the energyeigenvalues E m of the quantum problem being replacedby the Laplacian eigenvalues λ m [37]. Under the hypoth-esis of Lifshitz tails, the integrated density of states of L (i.e. the cumulative eigenvalue density distribution) isexpected to exhibit a tail of the general form [35] N ( λ ) = c exp (cid:104) c ( λ − λ ) − d L2 (cid:105) , (5)where λ is the lower bound of the continuum spectrum,which we can set equal to 0 in the case of hierarchical-modular networks, as they possess vanishing spectralgaps (0 < λ (cid:28)
1) [17], and c and c are constants.The real number d L coincides with the space Euclideandimension in the original Lifshitz argument for contin-uum quantum problems. In the present discrete classicalcase, since that we cannot yet provide a Lifshitz-like ar-gument, we simply name d L the Lifshitz dimension ofthe problem. In the light of the above considerations,the density of states ρ ( λ ) = d N /dλ is dominated by thefollowing low- λ tail ρ ( λ ) ∼ exp( c λ − d L2 ) . (6)Observe that Eq.6 differs significantly from the abovepower-law relationship for lattices ρ ( λ ) ∼ λ ( d s / − . Issuch a difference a spectral signature of the anomalousbehaviour (i.e. the stretched-exponential tail of the re-turn probabilities R ( t ) and the lack of a well-definedspectral dimension d s ) encountered in RW simulations?As the two representations connect through Eq.4, onecan compute R ( t ) as in Eq.4, using the hypothesis ofa Lifshitz tail from Eq.6. While the integral involvedin the calculation, of the form w ( t ) = (cid:82) e g ( λ,t ) dλ with g ( λ, t ) = c λ − d L / − λt , is highly non-trivial in the caseof real d L , here we are only interested in its asymptotic t → ∞ behavior, which is captured by the values of λ for which g ( λ, t ) is maximum. This is readily obtainedthrough the saddle-point approximation w ( t ) ≈ e g ( λ ∗ ,t ) ,with λ ∗ the location of such maximum, leading to thefinal result R ( t ) ∼ exp (cid:18) − t d L2+ d L (cid:19) . (7)Eq.7 remarkably recovers a stretched exponential tail be-havior, as reported above for computer simulations, con-firming for the first time that the dynamical slowing andlack of spectral dimension can be attributed to the exis-tence of Lifshitz tails and providing us with an interpre-tation of the anomalous exponent in terms of the Lifshitzdimension, d L β = d L d L . (8)We can conclude that in the present hierarchical-modularnetwork model, not only Lifshitz tails explain the anoma-lous dynamics and the lack of a well-defined spectral di-mension, but also d L provides us with a meaningful di-mensionality measure, generalizing the behavior of quan-tum systems in the continuum, where the Lifshitz dimen-sion identifies the spatial dimension.
10 12 14ln1/0.20.00.20.40.6 ( / d L ) l n [ l n ()] = 1.00= 1.25= 1.50= 1.75= 2.00 FIG. 3. Lower spectral edge of L . Using the rescaling fromEq.9, Lifshitz tails appear as straight lines. By choosing val-ues of d L obtained from the RW simulation results throughEq.8, Lifshitz tails collapse in a single curve, confirming thatthe anomalous dynamical behavior has its origin in the spec-tral properties of L . So far, we have only hypothesized that the lower spec-tral edge of L exhibits a Lifshitz tail of the form given byEqs.5 and 6. Now we corroborate such a hypothesis byverifying, not only that the integrated density of states N ( λ ) obeys the tail behavior in Eq.5, but also that theexponent d L / β of the dynamical simulation, as predicted bythe result in Eq.8. To this end, we notice that accordingto the prediction above, one expectsln[ − ln N ( λ )] ∼ ( d L /
2) ln 1 /λ (9)for large 1 /λ . We obtain d L = 2 β/ (1 − β ) from Eq.8, andusing the values of β obtained from the initial randomwalk simulations one can easily verify Eq.9 by comput-ing the lower spectral edges of hierarchical-modular net-works of the same type. Computational results, shownin Fig. 3 clearly confirm the linear dependence predictedby Eq.9 for small values of λ . In other words, the predic-tion based on the Lifshitz tails assumption is correct: thespectra of hierarchical-modular networks exhibit Lifshitztails, with an associated Lifshitz dimension d L , and theirconcomitant anomalous dynamical behavior is controlledby d L .Moreover, even if not explicitly analyzed here, theeigenvalues in the Lifshitz tail have strongly localizedeigenvectors, meaning that their components vanish al-most everywhere except in specific network locations suchas moduli [6, 17]. This property, known as eigenvectorlocalization [35] is analogous to the case of disorderedquantum systems, where localization stands for absenceof diffusion, and has been observed in models of epidemicspreading on networks [23, 47–49], as well in problemsinspired by brain connectivity [17, 44] and biological ma-terials [50]. In the particular case of brain dynamics,localization can play a key role in allowing for task seg-regation.It is noteworthy that the emergence of classical Lif-shitz tails in network spectra has been rigorously provedin Erd˝os R´enyi graphs below the percolation threshold[37], i.e. for networks that have not yet developed a giantconnected component. Our results suggest a remarkableproperty of hierarchical-modular networks, which exhibitLifshitz tails while being connected, i.e. while possessinga single connected component. We believe that Lifshitztails, and the resulting anomalous dynamical behavior,may be observed in general in network models exhibitingsimilar localization properties, such as hierarchical treesdisplaying patchy percolation [51], and dense hierarchicalDyson networks [34], where ergodicity is known to breakdown in the thermodynamic limit [32, 33]. In fact, wepropose the emergence of Lifshitz tails and their associ-ated Lifshitz dimension as a criterion for the existence oflocalization in networks.Our focus on the spectral dimension and its unde-fined nature in hierarchical networks allows us to estab-lish a connection between network structural propertiesand anomalous dynamics in systems such as brain net-works, where the ongoing structure vs. function debatehas long dealt with the issues of relating activity pattersto specific anatomical arrangements, or alternatively pre-senting them as emergent , or self-organized . Our resultsclearly show a connection between dynamical slowing-down, localization properties, and Laplacian spectra; letus emphasize that such a correspondence is clean-cut be-cause of the simplicity of the random walk model, andpossibly because of the simplifying assumptions in thechoice of our network model. While diffusion –lackingany form of non-linearity– is arguably a very crude sim-plification of neural dynamics on the structurally com-plex human connectome [19], it has been found that theeigenvectors of a diffusion problem on the connectomeare relevant in predicting functional patterns of neuralactivity [52, 53]. More in general, the Laplacian matrixprovides the linearization of oscillator models near a syn-chronization transition [54] and the relevance of its spec-trum in the problem of brain synchronization has beendiscussed in the literature and related to the observationof frustrated synchronization [26–28, 41]. While some ofthose results were based on the hypothesis of the exis-tence of Lifshitz tails, here we are able to prove such ahypothesis and rationalize those results within a propertheoretical framework, where the slowing down of syn-chronization processes is governed by the Lifshitz dimen-sion d L , which effectively tunes the dynamical anoma-lies. Let us also note that, while the approach here re-sorts to unweighted networks, the Laplacian formalismlends itself to the introduction of weights [32, 33], a quan-tity that in neuroimaging encodes the number of connec-tions between pairs of brain regions. Beyond the case ofpairwise interactions, recent advances in integrating theconcepts of diffusion and spectral dimension within thebroader field of algebraic topology and simplicial com- plexes [55–57] provide a promising avenue to strengthenthe theoretical framework for the localization phenom-ena that we discuss here to describe, for instance, sys-tems with higher-order interactions between their com-ponents/nodes.In conclusion, we have established a theoretical frame-work for the prediction of anomalous dynamics in hierar-chical network models of interest in brain modelling. Toour knowledge, hierarchical-modular networks constitutethe first heterogeneous network model displaying Lifshitztails above the percolation threshold, and the first notto exhibit a power-law behavior for the average returnprobability and a well-defined spectral dimension. Beingable to connect these two singular features allows us torationalize previous experimental observations of activ-ity localization in the brain and their numerical models,where spectral anomalies and Lifshitz tails where onlyhypothesized. We believe that these results will stim-ulate interest and further work, in e.g. computationalneuroscience, as a way to advance the knowledge on howthe brain achieves an optimal balance between segrega-tion (localization on specific moduli) and integration. Inparticular, we plan to extend our approach to novel net-work models of brain connectivity, including importantarchitectural features such as a prominent core-peripheryor rich-club organization [58–60]. Finally, we are con-fident that the present framework will provide us withmore powerful tools for the tunability and controllabilityof network models exhibiting strong localization, relevantin the design of synthetic networks for brain-inspired neu-romorphic computing.We acknowledge the Deutsches Forschungsgemein-schaft through grants MO 3049/1-1, MO 3049/3-1and GRK 2423 FRASCAL, the Spanish Ministryand Agencia Estatal de investigaci´on (AEI) throughgrants FIS2017-84256-P (European Regional Develop-ment Fund(ERDF), as well as the Consejer´ıa deConocimiento, Investigaci´on y Universidad, Junta deAndaluc´ıa and ERDF, Ref. A-FQM-175-UGR18 andSOMM17/6105/UGR and for financial support. [1] T. M. Liggett, Interacting particle systems , Vol. 276(Springer Science & Business Media, 2012).[2] A. Barrat, M. Barthelemy, and A. Vespignani,
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