Evaluating topology quality through random walks

Computing Research Repository, 2004

Mapping the Internet generally consists in sampling the network from a limited set of sources by using traceroute-like probes. This methodology, akin to the merging of different spanning trees to a set of destinations, has been argued to introduce uncontrolled sampling biases that might produce statistical properties of the sampled graph which sharply differ from the original ones [7-9]. Here we explore these biases and provide a statistical analysis of their origin. We derive a mean-field analytical approximation for the probability of edge and vertex detection that exploits the role of the number of sources and targets and allows us to relate the global topological properties of the underlying network with the statistical accuracy of the sampled graph. In particular we find that the edge and vertex detection probability is depending on the betweenness centrality of each element. This allows us to show that shortest path routed sampling provides a better characterization of underlying graphs with scale-free topology. We complement the analytical discussion with a throughout numerical investigation of simulated mapping strategies in different network models. We show that sampled graphs provide a fair qualitative characterization of the statistical properties of the original networks in a fair range of different strategies and exploration parameters. The numerical study also allows the identification of intervals of the exploration parameters that optimize the fraction of nodes and edges discovered in the sampled graph. This finding might hint the steps toward more efficient mapping strategies.

Environment and Planning B: Urban Analytics and City Science

A framework for calculating a weighted random walk on an urban street segment network is described, and tested as a predictor of pedestrian and vehicle movement in London and the wider region. This paper has three aims. First, it proposes the simplest possible model of agency in that individuals have neither memory, goals nor knowledge of the network beyond street segments immediately visible at an intersection. Second, it attempts to reconcile two divergent approaches to urban analysis, graph centrality measures and agent simulation, by demonstrating properties of topological graphs emerge from the lowest level agent behaviour. Third, it aims for far faster computation of relevant features such as the foreground street network and prediction of movement than currently exists. The results show that the idealised random walk predicts observed movement as well as the best existing centrality measures, is several orders of magnitude faster to calculate, and may help to explain movement...

International Conference on Computational Science, 2004

The networks considered here consist of sets of interconnected vertices, examples of which include social networks, technological networks, and biological networks. Two important issues are to measure the extent of proximity between vertices and to identify the community structure of a network. In this paper, the proximity index between two nearest-neighboring vertices of a network is measured by a biased

Physical Review E, 2010

We present a new approach of topology biased random walks for undirected networks. We focus on a one parameter family of biases and by using a formal analogy with perturbation theory in quantum mechanics we investigate the features of biased random walks. This analogy is extended through the use of parametric equations of motion (PEM) to study the features of random walks {\em vs.} parameter values. Furthermore, we show an analysis of the spectral gap maximum associated to the value of the second eigenvalue of the transition matrix related to the relaxation rate to the stationary state. Applications of these studies allow {\em ad hoc} algorithms for the exploration of complex networks and their communities.

AIP Conference Proceedings, 2003

We present an algorithm [1] to grow a graph with scale-free structure of in-and out-links and variable wiring diagram in the class of the worldwide Web. We then explore the graph by intentional random walks using local next-near-neighbor search algorithm to navigate through the graph. The topological properties such as betweenness are determined by an ensemble of independent walkers and efficiency of the search is compared on three different graph topologies. In addition we simulate interacting random walks which are created by given rate and navigated in parallel, representing transport with queueing of information packets on the graph.

Scientific Reports, 2013

Disclosing the main features of the structure of a network is crucial to understand a number of static and dynamic properties, such as robustness to failures, spreading dynamics, or collective behaviours. Among the possible characterizations, the core-periphery paradigm models the network as the union of a dense core with a sparsely connected periphery, highlighting the role of each node on the basis of its topological position. Here we show that the core-periphery structure can effectively be profiled by elaborating the behaviour of a random walker. A curve-the core-periphery profile-and a numerical indicator are derived, providing a global topological portrait. Simultaneously, a coreness value is attributed to each node, qualifying its position and role. The application to social, technological, economical, and biological networks reveals the power of this technique in disclosing the overall network structure and the peculiar role of some specific nodes.

2011

Journal of Physics A: Mathematical and General, 2005

Random walks on graphs are widely used in all sciences to describe a great variety of phenomena where dynamical random processes are affected by topology. In recent years, relevant mathematical results have been obtained in this field, and new ideas have been introduced, which can be fruitfully extended to different areas and disciplines. Here we aim at giving a brief but comprehensive perspective of these progresses, with a particular emphasis on physical aspects. Contents 1 Introduction 2 Mathematical description of graphs 3 The random walk problem 4 The generating functions 5 Random walks on finite graphs 6 Infinite graphs 7 Random walks on infinite graphs 8 Recurrence and transience: the type problem 9 The local spectral dimension 10 Averages on infinite graphs 11 The type problem on the average 1 12 The average spectral dimension 21 13 A survey of analytical results on specific networks 23 13.1 Renormalization techniques. .

IEEE Transactions on Pattern Analysis and Machine Intelligence, 1980

A formal definition of random graphs is introduced which is applicable to graphical pattern recognition problems. The definition is used to formulate rigorously the structural-contextual dichotomy of random graphs. The probability of outcome graphs is expressed as the product of two terms, one due to the statistical variability of structure among the outcome graphs and the other due to their contextual variability. Expressions are obtained to estimate the various probability, typicality, and entropy measures. The members in an ensemble of signed digraphs are interpreted as outcome graphs of a random graph. The synthesized random graph is used to quantify the structural, contextual, and overall typicality of the outcome graphs with respect to the random graph.

Physical Review E, 2012

In this work, we employed the concept of the first-passage time in stochastic processes to estimate node degrees and the degree distribution of a network. A statistical exploration of the coupling reveals the relation between the node degree and the coupling term. In practical terms, an effective way to reveal the statistical property is to investigate the differences between coupled oscillators in a network and uncoupled ones with the same initial states. We discovered a monotonically decreasing relation between the node degree and the mean first-passage time (MFPT) for the evolution of the coupled node deviating from the uncoupled one. Moreover, this relation can be understood as the competition of different relaxational time scales. The MFPT method is independent of both the dynamics of the nodes and the topological properties of the network. This might be advantageous in our efforts to build a bridge between the topological property and the dynamics of a network.

2012

First and foremost, I would like to express my deep and sincere gratitude to my supervisor, Dr. Piotr Bia las, for introducing me into the fascinating topic of random graphs, scientific guidance, and insightful discussions. Without his constant help, inspiring comments, and endless patience this dissertation would not be possible. I would like to heartily thank my dear wife Ma lgosia for her love and encouragement throughout, particularly during the last hectic stage of writing. Last but not least, I am deeply indebted to my parents, who nurtured my interest in physics and mathematics from the early school years, and who stood by me through the many trials and decisions along my educational career. This research was supported in part by the PL-Grid Infrastructure. Monte Carlo simulations were performed on the Shiva computing cluster at the Faculty of Physics, Astronomy and Applied Computer Science, Jagiellonian University, and at the Academic Computer Centre CYFRONET AGH using the Zeus cluster.

Scientific Reports, 2016

Complex systems made of interacting elements are commonly abstracted as networks, in which nodes are associated with dynamic state variables, whose evolution is driven by interactions mediated by the edges. Markov processes have been the prevailing paradigm to model such a network-based dynamics, for instance in the form of random walks or other types of diffusions. Despite the success of this modelling perspective for numerous applications, it represents an oversimplification of several real-world systems. Importantly, simple Markov models lack memory in their dynamics, an assumption often not realistic in practice. Here, we explore possibilities to enrich the system description by means of second-order Markov models, exploiting empirical pathway information. We focus on the problem of community detection and show that standard network algorithms can be generalized in order to extract novel temporal information about the system under investigation. We also apply our methodology to temporal networks, where we can uncover communities shaped by the temporal correlations in the system. Finally, we discuss relations of the framework of second order Markov processes and the recently proposed formalism of using non-backtracking matrices for community detection. Dynamics on complex networks, such as the diffusion of information in social networks, are commonly modelled as Markov processes. An advantage of this approach is that for every (static) network with positive edge-weights we can define a corresponding Markov process by interpreting the network as the state space of a random walker, and assigning the state-transition probabilities according to the link weights. This direct correspondence between the state space of the Markov process and the network enables us to examine the interplay between structure and dynamics from two sides. On the one hand, one can assess how the topological properties of a network influence the dynamical process. On the other hand, this coupling between topology and dynamics allows us to explore the structure of a network by means of a dynamical process. Specifically, for a linear Markov process the impact of the network structure on the dynamics will be mediated by the spectral properties of the matrix governing the time-evolution of the process, e.g. the adjacency matrix or the Laplacian 1,2. Vice versa, spectral properties can be used to uncover salient structural properties of a network, such as modular organisation 3-5. While simple Markov models have been very successful in modelling dynamics of complex systems and found many applications, they have one obvious disadvantage. In this class of models, the future state of the system only depends on its current state and does not account for its history. In a diffusion process, for instance, the next position of a random walker only depends on the currently occupied node and its outgoing links, but not on any of the previously visited nodes. However, as it has been emphasised recently, for a broad range of networked systems, flows tend to exhibit a temporal path dependence 6,7. Think of human mobility: the places a person is likely to visit next, depend in most cases strongly on where the person came from. For instance, a person coming to work from home is likely to return home afterwards 8. Other examples of processes with temporal memory include web traffic, journal citation flows and email cascades. Such processes therefore cannot be reproduced accurately by simple Markov models. However, the impact of this temporal correlations can often be already well-approximated by second-order Markov (2 ) models 6. Importantly, the transition probabilities to define these models can be obtained empirically by measuring pathways of interaction cascades, rendering such an approach suitable for applications like information spreading or human mobility.

2020

Supplemental material, sj-pdf-1-epb-10.1177_2399808320946766 for Random walks in urban graphs: A minimal model of movement by Sean Hanna in Environment and Planning B: Urban Analytics and City Science

Physical review. E, Statistical, nonlinear, and soft matter physics, 2014

We generate time series from scale-free networks based on a finite-memory random walk traversing the network. These time series reveal topological and functional properties of networks via their temporal correlations. Remarkably, networks with different node-degree mixing patterns exhibit distinct self-similar characteristics. In particular, assortative networks are transformed into time series with long-range correlation, while disassortative networks are transformed into time series exhibiting anticorrelation. These relationships are consistent across a diverse variety of real networks. Moreover, we show that multiscale analysis of these time series can describe and classify various physical networks ranging from social and technological to biological networks according to their functional origin. These results suggest that there is a unified dynamical mechanism that governs the structural organization of many seemingly different networks.

Most methods proposed to uncover communities in complex networks rely on combinatorial graph properties. Usually an edge-counting quality function, such as modularity, is optimized over all partitions of the graph compared against a null random graph model. Here we introduce a systematic dynamical framework to design and analyze a wide variety of quality functions for community detection. The quality of a partition is measured by its Markov Stability, a time-parametrized function defined in terms of the statistical properties of a Markov process taking place on the graph. The Markov process provides a dynamical sweeping across all scales in the graph, and the time scale is an intrinsic parameter that uncovers communities at different resolutions. This dynamic-based community detection leads to a compound optimization, which favours communities of comparable centrality (as defined by the stationary distribution), and provides a unifying framework for spectral algorithms, as well as d...