Robustness of community structure in networks

Karrer, Brian; Levina, Elizaveta; Newman, M. E. J.

doi:10.1103/physreve.77.046119

Cited by 299 publications

(291 citation statements)

References 42 publications

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“…On the other hand, Guimerà, Sales-Pardo, and Amaral [85] and Fortunato and Barthélemy [73] showed that random graphs have high-modularity subsets and that there exists a size scale below which communities cannot be identified. In part as a response to this, some recent work has had a more statistical flavor [86,140,144,94,133]. In light of our results, this work seems promising, both due to potential "overfitting" issues arising from the extreme sparsity of the networks, and also due to the empirically-promising regularization properties exhibited by local spectral methods.…”

Section: Relationship With Community Identification Methodsmentioning

confidence: 78%

“…For these networks, the interpretation is similar to that for the low-dimensional networks: the downward slope indicates that as potential communities get larger and larger, there are relatively more intra-edges than inter-edges; and empirically we observe that local minima in the NCP plot correspond to sets of nodes that are plausible communities. Consider, e.g., Zachary's karate club [160] network (ZacharyKarate), an extensivelyanalyzed social network [128,131,94]. The network has 34 nodes, each of which represents a member of There are two local minima in the plot: the first dip at k = 5 corresponds to the Cut A, and the second dip at k = 17 corresponds to Cut B.…”

Section: Observation 1 If the Network Under Consideration Correspondsmentioning

confidence: 99%

“…Intuitively, random graphs have no community structure, but there can still exist sets of nodes with good community scores, at least as measured by modularity (due to random fluctuations about the mean). Moreover, very recently a study of robustness of community structure showed that the canonical example of presence of community structure in networks [160] may have no significant community structure [94].…”

Section: Connections and Broader Implicationsmentioning

confidence: 99%

See 2 more Smart Citations

Community Structure in Large Networks: Natural Cluster Sizes and the Absence of Large Well-Defined Clusters

Leskovec¹,

Lang²,

Dasgupta³

et al. 2009

Internet Mathematics

1,512

1,082

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A large body of work has been devoted to defining and identifying clusters or communities in social and information networks, i.e., in graphs in which the nodes represent underlying social entities and the edges represent some sort of interaction between pairs of nodes. Most such research begins with the premise that a community or a cluster should be thought of as a set of nodes that has more and/or better connections between its members than to the remainder of the network. In this paper, we explore from a novel perspective several questions related to identifying meaningful communities in large social and information networks, and we come to several striking conclusions.Rather than defining a procedure to extract sets of nodes from a graph and then attempt to interpret these sets as a "real" communities, we employ approximation algorithms for the graph partitioning problem to characterize as a function of size the statistical and structural properties of partitions of graphs that could plausibly be interpreted as communities. In particular, we define the network community profile plot, which characterizes the "best" possible community-according to the conductance measure-over a wide range of size scales. We study over 100 large real-world networks, ranging from traditional and on-line social networks, to technological and information networks and web graphs, and ranging in size from thousands up to tens of millions of nodes.Our results suggest a significantly more refined picture of community structure in large networks than has been appreciated previously. Our observations agree with previous work on small networks, but we show that large networks have a very different structure. In particular, we observe tight communities that are barely connected to the rest of the network at very small size scales (up to ≈ 100 nodes); and communities of size scale beyond ≈ 100 nodes gradually "blend into" the expanderlike core of the network and thus become less "community-like," with a roughly inverse relationship between community size and optimal community quality. This observation agrees well with the so-called Dunbar number which gives a limit to the size of a well-functioning community.However, this behavior is not explained, even at a qualitative level, by any of the commonly-used network generation models. Moreover, it is exactly the opposite of what one would expect based on intuition from expander graphs, low-dimensional or manifold-like graphs, and from small social networks that have served as testbeds of community detection algorithms. The relatively gradual increase of the network community profile plot as a function of increasing community size depends in a subtle manner on the way in which local clustering information is propagated from smaller to larger size scales in the network. We have found that a generative graph model, in which new edges are added via an iterative "forest fire" burning process, is able to produce graphs exhibiting a network community profile plot similar to what we observe i...

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Section: Relationship With Community Identification Methodsmentioning

confidence: 78%

Section: Observation 1 If the Network Under Consideration Correspondsmentioning

confidence: 99%

Section: Connections and Broader Implicationsmentioning

confidence: 99%

See 1 more Smart Citation

Community Structure in Large Networks: Natural Cluster Sizes and the Absence of Large Well-Defined Clusters

Leskovec¹,

Lang²,

Dasgupta³

et al. 2009

Internet Mathematics

1,512

1,082

View full text Add to dashboard Cite

show abstract

“…In the analysis of network data, many methods have been proposed for finding communities, but few have been proposed for determining whether the distribution breakdown found is statistically significant or a result of chance. With the method used here, the significance of community structure can be effectively quantified by measuring its robustness to small perturbations in network structure (Karrer et al, 2008), using the modularity Q. In principle, the maximum value of the modularity is Q max 5 1 and such a network could be considered as highly modular.…”

Section: Discussionmentioning

confidence: 99%

Structural vulnerability of the French swine industry trade network to the spread of infectious diseases

Rautureau

Dufour

Durand

2012

Animal

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The networks generated by live animal movements are the principal vector for the propagation of infectious agents between farms, and their topology strongly affects how fast a disease may spread. The structural characteristics of networks may thus provide indicators of network vulnerability to the spread of infectious disease. This study applied social network analysis methods to describe the French swine trade network. Initial analysis involved calculating several parameters to characterize networks and then identifying high-risk subgroups of holdings for different time scales. Holding-specific centrality measurements (‘degree’, ‘betweenness’ and ‘ingoing infection chain’), which summarize the place and the role of holdings in the network, were compared according to the production type. In addition, network components and communities, areas where connectedness is particularly high and could influence the speed and the extent of a disease, were identified and analysed. Dealer holdings stood out because of their high centrality values suggesting that these holdings may control the flow of animals in part of the network. Herds with growing units had higher values for degree and betweenness centrality, representing central positions for both spreading and receiving disease, whereas herds with finishing units had higher values for in-degree and ingoing infection chain centrality values and appeared more vulnerable with many contacts through live animal movements and thus at potentially higher risk for introduction of contagious diseases. This reflects the dynamics of the swine trade with downward movements along the production chain. But, the significant heterogeneity of farms with several production units did not reveal any particular type of production for targeting disease surveillance or control. Besides, no giant strong connected component was observed, the network being rather organized according to communities of small or medium size (<20% of network size). Because of this fragmentation, the swine trade network appeared less structurally vulnerable than ruminant trade networks. This fragmentation is explained by the hierarchical structure, which thus limits the structural vulnerability of the global trade network. However, inside communities, the hierarchical structure of the swine production system would favour the spread of an infectious agent (especially if introduced in breeding herds).

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“…Through the use of surrogate chemical randomisations 37 , we have shown that Markov Stability is able to detect chemical groups, biochemical units, as well as structural features such as helical turns. At long Markov times, we look for significant partitions with the defining feature of being robust to perturbations 42 . We use two measures to quantify this robustness:…”

Section: Construction Of the Atomistic Networkmentioning

confidence: 99%

Uncovering allosteric pathways in caspase-1 using Markov transient analysis and multiscale community detection

et al. 2014

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Allosteric regulation at distant sites is central to many cellular processes. In particular, allosteric sites in proteins are a major target to increase the range and selectivity of new drugs, and there is a need for methods capable of identifying intra-molecular signalling pathways leading to allosteric effects. Here, we use an atomistic graph-theoretical approach that exploits Markov transients to extract such pathways and exemplify our results in an important allosteric protein, caspase-1. Firstly, we use Markov Stability community detection to perform a multiscale analysis of the structure of caspase-1 which reveals that the active conformation has a weaker, less compartmentalised large-scale structure as compared to the inactive conformation, resulting in greater intra-protein coherence and signal propagation. We also carry out a full computational point mutagenesis and identify that only a few residues are critical to such structural coherence. Secondly, we characterise explicitly the transients of random walks originating at the active site and predict the location of a known allosteric site in this protein quantifying the contribution of individual bonds to the communication pathway between the active and allosteric sites. Several of the bonds we find have been shown experimentally to be functionally critical, but we also predict a number of as yet unidentified bonds which may contribute to the pathway. Our approach offers a computationally inexpensive method for the identification of allosteric sites and communication pathways in proteins using a fully atomistic description.

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Robustness of community structure in networks

Cited by 299 publications

References 42 publications

Community Structure in Large Networks: Natural Cluster Sizes and the Absence of Large Well-Defined Clusters

Community Structure in Large Networks: Natural Cluster Sizes and the Absence of Large Well-Defined Clusters

Structural vulnerability of the French swine industry trade network to the spread of infectious diseases

Uncovering allosteric pathways in caspase-1 using Markov transient analysis and multiscale community detection

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