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Random and Longest Paths: Unnoticed Motifs of Complex Networks

Luciano da Fontoura Costa

arXiv:0712.0415v1physics.soc-phcond-mat.dis-nnphysics.comp-ph

TL;DR

The paper addresses the limited study of random and longest paths in complex networks and their structural significance. It samples self-avoiding random paths across network models and estimates the longest path by taking the longest sampled path. Random-path distributions differ markedly across models, while the estimator converges quickly for the considered networks and settings.

Abstract

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Paths are important structural elements in complex networks because they are finite (unlike walks), related to effective node coverage (minimum spanning trees), and can be understood as being dual to star connectivity. This article introduces the concept of random path applies it for the investigation of structural properties of complex networks and as the means to estimate the longest path. Random paths are obtained by selecting one of the network nodes at random and performing a random self-avoiding walk (here called path-walk) until its termination. It is shown that the distribution of random paths are markedly different for diverse complex network models (i.e. Erdos-Renyi, Barabasi-Albert, Watts-Strogatz, a geographical model, as well as two recently introduced path-based network types), with the BA structures yielding the shortest random walks, while the longest paths are produced by WS networks. Random paths are also explored as the means to estimate the longest paths (i.e. several random paths are obtained and the longest taken). The convergence to the longest path and its properties ire characterized with respect to several networks models. Several results are reported and discussed, including the markedly distinct lengths of the longest paths obtained for the different network models.

I. INTRODUCTION

The paper motivates studying random and longest paths as underexamined network structures with finite length, coverage relevance, and a dual relationship to stars. It introduces random path-walks and investigates these paths across six network models, including a stochastic longest-path estimator.

  • Random and longest paths have received less attention than node degree and shortest paths despite their finite length, coverage relevance, and duality with stars.Source: §I · p. 1
  • A random path starts from a randomly selected node and follows a self-avoiding path-walk until no unvisited edges or nodes remain.These paths relate to network exploration, sampling, coverage, and possible bottleneck detection.Source: §I · p. 1
  • Long paths can indicate structures that are important to network dynamics or that were incorporated into network construction.Source: §I · pp. 1–2
  • The article investigates random and longest paths across six theoretical network models and introduces a simple stochastic method for estimating the longest path.Source: §I · p. 2

II. BASIC CONCEPTS

The paper formalizes network paths and related local motifs, then compares six theoretical network models under fixed size and connectivity settings. Analyses are restricted to the largest connected component.

  • A path is a walk that repeats neither edges nor nodes, and its length equals the number of constituent edges.A Hamiltonian path visits all network nodes.Source: §II · p. 2
  • The framework also defines stars, node degree, average degree, and clustering coefficient to characterize local connectivity.Source: §II · p. 2
  • The study considers ER, BA, WS, geographical, path-transformed BA, and path-regular network models.PA networks result from star-path transforming BA networks, whereas PN networks use paths containing all nodes in random order without repetition.Source: §II · p. 2
  • All networks contain N = 100 nodes with m = 3 or m = 5, implying average degree ⟨k⟩ = 2m.Because these average degrees exceed the percolation threshold, analyses use the largest connected component.Source: §II · p. 2

III. METHODOLOGY

The methodology samples self-avoiding random paths and estimates the longest path by retaining the longest sampled path. It evaluates convergence because exact longest-path determination is NP-complete.

  • The longest-path estimator generates many random paths and keeps the largest observed path length.The method is experimentally evaluated for convergence despite the NP-completeness of exact longest-path determination.Source: §III · pp. 2–3
  • Each random path begins at a randomly selected node and extends through self-avoiding path-walks from up to two outbound edges until both sides terminate.Source: §III · p. 2
  • The sampling procedure selects a node, chooses two incident edges when possible, and follows both edges until no unvisited node or edge remains.Source: §III · p. 2
  • The length of each random path is denoted by p.Source: §III · p. 2

IV. RESULTS AND DISCUSSION

The experiments use 50 realizations of six network models with N = 100 nodes and m = 3 or m = 5. These settings define the empirical comparison reported in the results section.

  • The experiments analyze 50 realizations of each PA, PN, ER, BA, WE, and GG model with N = 100 nodes and m = 3 and m = 5.Source: §IV · p. 3

A. Random Paths in Network Models

Random path lengths differ substantially across network models, reflecting structural differences in connectivity, regularity, hubs, and spatial organization. Increasing average degree generally lengthens random paths and changes their dispersion.

A. Random Paths in Network Models

  • BA networks produced the shortest random paths, whereas WS networks produced the longest and most variable paths.For m = 3, BA paths were about 30 edges long, while WS paths ranged from 70 to 100 edges.Source: A. Random Paths in Network Models · p. 3
  • Higher average degree increased random path lengths across all models, with the largest increase occurring in GG networks.For m = 5, GG paths reached 100 edges and beyond, while BA values shifted from [15, 45] to [25, 65].Source: A. Random Paths in Network Models · p. 3
  • BA paths become short because path-walks rapidly encounter hubs whose many incident edges become unusable after the hubs are visited.This hub saturation limits continued traversal.Source: A. Random Paths in Network Models · p. 3
  • WS networks support long random paths because near-uniform connectivity increases the chance of continuing through unvisited nodes and edges, despite their small-world structure.Rewiring accounts for the large variance in WS path lengths.Source: A. Random Paths in Network Models · p. 3
  • PN networks showed unusually uniform random path lengths, while PA and ER networks had remarkably similar measurements for m = 3.The PA–ER similarity corresponds to similarities previously observed in other topological properties.Source: A. Random Paths in Network Models · p. 4
  • GG networks exhibited the widest dispersion, ranging from roughly 5 to over 80 edges, making coverage by a moving agent difficult to foresee.The authors suggest examining larger d values to better understand this variance.Source: A. Random Paths in Network Models · pp. 4–5

B. Convergence to the Maximum Random Path

Successive random paths rapidly identify a stable maximum length, providing a simple estimate of the longest path. Higher average degree increases this maximum, especially in geographical networks.

B. Convergence to the Maximum Random Path

  • After about 200 random paths, the currently longest path generally reached a stable plateau, suggesting rapid convergence to the maximum random path.Experiments with 2500 steps confirmed that the plateaux usually remained stable, except for greater variation in GG networks.Source: B. Convergence to the Maximum Random… · p. 5
  • Increasing m increased longest-path lengths in every model, with the strongest increase in GG networks where the length nearly doubled.Standard deviations also tended to decrease at the higher m value.Source: B. Convergence to the Maximum Random… · p. 5
  • The method generally converged quickly to the longest path, or to a path with very similar length, for the tested N and m values.Most models contained longest paths through most network nodes, and WS networks yielded several Hamiltonian paths.Source: B. Convergence to the Maximum Random… · p. 5

C. Similarity between random paths and the maximum random path

The study compares random paths with a reference longest path using node and edge dissimilarity, revealing model-specific overlap patterns.

C. Similarity between random paths and the maximum random path

  • Random paths are compared with a reference longest path obtained after 2500 steps to assess how subsequent paths resemble a long-path benchmark.The analysis quantifies differences through node and edge overlap across algorithm steps.Source: C. Similarity between random paths and… · p. 5
  • The comparison uses node and edge dissimilarity indices because equal node sets or edge sets do not guarantee identical paths.The indices ignore edge order and therefore distinguish overlap from exact path identity.Source: C. Similarity between random paths and… · p. 5
  • Node dissimilarity differs across models: WS paths are closest to the reference longest path, whereas GG paths differ most.For m = 3, WS dissimilarity is about 7%; for m = 5, node dissimilarities are smaller across all models.Source: C. Similarity between random paths and… · p. 6
  • Edge dissimilarity is near 80% for most models except WS, showing that shared nodes often correspond to substantially different edges.Increasing m changes edge dissimilarity little at N = 100, although WS shows lower dissimilarity for m = 5.Source: C. Similarity between random paths and… · p. 6

V. CONCLUDING REMARKS

The paper argues that random and longest paths reveal structural differences across network models beyond degree and shortest-path statistics. A random-path procedure estimates long paths quickly, with BA producing the shortest and WS the longest paths under the studied conditions.

V. CONCLUDING REMARKS

  • Random paths distinguish ER, BA, WS, GG, PA, and PN networks, with BA paths shortest and WS paths longest.BA hubs can block access to many neighboring nodes, while PN paths tend to have remarkably similar lengths.Source: §V · pp. 6–7
  • The random-path method estimates a long path quickly and steadily despite longest-path identification being NP-complete.The evidence is limited to relatively small networks with N = 100 and the considered average degrees.Source: §V · p. 7
  • Longest paths increase with average degree in all models, while their lengths have small standard deviations except in GG networks.For m = 5, the increase particularly extends GG longest paths, and the longest-path curves converge toward a stable plateau.Source: §V · p. 7, Figure 4 · p. 8
  • The work motivates future studies of alternative network models, additional structural measurements, random and longest cycles, and refined path-similarity measures.These extensions are proposed as ways to investigate path diversity and transient network dynamics.Source: §V · pp. 7–8
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