Unweighted Graphs Research Articles

A hybrid GA-PSO strategy for computing task offloading towards MES scenarios.

As a new type of computing paradigm closer to service terminals, mobile edge computing (MEC), can meet the requirements of computing-intensive and delay-sensitive applications. In addition, it can also reduce the burden on mobile terminals by offloading computing. Due to cost issues, results in the deployment density of mobile edge servers (MES) is restricted in real scenario, whereas the suitable MES should be chosen for better performance. Therefore, this article proposes a task offloading strategy under the sparse MES density deployment scenario. Commonly, mobile terminals may reach MES through varied access points (AP) based on multi-hop transmitting mode. The transmission delay and processing delay caused by the selection of AP and MES will affect the performance of MEC. For the purpose of reducing the transmission delay due to system load balancing and superfluous multi-hop, we formulated the multi-objective optimization problem. The optimization goals are the workload balancing of edge servers and the completion delay of all task offloading. We express the formulated system as an undirected and unweighted graph, and we propose a hybrid genetic particle swarm algorithm based on two-dimensional genes (GA-PSO). Simulation results show that the hybrid GA-PSO algorithm does not outperform state-of-the-art GA and NSA algorithms in obtaining all task offloading delays. However, the workload by standard deviation approach is about 90% lower than that of the GA and NSA algorithms, which effectively optimizes the performance of load balancing and verifies the effectiveness of the proposed algorithm.

Interlayer Link Prediction in Multiplex Social Networks Based on Multiple Types of Consistency Between Embedding Vectors.

Online users are typically active on multiple social media networks (SMNs), which constitute a multiplex social network. With improvements in cybersecurity awareness, users increasingly choose different usernames and provide different profiles on different SMNs. Thus, it is becoming increasingly challenging to determine whether given accounts on different SMNs belong to the same user; this can be expressed as an interlayer link prediction problem in a multiplex network. To address the challenge of predicting interlayer links, feature or structure information is leveraged. Existing methods that use network embedding techniques to address this problem focus on learning a mapping function to unify all nodes into a common latent representation space for prediction; positional relationships between unmatched nodes and their common matched neighbors (CMNs) are not utilized. Furthermore, the layers are often modeled as unweighted graphs, ignoring the strengths of the relationships between nodes. To address these limitations, we propose a framework based on multiple types of consistency between embedding vectors (MulCEVs). In MulCEV, the traditional embedding-based method is applied to obtain the degree of consistency between the vectors representing the unmatched nodes, and a proposed distance consistency index based on the positions of nodes in each latent space provides additional clues for prediction. By associating these two types of consistency, the effective information in the latent spaces is fully utilized. In addition, MulCEV models the layers as weighted graphs to obtain representation. In this way, the higher the strength of the relationship between nodes, the more similar their embedding vectors in the latent representation space will be. The results of our experiments on several real-world and synthetic datasets demonstrate that the proposed MulCEV framework markedly outperforms current embedding-based methods, especially when the number of training iterations is small.

Detecting Anomalous Graphs in Labeled Multi-Graph Databases

Within a large database 𝒢 containing graphs with labeled nodes and directed, multi-edges; how can we detect the anomalous graphs? Most existing work are designed for plain (unlabeled) and/or simple (unweighted) graphs. We introduce CODEtect , the first approach that addresses the anomaly detection task for graph databases with such complex nature. To this end, it identifies a small representative set 𝒮 of structural patterns (i.e., node-labeled network motifs) that losslessly compress database 𝒢 as concisely as possible. Graphs that do not compress well are flagged as anomalous. CODEtect exhibits two novel building blocks: (i) a motif-based lossless graph encoding scheme, and (ii) fast memory-efficient search algorithms for 𝒮. We show the effectiveness of CODEtect on transaction graph databases from three different corporations and statistically similar synthetic datasets, where existing baselines adjusted for the task fall behind significantly, across different types of anomalies and performance metrics.

Improved Merlin–Arthur Protocols for Central Problems in Fine-Grained Complexity

In a Merlin–Arthur proof system, the proof verifier (Arthur) accepts valid proofs (from Merlin) with probability 1, and rejects invalid proofs with probability arbitrarily close to 1. The running time of such a system is defined to be the length of Merlin’s proof plus the running time of Arthur. We provide new Merlin–Arthur proof systems for some key problems in fine-grained complexity. In several cases our proof systems have optimal running time. Our main results include:Certifying that a list of n integers has no 3-SUM solution can be done in Merlin–Arthur time tilde{O}(n). Previously, Carmosino et al. [ITCS 2016] showed that the problem has a nondeterministic algorithm running in tilde{O}(n^{1.5}) time (that is, there is a proof system with proofs of length tilde{O}(n^{1.5}) and a deterministic verifier running in tilde{O}(n^{1.5}) time).Counting the number of k-cliques with total edge weight equal to zero in an n-node graph can be done in Merlin–Arthur time {tilde{O}}(n^{lceil k/2rceil }) (where kge 3). For odd k, this bound can be further improved for sparse graphs: for example, counting the number of zero-weight triangles in an m-edge graph can be done in Merlin–Arthur time {tilde{O}}(m). Previous Merlin–Arthur protocols by Williams [CCC’16] and Björklund and Kaski [PODC’16] could only count k-cliques in unweighted graphs, and had worse running times for small k.Computing the All-Pairs Shortest Distances matrix for an n-node graph can be done in Merlin–Arthur time tilde{O}(n^2). Note this is optimal, as the matrix can have Omega (n^2) nonzero entries in general. Previously, Carmosino et al. [ITCS 2016] showed that this problem has an tilde{O}(n^{2.94}) nondeterministic time algorithm.Certifying that an n-variable k-CNF is unsatisfiable can be done in Merlin–Arthur time 2^{n/2 - n/O(k)}. We also observe an algebrization barrier for the previous 2^{n/2}cdot textrm{poly}(n)-time Merlin–Arthur protocol of R. Williams [CCC’16] for #SAT: in particular, his protocol algebrizes, and we observe there is no algebrizing protocol for k-UNSAT running in 2^{n/2}/n^{omega (1)} time. Therefore we have to exploit non-algebrizing properties to obtain our new protocol.Certifying a Quantified Boolean Formula is true can be done in Merlin–Arthur time 2^{4n/5}cdot textrm{poly}(n). Previously, the only nontrivial result known along these lines was an Arthur–Merlin–Arthur protocol (where Merlin’s proof depends on some of Arthur’s coins) running in 2^{2n/3}cdot textrm{poly}(n) time. Due to the centrality of these problems in fine-grained complexity, our results have consequences for many other problems of interest. For example, our work implies that certifying there is no Subset Sum solution to n integers can be done in Merlin–Arthur time 2^{n/3}cdot textrm{poly}(n), improving on the previous best protocol by Nederlof [IPL 2017] which took 2^{0.49991n}cdot textrm{poly}(n) time.

Min-max coverage problems on tree-like metrics

We consider a number of min-max coverage problems. In each problem, the input is an unweighted graph G and an integer k, and possibly some additional information, such as a root vertex r. In the Min-Max Path Cover problem, the task is to cover all vertices of the graph by k walks, minimizing the length of the longest walk. The variant of Min-Max Path Cover in which all walks start and end at the same prescribed root vertex r is called the k-Traveling Salesmen Problem. In the Min-Max Tree Cover problem, the task is to cover all vertices of the graph by k trees, minimizing the size (number of edges) of the largest tree. In the rooted version, Min-Max k-Rooted Tree Cover, the input also contains k roots r1, ., rk, and the ith tree must contain the root ri. These four problems are all known to be APX-hard and to admit a constant-factor approximation. In this paper, we initiate the systematic study of these problems on trees and, more generally, on graphs of constant treewidth. As opposed to most graph problems, all four of the above coverage problems remain NP-hard even when G is a tree. We obtain an nO(k)-time exact algorithm for all four problems on graphs of bounded treewidth. Our main contribution is a quasi-polynomial-time approximation scheme (QPTAS) for the k-Traveling Salesmen Problem, Min-Max Path Cover, and Min-Max Tree Cover on graphs of bounded treewidth.

Accurately modeling biased random walks on weighted networks using node2vec.

Accurately representing biological networks in a low-dimensional space, also known as network embedding, is a critical step in network-based machine learning and is carried out widely using node2vec, an unsupervised method based on biased random walks. However, while many networks, including functional gene interaction networks, are dense, weighted graphs, node2vec is fundamentally limited in its ability to use edge weights during the biased random walk generation process, thus under-using all the information in the network. Here, we present node2vec+, a natural extension of node2vec that accounts for edge weights when calculating walk biases and reduces to node2vec in the cases of unweighted graphs or unbiased walks. Using two synthetic datasets, we empirically show that node2vec+ is more robust to additive noise than node2vec in weighted graphs. Then, using genome-scale functional gene networks to solve a wide range of gene function and disease prediction tasks, we demonstrate the superior performance of node2vec+ over node2vec in the case of weighted graphs. Notably, due to the limited amount of training data in the gene classification tasks, graph neural networks such as GCN and GraphSAGE are outperformed by both node2vec and node2vec+. The data and code are available on GitHub at https://github.com/krishnanlab/node2vecplus_benchmarks. All additional data underlying this article are available on Zenodo at https://doi.org/10.5281/zenodo.7007164. Supplementary data are available at Bioinformatics online.

Structural analysis of SARS-CoV-2 Spike protein variants through graph embedding.

Since December 2019, severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) has affected almost all countries. The unprecedented spreading of this virus has led to the insurgence of many variants that impact protein sequence and structure that need continuous monitoring and analysis of the sequences to understand the genetic evolution and to prevent possible dangerous outcomes. Some variants causing the modification of the structure of the proteins, such as the Spike protein S, need to be monitored. Protein contact networks (PCNs) have been recently proposed as a modelling framework for protein structures. In such a framework, the protein structure is represented as an unweighted graph whose nodes are the central atoms of the backbones (C- ), and edges connect two atoms falling in the spatial distance between 4 and 7Å. PCN may also be a data-rich representation since we may add to each node/atom biological and topological information. Such formalism enables the possibility of using algorithms from graph theory to analyze the graph. In particular, we refer to graph embedding methods enabling the analysis of such graphs with deep learning methods. In this work, we explore the possibility of embedding PCN using Graph Neural Networks and then analyze in the embedded space each residue to distinguish mutated residues from non-mutated ones. In particular, we analyzed the structure of the Spike protein of the coronavirus. First, we obtained the PCNs of the Spike protein for the wild-type, , , and variants. Then we used the GraphSage embedding algorithm to obtain an unsupervised embedding. Then we analyzed the point of mutation in the embedded space. Results show the characteristics of the mutation point in the embedding space.

Detection of d 1 ⨂ d 2 dimensional bipartite entangled state: a graph theoretical approach

Braunstein et al have started the study of entanglement properties of the quantum states through graph theoretical approach. Their idea was to start from a simple unweighted graph G and then they have defined the quantum state from the Laplacian of the graph G. A lot of research had already been done using the similar idea. We ask here the opposite one i.e can we generate a graph from the density matrix? To investigate this question, we have constructed a unital map ϕ such that ϕ(ρ) = L ρ + ρ, where the quantum state is described by the density operator ρ. The entries of L ρ depends on the entries of the quantum state ρ and the entries are taken in such a way that L ρ satisfies all the properties of the Laplacian. This make possible to design a simple connected weighted graph from the Laplacian L ρ . We show that the constructed unital map ϕ characterize the quantum state with respect to its purity by showing that if the determinant of the matrix ϕ(ρ) − I is positive then the quantum state ρ represent a mixed state. Moreover, we study the positive partial transpose (PPT) criterion in terms of the spectrum of the density matrix under investigation and the spectrum of the Laplacian associated with the given density matrix. Furthermore, we derive the inequality between the minimum eigenvalue of the density matrix and the weight of the edges of the connected subgraph of a simple weighted graph to detect the entanglement of d 1 ⨂ d 2 dimensional bipartite quantum states. Lastly, we have illustrated our results with few examples.

Approximate distance oracles with improved stretch for sparse graphs

Thorup and Zwick [1] introduced the notion of approximate distance oracles, a data structure that produces for an n-vertex, m-edge weighted undirected graph G=(V,E), distance estimations in constant query time. They presented a distance oracle of size O(kn1+1/k) that given a pair of vertices u,v∈V at distance d(u,v) produces in O(k) time an estimation that is bounded by (2k−1)d(u,v), i.e., a (2k−1)-multiplicative approximation (stretch). Thorup and Zwick [1] presented also a lower bound based on the girth conjecture of Erdős.For sparse unweighted graphs (i.e., m=O˜(n)) the lower bound does not apply. Pǎtraşcu and Roditty [2] used the sparsity of the graph and obtained a distance oracle that uses O˜(n5/3) space, has O(1) query time and a stretch of 2. Pǎtraşcu et al. [3] presented infinitely many distance oracles with fractional stretch factors that for graphs with m=O˜(n) converge exactly to the integral stretch factors and the corresponding space bound of Thorup and Zwick.It is not known, however, whether graph sparsity can help to get a stretch which is better than (2k−1) using only O˜(kn1+1/k) space. In this paper we answer this open question and prove a separation between sparse and dense graphs by showing that using sparsity it is possible to obtain better stretch/space tradeoffs than those of Thorup and Zwick. We show that for every k≥2 there is a distance oracle of size O˜(km1+1/k) that produces in O(k) time an estimation d⁎(u,v) that satisfies d(u,v)≤d⁎(u,v)≤(2k−1)d(u,v)−4, for k>2, and d(u,v)≤d⁎(u,v)≤3d(u,v)−2, for k=2.Another contribution of this paper is a refined stretch analysis of Thorup and Zwick distance oracles that allows us to obtain a better understanding of this important data structure. We present simple conditions for every w∈V that characterize the exact scenarios in which every query that involves w produces an estimation of stretch strictly better than 2k−1, even in the case of dense graphs. We complement this contribution with an experiment on real world graphs. The main finding in the experiment is that different real world graphs are likely to satisfy the required conditions and hence the stretch of Thorup and Zwick distance oracles is much better than its worst case bound in these real world graphs.

Faster Cut Sparsification of Weighted Graphs

A cut sparsifier is a reweighted subgraph that maintains the weights of the cuts of the original graph up to a multiplicative factor of (1pm epsilon ). This paper considers computing cut sparsifiers of weighted graphs of size O(nlog (n)/epsilon ^2). Our algorithm computes such a sparsifier in time O(mcdot min (alpha (n)log (m/n),log (n))), both for graphs with polynomially bounded and unbounded integer weights, where alpha (cdot ) is the functional inverse of Ackermann’s function. This improves upon the state of the art by Benczúr and Karger (SICOMP, 2015), which takes O(mlog ^2 (n)) time. For unbounded weights, this directly gives the best known result for cut sparsification. Together with preprocessing by an algorithm of Fung et al. (SICOMP, 2019), this also gives the best known result for polynomially-weighted graphs. Consequently, this implies the fastest approximate min-cut algorithm, both for graphs with polynomial and unbounded weights. In particular, we show that it is possible to adapt the state of the art algorithm of Fung et al. for unweighted graphs to weighted graphs, by letting the partial maximum spanning forest (MSF) packing take the place of the Nagamochi–Ibaraki forest packing. MSF packings have previously been used by Abraham et al. (FOCS, 2016) in the dynamic setting, and are defined as follows: an M-partial MSF packing of G is a set mathcal {F}={F_1, ldots , F_M}, where F_i is a maximum spanning forest in G{setminus } bigcup _{j=1}^{i-1}F_j. Our method for computing (a sufficient estimation of) the MSF packing is the bottleneck in the running time of our sparsification algorithm.

Diameter, Eccentricities and Distance Oracle Computations on H-Minor Free Graphs and Graphs of Bounded (Distance) Vapnik–Chervonenkis Dimension

Under the strong exponential-time hypothesis, the diameter of general unweighted graphs cannot be computed in truly subquadratic time (in the size of the input), as shown by Roditty and Williams. Nevertheless there are several graph classes for which this can be done such as bounded-treewidth graphs, interval graphs, and planar graphs, to name a few. We propose to study unweighted graphs of constant distance Vapnik–Chervonenkis (VC)-dimension as a broad generalization of many such classes—where the distance VC-dimension of a graph is defined as the VC-dimension of its ball hypergraph whose hyperedges are the balls of all possible radii and centers in . In particular for any fixed , the class of -minor free graphs has distance VC-dimension at most . Our first main result is a Monte Carlo algorithm that on graphs of distance VC-dimension at most , for any fixed , either computes the diameter or concludes that it is larger than in time , where only depends on and the notation suppresses polylogarithmic factors. We thus obtain a truly subquadratic-time parameterized algorithm for computing the diameter on such graphs. Then as a byproduct of our approach, we get a truly subquadratic-time randomized algorithm for constant diameter computation on all the nowhere dense graph classes. The latter classes include all proper minor-closed graph classes, bounded-degree graphs, and graphs of bounded expansion. Before our work, the only known such algorithm was resulting from an application of Courcelle’s theorem; see Grohe, Kreutzer, and Siebertz [J. ACM, 64 (2017), pp. 1–32]. For any graph of constant distance VC-dimension, we further prove the existence of an exact distance oracle in truly subquadratic space, that answers distance queries in truly sublinear time (in the number of vertices). The latter generalizes prior results on proper minor-closed graph classes to a much larger graph class. Finally, we show how to remove the dependency on for any graph class that excludes a fixed graph as a minor. More generally, our techniques apply to any graph with constant distance VC-dimension and polynomial expansion (or equivalently having strongly sublinear balanced separators). As a result for all such graphs one obtains a truly subquadratic-time deterministic algorithm for computing all the eccentricities, and thus both the diameter and the radius. Our approach can be generalized to the -minor free graphs with bounded positive integer weights. We note that all our algorithms for the diameter problem can be adapted for computing the radius, and more generally all the eccentricities. Our approach is based on the work of Chazelle and Welzl who proved the existence of spanning paths with strongly sublinear stabbing number for every hypergraph of constant VC-dimension. We show how to compute such paths efficiently by combining known algorithms for the stabbing number problem with a clever use of -nets, region decomposition, and other partition techniques.

A new network-based community detection algorithm for disjoint communities

A community is a group of people that shares something in common. The definition of the community can be generalized as things that have common properties. By using this definition, community detection can be used to solve different problems in various areas. In this study, we propose a new network-based community detection algorithm that can work on different types of datasets. The proposed algorithm works on unweighted graphs and determines the weight by using cosine similarity. We apply a bottom-up approach and find the disjoint communities. First, we accept each node as an independent community. Then, the merging process is applied by using the modularity value as a stopping criterion. We use real datasets and evaluate the algorithm with modularity, normalized mutual information, and performance metrics. In addition, we test our algorithm by using central nodes. We also take into consideration the number of communities in the case they are known. The proposed algorithm has high modularity and accuracy in different datasets.

Exponentially Faster Shortest Paths in the Congested Clique

We present improved deterministic algorithms for approximating shortest paths in the Congested Clique model of distributed computing. We obtain poly(log log n )-round algorithms for the following problems in unweighted undirected n -vertex graphs: ( 1 + ϵ )-approximation of multi-source shortest paths (MSSP) from O (√ n ) sources. (2 + ϵ )-approximation of all pairs shortest paths (APSP). (1 + ϵ , β)-approximation of APSP where β = O (log log n / ϵ ) log log n . These bounds improve exponentially over the state-of-the-art poly-logarithmic bounds due to [Censor-Hillel et al., PODC19]. It also provides the first nearly-additive bounds for the APSP problem in sub-polynomial time. Our approach is based on distinguishing between short and long distances based on some distance threshold t = O ( β / ϵ ) where β = O (log log n / ϵ ) log log n . Handling the long distances is done by devising a new algorithm for computing a sparse (1 + ϵ , β ) emulator with O ( n log log n ) edges. For the short distances, we provide distance-sensitive variants for the distance tool-kit of [Censor-Hillel et al., PODC19]. By exploiting the fact that this tool-kit should be applied only on local balls of radius t , their round complexities get improved from poly (log n ) to poly (log t ). Finally, our deterministic solutions for these problems are based on a derandomization scheme of a novel variant of the hitting set problem, which might be of independent interest.

Linear-time parameterized algorithms with limited local resources

We propose a new computational model for the study of massive data processing. Our model measures the complexity of reading the input data in terms of their very large size N and analyzes the computational cost in terms of a parameter k that characterizes the computational power provided by limited local computing resources. We develop new algorithmic techniques for solving well-known computational problems on the model. In particular, randomized algorithms of running time O(N+g1(k)) and space O(k2), with very high probability, are developed for the famous graph matching problem on unweighted and weighted graphs. More specifically, our algorithm for unweighted graphs finds a k-matching (i.e., a matching of k edges) in a general unweighted graph in time O(N+k2.5), and our algorithm for weighted graphs finds a maximum weighted k-matching in a general weighted graph in time O(N+k3log⁡k).

Fractional revival on non-cospectral vertices

Perfect state transfer and fractional revival can be used to move information between pairs of vertices in a quantum network. While perfect state transfer has received a lot of attention, fractional revival is newer and less studied. One problem is to determine the differences between perfect state transfer and fractional revival. If perfect state transfer occurs between two vertices in a graph, the vertices must be cospectral. Further if there is perfect state transfer between vertices a and b in a graph, there cannot be perfect state transfer from a to any other vertex. No examples of unweighted graphs with fractional revival between non-cospectral vertices were known; here we give an infinite family of such graphs. No examples of unweighted graphs where the pairs involved in fractional revival overlapped were known; we give examples of such graphs as well.

Generating target graph couplings for the quantum approximate optimization algorithm from native quantum hardware couplings

We present methods for constructing any target coupling graph using limited global controls in an Ising-like quantum spin system. Our approach is motivated by implementing the quantum approximate optimization algorithm (QAOA) on trapped-ion quantum hardware to find approximate solutions to MaxCut. We present a mathematical description of the problem and provide approximately optimal algorithmic constructions that generate arbitrary unweighted coupling graphs with $n$ nodes in $O(n)$ global entangling operations and weighted graphs with $m$ edges in $O(m)$ operations. These upper bounds are not tight in general, and we formulate a mixed-integer program to solve the graph coupling problem to optimality. We perform numeric experiments on small graphs with $n\ensuremath{\le}8$ and show that optimal sequences, which use fewer operations, can be found using mixed-integer programs. Noisy simulations of MaxCut QAOA show that our implementation is less susceptible to noise than the standard gate-based compilation.

A Game Theory Proof of Optimal Colorings Resilience to Strong Deviations

This paper provides a formal proof of the conjecture stating that optimal colorings in max k-cut games over unweighted and undirected graphs do not allow the formation of any strongly divergent coalition, i.e., a subset of nodes able to increase their own payoffs simultaneously. The result is obtained by means of a new method grounded on game theory, which consists in splitting the nodes of the graph into three subsets: the coalition itself, the coalition boundary and the nodes without relationship with the coalition. Moreover, we find additional results concerning the properties of optimal colorings.

Eccentricity queries and beyond using hub labels

Hub labeling schemes are popular methods for computing distances on road networks and other large complex networks, often answering to a query within a few microseconds for graphs with millions of edges. In this work, we study their algorithmic applications beyond distance queries. Indeed, several implementations of hub labels were reported to have good practical performances, both in terms of pre-processing time and maximum label size. There are also a few relevant graph classes for applications for which we know how to compute hub labelings with sublinear labels in quasi linear time. These positive results raise the question of what can be computed efficiently given a graph and a hub labeling of small maximum label size as input. We focus on eccentricity queries and distance-sum queries, for several versions of these problems on directed weighted graphs, that is in part motivated by their importance in facility location problems. On the negative side, we show conditional lower bounds for these above problems on unweighted undirected sparse graphs, via standard constructions from “Fine-grained” complexity. Specifically, under the Strong Exponential-Time Hypothesis (SETH), answering to these above queries requires Ω(|V|2−o(1)) pre-processing time or Ω(|V|1−o(1)) query time, even if we are given as input a hub labeling of maximum label size in ω(log⁡|V|). However, things take a different turn when the hub labels have a sublogarithmic size. Indeed, for every ε>0 there exists a δε>0 such that the following hold, being given a hub labeling of maximum label size ≤k: after pre-processing the labels in total O(2δε⋅k⋅|V|1+ε) time, we can compute both the eccentricity and the distance-sum of any vertex in O(2δε⋅k⋅|V|ε) time. Our data structure is a novel application of the orthogonal range query framework of Cabello and Knauer (2009) [17]. It can also be applied to the fast global computation of some topological indices. Finally, as a by-product of our approach, on any fixed class of unweighted graphs with bounded expansion, we can decide whether the diameter of an n-vertex graph in the class is at most k in fε(k)⋅n1+ε time for any ε>0, for some “explicit” function fε that depends on ε and the graph class considered. This result is further motivated by the empirical evidence that many classes of complex networks have bounded expansion (Demaine et al., 2019 [24]), and that some of these networks, such as the online social networks, have a relatively small diameter.

The structure of the genetic code as an optimal graph clustering problem.

The standard genetic code (SGC) is the set of rules by which genetic information is translated into proteins, from codons, i.e. triplets of nucleotides, to amino acids. The questions about the origin and the main factor responsible for the present structure of the code are still under a hot debate. Various methodologies have been used to study the features of the code and assess the level of its potential optimality. Here, we introduced a new general approach to evaluate the quality of the genetic code structure. This methodology comes from graph theory and allows us to describe new properties of the genetic code in terms of conductance. This parameter measures the robustness of codon groups against the potential changes in translation of the protein-coding sequences generated by single nucleotide substitutions. We described the genetic code as a partition of an undirected and unweighted graph, which makes the model general and universal. Using this approach, we showed that the structure of the genetic code is a solution to the graph clustering problem. We presented and discussed the structure of the codes that are optimal according to the conductance. Despite the fact that the standard genetic code is far from being optimal according to the conductance, its structure is characterised by many codon groups reaching the minimum conductance for their size. The SGC represents most likely a local minimum in terms of errors occurring in protein-coding sequences and their translation.

Non-Crossing Shortest Paths in Undirected Unweighted Planar Graphs in Linear Time

Given a set of terminal pairs on the external face of an undirected unweighted planar graph, we give a linear-time algorithm for computing the union of non-crossing shortest paths joining each terminal pair, if such paths exist. This allows us to compute distances between each terminal pair, within the same time bound. We also give a novel concept of incremental shortest path subgraph of a planar graph, i.e., a partition of the planar embedding in subregions that preserve distances, that can be of interest itself.