A scaling law for random walks on networks

Perkins, Theodore J.; Foxall, Eric; Glass, Leon; Edwards, Roderick

doi:10.1038/ncomms6121

Cited by 30 publications

(30 citation statements)

References 39 publications

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“…SSR processes can be related to diffusion processes on directed networks. For a specific example we demonstrated that the visiting times of nodes follow a Zipf's law, and could further reproduce very general recent findings of path-visit distributions in random walks on networks [27]. Here we presented results for a completed directed graph, however we conjecture that SSR processes on networks and the associated Zipf's law of node-visiting distributions are tightly related and are valid for much more general directed networks.…”

Section: Discussionsupporting

confidence: 86%

“…5 (b) (black dashed line), which shows the observed path rank distribution for the 2 N −1 = 16 paths. For cyclic processes, where at least one node participates in at least two distinct cycles, [27] predicts power-laws, which we clearly confirm for the cyclic Φ mix process with p exit = 0.3 (red line). Note that in our example node 1 alone is involved in 5 distinct cycles.…”

Section: Examplessupporting

confidence: 71%

“…The path that is most often taken through the network has rank 1, the second most popular path has rank 2, etc. Recent theoretical work [27] predicts a difference in the corresponding distributions for different values of p exit . According to [27] acyclic process are expected to show finite path rank distributions of no particular shape.…”

Section: Examplesmentioning

confidence: 98%

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Understanding scaling through history-dependent processes with collapsing sample space

Corominas-Murtra

Hanel

Thurner

2015

Proc. Natl. Acad. Sci. U.S.A.

107

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History-dependent processes are ubiquitous in natural and social systems. Many such stochastic processes, especially those that are associated with complex systems, become more constrained as they unfold, meaning that their sample space, or their set of possible outcomes, reduces as they age. We demonstrate that these samplespace-reducing (SSR) processes necessarily lead to Zipf's law in the rank distributions of their outcomes. We show that by adding noise to SSR processes the corresponding rank distributions remain exact power laws, p(x) ∼ x −λ , where the exponent directly corresponds to the mixing ratio of the SSR process and noise. This allows us to give a precise meaning to the scaling exponent in terms of the degree to which a given process reduces its sample space as it unfolds. Noisy SSR processes further allow us to explain a wide range of scaling exponents in frequency distributions ranging from α = 2 to ∞. We discuss several applications showing how SSR processes can be used to understand Zipf's law in word frequencies, and how they are related to diffusion processes in directed networks, or aging processes such as in fragmentation processes. SSR processes provide a new alternative to understand the origin of scaling in complex systems without the recourse to multiplicative, preferential, or self-organized critical processes.scaling laws | Zipf's law | random walks | path dependence | network diffusion A typical feature of aging is that the number of possible states in a system reduces as it ages. Whereas a newborn can become a composer, politician, physicist, actor, or anything else, the chances for a 65-y-old physics professor to become a concert pianist are practically zero. A characteristic feature of historydependent systems is that their sample space, defined as the set of all possible outcomes, changes over time. Many aging stochastic systems (such as career paths) become more constrained in their dynamics as they unfold (i.e., their sample space becomes smaller over time). An example for a sample-space-reducing (SSR) process is the formation of sentences. The first word in a sentence can be sampled from the sample space of all existing words. The choice of subsequent words is constrained by grammar and context, so that the second word can only be sampled from a smaller sample space. As the length of a sentence increases, the size of the sample space of word use typically reduces.Many history-dependent processes are characterized by powerlaw distribution functions in their frequency and rank distributions of their outcomes. The most famous example is the rank distribution of word frequencies in texts, which follows a power law with an approximate exponent of −1, the so-called Zipf's law (1). Zipf's law has been found in countless natural and social phenomena, including gene expression patterns (2), human behavioral sequences (3), fluctuations in financial markets (4), scientific citations (5, 6), distributions of city (7) and firm sizes (8, 9), and many more (see, e.g., ref. 10). (Some of thes...

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Section: Discussionsupporting

confidence: 86%

Section: Examplessupporting

confidence: 71%

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Understanding scaling through history-dependent processes with collapsing sample space

Corominas-Murtra

Hanel

Thurner

2015

Proc. Natl. Acad. Sci. U.S.A.

107

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“…We call diffusion processes on DAG structures targeted diffusion, since, in this type network, diffusion is targeted towards a set of target or sink nodes, see figure 1(e). The targeted diffusion results we present here are in line with recent findings reported in [29].…”

supporting

confidence: 93%

Extreme robustness of scaling in sample space reducing processes explains Zipf’s law in diffusion on directed networks

2016

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It has been shown recently that a specific class of path-dependent stochastic processes, which reduce their sample space as they unfold, lead to exact scaling laws in frequency and rank distributions. Such sample space reducing processes offer an alternative new mechanism to understand the emergence of scaling in countless processes. The corresponding power law exponents were shown to be related to noise levels in the process. Here we show that the emergence of scaling is not limited to the simplest SSRPs, but holds for a huge domain of stochastic processes that are characterised by non-uniform prior distributions. We demonstrate mathematically that in the absence of noise the scaling exponents converge to −1 (Zipf's law) for almost all prior distributions. As a consequence it becomes possible to fully understand targeted diffusion on weighted directed networks and its associated scaling laws in node visit distributions. The presence of cycles can be properly interpreted as playing the same role as noise in SSRPs and, accordingly, determine the scaling exponents. The result that Zipf's law emerges as a generic feature of diffusion on networks, regardless of its details, and that the exponent of visiting times is related to the amount of cycles in a network could be relevant for a series of applications in traffic-, transport-and supply chain management. intuitive way. SSRPs add an alternative and independent route to understand the origin of scaling (Zipf's law in particular) to the well known classical ways [14,15], criticality [16], self-organised criticality [17,18], multiplicative processes with constraints [19][20][21], and preferential attachment models [22,23]. Beyond their transparent mathematical tractability, SSRPs seem to have a wide applicability, including diffusion on complete directed acyclical graphs [13], quantitative linguistics [24], record statistics [25,26], and fragmentation processes [27].SSRPs can be seen as very specific non-standard sampling processes, with a directional bias or a symmetry breaking mechanism. In the same pictorial view as above a standard sampling processes can be depicted as a ball bouncing randomly to the left and to the right (without a directional bias as in the SSRP) over a set of states, see figure 1(b). The ball samples the states with a uniform prior probability, meaning that all states are sampled with equal probability. A situation with non-uniform priors is shown in figure 1(c) where the different widths of boxes represent the probability to hit a particular state. In a standard sampling process exactly this non-uniform prior distribution will be recovered.So far, SSRPs have been studied for the simplest case only, where the potential outcomes or states are sampled from an underlying uniform prior distribution [13]. In this paper we demonstrate that a much wider class of SSRPs leads to exact scaling laws. In particular we will show that SSRPs lead to Zipf's law irrespective of the underlying prior distributions. This is schematically shown in figure 1(d), wher...

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“…Our current statistical inference methods in modelling cancer Waddington landscapes (network state-space) mainly comprise of graph theory, classical information theory, Bayesian networks, cluster analysis and machine learning algorithms (9)(10)(11). Due to the stochastic nature of molecules, the transitions between the attractors of the landscape are defined by random walks on a network (i.e., Brownian motion) (12). That is, the fluctuations around the epigenetic barriers of the landscapes are governed by diffusion equations (13)(14)(15).…”

mentioning

confidence: 99%

Cancer: A Turbulence Problem

Uthamacumaran

2019

Preprint

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As we transition towards an era of Computational Medicine, our mathematical models of cancer dynamics must be revised. As such, recent evidence supports the perspective that cancermicroenvironment interactions consist of turbulent flows and strange attractor dynamics. Cancer pattern formation, invasion-growth dynamics, protein folding kinetics, stem cell fate bifurcations and metastatic invasion are discussed within the context of hydrodynamical turbulence. Cancer is presented as a three-dimensional Navier-Stokes equations global regularity, smoothness and existence problem. POSTULATE:Cancer stem cells are strange attractors on the Waddington energy landscape governed by turbulence dynamics. EMERGENCECancers are complex systems. The Hanahan-Weinberg 'hallmarks' are an inadequate representation of these dynamical systems. Complex systems are systems in which the interactions of its elements and parts cannot be deduced due to interdependence. Some general characteristics of complex systems include nonlinearity, emergence, computational irreducibility, unpredictability, nonequilibrium statistical mechanics and intractability. In simple terms, the concerted whole cannot be defined by the sum of its interacting parts. For example, the flow of exosomes between cancer cells are emergent characteristics of tumor ecosystems. It is impossible to understand exosomes without complex systems approaches. Exosomes are heterogeneous nano-scaled packets of information forming complex long-range communication networks. Along with ctDNA (circulating tumor DNA), exosomes are emerging targets for early tumor detection from liquid biopsies of patients (1, 2). Human embryonic stem cells-derived exosomes have shown capability of reprogramming malignant cancer phenotypes to benign-like fates, indicating exosomes are potent phenotype reprogramming machineries (3). However, the identity of cancer stem cells remains ambiguous. It is debated whether cancer stem cells conform a small subset of the tumor hierarchy or whether all cancer cells are potentially stem cells (i.e., have the potency of phenotypic plasticity and unlimited replication). In attempt to reconcile such problems, precision oncology is transitioning towards the use of artificial intelligence and machine learning algorithms in guiding clinical decision-making (4). Machine intelligence is emerging as a powerful tool in deciphering cancer ecosystems.Each tumor exhibits spatio-temporal heterogeneity and emergent properties that are unpredictable. For example, certain malignant melanomas spontaneously regress, a property not recognized as a hallmark (5). Another example, PEDF (pigment epithelium-derived factor) upregulated by extracellular vesicle bodies-mediated EGFRvIII promotes the self-renewal and propagation of GSCs (glioma stem cells) (6, 7). However, PEDF is an anti-angiogenic factor and GSCs require vascular interactions for development. Such complex feedback loops are emergent properties of complex systems. Certain tumors exhibit vasculogenic mimicry, the spontaneous form...

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A scaling law for random walks on networks

Cited by 30 publications

References 39 publications

Understanding scaling through history-dependent processes with collapsing sample space

Understanding scaling through history-dependent processes with collapsing sample space

Extreme robustness of scaling in sample space reducing processes explains Zipf’s law in diffusion on directed networks

Cancer: A Turbulence Problem

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