Random Graphs
Abstract
In this second edition of the now classic text, the already extensive treatment given in the first edition has been heavily revised by the author. The addition of two new sections, numerous new results and 150 references means that this represents a comprehensive account of random graph theory. The theory (founded by Erdös and Rényi in the late fifties) aims to estimate the number of graphs of a given degree that exhibit certain properties. It not only has numerous combinatorial applications, but also serves as a model for the probabilistic treatment of more complicated random structures. This book, written by an acknowledged expert in the field, can be used by mathematicians, computer scientists and electrical engineers, as well as people working in biomathematics. It is self-contained, and with numerous exercises in each chapter, is ideal for advanced courses or self study.
Integrity signals
Registry
Cites
Statistics
Claims the paper makes
Claims other papers make about this one
Each is checked against this paper’s own text: does it say what the citing paper says it does?
The mouse brain has a higher mean clustering coefficient, 0.42, than expected by a random network with identical sparseness, 0.12.
“Network analyses (Fig. 4c, see also Supplementary Notes) reveal that the mouse brain has a higher mean clustering coefficient (which gives the ratio of existing over possible connections), 0.42, than expected by a random network29,30 with identical sparseness, 0.12.”
From A mesoscale connectome of the mouse brain (2014) · cites this paper as [30]
Cited by
Sorted by citations
The Structure and Function of Complex Networks
Newman, M. E. J. (2003) · SIAM Review · doi:10.1137/s003614450342480 · Cited by 18,719 · see citing works
Coordination of groups of mobile autonomous agents using nearest neighbor rules
Jadbabaie, A., Lin, J., & Morse, A. S. (2003) · IEEE Transactions on Automatic Control · doi:10.1109/tac.2003.812781 · Cited by 8,339 · see citing works
Opinion mining and sentiment analysis
Pang, B., & Lee, L. (2008) · Foundations and Trends® in Information Retrieval · doi:10.1561/1500000011 · Cited by 6,372 · see citing works
Sampling-based algorithms for optimal motion planning
Karaman, S., & Frazzoli, E. (2011) · The International Journal of Robotics Research · doi:10.1177/0278364911406761 · Cited by 5,368 · see citing works
“The authors are grateful to Professor MS Branicky, Professor GJ Gordon, and Professor S LaValle, as well as the anonymous reviewers, for their insightful comments on draft versions of this paper.”
BackgroundExact Matrix Completion via Convex Optimization
Candès, E. J., & Recht, B. (2009) · Foundations of Computational Mathematics · doi:10.1007/s10208-009-9045-5 · Cited by 5,102 · see citing works
Assortative Mixing in Networks
Newman, M. E. J. (2002) · Physical Review Letters · doi:10.1103/physrevlett.89.208701 · Cited by 5,034 · see citing works
“model networks in the limit of large network size: (u) the random graph of Erdős and Rényi [14]; (v) the grown graph model of Callaway et al.”
InfluentialControllability of complex networks
Liu, Y., Slotine, J.-J. E., & Barabási, A.-Ĺ. (2011) · Nature · doi:10.1038/nature10011 · Cited by 3,238 · see citing works
A mesoscale connectome of the mouse brain
Oh, S. W., Harris, J. A., Ng, L., Winslow, B., Cain, N., Mihalas, S., Wang, Q., Lau, C., Kuan, L., Henry, A. M., et al. (2014) · Nature · doi:10.1038/nature13186 · Cited by 2,612 · see citing works
Scientific collaboration networks. II. Shortest paths, weighted networks, and centrality
Newman, M. E. J. (2001) · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics · doi:10.1103/physreve.64.016132 · Cited by 2,631 · see citing works
“In a standard random graph (Erdős and Rényi, 1960; Bollobás, 2001), for instance, ` = log n/ log z, where z is the average degree of a vertex, the average number of collaborators in our terminology.”
Information, Physics, and Computation
Mézard, M., & Andrea Montanari, A. (2009) · doi:10.1093/acprof:oso/9780198570837.001.0001 · Cited by 1,941 · see citing works
Guimerà, R., Mossa, S., Turtschi, A., & Amaral, L. A. N. (2005) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.0407994102 · Cited by 1,772 · see citing works
Structure and evolution of online social networks
Kumar, R., Novak, J., & Tomkins, A. (2006) · ACM SIGKDD Conference on Knowledge Discovery and Data Mining (KDD) · doi:10.1145/1150402.1150476 · Cited by 1,091 · see citing works
Modularity from fluctuations in random graphs and complex networks
Guimerà, R., Sales-Pardo, M., & Amaral, L. A. N. (2004) · Physical Review E · doi:10.1103/physreve.70.025101 · Cited by 894 · see citing works
Bäckström, L., Dwork, C., & Kleinberg, J. M. (2007) · International World Wide Web Conference (WWW) · doi:10.1145/1242572.1242598 · Cited by 857 · see citing works
“It is known that for a random graph H such as we have constructed, the following properties hold with probability going to 1 exponentially quickly in k [9]: first, that γ(H) = δ(H); second, that δ(H) ≥ (1/2− ε)k for any constant ε > 0; and third, that H has no non-trivial automorphisms.”
Analyzing and modeling real-world phenomena with complex networks: a survey of applications
Costa, L. D. F., Oliveira, O. N., Travieso, G., Rodrigues, F. A., Villas-Boas, P. R., Antiqueira, L., Viana, M. P., & Rocha, L. E. C. (2011) · Advances In Physics · doi:10.1080/00018732.2011.572452 · Cited by 804 · see citing works
The effect of network topology on the spread of epidemics
Ganesh, A. J., Massoulié, L., & Towsley, D. (2005) · doi:10.1109/infcom.2005.1498374 · Cited by 761 · see citing works
Cartography of complex networks: modules and universal roles
Guimerà, R., & Amaral, L. A. N. (2005) · Journal of Statistical Mechanics Theory and Experiment · doi:10.1088/1742-5468/2005/02/p02001 · Cited by 757 · see citing works
Structural diversity in social contagion
Ugander, J., Backstrom, L., Marlow, C. A., & Kleinberg, J. (2012) · Proceedings of the National Academy of Sciences of the United States of America · doi:10.1073/pnas.1116502109 · Cited by 684 · see citing works
Control principles of complex systems
Liu, Y., & Barabási, A.-Ĺ. (2016) · Reviews of Modern Physics · doi:10.1103/revmodphys.88.035006 · Cited by 636 · see citing works
“For example, for directed Erdős-Rényi random networks (Bollobás, 2001; Erdős and Rényi, 1960), both P (kin) and P (kout) follow a Poisson distribution, i.e. e−〈k〉/2(〈k〉/2)k/k!.”
On k-coverage in a mostly sleeping sensor network
Kumar, S. D., Lai, T. H., & Balogh, J. (2004) · Proceedings of the 28th Annual International Conference on Mobile Computing And Networking · doi:10.1145/1023720.1023735 · Cited by 644 · see citing works
Agreement over random networks
Hatano, N., & Mesbahi, M. (2004) · IEEE Conference on Decision and Control (CDC) · doi:10.1109/cdc.2004.1430343 · Cited by 545 · see citing works
Clique Percolation in Random Networks
Derényi, I., Palla, G., & Vicsek, T. (2005) · Physical Review Letters · doi:10.1103/physrevlett.94.160202 · Cited by 563 · see citing works
The phase transition in inhomogeneous random graphs
Bollobás, B., Janson, S., & Riordan, O. M. (2007) · Random Structures and Algorithms · doi:10.1002/rsa.20168 · Cited by 548 · see citing works
Self-citationNetwork synchronization, diffusion, and the paradox of heterogeneity
Motter, A. E., Zhou, C., & Kurths, J. (2005) · Physical Review E · doi:10.1103/physreve.71.016116 · Cited by 547 · see citing works
Extracting the hierarchical organization of complex systems
Sales‐Pardo, M., Guimerà, R., Moreira, A. A., & Amaral, L. A. N. (2007) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.0703740104 · Cited by 539 · see citing works
CYCLON: Inexpensive Membership Management for Unstructured P2P Overlays
Voulgaris, S., Gavidia, D., & van Steen, M. R. (2005) · Journal of Network and Systems Management · doi:10.1007/s10922-005-4441-x · Cited by 512 · see citing works
Agreement over random networks
Hatano, N., & Mesbahi, M. (2005) · IEEE Transactions on Automatic Control · doi:10.1109/tac.2005.858670 · Cited by 503 · see citing works
Epidemic information dissemination in distributed systems
Eugster, P. T., Guerraoui, R., Kermarrec, A.-M., & Massoulié, L. (2004) · Computer · doi:10.1109/mc.2004.1297243 · Cited by 500 · see citing works
Cognitive Network Neuroscience
Medaglia, J. D., Lynall, M.-E., & Bassett, D. S. (2015) · Journal of Cognitive Neuroscience · doi:10.1162/jocn_a_00810 · Cited by 419 · see citing works
Statistical physics of inference: thresholds and algorithms
Zdeborová, L., & Krząkała, F. (2016) · Advances In Physics · doi:10.1080/00018732.2016.1211393 · Cited by 452 · see citing works
“The key point is that, in this limit, such random graphs can be considered locally as trees [36, 133].”
Structure and Evolution of Online Social Networks
Kumar, R., Novak, J., & Tomkins, A. (2010) · Springer eBooks · doi:10.1007/978-1-4419-6515-8_13 · Cited by 438 · see citing works
Toward link predictability of complex networks
Lü, L., Pan, L., Zhou, T., Zhang, Y., & Stanley, E. (2015) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.1424644112 · Cited by 406 · see citing works
Graph theory and networks in Biology
Mason, O., & Verwoerd, M. (2007) · IET Systems Biology · doi:10.1049/iet-syb:20060038 · Cited by 399 · see citing works
Who Is the Best Connected Scientist?A Study of Scientific Coauthorship Networks
Newman, M. E. J. (2004) · Lecture notes in physics · doi:10.1007/978-3-540-44485-5_16 · Cited by 401 · see citing works
Spatially Coupled Ensembles Universally Achieve Capacity Under Belief Propagation
Kudekar, S., Richardson, T., & Urbanke, R. L. (2013) · IEEE Transactions on Information Theory · doi:10.1109/tit.2013.2280915 · Cited by 361 · see citing works
Analysis of a large-scale weighted network of one-to-one human communication
Onnela, J.-P., Saramäki, J., Hyvönen, J., Szabó, G., de Menezes, M. A., Kaski, K., Barabási, A.-Ĺ., & Kertész, J. (2007) · New Journal of Physics · doi:10.1088/1367-2630/9/6/179 · Cited by 347 · see citing works
“, automorphic, defined as adjacency-preserving permutation of the vertices of the graph [31].”
Analysis of a large-scale weighted network of one-to-one human communication
Onnela, J. P., Barabási, A.-Ĺ., Ferreira de Menezes, M. A., Hyvönen, J., Kaski, K., Kertész, J., Saramäki, J., & Szabó, G. (2007) · Digital Repository Service (Northeastern University) · Cited by 324 · see citing works
Generating Simple Random Graphs with Prescribed Degree Distribution
Britton, T., Deijfen, M., & Martin‐Löf, A. (2006) · Journal of Statistical Physics · doi:10.1007/s10955-006-9168-x · Cited by 309 · see citing works
“parameters n− 1 and pand, if pis scaled by 1/n, we get a Poisson distribution in the limit as n→ ∞. Erdo˝s-R´enyi graphs have been widely studied and thorough descriptions of the field can be found in [4, 12]. An area that has received a lot of attention within statistical physics during the last few years is the use o…”
Enhancing complex-network synchronization
Motter, A. E., Zhou, C. S., & Kurths, J. (2005) · Europhysics Letters (EPL) · doi:10.1209/epl/i2004-10365-4 · Cited by 324 · see citing works
Population Graphs: the graph theoretic shape of genetic structure
Dyer, R. J., & Nason, J. D. (2004) · Molecular Ecology · doi:10.1111/j.1365-294x.2004.02177.x · Cited by 315 · see citing works
“A graph ( G ) is a mathematical mapping of two disjoint sets, a set of nodes ( V ) and a set of edges ( E ), and is denoted as G = { V, E } (Bollobás 2001).”
BackgroundReconstruction and estimation in the planted partition model
Mossel, E., Neeman, J., & Sly, A. (2014) · Probability Theory and Related Fields · doi:10.1007/s00440-014-0576-6 · Cited by 304 · see citing works
Budget-Optimal Task Allocation for Reliable Crowdsourcing Systems
Karger, D. R., Oh, S., & Shah, D. (2014) · Operations Research · doi:10.1287/opre.2013.1235 · Cited by 311 · see citing works
Phase transitions in the coloring of random graphs
Zdeborová, L., & Krząkała, F. (2007) · Physical Review E · doi:10.1103/physreve.76.031131 · Cited by 298 · see citing works
Blum, A. L., Hopcroft, J. E., & Kannan, R. (2020) · Cambridge University Press eBooks · doi:10.1017/9781108755528 · Cited by 320 · see citing works
Topological patterns in street networks of self-organized urban settlements
Buhl, C., Jacques Gautrais, A., Reeves, N. P., Solé, R. V., Valverde, S., Kuntz, P., & Guy Theraulaz, A. (2006) · The European Physical Journal B · doi:10.1140/epjb/e2006-00085-1 · Cited by 300 · see citing works
Local search of communities in large graphs
Cui, W., Xiao, Y., Wang, H., & Wang, W. (2014) · ACM SIGMOD International Conference on Management of Data (SIGMOD) · doi:10.1145/2588555.2612179 · Cited by 305 · see citing works
Finding a Better Immunization Strategy
Chen, Y., Paul, G., Havlin, S., Liljeros, F., & Stanley, E. (2008) · Physical Review Letters · doi:10.1103/physrevlett.101.058701 · Cited by 299 · see citing works
Krivelevich, M., & Sudakov, B. (2006) · Bolyai Society mathematical studies · doi:10.1007/978-3-540-32439-3_10 · Cited by 301 · see citing works
Performance Evaluation of Blockchain Systems: A Systematic Survey
Fan, C., Ghaemi, S., Khazaei, H., & Musı́lek, P. (2020) · IEEE Access · doi:10.1109/access.2020.3006078 · Cited by 304 · see citing works
Network Topology Inference from Spectral Templates
Segarra, S., Marques, A. G., Mateos, G., & Ribeiro, A. (2017) · IEEE Transactions on Signal and Information Processing over Networks · doi:10.1109/tsipn.2017.2731051 · Cited by 288 · see citing works
Random hypergraphs and their applications
Ghoshal, G., Zlatić, V., Caldarelli, G., & Newman, M. E. J. (2009) · Physical Review E · doi:10.1103/physreve.79.066118 · Cited by 273 · see citing works
“A variety of random graph models have been studied, from models of simple undirected or directed graphs to more complicated examples with correlations, communities, or bipartite structure [10, 11, 12, 13, 14].”
BackgroundEnhancing complex-network synchronization
Motter, A. E., Zhou, C., & Kurths, J. (2005) · Cited by 277 · see citing works
Network link prediction by global silencing of indirect correlations
Barzel, B., & Barabási, A.-Ĺ. (2013) · Nature Biotechnology · doi:10.1038/nbt.2601 · Cited by 269 · see citing works
A Guide to Conquer the Biological Network Era Using Graph Theory
Koutrouli, M., Karatzas, E., Páez-Espino, D., & Pavlopoulos, G. A. (2020) · Frontiers in Bioengineering and Biotechnology · doi:10.3389/fbioe.2020.00034 · Cited by 265 · see citing works
“In order to better understand a network’s topology and come to the conclusion of whether observed features are network-specific or not, several models such as the Erdos–Rényi (Bollobás, 2001), Watts-Strogatz (Watts and Strogatz, 1998), and Barabási– Albert (Barabasi and Albert, 1999) have been introduced ( Figure 5 ).”
BackgroundInfluentialConcepts of Exact QoS Routing Algorithms
VanMieghem, P., & Kuipers, F. A. (2004) · IEEE/ACM Transactions on Networking · doi:10.1109/tnet.2004.836112 · Cited by 266 · see citing works
Hierarchy Measure for Complex Networks
Mones, E., Vicsek, L., & Vicsek, T. (2012) · PLoS ONE · doi:10.1371/journal.pone.0033799 · Cited by 259 · see citing works
Decentralized erasure codes for distributed networked storage
Dimakis, A. G., Prabhakaran, V. M., & Ramchandran, K. (2006) · IEEE Transactions on Information Theory · doi:10.1109/tit.2006.874535 · Cited by 259 · see citing works
Using Gossips to Spread Information: Theory and Evidence from Two Randomized Controlled Trials
Banerjee, A., Chandrasekhar, A. G., Duflo, E., & Jackson, M. O. (2019) · The Review of Economic Studies · doi:10.1093/restud/rdz008 · Cited by 280 · see citing works
Synchronization and Convergence of Linear Dynamics in Random Directed Networks
Wu, C. W. (2006) · IEEE Transactions on Automatic Control · doi:10.1109/tac.2006.878783 · Cited by 245 · see citing works
RANDOM MATRICES: THE CIRCULAR LAW
Tao, T., & Vu, V. (2008) · Communications in Contemporary Mathematics · doi:10.1142/s0219199708002788 · Cited by 256 · see citing works
(2013) · Discrete mathematics and its applications · doi:10.1201/b16132-44 · Cited by 253 · see citing works
Heat Conduction Process on Community Networks as a Recommendation Model
Zhang, Y., Blattner, M., & Yu, Y. (2007) · Physical Review Letters · doi:10.1103/physrevlett.99.154301 · Cited by 252 · see citing works
Percolation in multiplex networks with overlap
Cellai, D., López, E., Zhou, J., Gleeson, J. P., & Ginestra Bianconi, A. (2013) · Physical Review E · doi:10.1103/physreve.88.052811 · Cited by 244 · see citing works
RESILIENCE TO CONTAGION IN FINANCIAL NETWORKS
Amini, H., Rama Cont, A., & Minca, A. (2013) · Mathematical Finance · doi:10.1111/mafi.12051 · Cited by 312 · see citing works
“…Configuration Model A standard method for studying random graphs with prescribed degree sequence is to consider (see, e.g., Molloy and Reed 1998; Bollobás 2001; Janson 2009a) a related random multigraph with the same degree sequence, known as the configuration model, then condition on this…”
Slim Fly: A Cost Effective Low-Diameter Network Topology
Besta, M., & Hoefler, T. (2014) · 2012 International Conference for High Performance Computing, Networking, Storage and Analysis · doi:10.1109/sc.2014.34 · Cited by 243 · see citing works
Pathways towards instability in financial networks
Bardoscia, M., Battiston, S., Caccioli, F., & Caldarelli, G. (2017) · Nature Communications · doi:10.1038/ncomms14416 · Cited by 229 · see citing works
Semantic Networks: Structure and Dynamics
Borge‐Holthoefer, J., & Arenas, À. (2010) · Entropy · doi:10.3390/e12051264 · Cited by 238 · see citing works
Jamming is limited in scale-free systems
Toroczkai, Z., & Bassler, K. E. (2004) · Nature · doi:10.1038/428716a · Cited by 230 · see citing works
The Transport Capacity of Wireless Networks Over Fading Channels
Xue, F., Xie, L.-L., & Kumar, P. R. (2005) · IEEE Transactions on Information Theory · doi:10.1109/tit.2004.842628 · Cited by 223 · see citing works
Fortunato, S., & Castellano, C. (2011) · Springer eBooks · doi:10.1007/978-1-4614-1800-9_33 · Cited by 221 · see citing works
Nahum, A., Roy, S., Skinner, B. J., & Ruhman, J. (2021) · PRX Quantum · doi:10.1103/prxquantum.2.010352 · Cited by 221 · see citing works
Effect of correlations on network controllability
Pósfai, M., Liu, Y.-Y., Slotine, J.-J. E., & Barabási, A.-Ĺ. (2013) · Scientific Reports · doi:10.1038/srep01067 · Cited by 203 · see citing works
“We start from a directed network with Poisson [29, 30] or scale-free degree distribution [31, 32].”
MethodYağan, O., Qian, D., Zhang, J., et al. (2012) · IEEE Transactions on Parallel and Distributed Systems · doi:10.1109/tpds.2012.62 · Cited by 203 · see citing works
The Repulsive Lattice Gas, the Independent-Set Polynomial, and the Lov�sz Local Lemma
Scott, A. D., & Sokal, A. D. (2005) · Journal of Statistical Physics · doi:10.1007/s10955-004-2055-4 · Cited by 203 · see citing works
Ginestra Bianconi, A. (2021) · Cambridge University Press eBooks · doi:10.1017/9781108770996 · Cited by 207 · see citing works
Energy Landscape of Social Balance
Marvel, S. A., Strogatz, S. H., & Kleinberg, J. M. (2009) · Physical Review Letters · doi:10.1103/physrevlett.103.198701 · Cited by 197 · see citing works
Metrics for graph comparison: A practitioner’s guide
Wills, P., & Meyer, F. G. (2020) · PLoS ONE · doi:10.1371/journal.pone.0228728 · Cited by 199 · see citing works
Explosive phenomena in complex networks
D’Souza, R. M., Gómez‐Gardeñes, J., Nagler, J., & Arenas, A. (2019) · Advances In Physics · doi:10.1080/00018732.2019.1650450 · Cited by 201 · see citing works
“In fact, for t slightly greater than tc the size of the largest component is described by the function |C| ∼ (4t− 2)N [Bollobás 2001].”
BackgroundInfluentialGossiping in distributed systems
Kermarrec, A.-M., & van Steen, M. R. (2007) · ACM SIGOPS Operating Systems Review · doi:10.1145/1317379.1317381 · Cited by 190 · see citing works
Fault tolerant deployment and topology control in wireless networks
Li, X., Wan, P., Wang, Y., et al. (2003) · doi:10.1145/778415.778431 · Cited by 188 · see citing works
The Probability That a Random Multigraph is Simple
Janson, S. (2009) · Combinatorics Probability Computing · doi:10.1017/s0963548308009644 · Cited by 185 · see citing works
Graph cluster randomization: network exposure to multiple universes
Ugander, J., Karrer, B., Backstrom, L., & Kleinberg, J. (2013) · Proceedings of the 19th ACM SIGKDD international conference on Knowledge discovery and data mining · doi:10.1145/2487575.2487695 · Cited by 183 · see citing works
Spread of infectious disease through clustered populations
Miller, J. C. (2009) · Journal of The Royal Society Interface · doi:10.1098/rsif.2008.0524 · Cited by 182 · see citing works
Analysis of complex contagions in random multiplex networks
Yağan, O., & Gligor, V. D. (2012) · Physical Review E · doi:10.1103/physreve.86.036103 · Cited by 174 · see citing works
On the privacy of anonymized networks
Pedarsani, P., & Grossglauser, M. (2011) · ACM SIGKDD Conference on Knowledge Discovery and Data Mining (KDD) · doi:10.1145/2020408.2020596 · Cited by 173 · see citing works
A discussion on: Sparse graphs using exchangeable random measures
CASARIN, R., Iacopini, M., & ROSSINI, L. (2017) · Bond University Research Portal (Bond University) · doi:10.1111/rssb.12233 · Cited by 169 · see citing works
“E-mail: caron@stats.ox.ac.uk 1296 F. Caron and E. B. Fox Bollobás (2001), Durrett (2007), Goldenberg et al. (2010) and Fienberg (2012).”
Epidemic-style proactive aggregation in large overlay networks
Jelasity, M., & Montresor, A. (2004) · International Conference on Distributed Computing Systems · doi:10.1109/icdcs.2004.1281573 · Cited by 176 · see citing works
Weight Distribution of Low-Density Parity-Check Codes
Di, C., Richardson, T., & Urbanke, R. L. (2006) · IEEE Transactions on Information Theory · doi:10.1109/tit.2006.883541 · Cited by 176 · see citing works
An ad hoc mobility model founded on social network theory
Musolesi, M., Hailes, S. M. V., & Mascolo, C. (2004) · ACM International Symposium on Modeling, Analysis and Simulation of Wireless and Mobile Systems (MSWiM) · doi:10.1145/1023663.1023669 · Cited by 175 · see citing works
Dynamic interdependence and competition in multilayer networks
Danziger, M. M., Bonamassa, I., Boccaletti, S., & Havlin, S. (2018) · Nature Physics · doi:10.1038/s41567-018-0343-1 · Cited by 171 · see citing works
Conjoining Speeds up Information Diffusion in Overlaying Social-Physical Networks
Yağan, O., Qian, D., Zhang, J., et al. (2013) · IEEE Journal on Selected Areas in Communications · doi:10.1109/jsac.2013.130606 · Cited by 171 · see citing works
Effect of correlations on network controllability
Liu, Y., Pósfai, M., Slotine, J.-J. E., & Barabási, A.-Ĺ. (2012) · DSpace@MIT (Massachusetts Institute of Technology) · Cited by 168 · see citing works
Dimakis, A. G., Prabhakaran, V. M., & Ramchandran, K. (2005) · doi:10.5555/1147685.1147708 · Cited by 166 · see citing works
An Introduction to Temporal Graphs: An Algorithmic Perspective *
Michail, O. (2016) · Internet Mathematics · doi:10.1080/15427951.2016.1177801 · Cited by 166 · see citing works
“[13]).”
The Small World of Corporate Boards
Conyon, M. J., & Muldoon, M. R. (2006) · Journal of Business Finance & Accounting · doi:10.1111/j.1468-5957.2006.00634.x · Cited by 164 · see citing works
Griffiths Phases on Complex Networks
Muñoz, M. A., Juhász, R., Castellano, C., & Ódor, G. (2010) · Physical Review Letters · doi:10.1103/physrevlett.105.128701 · Cited by 162 · see citing works
“, where mean-field holds) and diverges at the percolation threshold hki 1⁄4 1 below which the network is fragmented [2,10], it cannot sustain activity and the phase transition disappears.”
BackgroundInfluentialBehavioral experiments on biased voting in networks
Kearns, M. J., Judd, S. J., Tan, J., & Wortman, J. (2009) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.0808147106 · Cited by 157 · see citing works
Anomalous critical and supercritical phenomena in explosive percolation
D’Souza, R. M., & Nagler, J. (2015) · Nature Physics · doi:10.1038/nphys3378 · Cited by 161 · see citing works
Coverage and connectivity of ad hoc networks in presence of channel randomness
Miorandi, D., & Altman, E. (2005) · doi:10.1109/infcom.2005.1497917 · Cited by 161 · see citing works
Epidemic dynamics on complex networks*
Zhou, T., Fu, Z., & Binghong, W. (2006) · Progress in Natural Science Materials International · doi:10.1080/10020070612330019 · Cited by 160 · see citing works
- Authors
- Bollobás, B.
- Affiliations
- University of Memphis (US)
- Corresponding author
- Bollobás, Béla
- ORCID
- Bollobás, Béla
- Venue
- Cambridge University Press eBooks
- Publisher
- Cambridge University Press
- Published
- 2001-08-30
- OpenAlex
- W390146837
- Cited by
- 2,639
- References
- 0
- Citations by year
- 2026: 37 · 2025: 68 · 2024: 79 · 2023: 82 · 2022: 126 · 2021: 136 · 2020: 137 · 2019: 111 · 2018: 108 · 2017: 138 · 2016: 138 · 2015: 129 · 2014: 136 · 2013: 135 · 2012: 151
- FWCI
- 6.29
- Type
- book
- Language
- en
- Open access
- closed
- Locations
- Cambridge University Press eBooks (publishedVersion)
- Topics
- Physical Sciences › Mathematics › Discrete Mathematics and Combinatorics › Limits and Structures in Graph Theory
- Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Complex Network Analysis Techniques
- Physical Sciences › Mathematics › Mathematical Physics › Stochastic processes and statistical mechanics
- Keywords
- random graphs, Erdős-Rényi random graphs, degree sequences, probabilistic combinatorics, random structures, graph enumeration
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
Are you an author, or do you see a problem with this page? Request a review
