Exploring the topological sources of robustness against invasion in biological and technological networks

Cuesta, Fernando Alcalde; Sequeiros, Pablo González; Rojo, Álvaro Lozano

doi:10.1038/srep20666

Cited by 9 publications

(13 citation statements)

References 51 publications

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“…Initially motivated by our interest in the robustness of biological and technological networks against invasion [ 16 ], we decided to compute the fixation probability of all graphs with 10 vertices or less, totaling 11,989,764 graphs, to facilitate general searches without specific aims. As explained in Materials and Methods, we solved (by Gaussian elimination) the system of linear equations Eq 6 for each graph and for each fitness value varying from 0.25 to 10 with step size of 0.25.…”

Section: Discussionmentioning

confidence: 99%

Evolutionary regime transitions in structured populations

2018

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The evolutionary dynamics of a finite population where resident individuals are replaced by mutant ones depends on its spatial structure. Usually, the population adopts the form of an undirected graph where the place occupied by each individual is represented by a vertex and it is bidirectionally linked to the places that can be occupied by its offspring. There are undirected graph structures that act as amplifiers of selection increasing the probability that the offspring of an advantageous mutant spreads through the graph reaching any vertex. But there also are undirected graph structures acting as suppressors of selection where this probability is less than that of the same individual placed in a homogeneous population. Here, firstly, we present the distribution of these evolutionary regimes for all undirected graphs with N ≤ 10 vertices. Some of them exhibit transitions between different regimes when the mutant fitness increases. In particular, as it has been already observed for small-order random graphs, we show that most graphs of order N ≤ 10 are amplifiers of selection. Secondly, we describe examples of amplifiers of order 7 that become suppressors from some critical value. In fact, for graphs of order N ≤ 7, we apply computer-aided techniques to symbolically compute their fixation probability and then their evolutionary regime, as well as the critical values for which they change their regime. Thirdly, the same technique is applied to some families of highly symmetrical graphs as a mean to explore methods of suppressing selection. The existence of suppression mechanisms that reverse an amplification regime when fitness increases could have a great interest in biology and network science. Finally, the analysis of all graphs from order 8 to order 10 reveals a complex and rich evolutionary dynamics, with multiple transitions between different regimes, which have not been examined in detail until now.

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Section: Discussionmentioning

confidence: 99%

Evolutionary regime transitions in structured populations

2018

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“…Since VLSI circuits are disposed in the plane, it would be reasonable to derive Rent’s rule from placement-based algorithms making use of the geometrical information of the placed circuit. A similar argument could be applied to some networks studied in a previous paper 9 , namely the US Power Grid network 13 and the Internet2 academic network. However, it does not apply to neural networks, like that of the nematode Caenorhabditis elegans 5 , 9 , hierarchical modular graphs 8 , 9 or geometrical random graphs 8 , 9 .…”

Section: Introductionmentioning

confidence: 84%

“…It relates the average number of external connections or pins on a module and the average number of blocks within the module for partitions of computer logic graphs 2 . Besides its extensive application in Circuit Design at all scales, from SSI to GSI passing through VLSI, Rent’s rule has been used to study the interconnection complexity of biological networks 3 – 6 , as well some benchmark models and technological networks 7 – 9 . At origin, given a logic circuit, the relationship between the average number of blocks or cells B in a module in a given partition and the average number of pins P connecting each module with the others is where k is the average number of pins per logic block (also called Rent coefficient ) and p is the Rent exponent describing proportionality in a log-log scale.…”

Section: Introductionmentioning

confidence: 99%

“…However, it does not apply to neural networks, like that of the nematode Caenorhabditis elegans 5 , 9 , hierarchical modular graphs 8 , 9 or geometrical random graphs 8 , 9 . Accordingly, we adopt a topological point of view, which is also present in other authors’ works 5 , 8 , 9 , where Rent’s rule is directly deduced from graph partitioning without distinguishing where and how it is embedded. As explained by Bassett et al .…”

Section: Introductionmentioning

confidence: 99%

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A method for validating Rent’s rule for technological and biological networks

Cuesta

Sequeiros

Rojo

2017

Sci Rep

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Rent’s rule is empirical power law introduced in an effort to describe and optimize the wiring complexity of computer logic graphs. It is known that brain and neuronal networks also obey Rent’s rule, which is consistent with the idea that wiring costs play a fundamental role in brain evolution and development. Here we propose a method to validate this power law for a certain range of network partitions. This method is based on the bifurcation phenomenon that appears when the network is subjected to random alterations preserving its degree distribution. It has been tested on a set of VLSI circuits and real networks, including biological and technological ones. We also analyzed the effect of different types of random alterations on the Rentian scaling in order to test the influence of the degree distribution. There are network architectures quite sensitive to these randomization procedures with significant increases in the values of the Rent exponents.

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“…Initially motivated by our interest in the robustness of biological and technological networks against invasion [16], we found in [7] some graph structures suppressing the advantage of mutant individuals occupying their nodes for any fitness value. This seems particularly appealing for biological networks like brain and protein-protein interaction networks, but also in the tumor initiation process within healthy tissues as proposed in [17].…”

Section: Discussionmentioning

confidence: 99%

Phase transitions in evolutionary dynamics

Ál

2018

Preprint

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The evolutionary dynamics of a finite population where resident individuals are replaced by mutant ones depends on its spatial structure. The population adopts the form of an undirected graph where the place occupied by each individual is represented by a node and it is bidirectionally linked to the places that can be occupied by its clonal offspring. There are undirected graph structures that act as amplifiers of selection increasing the probability that the offspring of an advantageous mutant spreads through the graph reaching any node. But there also are undirected graph structures acting as suppressors of selection where, on the contrary, the fixation probability of an advantageous mutant is less than that of the same individual placed in a homogeneous population. Here, we show that some undirected graphs exhibit phase transitions between both evolutionary regimes when the mutant fitness varies. Firstly, as was already observed for small order random graphs, we show that most graphs of order 10 or less are amplifiers of selection or suppressors that become amplifiers from a unique transition phase. Secondly, we give examples of amplifiers of order 7 that become suppressors from some critical value. For graphs of order 6 and 7, we apply computer aided techniques to exactly determine their fixation probabilities and then their evolutionary regimes, as well as the exact critical values for which each graph changes its regime. A similar technique is used to explore a general method to suppress selection in bigger orders, namely from 8 to 15, up to some critical fitness value. The analysis of all graphs from order 8 to order 10 reveals a complex and rich evolutionary dynamics, which have not been examined in detail until now, and poses some new challenges in computing fixation probabilities and times of evolutionary graphs.

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Exploring the topological sources of robustness against invasion in biological and technological networks

Cited by 9 publications

References 51 publications

Evolutionary regime transitions in structured populations

Evolutionary regime transitions in structured populations

A method for validating Rent’s rule for technological and biological networks

Phase transitions in evolutionary dynamics

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