Network Analysis and Optimization of Animal Transports
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Optimal Subgraph Structures in Scale-Free Configuration
Submitted to the Annals of Applied Probability OPTIMAL SUBGRAPH STRUCTURES IN SCALE-FREE CONFIGURATION MODELS , By Remco van der Hofstad∗ §, Johan S. H. van , , Leeuwaarden∗ ¶ and Clara Stegehuis∗ k Eindhoven University of Technology§, Tilburg University¶ and Twente Universityk Subgraphs reveal information about the geometry and function- alities of complex networks. For scale-free networks with unbounded degree fluctuations, we obtain the asymptotics of the number of times a small connected graph occurs as a subgraph or as an induced sub- graph. We obtain these results by analyzing the configuration model with degree exponent τ (2, 3) and introducing a novel class of op- ∈ timization problems. For any given subgraph, the unique optimizer describes the degrees of the vertices that together span the subgraph. We find that subgraphs typically occur between vertices with specific degree ranges. In this way, we can count and characterize all sub- graphs. We refrain from double counting in the case of multi-edges, essentially counting the subgraphs in the erased configuration model. 1. Introduction Scale-free networks often have degree distributions that follow power laws with exponent τ (2, 3) [1, 11, 22, 34]. Many net- ∈ works have been reported to satisfy these conditions, including metabolic networks, the internet and social networks. Scale-free networks come with the presence of hubs, i.e., vertices of extremely high degrees. Another property of real-world scale-free networks is that the clustering coefficient (the probability that two uniformly chosen neighbors of a vertex are neighbors themselves) decreases with the vertex degree [4, 10, 24, 32, 34], again following a power law. -
Detecting Statistically Significant Communities
1 Detecting Statistically Significant Communities Zengyou He, Hao Liang, Zheng Chen, Can Zhao, Yan Liu Abstract—Community detection is a key data analysis problem across different fields. During the past decades, numerous algorithms have been proposed to address this issue. However, most work on community detection does not address the issue of statistical significance. Although some research efforts have been made towards mining statistically significant communities, deriving an analytical solution of p-value for one community under the configuration model is still a challenging mission that remains unsolved. The configuration model is a widely used random graph model in community detection, in which the degree of each node is preserved in the generated random networks. To partially fulfill this void, we present a tight upper bound on the p-value of a single community under the configuration model, which can be used for quantifying the statistical significance of each community analytically. Meanwhile, we present a local search method to detect statistically significant communities in an iterative manner. Experimental results demonstrate that our method is comparable with the competing methods on detecting statistically significant communities. Index Terms—Community Detection, Random Graphs, Configuration Model, Statistical Significance. F 1 INTRODUCTION ETWORKS are widely used for modeling the structure function that is able to always achieve the best performance N of complex systems in many fields, such as biology, in all possible scenarios. engineering, and social science. Within the networks, some However, most of these objective functions (metrics) do vertices with similar properties will form communities that not address the issue of statistical significance of commu- have more internal connections and less external links. -
Processes on Complex Networks. Percolation
Chapter 5 Processes on complex networks. Percolation 77 Up till now we discussed the structure of the complex networks. The actual reason to study this structure is to understand how this structure influences the behavior of random processes on networks. I will talk about two such processes. The first one is the percolation process. The second one is the spread of epidemics. There are a lot of open problems in this area, the main of which can be innocently formulated as: How the network topology influences the dynamics of random processes on this network. We are still quite far from a definite answer to this question. 5.1 Percolation 5.1.1 Introduction to percolation Percolation is one of the simplest processes that exhibit the critical phenomena or phase transition. This means that there is a parameter in the system, whose small change yields a large change in the system behavior. To define the percolation process, consider a graph, that has a large connected component. In the classical settings, percolation was actually studied on infinite graphs, whose vertices constitute the set Zd, and edges connect each vertex with nearest neighbors, but we consider general random graphs. We have parameter ϕ, which is the probability that any edge present in the underlying graph is open or closed (an event with probability 1 − ϕ) independently of the other edges. Actually, if we talk about edges being open or closed, this means that we discuss bond percolation. It is also possible to talk about the vertices being open or closed, and this is called site percolation. -
Percolation Thresholds for Robust Network Connectivity
Percolation Thresholds for Robust Network Connectivity Arman Mohseni-Kabir1, Mihir Pant2, Don Towsley3, Saikat Guha4, Ananthram Swami5 1. Physics Department, University of Massachusetts Amherst, [email protected] 2. Massachusetts Institute of Technology 3. College of Information and Computer Sciences, University of Massachusetts Amherst 4. College of Optical Sciences, University of Arizona 5. Army Research Laboratory Abstract Communication networks, power grids, and transportation networks are all examples of networks whose performance depends on reliable connectivity of their underlying network components even in the presence of usual network dynamics due to mobility, node or edge failures, and varying traffic loads. Percolation theory quantifies the threshold value of a local control parameter such as a node occupation (resp., deletion) probability or an edge activation (resp., removal) probability above (resp., below) which there exists a giant connected component (GCC), a connected component comprising of a number of occupied nodes and active edges whose size is proportional to the size of the network itself. Any pair of occupied nodes in the GCC is connected via at least one path comprised of active edges and occupied nodes. The mere existence of the GCC itself does not guarantee that the long-range connectivity would be robust, e.g., to random link or node failures due to network dynamics. In this paper, we explore new percolation thresholds that guarantee not only spanning network connectivity, but also robustness. We define and analyze four measures of robust network connectivity, explore their interrelationships, and numerically evaluate the respective robust percolation thresholds arXiv:2006.14496v1 [physics.soc-ph] 25 Jun 2020 for the 2D square lattice. -
Robustness and Assortativity for Diffusion-Like Processes in Scale
epl draft Robustness and Assortativity for Diffusion-like Processes in Scale- free Networks G. D’Agostino1, A. Scala2,3, V. Zlatic´4 and G. Caldarelli5,3 1 ENEA - CR ”Casaccia” - Via Anguillarese 301 00123, Roma - Italy 2 CNR-ISC and Department of Physics, University of Rome “Sapienza” P.le Aldo Moro 5 00185 Rome, Italy 3 London Institute of Mathematical Sciences, 22 South Audley St Mayfair London W1K 2NY, UK 4 Theoretical Physics Division, Rudjer Boˇskovi´cInstitute, P.O.Box 180, HR-10002 Zagreb, Croatia 5 IMT Lucca Institute for Advanced Studies, Piazza S. Ponziano 6, Lucca, 55100, Italy PACS 89.75.Hc – Networks and genealogical trees PACS 05.70.Ln – Nonequilibrium and irreversible thermodynamics PACS 87.23.Ge – Dynamics of social systems Abstract – By analysing the diffusive dynamics of epidemics and of distress in complex networks, we study the effect of the assortativity on the robustness of the networks. We first determine by spectral analysis the thresholds above which epidemics/failures can spread; we then calculate the slowest diffusional times. Our results shows that disassortative networks exhibit a higher epidemi- ological threshold and are therefore easier to immunize, while in assortative networks there is a longer time for intervention before epidemic/failure spreads. Moreover, we study by computer sim- ulations the sandpile cascade model, a diffusive model of distress propagation (financial contagion). We show that, while assortative networks are more prone to the propagation of epidemic/failures, degree-targeted immunization policies increases their resilience to systemic risk. Introduction. – The heterogeneity in the distribu- propagation of an epidemic [13–15]. -
Ollivier Ricci Curvature of Directed Hypergraphs Marzieh Eidi1* & Jürgen Jost1,2
www.nature.com/scientificreports OPEN Ollivier Ricci curvature of directed hypergraphs Marzieh Eidi1* & Jürgen Jost1,2 Many empirical networks incorporate higher order relations between elements and therefore are naturally modelled as, possibly directed and/or weighted, hypergraphs, rather than merely as graphs. In order to develop a systematic tool for the statistical analysis of such hypergraph, we propose a general defnition of Ricci curvature on directed hypergraphs and explore the consequences of that defnition. The defnition generalizes Ollivier’s defnition for graphs. It involves a carefully designed optimal transport problem between sets of vertices. While the defnition looks somewhat complex, in the end we shall be able to express our curvature in a very simple formula, κ = µ0 − µ2 − 2µ3 . This formula simply counts the fraction of vertices that have to be moved by distances 0, 2 or 3 in an optimal transport plan. We can then characterize various classes of hypergraphs by their curvature. Principles of network analysis. Network analysis constitutes one of the success stories in the study of complex systems3,6,11. For the mathematical analysis, a network is modelled as a (perhaps weighted and/or directed) graph. One can then look at certain graph theoretical properties of an empirical network, like its degree or motiv distribution, its assortativity or clustering coefcient, the spectrum of its Laplacian, and so on. One can also compare an empirical network with certain deterministic or random theoretical models. Successful as this analysis clearly is, we nevertheless see two important limitations. One is that many of the prominent concepts and quantities used in the analysis of empirical networks are node based, like the degree sequence. -
Assortativity and Mixing
Assortativity and Assortativity and Mixing General mixing between node categories Mixing Assortativity and Mixing Definition Definition I Assume types of nodes are countable, and are Complex Networks General mixing General mixing Assortativity by assigned numbers 1, 2, 3, . Assortativity by CSYS/MATH 303, Spring, 2011 degree degree I Consider networks with directed edges. Contagion Contagion References an edge connects a node of type µ References e = Pr Prof. Peter Dodds µν to a node of type ν Department of Mathematics & Statistics Center for Complex Systems aµ = Pr(an edge comes from a node of type µ) Vermont Advanced Computing Center University of Vermont bν = Pr(an edge leads to a node of type ν) ~ I Write E = [eµν], ~a = [aµ], and b = [bν]. I Requirements: X X X eµν = 1, eµν = aµ, and eµν = bν. µ ν ν µ Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. 1 of 26 4 of 26 Assortativity and Assortativity and Outline Mixing Notes: Mixing Definition Definition General mixing General mixing Assortativity by I Varying eµν allows us to move between the following: Assortativity by degree degree Definition Contagion 1. Perfectly assortative networks where nodes only Contagion References connect to like nodes, and the network breaks into References subnetworks. General mixing P Requires eµν = 0 if µ 6= ν and µ eµµ = 1. 2. Uncorrelated networks (as we have studied so far) Assortativity by degree For these we must have independence: eµν = aµbν . 3. Disassortative networks where nodes connect to nodes distinct from themselves. Contagion I Disassortative networks can be hard to build and may require constraints on the eµν. -
15 Netsci Configuration Model.Key
Configuring random graph models with fixed degree sequences Daniel B. Larremore Santa Fe Institute June 21, 2017 NetSci [email protected] @danlarremore Brief note on references: This talk does not include references to literature, which are numerous and important. Most (but not all) references are included in the arXiv paper: arxiv.org/abs/1608.00607 Stochastic models, sets, and distributions • a generative model is just a recipe: choose parameters→make the network • a stochastic generative model is also just a recipe: choose parameters→draw a network • since a single stochastic generative model can generate many networks, the model itself corresponds to a set of networks. • and since the generative model itself is some combination or composition of random variables, a random graph model is a set of possible networks, each with an associated probability, i.e., a distribution. this talk: configuration models: uniform distributions over networks w/ fixed deg. seq. Why care about random graphs w/ fixed degree sequence? Since many networks have broad or peculiar degree sequences, these random graph distributions are commonly used for: Hypothesis testing: Can a particular network’s properties be explained by the degree sequence alone? Modeling: How does the degree distribution affect the epidemic threshold for disease transmission? Null model for Modularity, Stochastic Block Model: Compare an empirical graph with (possibly) community structure to the ensemble of random graphs with the same vertex degrees. e a loopy multigraphs c d simple simple loopy graphs multigraphs graphs Stub Matching tono multiedgesdraw from theno config. self-loops model ~k = 1, 2, 2, 1 b { } 3 3 4 3 4 3 4 4 multigraphs 2 5 2 5 2 5 2 5 1 1 6 6 1 6 1 6 3 3 the 4standard algorithm:4 3 4 4 loopy and/or 5 5 draw2 from5 the distribution2 5 by sequential2 “Stub Matching”2 1 1 6 6 1 6 1 6 1. -
Extraction and Analysis of Facebook Friendship Relations
Chapter 12 Extraction and Analysis of Facebook Friendship Relations Salvatore Catanese, Pasquale De Meo, Emilio Ferrara, Giacomo Fiumara, and Alessandro Provetti Abstract Online social networks (OSNs) are a unique web and social phenomenon, affecting tastes and behaviors of their users and helping them to maintain/create friendships. It is interesting to analyze the growth and evolution of online social networks both from the point of view of marketing and offer of new services and from a scientific viewpoint, since their structure and evolution may share similarities with real-life social networks. In social sciences, several techniques for analyzing (off-line) social networks have been developed, to evaluate quantitative properties (e.g., defining metrics and measures of structural characteristics of the networks) or qualitative aspects (e.g., studying the attachment model for the network evolution, the binary trust relationships, and the link prediction problem). However, OSN analysis poses novel challenges both to computer and Social scientists. We present our long-term research effort in analyzing Facebook, the largest and arguably most successful OSN today: it gathers more than 500 million users. Access to data about Facebook users and their friendship relations is restricted; thus, we acquired the necessary information directly from the front end of the website, in order to reconstruct a subgraph representing anonymous interconnections among a significant subset of users. We describe our ad hoc, privacy-compliant crawler for Facebook data extraction. To minimize bias, we adopt two different graph mining techniques: breadth-first-search (BFS) and rejection sampling. To analyze the structural properties of samples consisting of millions of nodes, we developed a S.Catanese•P.DeMeo•G.Fiumara Department of Physics, Informatics Section. -
A Multigraph Approach to Social Network Analysis
1 Introduction Network data involving relational structure representing interactions between actors are commonly represented by graphs where the actors are referred to as vertices or nodes, and the relations are referred to as edges or ties connecting pairs of actors. Research on social networks is a well established branch of study and many issues concerning social network analysis can be found in Wasserman and Faust (1994), Carrington et al. (2005), Butts (2008), Frank (2009), Kolaczyk (2009), Scott and Carrington (2011), Snijders (2011), and Robins (2013). A common approach to social network analysis is to only consider binary relations, i.e. edges between pairs of vertices are either present or not. These simple graphs only consider one type of relation and exclude the possibility for self relations where a vertex is both the sender and receiver of an edge (also called edge loops or just shortly loops). In contrast, a complex graph is defined according to Wasserman and Faust (1994): If a graph contains loops and/or any pairs of nodes is adjacent via more than one line the graph is complex. [p. 146] In practice, simple graphs can be derived from complex graphs by collapsing the multiple edges into single ones and removing the loops. However, this approach discards information inherent in the original network. In order to use all available network information, we must allow for multiple relations and the possibility for loops. This leads us to the study of multigraphs which has not been treated as extensively as simple graphs in the literature. As an example, consider a network with vertices representing different branches of an organ- isation. -
The Perceived Assortativity of Social Networks: Methodological Problems and Solutions
The perceived assortativity of social networks: Methodological problems and solutions David N. Fisher1, Matthew J. Silk2 and Daniel W. Franks3 Abstract Networks describe a range of social, biological and technical phenomena. An important property of a network is its degree correlation or assortativity, describing how nodes in the network associate based on their number of connections. Social networks are typically thought to be distinct from other networks in being assortative (possessing positive degree correlations); well-connected individuals associate with other well-connected individuals, and poorly-connected individuals associate with each other. We review the evidence for this in the literature and find that, while social networks are more assortative than non-social networks, only when they are built using group-based methods do they tend to be positively assortative. Non-social networks tend to be disassortative. We go on to show that connecting individuals due to shared membership of a group, a commonly used method, biases towards assortativity unless a large enough number of censuses of the network are taken. We present a number of solutions to overcoming this bias by drawing on advances in sociological and biological fields. Adoption of these methods across all fields can greatly enhance our understanding of social networks and networks in general. 1 1. Department of Integrative Biology, University of Guelph, Guelph, ON, N1G 2W1, Canada. Email: [email protected] (primary corresponding author) 2. Environment and Sustainability Institute, University of Exeter, Penryn Campus, Penryn, TR10 9FE, UK Email: [email protected] 3. Department of Biology & Department of Computer Science, University of York, York, YO10 5DD, UK Email: [email protected] Keywords: Assortativity; Degree assortativity; Degree correlation; Null models; Social networks Introduction Network theory is a useful tool that can help us explain a range of social, biological and technical phenomena (Pastor-Satorras et al 2001; Girvan and Newman 2002; Krause et al 2007). -
Critical Window for Connectivity in the Configuration Model 3
CRITICAL WINDOW FOR CONNECTIVITY IN THE CONFIGURATION MODEL LORENZO FEDERICO AND REMCO VAN DER HOFSTAD Abstract. We identify the asymptotic probability of a configuration model CMn(d) to produce a connected graph within its critical window for connectivity that is identified by the number of vertices of degree 1 and 2, as well as the expected degree. In this window, the probability that the graph is connected converges to a non-trivial value, and the size of the complement of the giant component weakly converges to a finite random variable. Under a finite second moment con- dition we also derive the asymptotics of the connectivity probability conditioned on simplicity, from which the asymptotic number of simple connected graphs with a prescribed degree sequence follows. Keywords: configuration model, connectivity threshold, degree sequence MSC 2010: 05A16, 05C40, 60C05 1. Introduction In this paper we consider the configuration model CMn(d) on n vertices with a prescribed degree sequence d = (d1,d2, ..., dn). We investigate the condition on d for CMn(d) to be with high probability connected or disconnected in the limit as n , and we analyse the behaviour of the model in the critical window for connectivity→ ∞ (i.e., when the asymptotic probability of producing a connected graph is in the interval (0, 1)). Given a vertex v [n]= 1, 2, ..., n , we call d its degree. ∈ { } v The configuration model is constructed by assigning dv half-edges to each vertex v, after which the half-edges are paired randomly: first we pick two half-edges at arXiv:1603.03254v1 [math.PR] 10 Mar 2016 random and create an edge out of them, then we pick two half-edges at random from the set of remaining half-edges and pair them into an edge, etc.